गणितसारसङ्ग्रहः
| गणितसारसङ्ग्रहः महावीराचार्य १९१२ |
THE
GAŅITA-SĀRA-SAŃGRAHA
OF
MAHĀVĪRĀCĀRYA
WITH
ENGLISH TRANSLATION AND NOTES
BY
M. RANGACARYA, M.A., Rao Bahadur,
Professor of Sanskrit and Comparitive Philology, Presidency College,
and Curator, Government Oriental Manuscripts Library, Madras.
Published under the orders of the Government of Madras.
M A D R A S :
PRINTED BY THE SUPERINTENDENTGOVERNMENT PRESS,
1912.
महावीराचार्यप्रणीतः
ग णि त सा र स ङ् ग्र ह:
म. रङ्गाचार्येण
परिशोधितः
आङ्गलभाषानुवादटीकाभ्यां सह राजकीयाज्ञानुसारेण प्रकाशितश्च।
चेन्नपुर्यां
राजकीयमुत्राक्षरशालायां (सूपरिन्टेण्डेण्टाख्यैन) तन्निर्वाहकेण मुद्रितः ।
१९१२
TABLE OF TRANSLITERATION.
| ---- | Consonants | Vowels. | Dipthongs. | ||
|---|---|---|---|---|---|
| Gutturals .. | k, kh, g, gh, ń, h, h. .. क, स्व, ग, घ, ङ्, ह, : .. |
a, ā .. अ आ |
| ||
| Palatals .. | c, ch, j, jh, ñ, y, ś. .. च, छ, ज, झ, ज्ञ, य, श .. |
i, ī.. इ, ई | |||
| Linguals .. | ț, țh, ḍ, ḍh, ņ, r, ș. .. ट, ठ, ड, ढ, ण, र, ष .. |
ŗ, ř .. ऋ, ॠ |
.. | ||
| Dentals .. | t, țh, ḍ, dh, n, l, s. .. त, थ, द, ध, न, ल, स .. |
ļ .. ल्र् |
.. | ||
| Labials .. | p, ph, b, bh, m, v, .. प, फ, ब, भ म, व .. |
u, ū .. उ, ऊ |
ō (o) au. ओ ऑ |
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P R E F A C E
Soon after I was appointed Professor of Sanskrit and Comparative Philology in the Presidency College at Madras, and in that capacity took charge of the office of the Curator of the Government Oriental Manuscripts Library, the late Mr. G. H. Stuart, who was then the Director of Public Instruction, asked me to find out if in the Manuscripts Library in my charge there was any work of value capable of throwing new light on the history of Hindu mathematics, and to publish it, if found, with an English translation and with such notes as were necessary for the elucidation of its contents. Accordingly the mathematical manuscripts in the Library were examined with this object in view; and the examination revealed the existence of three incomplete manuscripts of Mahāvērācārya's Gaṇita-sāra-saṅgraha.. A cursory perusal of these manuscripts made the value of this work evident in relation to the history of Hindu Mathematics. The late Mr. G. H. Stuart's interest in working out this history was so great that, when the existence of the manuscripts and the historical value of the work were brought to his notice, he at once urged me to try to procure other manuscripts and to do all else that was necessary for its proper publication. He gave me much advice and encouragement in the early stages of my endeavour to publish it; and I can well guess how it would have gladdened his heart to see the work published in the form he desired. It has been to me a source of very keen regret that it did not please Providence to allow him to live long enough to enable me to enhance the value of the publication by means of his continued guidance and advice; and my consolation now is that it is something to have been able to carry out what he with scholarly delight imposed upon me as a duty.
Of the three manuscripts found in the library one is written on paper in Grantha characters, and contains the first five chapters of the work with a running commentary in Sanskrit: it has been denoted here by the letter P. The remaining two are palm-leaf manuscripts in Kanarese characters, one of them containing, like P, the first five chapters, and the other the seventh chapter dealing with the geometrical measurement of areas. In both these manuscripts there is to be found, in addition to the Sanskrit text of the original work, a brief statement in the Kanarese language of the figures relating to the various illustrative problems as also of the answers to those same problems. Owing to the common characteristics of these manuscripts and also owing to their no overlapping one another in respect of their contents, it has been thought advisable to look upon them as one manuscript and denote them by K. Another manuscript, denoted by M, belongs to the Government Oriental Library at Mysore, and was received on loan from Mr. A. Mahadeva Sastri, B.A., the Curator of that institution. This manuscript is a transcription on paper in Kanarese characters of an original palm-leaf manuscript belonging to a Jaina Pandit, and contains the whole of the work with a short commentary in the Kanarese language by one Vallabha, who claims to be the author of also a Telugu commentary on the same work. Although incorrect in many places, it proved to be of great value on account of its being complete and containing the Kanarese commentary; and my thanks are specially due to Mr.A.Mahadeva Sastri for his leaving it suficiently long at my disposal. A fifth manuscript, denoted by B, is a transcription on paper in Kanarese characters of a palm-leaf manuscript found in a Jaina monastery at Mudbidri in South Canara, and was obtained through the kind effort of Mr.R.Krishnamacharyar, M.A., the Sub-assistant Inspector of Sanskrit Schools in Madras, and Mr. U.B.Venkataramanaiya, of Mudbidri. This manuscript also contains the whole work, and gives, like K, in Kanarese a brief statement of the problems and their answers. The endeavour to secure more manuscripts having proved fruitless, the work has had to be brought out with the aid of these five manuscripts; and owing to the technical character of the work and its elliptical and often riddle-like language and the inaccuracy of the manuscripts, the labour involved in ringing it out with the translation and the requisite notes has been heavy and trying. There is, however, the satisfaction that all this labour has been bestowed on a worthy work of considerable historical value.
It is a fortunate circumstance about the Gaṇita-sāra-saṅgraha that the time when its author Mahaviracarya lived may be made out with fair accuracy. In the very first chapter of the work, we have, immediately after the two introductory stanzas of salutation to Jina Mahāvīra, six stanzas describing the greatness of a king, whose name is said to have been Cakrikā-bhañjana, and who appears to have been commonly known by the title of Amōghavarṣa Nṛpatuṅga ; and in the last of these six stanzas there is a benediction wishing progressive prosperity to the rule of this king. The results of modern Indian epigraphical research show that this king Amōghavarṣa Nṛupatuṅga reigned from A.D.814 or 815 to A.D. 577 or 878.[*] Since it appears probable that the author of the Gaṇita-sāra-saṅgraha was in some way attached to the court of this Rāṣṭrakūta king Amōghavarsa Nṛupatuṅga, we may consider the work to belong to the middle of the ninth century of the Christian era. It is now generally accepted that, among well-known early Indian mathematicians Āryabhata, lived in the fifth, Varāhamihira in the sixth, Brahmagupta in the seventh and Bhaskaracarya, in the twelfth century of the Christian era; and chronologically, therefore, Mahāvīrācārya comes between Brahmagupta and Bhaskaracarya . This in itself is a point of historical noteworthiness ; and the further fact that the author of the Gaṇita-sāra-saṅgraha belonged to the Kanarese speaking portion of South India in his days and was a Jaina in religion is calculated to give an additional importance to the historical value of his work . Like the other mathematicians mentioned above, Mahāvīrācārya was not primarily an astronomer, although he knew well and has himself remarked about the usefulness of mathematics for the study of astronomy. The study of mathematics seems to have been popular among Jaina scholars; it forms, in fact, one of their four anuyōgas or auxiliary sciences indirectly serviceable for the attainment of the salvation of soul-liberation known as mōksa.
A comparison of the Gaṇita-sāra-saṅgraha , with the corresponding portions in the Brahmasphuṭa-siddhānta of
^* Vide Nilgunā Inscription of the time of Amōgovarsa I, A.D. 806; edited by J.F.Fleet, PH.D., C.I.E., in Epigraphic Indica, vol. VI, pp. 98–108,
Brahmagupta is calculated to lead to the conclusion that, in all probability, Mahāvīrācārya was familiar with the work of Brahmagupta and endeavoured to improve upon it to the extent to which the scope of his Gaṇita-sāra-saṅgraha permitted such improvement. Mahāvīrācārya's classification of arithmetical operations is simpler, his rules are fuller and he gives a large number of examples for illustration and exercise. Pṛthūdakasvāmin, the well known commentator on the Brahmasphuṭa-siddhānta, could not have been chronologically far removed from Mahāvīrācārya, and the similarity of some of the examples given by the former with some of those of the latter naturally arrests attention. In any case it cannot be wrong to believe, that, at the time, when Mahāvīrācārya wrote his Gaṇita-sāra-saṅgraha, Brahmagupta , must have been widely recognized as a writer of authority in the field of Hindu astronomy and mathematics. Whether Bhāskarācārya was at all acquainted with the Gaṇita-sāra-saṅgraha of Mahāvīrācārya, it is not quite easy to say. Since neither Bhāskarācārya nor any of his known commentators seem to quote from him or mention him by name, the natural conclusion appears to be that Bhāskarācārya's Siddhānta-śirōmaṇi including his Līlāvatī and Bījagaṇita, was intended to be an improvement in the main upon the Brahmasphuṭa-siddhānta of Brahmagupta. The fact that Mahāvīrācārya was a Jaina, might have prevented Bhāskarācārya from taking note of him; or it may be that the Jaina mathematician's fame had not spread far to the north in the twelfth century of the Christian era. His work, however, seems to have been widely known and appreciated in Southern India. So early as in the course of the eleventh century and perhaps under the stimulating influence of the enlightened rule of Rājarājanarēndra, of Rajahmundry, it was translated into Telugu in verse by Pāvulūri Mallana ; and some manuscripts of this Telugu translation are now to be found in the Government Oriental Manuscripts Library here at Madras. It appeared to me that to draw suitable attention to the historical value of Mahāvīrācārya's Gaṇita-sāra-saṅgraha, I could not do better than seek the help of Dr.David Eugene Smith of the Columbia University of New York, whose knowledge of the history of mathematics in the West and in the East is known to be wide and comprehensive, and who on the occasion when he met me in person at Madras showed great interest in the contemplated publication of the Gaṇita-sāra-saṅgraha and thereafter read a paper on that work at the Fourth International Congress of Mathematicians held at Rome in April 1908. Accordingly I requested him to write an introduction to this edition of the Gaṇita-sāra-saṅgraha, giving in brief outline what he considers to be its value in building up the history of Hindu mathematics. My thanks as well as the thanks of all those who may as scholars become interested in this publication are therefore due to him for his kindness in having readily complied with my request; and I feel no doubt that his introduction will be read with great appreciation.
Since the origin of the decimal system of notation and of the conception and symbolic representation of zero are considered to be important questions connected with the history of Hindu mathematics, it is well to point out here that in the Gaṇita-sāra-saṅgraha twenty-four notational places are mentioned, commencing with the units place and ending with the place called mahāksōbha, and that the value of each succeeding place is taken to be ten times the value of the immediately preceding place. Although certain words forming the names of certain things are utilized in this work to represent various numerical figures, still in the numeration of numbers with the aid of such words the decimal system of notation is almost invariably followed. If we took the words moon, eye, fire, and sky to represent respectively 1, 2, 3 and 0, as their Sanskrit equivalants are understood in this work, then, fire-sky-mōn-eye would denote the number 2103,and moon-eye-fire-sky fire would denote 3021, since these nominal numerals denoting numbers are generally repeater in order from the units place upwards. This combination of nominal numerals and the decimal system of notation has been adopted obviously for the sake of securing metrical convenience and avoiding at the same time cumbrous ways of mentioning numerical expressions; and it may well be taken for granted that for the use of such nominal numerals as well as the decimal system of notation Mahāvīrācārya, was indebted to his predecessors. The decimal system of notation is distinctly described by Āryabhaṭa,and there is evidence in his writings to show that he was familiar with nominal numerals. Even in his brief mnemonic method of representing numbers by certain combinations of the consonants and vowels found in the Sanskrit language, the decimal system of notation is taken for granted; and ordinarily 19 notational places are provided for therein. Similarly in Brahmagupta's writings also there is evidence to show that he was acquainted with the use of nominal numerals and the decimal system of notation. Both Āryabhaṭa and Brahmagupta, claim that their astronomical works <poem> xiv GANITASARASANGRAHA.
are related to the Brahma-siddhānta; and in a work of this name, which is said to form a part of what is called Śakalya-sāṁhitā and of which a manuscript copy is to be found in the Government Oriental Manuscripts Library here, numbers are expressed mainly by nominal numerals used in accordance with the decimal system of notation. It is not of course meant to convey that this work is necessarily the same as what was known to Ārayabhaṭa and Brahmagupta; and the fact of its using nominal numerals and the decimal system of notation is mentioned here for nothing more than what it may be worth.
It is generally recognized that the origin of the conception of zero is primarily due to the invention and practical utilization of a system of notation wherein the several numerical figures used have place-values apart from what is called their intrinsic value. In writing out a number according to such a system of notation, any notational place may be left empty when no figure with an intrinsic value is wanted there. It is probable that owing to this very reason the Sanskrit word śūnya, meaning fempty ', came to denote the zero ; and when it is borne in mind that the English word 'cipher' is derived from an Arabic word having the same meaning as the Sanskrit śūnya, we may safely arrive at the conclusion that in this country the conception of the zero came naturally in the wake of the decimal system of notation: and so early as in the fifth century of the Christian era, Āryabhaṭa is known to have been fully aware of this valuable mathematical conception. And in regard to the question of a symbol to represent this conception, it is well worth bearing in mind that opera tions with the zero cannot be carried on-not to say cannot be even thought of easily—without a symbol of some sort to represent it. Mahāvīrācārya gives, in the very first chapter of his Gaṇita-sāra-saṅgraha, the results of the operations of addition, subtraction, multiplication and division carried on in relation to the zero quantity; and although he is wrong in saying that a quantity, when divided by zero, remains unaltered, and should have said, like Bhāskarācārya, that the quotient in such a case is infinity, still the very mention of operations in relation to zero is enough to show that Mahāvīrācārya must have been aware of some symbolic representation of the zero quantity. Since Brahmagupta, who must have lived at least 150 years before Mahāvīrācārya, mentions in his work the results of operations in relation to the zero quantity, it is not unreasonable to suppose that before his time the zero must have had a symbol to represent it in written calculations. That even Āryabhaṭa knew such a symbol is not at all improbable. It is worthy of note in this connection that in enumerating the nominal numerals in the first chapter of his work, Mahāvīrācārya mentions the names denoting the nine figures from 1 to 9, and then gives in the end the names denoting zero, calling all the ten by the name of saṅkhya: and from this fact also, the inference may well be drawn that the zero had a symbol, and that it was well known that with the aid of the ten digits and the decimal system of notation numerical quantities of all values may be definitely and accurately expressed. What this known zero-symbol was, is, however, a different question.
The labour and attention bestowed upon the study and translation and annotation of the Gaṇita-sāra-saṅgraha have made it clear to me that I was justified in thinking that its publication might prove useful in elucidating the condition of mathematical studies as they flourished in South India, among the Jainas in the ninth century of the Christian era; and it has been to me a source of no small satisfaction to feel that in bringing out this work in this form, I have not wasted my time and thought on an unprofitable undertaking. The value of the work is undoubtedly more historical than mathematical. But it cannot be denied that the step by step construction of the history of Hindu culture is a worthy endeavour, and that even the most insignificant labourer in the field of such an endeavor deserves to be looked upon as a useful worker. Although the editing of the Ganita-sāra-saṅgraha has been to me a labour of love and duty, it has often been felt to be heavy and taxing; and I, therefore, consider that I am specially bound to acknowledge with gratitude the help which I have received in relation to it. In the early stage when conning and collating and interpreting the manuscripts was the chief work to be done, Mr.B.Varadaraja Aiyangar, B.A., B.Ḷ., who is an Advocate of the Chief Court at Bangalore, co-operated with me and gave me an amount of aid for which I now offer him my thanks. Mr.K.Krishnaswami Aiyangar, B.A., of the Madras Christian College, has also rendered considerable assistance in this manner; and to him also I offer my thanks. Latterly I have has to consult on a few occasions Mr.P. V. Seshu Aiyar, B.A., L.T., Professor of Mathematical Physics in the Presidency College here, in trying to explain the rationale of some of the rules given in the work; and I am much obliged to him for his ready willingness in allowing me thus to take advantage of his expert knowledge of mathematics. My thanks are, I have to say in conclusion, very particularly due to Mr.P.Varadacharyar, B.A., Librarian of the Government Oriental Manuscripts Library at Madras, but for whose Zealous and steady co-operation with me throughout and careful and continued attention to details, it would indeed have been much harder for me to bring out this edition of the Ganita-sāra-saṅgraha.
| February 1912, Madras. |
M. RANGACHARYA. |
गणितसारसङ्ग्रहः
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| तृतीयः प्रकीर्णकव्यवहारः | ||||
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| चतुर्थः त्रैराशिकव्यवहारः | ||||
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| पञ्चमः मिश्रकव्यवहारः | ||||
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| षष्टः क्षेत्रगणितव्यवहारः | ||||
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| सप्तमः खातव्यवहारः | ||||
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ग णि त सा र स स ङ्ग्र हः[सम्पाद्यताम्]
महावीराचार्यप्रणीतः ।
---
संज्ञाधिकारः ।
---
मङ्गलाचरणम् ।
अलङ्घ्यं त्रिजगत्सारं यस्यानन्तचतुष्टयम् ।
नमस्तस्मै जिनेन्द्राय महावीराय तायिने ॥ १ ॥
सङ्ख्याज्ञानप्रदीपेन जैनेन्द्रेण महा1त्विषा ।
प्रकाशीितं जगत्सर्वं येन तं प्रणमाम्यहम् ॥ २ ॥
प्री2णितः प्राणिस3स्यौधो निरीतिर्निरवग्रहः।
श्रीमतामोघवर्षेण येन स्वेष्टहितैषिणा ॥ ३ ॥
पापरूपाः परा यस्य चित्तवृत्तिहविर्भुजि ।
भस्मसा4द्भवमीयुस्तेऽवन्ध्यकोपोऽभ5वत्ततः ॥ ४ ॥
वशीकुर्वन् जगत्सर्वं स्वयं6नानुवशः परैः।
नाभिभूतः प्रभुस्तस्मादपूर्वमकरध्वजः ॥ ५ ॥
यो विक्रमक्रमाक्रान्तचक्रि7च8क्रकृतक्रियः ।
चक्रिकाभञ्जनो नाम्ना चक्रिकाभञ्जनोऽञ्जसा ॥ ६ ॥
यो विद्यानद्याधिष्ठानो मर्यादावज्रवेदिकः।
रत्नगर्भो यथाख्यातचारित्रजलधिर्महान् ॥ ७ ॥
विध्वस्तैकान्तपक्षस्य स्याद्वादन्यायवादिनः9 ।
देवस्य नृपतुङ्गस्य वर्धतां तस्य शासनम् ॥ ८ ॥
1 M and B मह°. 2 M प्रणीतः . 3 सर्गौ° .
4M and K सद्भा° . 5 K, P and B भवेत् . 6 B योऽयं .
7 M क्री° . 8 M and B श° . 9 P वेदिनः .
गणितशास्त्रप्रशंसा।
लौकिके वैदिके वापि तथा सामायिकेऽपि यः।
व्यापारस्तत्र सर्वत्र सङ्ख्यानमुपयुज्यते ॥ ९ ॥
कामतन्त्रेऽर्थशास्त्रे च गान्धर्वे नाटकेऽपि वा।
सूपशास्त्रे तथा वैद्ये वास्तुविद्यादिवस्तुषु ॥ १० ॥
छन्दोऽलङ्कारकाव्येषु तर्कव्याकरणादिषु ।
कलागुणेषु सर्वेषु प्रस्तुतं गणितं परम् ॥ ११ ॥
सूर्यादिग्रहचारेषु ग्रहणे प्रहसंयुतौ ।
त्रिप्रश्ने चन्द्रवृत्तौ च सर्वत्रङ्गीकृतं हि तत् ।। १२ ॥
द्वीपसागरशैलानां सङ्ख्याव्यासपरिक्षिपः।
भवनव्यन्तरज्योतिर्लोककल्पार्थिवासिनाम् ॥ १३ ॥
नारकाणां च सर्वेषां श्रेणीबन्धेन्द्रकोत्कराः ।
प्रकीर्णकप्रमाणाद्या बुध्यन्ते गणितेन ते ॥ १४ ॥
प्राणिनां तत्र संस्थानमायुरष्टगुणादयः।
यात्राद्यास्संहिताद्याश्च सर्वे ते गणिताश्रयाः ॥ १५ ॥
बहुभिर्विप्रलापैः किं त्रैलोक्ये सचराचरे ।
यत्किचिद्वस्तु तत्सर्वं गणितेन विना न हि ॥ १६ ॥
तीर्थकृद्भ्यः कृतार्थेभ्यः पूज्येभ्यो जगदीश्वरैः।
तेषां शिष्यप्रशिष्येभ्यः प्रसिद्धाद्गुरुपर्वतः ॥ १७ ॥
जलधेरिव रत्नानि पाषाणादिव काञ्चनम्।
शुक्तेर्मुक्ताफलानीव सङ्ख्याज्ञान महोदधे ॥ १८ ॥
- M स्यात् ; B चापि.
A B च.
- M and B दण्डा°.
5 M and B पुरा. 3 K and M महा' ७ K a nन'M क्षिपाः
K and P मव for ज्ञान. 8 M 1 K, M and B वदे°.
|
किशद्द्त्य तत्सारं वक्ष्येऽहं मतिरक्तितिः। |
=-= -------~-
1 ह ऋत ह अल्प. : £ सज्ञातायसमा . „+ „+ ° दु (णा ४ शल ९8 न्ह ग त्थ). ५ भ क ए सङ्ूटे* 5 दयु. ० ४०१८ णु. 2 7 त 8 वृ.
° ४ 9१ ए- श्य. ० ए. ध, गणितसारसङ्ग्रहः
तपञ्चकशतं प्रोक्तं प्रमाणं मानवेदिभिः। वर्तमाननराणामङ्गलमात्माङ्गलं भवेत् ॥ २८ ॥
यवहारप्रमाणे हे ' राद्धान्ते लौकिके विदुः ।
आत्माङ्गलमिति त्रेधा तिर्यक्पादः षडङ्गलैः ॥ २९ ॥
पादद्वयं वितस्तिस्स्यात्ततो हस्तो द्विसङ्गणः।
दण्डो हस्तचतुषेण क्रशस्तद्विसहस्रकम् ॥ ३० ॥
योजनं चतुरः क्रोशान्प्राहुः क्षेत्रविचक्षणाः ।
वक्ष्यतेऽतः परं कालपरिभाषा यथाक्रमम् ॥ ३१ ॥
- अथ कालपरिभाषा ।
अणुरण्वन्तरं काले व्यतिक्रामति यावति ।
स कालस्समयोऽसङ्ख्यैस्समयैरावलिर्भवेत् ॥ ३२ ॥
सङ्ख्यातावलरुच्छासः स्तोकस्तूच्छाससप्तकः ।
स्तोकास्सप्त लवसेषां साधीष्टात्रिंशता घठी ॥ ३३ ॥
घठीद्वयं मुहूर्ताऽत्र मुहूतैस्त्रिशता दिनम् ।
पथनौस्त्रिदिनैः पक्षः पक्षौ द्वौ मास इष्यते ।। ३४ ॥
अतुर्मासद्वयेन स्यात्रिभिस्तैरयनं मतम् ।
तदयं वत्सरो वक्ष्ये धान्यमानमतः परम् ॥ ३९ ॥
- अथ धान्यपरभाष ।
विद्धि षोडशिकास्तत्र चतस्रः कुडहो भवेत्।
कुडहांश्चतुरः प्रस्थश्चतुः प्रस्थानथाढकम् ॥ ३१ ॥
चतुर्भिराढकैद्रणो मानी द्रोणैश्चतुर्गुणैः ।
वारी मानचतुषेण रत्वार्यः पञ्च प्रवर्तिका ॥ ३७ ॥
नये. • K and B वा. • £ वा. संज्ञाधिकारः
सेयं चतुर्गुणा वाहः कुम्भः पञ्च प्रवर्तिकाः ।
इतः परं सुवर्णस्य परिभाषा विभाष्यते ॥ ३८ ॥
अथ सुवर्णपरिभाषा ।
चतुर्भिर्गण्डकैगुञ्जा गुञ्जाः पञ्च पणोऽष्ट ते ।
धरणं धरणे कर्षः पलं कर्षचतुष्टयम् ॥ ३९ ॥
अथ रजतपरिभाषा ।
धान्यद्वयेन गुफैका गुजयुग्मेन माषकः ।
माष षोडशकनात्र धरणे परभाष्यते। ॥ ४० ॥
तद्वयं सार्धकं कर्षः पुराणांश्चतुरः पलम् ।
रूप्ये मागधमानेन प्राहुस्सङ्ख्यानकोविदाः ॥ ४१ ॥
अथ लोहपरिभाषा ।
कला नाम चतुष्पादाः सपदार्षकल यवः ।
थवैश्चतुर्भिरंशस्स्याद्भागोंऽशानां चतुष्टयम् ।। ४२ ॥
द्रणो भागषट्रेन दीनारोऽस्माद्विसङ्गणः ।
द्वौ दीनारौ सतेरं स्यात्प्राहुलेहेऽत्र सूरयः ॥ ४३ ॥
- For the whole of धान्यपरिभाषा, P and B add what is given below as another
reading and M has it in the original with the variations which are enclosed in trackets.
भाद्या षोडशिका तत्र कुड(ङ)बः प्रस्थ आढक :।
द्रोणो मानी तत: खारी क्रमेण (मश:*) चतुराहताः ॥
(सहत्रैश्च त्रिभिष्यङ्गितैश्च त्रीहिभिस्समम्।
यस्सम्पूर्णेऽभवत्सोऽयं कुडुबः परिभाष्यते ॥ )
प्रवतकात्र ताः पञ्च वाहस्तस्याश्चतुर्गुणः ।
कुम्भस्सपादवाहस्स्यात् (पञ्च प्रवर्तकः कुम्भ:) स्वर्णसंज्ञाथ वर्यते ॥
ॐ सतेराख्यम्,
• In Balco.
- गणितसारसङ्गहः
पलैर्द्वादशभिस्सार्धैः प्रस्थः पलशतद्वयम ।
तुला दश तुला भारः सङ्ख्यादक्षाः प्रचक्षते ॥ ४४ ॥
वस्त्राभरणवेत्राणां युगळान्यत्र विंशतिः ।
कोटिकानन्तरं भाष्ये परिकर्माणि नामत : ॥ ४५ ॥
- अथ पारिकर्मनामानि ।
आदिमं गुणकारोऽत्र प्रत्युत्पन्नोऽपि तद्भवेत्।
द्वितीयं भागहाराख्यं तृतीयं कृतिरुच्यते ॥ ४६ ॥
चतुर्थे वर्गमूले हि भाष्यते पञ्चमं घनः ।
घनमूलं ततष्षष्ठं सप्तमं च चितिस्स्मृतम् ।। ४७ ॥
तत्सङ्कलितमप्युक्तं व्युत्कलितमतोऽष्टमम ।
तच शेषमिति प्रोक्तं क्षिन्नान्यष्टावन्यपि ॥ ४८ ॥
अथ धनणन्यावषयकसामान्यानियमः ।
ताडितः वेन राशिः र्वं सोऽविकारी हतो युतः ।
हीनोऽपि रववधादिः रवं योगे रवं योज्यरूपकम् ॥ ४९ ॥
ऋणयोर्धनयोर्धाते भजने च फलं धनम् ।
ऋणं धनर्णयोस्तु स्यात्स्वर्णयोर्विवरं युतौ ॥ ५० ॥
ऋणयोर्धनयोर्योगो यथासङ्ख्यमृणं धनम् । ।
शोध्यं धनमृणं राशेः ऋणं शोध्यं धनं भवेत् ।। ५१ ।
धनं धनर्णयोर्वर्गे मूले स्वर्गे तयोः क्रमात् ।
अणं चरूपतोऽवतुं यतस्तस्मान्न तत्पदम् ॥ ५२ ॥
- अथ सङ्ख्यासंज्ञाः ।
शशी सोमश्च चन्द्रेन्दू प्रालेयांशू रजनीकरः ।
श्वेतं हिमगु रूपञ्च मृगाङ्कश्च कलाधरः । ५३ ॥
• ४ रं. 2 M डि. * M विद्यात्कलासवर्णस्य.
- Stanzas 58 to 68 occur only in M, and are given here, though erroneous
here and there, as found in the original.
• Used here in the 4th conjugation, active voice,
- संज्ञाधिकारः
द्वे द्वे द्वावुभौ युगलयुग्मं च लोचनं द्वयम् ।
दृष्टिर्नेत्राम्बकं द्वन्द्वमभिचक्षुर्नयं दृशौ ॥ ५४ ॥
हरनेत्रं पुरं लोकं त्रै(त्रि)रत्नं भुवनत्रयम् ।
गुणो वह्निः शिखी ज्वलनः पावकश्च हुताशनः ॥ ५५ ॥
अभ्युधिर्विषधिर्वर्धिः पयोधिसागरो गतिः ।
जलधिर्बन्धश्चतुर्वेदः कषायस्सलिलाकरः ॥ ५६ ॥
इषुर्वाणं शरं शस्त्रं भूतमिन्द्रियसायकम् ।
पञ्च व्रतानि विषयः करणीयस्तन्तुसायकः ॥ ५७ ॥
ऋतुजीव रस लंख्या द्रव्यञ्च षट्कं खरन ।
कुमारवदनं वर्ण शिलीमुरवपदानि च ॥ ५८ ॥
शैलमद्रिर्भयं भूभो नगाचलमुनिर्गिरि : ।
अवश्वपन्नगा द्वीप धातुव्यसनमातृकम् ॥ ५९ ॥
अष्टौ तनुर्गजः कर्म वसु वारणपुरम् ।
द्विरदं दन्ती दिग्दुरितं नागानीकं करी यथा ॥ ६० ॥
नव नन्दं च रन्ध्रञ्च पदार्थे लधकेशवौ ।
निधिरत्नं ग्रहाणां च दुर्गनाम च सङ्ख्यया ॥ ६१ ॥
आकाशं गगनं शून्यमम्बरं रवं नभो वियत् ।
अनन्तमन्तरिक्षे च विष्णुपादं दिवि स्मरेत् ॥ ६२ ॥
- अथ स्थाननामानि ।
एकं तु प्रथमस्थानं द्वितीयं दशसंज्ञिकम् ।
तृतीयं शतमित्याहुः चतुर्थं तु सहस्रकम् ॥ ६३ ॥
पञ्चमं दशसाहस्रं षष्ठं स्याल्लक्षमेव च ।
सप्तमं दशलक्षे तु अष्टमं कोटिरुच्यते ॥ ६४ ॥
- गणितसारसङ्गहः
नवमं दशकोट्यस्तु दशमं शतकोटयः ।
अर्धदं रुद्रसंयुक्तं न्यर्बुदं द्वादशं भवेत् ॥ ६५ ॥
खर्व त्रयोदशस्थानं महाखर्व चतुर्दशम् ।
पद्मं पञ्चदशं चैव महापद्मं तु षोडशम् ॥ ६६ ॥
क्षोणी सप्तदशं चैव महाक्षोणी दशाष्टकम् ।
शङ्खं नवदशं स्थानं महाशङ्खं तु विंशकम् ॥ ६७ ॥
क्षित्यैकविंशतिस्थानं महाक्षित्या द्विविंशकम् ।
त्रिविंशकमथ क्षोभं महाक्षोभं चतुर्नयम् ॥ ६८ ॥
- अथ गणकगुणनिरूपणम् ।
लघुकरणोहापोहानालस्यग्रहणधारणोपायैः।
व्याक्तिकराङ्गविशिष्टैर्गणकोऽष्टाभिर्गुणैर्ज्ञेयः ॥ ६९ ।।
इति संज्ञा समासेन भाषिता मुनिपुङ्गवैः।
विस्तरेणागमाद्वेद्यं वक्तव्यं यदितः परम् ॥ ७० ॥
इति सारसङ्गहे गणितशास्त्रे महावीराचार्यस्य कृतौ संज्ञाधिकारस्समाप्तः ॥
- प्रथमः परिकर्मव्यवहारः
इतः परं परिकर्माभिधानं प्रथमव्यवहारमुदाहरिष्यामः ।
प्रत्युत्पन्नः
तत्र प्रथमे प्रत्युत्पन्नपरिकर्मणि करणसूत्रं यथा-
गुणयेद्गुणेन गुण्यं कवाटसन्धिक्रमेण संस्थाप्य ।
राश्यर्धखण्डतत्स्थैरनुलोमविलोममार्गाभ्याम् ॥ १ ॥
- अत्रोद्देशकः ।
दत्तान्येकैकस्मै 'जिनभवनयाम्बुजानि तान्यष्टौ ।
वसतीनां चतुरुत्तरचत्वारिंशच्छताय कति ॥ । २ ॥
नव पद्मरागमणयस्समर्चिता एकजिनगृहे दृष्टाः ।
साष्टाशीतिद्विशतीमितवसतिषु ते कियन्तस्स्यु ॥ ३ ॥ }
चत्वारिंशच्चैकोनशताधिकपुष्यरागमणयोऽच्यः ।
एकस्मिन् जिनभवने सनवशते ब्रूहि कति ममयः ॥ ४ ॥
पझानि सप्तविंशतिरे कस्मिन् जिनगृहे प्रदत्तानि ।
साष्टानवतिसह "सनवशते तानि कति कथय ॥ ५ ॥
12
एकैकस्यां वसतोवष्टोत्तरशतसुवर्णपद्मानि ।
एकाष्टचतुस्सप्तकनवषयाष्टकानां किम् ॥ ६ ॥ -
1 K तत्र च. K and B विन्यस्योभौ राशी. 3 K and B सङ्गणयेत्.
+ B स्य हि. 5 B नस्या. ७ B शतस्य कति भवनानाम्.
• • • M and B चत्वारिंशद्यका शताधिका. 8 M ऽच्छाः .
५ M ते कियन्तस्स्युः . 0 M एकैकजिनालयय दत्तानि.
"M प्रयुक्तनवशतगृहाणां किम्, TE This stanza is found only in M and B. 10
- गणितसारसङ्गहः
शशिवसुरवरजलनिधिनवपदार्थभयनयसमूहमास्थाप्य ।
हिमकरविषनिधिगतिभिर्गुणिते कि ' राशिपरिमाणम् ॥ ७ ॥
हिमगुपयोनिधिगतिशशिवह्निव्रतनिचयमत्र संस्थाप्य ।
सैकाशीत्या त्वं मे गुणयित्वाचक्ष्व तत्सद्भयाम् ॥ ८ ॥
अग्निवसुवरभयेन्द्रियशशलाञ्छनराशिमत्र संस्थाप्य ।
रन्नैर्गुणयित्वा मे कथय सरवे राशिपरिमाणम् ॥ ९ ॥
नन्दादृतुशरचतुस्त्रिद्वन्द्वैकं स्थाप्य मत्र नवगुणितम् ।
आचार्यमहावीरैः कथितं नरपालकण्ठिकाभरणम् ॥ १० ॥
षट्रत्रिकं पञ्चषट्च सप्त चादौ प्रतिष्ठितम् ।
त्रयस्त्रिंशत्सङ्गणतं कण्ठाभरणमादिशत् ॥ ११ ॥
हुतवहगतिशशिमुनिभिर्वसुनयगतिचन्द्रमत्र संस्थाप्य ।
शैलेन तु गुणायत्वा कथयेदं रत्नकण्ठिकाभरणम् ॥ १२ ॥
अनलाब्धिहिमगुमुनिशरदुरिताक्षिपयोधिसोममास्थाप्य ।
शैलेन तु गुणयित्वा कथय त्वं राजकण्ठिकाभरणम् ॥ १३ ॥
गिरिगुणदिवीिगिरिगुणदिविगिरिगुणनिकरं तथैव गुणगुणितम्।
पुनरेवं गुणगुणितम् एकादिनवोत्तरं विद्धि ॥ १४ ॥
सप्त शून्यं द्वयं इन्डै पचैकश्च प्रतिष्ठितम् ।
त्रयःसप्ततिसङ्गण्यं "कण्ठाभरणमादिशेत् ॥ १५ ॥
जलनिधिपयोधिशशधरनयनद्रव्याक्षिनिकरमास्थाप्य ।
गुणिते तु चतुष्षष्टया का सर्वांचा गणितविद्रुहि ॥ १६॥
- M and B किन्तस्य. 2 3 प्यम्. अM अहं. * M मे शीघ्रम्.
• B विन्यस्य a Stanzas from 10 to 15 are found only in M and B.
१ A1] the MES. read स्थाप्य तत्र.6 B शे . 3 B नय
All the Mrs. give the metrically erroneous reading कण्ठाभरणं विनिदिशेत् ।
- परिकर्मव्यवहारः 11
शशाङ्कदुवैकेन्दुशन्यैकरूपं
निधाय क्रमेणात्र राशिप्रमाणम् ।
हिमांश्वग्ररन्यैः प्रसन्ताडितेऽस्मिन्
• भवेत्कण्ठिका राजपुत्रस्य योग्या ।। १७ ॥
इति परिकर्मविधौ प्रथमः प्रत्युत्पन्नः समाप्तः ।
- भागहारः ।
द्वितीये भागहारपरिकर्मणि करणसूत्रं यथा
विन्यस्य भाज्यमानं तस्याधस्स्थेन भागहारेण ।
सदृशपवर्तविधिना भागं कृत्वा फलं प्रवदेत् ॥ १८॥
अथ वा
प्रतिलोमपथेन भजेद्राज्यमधस्स्थेन भागहारेण ।
सदृशापवर्तनविधिर्यद्यस्ति विधाय तमपि तयोः ॥ १९ ।
अत्रोद्देशकः ।
दीनाराष्टसहवं द्वनवतियुतं शतेन संयुक्तम् ।
चतुरुत्तरषष्टिनरैर्भक्तं कऽशो नुरेकस्य ॥ २० ॥
रूपाग्रसप्तविंशतिशतानि कनकानि यत्र भाज्यन्ते ।
सप्तत्रिंशत्पुरुषेरेकस्यांशं ‘ममाचक्ष्व ॥ २१ ॥
दीनारदशसहखं त्रिशतयुतं सप्तवर्गसम्मिश्रम् ।
नवसप्तत्या पुरुषेर्भक्तं किं के लब्धमेकस्य ॥ २२ ॥
अयुतं चत्वारिंशच्चतुस्सहत्रैकशतयुतं ‘हेम्नाम् ।
नवसप्ततिवसतीनां दत्तं वित्तं किमेकस्याः ॥ २३ ॥
- This stanza is not found in P.
- K स. . , " M कोंऽशो नुरेकस्य.
5 B and K हेमम्
- This stanya is not found in P.
12
गणितसारसङ्गहः
सप्तदशत्रिशतयुतान्येकात्रिंशत्सहस्रजम्बूनि ।
भक्तानि नवत्रिंशन्नरैर्वदैकस्य भाग त्वम् ॥ २४ ॥
यधिकदशत्रिशतयुतान्येकत्रिंशत्सहस्रजम्बूनि ।
सैकाशीतिशतेन प्रहृतानि नरैर्वेदैकांशम ॥ २९ ॥
त्रिदशसह सैका षष्टिद्विशतीसहस्रपङ्कयुता ।
रत्नानां नवपुंसां दत्तैकनरोऽत्र किं लभते ॥ २६ ॥
एकादिषडन्तानि क्रमेण हीनानि हाटकानि सरवे ।
विधुजलधिबन्धसङ्घचैर्नरैर्हतान्येकभागः कः ॥ २७ ॥
यशीतिमिश्राणि चतुश्शतानि
चतुस्सहस्रन्ननगान्वितानि ।
रत्नानि दत्तानि जिनालयानां
त्रयोदशानां कथयैकभागम् ॥ २८ ॥
इति परिकर्मविधौ द्वितीयो भागहारः समाप्तः ।
वर्गः
तृतीये वर्गपरिकर्मणि करणसूत्रं यथा-
द्विसमवधो घातो वा स्वेटोनयुतद्वयस्य सेटकृतिः ।
एकादिद्विचयेच्छागच्छयुतिर्वा भवेद्वर्गः॥ २९ ॥
1 M reads the problem contained in this stanza thus :-
त्रिशतयुतैकत्रिंशत्सहस्रयुक्ता दशाधिकाः सप्त ।
भक्ताश्चत्वारिंशत्पुरुषेरेकोनैस्तत्र दीनारम् ॥
१ This stanza is found only in M.
- एकद्वित्रिचतुःपञ्चषहीनाः क्रमेण सम्भक्ताः ।
सैकचतुःशत संयुतचत्वारिंशज्जनालयानां किम् ॥
परिकर्मव्यवहारः 13
द्विस्थानप्रभृतीनां राशीनां सर्ववर्गसंयोगः ।
तेषां क्रमघातेन द्विगुणेन विमिश्रितो वर्गः ॥ ३० ॥
कृत्वान्त्यांत हन्याच्छषपदद्विगुणमन्त्यमुत्सायै ।
- शेषानुत्सयैवं करणीयो विधिरयं वर्गे ॥ ३१ ॥
अत्रोद्देशकः ।
एकादिनवान्तानां पञ्चदशानां द्विसङ्गणाष्टानाम् ।
व्रतयुगयोश्च रसाग्न्योश्शरनगयोर्वर्गमाचक्ष्व ॥ ३२ ॥
साष्टात्रिंशत्रिशती चतुस्सहमैकषष्टिषट्छतिका ।
द्विशती षट्पञ्चशान्मिश्रा वर्गीकृता किं स्यात् ॥ ३३ ॥
लेख्यागुणेषुबाणद्रव्याणां शरगातित्रिसूर्याणाम् ।
गुणरत्नाग्निपुराणां वर्णं भण गणक यदि वेसि ॥ १४ ॥
सप्ताशीतित्रिशतसहितं षट्सहवं पुनश्च
पञ्चत्रिंशच्छतसमधिकं सप्तनिघ्नं सहस्रम् ।
द्वाविंशत्या युतदशशतं वर्गितं तत्रयाणां
ब्रूहि त्वं मे गणक गुणवन्सङ्गणय्य प्रमाणम् ॥ ३९ ॥
इति परिकर्मविधौ तृतीयो वर्गस्समाप्तः ।।
वर्गमूलम् ।
चतुर्थे वर्गमूलपरिकर्मणि करणसूत्रं यथा
अन्त्यौजदपह्नरुतिमूलेन द्विगुणितेन युग्महतौ ।
लब्धकृतिस्त्याज्यौजे द्विगुणदलं वर्गमूलफलम् ।। ३६ ॥
P, K and B राशिरेतकृतीनाम्
14. गणितसारसङ्गहः
अत्रदशक ।
एकादिनवान्तानां वर्गगतानां वदाशु मे मूलम् ।
ऋतुविषयलोचनानां द्रव्यमहीनेन्द्रियाणाञ्च ॥ ३७ ॥
एकाश्रषष्टिसमधिकपञ्चशतोपेतषट्सहस्राणाम् ।
घदुर्गपञ्चपञ्चकषण्णामपि मूलमाकलय ॥ ३८ ॥
द्रव्यपदार्थनयाचललेख्यालब्ध्यब्धि निधिनयाब्धीनाम् ।
शशिनेत्रेन्द्रिययुगनयजीवानञ्चापि किं मूलम् ॥ ३९ ॥
चन्द्राब्धिगतिकषायद्रव्यप्तहुताशनतुराशीनाम् ।
विधुलेख्येन्द्रियहिमकरमुनिगिरिशशिनां च मूलं किम् ॥ ४० ॥
द्वादशशतस्य मूलं षण्णवतियुतस्य कथय सञ्चिन्त्य ।
शतषट्कस्यापि सरवे पञ्चकवर्गेण युक्तस्य ॥ ४१ ॥
अङ्गुभकर्माम्बरशङ्कराणां
सोमाक्षिवैश्वानरभास्कराणाम् ।
चन्द्रव्रबाणाब्धिगतिद्विपानामाचक्ष्व ।
मूलं गणकाग्रणीस्वम् ॥ ४२ ॥
इति परिकर्मविधौ चतुर्थे वर्गमूलं समाप्तम् ।
घनः ।
पञ्चमे घन पारेकमणे करणसूत्र यथा
त्रिसमाहतिर्घनस्स्यादिष्टोनयुतान्यराशिघातो वा ।
अल्पगुणितष्टकृत्या कलितो वृन्देन चेष्टस्य ॥ ४३ ॥
इष्टादिद्विगुणेष्टप्रचयेष्टपदन्वयोऽथ वेष्टकृतिः।
ज्येके ष्टहतैकादिद्विचयेष्टपदैक्ययुक्ता वा ।। ४४ ।।
एकादिचयेष्ठपदे पूर्व राशिं परेण सङ्गणयेत् ।
गुणितसमासांस्त्रगुणश्चरमेण युतां घनो भवति ॥ ४९ ॥
P and My वर्गगतानां शीघ्र रूपादिनवावसानराशीनाम् । मलं कथय सखें स्वं ।
- 'his stanza i+ not found in P.
•
A नव
परिकर्मव्यवहारः 15
अन्त्यान्यस्थानतः परस्परस्थानसङ्गण हिता ।
पुन रेवं तद्यो गस्सर्वपदघनान्वितो वृन्दम् ॥ ४६ ॥
भृत्यस्य धेनः कुतिरपि सा त्रिहतोत्सार्य शेषगुणिता वा ।
शेषकृतस्यन्त्यहत स्थाप्यत्सायेवमत्र विधिः ॥ ४७ !
अत्रादशकः ।
एकादिनवान्तानां पञ्चदशनां शरेक्षणस्याप ।
रसवह्नयोर्गिरिनणयोः कथय घन द्रव्यल3ध्योश्च ।। ४८ ।।
हिमकरगगनेन्दूनां नयगिरिशशिनां वरेन्दुमणनाम् ।
वद मुनिचयनां वृन्द चतुरुदधिगुणशशिना ॥ ४९ ॥
राशर्धनीकृतोऽयं शतद्वयं मिश्रितं त्रयोदशभिः ।
तद्दिगुणोऽस्मात्रिगुणश्चतुर्गुणः पवगुणेतश्च ॥ ६० ॥
शतमष्टषष्टियुक्तं दृष्टमभीष्ट घन विशिष्टतमैः ।
एककादिभिरष्टान्यैर्गुणितं वद तद्धनं शीघ्रम् ॥ ११ ॥
बन्धचतुगगनन्द्यकवनां
सह्याः क्रमेण विनिधाय घनं गृहीत्वा ।
आचक्ष्व ल5धमधुना करणनुयोग
गम्भरिसरतरसगरपरिदृश्वन् । १२ ॥
इति परिकर्मविधौ पञ्चमो घनस्समाप्तः ।।
11 °रपि. २ A ‘गो व .
3 This stanza is omitted in M. the following sta22 is found as a पाठान्तर
in P, K and B; though not quito exploit, it mention8 two of the processee above
&bscribed :-
त्रिसमगुणोऽन्यस्य घनस्तद्वर्गात्रिगुणितों हतशेधैः।
उत्सार्य शेषकृतिरथ निष्ठा त्रिगुणा घनस्तथाग्रे वा ।
P
Instead of stanzas 48 and 49, M{ reads
एकादिनवान्तानां रुद्राणां हिमकरेन्दूनाम् ।
वद मुनिचन्द्रयतीनां वृन्दं चतुरुदाधिगुणशशिनाम्।
16 गणितसारसङ्ग्रहः
घनमूलम् ।
षष्ठे घनमूलपरिकर्मणि करणसूत्रे यथा
अन्यघनादपहृतघनमूलकृतित्रिहतिभाजिते भाज्ये ।
प्राक्त्रिहताप्तस्य कृतिशोध्या शोध्ये घनेऽथ घनम् ॥ ५३ ॥
घनमेकं वे अघने घनपदकृत्या भजेत्रिगुणयाघनतः ।
पूर्वत्रिगुणाप्तऋतिस्त्याज्यान्नघनश्च पूर्ववछब्धपदैः ॥ १४ ॥
अत्रोद्देशकः ।
एकादिनवान्तानां घनात्मनां रत्नशशिनवाब्धीनाम् ।
‘नगरसवसरवतगजक्षपाकराणाञ्च मूलं किम् ॥ ९५ ॥
गतिनयमदशिविशशिनां मुनिगुणरववक्षिनव'खराग्नीनाम् ।
वसुवयुगवद्विगतिकरिचन्द्रतूनां गृहाण पदम् ॥ ५६ ॥
चतुःपयोध्यग्निशराक्षिदृष्टि
हयेभरवव्यांमभयंक्षणस्य ।
वदाष्टकमब्धिरवघातिभाव
द्विवह्निरलतुनगस्य मूलम् ।। १७ ॥
द्रव्याश्श्वशैलदुरतरववह्यद्भयस्य वदत घनमूलम् ।
नवचन्द्रहिमगुमुनिशशिल5ध्यम्घररवरयुगस्यापि ॥ १८ ॥
गतिगजविषयेषुविधुराद्विकरगतयुगस्य भण मूलम् ।
लेख्याश्वनगनवाचलपुरवरनयजीवचन्द्रमसम् ॥ १९ ॥
गतिवरदुरितेभाम्भोधितार्थध्वजाक्ष
द्विकुतिनवपदार्थद्रव्यवहन्दुचन्द्र-
जलधरपथरन्ध्रस्वष्टकानां घनानां
गणक गणितदक्षचक्ष्व मूलं परीक्ष्य ॥ ६० ॥
इति परिकर्मविधौ षष्ठे घनमूलं समाप्तम् ॥
1 This stanza is not found in 1. • M गिरि S M ग्रस.
के विधुपुरखरस्वनृज्वलनधराणां '. २ 'This ston2 is not found in M
गणितसारसङ्ग्रहः 17
सङ्कलितम् ।
सप्तमे सङ्कलितपरिकर्मणि करणसूत्रं यथा --
रूपेणोपे गच्छो दलीकुतः प्रचयताडितो मिश्रः ।
प्रभवेण पदाभ्यस्तस्सङ्कलितं भवति सर्वेषाम् ॥ ६१ ॥
प्रकारान्तरेण धनानयनसूत्रम् --
'एकविहीनो गच्छः प्रचयगुणो द्विगुणितादिसंयुक्तः ।
गच्छाभ्यस्तो द्विहतः प्रभवेत्सर्वत्र सङ्कलितम् ॥ ६२ ॥
आधुत्तरसवधनानयनसूत्रम्--
पदहतमुखमादिधनं व्येकपदाम्नचयगुण गच्छः ।
उत्तरधनं तयोयगो धनमूनोत्तरं मुरवेऽन्त्यधने ॥ ६३ ॥
अन्यधनमध्यधनसर्वधनानयनसूत्रम्-
चयगुणितैकोनपदं सद्यन्त्यधनं तदादियोगार्धम् ।
मध्यधनं तत्पदवधमुद्दिष्टं सर्वसङ्कलितम् ॥ ६४ ॥
अत्रोद्देशकः ।
एकादिदशान्ताद्यास्तावत्प्रचथास्समर्चयन्ति धनम् ।
वणिजो दश दश गच्छास्तेषां सङ्कलितमाकलय ॥ ६५ ॥
द्विमुवत्रिचयैर्मणिभिः प्रनखी श्रावकोत्तमः कश्चित् ।
पञ्चवसतीरमीषां का सह्या बृहि गणितज्ञ ॥ ६६ ॥
आदित्रयश्चयोऽर्थ द्वादश गच्छत्रयाऽप रूपेण ।
आ सप्तकात्प्रवुडस्सर्वेषां गणक भण गणेतम् ॥ ६ ७ ॥
द्विकृतिभुवं चयोऽर्थो नगरहवें समाचितं गणितम् ।
गणिताब्धिसमुत्तरणे बाहुबलिन् त्वं समाचक्ष्व ॥ ६८ ॥
A तदूना सैक(व ?)पदासा युतिः प्रभवः ।
This stanza is omitted in M. • M बली.
18 गणितसारसङ्गहः
गच्छानयनसूत्रम्
अष्टोत्तरगुणराशेर्डिगुणाद्युत्तरविशेषातिसहितात् ।
मूलं चययुन मर्धितमाङ्कनं चयहतं गच्छः i ६९॥
प्रकारान्तरेण गच्छनयनसूत्रम्
अष्टोत्तरगुणराशेर्डिगुणावुत्तरविशेषकृतिसाहितात् ।
मूलं क्षेपपदोनं दलितं चयभाजितं गच्छः ॥ ७० ॥
अत्रोद्देशकः ॥
आदिदं प्रचयोऽर्थौ द्वौ रूपेणा त्रयात्क्रमाइडौ ।
रवाजे रसाद्रिनेत्रं रवेन्दुझरा वित्तमत्र को गच्छ ॥ ७१ ॥
आदिः पञ्च चयोऽष्टं गुणराग्निधनमत्र को गच्छः ।
षट् प्रभवश्च चयोऽष्टौ वह्निचतुस्त्वं पदं किं स्यात् ॥ ७२ ॥
उत्तराद्यानयनसूत्रम्-
आदिधनोनं गणितं पदोनपदछतिदलेन सम्भजितम् ।
प्रचयस्तद्धनहीनं गणि पदभाजितं प्रभवः ॥ ७३ ॥
आधुत्तरानयनसूत्रम् =
प्रभवो गच्छाधनं विगतैकपदार्थगुणितचयहीनम् ।
पदहतधनमथुनं निरेकपददलहतं प्रचयः ॥ ७४ ॥
प्रकारान्तरणात्तराद्यानयनसूत्रद्वयम्
द्विहतं सङ्कलितधनं गच्छहृतं द्विगुणितादिना रहितम् ।
विगतैकपदविभक्तं प्रचयस्स्यादिति विजानीहि ॥ ७१ ॥
द्विगुणितसङ्कलितधनं गच्छहतं रूपरहितगच्छेन ।
ताडितचयेन रहितं द्वयेन सम्भाजितं प्रभवः ॥ ७६ ॥
परिकर्मव्यवहारः 19
अत्रोद्देशकः ।
नव वदनं तस्वपदं भावाधिकशतधनं कियान्प्रचयः ।
पञ्च चमोऽष्ट पदं षट्पञ्चाशच्छतधनं मुरवं कथय ॥ ७७ ॥
स्वेष्टाद्युत्तरगच्छानयनसूत्रम्-
सङ्कलिते वेष्टहृते हारो गच्छोऽत्र लब्ध इष्टोने ।
अनितमादिशेषे व्येकपदाखूर्द्धते प्रचयः ॥ ७८ ॥
अत्रोद्देशकः ।
चत्वारिंशत्सहिता पञ्चशती गणतमत्र सन्दृष्टम् ।
गच्छप्रचयप्रभवान् 'गणितज्ञशिरोमणे कथय । ७९ ॥
आधुत्तरगच्छसर्वमिश्रधनविश्लेषणे सूत्रत्रयम् -
उत्तरधनेन रहितं गच्छेनैकेन संयुतेन हतम् ।
मिश्रधनं प्रभवस्स्यादिति गणकशिरोमणे विद्धि ॥ ८० ॥
आदिधनोनं मित्रं रूपोनपदार्थगुणितगच्छेन ।
सैकेन हृते प्रचयो गच्छविधानात्पदं मुरवे सैके ॥ १ ॥
मिश्रादपनीतेथे मुरवगच्चैौ प्रचयमिश्रविधिलब्धः।
यो राशिस्स चयस्स्यात्करणमिदं सर्वसंयोगे ॥ ८२ ॥
अत्रोद्देशकः ।
द्वित्रिकपञ्चदशाग्रा चत्वारिंशन्मुखादिमिश्रधनम् ।
तत्र प्रभवं प्रचयं गच्छं सर्वं च मे ब्रूहि ॥ ३ ॥
M विगणय्य सखे ममाचक्ष्व.
M पदोनपदकृतिदलैन सैकेन ।
भक्तं प्रचयोऽत्र पदं गच्छविधानान्मुखे सैके ।
20 गणितसारसङ्गहः
दृष्टधनाद्युत्तरतो द्विगुणत्रिगुणद्विजागत्रिभागादीष्टधनाद्युत्तरानयन
सूत्रम् दृष्टविभक्तेष्टधनं द्विष्टं नत्नचयताडितं 'प्रचयः।
तत्प्रभवगुणं प्रभवो 'गुणभागस्येष्टवित्तस्य ॥ ८४ ॥
अत्रोद्देशकः ।
समगच्छश्चत्वारष्षष्टिर्मुखमुत्तरं ततो द्विगुणम् ।
तद्दयादि हतविभक्तस्वेष्टस्याद्युत्तरे ब्रूहि ॥ ८५ ॥
इष्टगच्छयोर्यस्ताद्युत्तरसमधनद्विगुणत्रिगुणद्विभागत्रिभागादिधनान-
यनसूत्रम् व्येकात्महतो गच्छस्वेष्टनो द्विगुणितान्यपदहीनः ।
मुवमात्मोनान्यकृतिर्दिकेष्टपदघातवर्जिता। प्रचयः ॥ ८६ ॥
अत्रोद्देशकः ।
पश्चाष्टगच्छपुस व्र्यस्तप्रभवोत्तरे समानधनम् ।
द्वित्रिगुणादेधनं वा ब्रूहि त्वं गणक विगणय्य ॥ ८७ ॥
द्वादशषोडशपदयोर्यस्तप्रभवोत्तरे समानधनम् ।
व्यादिगुणभागघनमपि कथय त्वं गणितशास्त्रज्ञ ॥ ८८ ॥
असमानोत्तरसमगच्छसमघनस्याद्युत्तरानयनसूत्रम्--
अधिकचयस्यैकादिश्चाधिकचयशेषचयविशेषो गुणितः।
विगतैकपदार्थेन सरूपश्च मुरवानि मित्र शेषचयानाम् ॥ ८९ ॥
अत्रोद्देशकः ।
एकादे षडन्तचयानामेकांत्रतयपञ्चसप्तचयानाम् ।
नवनवगच्छानां समवित्तानां चाशु वद मुरवानि सखे ।। ९० ॥
1 M गुणभागावुत्तरेच्छायाः . १ M गुण '. • A गणकमुखतिलक ।
परिकर्मव्यवहारः 21
विसदृशादसदृशगच्छसमधनानामुत्तरानयनसूत्रम्--
अधिकमुखस्यैकचयश्चाधिकमुखशेषमुखविशेषो भक्तः ।
विगतैकअदाधेन सरूपश्च चया भवन्ति शेषमुवानम् ॥ ११ ॥
अत्रोद्देशकः।
एकत्रिपञ्चसप्तनवैकादशवदनपञ्चपञ्चपदानम् ।
समांवत्तानां कथयात्तराणं गणेताब्धिपारदृश्वन् गणक ॥ ९२ ॥
अथ गुणधनगुणसङ्कलितधनयोस्सूत्रम्
पदमितगुणहतिगुणितप्रभवस्याह्नणधनं तदायूनम् ।
एकोनगुणविभक्तं गुणसङ्कलितं विजानीयात् ॥ ९३ ॥
गुणसङ्कलिते अन्यदपि सूत्रम् –
समदलविषमखरूपो गुणगुणितो वर्गतडितो गच्छ ।
रूपोनः प्रभवघ्नो व्येकोत्तरभाजितस्सारम् ॥ ९४ ॥
गुणसङ्कलितान्यधनानयने तत्सङ्कलितधनानयने च सूत्रम्-
गुणसङ्कलितान्यधनं विगतैकपदय गुणधनं भवति ।
ततृणगुणं मुखोनं व्येकोत्तरभाजितं सारम् ॥ ९१ ॥
गुणधनस्यादाहरणम् ।
स्वर्णद्वयं गृहीत्वा त्रिगुणधनं प्रतिपुरं 'समार्जयति ।
यः पुरुषोऽष्टनगर्यां तस्य कियद्वित्तमाचक्ष्व ॥ ९६ ॥
गुणधनस्याद्युत्तरानयनसूत्रम् –
गुणधनमादिविभक्तं यत्पदमितवधसमं स एव चयः।
न गच्छप्रमगुणघातप्रहतं गुणितं भवेत्प्रभवः ॥ ९७ ॥
- % समर्चयति.
22 गणितसारसङ्गहः
गुणधनस्य गुच्छानयनसूत्रम्
मुखक्षते गुणवत्ते यथा निग्रं तथा गुणेन हृते ।
यावत्योऽत्र शलाकास्तावान् गच्छो गुणधनस्य ॥ ९ ॥
गुणसङ्कलितोदाहरणम् ।
दीनारपञ्चकादिद्विगुणं धनमर्जयन्नरः कश्चित् ।
प्राविशदष्टनगरीः कति जातास्तस्य दीनाराः ॥ ९९ ॥
सप्तमुखात्रिगुणचयत्रिवर्गगच्छस्य किं धनं वणिजः ।
त्रिकपञ्चकपञ्चदशप्रभवगुणोत्तरपदस्यापि ॥ १०० ॥
गुणसङ्कलितोत्तराद्यानयनसूत्रम् –
असकृचेकं मुखहतवित्तं येनोद्धृतं भवेत्स चयः ।
व्येकगुणगुणितगणीतं निरेकपदमात्रगुणवधातुं प्रभवः ।। १०१ ॥
अत्रोद्देशकः ।
त्रिमुवर्तुगच्छणाङ्काम्वरजलनिधिधने कियान्प्रचयः ।
षङ्गणचयपञ्चपदाम्वरशशिहिमचुत्रवित्तमत्र मुखं किम् ॥ १०२ ॥
गुणसङ्कलितगच्छानयनसूत्रम् .
एकोनगुणाभ्यस्ते प्रभवहृतं रूपसंयुतं वित्तम् ।
यावत्कृत्वो भक्तं गुणेन तद्रसम्मितिर्गच्छः ॥ १०३ ॥
अत्रोद्देशकः ।
त्रिप्रभवं पटुगुणं सारं सप्तत्युपेतसप्तशती ।
सप्ताग्रा ब्रूहि सरवे कियत्पदं गणक गुणनिपुण ॥ १०४ ॥
पधादिद्विगुणोत्तरे शरगिरिर्थकप्रमाणे धने
सप्तादि' त्रिगुणे नगेभदुरितस्तम्बेरमतृप्रमे ।
1 M .
परिकर्मव्यवहारः 28
यास्ये पञ्चगुणाधिके हुतवहोपेन्द्राक्षवह्निद्विप
श्वेतांशुद्रिदेभकर्मकरढञ्यानेऽपि गच्छः कियान् ॥ १०५ ॥
इति परिकर्मन्निधौ सप्तमं सङ्कलितं समाप्तम् ।
व्युत्कलतम् ।
अष्टमे व्युत्कलितपरिकर्मणि करणसूत्रं यथा -
सपर्दष्टं वेष्टमपि व्येकं दलितं चयाहतं समुरवम् ।
शेषेष्टगच्छगुणितं व्युत्कलितं स्वेष्टचित्तं च ॥ १०६ ॥
प्रकारान्तरेण व्युत्कालितधनस्वष्टधनानयनसूत्रम् -
गच्छसहितेष्टमिष्टं चैकोनं चयहतं द्विहादियुतम् ।
शेषेष्टपदार्धगुणं व्युत्कलितं स्वेष्टवित्तमपि ॥ १०७ ॥
चयगुणक्षवव्युत्कलिनधनानयने व्युत्कलितधनस्थ शेषेष्टगच्छान
यने च सूत्रम् -
इष्टधनोनं गणितं व्यवकलितं चयभवं गुणोत्थं च ।
सर्वेष्टगच्छशेषे शेषपदं जायते तस्य ॥ १०८ ॥
शेषगच्छस्याद्यानयनसूत्रम्
प्रचयगुणतष्टगच्छस्सादः प्रभवः पदस्य शषस्य ।
प्राक्तन एव चयस्याङ्गच्छस्यष्टस्यं तावव ॥ १०९ ॥
गुणव्युत्कालितशेषगच्छस्थाद्यानयनसूत्रम्-
गुणगुणितेऽपि चयादी तथैव भेदोऽयमत्रशेषपदे ।
इष्टपदमितिगुणाहतिगुणितप्रभवो भवेद्दक्रम् ॥ १११ ॥
- MI गणितं.
24 गणितसारसङ्गहः
अत्रोद्देशकः ।
द्विमुखस्त्रिचयो गच्छश्चतुर्दश खेग्मितं पदं सप्त ।
अष्टनवषट्पञ्च च किंव्युत्कलतं समाकलय ।। १११ ॥
षडादिरष्टौ प्रचयोऽत्र षटूतिः
पदं दश द्वादश षोडशेप्सितम् ।
मुखादिरन्यस्य तु पञ्चपञ्चकं
शतद्वयं हि शतं व्ययः कियान् ॥ ११२ ॥
षडुनमानो गच्छः प्रचयोऽर्थं द्विगुणसप्तकं वक्रम् ।
सप्तत्रिंशत्वेष्टं पदं समाचक्ष्व फलमुभयम ॥ ११३ ॥
अष्टकृतिरादिरुत्तरसूनं चत्वारि षोडशात्र पदम्।
• इष्टानि तत्वकेशवरुद्रार्कपदनि कि शेषम् ॥ ११४ ॥
गुणव्युत्कलिकस्योदाहरणम्
चतुरादिद्विगुणात्मकोत्तरयुतो गच्छश्चतुर्ण छति-
देश वाञ्छापदमङ्गसिन्धुरगिरिद्रव्येन्द्रियाम्भोधयः।
कथय व्युत्कलितं फलं सकलसद्भजामिमं व्याप्तवान्
करणस्कन्धवनान्तरं गणितविन्मत्तेभविक्रीडितम् ॥ ११५ ॥
इति परिकर्मविधावष्टमं व्युत्कलितं समाप्तम् ॥
इति सारसङ्गहे गणितशास्त्रे महावीराचार्यस्य कृतौ परिकर्मनामा प्रथम व्यवहारः समाप्तः ।
- M प्रा.
25 अथ द्वितीयः कलासवर्णव्यवहारः ।
त्रिलोकराजेन्द्रकिरीटकोटिप्रभाभिरालीदपदारवन्दम् ।
निर्मूलमुन्मूलितकर्मवृक्ष जिनेन्द्रचन्द्रं प्रणमामि भक्त्या ॥ १ ॥
इतः परं कलासवर्ण द्वितीयव्यवहारमुदाहरिष्यामः ॥
भिन्नप्रत्युत्पन्नः ।
तत्र भिन्नप्रत्युत्पन्न करणसूत्र यथा
गुणयेदंशानंशैर्हरान् हरैर्धठेत यदि तेषाम् ।
वत्रापवर्तनविधिविधाय तं भिन्नगुणकारे ॥ २ ॥
अत्रोद्देशकः ।
शुण्ठ्याः पलेन लभते चतुर्नवांशं पणस्य यः पुरुषः।
किमसौ बृह सरवे त्वं त्रिगुणेन पलाष्टभागेन ॥ ३ ॥
मरिचस्य पलस्यार्घः पणस्य समष्टमांशको यत्र।
तत्र भवेतक मूल्यं पलषट्पन्नांशकस्य बद ॥ ४ ॥
कथपणन लभते त्रपञ्चभाग पलस्य पप्परयाः ।
नवभिः परैर्दिक्षक्तैः किं गणकाचक्ष्व गुणयित्वा ॥ १ ॥
क्रीणाति पणेन वणिजीरकपलनवदशशकं यत्र ।
तत्र पणैः पञ्चधः कथय त्वं कि समग्रमते ॥ ६ ॥
द्यादयो द्वितयवृद्धयोंऽशका
स्यादयो इयचया हराः पुनः ।
ते द्वये दशपदाः कियत्फलं
ब्रूहि तत्र गुणने द्वयोर्दयोः ॥ ७ ॥
इति भिन्नगुणकारः।
This stanza is omitted in P.
x मौ.
28 गणितसारसङ्ग्रहः
भिन्नभागहारः।
भिन्नभागहारे करणसूत्रं यथा
अशकृत्यच्छेद प्रमाणराशस्ततः क्रिया गुणवत् ।
प्रमितफलेऽन्यहरने विच्छिदि वा सकलवच्च भागहृतौ ॥ ८ ॥
अत्रोद्देशक ।
हिङ्गोः पलार्धमौल्यं पणत्रिपादांशको भवेद्यत्र ।
तत्रायै विक्रीणन् पलमेकं किं नरो लक्षते ॥ ९ ॥
अगरेः पलाष्टमेन त्रिगुणेन पणस्य विंशतिवंशान् ।
उपलभने यत्र पुमानेकेन पलेन किं तत्र ॥ १० ॥
पणपञ्चमैश्चतुर्भिर्नरवस्य पलसप्तमो ह्यशीतिगुणः।
संप्राप्यो यत्र स्यादकेन पणेन किं तत्र ॥ ११ ॥
व्यादिरूपपरिवृद्धियुजोंऽश
यावदष्टपदमेकाविहीनाः ।
हरकाशत इह द्वितयाद्यः
किं क फलं वद परेषु हतेषु ॥ १२ ॥
इति भिन्नभागहारः ।
भिन्नवर्गवर्गमूलघनघनमूलानि ।
भिन्नवर्गवर्गमूलघनघनमूलेषु करणसूत्रं यथा
कृत्वाच्छेदांशकयोः कृतिकृतिमूले घनं च घनमूलम् ।
तच्छेदैशहूतौ वर्गादिफलं भवेद्दिने ॥ १३ ॥
- M भिन्नवर्गभित्रवर्गमूलभिन्नघनतन्मूलेषु
कलासवर्णव्यवहारः 27
अत्रोद्देशकः ।
पञ्चकसप्तनवानां दलितानां कथय गणक वर्गे त्वम्।
षोडशविंशतिशतकद्विशतानां च त्रिभक्तानाम् ॥ १४ !
त्रिकादिरूपद्वयवृद्धयोंऽशा
दैकादरूपांतरका हराया ।
पद मत द्वादश वगेमष
वदाशु मे त्वं गणकाग्रगण्य ॥ १५ ॥
पादनवांशकषोडशभागानां पञ्चविंशतितमस्य ।
षट्त्रशद्भागस्य च कृतिमूलं गणक भण शीघ्रम् ॥ १६ ॥
भन्न वर्गों राशयो वर्गिता ये
तेषां मूलं सप्तशत्याश्च कि स्यात् ।
च्यटोनायाः पञ्चवगद्धतया
ब्रूहि त्वं मे वर्गमूळे प्रवीण ॥ १७ ॥
अधीत्रिभागपादाः पञ्चांशक षष्ठसप्तमाष्टांशाः ।
दृष्टानवमश्चैषां पृथक् पृथगब्रूहि गणक घनम् ॥ १ ॥
त्रितयादेचतुश्चयकाऽशगण
द्विमुखद्विचयोऽत्र हरप्रचयः।
दशकं पदमाशु तदीयघनं
कथय प्रिय सूक्ष्ममते गणिते ।। १९ ॥
शतकरय पञ्चविंशस्याष्टविभक्तस्य कथय घनमूलम् ।
नवयुतसप्तशतानां विंशानामष्टभक्तानाम् ॥ २० ॥
- M सप्तशतस्यापि सखें ध्येकोनत्रिंशकाष्टकाप्तस्य ।
28 गणितसारसङ्ग्रहः
भिन्नघने परिदृष्टघनानां
मूलमुदग्रमते वद मित्र ।
यूनशतद्वययुर्द्वसहया
थापि नवप्रहतात्रिहृतायाः ॥ २१ ॥
इति भिन्नवर्गवर्गमूलघनघनमूलानि ॥
भिन्नसङ्कलितम् ।
भिन्नसङ्कलिते करणसूत्र यथा --
पदमिष्टं प्रचयहतं द्विगुणप्रभवान्वितं चयेनोनम् ।
गच्छाधेनाभ्यस्तं भवति फलं भिन्नसङ्कलिते ॥ २२ ॥
अत्रोद्देशकः ।
द्विध्र्यशष्षड्भागस्त्रिचरणगो मुरवं चयो गच्छः ।
द्वौ पञ्चमी त्रिपादो द्विपंशोऽन्यस्य कथय कि वित्तम् ।। २३ ॥
आदः प्रचय गच्छस्त्रपञ्चमः पञ्चमस्त्रपदांशः ।
सर्वांशहरौ वृद्धौ द्वित्रिभिरा सप्तकाच का चितिः ॥ २४ ॥
इष्टगच्छस्याद्युत्तरवगरूपघनरूपधनानयनसूत्रम् –
पदमिष्टमकमादव्यंकष्टदलङ्कत मुवनिपदम् ।
प्रचयो वित्त तेषां वगों गच्छाहतं बृन्दम् ॥ । २१ ॥
अत्रोद्देशकः।
'पदमिष्टं द्वित्र्यंशो रूपेणांशो हरश्च संवृद्धः।
यावद्दशपदमेषां वद मुरवचपवर्गवृन्दानि ॥ २६ ॥
1 This stanya is not found in M
कलासवर्णव्यवहारः 29
इष्टघनधनाद्युत्तरगच्छानयनसूत्रम् --
इष्टचतुर्थः प्रभवः प्रभवत्प्रचय भवेद्दिसङ्गणितः ।
प्रचयश्चतुरभ्यस्तो गच्छस्तेषां युतिबृन्दम् ॥ २७ ॥
अत्रोद्देशकः ।
द्विमुवैकचया अंशास्त्रिप्रभवैकोत्तरा हरा उभय ।
पञ्चपदा वद तेषां घनधनमुवचयपदानि सरवे ॥ २८ ॥
दृष्टधन्युत्तरतो द्विगुणत्रिगुणद्विभागत्रिभागादीष्टधनाद्युत्तरानयन सूत्रम्
दृष्टविभक्तेष्टधनं द्विष्टं तत्प्रचयताडितं प्रचयः ।
तत्प्रभवगुणं प्रभवो गुणभागस्येष्टवित्तस्य ॥ २९ ॥
अत्रोद्देशकः ।
प्रभवस्थध रूपं प्रचयः पञ्चाष्टमस्समानपदम् ।
इच्छधनमाप तावत्कथय सरव क मुखपचयौ ॥ ३० ॥
प्रचयादादिद्विगुणत्रयादशाष्टादश पदं दृष्टम् ।
वित्तं तु सप्तषष्टिः षड्नभक्ता वदादिचयौ ॥ ३१ ॥
मुखमेकं द्वित्र्यंशः प्रचयो गच्छस्समश्चतुर्नवमः ।
धनमिष्टं द्वाविंशातरेकाशीत्याि वदादिचयौ ॥ ३२ ॥
गच्छानयनसूत्रम् -
द्विगुणचयगुणितावित्तादुत्तरदलमुखविशेषकृतिसहितात् ।
मूलं प्रचयार्धयुतं प्रभवोनं चयहतं गच्छः ॥ ३३ ॥
र A गुणभागयुत्तरानयनसूत्रम् । १ 1 प्रचयेन
२ MI गुणभागयुत्तरेच्छयाः
'A This stanza takes the place of stanya No. 31 in M and is omitted in B.
- Instead of the following अष्टोत्तरगुणराशीत्यादिना इष्ट
two stanzas Mreads
धनगच्छ आनेतव्यः and repeats stanza No. 70 given under परिक्रमव्यवहार
30 गणितसारसङ्गहः
प्रकारान्तरेण तदेवाह –
द्विगुणचयगुणितवित्तादुत्तरदलमुखविशेषकृतिसहितात् ।
मूलं क्षेपपदोनं प्रचयेन हृतं च गच्छस्स्यात् ॥ । ३४ ॥
अत्रोद्देशकः ।
द्विपचांश वक्र त्रिगुणचरणस्स्यादिह चयः
पङशस्सप्तन्नास्त्रिछतिविहृतो वित्तमुदितम् ।
चयः पशष्टांशः पुनराप मुखं यष्टममिति
त्रिचत्वारिंशास्वं प्रिय वद पदं शीघ्रमनयोः ॥ ३१ ॥
आद्युत्तरानयनसूत्रम्
गच्छाप्तगणितमादिर्विगतैकपदार्थगुणितचयहीनम् ।
पदहृतधनमधून निरेकपद्दलहुत प्रचयः ॥ ३६ ॥
अत्रोद्देशकः ।
त्रिचतुर्थचतुःपञ्चमचयगच्छे खेषुशशिहतैकत्रिंशद् ।
वित्ते च्यंशचतुःपञ्चममुखगच्छे च वद मुखं प्रचयं च ॥ ३७ ॥
इष्टगच्छयोव्र्यस्ताद्युत्तरसभधनद्विगुणत्रिगुणद्विभाणत्रिभागधनानयः
नसूत्रम् येकात्भहतो गच्छस्वेष्टनो द्विगुणितान्यपदहीनः ।
मुखमात्मनन्यकृतिर्दिकेष्टपदधातवर्जिता प्रचयः ॥ ३८ ॥
अत्रोद्देशकः।
एकादिगुणविभागस्वं व्यस्तावुत्तरे हि वद मित्र ।
द्वित्र्यंशेनैकादशपञ्चांशकमिश्रनवपदयोः ॥ ३९ ॥
A K and JB प्रभवो गच्छाप्तधनम्
31 कलासवर्णव्यवहारः
गुणधनगुणसङ्कलितधनयोः सूत्रम् --
पदमितगुणहतिगुणितप्रभवः स्याह्नणधनं तदायूनम् ।
एकोनगुणविभक्तं गुणसङ्कलितं विजानीयात् ।। ४० ।।
गुणसङ्कलितान्त्यधनानयने तसङ्कलितानयने च सूत्रम्--
गुणसङ्कलितान्त्यधनं विगतैकपदस्य गुणधनं भवति ।
तदृणगुणं भुवनं व्येकोत्तरभाजितं सारम् ॥ ४१ ॥
अत्रोद्देशकः ।
प्रभवोऽष्टमश्चतुर्थः प्रचधः पञ्च पदमत्र गुणगुणितम् ।
गुणसङ्कलितं तस्यान्यधनं चाचक्ष्व मे शीघ्रम् ॥ ४२ ॥
'गुणधनसङ्कलित धनयोराद्युत्तरपदान्यपि पूर्वोक्तसूत्रैरानयेत् ।
समानेष्टोत्तरगच्छसङ्कलितगुणसङ्कलितसमधनस्याद्यानयनसूत्रम-
मुखमेकं चयगच्छाविौ मुखवितराहितगुणचित्या ।
हृतचयधनमादगुणं मुखं भवेदिचतधनसाम्ये ॥ ४३ ॥
अत्राद्देशकः ।
भाववार्धिभुवनानि पदान्य
क्षोधिपथमुनयस्त्रिहृतास्ते ।
उत्तरााण वदनानि काति स्यु-
युग्मसङ्कलेतांवत्तसमेषु ॥ ४४ ॥
इति भिन्नसङ्कलितं समाप्तम् ।
भिन्नव्युत्कलितम् ।
भिन्नव्युत्कलिने करणसूत्र यथा --
गच्छाधिकेष्टमिष्टं चयहतमूनोत्तरं द्विहादियुतम् ।
शेषेष्टपदार्थगुणं व्युत्कलितं स्वेष्टवित्तं च ॥ ४९ ॥
- Found cnly in B.
32 गणितसारसङ्गहः
शेषगच्छस्याद्यानयनसूत्रम्
प्रचयाधनः प्रभवो युतश्चयनष्टपदचयाञ्चम्याम् ।
शेषस्य पदस्यादिश्चयस्तु पूवक्त एव भवेत् ॥ ४६ ॥
गुणगुणितेऽपि चयादी तथैव भेदोऽयमत्र शेषपदे ।
इष्टपदमितगुणाहतिगुणितप्रभवो भवेद्वक्रम् ।। ४७ ।।
अत्रोद्देशकः ।
पादोत्तरं दलास्यं पदं त्रिपादांशकस्समुद्दिष्टः ।
स्वेष्टं चतुर्थभागः किं व्युत्कलितं समाकलय ॥ ४८ ॥
प्रभवोऽर्ध पञ्चांशः प्रचय द्विध्यंशको भवेद्च्छः ।
पचेष्टांशस्वंष्ट पदमृणमाचक्ष्व गणितज्ञ ॥ ४९ ॥
आदिश्चतुर्थभागः प्रचयः पञ्चांशकास्त्रिपधांशः ।
गच्छ वाञ्छागच्छो दशमो व्यवकलितमानं किम् ॥ ५० ॥
त्रिभागौ द्वौ वक्र पञ्चमांशश्चयस्स्यात् ।
पदं त्रिन्नः पादः पवमवष्टगच्छः ।
षडशस्सप्तांश व व्ययः का वद त्वम्
कलावास प्रज्ञाचान्द्रकास्वादिन्दो ॥ ५१ ॥
द्वादशपदं चतुर्थणोत्तरमधेनपञ्चकं वदनम् ।
त्रिचतुःपचाष्टष्टपदान व्युत्कालतमाकलय ॥ ५२ ॥
- 2 प्रचयगुणितेष्टगच्छस्सादिः प्रभवः पदस्य शेषस्य ।
पूर्वोक्तः प्रचयस्स्यादिष्टस्य प्राक्तनादेव ॥
- M [ च चतुभाग .
3 M कि व्युत्कलतं समाकलय.
कलासवर्णव्यवहारः 33
गुणसङ्कलितव्युत्कालितोदाहरणम् ।
द्वित्रिभागरहिताष्टमुखं द्वि-
त्र्यंशको गुणचयोऽष्ट पदं भोः।
मंत्र रत्नगतिपञ्चपदानी-
ष्टानि शषमुखवित्तपदं कम् ॥ ५३ ॥
इति भिन्नव्युत्कालतं समाप्तम् ।।
कलासवर्णषड्जातिः ॥
इतः परं कलासवणें षड्जातिमुदाहरिष्यामः
गप्रभागवथ भागभाग
भागानुबन्धः परिकीर्तितोऽतः ।
फागपवाहस्सह भागमात्रा
षड़जातयोऽमुत्र कलासवणें ॥ ५४ ॥
भागजातः ।
तत्र आगजात करणसूत्र यथा
सदृशहतच्छेदहतौ मिथोंऽशहारौ समच्छिदावंशौ ।
लुप्तकहरा योज्यौ त्याज्यौ वा आगजातिविधौ ॥ ५५ ॥
प्रकारान्तरेण समानच्छदमुद्वर्तुमुत्तरसूत्रम्
छेदापवर्तकानां लब्धानां चाहतौ निरुद्धः स्यात् ।
हरहृतानिरुद्वगुणिते हारांशगुणे समो हारः ॥ ५६ ॥
( K and Mr add after this इति सारसङ्गहे महावीराचार्यस्य कृतौ द्वितीयस्य -
रस्सामप्तः• This, however, seems to be a mistake .
१ This and the stanza following are not found in MN
34 गणितसारसङ्गहः
अत्रोद्देशकः ।
जम्बूजस्वीरनारङ्गचोचमोचाम्रदाडिमम् । ।
अकैषीद्दळघइफागद्वादशांशकविशकैः ॥ ५७ ॥
हेम्नस्त्रिशचतुर्विधानाष्टमेन यथाक्रमम् ।
श्रावको जिनपूजायै तद्योगे किं फलं वद ॥ ५८ ॥
अष्टपञ्चदश विंशं सप्तषत्रिंशदंशकम् । ।
एकादशत्रिषष्ट्यंशमेकविंशं च स इक्षिप ॥ ५९ ॥
एकद्विकत्रिकाद्येकोत्तरनवदशकषोडशान्त्यहराः।
निजनिजमुरवप्रमांशाखपराभ्यस्ताश्च किं फलं तेषाम् ॥ ६० ॥
एकद्विकत्रिकाद्याश्चतुराद्याश्चैकवुद्धिका हाराः ।
निजनिजमुखप्तमांशः स्वसन्नपराहताः क्रमशः ॥ ६१ ॥
विंशत्यन्ताः षड्गुणसप्तान्ताः पञ्चवर्गपश्चिमकाः ।
षट्त्रिंशत्पाश्चात्याः सङ्क्षेपे किं फलं तेषाम् ॥ ६२ ॥
चन्दनघनसारागरुकुङ्कममकृष्ट जिनमहाय नरः।
चरणदळविंशपक्षमभागैः कनकस्य किं शेषम् ॥ ६३ ॥
पादं पञ्चांशमधं त्रिगुणितद्रशमं सप्तविंशांशकञ्च
स्वर्णद्वन्द्वं प्रदाय स्मितसितकमलं स्यानदध्याज्यदुग्धम् ।
1 stanzas Nok, 67 and 58 are omitted in P.
- This stanza is found in K and B.
3 anza Bio . 68 and 64 are found in K and B.
- M मृ.
कलासवर्णव्यवहारः 35
श्रीखण्डं त्वं गृहीत्वनय जिनसदनम्नार्चनायाब्रवीन्मा-
मित्यद्य श्रावकाय भण गणक क्रियच्छेषमंशान्विशोध्य ॥ ६ ४ ॥
अष्टपञ्चमुच हारावुभयेऽप्येकद्युद्विकाः ।
त्रिंशदन्ताः पराभ्यस्ताश्चतुणितपश्चिमाः ॥ ६५॥
खववत्प्रमाणांश रूपात्सशTध्य तद्द्वयम् ।
शेषं सरवे समाचक्ष्व प्रोत्तीर्णगणितार्णव ।। ६६ ।।
एकोनविंशतिरथ क्रमात् त्रयोविंशतिर्दिषष्टिश्च ।
रूपविहीन त्रिंशत्ततस्त्रयोविंशतिशतं स्यात् ॥ ६७ ॥
पञ्चत्रिंशत्तस्मादष्टाशतकशतं विनिर्दिष्टम् ।
सप्तत्रिंशदमुष्मादष्टानवताित्रिकोनपञ्चाशत् ॥ ६८ ॥
चत्वारिंशच्छतिका सैका च पुन१२Tतं सषोडशकम् ।
एकत्रिंशदतस्स्याद्दानवतिः सप्तपञ्चाशत् ॥ ६९ ॥
यधिका. सप्ततिरस्मात्सपञ्चपञ्चाशदपि च सा द्विगुणा ।
सप्तकृतिः सचतुषु सप्ततिरेकोनविंशतिद्विशतम् ॥ ७० ॥
हारा निरूपिता अंशा एकाद्यकांतरा अमून् ।
प्रक्षेप्य फलमाचक्ष्व भागजात्यब्धिपारग ।। ७१ ।।
अत्रशत्पत्ता सूत्रम् --
एकं परिकल्प्यांशं तैरिष्टैस्समहरांशकान् हन्यात् ।
यदुणितांशसमासः फलसदृशोंऽशात एवेष्टाः ॥ ७२ ॥
I'his stanza is omitted in M.
? B विंशत्य.
+ K and B भागजात्यधिपारगः
3 his stanza is not found in M.
७ 3 प्रोत्तीर्णगणितार्णव .
36 गणितसारसङ्ग्रहः
एकांशवृद्धीनां राशीनां युगवंशाहारस्याधिक्ये सत्यंशोत्पादक सूत्रम् -
समहारौकांशकयुतिहतयुत्यंशोंऽश एकर्टीनाम् ।
शेषमितरांशयुतिहतमन्यांशोऽस्त्येवमा चरमत् ॥ ७३ fi
अत्रोद्देशकः ।
नवकदशैकादशहृतराशनां नवतिनवशतीभक्ता ।
यूनाशीत्यष्टशती संयोगः केंऽशकाः कथय ॥ ७४ ॥
छेदोत्पत्तौ सूत्रम्--
रूपांशकराशीनां रूपाद्यास्त्रिगुणिता हराः क्रमशः।
द्विद्वियंशाभ्यस्तावादिमचरमौ फले रूपे ॥ ७५ ॥
अत्रोद्देशकः।
पधानां राशीनां रूपांशानां युतिर्भवेद्पम् ।
षण्णां सप्तानां वा के हाराः कथय गणितज्ञ ॥ ७६ ॥
विषमस्थानानां छेदोत्पत्ता सूत्रम् --
एकांशकरानां द्याद्या रूपोत्तरा भवन्ति हराः ।
स्वासन्नपराभ्यस्तास्सर्वे दलिताः फले रूपे ॥ ७७ ॥
एकाशानामनेकांशानां चैकांशे फलं छेदोत्पत्तौ सूत्रम्
लब्धहरः प्रथमस्यच्छेदः सस्वांशकोऽयमपरस्य।
प्राक स्वपरेण हतोऽन्त्यः स्वांशेनैकांशके योगे ॥ ७८ ॥
अत्रोद्देशकः ।
सप्तकनवकत्रितयत्रयोदशांशप्रयुक्तराशीनाम् ।
रूपं पादः षष्टः संयोगाः के हराः कथय ।। ७९ ॥
• B सदृशवृष्यंशराशीनां अंशोत्पादकसूत्रम् ।
कलासवर्णव्यवहारः 37
एकांशकानामेकांशेऽनेकांशे च फले छेदोत्पत्तौ सूत्रम्--
सष्टो हारो भक्तः स्वांशेन निरग्रमादिमांशहरः ।
तद्युतिहाराप्तेष्टः शेषोऽस्मादित्थमितरेषाम् । ८० ॥
अत्रोद्देशकः ।
त्रयाणां रूपकांशTान राशीनां के हरा वद ।
फलं चतुर्थभागस्स्याच्चतुर्णा च त्रिसप्तमम् ॥ ८१ ।।
ऐकांशानामनेकांशानां चानेकांशे फले छेदोत्पत्तौ सूत्रम्-
इष्टहता दृष्टांशाः फलांशसदृशो यथा हि तद्योगः ।
निजगुणहतफलहारस्तद्धारो भवति निर्दिष्टः ॥ ८२ ॥
अत्रोद्देशकः ।
एककांशेन राशीनां त्रयाणां के हरा वद ।
द्वादशाप्त त्रयोविंशत्यशंका च युतिर्भवेत् ।। ८३ ।।
त्रिसप्तकनवांशानां त्रयाणां क हरा वद ।
यूनपञ्चाशदान त्रिसप्तत्यंशा युतिर्भवेत् ॥ ८४ ॥
एकांशशकयो राश्योरेकांशे फले छेदोत्पत्तौ सूत्र
वाञ्छाहतयुतिहारश्छेदः स व्येकवाञ्छयाप्तोऽन्यः ।
फलहारहारलब्धे स्वयोगगुणिते हरौ वा स्तः ॥ ८५ /
अत्रोद्देशकः ।
राश्योरेकांशयोश्छेद कौ भवेतां तयोर्युतिः।
षडंशो दशभागे वा ब्रूहि त्वं गणितार्थवित् ।। ८६ ॥ ।
| Stanzas 88 and 84 are omitted in B.
0D
38 गणितसारसङ्ग्रहः
एकांशकयोरनेकशयाश्च एकांशेऽनेकांशेऽपि फले छेदोत्पत्तौ प्रथम सूत्रम्--
इष्टगुणांशोऽन्यांशप्रयुतः शुङ हृतः फलांशेन ।
इष्टाप्तयुतिहरनो हरः परस्य तु तदिष्टहृतिः ॥ ७ ॥ ६
अत्रोद्देशकः ।
रूपांशकयो राश्योः कौ स्यातां हारकौ युतिः पादः ।
पञ्चांशो वा द्विहतस्सप्तकनवकांशयोश्च वद ॥ ८ ॥
द्वितीयसूत्रम् --
फलहारताडितांशः परांशसहितः फलांशकेन हृतः ।
स्यादेकस्यच्छेदः फलहरगुणितोऽयमन्यस्य ॥ ८९ ॥
अत्रोद्देशकः ।
राशिद्वयस्य कौ हारावकांशस्यास्य संयुतिः ।
द्विसप्तांशो भवेद्राहि षडष्टांशस्य च प्रिय ॥ ९० ॥
अर्धत्र्यंशदशांशकपञ्चदशांशुकयुतिर्भवेद्पम् ।
त्यक्ते पञ्चदशांश रूपांशावत्र कौ योज्यौ ॥ ९१ ॥
दलपादपञ्चमांशकविंशानां भवति संयुती रूपम् ।
सप्तैकादशकांशौ क योज्याविह विना विंशम् ॥ ९२ ॥
युग्मान्याश्रित्यच्छेदोत्पत्तौ सूत्रम्-
युग्मप्रमितान् भागानेकैकांशान् प्रकल्प्य फलराशः ।
तेभ्यः फलात्मकेभ्यो द्विराशिविधिना हरास्साध्याः ॥ ९३ ॥
P and B add as another reading
शुद्धं फलांशभक्तः स्वान्यांशयुतो निजेष्टगुणितांशः ।
कलासवर्णव्यवहारः 39
अत्रोद्देशकः ।
त्रिकपञ्चकत्रयोदशसप्तनवैकादशांशानाम् ।
के हाराः फलमकं पचांश वा। चतुर्गुणितः ॥ ९४ ॥
एकसूत्रोत्पन्नरूपांशहस्सूत्रान्तरोत्पन्नरूपांशहारैश्च फले रूपे छेदोत्पत्तौ नष्टभागानयने च सूत्रम्
वाञ्छितसूत्रजहारा हरा भवन्त्यन्यसूत्रजहरन्नाः ।
दृष्टांशैक्योनं फलमभीष्टनष्टांशमानं स्यात् ।। ९५ ॥
अत्रोद्देशकः ।
परहतिदलनविधानात्रयोदश स्वपरसङ्गणविधानात् ।
भागाश्चत्वारोऽतः कांत भागायुः फले रूपे ॥ ९६ ।।
प्राक्स्वपरहतविधानात्सप्तस्यासन्नपरगुणार्धविधानात् ।
भागास्त्रितयश्चातः कति भागास्स्युः फले रूप ॥ ९७ ॥
रूपका द्विषट्कद्वादशवTतहरा विनष्टोऽत्र ।
पञ्चमराशी रूपं सर्वसमासस्स राशिः कः ॥ ९८ ॥
इति भागजातिः ।
प्रभागभागभागजात्योस्सूत्रम्-
अंशानां सङ्गणन हाराणां च प्रभागजातों स्यात् ।
गुणकारांऽशकराशहरहरो भागभागजातिविधौ ॥ ९९ ॥
प्रभागजातावुद्देशकः ।
रूपार्थं त्र्यंशधं त्र्यंशार्धार्ध दलार्धपञ्चांशम् ।
पञ्चांशार्धत्र्यंशं तृतीयभागार्धसप्तांशम् ॥ १०० ॥
40 गणितसारसङ्गहः
दलदलदलसप्तांशं त्र्यंशयंशकदलार्धदलभागम् ।
अर्धज्यंशयंशकपञ्चांशं पञ्चमांशदलम् ॥ १०१ ॥
क्रीतं पणस्य दवा कोकनदं कुन्दकेतककुमुदम् ।
जिनचरणं प्रार्चयितुं प्रक्षिप्यैतान् फलं ब्रूहि ॥ १०२ ॥
रूपाधे त्र्यंशकार्बोधं पादसप्तनवांशकम् ।
द्वित्रिभागद्विसप्तांशं द्विसप्तांशनवांशकम् ॥ १०३ ॥
दवा पणद्वयं काश्चिदानैर्घनूतनं घृतम् ।
जिनालयस्य दीपार्थं शेषं किं कथय प्रिय ॥ १०४ ॥
त्र्यंशाद्विपञ्चमांशस्तृतीयभागात् त्रयादेशषडशः ।
पचाष्टादशभागात् त्रयोदशांशोऽष्टमान्नवमः ॥ १०५ ॥
नवमाच्चतुस्त्रयोदशभागः पञ्चांशकात् त्रिपादार्धम् ।
सक्षिप्याचक्ष्वैतान् प्रभागजातौ श्रमोऽस्ति यदि ॥ १०६ ॥
अत्रकाव्यक्तानयनसूत्रम्-
रूपं न्यस्याव्यक्तं प्राग्विधिना यत्फलं भवेत्तेन ।
भक्तं परिदृष्टफलं प्रभागजातौ तदज्ञातम् ॥ १०७ ॥
अत्रोद्देशकः ।
राशेः कुतश्चिदष्टांशस्यंशपदोऽर्धपञ्चमः।
षष्ठत्रिपादपघांशः किमव्यक्तं फलं दलम् ॥ १०८ ॥
अनेकाव्यक्तानयनसूत्रम्-
कृत्वाज्ञातानिष्टान् फलसदृशी तद्युतिर्यथा भवति ।
विभजेत पृथग्व्यक्तैरविदितराशिप्रमाणानि ॥ १०९ ॥
41 अत्रोदेशकः ।
राशेः कुतश्चिदर्घ कुतश्चिदष्टांशक त्रिपञ्चांशः ।
कस्मादियंशrधं फलमट्टी के स्युरज्ञाताः ॥ ११० ॥
भागभागजातखुट्शकः ।
षट्सप्तभागभागस्यष्टांशश्चतुनवशः ।
त्रिचतुर्थभागभागः किं फलमेतद्युतौ बूहि ॥ १११ ॥
द्वित्र्यंशाप्तं रूपं त्रिपादभक्तं द्विकं द्वयं चापि ।
द्वित्र्यशङ्कतमेकं नवकासशTIध्य वद शेषम् ॥ ११२ ॥
इति प्रभागभगभागजातं ।
भागानुबन्धजातों सूत्रम् –
हहतरूपेष्वशान् स क्षिप भागानुबन्धजातिविधौ ।
गुणयागांशच्छेदावंशयुतच्छदहारभ्याम् ॥ ११३ ॥
रूपभागनुवन्ध उद्देशकः ।
'द्वित्रिषङ्कष्टनिष्काणि द्वादशाष्टषडंशकैः ।
पोष्टमैस्समेतानि विंशतेश्शोधय प्रिय ॥ ११४ ॥
साधेनैकेन पङ्कजं साष्टांशैर्दशभिर्हिमम ।
साधाभ्यां कुञ्जमं द्वाभ्यां क्रीतं योगे कियद्भवेत् ॥ । ११५ ॥
साष्टमाष्टों षडंशान् षड्द्वादशांशयुत द्वयम् ।
त्रयं पधाष्टमोपेतं वंशतशोधय प्रिय ॥ ११६ ॥
1B readia गुणयेदप्रांशहरों सहितांशच्छेद. 8 1 [ द्वदेत्
- This stanza is not found in P. * This stanza is found only in P. ,
42 गणितसारसङ्गह
सप्ताष्टौ नवदशमाषकान् सपादान्
दवा न जनानलये चकार पूजाम् ।
उन्मीलत्कुरवककुन्दजातिमल्ली-
मालाभिर्गणक वदाशु तान् समस्य ।। ११७ ॥
भागभागानुवन्ध उद्देशकः ।
चत्र्यंशपादसंयुक्तं दलं पञ्चांशकोऽपि च ।
यंशस्वकीयषष्ठार्धसहिनस्तद्युतौ कियत् ।। ११८ ॥
त्र्यंशाद्ययंशकसप्तमांशचरमैर्वैरन्वितादर्धतः
पुष्पाण्यर्धतुरीयपञ्चनवगैरेखीयैर्युतासप्तमात् ।
गन्धं पञ्चमभागोऽर्धचरणयंशांशकैर्मिश्रिता
धूपं चर्चायतं नरो जिनवराननष्ट कि तद्युतौ ॥ ११९ ॥
वदलसहितं पादं स्वयंशकेन समन्वित
द्विगुणनवमं स्वाशयंशकार्धविमिश्रितम् ।
नवममपि च स्वाष्टशाद्यर्धपश्चिमसयुतं
निजदलयुतं ये संशोधय त्रितयात्प्रिय ॥ १२० ॥
स्वदलसहतपदं सखपादं दशांशं
नजदलतषष्ठ सस्वकथ्यमधम् ।
चरणमाप समतवात्रभाग समस्य
प्रिय कथय समग्रप्रज्ञ भागनुबन्ध ॥ १२१ ॥
अत्राग्राव्यक्तानयनसूत्रम्-
लब्धात्कार्पितभाग रूपानतनुबन्धफलभक्ता ।
क्रमशः रचण्डसमानास्तेऽज्ञातांशप्रमाणानि ॥ १२२ ॥
B स्वचरण।द्यधान्तमः
43 कलासवर्णव्यवहारः
अत्रोद्देशकः ।
कश्चित्स्वकैरर्धतृतीयपादै-
रंशोऽपरः पञ्चचतुर्नवांशैः ।
अन्यस्त्रिपञ्चांशनवांशकार्ध-
धृतो युती रूपमिहांशकाः के ॥ १२३ ॥
कोऽप्यंशस्वार्थपघांशत्रिपादनवमैर्युतः।
भधं प्रजायते शत्रं वदव्यक्तप्रमां प्रिय ॥ १२४ ॥
शेषेष्टस्थानाव्यक्तभागानयनसूत्रम्-
लब्धात्कल्पितभागास्सवर्णितेय्क्तराशिभिर्भक्ताः ।
क्रमशो रूपविहीनास्वेष्टपदेष्घविदितांशास्स्युः ॥ १२५. ॥ ।
इति भागानुवन्धजातिः ।
अथ भागापवाहजतौ सूत्रम्--
हरहतरूपेष्वनपनय भाग|पवाहजानवधा ।
गुणयागांशच्छेदावंशोनच्छेदहाराभ्याम् ।। १२६ ॥
रूपभागापवाह उद्देशकः ।
यष्टचतुदशकषोः पदाबूद्वादशांशषष्ठनः ।
सवनाय नरैर्दत्तास्र्थळनां तद्युतौ किं स्यात् ॥ १२७ ॥
त्रिगुणपाददलत्रिहाष्टमैर्विरहिता नव सप्त नव क्रमात् ।
- ॐ गुणयेदग्रांशहरौ राहितांशच्छेदहाराभ्याम् ।
6-A
44 गणितसारसङ्ग्रहः
प्रिय विशोध्य चतुर्गुणषट्तः
कथय शेषधनप्रमित द्रुतम् ॥ १२८ ॥
भागभागापवाह उद्देशकः ।
द्विगुणितपञ्चमनवमत्र्यंशाष्टांशाद्विसप्तमान् क्रमशः ।
चषडंशपादचरणयंशाष्टमवर्जितान् समस्य वद ॥ १२९ ॥
षट्सप्तांशस्वषष्ठाष्टमनवमदशशवयुक्तः पणस्य
स्यात्पञ्चद्वादशांशस्वकचरणतृतीयांशपधांशकनः ।
खद्वियंशाद्विपञ्चांशकदलवियुतः पञ्चषड्भागराशि-
ट्टीिच्यंशोऽन्यस्वपोष्टमपरिरहितस्तत्समासे फलं किम् ॥ १३० ॥
अर्द्ध व्यष्टमभागपादनवमैयौर्विहीनं पुनः
स्वैरष्टांशकसप्तमांशचरणैरूनं तृतीयांशकम् ।
अध्यधत्परिशोध्य सप्तममपि चाष्टांशषष्ठोनितं
शेषं ब्रूहि परिश्रमोऽस्ति यदि ते भागापवाहे सखे ॥ १३१ ॥ .
अत्राग्राव्यक्तभागानयनसूत्रम्
लब्धात्काल्पित भाग रूपानंतापवाहफलभक्ताः ।
क्रमशः खण्डसमानास्तेऽज्ञातांशप्रमाणानि ॥ । १३२ ॥
अत्रोद्देशकः ।
कश्चित्स्वकैश्चरणपञ्चमभागषत्रैः
कोऽप्यंशको दलषडंशकपञ्चमः ।
हीनोऽपरो द्विगुणपथमपादषष्ठेः
तत्संयुतिर्दलमिहाविदितांशकाः के ॥ १३३ ॥
45 कलासवर्णव्यवहारः
कोऽप्यंशस्वार्धषड्भागपञ्चमाष्टमसप्तमैः ।
विहीनो जायते षष्ठस्स कोंऽश गणितार्थवित् ।। १३५ ॥
शेषेष्ठस्थानाव्यक्तभोगानयनसूत्रम्-
लब्धात्कल्पितभागास्सवर्णितैव्पक्तराशिभिर्भक्ताः ।
रूपात्थापनीतास्वेष्टपदेष्वविदितांशास्स्युः ॥ १३१ ॥
इति भागापवाहजातिः ।
भागानुबन्धभागपवाहयोस्सवोव्यक्तभागानयनसूत्रम् –
त्यक्त्वैकं स्वेष्टांशान् प्रकल्पयेदविदितेषु सर्वेषु ।
एतस्तं पुनरंशं प्रागुक्रानयेत्सूत्रैः ॥ १३६ ॥
अत्रोद्देशकः ।
कश्चिदंशोंऽशकैः कैश्चित्पञ्चभिस्वैद्युतो दलम् ।
वियुक्तो वा भवेत्पादस्तानंशान् कथय प्रिय ॥ १३७ ॥
भागमातृजात सूत्रम्--
भागादिमजातीनां स्वस्वविधिर्भागमातृजातौ स्यात् ।
सा षड्रेिशातिभेदा रूपं छेदोऽच्छिदो राशेः ॥ १३८ ॥
अत्रोद्देशकः ।
यंशः पादो ऽधोधे पञ्चमषष्ठांस्त्रपादहमेकम् ।
पञ्चधहृतं रूप सषष्ठमेकं सपञ्चमं रूपम् ।। १३९ ॥
स्वीयतृतीययुदलमतो निजपष्टयुतो द्विसप्तमो
हीननवांशमेकमपनीतदशांशकरूपमष्टमः ।
P, K and B तयुतैः for जायते.
1D
46 गणितसारसङ्ग्रहः॒
चैन नवांशकेन रहितश्चरणस्वकपञ्चमोज्झितों
ब्रूहि समस्य तान् प्रिय कलासमकोत्पलमालिकाविधौ ।। १४० ।।
इति भागमातृजातिः ।
इति सारसङ्ग्रहे गणितशास्त्रे महावीराचार्यस्य कृतौ कलासवर्णो
नाम द्वितीयव्यवहारस्समाप्तः ।
47
तृ ती यः
प्रकीर्णकव्यवहारः
पुणुतानन्तगणोघ प्रणिपत्य जिनेश्वरं महावीरम् ।
प्रणतजगत्रयवरदं प्रकीर्णक गणितमभिधास्ये ॥ १ ॥
विध्वसदुर्नयध्वान्तः सिद्धः स्याद्वादशासनः ।
विद्यानन्दो जिनो जीयाद्यदीन्द्रो मुनिपुङ्गवः॥ २ ॥
इतः परं प्रकीर्णकं तृतीयव्यवहारमुदाहरिष्यामः
भागश्शेषो मूलकं शेषमूलं
स्यातां जाती है द्विरश्रांशमूले ।
अगाभ्यासाऽनऽशTवगऽथ मूल-
मित्रं तस्माद्भिन्नदृश्यं दशामूः ॥ ३ ॥
तत्र भागजातशTषजात्यरसूत्रम् --
भागोनरूपभक्तं दृश्यं फलमत्र भागजातिविधौ ।
अशTानितरूपाहतहृतमणं शेषजातिविधौ ॥ ४ ॥
भागजातखुद्दकः
दृष्टाऽष्टम पृथिव्यां सम्भस्य त्रयशक मया तये ।
पादांशः शैवाले कः स्तम्भः सप्त हस्ताः वे ॥ १ ॥
षडुागः पाटलीषु भ्रमरवरततेस्तत्रिभगः कदम्
पादधूतद्रुमषु प्रदालतकुसुमे चम्पक पञ्चमांशः ।
प्रोत्फुञ्चाम्भोजषण्डे रविकरदलिते त्रिंशदंशोऽभिरेमे
तत्रैको मत्तभृङ्ग भ्रमाति नभसि का तस्य वृन्दस्य सङ्ख्या ।। ६ ॥
+ B and M omit this stanza
48 गणितसारसङ्गहः
आदायाम्भोरुहाणि स्तुतिशतमुखरः श्रावकस्तीर्थकृद्यः
पूजां चक्रे चतुभ्यं वृषभजिनवरात इयंशमेषाममुष्य ।
त्र्यंशं तुर्यं षडंशं तदनु सुमतये तन्नवद्वादशांशौ
शेषेभ्यो द्विद्विपङ्गं प्रमुदितमनसादत्त कि तत्प्रमाणम् ॥ ७ ॥
स्ववशीकृतेन्द्रियाणां दूरीकृतविषकषायदोषाणाम् ।
शीलगुणाभरणानां दयाङ्गनालेiङ्गताङ्गानाम् ॥ ८ ॥
साधूनां सद्वन्दं सन्दृष्टं द्वादशोऽस्य तर्कज्ञः ।
चत्र्यंशवर्जितोऽयं सैद्धान्तश्छान्दसस्तयोश्शेषः ॥ ९ ॥
षडुन्नऽयं धर्मकथी स एव नैमित्तिकः स्वपादोनः ।
वादी तयोर्विशेषः षड्रणितोऽयं तपस्वी स्यात् ॥ १० ॥
गांराशरवरती मयापदृष्ट
यांतपतयां नवसङ्गणसङ्घाः ।
रविकरपरितापितोज्ज्वलाङ्गाः
कथय मुनीन्द्रसमूहमाशु मे त्वम् ॥ ११ ॥
फलभारनम्रकने शालिक्षेत्रे शुकास्समुपविष्टः।
सहसोत्थिता मनुष्यैः सर्वे सन्त्रासितास्सन्सः ॥ १२ ॥
तेषामर्थे प्राचीनादेयीं प्रति जगाम षड्भागः।
पूर्वानेयशिषः स्खदलनः स्वार्थवर्जितो यामीम् ॥ १३ ॥
यास्याग्रेयीशोषः स नैीतिं स्खद्विपञ्चभागोनः ।
यामीनैर्नत्यंशकपरिशेषो वारुणीमाशाम् ॥ १४ ॥
नैनीत्यपरविशेषो वायव्यां सस्खकत्रिसप्तशः ।
वायव्यपरविशेषो युतस्वसप्ताष्टमः सौमीम् ।। १५ ॥
प्रकीर्णकव्यवहारः 49
वायव्युत्तरयोर्युतिरैशानीं स्वत्रिभागयुगहीना ।
दशगुणिताष्टाविंशतिरवशिष्टा व्योम्नि कति कीराः ॥ १६ ॥
काचिद्वसन्समासे प्रसूनफलगुच्छभारनम्रोद्याने ।
कुसुमासवरसरञ्जितशुककोकिलमधुपमधुरानिवननिचिते ॥ १७ ॥
हिमकरधवले पृथुले सौधतले सान्द्ररुन्द्रमृदुतल्पे ।
फणिफणनितम्बबिम्बा कनदमलाभरणशोभाङ्गी ॥ १८ ॥
पाठीनजठरनयना कठिनस्तनहरनम्रतनुमध्या ।
सह निजपतिना युवती रात्रौ प्रीत्यानुरममाणा ।। १९ ॥
प्रणयकलहे समुत्थं मुक्तामयकण्ठिका तदबलायाः ।
छिन्नावनौ निपतिता तत्रर्यशठिकां प्रापत् ।। २० ॥
षड्भागः शय्यायामनन्तरानन्तरार्धमितिभागाः ।
षट्सह्यानास्तस्याः सर्वे सर्वत्र सम्पतिताः ॥ २१ ॥
एकाग्रषष्टिशतयुतसहस्रमुक्ताफलानि दृष्टानि ।
तन्मौक्तिकप्रमाणं प्रकीर्णकं वेत्सि चेत् कथय ॥ २२ ॥
स्फुरदिन्द्रनीलवर्णं षट्पदवृन्दं प्रफुछिनोद्याने ।
दृष्टं तस्याष्टांशोऽशोके कुटजे षडेशको । लीनः ॥ २३ ॥
कुटजाशोकविशेषः षड्गुणितो विपुलपाटलीषण्डे ।
पाटल्यशोकशेषः स्वनवांशोनो विशालसालवने ॥ २४ ॥
पाठल्यशोकशेषो युतस्वसप्तांशकेन मधुकवने ।
पञ्चांशस्सन्दृष्टां वकुलपूर्छमुकुलेषु ॥ २५ ॥
तिलकेषु कुरवकेषु च सरलेखामेषु पद्मषण्डषु ।
वनकरिकपोलमूलेषापि सन्तस्थे स एवांशः ॥ २६ ॥
M reads स् रेत' द् ' .
50 गणितसारसङ्ह:
किञ्जल्कपुञ्जपिञ्जरकजवने मधुकरास्त्रयस्त्रिंशत् ।
दृष्टा भ्रमरकुलस्य प्रमाणमचक्ष्व गणक त्वम् ॥ २७ ॥
गोयूथस्य क्षितिभूति दलं तदलं शैलमूले
षट तस्यांशा विपुलांवपन पूवपूवर्धमानाः ।
सन्तिष्ठन्ते नगरनिकटे धेनवो दृश्यमान
वृत्रशत् त्वं वद मम सर्वं गोकुलस्य प्रमाणम् ॥ २८ ॥
इति भागजातात्युद्देशकः ।
शेषजातावुद्देशकः ।
षड्भागमामराशे राजा शेषस्य पचम राज्ञा ।
तुर्यत्र्यंशदलानि त्रयोऽग्रहीषुः कुमारवराः ॥ २९ ॥
शेषाणि त्रीणि चूनानि कनिष्ठा दारकोऽग्रहीत् ।
तस्य प्रमाणमाचक्ष्व प्रकणकावशारद ॥ ३० ॥
चरति गिरौ सप्तांशः करिणां षष्ठादिमार्धपाश्चात्यः ।
प्रतिशेषांशा विपिने घइदृष्टास्सरासि कति त स्युः ॥ ३१ ॥
कोष्ठस्य लेभे नवमांशमेकः परेऽष्टभागादिदलान्तिमांशान् ।
शषस्य शेषस्य पुनः पुराण दृष्टा मया द्वादश तत्प्रम का ।। ३२ ।।
इति शेषजात्युदंशकः ।
अथ मूलजात सूत्रम्-
मूलाधारे छिन्द्यादंशनैकेन युक्तमूलकृतेः ।
दृश्यस्य पदं सपदं वर्गितामह मूलजातौ स्वम् ॥ ३३ ॥
प्रकीर्णकव्यवहारः 51
अत्रोद्देशकः |
दृष्टोऽटव्यामुष्ट्रयूथस्य पादो
मूले च द्वे शैलसानौ निविष्टे ।
उष्ट्रास्त्रिनाः पच नद्यारतु तीरे
किं तस्य स्यादुष्टकस्य प्रमाणम् ।। ३४ ॥
श्रुत्वा वषभ्रमालापटहपटुरवं श्लङ्गरुरङ्ग
नायं चक्रे प्रमोदप्रमुदिताशिखिनां षडशांशTIऽष्टमश्च ।।
यशः शेषस्य षष्ठो वरवकुलवने पञ्च मूलानि तस्थुः
पुन्नागे पञ्च दृष्टा भण गणक गणं बर्हिणां सङ्गणय्य ॥ ३९ ॥
चरति कमलषण्डे सारसानां चतथै
नवमचरणभाग सप्त मूलानि चाद्रौ ।
विकचवकलमध्ये सप्तनिम्नाष्टमानाः
कति कथय सरवे त्वं पक्षिणों दक्ष साक्षात् ।। ३६ ॥
न भागः कांपवृन्दस्य त्रण मूलानि पर्वते ।
चत्वारिंशद्वने दृष्टा वानरास्तङ्गणः कियान् ॥ ३७ ॥
कलकण्ठानामधं सहकारतराः प्रफुछशाखायाम् ।
तिलकेऽष्टादश तस्थुनं मूलं कथय पिकनिकरम् ॥ ३८ ॥
हंसकुलस्य दलं वकुलेऽस्थात्
पञ्च पदानि तमालकुजाग्रे ।
• B reads हस्त.
3 B reads किं स्यात्तेषां कुञ्जराणां प्रमाणम् ।
१ B reade नागः
D
52 गणितसारसङ्गहः
अत्र न किञ्चिदपि प्रतिदृष्टं
तत्प्रमितिं कथय प्रिय शीघ्रम् ॥ ३९ ॥
इति मूलजातिः ।
अथ शेषमूलजातौ सूत्रम् –
पददलवर्गयुताग्रान्मूलं सप्राक्पदामस्य कृतिः।
दृश्ये मूलं प्राप्ते फलमिह भागं तु भागजातिविधिः ॥ ४० ॥
अत्रोद्देशकः ॥
गजयूथस्य त्र्यंश '१३षपदं च त्रिसङ्गणं सान ।
सरसि त्रिहस्तिनीभिर्नागो दृष्टः कतीह गजाः ॥ ४१ ॥
निर्जन्तुकप्रदेशे नानाह्मषण्डमण्डितोद्याने ।
आसीनानां यमिनां मूलं तरुमूलयोगयुतम् ॥ ४२ ॥
शेषस्य दशमभाग मूलं नवमऽथ मूलमष्टांशः ।
मूलं सप्तममूलं षष्ठो मूलं च पञ्चमो मूलं ॥ ४३ ॥
एते भागाः काव्यप्रवचनधमप्रमाणनयविद्याः ।
वादच्छन्दोज्यौतिषमन्त्रालङ्कारशब्दज्ञाः ॥ ४४ ॥
इदशतपःप्रभावा द्वादशभदाङ्गशास्त्रकुशलधियः ।
द्वादश मुनया दृष्टाः कियती मुनिचन्द्र यतस मितिः ॥ ४५ ॥
मूलानि पञ्च चरणेन युतानि सानौ
शषस्य पञ्चनवमः करिणां नगाग्रे ।
मूलान पचे सरसाजवन रमन्ते
नद्यास्तटे षडिह ते द्विरदाः कियन्तः ॥ ४६ ॥
इति शेषमूलजातिः ।
स B geade शैघस्य पदे त्रिसंगुणं.
प्रकीर्णकव्यवहारः 53
अथ द्विरग्रशेषमूलजातौ सूत्रम् –
मूलं दृश्यं च भजेदंशकपरिहाणरूपघान ।
पूर्वाग्रमग्रराशौ क्षिपेदतश्शेषमूलविधिः ॥ ४७ ॥
अत्राद्देशकः ।
मधुकर एको दृष्टः रवे पदं शेषपञ्चमचतुर्थे ।
शेषन्यंशो मूलं द्वावाने ते कियन्तः स्युः ॥ ४८ ॥
सिंहाश्चत्वारोऽद्रौ प्रतिषषडंशकादिमर्धान्ताः ।
मूल चत्वारोऽपि च विपिने दृष्टाः कियन्तस्ते ॥ ४९ ॥
तरुणहरिणीयुग्मं दृष्टं द्विसङ्गणत वने ।
कुधरनिकटे शेषाः पञ्चांशकादिदलान्तिमाः।
विपुलकलमक्षेत्रे तासां पदं त्रिभिराहतं
कमलसरसीतीरे तस्थुदशव गणः कियान् ॥ । ५० ॥ ।
इति द्विरग्रशेषमूलजातिः ॥
अथांशमूलजातौ सूत्रम् –
भागगुणे मूलाग्रे न्यस्य पदप्राप्तदृश्यकरणेन ।
यछब्धं भागहतं धनं भवेदंशमूलविधौ ॥ ५१ ॥
अन्यदपि मूत्रम् --
दृश्यादंशकभक्ताच्चतुर्गुणान्मूलकृतियुतान्मूलम् ।
सपदं दलितं वर्णितमंशाभ्यस्तं भवेत् सारम् ॥ १२ ॥
1 B reads द्वौ चामे.
54 गगितसारसङ्ग्रहः
अत्रोद्देशकः ।
पद्मनालत्रिभागस्य जले मूलाष्टकं स्थितम् ।
षोडशाङ्गुलमाकाशे जलनालोदयं वद ॥ ५३ ॥
द्वित्रिभागस्य यन्मतं नवनं हस्तिनां पुनः ।
शेषत्रिपथमांशस्य मूलं पङिसमाहतम् ॥ ५४ ॥
विगलद्दनधाराऐंगण्डमण्डलदन्तिनः।
चतुर्विंशतिरदृष्टा मयाटव्यां कति द्विपाः । ५५ ॥
क्रोडघार्धचतुःपदानि विपिनं शार्दूलविक्रीडिते
प्रापुःशेषदशांशमूलयुगलं शैलं चतुस्ताडितम् ।
शषार्धस्य पदं त्रिवर्गगुणितं वप्र वराह वने
दृष्टास्सप्तगुणाष्टकप्रमितयस्तेषां प्रमाणं वद ॥ ५६ ॥
इत्यंशमूलजातिः ॥
अथ भागसंवर्गजातौ सूत्रम् --
स्वांशप्तहरादूनाच्चतुर्गुणाग्रेण तद्धरेण हतात् ।
मूलं योज्यं त्याज्यं तच्छेद तद्दल वित्तम् ॥ ५७ ॥
अत्रोद्देशकः ।
अष्टमं षोडशांशन शालिराशेः कृषीबलः ।
चतुर्विंशतिवाहांश्च लेभे राशिः कियान् वद ॥ ९ ॥
1 B read8 वाराद्•
2 After this stanzh all the MIss. 1avo the following stanza; but it is
simply a paraphrase of stanya No. 57 :- -
अन्यत्र
चतुर्हतदृष्टेनोनाद्राग'हन्यंशहृतहारात् ।
तच्छेदेन हृत।मूलं योज्यं त्याज्यं तेढछेदे तदर्ध चित्तम् ॥
प्रकीर्णकव्यवहारः 55
शिखिनां षोडशभागः स्वगुणभूते तमालषण्डंऽस्थात् ।
शेषनवांशः वहतश्चतुरग्रदशापि काति ते स्युः ॥ ५९ ॥
जलं त्रिशदशाहत द्वादशशः
स्थितश्शेषविंशो हतः षोडशेन ।
त्रिनिनेन पद्धं करा विंशतिः वे
सरवं स्तम्भदेव्येस्य मानं वद त्वम् । ६० ॥
इति भागसंवर्गजातिः ।
अथानधिकांशवर्गजातौ सूत्रम्--
स्वांशकअक्तहरार्ध न्यूनयुगधिकोनितं च तद्वर्गात् ।
व्यूनाधिकवगोग्रान्मूलं स्वर्गं फलं पहुंऽशहृतम् ॥ ६१ ॥
हीनालाप उदाहरणम्।
महिषीणामष्टांशो व्येको वर्गीकृतो वने रमते ।
पञ्चदशाद्रौ दृष्टारतृणं चरन्त्यः कियन्त्यस्ताः ॥ ६२ ॥
अनेकपानां दशमो द्विवर्जितः
स्वसङ्गुणः क्रीडति सल्लकीवने ।
चरन्ति षड़गमत गजा गिरौ
कियन्त एतेऽत्र भवन्ति दन्तिनः ॥ ६३ ॥
आधिकालाप उदाहरणम् ।
जम्बूवृक्षे पचदशांषो द्विकयुक्तः
स्वेनाभ्यस्तः केकिकुलस्य द्विकृतिघ्नाः ।
•M[ omita हीन.
- M out8 this as well as the following stanza.
56 गणितसारसङ्ग्रहः
पञ्चप्यन्ये मत्तमयूरास्सहकारे
रंरम्यन्ते मित्र वदंषां परिमाणम् ।। ६४ ॥
इत्यूनाधिकांशवर्णजातिः ॥
अथ मूलमिश्रजातौ सूत्रम्--
मिश्रकृतिरूनयुका व्यधिका च द्विगुणमिश्रसम्भक्ता ।
वर्गीकृता फलं स्यात्करणमिदं मूलमश्नविधौ ॥ ६५ ॥
हीनालाप उद्देशकः ।
मूलं कपोतवृन्दस्य द्वादशानस्य चापि यत् ।
तयोयोगं कपोताष्षड् दृष्टास्तन्निकरः कियान् ॥ ६६ ॥
पारावतीयसवें चतुर्घनोनेऽपि तत्र यन्मूलम् ।
तह्ययोगः षोडश तद्वन्दे कति विहङ्गाः स्युः ॥ ६७ ॥
अधिकालाप उद्देशकः ।
राजहसनकरस्य यत्पदं
साष्टषष्टिसहितस्य चैतयोः ।
संयुतिर्दकविहीनषद्कृति-
स्तद्रणे कति मरालका वद ।। ६८ ॥
इति मूलमिश्रजाति ।
अथ भिन्नदृश्यजातौ सूत्रम्-
दृश्यांशोने रूपे भागाभ्यासेन भाजिते तत्र ।
यल्लब्धं तत्सरं प्रजायते भिन्नदृश्यविधौ ॥ ६९ ॥
B read8 योगः.
अत्रोद्देशकः ।। सिफतायामष्टांशस्सन्दृष्टोऽष्टादशांशसङ्गणितः । स्तम्भस्यर्थं 'दृष्टं स्तम्भायामः कियान् कथय ।। ७० || डिभक्तनवमांशकप्रहतसप्तविंशांशकः प्रमोदमवतिष्ठते करिकुलस्य पृथ्वीतले । • विनीलजलदारुतिर्विहरति त्रिभागो नगे वद त्वमधुना सरवे करिकुलप्रमाणं मम ।। ७१ ।। साधूत्कृतोर्नवसति षोडशांशक- स्विभाजितः स्वकगुणितो वनान्तरे । पादो गिरौ मम कथयाशु तन्मित प्रोत्तीर्णवान् जलधिसमं प्रकीर्णकम् ।। २ ।। इति भिन्नदृश्यजातिः ॥ इति सारसमन्हे गणितशास्त्रे महावीराचार्थस्य कृतै। प्रकीर्णको नाम तृतीयव्यवहारः समाप्तः ॥ । 'B, M and K read गगने. चतुर्थः त्रैराशिकव्यवहारः ।। त्रिलोकबन्धवे तस्मै केवलज्ञानभानवे ।। नमः श्रीवर्धमानाय निर्धेतारिवलकर्मणे ।। १ ।। इतः परं वैराशिकं चतुर्थव्यवहारमदाहरिष्यामः ।। तत्र करणसूत्रं यथा- त्रैराशिकेऽत्र सारं फलमिच्छासङ्गणं प्रमाणाप्तम् । इच्छाप्रमयोस्साम्ये विपरीतेयं क्रिया व्यस्ते ॥ २ ॥ पूर्वाधादेशकः । दिवसैषिभिस्सपादैयोजनषटू चतुर्थभागोनम् । गच्छति यः पुरुषोऽसौ दिनयुतवर्षेण किं कथय ।। ३ ।। व्यर्धाष्टाभिरहोभिः क्रोशाष्टांशं स्वपञ्चमं याति । पङ्गस्सपञ्चभागैर्वर्षेत्रिभिरत्र किं ब्रूहि ॥ ४ ॥ अङ्गलचतुर्थभागं प्रयाति कीटो दिनाष्टभागेन । मेरोमूलाच्छिखरं कतिभिरहोभिस्समाप्नोति ।। ५ ।।। कार्षापणं सपाद निर्विशति त्रिभिरहोभिरर्धयुतैः । । यो ना पुराणशतकं सपणं कालेन केनासौ ।। ६ ।। कृष्णागरुसवण्डं द्वादशहस्तायतं त्रिविस्तारम् । क्षयमेत्यङ्गलमह्नः क्षयकालः कोऽस्य वृत्तस्य ।। ७ ।। स्वर्णैर्दशभिस्सार्धेद्रणाढककुडब मिश्रितः क्रीतः ।। वरराजमाषवाहः किं हेमशतेन साधेन ।। ८ ॥ "P, K and M read A for $9.
- B reads सत्कृष्णागरुघण्रं. त्रैराशिकव्यवहारः
सार्धेखिभिः पुराणैः कुङ्कुम पलमष्टभागसंयुक्तम् ।। संप्राप्यं यत्र स्यात् पुराणशतकेन किं तत्र ॥ ९ ॥ साधर्दिकसप्तपलेश्चतुर्दशाधॉनिताः पणा ' लब्धाः ।। द्वात्रिंशदाईकपलैस्सपञ्चमैः किं सरवे ब्रूहि ॥ १० ॥ काषपणैश्चतुर्भिः पञ्चाशयुतैः पलानि रजतस्य । षोडश साधनि नरो लभते किं कर्षनियुतेन ।। ११ ।। कप्रस्याष्टपलैस्व्यंशोनैनत्र पञ्च दीनारान् । भागांशकलायुक्तान् लभते किं पलसहस्रेण ।। १२ ।। साधैत्रिभिः परिह घृतस्य पलपञ्चकं सपञ्चांशम् । क्रीणाति यो नरोऽयं किं साष्टमकर्षशतकेन ।। १३ ।। साधैः पञ्चपुराणैः षोडश सदलानि वस्त्रयुगलानि । लब्धानि सैकषष्ट्या कर्षाणां किं सरवे कथय ।। १४ ।। वापी समचतुरश्रा सलिलवियुक्ताष्टहस्तघनमाना । शैलस्तस्यास्तीरे समुत्थिताश्शखरतस्तस्य ।। १५ ।। वृत्ताङ्गुलविषुम्भा जलधारा स्फठिकनिर्मला पतिता । वाप्यन्तरजलपूर्णा नगोच्छूितिः का च जलसङ्ख्या ॥ १६ ॥ 'मुद्रद्रोणयुगं नवाज्यकुडवान् षट् तण्डुलद्रोणका- नष्टौ वस्त्रयुगानि वत्ससहिता गाप्षट् सुवर्णत्रयम् । 1 M and B read लभ्याः ,
- B reads समुत्थिता शि.
& B and K read the following for this stanza : दुग्वद्रोणयुगं नवाज्यकुडवान् षट् शर्कराद्रोणका- नष्टौ चोचफलानि सान्द्रदधिखार्यष्षट् पुराणत्रयम् । श्रीखण्डं ददता नृपेण सवनार्थ षड्जिनागारके पत्रिंशत्रिशतेषु मित्र वद मे तद्दत्तदुग्धादिकम् ॥ 1-A गणितसारसङ्ग्रहः सङ्क्रान्तौ ददता नराधिपतिना षड्भ्यो द्विजेभ्यस्सखे षत्रिशत्रिशतेभ्य आशु वद किं तद्दत्तमुद्रादिकम् ।। १७ ।। इति त्रैराशिकः । । व्यस्तत्रैराशिके तुरीयपादस्योद्देशकः । कल्याणकनकनवतेः कियन्ति नववर्णकानि कनकानि । साष्टांशकदशवर्णकसगुञ्जहेम्नां शतस्यापि ।। १८ ॥ व्यासेन दैध्येण च षट्कराणां चीनाम्बराणां त्रिशतानि तानि । त्रिपञ्चहस्तानं कियन्ति सन्ति व्यस्तानुपातक्रमावइद त्वम् ॥ १९ ॥ इति व्यस्तत्रैराशिकः ।। व्यस्तपञ्चराशिक उद्देशकः । पञ्चनवहस्तविस्तृतदैयां चीनवस्त्रसप्तत्याम् । द्वित्रिकरव्यासायति तच्छृतवस्त्राणि कति कथय ।। २० ॥ व्यस्तसप्तराशिक उद्देशकः । व्यासायामोदयतो बहुमाणिक्ये चतुर्नवाष्टकरे ।। द्विषडेकहस्तमितयः प्रतिमाः कति कथय तीर्थकृताम् ।। २१ ।। व्यस्तनवराशिक उद्देशकः ।। विस्तारदैच्योदयतः करस्य षट्त्रिंशदष्टप्रमिता नवाघ । शिला तया तु द्विषडेकमानास्ताः पञ्चकाघः कति चैत्ययोग्याः ।। २९ ।। इति व्यस्तपथसप्तनवराशिकाः ।। त्रैराशिकव्यवहारः । गतिनिवृत्तौ सूत्रम्-- निजनिजकालोड़तयोर्गमननिवृत्त्योर्विशेषणाजाताम् । दिनशुद्धगतिं न्यस्य त्रैराशिकविधिमतः कुर्यात् ।। २३ ।।। अत्रोद्देशकः । क्रोशस्य पञ्चभागं नौयाति दिनत्रिसप्तभागेन । 'वार्धी वातावडा प्रत्येति क्रोशनवमांशम् ।। २४ ॥ कालेन केन गच्छेत् त्रिपञ्चभागोनयोजनशतं सा । सङ्ख्याब्धिसमुत्तरणे बाहुबलिंस्त्वं समाचक्ष्व ॥ २५ ॥ सपादहेम त्रिदिनैस्सपञ्चमैनरोऽर्जयन् व्येति सुवर्णतुर्यकम् । निजाष्टमं पञ्चदिनैदलोनितैः स केन कालेन लभेत सप्ततिम् ॥ २६ ।।। गन्धेभो मदलुब्धषट्पदपदप्रोद्भिन्नगण्डस्थलः साधं योजनपञ्चमं व्रजति यष्षभिर्दलोनैर्दिनः ।। प्रत्यायाति दिनखिभिश्च सदलैः क्रोशद्विपञ्चांशकं ब्रूहि क्रोशदलोनयोजनशतं कालेन केनाप्नुयात् ॥ २७ ॥ वापी पयःप्रपूर्णा दशदण्डसमुच्छ्रिताब्जमिह जातम् । अङ्गलयुगलं सदलं प्रवर्धते सार्धदिवसेन ॥ २८ ॥ निस्सरति यन्त्रतोऽम्भः साधेनाह्वाङ्गले सविशे हे। शुष्यति दिनेन सलिलं सपञ्चमाहुलकमिनकिरणैः ।। २९ ॥ कूम नालमधस्तात् सपादपञ्चाङ्गलानि चाकृषति ।। साधैत्रिदिनैः पद्म तोयसमें केन कालेन ॥ ३० ॥ 1 B and K read तस्मिन्काले वाध. गणितसारसङ्ग्रहः द्वात्रिंशद्धस्तदीर्घः प्रविशति विवरे पञ्चभिस्ससमाधैः कृष्णाहीन्द्रो दिनस्यासुरवपुरजितः सार्धसप्ताङ्गलानि । पानाहोऽङ्गले हे त्रिचरणसहिते वर्धते तस्य पुच्छे रन्धं कालेन केन प्रविशति गणकोत्तंस में ब्रूहि सोऽयम् ॥ ३१ ॥ इति गतिनिवृत्तिः ॥ पञ्चसप्तनवराशिकेषु करणसूत्रम्- 'लाभं नीत्वान्योन्यं विभजेत् पृथुपङ्मिल्पया पङ्कया। गुणयित्वा जीवानां क्रयविक्रययोस्तु तानेव ॥ ३२ ॥ । अत्रोद्देशकः ।। द्वित्रिचतुश्शतय.गे पञ्चाशत्षष्टिसप्ततिपुराणाः । लाभार्थिना प्रयुक्ता दशमासेष्वस्य का वृद्धिः ॥ ३३ ॥ हेम्नां सार्धाशीतेमसञ्यंशेन वृद्धिरध्यर्धा ।। सत्रिचतुर्थनवत्याः कियती पादोन षण्मासैः ।। ३४ ।। षोडशवर्णककाञ्चनशतेन यो रत्नविंशतिं लभते ।। दशवर्णसुवर्णानामष्टाशीतिद्विशत्या किम् ।। ३५ ॥ 1 P reads as variations the following: प्रकारान्तरेण सतम्-- सङ्क्रम्य फलं छिन्द्याल्लघुपङ्क्त्यानेकराशिकां पङ्क्तम् ।। स्वगुणामश्वादीनां क्रयविक्रययोस्तु तानेव ॥ अन्यदपि सूत्रम्- सङक्रम्य फलं छिन्द्यात् पृथुपङ्क्त्यभ्यासमल्पया पङ्क्त्या । अश्वादीनां क्रयविक्रययोरश्वादिकांश्च सङ्क्रम्य ॥ B gives only the latter of these stanzas with the following variation in whe bacond uarter: पृथुपक्यभ्यासमल्पपत्याहत्या. 63 गोधूमानां मानीर्नव नयता योजनत्रयं लब्धाः । षष्टिः पणाः सवाहं कुम्भं दशयोजनानि कति ॥ ३६ ॥ भाण्डप्रतिभाण्डस्योद्देशकः । 1 कस्तूरीकर्षत्रयमुपलभते दशभिरष्टभिः कनकैः । कर्षद्वयकर्ररं मृगनाभित्रिशतकर्षकैः कति न ॥ ३७ ॥ पनसानि षष्टिमष्टभिरुपलभतेऽशतिमातुलुङ्गानि । दशभिर्मापैर्नवशतपनसैः कति मातुलुङ्गानि ॥ ३८ ॥ जीवक्रयविक्रययोरुद्देशकः । षोडशवर्षांस्तुरगा विंशतिरर्हन्ति नियुतकनकानि । दशवर्षसप्तिसप्ततिरिह कति गणकाग्रणीः कथय ॥ ३९ ॥ स्वर्णत्रिशती मूल्यं दशवर्षाणां नवाङ्गनानां स्यात् । षत्रिंशन्नारीणां षोडशसंवत्सराणां किम् ॥ ४० ॥ षट्कशतयुक्तनवतेर्दशमासैवृद्धिरत्र का तस्याः । कः कालः किं वित्तं विदिताभ्यां भण गणकमुखमुकुर ॥ । ४१॥ सप्तराशिक उद्देशकः। त्रिचतुष्यसायामौ श्रीखण्डार्हतोऽष्टहेमानि । षण्णवविस्तृतिदैव्यं हस्तेन चतुर्दशात्र कति ॥ ४२ ॥ इति सप्तराशिकः । • B add ना at the end 2 K, M and B read हेमकर्घः for ना.
AD64 गणितसारसङ्गहः
नवराशिक उद्देशकः ।
पोष्टपत्रव्यासदैव्योदयाम्भ
धत्ते वापी शालिनी वाहक्षकम् ।
सप्तव्यासा हस्ततः षष्टिदैर्याः
पासधोः किं नवाचक्ष्व विद्वन् ॥ ४३ ॥
इति सारसङ्गहे गणितशास्त्रे महावीराचार्यस्य कृतौ त्रैराशिको
नाम चतुर्थव्यवहारः
Tho fllowing stanza is found in K and B in addition to stanza No 48.
द्वयष्टाशीतिव्यासदैध्येंन्नताम्भो
धत्ते वापी शालिनी सार्थवाहौ ।
हस्तादष्टायामकाः षोडशच्छाः
षट्कव्यासाः कि चतस्र वद त्वम् ।
पञ्चमः
मिश्रकव्यवहारः
प्राप्तानन्तचतुष्टयान् भगवतस्तीर्थस्य कर्तृन् जिनान्
सिद्धान् शुद्धगुणांस्त्रिलोकमहितानाचार्यवर्यानपि ।
सिद्धान्तार्णवपारगान् भवभूतां नेतृनुपाध्यायकान्
साधून् सर्वगुणाकरान् हितकरान् वन्दामहे श्रेयसे ॥ १ ॥
इतः परं मिश्रगणितं नाम पञ्चमव्यवहारमुदाहरिष्यामः। तद्यथा
सङ्क्रमणसंज्ञया विषमसङ्क्रमणसज्ञायाश्च सूत्रम्--
युतिवियुतिदलनकरणं सङ्क्रमणं छेदलब्धयो राश्योः ।
सङ्क्रमणं विषममिदं प्राहुर्गणितार्णवान्तगताः ॥ २ ॥
अत्रोद्देशकः ।
द्वादशसर्वचाराशेर्दाभ्यां सङ्क्रमणमत्र किं के भवति ।
तस्माद्राशेर्भक्तं विषमं वा कि तु सङ्क्रमणम ॥ ३ ॥
पञ्चराशिकविधिः ॥
पराशकस्वरूपवुद्यानयनसूत्रम्--
इच्छाराशिः स्वस्य हि कालेन गुणः प्रमाणफलगुणितः ।
कालप्रमाणभक्तो भवति तदिच्छाफलं गणिते ॥ ४ ॥
अत्रोद्देशकः ।
त्रिकपञ्चकषट्शते पञ्चशत्षष्टिसप्ततिपुराणाः।
लाभार्थतः प्रयुक्ताः का वृद्धिर्मासषट्स्य ॥ ५॥
66 गणितसारसङ्ग्रहः
व्यधृष्टकशतयुक्तास्त्रिशत्कार्षापणाः पणाश्चाष्टौ ।
मासाष्टकेन जाता दलहीनेनैव का वृद्धिः ॥ ६ ॥
षया वाडिर्टष्टा पञ्च पुराणाः पणत्रयविमिश्राः ।
मासद्वयेन लब्ध शतऋद्धिः का त वर्षस्य ।। ७ ॥
सार्धशतकप्रयोगे सार्धकमासेन पञ्चदश् लाभः ।
मासदशकेन लब्धा शतत्रयस्यात्र का वृद्धिः ॥ । ८ ॥
साष्टशतकाष्टयोगे त्रिषष्टिकार्षापणा विशा दत्ताः ।
सप्तानां मासानां पञ्चमभागान्वितानां किम् ॥ ९ ॥
मूलानयनसूत्रम्--
मूलं स्वकालगुणितं खफलेन विभाजितं तदिच्छायाः।
कालेन भजेल्लब्धं फलेन गुणितं तादिच्छा स्यात् ॥ १० ॥
अत्रोद्देशकः ।
पञ्चधैकशतयोगे पञ्च पुराणान्दलोनमासीौ द्वौ ।
खुडि लभते कश्चित् किं मूलं तस्य मे कथय ॥ ११ ॥
सप्तत्याः सार्धमासेन फलं पञ्चधमेव च ।
ध्यर्धाष्टमासे मूलं किं फलयोस्साधयेद्दयोः ॥ १२ ॥
त्रिकपञ्चकषट्शते यथा नवाष्टादशाथ पञ्चक्रुतिः।
पञ्चांशकेन मिश्रा षट्सु हि मासेषु कानि मूलानि ॥ १३ ॥
कालानयनसूत्रम्
कालगुणितप्रमाणं स्वफलेच्छाभ्यां हृतं ततः कृत्वा ।
तदिहेच्छाफलगुणितं लब्धं कालं बुधाः प्राहुः ॥ १४ ॥
मिश्रकव्यवहारः 67
अत्रोद्देशकः ।
सप्तार्धशतकयोगे वृद्धिस्त्वष्टाग्रविंशतिरशीत्या ।
कालेन केन लब्धा कालं विगणय्य कथय सरवे ॥ १९ ॥
विंशतिषट्शतकस्य प्रयोगतः सप्तगुणषष्टिः ।
वृद्धिरपि चतुरशीतिः कथय सर्वे कालमाशु त्वम् ॥ १६ ॥
षट्शतेन हि युक्ताः षण्णवतिवृद्धिरत्र सन्दृष्टा ।
• सप्तोत्तरपञ्चाशत् त्रिपञ्चभागश्च कः कालः ॥ १७ ॥
भाण्डप्रतिभाण्डसूत्रम्-
भाण्डस्वमूल्यभक्तं प्रतिभाण्डं भाण्डमूल्यसङ्गणितम् ।
स्वेच्छाभाण्डाभ्यस्तं भाण्डप्रतिभाण्डमूल्यफलमेतत् ॥ १८ ॥
अत्रोद्देशकः ।
क्रीतान्यष्टौ शुण्ठ्याः पलानि षभिः परैः सपादांशैः।
पिप्पल्याः पलपञ्चकमथ पादोनैः परैर्नवभिः ॥ १९ ॥
शुण्ठ्याः पलैश्च 'केनचिदशीतिभिः कति पलानि पिप्पल्याः।
क्रीतानि विचिन्त्य त्वं गणितबिदाचक्ष्व मे शीघ्रम् ॥ २० ॥
इति मिश्रकव्यवहारे पञ्चराशिकविधिः समाप्तः ।
वृद्धिविधानम् ॥
इतः परं मिश्रकव्यवहारे वृद्धिविधानं व्याख्यास्यामः।
मूलघडिमिश्रविभागानयनसूत्रम्--
रूपेण कालहृद्या युतेन मिश्रस्य भागहारविधिम् ।
कृत्वा लब्धं मूल्यं वृद्धिमूलोनमिश्रधनम् ॥ २१ ॥
- Both M and B have the erroneous reading कश्चित् वशीतिभि: स च
पलानि पिप्पल्याः
68 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
पधकशतप्रयोगे द्वादशमासैर्घनं प्रयुक्ते चेत् ।
साष्टा चत्वारिंशन्मिथं तन्मूलठी के ॥ २२ ॥
पुनरपि मूलचडिमिश्रावभागसूत्रम् –
इच्छाकालफलग्नं स्वकालमूलेन भाजितं सैकम् ।
सम्मिश्रस्य विभक्तं लब्धं मूलं विजानीयात् ॥ २३ ॥
अत्रोद्देशकः ।
सार्धद्विशतकयोगे मासचतुषेण किमपि धनमेकः ।
दत्वा मित्रं लभते क मूल्यं स्यात् त्रयस्त्रिशत् ॥ २४ ॥
कालवृद्धिमिश्रविभागानयनसूत्रम्--
मूलं स्वकालगुणितं स्वफलेच्छाभ्यां हृतं ततः कृत्वा।
सक तेनाप्तस्य च मिश्रस्य फलं हेि बुद्धिः स्यात् ।। २५॥
अत्रोद्देशकः ।
पञ्चकशतप्रयोगे फलार्थना योजितैव धनधाष्टिः ।
कालः स्ववृद्धिसहितो विंशतिरत्रापि कः कालः ॥ २६ ॥
अर्धत्रिकसप्तत्यः साधया योगयोजितं मूलम् ।
पञ्चोत्तरसप्तशतं मिश्रमशीतिः स्वकालवृद्योर्हि ॥ २७ ॥
व्यर्थचतुष्टाशीत्या युक्ता मासद्वयेन सधेन ।
मूलं चतुश्शतं षट्त्रंशन्मित्रं हि कालवृध्द्योर्हि ॥ २८ ॥
मूलकालांमिश्रवभागानयनसूत्रम् --
स्वफलोद्धृतप्रमाणं कालचतुर्वेडिताडितं शोध्यम् ।
मिश्रकृतस्तन्मूलं मिश्रे क्रियते तु सङ्क्रमणम् ॥२९॥
मिश्रकव्यवहारः 89
अत्रोद्देशकः ।
सप्तत्या वृद्धिरिये ,चतुःपुराणाः फलं च पञ्चकृतिः ।
मिश्र नव पञ्चगुणाः पादेन युतास्तु किं मूलम् ॥ ३० ॥
त्रिकषण दत्वैकः कि मूलं केन कलन ।
प्राप्तोऽष्टादशवृडेि षट्षष्टिः कालमूलमित्रं हि ।। ३१ ॥
अध्यधमासेकफल षथाः पञ्चधमेव सन्दृष्टम् ।
वृद्धिस्तु चतुर्विंशतिरथ षष्टिर्दूलयुक्तकालश्च ॥ ३२ ॥
प्रमाणफलंच्छाकालांमश्रवभागानयनसूत्रम्--
मूलं स्वकालद्याद्विद्विकृतिगुणं छिन्नमितरमूलेन ।
मिश्रकृतिशेषमूलं मिश्रे क्रियते तु सङ्क्रमणम् ॥ ३३ ॥
अत्रोद्देशकः ।
अध्यर्धमासकस्य च शतस्य फलकालयोश्च मिश्रधनम् ।
द्वादश दलसंमिश्री मूलं त्रिंशत्फलं पञ्च ॥ ३४ ॥
मूलकालखद्विमिश्रविभागानयनसूत्रम्--
- मिश्रादूनितराशिः कालस्तस्यैव रूपलाभेन ।
सैकेन भजेन्मूलं स्वकालमूलनितं फलं मिश्रम् ॥ ३५ ॥
अत्रोद्देशकः।
पञ्चकशतप्रयोगे न ज्ञातः कालमूलफलराशिः ।
तन्मिश्र 'डशीतिमूलं किं कालठट्टी के ॥ ३६ ॥
This wrong form in the reading found in the MSS.; and the correct form
अशीति do not satisfy the exigencion of the metre.
70 गणितसारसङ्ग्रहः
बहुमूलकालचवृद्धिमिश्राविभागानयनसूत्रम्—
विभजेवकालताडितमूलसमासेन फलसमासहतम् ।
कालाभ्यस्तं मूलं पृथक् पृथक् चादिशवृद्धिम् ॥ ३७ ॥
अत्रोद्देशकः ।
चत्वारिंशत्रिंशविंशतिपञ्चाशदत्र मूलानि ।
मासाः पञ्चचतुस्त्रिकषट् फलपिण्डश्चतुस्त्रिंशत् ॥ ३८ ॥
बहुमूलमिश्राविभागानयनसूत्रम्--
स्वफलैस्स्वकालभक्तैस्तधृत्या मूलमिश्रधनराशिम् ।
'छिन्द्यादंशं गुणयेत् समागमो भवति मूलानाम् ॥ । ३९ ॥
अत्रोद्देशकः ।
दशषत्रिपञ्चदशका वृद्धय इषवश्चतुस्त्रिषण्मासाः ।
मूलसमास दृष्टश्चत्वारिंशच्छतेन संमिश्रा ॥ ४० ॥
पधार्धषड्दशापि च साधीः षोडश फलानि च त्रिंशत् ।
मासस्तु पञ्च षट् वलु सप्ताष्ट दशाप्यशीतिरथ पिण्डः ॥ ४१ ॥
बहुकालमिश्राविभागानयनसूत्रम्
स्वफलैः स्वमूलभक्तैस्तधृत्या कालमिश्रधनराशिम् ।
'छिन्द्यादंशं गुणयेत् समागम भवति कलानाम् ॥ ४२ ॥
अत्रोद्देशकः ।
चत्वारिंशत्रिंशद्दिशतिपञ्चशदत्र मूलानि ।
दशषत्रिपञ्चदश फलमष्टादश कालमिश्रधनराशिः ॥ ४३ ॥
प्रमाणराशौ फलेन तुल्यमिच्छाराशिमूलं च तदिच्छाराशौ वृद्धिं
The ASS read छिन्द्यादशान् which does not seem to be correct.
मिश्रकव्यवहारः 71
च संपीड्य तन्मिश्रराशौ प्रमाणराशेः वृद्धिविभागानयनसूत्रम्-
कालगुणितप्रमाणं परकालहृतं तदेकगुणमिश्रधनात् ।
इतरार्धछतियुतात् पदमितराधनं प्रमाणफलम् ॥ ४४ ॥
अत्रोद्देशकः ।
मासचतुष्कशतस्य प्रनष्टदृष्टिः प्रयोगमूलं तत् ।
स्वफलेन युतं द्वादश पञ्चकृतिस्तस्य कालोऽपि ॥ ४५ ॥
मासत्रितयात्यः प्रनष्टवृद्धिः स्वमूलफलराशेः ।
पञ्चमभागेनोनाश्चाष्टौ वर्षेण मूलवुडी के ॥ ४६ ॥
समानमूलवृद्धिमिश्रविभागसूत्रम् -
अन्योन्यकालविनिहतामिश्रविशेषस्य तस्य भागाख्यम् ।
कालविशेषेण हृते तेषां मूलं विजानीयात् ॥ ४७ ॥
अत्रोद्देशकः ।
पञ्चाशदष्टपञ्चशान्मिथं षषष्टिरेव च।
पञ्च सप्तैव नव हि मासाः किं फलमानय ॥ ४८ ॥
त्रिंशच्चैकत्रिंशद्दिव्यंशाः स्युः पुनत्रयात्रशत् ।
सज्यंशा मिश्रधनं पञ्चत्रिंशच्च गणकादात् ॥ ४९ ॥
कश्चिन्नरश्चतुर्णा त्रिभिश्चतुर्भिश्च पञ्चभिः षभिः।
मासैर्लब्धं किंस्यान्मूलं शीर्थो ममाचक्ष्व ॥ ५० ॥
समानमूलकालमिश्रविभागसूत्रम्-
अन्योन्यवृद्धिसङ्गणमिश्रविशेषस्य तस्य आगाख्यम् ।
वद्विविशेषेण हते लब्धं मूलं बुधाः प्राहुः ॥ ५१ ॥
72 गणितसारसङ्ग्रहः
एकात्रिपञ्चमिश्रितविंशतिरिह कालमूलयोर्मिश्रम ।
षड् दश चतुर्दश स्युलभाः किं मूलमत्र साम्यं स्यात् ॥ ५२ ॥
पञ्चत्रिंशन्मित्रं सप्तत्रिंशच्च नवयुतत्रिंशत् ।
विंशतिरष्टाविंशतिरथ षट्त्रिंशच्च यद्विधनम् ॥ ५३ ॥
उभयप्रयोगमूलानयनसूत्रम् -
रूपस्येच्छकालादुभयफले ये तयोर्विशेषेण।
लब्धं विभजेन्मूलं स्वपूर्वसङ्कल्पितं भवति ॥ ५४ ॥
अत्रोद्देशकः ।
उवृत्त्या षट्शते प्रयोजितोऽसौ पुनश्च नवकशते ।
मासैस्त्रिभिश्च लभते नैकाशीतिं क्रमेण मूलं किम् ॥ ५५ ॥
त्रिशुद्धचैव शते मासे प्रयुक्तश्चाष्टभिश्शते ।
लाभोऽशीतिः कियन्मूलं भवेत्तन्मासयोर्धयोः ॥ ५६ ॥
वृद्धिलविमोचनकालानयनसूत्रम्
मूलं स्वकालगुणितं फलगुणितं तत्प्रमाणकालाभ्याम् ।
भक्तं स्कन्धस्य फलं मूलं कालं फलात्प्राग्वत् ॥ ५७ ॥
अत्रोद्देशकः ।
मासे हि पश्यैव च सप्ततीनां
मासद्वयेऽष्टादशकं प्रदेयम् ।
---
This sane rule is somewhat dofeotively stated again with a modihotion
in eading thus:
पुनरप्युभयप्रयागमूलानयनसूत्रम्--
इच्छाकालादुभयप्रयोगवृद्धिं समानीय ।
तद्वदयन्तरभक्तं लब्धं मूलं विजानीयात् ॥
मिश्रकव्यवहारः 73
स्कन्ध चतुर्भिस्सहिता त्वशीतिः
मूलं भवेत्को नु विमुक्तिकालः ॥ ५८ ॥
पन्ना मासिकवृद्धिः पचैव हि मूलमपि च पञ्चत्रिंशत् ।
मासांत्रतयं स्कन्धे त्रिपञ्चकं तस्य कः कालः ।। ५९ ।।
समानदृहिमूलमिश्रविभागसूत्रम्--
मूलैः स्वकालगुणितैटीडिविभक्तैस्समासकौर्वभजेत् ।
मित्रं स्वकालनिनं वृद्धिर्तुलानि च प्राग्वत् ॥ ६० ॥
अत्रोद्देशकः ।
द्विकषट्चतुश्शतके चतुस्सहधं चतुश्शतं मिश्रम् ।
मासद्वयेन वृद्धया समनि कान्यत्र मूलानि ॥ ६१ ॥
त्रिकशतपञ्चकसप्ततिपादोनचतुष्कषष्टियोगेषु ।
नवशतसहस्रसङ्ख्या मासत्रितये समा युक्ता ॥ ६२ ॥
विमुक्तकालस्य मूलानयनसूत्रम्--
स्कन्धं स्वकालभक्तं विमुक्तकालेन ताडितं विभजेत् ।
निर्मुक्तकालयुद्धया रूपस्य हि सकया मूलम् ॥ ६३ ॥
अत्रोद्देशकः ।
पचकशतप्रयोगे मास डॉ स्कन्धमष्टकं दत्वा ।
मासैष्षष्टिभिरिह वै निर्मुक्तः कि भवेन्मूलम् ॥ ६४ ॥
दौ सत्रिपञ्चभागौ स्कन्धं द्वादशदिनैर्ददात्येकः ।
74 गणितसारसङ्ग्रहः॒
त्रिकशतयोगे दशभिर्मासैर्मुक्तं हि मूलं किम् ॥ ६५ ॥
बुद्धियुक्तहीनसमानमूलमिश्रविभागसूत्रम्--
कालवफलानाधिकरूपोहृतरूपयोगहतमिश्रे ।
प्रक्षेपो गुणकारः खफलोनाधकसमानमूलानि ॥ । ६६ ॥
अत्रोद्देशकः ।
त्रिकपञ्चकाष्टकशतैः प्रयोगतोऽष्टसहस्रपञ्चशतम् ।
विंशतिसहितं बुद्धिभिरुद्धृत्य समानि पञ्चभिर्मासैः ॥ ६० ॥
त्रिकषष्ट्राष्टकषया मासद्वितये चतुस्सहस्राणि ।
पञ्चाशद्विशतयुतान्यतोऽष्टमासकफलाढते सदृशानि ॥ ६८ ॥
द्विकपञ्चकनवकशते मासचतुष्क त्रयोदशसहस्रम् ।
सप्तशतेन च मिश्रा चत्वारिंशत्सह्याद्विसममूलानि ॥ ६९ ॥
सैकार्धकपर्धार्धकषडर्धकाशीतियोगयुक्तास्तु ।
माताष्टके षडधिका चत्वारिंशच्च षट्रतिशतानि ॥ ७० ॥
सङ्कलितस्कन्धमूलस्य मूलवृद्धिविमुक्तिकालानयनसूत्रम्--
स्कन्धाप्तमूलचितिगुणितस्कन्धेच्छाग्रघातियुतमूलं स्यात् ।
स्कन्धे कालेन फलं स्कन्धोद्धृतकालमूलहतकालः ॥ ७१ ॥
अत्रोद्देशक ।
केनापि संप्रयुक्ता षष्टिः पञ्चकशतप्रयोगेण ।
मासत्रिपञ्चभागात् सप्तोत्तरतश्च सप्तादिः ॥ ७२ ॥
तत्षष्टिसप्तमांशकपदमितिसङ्कलितधनमेव ।
दत्वा तत्सप्तकवृद्धि प्रादाच्च चितिमूलम् ।
• मिश्रः in the roading found in the MSS. ; मिश्रे is adopted as being core
maintactory capabioally.
मिश्रकव्यवहारः 75
किं तद्धिः का स्यात् कालस्तदृणस्य मौक्षिकों भवति ॥ ७३(१/२)॥
केनापि संप्रयुक्ताशुतिः पञ्चकशतप्रयोगेण ॥
अष्टाद्यष्टोत्तरतस्तदशीत्यष्टांशगच्छेन ।
मूलधनं दत्वाष्टाद्यष्टोत्तरतो धनस्य मासार्धात् ॥ । ७५ ॥
लुङि प्रादान्मूलं वृद्धश्च विमुक्तिकालश्च ।
एषां परिमाणं किं विगणय्य सर्वे ममाचक्ष्व ॥ ७६ ॥
एकीकरणसूत्रम्--
वृद्धिसमासं विभजेन्मासफील्येन लब्धमिष्टः कालः ।
कालप्रमाणगुणितस्तादष्टकालेन सम्भक्तः ।
वृद्धिसमासेन हतो मूलसमासेन भाजितो वृद्धिः ॥ ७७(१/२ )॥
अत्रोद्देशकः ।
युका चतुश्शतीह द्विकत्रिकपञ्चकचतुषशतेन ।
मासाः पञ्च चतुर्दित्रयः प्रयोगैककालः कः ॥ ७८(१/२) ॥
इति मिश्रकव्यवहारे वृद्धिविधानं समाप्तम् ॥
प्रक्षेपककुटीकारः ॥
इतः परं मिश्रकव्यवहारे प्रक्षेपककुद्वीकारगणितं व्याख्यास्यामः ।
प्रक्षेपककरणमिदं सवर्गविच्छेदनांशयुतिहृतमिश्रः ।
प्रक्षेपकगुणकारः कुटीकारो बुधैस्समुद्दिष्टम् ॥ ७९ (१/२)॥
अत्रोद्देशकः ।
द्वित्रिचतुष्षड्भागैर्विभाज्यते द्विगुणषष्टिरिह हेम्नाम ।
भृत्येभ्यो हि चतुभ्य गणकाचक्ष्वाशु मे भागान् ।। ८०(१/२) ॥
8-A
76 गणितसारसङ्ग्रहः॒
प्रथमस्यांशत्रितयं त्रिगुणोत्तरतश्च पञ्चभिर्भक्तम् ।
दीनाराणां त्रिशतं त्रिषष्टिसहितं क एकांशः ॥ ८१(१/२) ॥
आदाय चाम्बुजानि प्रविश्य सङ्कावकोऽथ जिननिलयम् ।
पूजां चकार भक्त्या पूजावैभ्यो जिनेन्द्रेभ्यः ॥ ८२(१/२) ॥
वृषभाय चतुर्थाशं षष्ठांशं शिष्टपाधेय ।
द्वादशमथ जिनपतये व्यंशं मुनिसुत्रताय ददौ ॥ ८३(१/२) ॥
नष्टाष्टकर्मणे जगदिष्टायारिष्टनेमयेऽष्टांशम ।
षष्ठन्नचतुर्भागं भक्त्या जिनशान्तये प्रददौ ॥ ८४(१/२) ॥
कमलान्यशीतिमिश्राण्यायातान्यथ शतानि चत्वारि ।
कुसुमानां आगाख्यं कथय प्रक्षेपकाख्यकरणेन ॥ ८५(१/२) ॥
चत्वारि शतानि सरवे युतान्यशीत्या नरैर्विभक्तानि ।
पञ्चभिराचक्ष्व त्वं द्वित्रिचतुःपञ्चषणितैः ॥ ८६(१/२) ॥
इष्टगुणफलानयनसूत्रम्--
भक्तं शेषेथूलं गुणगुणितं तेन योजितं प्रक्षेपम् ।
तद्द्रव्यं मूल्यन्नं क्षेपविभक्तं हि मूल्यं स्यात् ॥ ८७(१/२) ॥
अस्मिन्नर्थे पुनरपि सूत्रम् --
फलगुणकारैर्हत्वा पणान् फलैरेव आगमादाय ।
प्रक्षेपके गुणास्स्युस्त्रैराशिकतः फलं वदेन्मतिमान् ॥ ८८(१/२) ॥
अस्मिन्नर्थे पुनराप सूत्रम्--
स्वफलहूताः स्वगुणन्नाः पणास्तु तैर्भवति पूर्ववच्छेषः।
इष्टफलं निर्दिष्टं त्रैराशिकसाधितं सम्यक् ॥ ८९(१/२) ॥
मिश्रकव्यवहारः 77
अत्रोद्देशकः ।
द्वाभ्यां त्रीणि त्रिभिः पञ्च पञ्चभिस्सप्त मानकैः ।
दाडिमस्रकपित्थानां फलानि गणितार्थवित् ॥ ९०(१/२) ॥
कपित्थात् त्रिगुणं ह्यत्र दाडिमं षड्गुणं भवेत् ।
क्रीत्वानय सरवे शीघ्र त्वं षट्सप्ततािभिः परैः ॥ ९१(१/२) ॥
दध्याज्यवीरघटैर्जिनबिम्बस्याभिषेचनं कृतवान् ।
जिनपॅरुषो द्वासप्ततिपलैस्त्रयः पूरिताः कलशाः ॥ ९२(१/२) ॥
द्वात्रिंशत्प्रथमघटे पुनश्चतुर्विंशतिईितीयघटे ।
षोडश तृतीयकलशे पृथक् पृथक् कथय मे कृत्वा ॥ ९३(१/२)॥
तेषां दधिघृतपयसां ततश्चतुर्विंशतिर्दूतस्य पलानि ।
षोडश पयःपलानि द्वात्रिंशद् दाधिपलानीह ॥ ९४(१/२) ॥
वृत्तिस्त्रयः पुराणाः पुंसश्वारोहकस्य तत्राप ।
सर्वेऽपि पञ्चषष्टिः कोचिद्भग्ना धनं तेषाम् ॥ ९५(१/२) ॥
सन्निहितानां दत्तं लब्धं पुंसा दशैव चैकस्य ।
के सन्निहिता भग्नाः के मम सञ्चिन्त्य कथय त्वम ॥ ९६(१/२) ॥
इष्टरूपाधिकहीनप्रक्षेपककरणसूत्रम् -
पिण्डोऽधिकरूपोनो होनोत्तररूपसंयुतः शेषात् ।
प्रक्षेपककरणमतः कर्नव्यं तैर्युता हीनाः ॥ ९७(१/२) ॥
अत्रोद्देशकः ।
प्रथमस्यैकांशोऽतो द्विगुणद्विगुणोत्तराद्भजन्ति नराः।
चत्वारोऽसः कंस्स्यादेकस्य हि सप्तषष्टिरिह ॥ ९॥78 गणितसारसङ्ग्रहः
प्रथमादध्यर्धगुणात् त्रिगुणाढ्पोत्तराद्विभज्यन्ते ।
साष्टा सप्ततिरोभिश्चतुर्भिराप्तांशकान् ब्रूहि ॥ ९९(१/२) ॥
प्रथमादध्यर्धगुणाः पर्धगुणोत्तराणि रूपाणि ।
पञ्चानां पञ्चशत्सैका चरणत्रयाभ्यधिका ॥ १००(१/२) ॥
प्रथमात्पद्यर्धगुणाश्चतुर्गुणोत्तराविहीनभागेन ।
भक्तं नरैश्चतुर्भिः पञ्चदशोनं शतचतुष्कम् ॥ १०१(१/२) ॥
समधनाघोनयनतज्ज्येष्ठधनसङ्ख्यानयनसूत्रम्--
ज्येष्ठधनं सैकं स्यात् खविक्रयेऽन्त्यार्धगुणमपैकं तत् ।
क्रयणे ज्येष्ठानयनं समानयेत् करणविपरीतात् ॥ १०२(१/२) ॥
अत्रोद्देशकः ।
द्वावष्ट षट्त्रिंशन्मूलं नृणां षडेव चरमार्षः।
एकार्षेण क्रीत्वा विक्रीय च समधना जाताः ॥ १०३(१/२) ॥
सर्वेकमर्धमर्धद्वयं च सङ्गृह्य ते त्रयः पुरुषाः ।
क्रयविक्रयौ च कृत्वा षड़िः पश्चार्षीत्समधना जाताः ॥ १०४(१/२) ॥
चत्वारिंशत् सैका समधनसङ्ख्या षडेव चरमार्थः।
आचक्ष्व गणक शीघ्र ज्येष्ठधनं किं च कानि मूलानि ॥ १०५(१/२) ॥
समधनसङ्ख्या पञ्चत्रिंशद्भवन्ति यत्र दीनाराः ।
चत्वारश्चरमाधं ज्येष्ठधनं किं च गणक कथय त्वम् ॥ १०६(१/२) ॥
चरमार्षभिन्नजातौ समधनार्घनयनसूत्रम्--
तुझ्यापच्छेदधनान्यार्थाभ्यां विक्रयक्रयाघ्र प्राग्वत् ।
छेदच्छेदछतिनावनुपातात् समधनानि भिन्नेऽन्त्यार्षे ॥ १०७(१/२)॥
मिश्रकव्यवहारः 79
अर्धत्रिपादभागा धनानि षट्पञ्चमांशकाश्चरमार्थः।
एकार्षेण क्रीत्वा, विक्रीय च समधना जाताः ॥ १०८(१/२) ॥
पुनराप अन्त्यार्षे भिन्न सात समधनानयनसूत्रम्
ज्येष्ठांशदिहरहतिः सान्त्यहरा बिक्रयोऽन्त्यमूल्यन्नः ।
नैको परिवलहरन्नः स्यात्क्रयसङ्घचानुपातोऽथ ॥ १०९(१/२) ॥
अत्रदंशकः ।
अर्ध द्वौ त्र्यंशौ च त्रीन् पादांशांश्च सङ्गृह्य ।
विक्रीय क्रीत्वान्ते पञ्चभिरङ्यंशकैस्समानधनाः ॥ ११०(१/२) ॥
इष्टगुणष्टसख़्यायामेिष्टसर्वेयासमर्पणानयनसूत्रम्--
अन्त्यपदं स्वगुणहते क्षिपेदुपान्त्यं च तस्यान्तम् ।
तेनोपान्त्येन भजेद्यछब्धं तद्भवेन्मूलम् ॥ १११(१/२) ॥
अत्रोद्देशकः ।
कश्चिच्छावकपुरुषश्चतुर्मुरवं जिनगृहं समासाद्य ।
पूजां चकार भक्त्या सुरभीण्यादाय कुसुमानि ॥ ११२(१/२) ॥
द्विगुणमभूदाद्यमुखे त्रिगुणं च चतुर्गुणं च पञ्चगुणम् ।
सर्वत्र पञ्च पञ्च च तत्सङ्ख्याम्भोरुहाणि कानि स्युः ।। ११३(१/२) ॥
द्वित्रिचतुर्भागगुणः पद्यर्धगुणास्त्रिपञ्चसप्ताष्टौ ।
भक्तैर्भक्त्याद्रेभ्यो दत्तान्यादाय कुसुमानि ॥ ११४ (१/२) ॥
इति मिश्रकव्यवहारे प्रक्षेपककुट्टकारः समाप्तः ।
• * The following stanz१ is added in M after stanza No. 10}, but it is nob found
अर्धत्रिपादभागा धनानि षट्पञ्चमांशकान्त्यार्थः ।
एकाधंण क्रीत्वा विक्रीय च समधना जाता: ।
80 वछिकाकुट्टीकारः ।
इतः परं वछिकाकुइंकारगणितं व्याख्यास्यामः । कुट्टीकारे
वांछकागणतन्यायसूत्रम--
छित्वा छेदेन राशिं प्रथमफलमपोह्याप्तमन्योन्यभक्तं
स्थाप्योर्वाधर्यतोऽधो मतिगुणमयुजाल्पेऽवशिष्टे धनम् ।
छित्वाधः स्वपरिलोपर्युतहरभागोऽधिकाग्रस्य हारं
छित्वा छेदेन साग्रान्तरफलमधिकाग्रान्वितं हारघातम् । ११५(१/२) ॥
अत्रोद्देशकः ।
जम्बूजम्बीररम्भाकमुकपनसवर्द्धरहिन्तालताली-
पुन्नागानाद्यनेकद्रुमकुसुमफलैर्नम्रशाखाधिरूढम् ।
भ्राम्यदृङ्गाब्जवापीऽशुकपिककुलनानाध्वनिव्याप्तदिकं
पान्थाः श्रान्ता वनान्तं श्रमनुदममलं ते प्रविष्टाः प्रहृष्टाः॥ ११६(१/२) ॥
राशित्रिषष्टिः कदलीफलानां
सम्पीड्य संक्षिप्य च सप्तभिस्तैः ।
पान्यैस्त्रयोविंशतिभिर्विशुद्धा
राशेस्त्वमेकस्य वद प्रमाणम् ।। ११७(१/२) ॥
राशीन् पुनर्बादश दाडिमानां
समस्य संक्षिप्य च पञ्चभिस्तैः ।
पान्थैर्नरैर्विशतिभिर्निरेकै
भक्तांस्तथैकस्य वद प्रमाणम् ।। ११८(१/२) ॥
दृष्टाम्रराशीन् पथिको यथेक
त्रिशत्समूह कुरुते त्रिहीनम् ।
मिश्रकव्यवहारः 81
शेषे हृते सप्ततिभिस्त्रिमिश्रैर्नरैर्विशुद्धं थयैकसङ्ख्याम् ॥ ११९(१/२)॥
दृष्टास्सप्तत्रिंशत्कपित्थफलराशयो वने पथिकैः ।
सप्तदशापोह्य हते व्येकात्यांशकप्रमाणं किम् ॥ १२०(१/२)॥
दृषुषराशिमपहाय च सप्त पश्चा
इतोऽष्टभिः पुनरपि प्रविहाय तस्मात् ।
त्रीणि त्रयोदशभिरुद्दलिते विशुद्धः
पान्थैर्वने गणक मे कथयैकराशिम् ॥ १२१(१/२)॥
द्वाभ्यां त्रिभिश्चतुर्भिः पञ्चभिरेकः कपित्थफलराशिः ।
भक्तो रूपाग्रस्तत्प्रमाणमाचक्ष्व गणितज्ञ ॥ १२२(१/२)॥
द्वाभ्यामेकस्त्रिभिदं च चतुर्भिर्भाजिते त्रयः ।
चत्वारि पञ्चभिश्शेषः को राशिर्वद मे प्रिय ॥ १२३(१/२)॥
द्वाभ्यामेकस्त्रिभिश्शुद्धश्चतुर्भािजिते त्रयः ।
चत्वारि पञ्चभिश्शेषः को राशिर्वद मे प्रिय ॥ १२४(१/२)॥
द्वाभ्यां निरग्र एकाग्रस्त्रिभिर्नाग्रो विभाजितः ।
चतुर्भिः पशुभर्भक्तो रूपाग्रो राशिरेष कः ॥ १२५(१/२)॥
द्वाभ्यामेकस्त्रिभिश्चूडश्चतुर्भिर्भाजिते त्रयः ।
निरग्रः पञ्चभिर्भक्तः को राशिः कथयाधुना ॥ १२६(१/२)॥
दृष्टा जम्बूफलानां पथि पथिकजनै राशयस्तत्र राशी
डों यश्न तौ नवानां त्रय इति पुनरेकादशानां विभक्ताः ।
पञ्चग्रास्ते यतीनां चतुरधिकतराः पञ्च ते सप्तकानां
कुडीकारार्थविन्मे कथय गणक सञ्चिन्त्य राशिप्रमाणम्॥ १२७(१/२)॥
82 गणितसारसङ्ग्रहः
वनान्तरे दाडिमराशयस्ते पान्यैस्त्रयस्सप्तभिरेकशेषाः ।
सप्त त्रिशेषा नवभिर्विभक्ताः पञ्चाष्टभिः के गुणक द्विरग्राः ॥१२८(१/२)॥
भक्ता द्वियुक्ता नवभिस्तु पथ
युक्ताश्चतुर्भिश्च षडष्टभिस्तैः ।
पान्थैर्जनैस्सप्तभिरेकयुक्ता
श्रश्चत्वार एते कथय प्रमाणम् ॥१२९(१/२)॥
अग्रशेषविभागमूलानयनसूत्रम्-
शेषांशाग्रवधो युक् खानेपान्यस्तदंशकेन गुणः ।
पावन्नागास्तावाद्विच्छेदाः स्युस्तदग्रगुणाः ॥ १३०(१/२)॥
अत्रादशक ।
आनीतवत्याम्रफलानि पुंसि
प्रागेकमादाय पुनस्तदर्धम् ।
गतेऽग्रपुत्रे च तथा जघन्य
स्तत्रावशेषार्धमथो तमन्यः ॥ १३१(१/२) ॥
प्रविश्य जैनं भवनं त्रिपूरुषं
प्रागैकमभ्यच्र्य जिनस्य पादे' ।
शेषत्रिभागं प्रथमेऽनुमाने
तथा द्वितीये च तृतीयके तथा ॥ १३२(१/२) ॥
शेषत्रिभागद्वयतश्च शेष-
श्यंशद्वयं चापि ततस्त्रिभागान् ।
कृत्वा चतुर्विंशतितीर्थनाथान्
समर्चयित्वा गतवान् विशः ॥ १३३(१/२) ॥
इति मिश्रकव्यवहारे साधारणकुट्टीकारः समाप्तः ॥
Tho MBB gives पादौ, which does not ovem to booorreot hore. B reds केशान्
for पादे.
मिश्रकव्यवहारः 83
विषमकुट्टीकारः ॥
इतः पर विषमकुट्टीकारं व्याख्यास्यामः । विषमकुट्टीकारस्य सूत्रम्--
मतिसङ्गणितौ छेदौ योज्योनत्याज्यसंयुतौ राशिहतौ ।
भिन्ने कुट्टीकारे गुणकारोऽयं समुद्दिष्टः ॥ १३४(१/२) ॥
अत्रोद्देशकः ।
राशिःषट्रेन हतो दशान्वितो नवहतो निरवशेषः ।
दशभिर्हनश्च तथा तदुणकौ कौ ममाशु सङ्कथय ॥ १३५(१/२) ॥
सकलकुट्टीकारः ।
सकलकुह्वकारस्य सूत्रम्--
भाज्यच्छेदाग्रशेपैः प्रथमहतफलं त्यज्यमन्योन्यभक्तं
न्यस्यान्ते साग्रमूवैरुपरिगुणयुतं तैस्समानासमाने ।
वर्णनं व्याप्तहारौ गुणधनमृणयोश्चाधिकाप्रस्य हरं
हत्व हत्वा तु साप्तान्तरधनमाधकाप्रान्वितं हारघातम् ॥ । १३६(१/२) ॥
अत्रोद्देशकः ।
सप्तोत्तरसप्तत्य युत शत योज्यमानमष्टांत्रशत् ।
सैकशतद्वयभक्तं को गुणकारो भवेदत्र ॥ १३७(१/२) ॥
पञ्चत्रिंशत् युत्तरषोडशपदान्येव हाराश्च ।
द्वात्रिंशद्वधिकैका युत्तरतोऽग्राणि के धनर्णगुणाः ॥ १३८(१/२)॥
अधिकाल्पराश्योर्दूलमिश्रविभागसूत्रम्--
ज्येष्ठम्नमहाराशेर्जघन्यफलताडितोनमपनीय ।
फलवर्गशेषभागो ज्येष्ठाषऽन्यो गुणस्य विपरीतम् ॥ १३९(१/२) ॥
1B गुणकारौ .
84 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
नवानां मातुलुङ्गानां कपित्थानां सुगन्धिनाम् ।
सप्तानां मूल्यसम्मिथं सप्तोत्तरशतं पुनः ॥ १४०(१/२) ॥
सप्तानां मातुलुङ्गानां कपित्थानां सुगन्धिनाम् ।
नवानां मूल्यसम्मिश्रमेकोत्तरशतं पुनः ॥ १४१(१/२) ॥
मूल्ये ते वद मे शीघं मातुलुङ्गकपित्थयोः ।
अनयोर्गणक त्वं मे कृत्वा सम्यक् पृथक् पृथक् ॥ १४२(१/२) ॥
बहुराशिमिश्रतन्मूल्यमिश्रविभागसूत्रम् --
इष्टम्नफलैरूनितलाभादिष्टाप्तफलमसकृत् ।
तैरूनितफलपिण्डच्छेदा गुणयुतास्तदर्धस्स्युः ॥ १४३(१/२) ॥
अत्रोद्देशकः ।
अथ मातुलुङ्गकदलीकपित्थदाडिमफलानि मिश्राणि।
प्रथमस्य सैकविंशतिरथ द्विरग्रा द्वितीयस्य ।। १४४(१/२) ॥
विंशतिरथ सुरभीणि च पुनस्त्रयोविंशतिस्तृतीयस्य ।
तेषां मूल्यसमासस्त्रिसप्ततिः किं फलं कोऽर्घः ॥ १४५(१/२) ॥
जघन्योनमिलितराश्यानयनसूत्रम्-
पण्यहृताल्पफलोनैश्छिन्द्यादल्पन्नमूल्यहीनेष्टम् ।
कृत्वा तावत्खण्डं तदूनमूल्यं जघन्यपण्यं स्यात् ॥ १४६(१/२) ॥
अत्रोद्देशकः ।
द्वाभ्यां त्रयो मयूरास्त्रिभिश्च पारावताश्च चत्वारः ।
हंसाः पञ्च चतुर्भिः पञ्चभिरथ सारसाष्षट् च ॥ १४७(१/२) ॥
मिश्रकव्यवहारः 85
यत्रार्धस्तत्र सर्वे षट्पञ्चशत्पणैः खगान् क्रीत्वा ।
द्वासप्ततिमानयतामित्युक्त्वा मूलमेवादात् ।
कतिभिः परैस्तु विहगाः कति विगणय्याशु जानीयाः ॥ १४९ ॥
त्रिभिः परैः शुण्ठिपलानि पञ्च चतुर्भिरेकादश पिप्पलानाम् ।
अष्टाभिरेकं मरिचस्य मूल्यं षष्ठनयाष्टोत्तरषष्टिमशु ॥ १५० ॥
इष्टाघरष्टमूल्यॉरष्टवस्तुप्रमाणानयनसूत्रम्--
मूल्यन्नफलेच्छागुणपणान्तरेष्टनयुतिविपर्यासः ।
द्विष्टः स्वधनेष्टगुणः प्रक्षेपककरणमवशिष्टम् ॥ १५१ ॥
अत्रोद्देशकः ।
त्रिभिः पारावताः पञ्च पञ्चभिस्सप्त सारसाः ।
सप्तभिर्नव हंसाश्च नवभिश्शिाविनत्रयः ॥ १५२ ॥
क्रीडार्थं नृपपुत्रस्य शतेन शतमानय ।
इत्युक्तः प्रहितः कश्चित् तेन किं कस्य दीयते ॥ १५३ ॥
व्यस्तार्धपण्यप्रमाणानयनसूत्रम्'
पण्यैक्येन पणैक्यमन्तरमतः पण्येष्टपण्यन्तरे-
श्छिन्द्यात्सङ्क्रमणे कृते तदुभयोरघं भवेतां पुनः ।
पण्ये ते रवलु पण्ययोगविवरे व्यस्तं तयोरर्थयोः
प्रश्नानां विदुषां प्रसादनामिदं सूत्रं जिनेन्द्रोदितम् ॥ १५४ ॥
अत्रोद्देशकः ।
आद्यमूल्यं यदेकस्यं चन्दनस्यागरोस्तथा।
पलानि विंशतिर्मिश्रं चतुरग्रशतं पणाः ॥ १५५ ॥
1 Not found in any of the Mss. Consuired.
88 गणितसारसारसङ्ग्रहः
कालेन व्यत्ययार्धस्स्यात्सषोडशशतं पणाः ।
तयोरर्थफले चूहि त्वं षडष्ट पृथक् पृथक् ॥ १५६ ॥
सूर्यरथाश्वेष्टयोगयोजनानयनसूत्रम्-
अरिवलास्राविलयाजनसङ्ख्यापर्याययोजनानि स्युः।
तानीष्टयोगसङ्ख्यानिन्नन्येकैकगमनमानानि ॥ १५७ ॥
अत्रोद्देशकः।
रविरथतुरगास्सप्त हि चत्वारोऽश्व वहन्ति धूर ।
योजनसप्ततिगतयः के व्यूढः के चतुयोगाः ॥ १५८ ॥
सर्वधनेष्टहीनशेषपिण्डात् स्वस्वहस्तगतधनानयनसूत्रम् –
रूपोननरैर्विभजेत् पिण्डीकृतभाण्डसारमुपलब्धम् ।
सर्वधनं स्यात्तस्मादुक्तविहीनं तु हस्तगतम् ॥ १५९ ॥
अत्रोद्देशकः ।
वणिजस्ते चत्वारः पृथक् पृथक् शौल्किकेन परिपृष्टाः ।
किं भाण्डसारमिति वलं तत्राहैको वणिकृच्छेष्ठः ॥ १६० ॥
आत्मधनं विनिगृह्य द्वाविंशतिरिति ततः परोऽवोचत् ।
त्रिभिरुत्तरा तु विंशतिरथ चतुरधिकैव विंशतिस्तुर्यः ॥ १६१ ॥
सप्तोत्तरविंशतिरिति समानसारा निगृह्य सर्वेऽपि ।
ऊचुः कि ब्रूहि सरवे पृथक् पृथग्भाण्डसारं मे ॥ १६२ ॥
अन्योऽन्यमिष्टरलसद्वयां दवा समधनानयनसूत्रम्--
पुरुषसमासेन गुणं दातव्य तांदृशTद्य पण्यभ्यः ।
शेषपरस्परगुणितं खं वं हित्वा मणेर्मूल्यम् ॥ १६३ ॥
मिश्रकव्यवहारः 87
अत्रोद्देशकः ।
प्रथमस्य शक्रनीलाः षट् सप्त च मरकता द्वितीयस्य।
वब्राण्यपरस्याष्टावेकैकार्थं प्रदाय समाः ॥ १६४ ॥
प्रथमस्य शक्रनीलाः षोडश दश मरकता द्वितीयस्य ।
वज्ञास्तृतीयपुरुषस्याष्टौ ङौ तत्र दत्वैव ॥ १६५ ॥
तेषेकैकोऽन्याभ्यां समधनतां यान्ति ते त्रयः पुरुषाः ।
तच्छक्रनीलमरकतवज्ञाणां किविधा अधीः ॥ १६६ ॥
तथेविक्रयलाभेः मूलानयनसूत्रम्--
अन्योऽन्यमूल्यगुणिते विक्रयक्षक्ते क्रयं यदुपलब्धम् ।
तेनैकोनेन हतो लाभः पूर्वार्धेतं मूल्यम् ॥ १६७ ॥
अत्रोद्देशकः ।
त्रिभिः क्रीणाति सप्तैव विक्रीणाति च पञ्चभिः।
नव प्रस्थान् वणिक् किं स्याळाभो द्वासप्ततिर्धनम् ॥ १६८ ॥
इति मिश्रकव्यवहारे सकलङ्कीकारः समाप्तः ।
सुवर्णकुट्टीकारः ॥
इतः परं सुवर्णगणितरूपकुट्टीकारं व्याख्यास्यामः ।
समस्तेष्टवर्णेरेकीकरणेन सङ्करवर्णानयनसूत्रम्--
कनकक्षयसंवर्गे मिश्रवणहृतः क्षयो ज्ञेयः।
पवर्णप्रविभक्तं सुवर्णगुणितं फलं हेम्नः ॥ १६९ ॥
88 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
एकक्षयमेकं च द्विक्षयमेकं त्रिवर्णमेकं च ।
वर्णचतुष्के च वै पञ्चक्षयिकाश्च चत्वारः ॥ १७० ॥
सप्त चतुर्दशवर्णास्त्रिगुणितपञ्चक्षयाश्चष्टे ।
एतानेकीकृत्य ज्वलने क्षिप्त्वैव मिश्रवणं किम् ।
एतमिश्रसुवर्ण पूर्वैर्भक्तं च किं किमेकस्य ॥ १७१ ॥
इष्टवर्णानामिष्टस्ववर्णानयनसूत्रम्--
वैस्वैर्वर्णहतैर्मिश्री स्वर्णमिश्रेण भाजितम् ।
लब्धं वर्णं विजानीयात्तदिष्टातं पृथक् पृथक् ॥ १७२ ॥
अत्रोद्देशकः।
विंशतिपणास्तु षोडश वर्ण दशवर्णपरिमाणैः ।
परिवर्तिता वद त्वं कति हि पुराणा भवन्त्यधुना ॥ १७३ ॥
अष्टोत्तरशतकनक वणष्टांशत्रयेण सयुक्तम् ।
एकादशवर्णं चतुरुत्तरदशवर्णकैः कृतं च किं हेम ॥ १७४॥
अज्ञातवणनयनसूत्रम्-
कनकक्षयसंवर्गे मिी वर्णनमिश्रतः शोद्यम् ।
वर्णने हृतं वर्ण वर्णविशेषेण कनकं स्यात् ॥ १७५ ॥
अज्ञातवर्णस्य पुनरपि सूत्रम्
स्वर्णवर्णविनिहतयोगं खणैक्यदृढहताच्छोध्यम् ।
अज्ञातवर्णहेम्ना भक्तं वर्ण बुधाः प्राहुः ॥ १७६ ॥
अत्रोद्देशकः ।
षड्जलधिवह्निकनकैरस्त्रयोदशाष्टीवर्णकैः क्रमशः ॥
Here वह्नि is autabituted for रनल, and ष्टर्तुवर्णकैः /or ष्टवृतुक्षयेः as thereby
ho reading will be better grammatically.
मिश्रकव्यवहारः 89
अज्ञातवर्णहेम्नः पञ्च विमिश्रक्षयं च सैकदश।
अज्ञातवर्णसङ्कयां ब्रूहि सरवे गणिततज्ञ ॥ १७८ ॥
चतुर्दशैव वर्णानि सप्त स्वर्णानि तत्क्षये।
चतुस्वरों दशोत्पन्नमज्ञातक्षयकं वद ॥ १७९ ॥
अज्ञातखणीनयनसूत्रम्
स्वस्वर्णवर्णविनिहतयोगं खगैक्यगुणितदृढवर्णात् ।
त्यक्त्वाज्ञातस्वर्णक्षयदृढवर्णान्तराहतं कनकम् ॥ १८० ॥
अत्रोद्देशकः ।
द्वित्रिचतुःक्षयमानास्त्रिस्त्रिः कनकास्त्रयोदशक्षयिकः।
वर्णयुतिर्दश जाता ब्रूहि सखे कनकपरिमाणम् ॥ १८१ ॥
युग्मवणोमश्रसुवणोनयनसूत्रम्
ज्येष्ठारुपक्षयशोधितपक्कविशेषाप्तरूपकैः प्राग्वत् ।
प्रक्षेपमतः कुर्यादेवं बहुशोऽपि वा साध्यम् ॥ १८२ ॥
पुनरपि युग्मवर्णमिश्रवणनयनसूत्रम्-
इष्टाधिकान्तरं चैव हीनेष्टन्तरमेव च ।
उभे ते स्थापयेद्यस्तं स्वर्गं प्रक्षेपतः फलम् ॥ १८३ ॥
अत्रादकः
दशवर्णसुवर्णं यत् षोडशवर्णेन संयुतं पक्वम् ।
द्वादश चेत्कनकशतं द्विभेदकनके पृथक् पृथग्बृहि ॥ १८४ ॥
बहुस्रवणनयनसूत्रम्
व्येकपदानां क्रमशः स्वर्णानीष्टानि करपयेच्छषम् ।
अव्यक्तकनकाविधिना प्रसाधयेत् प्राक्तनायेव ॥ १८५ ॥
1 Phe reading in the MSS. i8 तत्क्षय, which is obviously erroneous.
90 गणितसारसङ्ग्रहः
अत्रोद्देशकः
वर्णाश्शरर्तुनगवसुमृडविश्वे नव च पक्ववर्ण हि।
कनकानां षष्टिश्चेत् पृथक् पृथक् कनकमा कि स्यात् ॥१८६ ॥
इयनष्टवणोनयनसूत्रम्
वणभ्यां हृतरूपे सुवर्णवर्णाहते दृषु ।
स्वर्णहृतैकेन च हीनयुते व्यस्तत हि वर्णफलम् ॥ १८७ ॥
अत्रोद्देशकः
षोडशदशकनकाभ्यां वर्णा न ज्ञायते’ पक्त्रम् ।
वर्णा चैकादश् चेद्दणं तत्कनकयोर्भवेतां कौ ॥ १८८ ॥
पुनरपि द्वयनष्टवर्णानयनसूत्रम् -
एकस्य क्षयमिष्टं प्रकल्प्य शेषं प्रसाधयेत् प्राग्वत् ।
बहुकनकानामिष्टं व्येकपदानां ततः प्राग्वत् । १८९ ॥
अत्रोद्देशकः ।
द्वादशचतुर्दशानां वर्णानां समरसीछते जातम् ।
वर्णानां दशकं स्यात् तदृणं ब्रूहि सञ्चिन्त्य॥ १९० ॥
अपरार्धस्योदाहरणम् ।
सप्तनवाशीविदशानां कनकानां संयुते पक्वम् ।
द्वादशवर्ण जातं किं ब्रूहि पृथक् पृथग्वर्णम् ॥ १९१ ॥
परीक्षणशलाकानयनसूत्रम्-
परमक्षयाप्तवर्णाः सर्वशलाकाः पृथक् पृथग्योज्याः ।
स्वर्णफलं तच्छोध्यं शलाकपिण्डात् प्रपूराणका ॥ १९२ ॥
B adds here यते ।
मिश्रकव्यवहारः 91
अत्रोद्देशकः।
वैश्याः स्वर्णशलाकाश्चिकीर्षवः स्वर्णवर्णज्ञाः।
चक्रुः स्वर्णशलाक द्वादशवर्णं तदाद्यस्य ॥ १९३॥
चतुरुत्तरदशवर्णा षोडशवर्णं तृतीयस्य ।
कनकं चास्ति प्रथमस्यैकोनं च द्वितीयस्य ॥ १९४ ॥
अर्धार्धन्यूनमथ तृतीयपुरुषस्य पादानम् ।
परवणादारभ्य प्रथमस्येकान्त्यमेव च व्यन्यम् ॥ १९५ ॥
यन्यं तृतीयवणिजः सर्वशलाकास्तु माषमिताः ।
भृढं कनक किं स्यात् प्रपूरणी का पृथक् पृथक् त्वं मे।
आचक्ष्व गणक शीघ्र सुवर्णगणितं हि यदि वेत्सि ॥ १९६(१/२) ॥
विनमयवणसुवणनयनसूत्रम् --
क्रयगुणसुवर्णविनिमयवर्गेष्टनान्तरं पुनः स्थाप्यम्।
व्यस्तं भवति हि विनिमयवर्णान्तरहृत्फलं कनक : ॥ १९७(१/२) ॥
अत्रोद्देशकः।
षोडशवणं कनकं सप्तशतं विनिमयं कृतं लभते ।
द्वादशदशवणोभ्यां साष्टसहस्त्रं तु कनकं किम् ।। १९८(१/२) ॥
बहुपदविनिमयसुवर्णकरणसूत्रम्-
वणन्नकनकमेष्टवर्णनात दृढक्षयो भवति ।
प्राग्वत्प्रसाथ लखधं विनिमयबहुपदसुवर्णानाम् ॥ १९९(१/२) ॥
अत्रोद्देशकः।
वर्णचतुर्दशकनकं शतत्रयं विनिमयं प्रकुर्वन्तः।
वर्णद्वादशदश्वसुनगैश्च शतपञ्चकं स्वर्णम् ।
9-A
92 गणितसारसङ्ग्रहः
एतेषां वर्णानां पृथक् पृथक स्वर्णभानं किम् ॥ २०१ ॥
विनिमयगुणवर्णकनकलाभानयनसूत्रम् –
खगैन्नवर्णयुतिकृतगुणयुतिमूलक्षयन्नरूपोनेन। आतं लब्धं शोध्यं मूलधनाच्छेषवित्तं स्यात् ।। २० २ ॥
तछब्धमूलयोगाद्विनिमयगुणयोगभाजितं लब्धम्। प्रक्षेपकेण गुणितं विनिमयगुणवर्णकनकं स्यात् ॥ २०३ ॥
अत्रोद्देशकः।
कश्चिद्वणिक् फलाथी षाडवणं शतद्वयं कनकम् । यत्किञ्चिद्विनिमयकृतमेकाचं द्विगुणितं यथा क्रमशः। २०४
द्वादशवसुनवदशकक्षयकं लाभ द्विरग्रशतम्। शोषं किं स्याद्विनिमयकांस्तेषां चापि मे कथय ।। २०५ ॥
दृश्यसुवर्णबािनमयसुवर्णामूलानयनसूत्रम्--
विनिमयवर्णनातं वांशं वेष्टक्षयन्नसंमिश्रात् । अंगैक्योनेनातं दृश्यं फलमत्र भवति मूलधनम् ॥ २०६ ॥
अत्राद्देशकः।
वणिजः कंचित् षोडशवर्णकसौवर्णगुलकमाहृत्य । त्रिचतुःपञ्चमभागान् क्रमेण तस्यैव विनिमयं कृत्वा ॥ २०७ ॥
द्वादशदशनववरैः संयुज्य च पूर्वशेषेण। मूलेन विना दृष्टं वर्णसहखं तु किं मूलम् ॥ २०८ ॥
इष्टांशादानेन इष्टवर्णानयनस्य तदिष्टांशकयोः सुवर्णानयनस्य च सूत्रम्--
अंशातैकं व्यस्तं क्षिप्त्वेष्टनं भवेत् सुवर्णमयी । सा गुलिका तस्या अपि परस्परांशाप्तकनकस्य ॥ २०९ ॥
<poem>मिश्रकव्यवहारः 93
स्वदृढक्षयेण वण प्रकल्पयेत्प्राग्वदव यथा।
एवं तद्द्वययोरप्युभ्यं साम्यं फलं भवेद्यदि चेत् ॥ २१०॥
प्रकल्पनंष्टवणों गुलिकाभ्यां निश्चयौ भवतः ।
न चेत्प्रथमस्य तदा किञ्चिन्यूनाधिकौ क्षयौ कृत्वा ॥ २११ ॥
तत्क्षयपूर्वक्षययोरन्तरिते शेषमत्र संस्थाप्य।
त्रैरशिकविधिलब्धं वर्णं तेनोनिताधिकौ स्पष्टौ॥ २१२ ॥
अत्रोद्देशकः ।
खर्णपरीक्षकवणि परस्परं याचितौ ततः प्रथमः ।
अर्ध प्रादात् तामपि गुलिकां वसुवर्ण आयोज्य।। २१३ ॥
वर्णदशकं करोमीत्यपरोऽवादीत् त्रिभागमात्रतया।
लब्धे तथैव पूर्ण द्वादशवर्ण करोमिं गुलिकाभ्याम् । २१४ ॥
उभयोः सुवर्णमाने वर्षों सञ्चिन्त्य गणेततवज्ञ ।
सौवर्णगणितकुशले यदि तेऽसि निगद्यतामाशु ॥ २१५ ॥
इति मिश्रकव्यवहारे सुवर्णकुट्टीकारः समाप्तः ॥
विचित्रकुट्टीकारः ।
इतः परं मिश्रकव्यवहारे विचित्रकुट्टीकारं व्याख्यास्यामः।
सत्यानृतसूत्रम्--
पुरुषाः सैकेष्टगुणा द्विगुणेष्टोना भवन्त्यसत्यानि ।
पुरुषकृतिस्तैरूना सत्यानि भवन्ति वचनानि ॥ २१६ ॥
अत्रोद्देशकः।
कामुकपुरुषाः पच हेि वश्यायाश्च प्रयास्त्रयस्तत्र ।
प्रत्येकं सा जूते त्वमिष्ट इति कानि सत्यानि ॥ २१७ ॥
94 गणितसारसङ्ग्रहः
प्रस्तारयोगभेदस्य सूत्रम् --
एकाद्योकोत्तरतः पदमूर्वधर्यतः क्रमोत्क्रमशः ।
स्थाप्य प्रतिलोमनं प्रतिलोमग्नेन भाजितं सारम् ॥ २१८ ॥
अत्रोद्देशकः।
वर्णाश्च रसानां कषायतिक्ताम्लकटुकलवणानाम्।
मधुररसन युतानां भेदान् कथयाधुना गणक ॥ २१९॥
वजेन्द्रनीलमरकतविद्ममुक्ताफलैस्त रचितमालायाः ।
कति भेदा युतिभेदात् कथय सरवे सम्यगाशु त्वम् ॥ २२०॥
केतक्यशोकचम्पकनीलोत्पलकुसुमराचितमालायाः ।
कति भेदा युतिभेदात्कथय सखे गणिततत्वज्ञ ॥ २२१ ॥
ज्ञाताज्ञातलाभमूलानयनसूत्रम्--
लाभानामअराशः प्रक्षपकतः फलानं ससाध्य ।
तेन हृतं तच्छब्ध मूल्यं त्वज्ञातपुरुषस्य ॥ २२२ ॥
अत्रोद्देशकः ।
समये केचिद्वणिजस्त्रयः क्रयं विक्रयं च कुर्वीरन् ।
प्रथमस्य षट् पुराण अष्टौ मूल्यं द्वितीयस्य ॥ २२३ ॥
न ज्ञायते तृतयस्य व्याप्तिस्तैर्नरैस्तु षण्णवतिः ।
अज्ञातस्यैव फलं चत्वारिंशद्वि तेनाप्तम् ॥ २२४ ॥
कस्तस्य प्रक्षेप वणिजोरुभयोर्भवेच्च को लाभः ।
प्रगणय्याचक्ष्व सरवे प्रक्षेपं यदि विजानासि ॥ २२५ ॥
भाठकानयनसूत्रम् –
भरभुतगतगम्यहाते त्यक्त्वा यजनदलनभारकृतेः ।
तन्मूलनं गम्यच्छन्न गन्तव्यभाजतं सारम् ॥ २२६ ॥
A M and B add त here ; netrically it is faulty.
95 मिश्रकव्यवहारः
अत्रोद्देशकः ।
पनसानि द्वात्रिंशतीत्वा योजनमसौ दलोना।
ग्रहात्यन्तर्भाष्टकमर्थं भग्नोऽस्य कि देयम् ॥ २२७ ॥
द्वितीयतृतीययोजनानयनस्य सूत्रम्--
भरभाठकसंवर्गाऽद्वितीयभूतिकृतिविवर्जतश्छेदः।।
तद्रुत्यन्तरभरगतिहतेरीतिः स्याद् द्वितीयस्य ॥ २२८ ॥
अत्रोद्देशकः ।
पनसान चतुर्विंशतिमा नीत्वा पञ्चयोजनानि नरः ।
लभते तवृतिमिह नव षड्भूतिवियुते द्वितीयनृगतिः का।। २२९॥
बहुपद भाटकानयनस्य सूत्रम्--
सन्निहितनरहतेषु प्रागुत्तरामिश्रितेषु मार्गेषु ।
व्यावृत्तनरगुणेषु प्रक्षेपकसाधत मूल्यम् ॥ २३० ॥
अत्रोद्देशकः ।
शिबिकां नयान्त पुरुषा विंशातिरथ ये जनद्वयं तेषाम् ।
वृत्तिर्वीनाराणां विंशत्याधकं च सप्तशतम् ॥ २३१ ॥
क्रोशद्वये निवृत्त द्वावुभयोः क्रोशयोस्त्रयश्चान्ये ।
पञ्च नरः शेषार्धाद्यावृत्ताः का भूतिस्तेषाम् ॥ २३२ ॥
इष्टगुणतपष्ठलकानयनसूत्रम्--
सैकगुणा स्वस्वेष्टं हित्वान्येन्यन्नशेषमितिः ।
अपवर्य योज्य मूलं विष्णोः) कृत्वा व्येकेन मूलेन ॥२३३॥
पूवोपवतेराशन् हत्वा पूर्वापवर्तराशियुतेः ।
पृथगेव पृथक् त्यक्त्वा हस्तगताः स्वधनसङ्ख्याः स्युः ।।२३४ ॥
ताः स्वस्वं हित्वैव त्वशेषयोगं पृथक् पृथक् स्थाप्य ।
स्वगुणन्नाः स्वकरगतंरूनाः पांडुलकसङ्ख्याः स्युः ।। २३५ ।।
B omits पद here.
96 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
मार्गे त्रिभिर्वाणिग्भिः पोट्लकं दृष्टमाह तत्रैकः ।
पेट्टलकमिदं प्राप्य द्विगुणधनोऽहं भविष्यामि । २३६ ॥
हस्तगताभ्यां युवयोस्त्रिगुणधनोऽहं द्वितीय आहोति ।
पधगुणोऽहं त्वपरः पोट्टलहस्तस्थमानं किम् ॥ २३७ ॥
सवतुल्यगुणकपाट्लकानयनहस्तगतनयनसूत्रम्--
व्येकपदतव्येकगुणेष्टांशवधोनितांशयुतिगुणघातः ।
हस्तगताः स्युर्भवति हि पूर्ववदिष्टांशभाजितं पोट्टलकम् ।। २३८ ॥
अत्रोद्देशकः ।
वैश्यैः पञ्चभिरेकं पट्टलक दृष्टमाह चैकैकः ।
पोट्टलकषष्ठसप्तमनवमाष्टमदशमभागमाप्त्वैव ।। २३९ ॥
खखकरस्थेन सह त्रिगुणं त्रिगुणं च शेषाणाम् ।
गणक त्वं में शत्रिं वद हस्तगतं च पोट्टलकम् । २४० ॥
इष्टांशेष्टगुणपोह्लकानयनसूत्रम-
इष्टगुणन्नन्यांशाः सेष्टांशाः सैकनिजगुणहता युक्ताः।
व्यूनपदनेष्टशन्यूनाः सैकेष्टगुणहता हस्तगताः ॥२४१॥
अत्रोद्देशकः ।
द्वाभ्यां पथि पथिकाभ्यां पोट्टलकं दृष्टमाह तत्रैकः ।
अस्यार्थी सम्प्राप्य द्विगुणधनोऽहं भविष्यामि ॥ २४२ ॥
अपरख्यंशद्वितयं त्रिगुणधनस्वकरस्थधनात् ।
मत्करधनेन सहितं हस्तगतं किं च पोट्टलकम् ॥ २४३ ॥
दृष्टं पथि पथिकाभ्यां पोट्टलकं तदृहीत्वा च ।
द्विगुणमभूदाद्यस्तु स्वकरस्थधनेन चान्यस्य ॥
मिश्रकव्यवहारः 97
हस्तस्थधनादन्यास्त्रिगुणं किं करगतं च पोष्टलकम् ॥ २४४(१/२) ॥
मार्गे नरै श्चतभिः पोट्टलकं दृष्टमाह तत्राद्यः ।
पोट्लकामदं लब्ध्वा ह्यष्टगुणोऽहं भविष्यामि ॥ २४५(१/२) ॥
खकरस्थधनेनान्यो नवसङ्गणितं च शेषधनात् ।
दशगुणधनवानपरस्त्वेकादशगुणतधनवान् स्यात् ।
पोट्टलकं किं करगतधनं कियद्वाहेि गणकाशु ।। २४७ ।।
मा नरैः पोह्लकं चतुभीष्टं हि तस्यैव तदा बभूवुः ।
पञ्चांशपादार्धतृतीयभागास्तद्वित्रिपञ्चस्रचतुर्गुणश्च ॥ २४८ ॥
मार्गे त्रिभिर्वणिग्भिः पोट्टलकं दृष्टमाह तत्राद्यः ।
यद्यस्य चतुर्भागं लभेऽहमित्याह स युवयोर्डिगुणः ॥ २४९ ॥
आह त्रिभागमपरः स्वहस्तधनसाहितमेव च त्रिगुणः ।
अस्यार्थं प्राप्याहं तृतीयपुरुषश्चतुर्नधनवान् स्याम् ।
आचक्ष्व गणक छं कि हस्तगतं च पोट्टलकम् ॥ २५०(१/२) ॥
याचितरूपैरिष्टगुणकहस्तगतानयनस्य सूत्रम्-
याचितरूपैक्यानि स्वसैकगुणवर्धितानि तैः प्राग्वत् ।
हस्तगताना नीत्वा चेष्टगुणनेति सूत्रेण ॥ २५१(१/२) ॥
सदृशच्छद कृत्वा सेकंष्टगुणाहृतष्टगुणयुत्या ।
रूपमेनितया भक्तान् तानेव करस्थितान् विजानीयात् ॥ २५२(१/२) ॥
अत्रोद्देशकः ।
वैश्यैस्त्रिभिः परस्परहस्तगतं याचितं धनं प्रथमः ।
चत्वार्यथ द्वितीयं पञ्च तृतीयं नरं प्रावें ॥ २५३(१/२) ॥
1 M and B read स्यु:; and it is obviously inappropriate.
98 गणितसारसङ्ग्रहः
द्विगुणोऽभवद्वितीयः प्रथमं चत्वारि षट् तृतीयमगात् ।
त्रिगुणं तृतीयपुरुषः प्रथमं पञ्च द्वितीयं च ॥ २५४(१/२) ॥
षट् प्रार्षीत्पञ्चकगुणः स्वहस्तस्थितानि कानि स्युः ।
कथयाशु चित्रकुटीमिझी जानासि यदि गणक ॥ २५५(१/२) ॥
पुरुषास्त्रयोऽतिङशलाश्चान्योन्यं याचितं धन प्रथमः ।
स द्वादश द्वितीयं त्रयोदश प्रार्थे तत्रिगुणः ॥ २५६(१/२) ॥
प्रथमं दश त्रयोदश तृतीयमभ्यर्थं च द्वितीयोऽभूत् ।
पञ्चगुणितो द्वितीयं द्वादश दश याचयित्वाद्यम् ॥ २५७(१/२) ॥
सप्तगुणितस्तृतीयोऽभवन्नरो वाञ्छितानि लब्धानि ।
कथय सखे विगणय्य च तेषां हस्तस्थितानि कानि स्युः॥ २५८(१/२) ॥
अन्त्यस्योपान्त्यतुल्यधनं दत्त्वा समधनानयनसूत्रम्—
वाञ्छाभक्तं रूपं स उपान्त्यगुणः सरूपसंयुक्तः ।
शेषाणां गुणकारः सैकोऽन्यः करणमेतत्स्यात् ॥ २५९(१/२) ॥
अत्रोद्देशकः ।
वैश्यात्मजास्त्रयस्ते मागेगत ज्यष्ठमध्यमकांनष्ठः ।
स्वधने ज्येष्ठो मध्यमधनमात्रं मध्यमाय ददौ ॥ २६०(१/२) ॥
स तु मध्यमो जघन्यजधनमात्रं यच्छति स्माय ।
समधनिकाः स्युस्तेषां हसगतं ब्रूहि गणक संचिन्त्य ॥ २६१(१/२) ॥
वैश्यात्मजाश्च पञ्च ज्येष्टादनुजः स्वकीयधनमात्रम् ।
लेभे सर्वेऽप्येवं समवित्ताः किं तु हस्तगतम् ॥ २६२(१/२) ॥
मिश्रकव्यवहारः 99
वणिजः पञ्च स्वस्वाद घी पूर्वस्य दत्त्वा तु ।
समवित्ताः सञ्चिन्त्य च किं तेषां बूहि हस्तगतम् ।। २६३(१/२) ॥
वणिजष्षट् खधनाद्वित्रिभागमात्रं क्रमेण तज्येष्ठाः ।
स्वस्वानुजाय दत्त्वा समवित्ताः कि च हस्तगतम् ॥ २६४(१/२) ॥
परस्परहस्तगतधनसङ्ख्यामात्रधनं दत्त्वा समधनानयनसूत्रम्—
वाञ्छाक्षक्तं रूपं पदयुतमादावुपयुपयेतत् ।
संस्थाप्य सैकवाञ्छागुणितं रूपोनमितरेषाम् ।। २६५(१/२) ॥
अत्रोद्देशकः ।
वणिजस्त्रयः परस्परकरस्थधनमेकतोऽन्योन्यम् ।
दत्त्वा समवित्ताः स्युः कि स्यादस्तास्थितं द्रव्यम ।। २६६(१/२) ॥
वणिजश्चत्वारस्तेऽप्यन्योन्यधनार्धमात्रमन्यस्मात् ।
खछित्य परस्परतः समवित्ताः स्युः कियत्करस्थधनम् ।। २६७(१/२) ॥
जयापजययोलोभानयनसूत्रम् -
स्वस्वच्छेदांशयुती स्थाप्यध्वधयतः क्रमात्क्रमशः ।
अन्योन्यच्छेदांशकगुणतौ वज्नापवर्तनक्रमशः ॥ २६८(१/२) ॥
छेदांशक्रमवत्स्थिततदन्तराभ्यां क्रमेण सम्भक्तौ ।
स्वांशहरन्नान्यहरौ वाञ्छाभौ व्यस्ततः करस्थमितिः ॥ २६९(१/२) ॥
अत्रोद्देशकः ।
दृष्ट्वा कुकुठयुद्धे प्रत्येकं तौ च कुक्कुटिकौ ।
उक्तौ रहस्यवाक्यैर्मन्त्रौषधशक्तिमन्महापुरुषेण ॥ २७०(१/२) ॥
100 गणितसारसङ्गहः
जयति हि पक्षी ते मे देहि वर्णा ह्यविजयोऽसि दद्यां ते।
तद्वियंशकमद्यत्यपरं च पुनः स संसृत्य ॥ । २७१(१/२) ॥
त्रिचतुर्थं प्रतिवाञ्छत्युभयस्माद्वादशैव लाभः स्यात् ।
तत्कुक्कुटिककरस्थं ब्रूहि त्वं गणकमुवतिलक ॥ २७२(१/२) ॥
राशिलब्धच्छेदमिश्रविभागसूत्रम् -
मिश्रादूनितसङ्ख्या छेदः सैकेन तेन शेषस्य ।
भागं हृत्वा लब्यं लाभोनितशेष एव राशिः स्यात् ।। २७३(१/२) ॥
अत्रोद्देशकः ।
केनापि किमपि भक्तं सच्छेदो राशिमिश्रितो लाभः ।
पञ्चशत्रिभिरधिका तच्छेदः किं भवेछब्धम् ॥ २७४(१/२) ॥
इष्टसङ्ख्यायोज्यत्याज्यवर्गमूलराश्यानयनसूत्रम्
योज्यत्याज्ययुतिः सरूपविषमाग्रश्नार्धिता वर्गिता
व्यग्रा बन्धहृता । च रूपसहिता त्याज्यैक्यशेषाग्रयोः ।
शेषेक्यार्धयुतोनित फलमिदं राशिर्भवेद्वाञ्छयोः
स्त्याज्यत्याज्यमहवयोरथ कृतेथूलं ददात्येव सः ।। २७५(१/२) ॥
अत्रोद्देशकः ।
राशिः कश्चिद्दशभिः संयुक्तः सप्तदशाभीरापि हीनः ।
मूलं ददाति शुदं तं राशिं स्यान्ममाशु वद गणक ॥ २७६(१/२) ॥
राशिस्सप्तभरूनो यः सोऽष्टादशभिरन्वितः कश्चित् ।
मूलं यच्छति शुद्धं विगणय्याचक्ष्व तं गणक ॥ २७७(१/२) ॥
मिश्रकव्यवहारः 101
राशद्येशांनांत्रसप्तर्भागान्वितस्स एव पुनः ।
मूलं यच्छति कोऽसौ कथय विचिन्त्याशु तं गणक ॥ २७८(१/२) ॥
इष्टसङ्ख्याहनियुक्तवर्गमूलानयनसूत्रम्--
उद्दिष्टो यो राशिस्वर्घछतवर्गितोऽथ रूपयुतः ।
यच्छति मूलं स्वेष्टात्संयुक्ते चापनीते च ॥ २७९(१/२) ॥
अत्रोद्देशकः ।
दर्शभिस्सम्मिश्रोऽयं दशभिस्तैर्वर्जितस्तु संशुद्धम् ।
यच्छति मूलं गणक प्रकथय सञ्चिन्त्य राशिं मे ।। २८०(१/२) ।।
इष्टवर्गीकृतराशिद्वयादिष्टम्नादन्तरमूलादिष्टानयनसूत्रम्--
सैकेटच्येकेटावर्धकृत्याथ वर्गितौ राशी ।
एताविष्टस्रावथ तद्विश्लेषस्य मूलमिष्टं स्यात् ॥ २८१(१/२) ॥
अत्रोद्देशकः ।
यौकौचिद्वर्गातराश गुणितौ तु सैकसप्तत्या ।
सद्विश्लेषपदं स्यादेकोत्तरसप्ततिश्च राशी कौ ।।
विगणय्य चित्रकुट्टकगणितं यदि वेत्सि गणकं मे ब्रूहि ॥ २८३ ॥
युतहीनप्रक्षेपकगुणकारानयनसूत्रम्--
संवर्गितेष्टशेषं द्विष्टं रूपेष्टयुतगुणाभ्यां तत् ।
विपरीतभ्यां विभजेत्प्रक्षेपौ तत्र हीनौ वा ॥ २८४ ॥
अत्रोद्देशक ।
त्रिक पञ्चकसंवर्गः पञ्चदशाष्टादशौव चेष्टमपि।
इष्टं चतुर्दशात्र प्रक्षेपः कोऽत्र हानिव ॥ २८५ ॥
102 गणितसारसङ्ग्रहः
विपरीतकरणानयनसूत्रम--
प्रत्युत्पन्न भागो भागे गुणितोंऽधिके पुनश्शोध्यः ।
वगै मूलं मूले वर्गे विपरीतकरणामिदम् ॥ २८६ ॥
अत्रोद्देशकः ।
सप्तदंते को राशित्रिगुणो वर्गीकृतः शरैर्युक्तः ।
त्रिगुणितपधांशहूतवर्धितमूलं च पञ्चरूपाणि ॥ २८७ ॥
साधारणशरपरिध्यानयनसूत्रम् -
शरपरिधित्रिकामिलनं वर्गितमेतत्पुनस्त्रिभिस्सहितम् ।
द्वादशहृतेऽपि लब्धं शरसह्या स्यात्कलापकाविष्टा ॥२८८॥
अत्रोद्देशकः ।
परिधिशरा अष्टादश तूणीरस्थाः शराः के स्युः ।
गणितज्ञ यदि विचित्रे कुटीकारे श्रमोऽस्ति ते कथय ।। २८९। ।
इति मिश्रकव्यवहारे विचित्रकुट्टीकारः समाप्तः ।
श्रेढीबद्धसङ्कलितम् ।
इतःपरं मिश्रकगणिते श्रेढीबद्धसङ्कलितं व्याख्यास्यामः ।
हीनाधिकचयसङ्कलितधनानयनसूत्रम् -
व्येकार्थपदोनाधिकचयघानान्चितः पुनः प्रभवः ।
गच्छाभ्यसो हीनाधिकचयसमुदायसङ्कलितम् ॥ २९० ॥
अत्रोद्देशकः ।
चतुरुत्तरदश चादिर्हनचयस्त्रीणि पञ्च गच्छः किम् ।
द्वावादिर्थंडिचयः षट् पदमष्टं धनं भवेदत्र ॥ २९१ ॥
मिश्रकव्यवहारः 103
अधिकहीनोत्तरसङ्कलितधने आद्युत्तरानयनसूत्रम्--
गच्छविभक्ते गणिते रूपोनपदार्थगुणतचयहीने ।
आदिः पदहृतवित्तं चाधूनं व्येकपददलहूतः प्रचयः ॥ २९२ ।।
अत्रोद्देशकः ।
चत्वारिंशद्भणितं गच्छः पञ्च त्रयः प्रचयः ।
न ज्ञायतेऽधुनादिः प्रभवो द्विः प्रचयमाचक्ष्व ॥ २९३ ॥
श्रेढसङ्कलितगच्छानयनसूत्रम्--
आदिविहीनो लाभः प्रचयार्धहृतस्स एव रूपयुतः ।
गच्छ लाभेन गुणो गच्छस्सङ्कलितधनं च सम्भवति ॥ २९४ ॥
अत्रोद्देशकः ।
त्रीण्युत्तरमादिॐ वनिताभिश्श्रोत्पलानि भक्तानि ।
एकस्या भागोऽौ कति वनिताः कति च कुसुमानि ॥ २९५ ॥
वर्गसङ्कलितानयनसूत्रम्—
सैकेष्टकृतिर्डिना सैकेटोनेष्टदलगुणिता ।
कृतिघनचितिसङ्घातास्त्रिकभक्तो वर्गसङ्कलितम् ॥ २९६ ॥
अत्रोद्देशकः ।
अष्टाष्टादशविंशतिषष्टयेकाशीतिषट्कृतीनां च ।
कृतिघनचितिसङ्कलितं वर्णचितिं चाशु मे कथय ॥ २९७ ॥
इष्टावुत्तरपदवर्गसङ्कलितधनानयनसूत्रम् –
द्विगुणैकोनपदोत्तरतिहातिषष्ठांशमुरवचयहतद्युतिः ।
व्येकपदम्न मुरवठतिसहिता पदताडितेषुछतिचितिका ॥ २९८ ॥
104 गणितसारसङ्ग्रहः
पुनरपि इष्टावुत्तरपदवर्णसङ्कलितानयनसूत्रम् -
द्विगुणैकोनपदोत्तरतिहतिरेकोनपदहताङ्गहृता ।
ठ्येकपदादिचयाहतिमुखकृतियुक्ता पदाहता सारम् ॥ २९९ ॥
अत्रोद्देशकः ।
त्रीण्यादिः पञ्च चयो गच्छः पचास्य कथय कतिचितिकाम् ।
पशदिखीणि चयो गच्छः सप्तास्य का च कृतिचितिका ।। ३०० ॥
घनसङ्कलेतानयनसूत्रम--
गच्छार्धवर्णराशी रूपाधिकगच्छवर्गसङ्गणितः ।
घनसङ्कलितं प्रोक्तं गणितेऽस्मिन् गणिततवनैः ॥ ३०१ ॥
अत्रोद्देशकः ।
षण्णामष्टानामपि सप्तानां पञ्चविंशतीनां च ।
षट्पञ्चशन्मिश्रितशतद्वयस्यापि कथय घनपिण्डम् ॥ ३०२ ॥
इष्टावुत्तरगच्छघनसङ्कलितानयनसूत्रम् -
चित्यादिहतिर्मुरवचयशेषम्ना प्रचयनिम्नचितिवर्गे ।
आदौ प्रचयादूने विद्युता युक्ताधिके तु घनचितिका ॥ ३०३ ॥
अत्रोद्देशकः ।
आदिस्त्रयश्चयो द्वौ गच्छः पचास्य घनचितिका ।
पञ्चदिस्सप्तचयो गच्छष्षट् का भवेच्च घनाचितिका ॥ ३०४ ॥
सङ्कलतसकलतानयनसूत्रम--
द्विगुणैकोनपदोत्तरतिहतिरङ्गहता चयार्धयुता ।
आदिचयाहतियुक्ता व्येकपदनदिगुणितेन ।
सैकमुभवेन युता पददलशुणितैव चितिचितिका ॥ ३०५(१/२) ॥
105 मिश्रकव्यवहारः
अत्रोद्देशकः ।
आदिष्षट् पञ्च चयः पदमप्यष्टादशथ सन्दृष्टम् ।
एकावेकोत्तरचितिसङ्कलितं किं पदाष्टदशकस्य ॥ ३०६(१/२) ॥
चतुस्सङ्कलितानयनसूत्रम--
सैकपदार्थपदाहतिरश्वैर्निहता पदोनिता व्याप्ता ।
सैकपदन चितिचितिचितिकृतिघनसंयुतिर्भवति ॥ ३०७(१/२) ॥
अत्रोद्देशकः ।
सतीष्टनवदशानां षोडशपञ्चशदेकषष्टीनम् ।
ब्रूहि चतुःसङ्कलितं सूत्राणि पृथक् पृथक् कृत्वा ॥ ३०८(१/२) ।।
सर्वतसङ्कालतानयनसूत्रम् --
गच्छस्त्रिरूपसहितो गच्छचतुर्भागताडितस्सैकः ।
सपदपदकृतिविनिघ्नो भवति हि सङ्घातसङ्कलितम् ॥ १०९(१/२) ॥
अत्रोद्देशकः ।
सप्तकृतेः षट्षष्टयास्त्रयोदशानां चतुर्दशानां च।
पञ्चगवंशतीनां किं स्यात् सङ्घातसङ्कलितम् ॥ ३१०(१/२) ॥
भिन्नगुणसङ्कलितानयनसूत्रम् --
समदलविषमखरूपं गुणगुणितं वर्गताडितं द्विष्ठम् ।
अंशातं व्येकं फलमाद्यन्घनं गुणोनरूपहृतम् ॥ ३११(१/२) ॥
अत्रोद्देशकः ।
दीनाराधं पञ्चसु नगरेषु चयस्त्रिभागोऽभूत् ।
आदित्रयंशः पादो गुणोत्तरं सप्त भिन्नगुणचितिका ।
10
106 गणितसारसङ्ग्रहः
का भवति कथय शीघ्र यदि तेऽस्ति परिश्रमो गणिते ॥ ३१३ ॥
अधिकहीनगुणसङ्कलितनयनसूत्रम्--
गुणचितिरन्यादिहृता विपदाधिकहीनसङ्कणा भक्ता।
व्येकगुणेनान्या फलरहिता हीनेऽधिके तु फलयुक्ता ॥ ३१४ ॥
अत्रोद्देशकः ।
पञ्च गुणोत्तरमादिदं त्रीण्यधिकं पदं हि चत्वारः ।
आधिकगुणोत्तरचितिका कथय विचिन्त्यािशु गणिततत्त्वज्ञ ॥ ३१५ ॥
आदिस्त्रीणि गुणोत्तरमष्टौ हीनं इयं च दश गच्छः ।
हीनगुणोत्तरचितिका का भवति विचिन्त्य कथय गणकाशु ॥ ३१६ ॥
आधुत्तरगच्छधनमिश्राद्युत्तरगच्छानयनसूत्रम्—-
मिश्रादुद्धत्य पदं रूपोनेच्छाधनेन सैकेन ।
लब्धं प्रचयः शेषः सरूपपदभाजितः प्रभवः ॥ ३१७ ॥
अत्रोद्देशकः ।
आद्युत्तरपदमित्रं पञ्चशडनमिहैव सन्दृष्टम् ।
गणितज्ञाचक्ष्व त्वं प्रभवोत्तरपदधनन्याशु ॥ । ३१८ ॥
सङ्कलितगतिषुवगतिभ्यां समानकालानयनसूत्रम् –-
ध्वगतिरादिविहीनश्चयदलभक्तस्सरूपकः कालः ।
द्विगुणो मार्गस्तद्रतियोगहृतो योगकालस्स्यात् ॥ ३१९ ॥
अत्रोद्देशकः ।
कश्चिन्नरः प्रयाति त्रिभिरादा उत्तरैस्तथाष्टाभिः।
नियतगतिरेकविंशतिरनयोः कः प्राप्तकालः स्यात् ॥ ३२० ॥
मिश्रकव्यवहारः 107
अपरार्धोदाहरणम् ।
षड् योजनानि कश्चित्पुरुषस्त्वपरः प्रयाति च त्रीणि ।
उभयोरभिमुखगत्योरष्टोत्तरशतकयोजनं गम्यम् ।
प्रत्येकं च तयोः स्यात्कालः किं गणक कथय मे शीघ्रम् ॥ ३२१(१/२) ॥
सङ्कलितसमागमकालयोजनानयनसूत्रम् –
उभयोराद्योश्शेषश्चयशेषहृतो द्विसङ्गणः सैकः ।
युगपत्प्रयाणयांस्स्यान्मार्गे तु समागमः कालः ॥ ३२२(१/२) ॥
अत्रोद्देशकः ।
चत्वार्याद्यष्टोत्तरमेको गच्छत्यथो द्वितीयो ना ।
द्वौ प्रचयश्च दशादिः समागमे कस्तयः कालः ॥ ३२३(१/२) ॥
वृद्धुत्तरहीनोत्तरयोस्समागमकालानयनसूत्रम् –
शेषश्चद्योरुभयोश्चययुतदलभक्तरूपयुतः ।
युगपत्प्रयाणकृतयोमोगें संयोगकालः स्यात् ॥ ३२४(१/२) ॥
अत्रोद्देशकः ।
पाद्यष्टोत्तरतः प्रथमा नाथ द्वैतायनरः ।
आदिः पञ्चघ्ननव प्रचयो हीनोऽष्ट योगकालः कः ॥ ३२५(१/२) ॥
शम्रिगतिमन्दगत्योस्समागमकालानयनसूत्रम्—-
मन्दगतिशीघ्रगत्योरेकाशागमनमत्र गम्यं यत् ।
तदात्यन्तरभक्तं लब्धदिनैसैः प्रयाति शीघ्रोऽल्पम् ॥ ३२६(१/२) ॥
अत्रोद्देशकः ।
नवयोजनानि कश्चित्प्रयाति योजनशतं गतं तेन ।
प्रतिदूतो व्रजति पुनस्त्रयोदशमोति कैर्दिवसैः ॥ ३२७(१/२) ॥
10-A
108 गणितसारसङ्ग्रहः
विषमबाणैस्सूणीरबाणपरिधिकरणसूत्रम् –-
परिणाहस्त्रिभिरधिको दलित वर्गीकृतास्त्रिभिर्भक्तः ।
सैक१शरास्तु परिधेरानयने तत्र विपरीतम् ॥ ३२३(१/२) ॥
अत्रोद्देशकः ।
नव परिधिस्तु शराणां सङ्ख्या न ज्ञायते पुनस्तेषाम् ।
युत्तरदशबाणास्तत्परिणाहशरांश्च कथय मे गणक ॥ ३२९(१/२) ॥
श्रेढीबद्धे इष्टकानयनसूत्रम्--
तरवग रूपोनस्त्रिभिर्विभक्तसरेण सङ्गणितः ।
तरसङ्कलिते वेष्टप्रताडिते मिश्रतः सारम् ॥ ३३०(१/२) ॥
अत्रोद्देशकः ।
पञ्चतरेकनाथं व्यवघटित गणितविन्मिने ।
समचतुरश्रश्रेढी कतीष्टकास्स्युर्ममाचक्ष्व || ३३१(१/२) ॥
नन्द्यावर्ताकारं चतुस्तराः षष्टिसमघटिताः ।
सर्वेष्टकाः कति स्युः श्रेढीबढं ममाचक्ष्व ॥ ३३२(१/२) ॥
छन्दश्शास्त्रोक्तषट्प्रत्ययानां सूत्राणि
समदलविषमरवरूपं द्विगुणं वर्गीकृतं च पदमङ्ख्या ।
सङ्ख्या विषमा सैका दलनो गुरुरेव समदलतः ॥ ३३३(१/२) ॥
स्याङघुरेवं तपशः प्रस्कारोऽयं विनिर्दिष्टः ।
नष्टाङ्कार्थं लघुरथ तत्सैकदले गुरुः पुनः पुनः स्थानम् ॥ ३३४(१/२) ॥
रूपादिगुणोत्तरतस्तद्दिष्टे लाङ्कसंयुतिः सैका ।
एकाद्यद्येकोत्तरतः पंदेसूध्धर्यतः क्रमोक्रमशः ॥ ३३५(१/२) ॥
मिश्रकव्यवहारः 109
स्थाप्य प्रतिलोमग्नं प्रतिलोमग्नेन जितं सारम् ।
स्याङघुगुरुक्रियेयं सङ्ख्या द्विगुणैकवर्जिता साध्वा ।। १३६(१/२) ।।
अत्रोद्देदेशकः ।
सङ्ख्यां प्रस्तारविधिं नोद्दिष्टे लगक्रियाध्वानौ
षट्प्रत्ययांश्च शीघ्र यक्षरवृत्तस्य मे कथय ॥ १३७(१/२) ॥
इति मिश्रकव्यवहारे श्रेढीबद्धसङ्कलितं समाप्तम् ॥
इति सारसङ्ग्रहे गणितशास्त्रे महावीराचार्यस्य कृतौ मिश्रकगणितं
नाम पञ्चमव्यवहारः समाप्तः ॥
षष्ठः
क्षेत्रगणितव्यवहारः
सिद्धेभ्यो निष्ठितायैभ्यो वरिष्ठेभ्यः कृतादरः ।
अभिप्रेतार्थसिद्ध्यर्थं नमस्कुर्वे पुनः पुनः ॥ १ ॥
इतः परं क्षेत्रगणितं नाम षष्ठगणितमुदाहरिष्यामः । तद्यथा —-
क्षेत्रं जिनप्रणीतं फलाश्रयाद्वयावहारिकं सूक्ष्ममिति ।
भेदाद् द्विधा विचिन्त्य व्यवहारं स्पष्टमेतदभिधास्ये ॥ २ ॥
त्रिभुजचतुर्भुजवृत्तक्षेत्राणि स्वस्वभेदभिन्नानि ।
गणितार्णवपारगतैराचार्यैस्सम्यगुक्तानि ॥ ३ ॥
त्रिभुजं त्रिधा विभिन्नं चतुर्भुजं पञ्चधाष्टधा वृत्तम् ।
अवशेषक्षेत्राणि ह्येतेषां भेदभिन्नानि ॥ ४ ॥
त्रिभुजं तु समं द्विसमं विषमं चतुरश्रमपि समं भवति ।
द्विद्विसमं द्विसमं स्यात्रिसमं विषमं बुधाः प्राहुः ॥ ५ ॥
समवृत्तमर्धवृत्तं चायतवृत्तं च कम्बुकावृत्तम् ।
निम्नोन्नतं च वृत्तं बहिरन्तश्चक्रवालवृत्तं च ॥ ६ ॥
व्यावहारिकगणितम् ।
त्रिभुजचतुर्भुजक्षेत्रफलानयनसूत्रम्--
त्रिभुजचतुर्भुजबाहुप्रतिबाहुसमासदलहतं गणितम् ।
नेमेर्भुजयुत्यर्धं व्यासगुणं तत्फलार्धमिह बालेन्दोः ॥ ७ ॥
अत्रोद्देशकः ।
त्रिभुजक्षेत्रस्याष्टौ बाहुप्रतिवाहुभूमयो दण्डाः ।
तद्व्यावहारिकफलं गणयित्वाचक्ष्व मे शीघ्रम् ॥ ८ ॥
क्षेत्रगणेतव्यवहारः 111
द्विसमत्रिभुजक्षेत्रस्यायामः सप्तसप्ततिर्दण्डाः ।
विस्तारो द्वाविंशतिरथ हस्ताभ्यां च सम्मिश्राः ॥ ९ ॥
त्रिभुनक्षेत्रस्य भुजस्त्रयोदश प्रतिभुजस्य पञ्चदश ।
भूमिश्चतुर्दशास्य हि दण्डा विषमस्य किं गणितम् ॥ १० ॥
गजदन्तक्षेत्रस्य च पृष्ठेऽष्टाशीतिरत्र सन्दृष्टाः ।
द्वासप्ततिरुदरे तन्मूलेऽपि त्रिंशदिह ' दण्डाः ॥ ११ ॥
क्षेत्रस्य, दण्डषष्टिबहुप्रतिबाहकस्य गणयित्वा ।
समचतुरश्रस्य त्वं कथय सर्वे गणितफलमाशु ॥ १२ ॥
आयतचतुरश्रस्य व्यायामः सैकषष्टिरिह दण्डाः ।
विस्तारो द्वात्रिंशद्यवहारं गणितमाचक्ष्व ॥ १३ ॥
दण्डास्तु सप्तषष्टिर्विसमचतुर्बाहुकस्य चायामः ।
व्यासश्चाष्टत्रिंशत् क्षेत्रस्यास्य त्रयात्रशत ॥ १४ ॥
क्षेत्रस्याष्टोत्तरशतदण्डा बाहुत्रये मुरवे चाष्टौ ।
हस्तैस्त्रिभिर्युतास्तात्रिसमचतुर्बाहुकस्य वद गणक ॥ १५ ॥
विषमक्षेत्रस्याष्टत्रिंशद्दण्डाः क्षितिभुवे द्वात्रिंशत् ।
पञ्चशप्रति बाहुः षष्टिस्वन्यः क्रिमस्य चतुरश्रे ॥ १६ ॥
परिघोदरस्तु दण्डात्रि इत्यष्ठं शतत्रयं दृष्टम् ।
नवपञ्चगुणो व्यासो नेमिक्षेत्रस्य किं गणितम् ॥ १७ ॥
हस्तौ दृौ विष्कम्भः पृष्ठेऽष्टाषष्टिरिह च सन्दृष्टाः ।
उदरे तु द्वात्रिंशद्वालेन्दोः कि फलं कथय ॥ १८ ॥
' * The reading in both B and V is fत्रंशातिः ; but as this is erroneour it it
sorrooted into त्रिंशदिह so as to meets the requirements of the metre also .
• B reads देक for त्प्रति .
112 गणितसारसङ्ग्रहः
वृत्तक्षेत्रफलानयनसूत्रम्--
त्रिगुणीकृतविष्कम्भः परिधिव्यसार्धवर्णराशिरयम् ।
त्रिगुणः फलं समेऽर्धे वृत्तेऽर्धे प्राहुराचार्याः ॥ १९ ॥
अत्रोद्देशकः ।
व्यासोऽष्टादश वृत्तस्य परिधिः कः फलं च किम् ।
व्यासोऽष्टादश वृत्तार्धे गणितं किं वदाशु मे ॥ २० ॥
आयतवृत्तक्षेत्रफलानयनसूत्रम्--
व्यासार्धयुतो द्विगुणित आयतवृत्तस्य परिधिरायामः ।
विष्कम्भचतुर्भागः परिवेषहतो भवेत्सारम् ॥ २१ ॥
अत्रोद्देशकः ।
क्षेत्रस्यायतवृत्तस्य विष्कम्भो द्वादशैव तु ।
आयामस्तत्र षट्त्रिंशत् परिधिः कः फलं च किम् ॥ २२ ॥
शंक्वाकारवृत्तस्य फलानयनसूत्रम्--
वदनाधनो व्यासस्त्रिगुणः परिधिस्तु कम्बुकाधत्ते ।
वलयार्धकृतियंशो मुरवधुवर्गत्रिपादयुतः ॥ २३ ॥
अत्रोद्देशकः ।
व्यासोऽष्टादश हस्ता मुरवविस्तारोऽयमपि च चत्वारः ।
कः परिधिः किं गणितं कथय त्वं कस्बुकावृत्ते ॥ २४ ॥
निम्नोन्नतवृत्तयोः फलानयनसूत्रम्--
परिधेश्च चतुर्भागो विष्कम्भगुणः स विद्धि गणितफलम् ।
चत्वाले कूर्मनिभे क्षत्रे निम्नोन्नते तस्मात् ॥ २५ ॥
क्षेत्रगणितव्यहारः 113
अत्रोद्देशकः ।
चत्वालक्षेत्रस्य व्यासस्तु भसह्यकः परिधिः ।
षट्पञ्चशदृष्टं गणितं तस्यैव किं भवति ॥ २६ ॥
कूर्मनिभस्योन्नतवृत्तस्योदाहरणम्--
विष्कम्भः पञ्चदश दृष्टः परिधिश्च षट्रत्रिंशत् ।
कूर्मनिभे क्षेत्रे किं तस्मिन् व्यवहारजं गणितम् ॥ २७ ॥
अन्तश्चक्रवालवृत्तक्षेत्रस्य बहिश्चक्रवालवृत्तक्षेत्रस्य व व्यवहारफलानयनसूत्रम्--
निर्गमसहितो व्यासस्त्रिगुणो निर्गमगुणो बहिर्गणितम् ।
रहिताधिगमञ्यासादभ्यन्तरचक्रवालवृत्तस्य ॥ २८ ॥
अत्रोद्देशकः ।
२थासोऽष्टादश हताः पुनर्बहिर्निर्गतास्त्रयस्तत्र ।
व्यासोऽष्टादश हस्ताश्चन्तः पुनरधिगतास्त्रयः किं स्यात् ॥ २९ ॥
समडत्तक्षेत्रस्य व्यावहारेकफलं च पराधप्रमाणं च व्यासप्रमाणं च
संयोज्य एतत्संयोगसह्यमेव स्वीकृत्य तत्संयोगप्रमाणराशेः सकाशात्
पृथक् परोिधव्यासफलानां सह्यानयनसूत्रम् --
गणिते द्वादशगुणिते मिश्रप्रक्षेपकं चतुःषष्टिः ।
तस्य च मूलं कृत्वा परिधिः प्रक्षेपकपदोनः ॥ ३० ॥
अत्रोद्देशकः ।
परिधिव्यासफलानां मित्रं षोडशशतं सहस्रयुतम् ।
कः परिधिः कि गणितं व्यासः को वा ममाचक्ष्व ॥ ३१ ॥
114 गणितसारसङ्ग्रहः
यवाकारमर्दलाकारपणवाकारवज्राकाराणां क्षेत्राणां व्यावहारिक
फलानयनसूत्रम्--
यवमुरजपणवशक्रायुधसंस्थानप्रतिष्ठितानां तु ।
मुरवमध्यसमासार्ध त्वायामगुणं फलं भवति ॥ ३२ ॥
अत्रोद्देशकः ।
यवसंस्थानक्षेत्रस्यायामोऽतिरस्य विष्कम्भः ।
मध्यश्चत्वारिंशत्फलं भवेकं ममाचक्ष्व ॥ ३३ ॥
आयामोऽशतिरयं दण्डा मुरवमस्य विंशतिर्मध्ये ।
चत्वारिंशत्क्षेत्रे मृदङ्गसंस्थानके बूहि ॥ ३४ ॥
पणवाकारक्षेत्रस्यायामः सप्तसप्ततिर्दण्डाः ।
मुरवयोर्विस्तारोऽष्टौ मध्ये दण्डास्तु चत्वारः ॥ ३५ ॥
वत्रकृतंस्तथास्य क्षत्रस्य षडग्रनवतरायामः ।
मध्ये सूचिभुवयोस्त्रयोदश त्र्यंशसंयुता दण्डाः ॥ ३६ ॥
उभयनिषेधादिक्षेत्रफलानयनसूत्रम्-
व्यासात्खायामगुणाद्विष्कम्भार्थनदीर्घमुत्सृज्य ।
त्वं वद निषेधमुभयोस्तदर्धपरिहीणमेकस्य ॥ ३७ ॥
अत्रोद्देशकः ।
आयामः षत्रिंशद्विस्तारोऽष्टादशैव दण्डास्तु ।
उभयनिषेधे किं फलमेकनिषेधे च किं गणितम् ॥ ३८ ॥
बहुविधवलकाराणां क्षेत्राणां व्यावहारिकफलानयनसूत्रम्--
रज्ज्वर्धकृतित्र्यंशो बाहुविभक्तो निरेकबाहुगुणः ।
सर्वेषामश्रवतां फलं हि बिम्बान्तरे चतुर्थाशः ॥ ३९ ॥
क्षेत्रगणितव्यवहारः 115
अत्रोद्देशकः ।
षड्बाहुकस्य बहविष्कम्भः पञ्च चान्यस्य ।
व्यासस्त्रयो भुजस्य त्वं षोडशबाहुकस्य वद ॥ ४० ॥
त्रिभुजक्षत्रस्य भुजः पञ्च प्रतिवाहुरपि च सप्त धरा षट् ।
अन्यस्य षडश्रस्य कादिषडन्तविस्तारः ॥ ४१ ॥
मण्डलचतुष्टयस्य हि नवविष्कम्भस्य मध्यफलम् ।
अपश्चतुर्यासा वृत्तत्रितयस्य मध्यफलम् ॥ ४२ ॥
धनुराकारक्षेत्रस्य व्यावहारिकफलानयनसूत्रम् —-
कुत्वेषुगुणसमासं बाणार्धगुणं शरासने गणितम् ।
शरवर्गात्पञ्चगुणाज्ज्यावर्गयुतात्पदं काष्ठम् ॥ ४३ ॥
अत्रोद्देशकः ।
ज्या षड्विंशतिरेषा त्रयोदशेषुश्च कार्मुकं दृष्टम् ।
किं गणितमस्य काष्ठं किं वाचक्ष्वाशु मे गणक ॥ ४४ ॥
वाणगुणप्रमाणानयनसूत्रम्--
गुणचापकृतिविशेषात् पञ्चह्नात्पदमिषुः समुद्दिष्टः ।
शरवर्गपञ्चगुणादूना धनुषः छतिः पदं जीवा ॥ ४५ ॥
अत्रोद्देशकः।
अस्य धनुःक्षेत्रस्य शरोऽत्र न ज्ञायते परस्यापि ।
न ज्ञायते च मौर्वी तह्यमाचक्ष्व गणितज्ञ ॥ ४६ ॥
बाहरन्तश्चतुरश्रकवृत्तस्य व्यावहारिकफलानयनसूत्रम्--
बारे वृत्तस्येदं क्षेत्रस्य फलं त्रिसंगुणं दलितम् ।
अभ्यन्तरे तदर्ध विपरीते तत्र चतुरश्रे ॥ ४७ ॥
116 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
पञ्चदशबाहुकस्य क्षेत्रस्याभ्यन्तरं बहिर्गणितम् ।
चतुरश्रस्य च वृत्तव्यवहारफलं ममाचक्ष्व ॥ ४८ ।।
इति व्यावहारिकगणितं समाप्तम् ।
अथ सूक्ष्मगणितम्
इतः परं क्षेत्रगणिते सूक्ष्मगणितव्यवहारमुदाहरिष्यामः । तद्यथा –
आबाधावलम्वकानयनसूत्रम्--
भुजकृत्यन्तरहृतभूसङ्कूमणं त्रिबाहुकाबाधे ।
तद्भभुजवर्गान्तरपदमवलम्वकमाहुराचार्याः ॥ ४९ ॥
सूक्ष्मगणितानयनसूत्रम्--
भुजयुत्यर्धचतुष्काद्भुजहीनाद्धातितात्पदं सूक्ष्मम् ।
अथवा मुरवतलयुतिदलमवलम्वगुणं न विषमचतुरश्रे ॥ ५० ॥
अत्रोद्देशकः ।
त्रिभुजक्षेत्रस्याष्टौ दण्डा भूर्वाहकौ समस्य त्वम् ।
सूक्ष्मं वद गणितं मे गणितविदवलम्बकाबाधे ॥ ५१ ॥
द्विसमत्रिभुजक्षेत्रे त्रयोदश स्युभुजद्वये दण्डाः ।
दश भूरयाबाधे अथावलम्बं च सूक्ष्मफलम् ॥ ५२ ॥
विषमत्रिभुजस्य भुजा त्रयोदश प्रतिभुजा तु पञ्चदश ।
भूमिश्चतुर्दशास्य हि किं गणितं चावलम्बकाबाधे ॥ ५३ ॥
After this M adds the following :- त्रिभुजक्षेत्रस्य भुजद्वयसंयोगस्थानमारभ्य
अधस्स्थितभूमिसंस्पृष्टरेखाय नाम अवलम्बकः स्यात् ।
117 क्षेत्रगणितव्यवहारः
इतः परं पश्चप्रकाराणां चतुरश्रक्षेत्राणां कर्णानयनसूत्रम्--
क्षितिहतविपरीतभुजौ मुरवगुणभुजमिश्रितौ गुणच्छेदौ ।
छेदगुणौ प्रतिभुजयोः संवर्गयुतेः पदं कर्णौ ॥ ५४ ॥
अत्रोद्देशकः ।
समचतुरश्रस्य त्वं समन्ततः पञ्चबाहुकस्याशु ।
कर्णं च सूक्ष्मफलमपि कथय सखे गणिततत्त्वज्ञ ॥ ५१ ॥
आथतचतुरश्रस्य द्वादश बाहुश्च कोटिरपि पञ्च ।
कर्णः कः सूक्ष्मं किं गणितं चाचक्ष्व मे शीघ्रम् ॥ ५६ ।
द्विसमचतुरश्रभूमिः षत्रिशद्वाहुरेकषाष्टिश्च ।
सोऽन्यश्चतुर्दशास्यं कर्णः कः सूक्ष्मगणितं किम् ॥ ५७ ॥
वर्गस्त्रयोदशानां त्रिसमचतुर्बाहुके पुनर्भूमिः ।
सप्त चतुश्शतयुक्तं कर्णाबाधावलम्बगणितं किम् ॥ ५८ ॥
विषमचतुरश्रबाहू त्रयोदशाभ्यस्तपञ्चदशविंशतिकौ ।
पञ्चघनो वदनमधस्त्रिशतं कान्यत्र कर्णमुखफलानि ॥ ५९ ॥
इतः परं वृत्तक्षेत्राणां सूक्ष्मफलानयनसूत्राणि । तत्र समवृत्तक्षेत्रस्य
सूक्ष्मफलानयनसूत्रम्--
वृत्तक्षेत्रव्यासो दशपदगुणितो भवेत्परिक्षेपः ।
व्यासचतुर्भागगुणः परिधिः फलमर्धमथै तत् ॥ ६० ॥
अत्रोद्देशकः ।
समवृत्तव्धासोऽष्टादश विष्कम्भश्च षष्टिरन्यस्य ।
द्वाविंशतिरपरस्य क्षेत्रस्य हि के च परिधिफले ।। ६१ ॥
द्वादशविष्कम्भस्य क्षेत्रस्य Iह चर्घवृत्तस्य ।
षष्टत्रिंशद्यासस्य कः परिधिः किं फलं भवति ॥ ६२ ॥
118 गणितसारसङ्ग्रहः
आयतवृत्तक्षेत्रस्य सूक्ष्मफलानयनसूत्रम्--
व्यासकृतिष्षड्गुणिता द्विसङ्गुणायामकृतियुता(पदं) परिधिः ।
व्यासचतुर्भागगुणश्चायतवृत्तस्य सूक्ष्मफलम् ॥ ६३ ॥
अत्रोद्देशकः।
आयतवृत्तायामः षट्त्रिंशद्वादशास्य विष्कम्भः ।
कः परिधिः किं गणितं सूक्ष्मं विगणय्य मे कथय ॥ ६४ ॥
शङ्खाकारक्षेत्रस्य सूक्ष्मफलानयनसूत्रम्--
वदनाखूनो व्यासो दशपदगुणितो भवेत्परिक्षेपः ।
मुखदलरहितव्यासार्थवर्गमुवचरणकृतियोगः ॥ ६५ ॥
दशपदगुणितः क्षेत्रे कम्युनिभे सूक्ष्मफलमेतत् ।। ६५(१/२) ॥
अत्रोद्देशकः ।
व्यासोऽष्टादश दण्डा मुरवविस्तारोऽयमपि च चत्वारः ।
कः परिधिः किं गणितं सूक्ष्मं तत्कम्बुकावृत्ते ॥ ६६(१/२) ॥
बहिश्चक्रवालवृत्तक्षेत्रस्य चान्तश्चक्रवालवृत्तक्षेत्रस्य च सूक्ष्मफलानय
नसूत्रम् निर्गमसहितो व्यासो दशपदनिर्गमगुणो बहिर्गणितम् ।
रहितोऽधिगमेनासावभ्यन्तरचक्रवालवृत्तस्य ॥ ६७ ॥
अत्रोद्देशकः ।
व्यासोऽष्टादश दण्डाः पुनर्बहिर्निर्गतास्त्रयो दण्डाः ।
सूक्ष्मगणितं वद त्वं बहिरन्तश्चक्रवालवृत्तस्य ॥ ६१ ॥
व्यासोऽष्टादश दण्डा अन्तःपुनरधिगताश्च चत्वारः ।
सूक्ष्मगणितं वद त्वं चाभ्यन्तरचक्रवालवृत्तस्य ॥ ६९ ॥
क्षेत्रगणितव्यवहारः 119
यवाकारक्षेत्रस्य च धनुराकारक्षेत्रस्य च सूक्ष्मफलानयनसूत्रम्--
इषुपादगुणश्च गुणो दशपदगुणितश्च भवति गणितफलम् ।
यवसस्थानक्षत्र धनुराकारे च विज्ञेयम् ।। ७०(१/२) ॥
अत्रोद्देशकः ।
द्वादशदण्डायामो मुखद्वयं सूचिरापि च विस्तारः ।
चत्वारो मध्येऽपि च यवसंस्थानस्य किं तु फलम् ॥ ७१(१/२) ॥
धनुराकारसंस्थाने ज्या चतुर्विंशतिः पुनः।
चत्वारोऽस्येषुरुद्दिष्टस्सूक्ष्मं किं तु फलं भवेत् ॥ ७२ ॥
धनुराकारक्षत्रस्य धनुःकाष्ठबाणप्रमाणानयनसूत्रम्--
शरवर्गः षङ्गुणितो ज्यावर्गसमन्वितस्तु यस्तस्य ।
मूल धनुर्गुणेषुप्रसाधने तत्र विपरीतम् ॥ ७३(१/२) ॥
विपरीतक्रियायां सूत्रम्--
गुणचापकृतिविशेषात्तर्कहतात्पदमिषुः समुद्दिष्टः ।
शरवर्गात् षड़णितादूनं धनुषः कृतेः पदं जीवा ॥ ७४(१/२) ॥
अत्रोदशकः।
धनुराकारक्षेत्रे ज्या द्वादश षट् शरः काष्ठम् ।
न ज्ञायते सरवं त्वं का जीवा क१२रस्तरस्य ॥ ७५(१/२) ॥
मृदङ्गनिभक्षेत्रस्य च पणवाकारक्षेत्रस्य च वत्राकारक्षेत्रस्य च
सूक्ष्मफलानयनसूत्रम्--
मुखगुणितायामफलं धनुःफलसंयुतं मृदङ्गनिभे ।
तत्पणववननिभयोर्धनुःफलोनं तयोरुभयोः ॥ ७६(१/२) ॥
1 The reading in both B and M is as given above; but षड्गुणितादूनाया धनुष्कृतेः
पदं जीवा gives the required meaning.
120 गणितसारसङ्ग्रहः
अत्रोद्देशकः ।
चतुर्विंशतिरायामो विस्तारोऽष्टं मुवद्वये ।
क्षेत्रे मृदङ्गसंस्थाने मध्ये षोडश किं फलम् ॥ ७७(१/२) ॥
चतुर्विंशतिरायामस्तथाष्टौ मुरवयोर्दयोः ।
चत्वारो मध्यविष्कम्भः किं फलं पणवाकृती ॥ ७८(१/२) ॥
चतुर्विंशतिरायामस्तथाष्टौ मुरवयोर्दयोः ।
मध्ये सूचस्तथाचक्ष्व वज्त्राकारस्य कि फलम् ॥ ७९(१/२) ॥
नेमिक्षेत्रस्य च बालेन्द्वकारक्षेत्रस्य च इभदन्ताकारक्षेत्रस्य च सूक्ष्म
फलानयनसूत्रम्--
पृष्ठोदरसंक्षेपः षड्भक्तो व्यासरूपसङ्गणितः ।
दशमूलगुणो नेमेबलेन्दिभदन्तयोश्च तस्यार्धम् ॥ ८०(१/२) ॥
अत्रोद्देशकः ।
पृष्ठं चतुर्दशोदरमष्टौ नेम्याकृतौ भूमौ ।
मध्ये चत्वारि च तद्वालेन्दोः किमिभदन्तस्य ॥ ८१(१/२) ॥
चतुर्मण्डलमध्यस्थितक्षेत्रस्य सूक्ष्मफलानयनसूत्रम् –-
विष्कम्भवर्गरावृत्तस्यैकस्य सूक्ष्मफलम् ।
त्यक्त्वा समवृत्तानामन्तरजफलं चतुर्णा स्यात् ॥ ८२(१/२) ॥
अत्रोद्देशकः ।
गोलकचतुष्टयस्य हि परस्परस्पर्शकस्य मध्यस्य ।
सूक्ष्मं गणितं किं स्याच्चतुष्कविष्कम्भयुक्तस्य ॥ ८३(१/२) ॥
क्षेत्रगणितव्यवहारः 121
वृसक्षेत्रत्रयस्थान्योऽन्यस्पर्शनाज़तस्यान्तरास्थनक्षेत्रस्य सूक्ष्मफलानयनसूत्रम्--
विष्कम्भमानसमकत्रिभुजक्षेत्रस्य सूक्ष्मफलम् ।
वृत्तफलार्धविहीनं फलमन्सरजं त्रयाणां स्यात् ॥ ८४(१/२) ॥
अत्रोद्देशकः ।
विष्कम्भचतुष्काणां वृत्तक्षेत्रत्रयाणां च ।
अन्योऽन्यस्ष्टष्टानामन्तरजक्षेत्रसूक्ष्मगणितं किम् ॥ ८५(१/२) ॥
षडश्रक्षेत्रस्य कर्णावलम्बकसूक्ष्मफलानयनसूत्रम्--
भुजभुजकृतिकृतिवर्णा द्वित्रित्रिगुणा यथाक्रमेणेव ।
भृत्यवलम्बकछतिधनकृतयश्च षडस्रके क्षेत्रे ॥ ८६(१/२) ॥
अत्रोद्देशकः ।
भुजषट्कक्षेत्रे द्वौ द्वौ दण्डौ प्रतिभुजं स्याताम् ।
अस्मिन् भृत्यवलम्बकसूक्ष्मफलानां च वर्गाः के ॥ ८७(१/२) ॥
वर्गस्वरूपकराणिराशीनां युतिसङ्ख्यायानयनस्य च तेषां वर्गस्वरूप-
करणिराशीनां यथाक्रमेण परस्परवियुतितः शेषसङ्ख्यानयनस्य च सूत्रम्--
केनाप्यपवर्तितफलपदयोगवियोगकृतिहताच्छेदात् ।
मूलं पदयुतिवियुती राशीनां विद्धि करणिगणितमिदम् ॥ ८८(१/२) ॥
अत्रोदेशकः।
षोडशषत्रिंशच्छतकरणीनां वर्गमूलपिण्डं मे ।
अथ चैतत्पदोषं कथय सरवे गणिततयज्ञ ॥ ८९(१/२) ॥
इति सूक्ष्मगणितं समाप्तम् ।
122 गणितसारसङ्ग्रहः
जन्यव्यवहारः
इतः परं क्षेत्रगणिते जन्यव्यवहारमुदाहरिष्यामः ।
इष्टसङ्ख्याबीजाभ्यामायतचतुरश्रक्षेत्रानयनसूत्रम्--
वर्गविशेषः कोटिसंबगों द्विगुणितो भवेद्वाहुः ।
वगेसमासः कर्णश्चायतचतुरश्रजन्यस्य ॥ ९०(१/२) ॥
अत्रोद्देशकः ।
एकविके तु बीजे क्षेत्रे जन्ये तु संस्थाप्य।
कथय विगणय्य शषिं कोटिभुजाकर्णमानानि ॥ ९१(१/२) ॥
बीजे हे त्रीणि सवे क्षेत्रे जन्ये तु संस्थाप्य ।
कथय विगणय्य शीषिं कोटिभुजाकर्णमानानि ॥ ९२(१/२) ॥
पुनरपि बीजसंज्ञाभ्यामायतचतुरश्रक्षेत्रकल्पनायाः सूत्रम्--
बीजयुतिवियुतिघातः कोटिस्तद्वर्गयोश्च सङ्क्रमणे ।
बाहुश्रुती भवेतां जन्यविधौ करणमेतदपि ॥ ९३(१/२) ॥
अत्रोद्देशकः
त्रिकपञ्चकबीजाभ्यां जन्यक्षेत्रं सरवे समुत्थाप्य ।
कोटिभुजाश्रुतिसङ्ख्याः कथय विचिन्त्याशु गणिततवज्ञ ॥ ९४(१/२) ॥
इष्टजन्यक्षेत्राद्वीजमंज्ञसङ्ख्ययोरानयनसूत्रम् –-
कोठिच्छेदावाप्त्योस्सङ्क्रमणे बाहुदलफलच्छेद
बीजे श्रुतीष्टकृत्योयोगवियोगार्धमूले ते ॥ ९५(१/२) ॥
अत्रोद्देशकः।
कस्यापि क्षेत्रस्य च षोडश कोटिश्च बीजे के ।
त्रिंशदथवान्यथबाहुबजे के ते श्रुतिश्चतुस्त्रिंशत् ॥ ९६(१/२) ॥
क्षेत्रगाणतव्यवहारः 123
कोटिसङ्ख्या ज्ञात्वा भुजाकर्णसङ्ख्यानयनस्य च भुजसख्य
ज्ञात्वा कोठिकर्णसङ्ख्यानयनस्य च कर्णसङ्ख्यां ज्ञात्वा कोटिभुजा
सङ्ख्यानयनस्य च सूत्रम् –
कोटिकृतेश्छेदाप्योस्सङ्गणे श्रुतिभुजौ भुजकृतेर्वा ।
अथवा भृतीष्टकृत्योरन्तरपदमिष्टमपि च कोटिभुजे ॥ ९७(१/२) ॥
अत्रोद्देशक ।
कस्यापि कोटिरेकादश बाहुष्षष्टिरन्यस्य ।
श्रुतिरेकषष्टिरन्यस्यानुक्तान्यत्र मे कथय ॥ ९८(१/२) ॥
द्विसंमचतुरश्रक्षेत्रस्यानयनप्रकारस्य सूत्रम्--
लन्यक्षेत्रभुजार्धहारफलजप्राग्जन्यकोट्योर्युति-
भूरास्यं वियुतिभुजा श्रुतिरथारुपाल्पा हि कोटिर्भवेत् ।
आबाधा महती श्रुतिः श्रुतिरभूज्येष्ठं फलं स्यात्फलं
बाहुस्स्यादवलम्बको द्विसमकक्षेत्रे चतुर्बाहुके ॥ ९९(१/२) ॥
अत्रोद्देशकः ।
चतुरश्रक्षेत्रस्य द्विसमस्य च पञ्च षटूबीजस्य ।
मुरवभूभुजावलम्बककर्णाबाधाधनानि वद ॥ १००(१/२) ॥
त्रिसमचतुरश्रक्षेत्रस्य मुखभूभुजावलम्बककर्णाबाधाधनानयनसूत्रम्--
भुजपदहतबीजान्तरहृतजन्यधनाप्तभागहाराभ्याम् ।
तद्वजकोटिभ्य च द्विसम इव त्रिसमचतुरश्रे ॥ १०१(१/२) ॥
अत्रोद्देशकः ।
चतुरश्रक्षेत्रस्य त्रिसमस्यास्य द्विकत्रिकखबीजस्य ।
मुरवधूभुजावलम्बककर्णाबाधाधनानि वद ।। १०२(१/२) ॥
11-A
124 गणितसारसङ्ग्रहः
विषमचतुरश्रक्षेत्रस्य मुरवभूभुजावलम्वककणबाधाधनानयनसूत्रम्--
ज्येष्ठारुपान्योन्यहीनश्रुतिहतभुजकोठी मुजे भूमुरवे ते
कोबोरन्योन्यदोभ्यं हतयुतिरथ दोघृतयुक्कोटिघातः ।
कर्णावरुपश्रुतिस्रावनधिकभुजकोव्याहतौ लवकौ ता
वाबाधे कोटिदोर्माववनिविवरके कर्णघातार्धमर्थः ॥ १०३(१/२) ॥
अत्रोद्देशकः ।
एकद्विकविकत्रिकजन्ये चोत्थाप्य विषमचतुरश्रे ।
मुरव भूभुजावलम्बककर्णाबाधाधनानि वद ॥ १०४(१/२) ॥
पुनरपि विषमचतुरश्रानयनसूत्रम्
द्वश्रुतिकृतिगुणितो ज्येष्ठभुजः कोठिरापि थरा वदनम् ।
कर्णाभ्यां सङ्गणितावुभयभुजावरुपभुजकोठी ॥ १०५(१/२) ॥
ज्येष्ठभुजकोटवियुतिर्दधातुपभुजकोठिताडिता युक्ता ।
देखभुजकोठियुतिगुणपृथुकोव्वरुपश्रुतिस्रको कर्णा ॥ १०६(१/२) ) ॥
अरुपश्रुतिहतकर्णाल्पकोटिभुजसंहती पृथग्लवौ ।
तदुजयुतिवियुतिगुणात्पदमावाधे फलं श्रुतिगुणार्धम् ॥ १०७(१/२) ॥
एकस्माज्जन्यायतचतुरश्राष्ट्रसमात्रभुजानयनसूत्रम् –
कथं भुजद्वयं स्याद्वाहुर्विगुणीकृतो भवेदूमिः ।
कोटिरवलबकोऽयं द्विसमत्रिभुजे धनं गणितम् ॥ १०८(१/२) ॥
अत्रोद्देशकः ।
त्रिकपञ्चकबीजोत्थद्विसमत्रिभुजस्य गणक वाहू द्वौ ।
भूमिमवलम्बकं च प्रगणय्याचक्ष्व मे शीघ्रम् ॥ १०९(१/२) ॥
क्षेत्रगणितव्यवहारः 125
विषमत्रिभुजक्षेत्रस्य कल्पनाप्रकारस्य सूत्रम्--
जन्यभुजाधे छित्वा केनापिच्छेदलब्धजं चाभ्याम् ।
कोठियुतिर्मुः कण भुजौ भुजा लम्बका विषमे ॥ ११०(१/२) ॥
अत्रोद्देशकः ।
हे द्वित्रिबीजकस्य क्षेत्रभुजाछैन चान्यमुत्थाप्य ।
तस्माद्विषमत्रिभुजे भुजभूम्यवलम्बकं ब्रूहि ॥ १११(१/२) ॥
इति जन्यव्यवहारः समाप्तः ॥
पैशाचिकव्यवहारः
इतः परं पैशाचिकव्यवहारमुदाहरिष्यामः ।
समचतुरश्रक्षेत्रे वा आयतचतुरश्रक्षेत्रे वा क्षेत्रफले रज्जुसङ्गचया समे
सति, क्षेत्रफले बाहुसह्यया समे सति, क्षेत्रफले कर्णसद्यया समे
सति, क्षेत्रफले रज्वर्धसङ्गचया समे सति, क्षेत्रफले बाहोस्तृतीयांश
सद्यया समे सति, क्षेत्रफले कर्णसङ्घचायाश्चतुर्थाशसद्यया समे
सति, द्विगुणितकर्णस्य त्रिगुणितबाहोश्च चतुर्गुणितकोठेश्च रजोस्संयो-
गसह्यां द्विगुणीकृत्य तद्विगुणितसङ्घचया क्षेत्रफले समाने सति, इत्येव-
मादीनां क्षेत्राणां कोटिभुजाकर्णक्षेत्रफलरज्जुषु इष्टराशिद्वयसाम्यस्य
चेष्टराशिद्वयस्यान्योन्यमिष्टगुणकारगुणितफलवत्क्षेत्रस्य भुजाकोटिसङ्
ख्यानयनस्य सूत्रम्--
स्वगुणेष्ठेन विभक्तारखेष्टानां गणक गणितगुणितेन ।
गुणिता मुजा भुजाः स्युः समचतुरश्रादिजन्यानाम् ॥ ११२(१/२) ॥
अत्रोद्देशकः ।
रज्जुर्गणितेन समा समचतुरश्रस्य का तु भुजसत्रया ।
अपरस्य बाहुसदृशं गणितं तस्यापि मे कथय ॥ ११३(१/२) ॥
126 गणितसारसङ्ग्रहः
कर्णो गणितेन समः समचतुरश्रस्य को भवेद्वाहुः ।
रज्जुर्द्विगुणोऽन्यस्य क्षेत्रस्य धनाच्च मे कथय ।। ११४(१/२) ॥
आयतचतुरश्रस्य क्षेत्रस्य च रज्जुतुल्यमिह गणितम् ।
गणितं कर्णेन समं क्षेत्रस्यान्यस्य को बाहुः ॥ ११५(१/२) ॥
कस्यापि क्षेत्रस्य त्रिगुणो बाहुर्धनाच्च को बाहुः ।
कर्णश्चतुर्गुणोऽन्यः समचतुरश्रस्य गणितफलात् ॥ ११६(१/२) ॥
आयतचतुरश्रस्य श्रवणं द्विगुणं त्रिसङ्गणो बाहुः ।
कोटिश्चतुर्गणा तै रज्जुयुतैfइगुणितं गणितम् ॥ ११७(१/२) ॥
आयतचतुरश्रस्य क्षेत्रस्य च रजुरत्र रूपसमः ।
कोटिः को बाहुर्वा शत्रं विगणय्य मे कथय ।। ११८(१/२) ॥
कणों द्विगुणो बाहुस्त्रिगुणः कोटिश्चतुर्गुणा मिश्रः ।
रज्ज्वा सह तत्क्षेत्रस्यायतचतुरश्रकस्य रूपसमः ॥ ११९(१/२) ॥
पुनरपि जन्यायतचतुरश्रक्षेत्रस्य बीजसङ्ख्यानयने करणसूत्रम्--
कान्नकणदलतत्कणोन्तरमुफययाध पदे ।
आयतचतुरश्रस्य क्षेत्रस्येयं क्रिया जन्ये ॥ १२०(१/२) ॥
अत्रोद्देशकः ।
आयतचतुरश्रस्य च काठः पचाशदधिकपच भुजा ।
साष्टाचत्वारिंशात्रिसप्ततिः श्रुतिरथात्र के बीजे ॥ १२१(१/२) ॥
इष्टकल्पितसङ्ख्याप्रमाणवत्कर्णसहितक्षेत्रानयनसूत्रम्--
यद्यत्क्षेत्रं जातं बीजैस्संस्थाप्य तस्य कर्णेन ।
इष्टं कथं विभजेछाभगुणाः कोटिदो:कर्णाः ॥ १२२(१/२) ॥
अत्रोद्देशकः ।
एकद्विकाद्विकत्रिकचतुष्कसप्तैकसाष्टकानां च ।
गणक चतुण शत्रिं बीजैरुत्थाप्य कोटिभुजाः ॥ १२३(१/२) ॥
127 क्षेत्रगणितव्यवहारः
आयतचतुरश्राणां क्षेत्राणां विषमबाहुकानां च ।
कर्णोऽत्र पञ्चषष्टिः क्षेत्राण्याचक्ष्व कानि स्युः ॥ १२४(१/२) ॥
इष्टजन्यायतचतुरश्रक्षेत्रस्य रज्जुसङ्ख्यां च कर्णसङ्ख्यां च ज्ञात्वा
तज्जन्यायतचतुरश्रक्षेत्रस्य भुजकाठसङ्ख्यानयनसूत्रम्--
कर्णकृतौ द्विगुणायां रज्वर्थछतिं विशोध्य तन्मूलम् ।
रवर्षे सकमणीकृते भुजा कोटिरपि भवति ॥ १२५(१/२) ॥
अत्रोद्देशकः ।
परािधिः स चतुस्त्रिंशत् कर्णश्चात्र त्रयोदशो दृष्टः ।
जन्यक्षेत्रस्यास्य प्रगणय्याचक्ष्व कोटिभुजौ ॥ १२६(१/२) ॥
क्षेत्रफलं कर्णसङ्ख्यां च ज्ञात्वा भुजकोठिसङ्ख्यानयनसूत्रम् –-
कर्णकृतौ द्विगुणीकृतगणितं हीनाधिकं कृत्वा ।
मूलं कोटिभुजौ हि ज्येष्ठे द्वेन सङ्कमणे ॥ १२७(१/२) ॥
अत्रोद्देशकः ।
आयतचतुरश्रस्य हि गणितं षष्टित्रयोदशास्यापि ।
कर्णस्तु कोटिभुजयोः परिमाणे श्रोतुमिच्छामि ॥ १२८(१/२) ॥
क्षेत्रफलसख्यां रज्जुसङ्ख्यां च ज्ञात्वा आयतचतुरश्रस्य भुज-
कोठिसङ्ख्यानयनसूत्रम्--
रज्वर्धवर्णराशेर्गणितं चतुराहतं विशोध्याथ ।
मूलन हि रज्व” सङ्कमणे सति भुजाकोठी ॥ १२९(१/२) ॥
अत्रोद्देशकः ।
सप्ततिशतं तु रज्जुः पञ्चशतोत्तरसहस्रमिष्टधनम् ।
जन्यायतचतुरश्रे कोटिभुज में समाचक्ष्व ॥ १३०(१/२)॥
128 गणितसारसङ्ग्रहः
आयतचतुरश्रक्षेत्रद्वये रज्जुसङ्ख्यायां सदृक्षाय सत्यां द्वितीयक्षेत्र
फलात् प्रथमक्षेत्रफले द्विगुणिते सति, अथवा क्षेत्रद्वयेऽपि क्षेत्रफल
सदृशे सति प्रथमक्षेत्रस्प रज्जुसङ्ख्याया अपि द्वितयिक्षेत्ररज्जुसङ्ख्या
यां द्विगुणायां सत्यम्, अथवा क्षेत्रद्वये प्रथमक्षेत्ररज्जुसङ्ख्याया अपि
द्वितीयक्षेत्रस्य रज्जुसङ्ख्यायां द्विगुणायां सत्यां द्वितीयक्षेत्रफलादपि ग्रंथ
मक्षेत्रफले द्विगुणे सति, तत्तत्क्षेत्रद्वयस्यानयनसूत्रम्--
स्वाल्पहृतरज्जुधनहतकृतिरिष्टघ्नैव कोटिस्स्यात् ।
व्येका दोस्तुल्यफलेऽन्यत्राधिकगणितगुणितेष्टम् ॥ १३१(१/२) ॥
व्येकं तदूनकोटिः त्रिगुणा दोः स्यादथान्यस्य ।
रवर्धवर्णराशेरिति पूर्वोक्तेन सूत्रेण ।
तद्भणितरनुमितितः समानयेत्तद्भजाकोटी ॥ १३३ ॥
अत्रोद्देशकः ।
असमव्यासायामक्षत्र हे द्वावथेष्टगुणकारः ।
प्रथमं गणितं द्विगुणं रज्जू तुल्ये किमत्र कोटिभुजे ॥ १३४ ॥
आयतचतुरश्रे द्वे क्षेत्रे द्वयमेव गुणकारः ।
गणितं सदृशं रज्जुर्द्विगुणा प्रथमात् द्वितीयस्य ॥ १३५ ॥
आयतचतुरश्रे हे क्षेत्रे प्रथमस्य धनामिह द्विगुणम् ।
द्विगुणा द्वितीयरजुस्तयोभुजां कोठिमपि कथय ॥ १३६ ॥
द्विसमत्रिभुजक्षेत्रयोः परस्पररजधनसमानसङ्ख्ययोरिष्टगुणकगुणि-
तरजुषनवतोर्वा द्विसमत्रिभुजक्षेत्रद्वयानयनसूत्रम्-–
रज्जुकृतिन्नान्योन्यधनारुपातं षट्त्रिमरुपमेकोनम् ।
तच्छेषं द्विगुणारुपं बीजे तज्जन्ययोर्युजादयः प्राग्वत् ॥ १३७ ॥
अत्रोद्देशकः ।
द्विसमत्रिभुजक्षेत्रद्वयं तयोः क्षेत्रयोस्समं गणितम् ।
रजू समे तयोस्स्यात् को कां बाहुः का भवेद्भमिः ॥ १३८ ॥
क्षेत्रगणितव्यवहारः 129
द्विसमत्रिभुजक्षेत्रे प्रथमस्य धनं द्विसङ्गुणितम् ।
रज्जुः समा द्वयोरपि को बाहुः का भवेद्भूमिः ॥ १३९ ॥
हिसमात्रिभुजक्षेत्र हैं रज्जुर्विगुणिता द्वितीयस्य ।
गणिते द्वयोस्समाने कां बाहुः का भवेद्भूमिः ॥ १४० ॥
द्विसमत्रिभुजक्षेत्रे प्रथमस्य धनं विसङ्गणितम् ।
द्विगुणा द्वितीयरज्जुः को बाहुः का भवेद्भूमिः ॥ १४१ ॥
एकद्वयादिगणनातीतसङ्ख्यासु इष्टसङ्ख्यामिष्टवस्तुनो भाग
सङ्ख्यां परिकल्प्य तदिष्टवस्तुओगसङ्ख्यायाः सकाशात् समचतुरश्र
क्षेत्रानयनस्य च समवृत्तक्षेत्रानयनस्य च समात्रिभुजक्षेत्रानयनस्य चायत-
चतुरश्रक्षेत्रनयनस्य च सूत्रम्--
स्वसमीकृतावधूतहृतधनं चतुर्न हि वृत्तसमचतुरश्रव्यासः ।
षड्गुणितं त्रिभुजायतचतुरश्रभुजार्धमपि कोटिः १४२ ॥
अत्रोद्देशकः ।
स्वान्तःपुरे नरेन्द्रः प्रसादतलं निजानामध्ये ।
दिव्यं स रनकम्बलमपीपतत्तच समवृत्तम् ॥ १४३ ॥
ताभिर्देवीभिर्युतमेभिर्मुजयोश्च मुष्टिभिर्लब्धम् ।
पञ्चदशैकस्याः स्युः कति वनिताः कोऽत्र विष्कम्भः॥ १४४ ॥
समचतुरश्रभुजाः कं समत्रिबाहौ भुजाश्चात्र ।
आयतचतुरश्रस्य हि तत्कोटिभुजो सरवे कथय ॥ १४५ ॥
क्षेत्रफलसख्यां ज्ञात्वा समचतुरश्रक्षेत्रानयनस्य चायतचतुरश्र
क्षेत्रानयनस्य च सूत्रम्--
सूक्ष्मगणितस्य मूलं समचतुरश्रस्य बाहुरिष्टहृतम् ।
धनमिष्टफले स्यातामायतचतुरश्रकोटिभुजों ॥ १४६ ॥
130 गणितसारसङ्ग्रहः
अत्रोद्देशकः।
कस्य हि समचतुरश्रक्षेत्रस्य फलं चतुष्षष्टिः ।
फलमायतस्य सूक्ष्मं षष्टिः के वात्र कोटिभुजे ॥ १४७ ॥
इष्टांडीसमचतुरश्रक्षेत्रस्य सूक्ष्मफलसङ्ख्यां ज्ञात्वा, इष्टसङ्ख्य
गुणकं परिकल्प्य, इष्टसङ्ख्याङ्कवीजाभ्यां जन्यायतचतुरश्रक्षेत्रं ‘परि
कल्प्य, तदिष्टद्विसमचतुरश्रक्षेत्रफलवादिष्टद्विसमचतुरश्रानयनसूत्रम्--
तदनगुणितेष्टऋतिर्जन्यधनोना भुजाहृता मुरवं कोटिः ।
द्विगुणा समुरवा भूदलस्वः कथं भुजे तदिष्टहृतः ॥ १४८ ॥
अत्रोद्देशकः।
सूक्ष्मधनं सतेष्टं त्रिकं हि बीजे डिके त्रिके दृष्टे ।
द्विसमचतुरश्रबाहू मुरवभूम्यवलम्बकान् ब्रूहि ॥ १४९ ॥
इष्टसूक्ष्मगणितफलवत्रिसमचतुरश्रक्षेत्रानयनसूत्रम्--
इष्टघनभक्तधनांतरष्टयुताधं भुजा द्विगुणितेष्टम् ।
विभुजं मुरवामिष्टातं गणितं ह्यवलम्बकं त्रिसमजन्ये ॥ १५० ॥
अत्रोद्देशकः ।
कस्यापि क्षत्रस्य त्रिसमचतुर्बाहुकस्य सूक्ष्मधनम् ।
षण्णवतिरिष्टमष्ट बाहुमुरवावलम्बकानि वद ॥ १५१ ॥
सूक्ष्मफलसङ्ख्यां ज्ञात्वा चतुर्भिरिष्टच्छेदैश्च विषमचतुरश्रक्षेत्रस्य
मुरवभूमुजाप्रमाणसङ्ख्यानयनसूत्रम्--
धनकृतिरिष्टच्छेदैश्चतुर्भिरातैव लब्धानाम् ।
युतिदलचतुष्टयं तैना विषमाख्यचतुरश्रभुजसङ्ख्या ॥ १५२ ॥
अत्रोद्देशकः ।
नवतिहिं सूक्ष्मगणितं छेदः पच्चैव नवगुणः ।
दशधृतिविंशातिषट्कृतिहतः क्रमाद्विषमचतुरश्रे ॥
मुरवभूमिथुनासत्या विगणय्य ममाशु सङ्कथय ॥ १५३(१/२) ॥
क्षेत्रणितव्यवहारः 131
सूक्ष्मगणितफलं ज्ञात्वा तत्सूक्ष्मगणितफलवत्समत्रिबाहुक्षेत्रस्य बाहुसङ्ख्यानयनसूत्रम्--
गणितं तु चतुर्गुणितंवर्गीकृत्वा भजेत् त्रिभिर्लब्धम् ।
त्रिभुजस्य क्षेत्रस्य च समस्य बाहोः कृतेर्वर्गम् ॥ १५४(१/२) ॥
अत्रोद्देशकः ।
कस्यापि समश्यश्रक्षेत्रस्य च गणितमुद्दिष्टम् ।
रूपाणि त्रीण्येव बृहि प्रगणय्य मे बाहुम् ॥ १५५(१/२) ॥
सूक्ष्मगणितंफलसङ्ख्यां ज्ञात्वा तत्सूक्ष्मगणितफलवह्निसमत्रिबाहु
क्षत्रस्य प्रजभूम्यवलम्बकसङ्ख्यानयनसूत्रम्--
इच्छाप्तधनेच्छाकृतियुतिमूलं दोः क्षितिर्टिगुणितेच्छा ।
इच्छाप्तधनं लवः क्षेत्रे द्विसमत्रिबाहुजन्ये स्यात् ।। १५६(१/२) ॥
अत्रोद्देशकः ।
कस्यापि क्षेत्रस्य द्विसमत्रिभुजम्य सूक्ष्मगणितमिनाः ।
त्रीणीच्छा कथय सरवे भुजभूम्यवलम्बकानाशु । १५७(१/२) ॥
सूक्ष्मगणितफलसख्यां ज्ञात्वा तत्सूक्ष्मगणितफलवद्विषमत्रिभुजान
यनस्य सूत्रम् –
अष्टगुणितेष्टकृतियुतधनपदघनमिष्टपदहदिष्टार्धम् ।
भूः स्यादूनं द्विपदाहृतेष्टवर्गे मुजे च सङ्कमणम् ।। १५८(१/२) ॥
अत्रोद्देशकः ।
कस्यापि विषमबाहोच्यश्रक्षेत्रस्य सूक्ष्मगणितामिदम् ।
हे रूपे निर्दिष्टे त्रीणीष्टं भूमिबाहवः के स्युः ॥ १५९(१/२) ॥
पुनरपि सूक्ष्मगणितफलसङ्ख्यां ज्ञात्वा तफलवद्विषमत्रिभुजानयनसूत्रम्-
'वर्गीकृत्वा ought to be वर्गीकृत्य ; but this form will not suit the require
ments of the metre.
32 गणितसारसङ्ग्रहः
वाष्टहतात्सेष्टकृतेः कृतिमूलं चेष्टमितरदितरहृतम् ।
ज्येष्ठं स्वरुपाञ्चनं स्वरुपधं तत्पदेन चेष्टेन ॥ १६०(१/२) ॥
क्रमशो हत्वा च तयोः सङ्कमणे भूभुजा भवतः ।
इष्टार्थमितरदः स्याद्विषमलैकोणके क्षेत्रे ॥ १६१(१/२) ॥ .
अत्रोद्देशकः ।
वे रूपे सूक्ष्मफलं विषमात्रभुजस्य रूपाण ।
त्रीणीषं दोषं कथय सरवे गणिततत्वज्ञ ॥ १६२(१/२) ॥
सूक्ष्मगणितफलं ज्ञात्वा तत्सूक्ष्मगणितफलवत्समवृत्तक्षेत्रानयनसूत्रम्--
गणितं चतुरभ्यस्तं दशपदभक्तं पदे भवेद्यासः ।
सूक्ष्मं समवृत्तस्य क्षेत्रस्य च पूर्ववत्फलं परिधिः ॥ १६३(१/२) ॥
अत्रोद्देशकः ।
समवृत्तक्षेत्रस्य च सूक्ष्मफलं पच निर्दिष्टम् ।
विष्कम्भः को वास्य प्रगणय्य ममाशु तं कथय ॥ १६४(१/२) ।।
व्यावहारिकगणितफलं च सूक्ष्मफलं च ज्ञात्वा तद्यावहारिकफलव
तत्सूक्ष्मगणितफलवह्निसमचतुरश्रक्षेत्रानयनस्य त्रिसमचतुरश्रक्षेत्राननस्य
च सूत्रम् --
धनवर्गान्तरपदयुतिवियुतीष्टं भूमुरवे भुजे स्थूलम् ।
डिसमे सपदस्थूलात्पदयुतिवियुतीष्टपदहतं त्रिसमे ॥१६५(१/२)॥
अत्रोद्देशकः ।
गणितं सूक्ष्मं पञ्च त्रयोदश व्यावहारिकं गणितम् ।
द्विसमचतुरश्रमूमुवदोषः के षोडशेच्छा च ॥ १६६(१/२) ॥
क्षेत्रगणितव्यवहारः 133
त्रिसमचतुरश्रस्योदाहरणम् ।
गणितं सूक्ष्मं पञ्च त्रयोदश व्यावहारिकं गणितम् ।
त्रिसमचतुरश्रबाहून् सञ्चित्य सखे ममाचक्ष्व ॥ १६७(१/२) ॥
व्यावहारिकस्थूलफलं सूक्ष्मफलं च ज्ञात्वा तद्यावहारिकस्थूलफलवत्
सूक्ष्मगणितफलवत्समत्रिभुजानयनस्य च समवृत्तक्षेत्रव्यासानयनस्य च सूत्रम्--
धनवगन्तरमूल यत्तन्मूलादिसङ्गणितम् ।
बहुस्त्रिसमात्रिभुजे समस्य वृत्तस्य विष्कम्भः ॥ । १६८(१/२) ॥
अत्रोद्देशकः ।
स्थूलं धनमष्टादश सूक्ष्मं त्रिघन नवाहतः करणेः ।
विगणय्य सरवे कथय त्रिसमत्रिभुजप्रमाणं मे ॥ १६९(१/२) ॥
पञ्चकृतेर्धगों दशगुणितः करणिर्भवेदिदं सूक्ष्मम् ।
स्थूलमपि पञ्चसप्ततिरेतको वृत्तविष्कम्भः ॥ १७०(१/२) ॥
व्यावहारिकस्थूलफलं च सूक्ष्मगणितफलं च ज्ञात्वा तद्यावहारिक
फलवत्तत्सूक्ष्मफलवह्निसमत्रिभुजक्षेत्रस्य भूभुजाप्रमाणसङ्ख्ययोरानयनस्य
सूत्रम् —
फलवगोन्तरमूल द्विगुणं भूव्यवहारिक बाहुः ।
भूम्यधेमूलभक्ते द्विसमत्रिभुजस्य करणमिदम् ॥ १७१(१/२) ॥
अत्रोद्देशकः ।
सूक्ष्मधनं षष्टिरिह स्थूलधनं पञ्चषष्टिरुद्दिष्टम् ।
गणयित्वा ब्रूहि सरवे द्विसमत्रिभुजस्य भुजसङ्ख्याम् ॥ १७२(१/२) ॥
134 गणितसारसङ्ग्रहः
इष्टसङ्ख्यावद्विसमचतुरश्रक्षेत्रं ज्ञात्वा तद्विसमचतुरश्रक्षेत्रस्य
सूक्ष्मगणितफलसमानसूक्ष्मफलवदन्यद्विसमचतुरश्रक्षेत्रस्य
भूभुजमुखससङ्ख्यानयनसूत्रम्--
लम्बछताविष्टेनासमसङ्कमणीछते भुजा ज्येष्ठा ।
द्वस्वयुतिवियुति मुरवधूयुतिदलितं तलमुरवे हिसमचतुरश्रे ॥ १७३(१/२) ॥
अत्रोद्देशकः।
भूरिन्द्रा दोर्विश्वे वनं गतयोऽवलम्बको रवयः।
इष्टं दिक् सूक्ष्मं तरफलवह्निसमचतुरश्रमन्यत् किम् ।। १७४(१/२)॥
द्विसमचतुरश्रक्षेत्रव्यावहारिकस्थूलफलसङ्ख्यां ज्ञात्वा तेद्यावहार
कस्थूलफले इष्टसङ्ख्याविभागे कृते सति तद्दीिसमचतुरश्रक्षेत्रमध्ये तत्त-
द्भागस्य भूमिसळख्यानयनेऽपि तत्तत्स्थानावलम्बकसङ्ख्यानयनेऽपि सूत्रम्--
रवण्डयुति भक्ततलमुवकृत्यन्तगुणितरवण्डमुखवर्णयुतम् ।
मूलमघतलमुवयुतदलहृतलब्धं च लम्बकः क्रमशः ॥ १७५(१/२) ॥
अत्रोद्देशकः।
वदनं सप्तक्तमधः क्षितिस्त्रयोविंशतिः पुनस्त्रिशत् ।
बाहू द्वाभ्यां भक्तं चैकैकं लब्धमत्र का भूमिः ॥ १७६(१/२) ॥
भूमिर्विषष्ठिशतमथ चाष्टादश वदनमत्र सन्दृष्टम् ।
लम्बश्चतुशतीदं क्षेत्रं भक्तं नरैश्चतुर्भिश्च ॥ १७७(१/२) ॥
एकद्विकत्रिकचतुःरवण्डान्येकैकपुरुषलब्धानि ।
प्रक्षेपतया गणितं तलमप्यवलम्बकं ब्रूहि ॥ १७८(१/२) ॥
भूमिरशीतिर्वदनं चत्वारिंशच्चतुर्गुणा षष्टिः ।
अवलम्बकप्रमाणं त्रीण्यष्टौ पञ्च वण्डानि ॥ १७९(१/२) ॥
क्षेत्रगणितव्यवहारः 135
स्तम्भद्वयप्रमाणसख्यां ज्ञात्वा तत्स्तम्भद्वयाग्रे सूत्रद्वयं बह्म तत्सू
त्रद्वयं कर्णाकारेण इतरेतरस्तम्भमूलं वा तत्स्तम्भमूलमतिक्रम्य वा संस्प.
श्य तत्कर्णाकारसूत्रद्वयस्पर्शनस्थानादारभ्य अधःस्थितभूमिपर्यन्तं तन्मध्ये
एकं सूत्रं प्रसार्य तत्सूत्रप्रमाणसङ्ख्यैव अन्तरावलम्बकसंज्ञा भवत ।
अन्तरावलम्बकस्पर्शनस्थानादरभ्य तस्यां भूम्यामुभयपार्श्वयोः कणका
रसूत्रद्वयस्पर्शनपर्यन्तमागधासंज्ञा स्यात् । तदन्तरावलम्बकसङ्ख्यानय-
नस्य आवाधासख्यानयनस्य च सूत्रम्--
स्तम्भौ रज्वन्तरभूहतौ स्वयोगहतौ च गुणितौ ।
आबाधे ते वामप्रक्षेपगुणोऽन्तरवलवः ॥ १८०(१/२) ॥
अत्रोद्देशकः ।
षोडशहस्तोच्छ्यौ स्तम्भाववनिश्च षोडशोद्दि टैौ ।
आबधान्तरसङ्गमित्राप्यवलम्बकं ब्रूहि ॥ १८१(१/२) ॥
स्तम्भेकस्यच्छायः षत्रिंशद्दिशतिर्दिनीयस्य ।
भूमिर्दादश हस्ताः काबाधा कोऽयमवलम् ॥ १८२(१/२) ॥
इदश च पञ्चदश च स्तम्भान्तरभूमेरापे च चत्वारः ।
द्वादशकस्तम्भाग्राद्रजुः पतितान्यतो मूलात् ॥ १८३(१/२) ॥
आक्रम्य चतुर्हस्तात्परस्य मूलं तथैकहस्ताच्च ।
पतिताग्रात्काबाधा कोऽस्मिन्नवलम्बको भवति ॥ १८४(१/२) ॥
बाहुप्रतिबाहू द्वौ त्रयोदशावनिरियं चतुर्दश च ।
वदनेऽपि चतुहेताः काबाधा कोऽन्तरावलम्बश्च । १८५(१/२) ॥
क्षेत्रमिदं मुरव भूम्योरेकैकानं परस्पराग्राच्च ।
रजुः पतिता मूलवं ब्रुह्यवलम्बकाबाधं ॥ १८६(१/२) ॥
136 गणितसारसङ्ग्रहः
बाहुयोदशैकः पञ्चदश प्रतिभुजा मुरवं सप्त ।
भूमिरियमेकविंशतिरस्मिन्नवलबकाबधे ॥ १८७(१/२) ॥
समचतुरश्रक्षेत्रं विंशतिहस्तायतं तस्य ।
कोणेभ्योऽथ चतुभ्यं विनिर्गता रज्जवस्तत्र ॥ १८८(१/२) ॥
भुजमध्यं द्वियुगभुजे' रज्जुः का स्यात्सुसंवीता।
को वावलम्बकः स्याद्बाधे केऽन्तरे तस्मिन् ॥ १८९(१/२) ॥
सम्भस्थोन्नतप्रमाणसख्यं ज्ञात्वा तस्मिन् स्तम्भे येनकेनचित्कार-
णेन भग्ने पतिते सति तत्स्तम्भाषमूलयोर्मध्ये स्थितौ सत्यां ज्ञात्वा
तत्स्तम्भमूलादारभ्य स्थितपरिमाणसङ्ख्यानयनस्य सूत्रम्--
निर्गमवर्णान्तरमितिवर्गविशेषस्य यद्भवेदर्धम् ।
निर्गमनेन विभक्तं तावत्स्थित्वाथ भग्नः स्यात् ॥ १९०(१/२) ॥
अत्रोद्देशकः ।
स्तमस्य पञ्चविंशतिरुच्छूयः काश्चिदन्तरे मनः ।
स्तम्भाश्रमूलमध्ये पञ्च स गत्वा कियान् भग्नः ॥ १९१(१/२) ॥
वेणूच्छाये हस्ताः सप्तकृतः कश्चिदन्तरे भग्नः ।
भूमिश्च सैकविंशतिरस्य स गत्वा कियान् भम' ॥ १९२(१/२) ॥
वृक्षोच्छायो विंशतिरग्रस्थः कोऽपि तत्फलं पुरुषः ।
कर्णाच्या व्यक्षिपदथ तरुमूलस्थितः पुरुषः ॥ १९३(१/२) ॥
तस्य फलस्याभिमुखं प्रमुजरूपेण गत्वा च ।
फलमग्रहीच्च तत्फलनरयोर्गतियोगसदैव ॥ १९४(१/२) ॥
पञ्चशदभूत्तत्फलगतिरूपा कर्णसङ्ख्या का ।
तदृक्षमूलगतनरगतिरूपा प्रतिभुजापि कियती स्यात ॥ १९५(१/२) ॥
‘भुजचतुर्षु च is the reading found in the MS., but it is not correct.
१ The Sandhi in केऽन्तरे is grammatically incorrect ; but the author seems
to have intended the phonetic fusion for the sake of the metre; vide stanza 204(1/2)
of this chapter.
त्रगणितव्यवहारः 137
ज्येष्ठस्तम्भसङ्ख्यां च अल्पस्तम्भसङ्ख्यां च ज्ञात्वा उभयस्त.
म्भान्तरभूमिसङ्ख्यां ज्ञात्वा तज्ज्येष्ठसख्ये भग्ने सति ज्येष्ठस्तम्भाग्रे
अल्पस्तम्भाग्रं स्पृशति सति ज्येष्ठस्तम्भस्य भग्नसङ्ख्यानयनस्य स्थित-
शेषसङ्ख्यानयनस्य च सूत्रम् –-
ज्येष्ठस्तम्भस्य कृतेर्ह्रस्वावनिवर्गयुतिमपोह्यार्धम् ।
स्तम्भविशेषेण हृतं लब्धं भग्नोन्नतिर्भवति ॥ १९६(१/२) ॥
अत्रोद्देशक: ।
स्तम्भः पचोच्छायः परत्रयोविंशतिस्तथा ज्येष्ठः ।
मध्यं द्वादश भर्ज्येष्ठाग्रं पतितमितराग्रे ॥ १९७(१/२) ॥
आयतचतुरश्रक्षेत्रकोठिसङ्ख्यायास्तृतीयांशद्वयं पर्वतोत्सेधं परि-
कल्प्य तत्पर्वतोत्सेधसङ्ख्यायाः सकाशात् तदायतचतुरश्रक्षेत्रस्य भुज.
सङ्ख्यानयनस्य कणसङ्ख्यानयनस्य च सूत्रम्--
गिर्यत्सेधो द्विगुणो गिरिपुरमथ्याक्षितिर्गिरेरर्धम् ।
गगनं तत्रोत्पतितं गिर्यर्धव्याससंयुतिः कर्णः ॥ १९८(१/२) ॥
अत्रोद्देशकः ।
षड्रयोजनोर्वशिखरिणि यतीश्वरौ तिष्ठतस्तत्र ।
एकोऽद्विचर्ययागात्तत्राप्याकाशचार्यपरः ॥ १९९(१/२) ॥
श्रुतिवशमुत्पत्य पुरं गिरोिशैिरवरान्मूलमधरुह्यन्यः ।
समगतिकौ सञ्जातौ नगरव्यासः किमुत्पतितम् ।। २००(१/२) ॥
डोलाकारक्षेत्रे सम्भद्वयस्य वा गिरिद्वयस्य वा उत्सेधपरिमाण-
सङ्ख्यामेव आयतचतुरश्रक्षेत्रद्वये भुजद्वयं परिकल्प्य तद्दिरिद्वयान्तर
भूम्यां वा तत्स्तम्भद्वयान्तरभूम्यां वा आबाधाद्वयं परिकल्प्य तदाबाधा-
I2
138 गणितसारसङ्ग्रहः
इयं व्युत्क्रमेण निक्षिप्य तद्युत्क्रमं न्यस्ताबाधाद्वयमेव आयतचतुरश्रक्षेत्र
द्वये कोटिद्वयं परिकल्प्य तत्कर्णद्वयस्य समानसङ्ख्यानयनसूत्रम्-–
डोलाकारक्षेत्रस्तम्भद्वितयोर्ध्वसङ्ख्ये वा।
शिखरिढयोर्वसङ्ख्ये परिकल्प्य भुजद्वयं त्रिकोणस्य ॥ २०१(१/२) ॥
तदोर्द्धितयान्तरगतसङ्ख्यायास्तदाबाधे ।
आनीय प्राग्वत्ते व्युत्क्रमतः स्थाप्य ते कोटी ॥ २०२(१/२) ॥
स्यातां तस्मिन्नायतचतुरश्रमंत्रयांश्च तद्दोभ्यम् ।
कोटिभ्यां कर्णं द्वौ प्राग्वत्स्यातां समानसङ्ख्यौ तौ॥ २०३(१/२) ॥
अत्रोद्देशकः ।
स्तम्भस्त्रयोदशैकः पञ्चदशान्यश्चतुर्दशान्तरितः ।
रजवेज़ शिरवरे भूमीपतिता क' आबाधे ॥
ते रज्जू समसङ्ख्ये स्यातां तद्रज्जुमानमपि कथय ॥ २०५(१/२) ॥
द्वाविंशतिरुत्सेधो गिरेस्तथाष्टादशान्यशैलस्य ।
विंशतिरुभयोर्मध्ये तयोश्च शिरवयोस्स्थितौ साधू ॥ २०६ ॥
आकाशचारिणौ तौ समागतौ नगरमत्र भिक्षायै ।
समगतिकौ सञ्जातौ तत्राबाधे कियत्सङ्ख्ये ।
समगतिसङ्ख्या कियती डोलाकारेऽत्र गणितज्ञ ॥ २०७(१/२) ॥
विंशतिरेकस्योन्नतिरत्रैश्च जिनास्तथान्यस्य ।
तन्मध्यं द्वाविंशतिरनयोरद्योश्च भृङ्गयोः स्थित्वा ॥ २०८(१/२) ॥
आकाशचारिणौ । तन्मध्यपुरं समायातौ।
भिक्षायै समगतिकौ स्यातां तन्मध्यशिवारिमध्यं किम् ॥ २०९(१/२) ॥
विषमात्रिकोणक्षेत्ररूपेण हीनाधिकगतिमतोर्नरयोः समागमदिनसङ्ख्यानयनसूत्रम्--
1. क आवाधे ib grammatioally incorrect, since there an be no sodhi between
के in the dual number and आबाधे; vide footnote on page 136.
क्षेत्रगणितव्यवहारः 139
दिनगतिकृतिसंयोगं दिनगतिछत्यन्तरेण हत्वाथ ।
हत्वोदग्गतिदिवसैस्तछब्धदिने समागमः स्यान्नोः ॥ २१०(१/२) ॥
अत्रोद्देशकः ।
dwe' योजने प्रयाति हि पूर्वगतिस्त्रीणि योजनान्यपरः ।
उत्तरतो गच्छति यो गत्वासौ तद्दिनानि पद्यथ ॥ २११(१/२) : ॥
गच्छन् कर्णाकृत्या कतिभिर्दवसैर्नरं समासोति ।
उभयोर्युगपद्मनं प्रस्थानदिनानि सदृशानि ॥ २१२(१/२) ।।
पचविधचतुरश्रक्षेत्राणां च त्रिविधत्रिकोणक्षेत्राणां चेत्यष्टविधवाद्य
सुत्तव्याससङ्ख्यानयनसूत्रम्--
श्रुतिरवलम्बकभक्ता पार्श्वभुजम्ना चतुभुजे त्रिभुजे ।
भुजघातो लस्बहूतो भवेद्वहिर्द्धतविष्कम्भः ॥ २१३(१/२) ॥
अत्रोद्देशकः ।
समचतुरश्रस्य त्रिकवाहुप्रतिबाहुकस्यं चान्यस्य ।
कोटिः पथ द्वादश भुजास्य किं वा बहिर्युतम् ॥ २१४(१/२) ॥
बाहू त्रयोदश मुरवं चत्वारि धरा चतुर्दश प्रोक्ता ।
द्विसमचतुरश्रबाहिरविष्कम्भः को भवेदत्र ॥ २१५(१/२)॥
पञ्चक्रुतिर्बदन भुजाश्चत्वारिंशच्च भूमिरेकोना ।
त्रसमचतुरश्रबाहिरवृत्तव्यास ममाचक्ष्व ॥ २१६(१/२) ॥
व्येका चत्वारिंशद्वाहुः प्रतिबाहुको द्विपञ्चाशत् ।
षष्टिर्भूमिर्वदनं पथछतिः कोऽत्र विष्कम्भः ॥ २१७(१/२) ॥
त्रिसमस्य च षड् बाहुरुत्रयोदश द्विसमबाहुकस्यापि ।
भूमिर्देश विष्कम्भावनयोः कौ बाह्यनुत्तयोः कथय।। २१८(१/२) ॥
13
140 गणितसारसङ्ग्रहः
बाहू पञ्चत्र्युत्तरदशकौ भूमिश्चतुर्दशो विषमे ।
त्रिभुजक्षेत्रे बाहिरवृत्तव्यासं ममाचक्ष्व ॥ २१९(१/२) ॥
डेिकबाहुषडश्रस्य क्षेत्रस्य भवेद्विचिन्त्य कथय त्वम् ।
बाहिरविष्कम्भं मे पैशाचिकमत्र यदि वेसि ।। २२०(१/२) ॥
इष्टसङख्याव्यासवत्समवृत्तक्षेत्रमध्ये समचतुरश्राद्यष्टक्षेत्राणां नुरव
भभुजसङख्यानयनसूत्रम्--
लब्धव्यासनेष्टव्यासां वृत्तस्य तस्य भक्तश्च ।
लब्धेन भुजा गुणयेद्भवेच्च जातस्य भुजसङ्ख्या ॥ २२१(१/२) ॥
अत्रोद्देशकः ।
वृत्तक्षेत्रव्यासस्त्रयोदशाभ्यन्तरेऽत्र सचिन्त्य ।
समचतुरश्राद्यष्टक्षेत्राणि सरवे ममाचक्ष्व ॥ २२२(१/२) ॥
आयतचतुरथं विना पूर्वकल्पितचतुरश्रादिक्षेत्राणां सूक्ष्मगणितं च
रज्जुसख्यां च ज्ञात्वा तत्तक्षेत्राभ्यन्तरावस्थितवृत्तक्षेत्रविष्कम्भानयन
परिधेः पादेन भजेदनायतक्षेत्रसूक्ष्मगाणितं तत् ।
क्षेत्राभ्यन्तरवृत्ते विष्कम्भोऽयं विनिर्दिष्टः ॥ २२३(१/२) ॥
अत्रोद्देशकः ।
समचतुरश्रादीनां क्षेत्राणां पूर्वकल्पितानां च ।
छत्वाभ्यन्तरवृत्तं गृह्यधुना गणिततत्त्वज्ञ ॥ २२४(१/२) ॥
समवृत्तव्याससङ्ख्यायामिष्टसङ्ख्यां बाणं परिकल्प्य तद्वाणपरि
माणस्य ज्यासयानयनसूत्रम--
क्षेत्रगणितव्यवहारः 141
व्यासाधिगमोनस्स च चतुर्गुणिताधिगमेन सङ्गुणितः ।
यत्तस्य वर्गमूलं ज्यारूपं निर्दिशेत्प्राज्ञः ॥ २२५(१/२) ॥
अत्रोद्देशकः ।
व्यासो दश वृत्तस्य द्वाभ्यां छिन्नो हि रूपाभ्याम् ।
छिन्नस्य ज्या का स्यात्प्रगणयाचक्ष्व तां गणक ॥ २२६(१/२) ।।
समवृत्तक्षत्रव्यासस्य च माव्योश्च सङ्ख्यां ज्ञात्वा बाणसया
नयनसूत्रम्--
व्यासज्यारूपकयोर्गविशेषस्य भवति यन्मूलम् ।
तद्विष्कम्भाच्छोध्यं शेषार्धमिषं विजानीयात् ।। २२७(१/२) ॥
अत्रोद्देशकः ।
दश वृत्तस्य विष्कम्भः शिञ्जिन्यभ्यन्तरे सरवे ।
डष्टाष्टो हि पुनस्तस्याः कः स्यादधिगमो वद ॥ २२८(१/२) ॥
ज्यासङ्ख्यां च बाणसङ्ख्यां च ज्ञात्वा समवृत्तक्षेत्रस्य मध्यभ्यास
सयानयनसूत्रम्--
अक्तश्चतुर्गुणेन च शरेण गुणवर्णराशिरिषुसहितः ।
समवृत्तमध्यमस्थितविष्कम्भोऽयं विनिर्दिष्टः ॥ २२९(१/२) ॥
अत्रोद्देशकः ।
कस्यापि च समवृत्तक्षेत्रस्याभ्यन्तराधिगमनं है।
ज्या दृष्टाष्टौ दण्डा मध्यव्यास अवेकोऽत्र ॥ २३०(१/२) ॥
समवृत्तद्वयसंयोगे एका मत्स्याकृतिर्भवति । तन्मत्स्यस्य मुरवपुच्छ-
विनिर्गतरेवा कर्तव्या । तया रेवया अन्योन्याभिमुरवधनुर्दयार्तिर्भि-
142 गणितसारसङ्ग्रहः
वति । तन्मुखपुच्छविनिर्गतरेवैव तद्धनुर्जयस्यापि ज्याकृतिर्भवति ।
तद्धनुर्द्वयस्य शरद्वयमेव वृत्तपरस्परसम्पातशरौ ज्ञेयौ । समवृत्तद्वयसंयोगे
तयः सम्पातरयांरानयनस्य सूत्रम्--
ग्रासानव्यासाभ्यां प्राज्ञ प्रक्षेपकः प्रकर्तव्यः ।
वृत्ते च परस्परतः सम्पातशरौ विनिर्दिष्टौ ॥ २३१(१/२) ॥
अत्रोद्देशकः ।
समवृत्तयोऽयोर्हि द्वात्रिंशदशतिहस्ताविस्तृतयोः ।
ग्रासेऽष्टों कौ बाणावन्योन्यभवों समाचक्ष्व ॥ २३२(१/२) ॥
इति पैशाचिकव्यवहारः समाप्तः ।
इति सारसङ्ग्रहे गणितशास्त्रे महवीराचार्यस्य कृतौ क्षेत्रगणितं
नाम षष्ठव्यवहारः समाप्तः ।
सप्तमः
खातव्यवहारः
सर्वामरेन्द्रमकुटार्चितपादपीठं
सर्वज्ञमव्ययमचिन्त्यमनन्तरूपम् ।
भव्यप्रजासरसिजाकरबालभानुं
भक्त्या नमामि शिरसा जिनवर्धमानम् ॥ १ ॥
क्षेत्राणि यानि विविधानि पुरोदितानि
तेषां फलानि गुणतान्यवगाहनानि (नेन) ।
कमन्तिकण्डूफलसूक्ष्मवकाल्पितान
वक्ष्यामि सप्तममिदं व्यवहारवातम् ॥ २ ॥
सूक्ष्मगणितम्
अत्र परिभाषाञ्चकः
हस्तघने पांसूनां द्वात्रिंशत्पलशतानि पूर्याणि ।
उत्कीर्यन्ते तस्मात् षत्रिंशत्पलशतानीह ॥ ३ ॥
खातगणितफलनयनसूत्रम्--
क्षेत्रफलं वेधगुणं समरवाते व्यावहारिकं गणितम् ।
मुरवतलयुतिदलमथ सत्सङ्ख्यातं स्यात्समीकरणम् ॥ ४ ॥
अत्रोद्दश्कः
समचतुरश्रस्याष्टौ बाहुः प्रतिबाहुकश्च वेधश्च ।
क्षेत्रस्य वातगणितं समरवाते किं भवेदत्र ॥ ५ ॥
त्रिभुजस्य क्षेत्रस्य द्वात्रिंशद्वाहुकस्य वेधे तु ।
षत्रिंशदृष्टास्ते षडलान्यस्य कि गणितम् ॥ ६ ॥
साष्टशतव्यासस्य क्षेत्रस्य हि पञ्चषष्टिसहितशतम् ।
वेधो वृत्तस्य त्वं समरवाते किं फलं कथय ॥ ७ ॥
आयतचतुरश्रस्य व्यासः पञ्चभर्विंशतिर्बाहुः।
षष्टिर्मेधोऽष्टशतं कथयाशु समस्य खातस्य ॥ ८ ॥
I4
144 गणितसारसङ्ग्रहः
अस्मिन् खातगणिते कर्मान्तिकसंज्ञफलं च औण्ड्रसंज्ञफलं च ज्ञात्वा
ताभ्यां कर्मान्तिकौण्ड्रसंज्ञफलाभ्यां सूक्ष्मखातफलानयनसूत्रम्--
बाह्याभ्यन्तरसंस्थिततत्तत्क्षेत्रस्थबाहुकोट्भुवः।
खप्रतिबाहुसमेता भक्तास्तत्क्षेत्रगणनयान्योन्यम् ॥ ९ ॥
गुणिताश्च वेधगुणिताः कर्मान्तिकसंज्ञगणितं स्यात् ।
तद्वाह्यान्तरसंस्थिततत्तत्क्षेत्रे फलं समानीय ॥ १० ॥
संयोज्य सङ्ख्ययातं क्षेत्राणां वेधगुणितं च ।
औण्ड्रफलं तत्फलयोर्विशेषकस्य त्रिभागेन ॥
संयुक्तं कर्मान्तिकफलमेव हि भवति सूक्ष्मफलम् ॥ ११(१/२) ॥
अत्रोद्देशकः ।
समचतुरश्रा वापी विंशतिरुपीह षोडशैव तले ।
वेधो नव किं गणितं गाणितविदाचक्ष्व मे शीघ्रम् ॥ १२(१/२) ॥
वापी समत्रिबाहुर्विंशतिरुपीह षोडशैव तले ।
वेषो नव किं गणितं कर्मान्तिकमौण्ड्रमपि च सूदमफलम् ॥ १३(१/२) ॥
समवृत्तासौ वापी विंशतिरुपीह षोडशैव तले ।
वेषो द्वादश दण्डाः किं क स्यात्कर्मान्तिकौण्ड्रसूदमफलम् ॥ १४(१/२) ॥
आयतचतुरश्रस्य त्वायामष्षष्टिरेव विस्तारः ।
द्वादश मुरवे तलेऽधं वेधोऽष्टौ किं फलं भवति ॥ १५(१/२) ॥
नवतिरशीतिः सप्ततिरायामथोर्वमभ्यमूलेषु ।
विस्तारो द्वात्रिंशत् षोडश दश सप्त वेधोऽयम् ॥ १६(१/२) ॥
यासः षाष्टिर्वदने मध्ये त्रिंशत्तले तु पञ्चदश।
समवृत्तस्य च वेधः षोडश किं तस्य गणितफलम् ॥ १७(१/२) ॥
खातव्यवहारः 145
त्रिभुजस्य मुखेऽशीतिः षष्टिर्मध्ये तले च पश्चात् ।
बाहुत्रयेऽपि वेधो नव किं तस्यापि भवति गणितफलम् ॥ १८(१/२) ॥
खातिकायाः खातगणितफलानयनस्य च खातिकाया मध्ये सूची
मुखाकारवत् उत्सेधे सति खातगणितफलानयनस्य च सूत्रम् -–
परिखामुखेन सहितो विष्कम्भत्रिभुजवृत्तयोस्त्रिगुणात् ।
आयामश्चतुरश्रे चतुर्गुणो व्याससङ्गणितः ॥ १९(१/२) ॥ ॥
सूचीमुखबड़ेघे परिखा मध्ये तु परिवार्धम् ।
मुरवसहितमथो करणं प्राग्वत्तलसूचिवेधे च ॥ २०(१/२) ॥
अत्रोदशकः
त्रिभुजचतुभुजवृत्तं पुरोदितं परिरवया परिक्षिप्तम् ।
दण्डाशीत्या व्यासः परिखाश्चतुरुर्विकास्त्रिवेधाः स्युः ॥ २१(१/२) ॥
आयतचतुरायामो विंशत्युत्तरशतं पुनव्यासः ।
चत्वारिंशत् परिरवा चतुरुवका त्रिवेधा स्यात् ॥ २२(१/२) ॥
उत्सेधे बहुप्रकारवति सति वातफलानयनस्य च, यस्य कस्यचित्
रवतफलं ज्ञात्वा तत्वातफलात् अन्यक्षेत्रस्य खातफलानयनस्य च सूत्रम्--
वेधयुतिः स्थानहृता वेधो मुखफलगुणः खरवातफलम् ।
त्रिचतुर्मुजवृत्तानां फलमन्यक्षेत्रफलहृतं वेघः ॥ २३(१/२) ॥
अत्रोद्देशकः ।
समचतुरश्रक्षेत्रे भूमिचतुर्हस्तमात्रविस्तारे ।
तत्रैकद्वित्रिचतुर्हस्तनिखाते कियान् हि समवेधः ॥ २४(१/२) ॥
14-A
146 गणितसारसङ्ग्रहः
समचतुरश्राष्टादशहस्तभुजा वापिका चतुर्वेधा ।
वापी तज्जलपूर्णान्या नववाहात्र को वेधः ॥ २५(१/२) ॥
यस्य कस्यचित्खातस्य ऊध्वीस्थितभुजासङ्ख्यां च अधस्स्थित
भुजासख्यां च उत्सेधप्रमाणं च ज्ञात्वा, तत्खाते इष्टोत्सेधसङ्ख्यायाः
भुजासङ्ख्यानयनस्य, अधस्सूचिवेधस्य च सङ्ख्यानयनस्य सूत्रम्-–
मुखगुणवेधो मुरवतलशोषहृतोऽत्रेव सूचिवेधः स्यात् ।
विपरीतवेधगुणमुरवतलयुत्यवलम्बहूव्यासः ॥ २६(१/२) ।।
अत्रोद्देशकः ।
समचतुरश्रा वापी विंशतिरूढे चतुर्दशाधश्च ।
वेधो मुखे नवाधस्त्रयो भुजाः केऽत्र सूचिवेधः कः ॥ २७(१/२) ॥
गोलकाकारक्षेत्रस्य फलानयनसूत्रम्--
व्यासार्धघनार्धगुणा नव गोलव्यावहारिक गणितम् ।
तद्दशमांशं नवगुणमशेषसूक्ष्मं फलं भवति ॥ २८(१/२) ॥
अत्रोद्देशकः ।
षोडशविष्कम्भस्य च गोलकवृत्तस्य विगणय्य ।
कि व्यावहारिकफलं सूक्ष्मफलं चापि मे कथय ॥ २९(१/२) ॥
शृङ्गाटकक्षेत्रस्य स्वातव्यावहारिकफलस्य रवातसूक्ष्मफलस्य च सूत्रम्--
भुजकुतिदलघनगुणदशपदनवहव्यवहारिकं गणितम् ।
त्रिगुणं दशपदभक्तं शृङ्गठकसूमघनगणितम् ॥ ३०(१/२) ॥
अत्रोद्देशकः ।
त्र्यश्रस्य च शृंङ्गाटकषड्बाहुघनस्य गणयित्वा ।
किं व्यावहारिकफलं गणितं सूक्ष्मं भवेत्कथय ॥ ३१(१/२) ॥
खातव्यवहारः 147
वापीप्रणालिकानां विमोचने तत्तदिष्टप्रणालिकासंयोगे तज्जलेन
वाप्यां पूणायां सत्यां तत्तत्कालानयनसूत्रम्--
वापीप्रणालिकाः स्वखकालभक्ताः सवर्णविच्छेदाः।
तद्युतिभक्तं रूपं दिनांशकः स्यात्प्रणालिकायुत्या ।
तद्दिनभागहतास्ते तज्जलगतयो भवन्ति तद्वष्याम् ॥ ३३ ॥
अत्रोद्देशकः ।
चतस्रः प्रणालिकाः स्युस्तत्रैकैका प्रपूरयातं वापीम् ।
द्वित्रिचतुःपञ्चांशैर्दिनस्य कतिभिर्दिनांशैस्ताः ॥ ३४ ॥
त्रैराशिंकाख्यचतुर्थगणितव्यवहारे सूचनामात्रोदाहरणमेव ; अत्र
सम्यग्विस्तार्य प्रवक्ष्यते--
समचतुरश्रा वापी नवहस्तघना नगस्य तले।
तच्छिरवराजलधारा चतुरश्राङ्गुलसमानविष्कम्भा ॥ ३५ ॥
पतिताग्रे विच्छिन्ना तथा घना सान्तरालजलपण ।
शैलोत्सेधे वाघ्यां जलप्रमाणं च मे ब्रूहि ॥ ३६ ॥
वापी समचतुरश्रा नवहस्तघना नगस्य तले ।
अङ्गलसमवृत्तघना जलधारा नेिपांतता च तच्छखरात् ॥ ३७ ॥
अग्रे विच्छिन्नाभूत्तस्या वाप्या मुखं प्रविष्टा हि ।
सा पूर्णान्तरगतजलधरोत्सेधेन शैलस्य ।
उत्सेधं कथय सवे जलप्रमाणं च विगणय्य ॥ ३८॥
समचतुरश्रा वापी नवहरुघन नगस्य तले।
तच्छिखराजलधारा पतिताङ्गलघनत्रिकोणा सा ॥ ३९ ॥
वापीमवप्रविष्ठ साग्रे छिन्नान्तरालजलपूर्णा ।
कथय सरवे विगणय्य च गिर्युत्सेधं जलप्रमाणं च ॥ ४० ॥
148 गणितसारसङ्ग्रहः
समचतुरश्रा वापा नवहस्तघना नगस्य तले ।
अङ्गुलविस्ताराङ्गुलखाताङ्गुलयुगलदीर्घजलधारा ॥ ४१(१/२) ॥
पतिताग्रे विच्छिन्ना वापीमुखसंस्थितान्तरालजलैः।
सम्पूर्णा स्याद्वापी गिर्युत्सेत्सेधो जलप्रमाणं किम् ॥ ४२(१/२) ॥
इति वातव्यवहारे सूक्ष्मगणितं सम्पूर्णम् ।
चितिगणितम्
इतः परं खातव्यवहारे चितिगणितमुदाहरिष्यामः । अत्र परिभाषा--
हतो दीर्घ व्यासस्तदधमङ्गलचतुष्कमुत्सेधः ।
दृष्टस्तथेष्टकायास्ताभिः कर्माणि कार्याणि ॥ ४३(१/२) ॥
इष्टक्षेत्रस्य स्वातफलानयने च तस्य रवातफलस्य इष्टकानयने च सूत्रम्--
मुरवफलमुदयेन गुण तदिष्टकागणितभक्तलब्धं यत् ।
चितिगणितं तद्विद्यात्तदेव भवतीष्टकासख्या ॥ ४४(१/२) ॥
अत्रोद्देशकः ।
वेदिः समचतुरश्रा साष्टभुजा हस्तनवकमुत्सेधः ।
घठिता तदिष्टकाभिः कतीष्टकाः कथय गणितज्ञ ।। ४५(१/२) ॥
अष्टकरसमत्रिकोणनवहस्तोत्सेधवेदिका रचिता।
पूर्वेष्टकाभिरस्य कतीष्टकाः कथय विगणय्य ।। ४६(१/२) ॥
समदुत्ताकृतिवेदिर्नवहसोध् कराष्टकव्यास ।
घटितेष्टकाभिरस्यां कतीष्टकाः कथय गणितज्ञ ॥ ४७(१/२) ॥
आयतचतुरश्रस्य त्वायामः षष्टिरेव विस्तारः ।
पञ्चकृतः षड् वेधस्तदिष्टकाचितिमिहाचक्ष्व ॥ ४८ ॥
खातव्यवहारः 149
प्राकारस्य व्यासः सप्त चतुर्विंशतिस्तदायामः ।
घठितेष्टकाः कति स्युश्चोच्छ्रायो विंशतिस्तस्य ॥ ४९(१/२) ॥
व्यासः प्राकारस्योध्वे षड धोऽथाष्ट तीर्थका दीर्घः ।
घठितेष्टकाः कति स्युश्चोच्छायो विंशतिस्तस्य ॥ ५०(१/२) ॥
द्वादश षोडश विंशतिरुत्सेधाः सप्त षट् च पञ्चधः ।
व्यासा मुवे चतुस्त्रिद्विकाश्चतुर्विंशतिर्दीर्घः ॥ ५१(१/२) ॥
इष्टवदेकभयां पतितायां सत्यां स्थितस्थाने इष्टकासख्यानयनस्य
व पतितस्थाने इष्टकासङ्ख्यानयनस्य च सूत्रम्--
मुरवतलशषः पतितांत्सेधगुणः सकलवधहृत्समुरवः ।
मुखभूम्योर्भूमिमुखे पूर्वोक्तं करणमवशिष्टम् ॥ ५२(१/२) ॥
अत्रांदंशकः ।
द्वादश दैर्यं व्यासः पञ्चधश्चोर्घमेकमुत्सेधः ।
दश तस्मिन् पच करा भग्नास्तत्रेष्टकाः काति स्युस्ताः ॥ ५३(१/२) ॥
प्राकारे कर्णाकारेण भने सति स्थितेष्टकानयनस्य च पतितेष्टकाः
नयनस्य च सूत्रम्--
भूमिमुखे द्विगुणे मुखभूमियुतेऽभमभूदययुतोने ।
दैव्योदयषष्ठांशमे स्थितपतितेष्टकाः क्रमेण स्युः ॥ ५४(१/२) ॥
अत्रोद्देश्कः ।
प्राकारोऽयं मूलान्मध्यावर्तेन वायुना विदुः।
कर्णाकृत्या भग्नस्तत्स्थितपतितेष्टकाः कियत्यः स्युः ॥ ५५(१/२) ॥
प्राकारोऽयं मूलान्मध्यावर्तेन चैकहस्तं गत्वा ।
कणोक्त्या भग्नः कतीष्टकाः स्युः स्थिताश्च पतिताः काः ॥ ५६(१/२) ॥
150
गणितसारसङ्ग्रहः
प्राकारमध्यप्रदेशोत्सेधे तरवृद्ध्यानयनस्य प्राकारस्य उभयपार्श्वयोः
तरहानेरानयनस्य च सूत्रम्--
इष्टेष्टकादयहृतो वेधश्च तरप्रमाणमे प्रभम् ।
मुखतलशेषेण हृतं फलमेव हि भवति तरहानिः ॥ ५७(१/२) ॥
अत्रोद्देशकः ।
प्राकारस्य व्यासः सप्त तले विंशतिसदुत्सेधः ।
एकेनाऐं घठितस्तरवृद्धयूने करोदयेथ्कया ॥ ५८(१/२) ॥
समवृत्तायां वाप्यां व्यासचतुष्केऽर्घयुक्तकरभूमिः ।
घटितेष्टकाभिरभितस्तस्यां वेधस्त्रयः काः स्युः ।
घटितेष्टकाः सर्वे भे विगणय्य ब्रूहि याद वेत्सि ॥ ६० ॥
इष्टकाघटितस्थले अधस्लब्यासे सति ऊध्र्वतलव्यासे सति च
गणितन्यायसूत्रम्—
द्विगुणनिवेशो व्यासायामयुतो द्विगुणितस्तदायामः ।
आयतचतुरश्रे स्यादुत्सेधव्याससङ्गणितः । ६१ ॥
अत्रोद्देशकः ।
विद्याधरनगरस्य व्यासोऽष्टौ द्वादशैव चायामः ।
पच प्रकारतले मुरवं तदेकं दशोत्सेधः ॥ ६२ ॥
इति वातव्यवहारे चितिगणितं समाप्तम् ।
क्रकचिकाव्यवहारः
इतः परं क्रकाचिकाव्यवहारमुदाहरिष्यामः । तत्र परिभाषा--
हस्तद्वये पडङ्गलहीनं किष्क्वाढ्यं भवति ।
इष्टाद्यन्तच्छेदनसङ्ख्यैव हि मर्गसंज्ञा स्यात् ॥ ६३ ॥
अथ शाकाख्यद्याद्द्रुमसमुदायेषु वक्ष्यमाणेषु ।
आसोदयमार्गणामडुलसंख्या परस्परन्नाप्ता ॥ १४ ॥
खातव्यवहारः 151
हस्ताङ्गुलवर्गेण क्राकचिके पट्टिकाप्रमाण स्यात् ।
शाकाह्वयद्रुमादिद्रुमेषु परिणाहदैर्घ्यहस्तानाम् ॥ ६५ ॥
संख्या परस्परम् मार्गाणां संख्यया गुणिता ।
तत्पाट्टिकासमाप्ता क्रकचऊ कमेसंख्या स्यात् ॥ ६६ ॥
शाकानुनलवंतससरलासेतसजेडुण्डुकाख्येषु ।
श्रीपर्णऽक्षाख्यद्रुमेष्वमी-कमार्गस्य ।
षण्णवतरङ्गलनामायामः किष्कुरेव विस्तारः ॥ ६७(१/२) ॥
अत्रोद्देशकः ।
शाकाख्यतरौ दीर्घः षोडश हतश्च विस्तारः ।
सार्धत्रयश्च मार्गाश्चाष्टौ कान्यत्र कर्माणि । ६८(१/२) ॥
इति रवव्यवहारं क्रकचकव्यवहारः समाप्तः ।
इति सारसङ्गहे गणितशास्त्रे भहवीराचार्यस्य कृतौ सप्तमः वातव्यवहारः समाप्तः ॥
अष्टमः
छायाव्यवहारः
शान्तिर्जिनः शान्तिकरः प्रजानां जगत्प्रभुर्ज्ञातसमस्तभावः ।
यः प्रातिहार्याष्टविवर्धमानो नमामि तं निर्जितशत्रुसङ्गम् ॥ १ ॥
आदौ प्राच्याद्यष्टदिक्साधनं प्रवक्ष्यामः
सलिलोपरितलवत्स्थितसमभूमितले लिवेट्टत्तम् ।
बिम्बं खेच्छाशङ्कद्विगुणितपरिणाहस्त्रेण ॥ २ ॥
तवृत्तमध्यस्थतदिष्टशको
श्छाया दिनादौ च दिनान्तकाले ।
तदृत्तरेवां स्पृशति क्रमेण
पश्चात्पुरस्ताच्च ककुप् प्रदिष्टा ॥ ३ ॥
तद्दिग्द्वयान्तर्गततन्तुना लिखे
न्मत्स्याक्रांत यान्यकुबेरदस्थाम् ।
तत्कणमध्य वदंशः प्रसाध्या
२छायैव याम्योत्तरदिग्दिशार्धजाः ॥ ४ ॥
अजधठरविसङ्कमणद्युदलजमैक्यार्धमेव विषुवद्भा ॥ ४(१/२) ॥
लङ्कायां यवकव्यां सिद्धपुरीरोमकापुयोः ।
विषुवद्भा नास्त्येव त्रिंशद्धठिकं दिनं भवेत्तस्मात् ॥ ५(१/२) ॥
देशेष्वितरेषु दिनं त्रिंशन्नाड्याधिकनं स्यात् ।
मेषधटायनदिनयोस्त्रिशदटिकं दिनं हि सर्वत्र ॥ ६(१/२) ॥
दिनमानं दिनदलभां ज्योतिश्शास्त्रोक्तमार्गेण ।
ज्ञात्वा छायागणितं विद्यादिह वक्ष्यमाणसूत्रधैः ॥ ७(१/२) ॥
1 M reads तत्वः
छायाव्यवहारः 153
विषुवच्छाया यत्रयत्र देशे नास्ति तत्रतत्र देशे इष्टशङ्कोरिष्टकालच्छायां
ज्ञात्वा तत्कालानयनसूत्रम्--
छाया सैका द्विगुणा तया हृतं दिनामितं च पूर्वाह्ने ।
अपराह्ने तच्छेषं विज्ञेयं सारसङ्ग्रहे गणिते ॥ ८(१/२) ॥
अत्रोद्देशकः ।
पूर्वाहे पौरुषी छाया त्रिगुणा वद किं गतम् ।
अपराहेऽवशेषं च दिनस्यांशं वद प्रिय ॥ ९(१/२) ॥
दिनांशे जाते सति घटिकानयनसूत्रम्-
अंशहतं दिनमानं छेदविभक्तं दिनांशके जाते ।
पूर्वादं गतनाड्यस्त्वपरावे शेषनाब्यस्तु ॥ १०(१/२) ॥
अत्रोद्देशकः ।
विषुवच्छायाविरहितदेशेऽष्टांशो दिनस्य गतः ।
शेषश्चाष्टांशः का घटिकाः स्युः खाग्निनाज्योऽङ्कः ॥ ११(१/२) ॥
मछयुद्धकालानयनसूत्रम्
कालानयनाद्दिनगतशेषसमासोनितः कालः ।
स्तम्भच्छाया स्तम्भप्रमाणभक्तैव पौरुषी छाया।। १२(१/२) ॥
अत्रोद्देशकः ।
पूर्वाहे शङ्करुपच्छायायां मछयुद्धमारब्धम् ।
अपराहे द्विगुणायां समाप्तिरासीच्च युद्धकालः कः ॥ १३(१/२)॥
अपरार्धस्योदाहरणम् ।
द्वादशहस्तस्तम्भच्छाया चतुरुत्तरैव विंशतिका ।
तत्कले पोषिकच्छया कियती भवेद्दणक ॥ १४(१/२) ॥
154 गणितसारसङ्ग्रहः
विषुवच्छायायुक्ते देशे इष्टच्छायां ज्ञात्वा कालानयनस्य सूत्रम्--
शङ्कुयुतेष्टच्छाया मध्यच्छायोनिता द्विगुणा ।
तदवाप्ता शङ्कुमितिः पूर्वापरयोर्दिनांशः स्यात् ॥ १५(१/२) ॥
अत्रोद्देशकः ।
द्वादशाङ्गुलशङ्खंदलच्छायाङ्गुलद्वयी ।
इष्टच्छायाष्टाङ्गलिका दिनांशः को गतः स्थितः।
च्यंशो दिनांशो घटिकाः कात्रिश्नाडिकं दिनम् ॥ १७ ॥
इष्टनाडिकानां छायानयनसूत्रम्-
द्विगुणितदिनभागहृता शङ्कमितिः शङ्कमानोना।
खुदलच्छायायुक्ता छाया तरवेष्टकालिका भवति ॥ १८॥
अत्रोद्देशकः ।
द्वादशाङ्गलशङ्कोर्छदलच्छायाङ्गुलद्वयी ।
दशानां घटिकानां मा का नृिशन्नाडिकं दिनम् ॥ १९ ॥
पदच्छयालक्षण पुरुषस्य पादप्रमाणस्य परिभाषासूत्रम्—-
पुरुषोन्नतिसप्तांशस्तत्पुरुषार्द्धस्तु देयं स्यात् ।
यद्येवं चेत्पुरुषः स भाग्यवानाङ्गिभा स्पष्ट ॥ २० ॥
आरूढच्छायायाः सङ्ख्यानयनसूत्रम्--
नृच्छायातिशङ्कभित्तिस्तम्भान्तरोनितो । भक्तः ।
नृच्छाययैव लब्धं शङ्कोभित्याश्रितच्छाया ॥ २१ ॥
अत्रोद्देशकः ।
विंशातिहसः स्तम्भो भित्तिस्तम्भान्तरं करा अर्थौ ।
पुरुषच्छाया द्विना भित्तिगता स्तम्भभा कि स्यात् ।। २२ ॥
1 Not found in any of the MSS.
छायाव्यवहारः 155
स्तम्भप्रमाणं च भित्त्यारूढस्तम्भच्छायासङ्ख्या च ज्ञात्वा
भित्तिस्तम्भान्तरसङ्ख्यानयनसूत्रम्--
पुरुषच्छायानिनं स्तम्भारूढान्तरं तयोर्मध्यम् ।
स्तम्भारूढान्तरहृततदन्तरं पौरुषी त्राया ॥ २३ ॥
अत्रदंशकः ।
विंशतिहस्तः स्तम्भः षोडश भित्याश्रितच्छाया।
द्विगुणा पुरुषच्छाया भित्तिस्तम्भान्तरं किं स्यात् ।। २४ ॥
अपराधस्योदाहरणम् ।
विंशतिहसः स्तम्भः षोडश भियाश्रितच्छाया ।
कियती पुरुषच्छाया भित्तिस्तम्भान्तरं चाष्टौ ॥ २५ ॥
आरूढच्छायायाः सङ्ख्यां च भित्तस्तम्भान्तरभूमसङ्ख्यं च
पुरुषच्छायायाः सङ्ख्यां च ज्ञात्वा स्तम्भप्रमाणसङ्ख्यानयनसूत्रम्--
नृच्छायामारूढा भित्तिस्तम्भान्तरेण सयुक्ता ।
पौरुषभाहृतलब्धं विंदुः प्रमाणं बुधाः स्तम्भे ॥ २६ ॥
अत्रोद्देशकः ।
षोडश भियारूढ़च्छाया द्विगुणेणैव पौरुषी छाया।
स्तम्भोत्सेधः कः स्यादित्तिस्तम्भान्तरं चाष्टौ ॥ २७ ॥
शङ्कप्रमाणशङ्कच्छायामिश्रविभक्तसूत्रम
शङ्कप्रमाणशङ्कच्छायामित्रं तु सैकपौरुष्या ।
भक्तं शङ्कमितिः स्याच्छङ्कच्छाया तदूनामित्रं हि ।। २८ ॥
अत्रोद्देशकः ।
शकुंप्रमाणशTङ्कच्छायामत्रं तु पचाशत् ।
शङ्कत्सेधः कः स्याच्चतुर्गुणा पौरुषी छाया ॥ २९ ॥
156 गणितसारसङ्ग्रहः
शङ्कुच्छायापुरुषच्छायामिश्रविभक्तसूत्रम्--
शङ्कुनरच्छाययुतिर्विभाजिता शङ्कुसैकमानेन ।
लब्धं पुरुषच्छाया शङ्कुच्छाया तदूनमिश्रं स्यात् ॥ ३० ॥
अत्रोद्देशकः ।
शङ्करुत्सेधो दश नृच्छायाशङ्कभामिश्रम् ।
पधोत्तरपञ्चशत्रुच्छाया भवति कियती च ॥ ३१ ॥
स्तम्भस्य अवनतसङ्ख्यानयनसूत्रम्--
छायावर्गाच्छोध्या नरभाकृतिगुणितशङ्कृतिः।
सैकनरच्छायाछतिगुणिता छायाछतेः शाध्या ॥ ३२ ॥
तन्मूलं छायायां शोध्यं नरभानवर्गरूपेण ।
भागं कृत्वा लब्धं स्तम्भस्यावनतिरेव स्यात् ॥ ३३ ॥
अत्रोद्देशकः ।
द्विगुणा पुरुषच्छाया युत्तरदशहस्तशङ्कोशं ।
एकोनत्रिंशत्सा स्तम्भावनतिश्च का तत्र ॥ ३४ ॥
काश्चिद्राजकुमारः प्रासादाभ्यन्तरस्थसन् ।
पूर्वीडे जिज्ञासुर्दिनगतकालं नरच्छायाम् ॥ ३५ ॥
द्वात्रिंशद्धस्तोध्वें जाले प्राग्भित्तिमध्य आयाता ।
रविशं पश्चाद्भित्तौ व्यकत्रिंशत्करोर्वदेशस्था ॥ ३६ ॥
तद्विात्तिद्वयमध्यं चतुरुत्तरवंशतिः करातास्मन् ।
काले दिनगतकालं नृच्छायां गणक विगणय्य ।
कथयच्छायागणिते यद्यस्ति परिश्रमसव चेत् ॥ ३७(१/२) ॥
समचतुरश्रायां दशहस्तघनायां न च्छाया ।
पुरुषोत्सेधद्विगुणा पूर्वाहे प्राक्तठच्छया ॥ ३८(१/२) ॥
1 नृभावर्ग is the reading given in the MSS for नरभान ;
but it is metrically incorrect.
157 छायाव्यवहारः
तस्मिन् काले पश्चात्तठाश्रिता का भवेद्दणक ।
आरूढच्छायाया आनयनं वेत्सि चेत्कथय ॥ ३९ ॥
शङ्कोदपच्छायानयनसूत्रम्-
शङ्कनितदीपोन्नतिराप्त शङ्कप्रमाणेन ।
तळब्धहृतं शक्रः प्रदीपशङन्तरं छाया ॥ १०३ ॥
अत्रोद्देश्कः.
शङमदीपयोर्मध्यं षण्णवत्यङ्गुलानि हि ।
द्वादशाङ्गलशकस्तु दीपच्छायां वदाशु मे ।
षष्टिदपशिरवोत्सेधो गणितार्णवपारग ।। ४२ ॥
दीपशङन्तरानयनसूत्रम्
शङ्कनितदपोन्नतिराप्ता झप्रामाणेन ।
तछब्धहता शङ्कच्छाया शङ्प्रदीपमध्यं स्यात् ॥ ४३ ॥
अत्रोद्देशकः ।
शङ्कच्छायाङ्गलान्यष्टौ षष्टिदपशिरवोदयः ।
शङदीपान्तरं ब्रूहि गणितार्णवपारग ॥ ४४ ॥
पोन्नतिसङ्ख्यानयनसूत्रम् –
शच्छायाभक्तं प्रदीपशङ्कन्तरं सेकम् ।
शकुंप्रमाणगुणितं लब्धं दीपोन्नतिर्भवति ।। ४५ ॥ ।
अत्रोद्देशकः ।
शङ्कुच्छाया द्विनयैव द्विशतं शङ्कदीपयोः ।
अन्तरं पङ्गलान्यत्र का दीपस्य समुन्नतिः ।। ४६ ।।
158 गणितसारसङ्ग्रहः
शङ्कुप्रमाणमत्रापि द्वादशाङ्गुलकं गते ।
ज्ञात्वोदाहरणे सम्याग्विद्यात्सूत्रार्थपद्धतिम् ॥ ४७ ॥
पुरुषस्य पादच्छायां च तत्पादप्रमाणेन वृक्षच्छायां च ज्ञात्वा वृक्षोन्नतेः
सङ्ख्यानयनस्य च, वृक्षोन्नतिसङ्ख्यां च पुरुषस्य पादच्छायायाः
सङ्ख्यां च ज्ञात्वा तत्पादप्रमाणेनैव वृक्षच्छायायाः सङ्ख्यानयनस्य च सूत्रम्--
स्वच्छायया भक्तनिजेष्टवृक्षच्छाया पुनस्सप्तभिराहता सा ।
वृक्षोन्नतिः साद्रिहृता स्वपादच्छायाहता स्याद्द्रुमभैव नूनम ॥ ४८ ।।
अत्रोद्देशकः ।
आत्मच्छया चतुःपादा वृक्षच्छाया शतं पदाम् ।
वृक्षोच्छ्रायः को भवेत्स्वपादमानेन तं वद ॥ ४९ ॥
वृक्षच्छायायाः सङ्ख्यानयनोदाहरणम् ।
आत्मच्छाया चतुःपादा पञ्चसप्ततिभिर्युतम् ।
शतं वृक्षोन्नतिर्वृक्षच्छाया स्यात्कियती तदा ॥ ५० ॥
पुरतो योजनान्यष्टौ गत्वा शैलो दशोदयः ।
स्थितः पुरे च गत्वान्यो योजनाशीतितस्ततः ॥ ५१ ॥
तदग्रस्थाः प्रदृश्यन्ते दीपा रात्रौ पुरे स्थितैः।
पुरमध्यस्थशैलस्यच्छाया पूर्वागमूलयुक् ।
अस्य शैलस्य वेधः को गणकाशु प्रकथ्यताम् ॥ ५२(१/२) ॥
इति सारसङ्ग्रहे गणितशास्त्रे महावीराचार्यस्य कृतौ छायाव्यवहारो नाम अष्टमः समाप्तः ।
- समाप्तोयं सारसङ्ग्रहः ॥
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GAŅIITA-SĀRA-SAŃGRAHA.
ENGLISH TRANSLATION.
CHAPTER I.
ON TERMINOLOGY
Salutation and Benediction.
1. Salutation to Mahàvìra, the Lord of the Jinas, the protector (of the faithful), whose four[१] infinite attributes, worthy to be esteemed in (all) the three worlds, are unsurpassable (in excellence).
2. I bow to that highly glorious Lord of the Jinas, by whom, as forming the shining lamp of the knowledge of numbers, the whole of the universe has been made to shine.
3. That blessed Amoghavarșa (i.e., one who showers down truly useful rain), who (ever) wishes to do good to those whom he loves, and by whom the whole body of animals and vegetables, having been freed from (the effects of) pests and drought, has been made to feel delighted:
4. He, in whose mental operations, conceived as fire, the enemies in the form of sins have all been turned into the condition of ashes, and who in consequence has become one whose anger is not futile:
5. He, who, having brought all the world under his control and being himself independent, has not been overcome by (any) opponents, and is therefore an absolute lord (like) a new God of Love:
6. He, to whom the work (of service) is rendered by a circle of kings, who have been overpowered by the progress of (his) heroism, and who, being Cakrikãbhñjana by name, is in reality a cakrikãbhñjana (i.e., the destroyer of the cycle of recurring re-births):
7. He, who, being the receptacle of the (numerous) rivers of learning, is characterised by the adamantine bank of propriety and holds the gems (of Jainism) within, and (so) is appropriately famous as the great ocean of moral excellence:
8. May (his rule)—the rule of that sovereign lord who has destroyed in philosophical controversy the position of single conclusions and propounds the logic of the syādvāda[२]—(may the rule) of that Nŗpatuńga prosper!
An Appreciation of the Science of Calculation.
9. In all those transactions which relate to worldly, Vedic or (other) similarly religious affairs, calculation is of use.
10. In the science of love, in the science of wealth, in music and in the drama, in the art of cooking, and similarly in medicine and in things like the knowledge of architecture:
11. In prosody, in poetics and poetry, in logic and grammar and such other things, and in relation to all that constitutes the peculiar value of (all) the (various) arts: the science of computation is held in high esteem.
12. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and the conjunction of planets, and in connection with the tripraśna[३] and the course of the moon—indeed in all these (connections) it is utilised.
13-14. The number, the diameter and the perimeter of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the (earthly) world, of the interspace (between the worlds), of the world of light, and of the world of the gods; (as also the dimensions of those belonging) to the dwellers in hell: and (other) miscellaneous measurements of all sorts -- all these are made out by means of computation.
15. The configuration of living being therein, the length of their lives, their eight attributes and other similar things, their progress and other such things, their staying together and such other things -- all these are dependent upon computation (for their due measurement and comprehension)
16. What is the good of saying much in vain ? Whatever 'there is' in all the three worlds, which are possessed of moving and non-moving beings -- all that indeed cannot exist as apart from measurement.
17-19. With the help of the accomplished holy sages who are worthy to be worshipped by the lords of the world, and of their disciples and disciples' disciples, who constitute the well-known jointed series of preceptors, I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are (picked up) from the sea, gold is from the stony rock and the pearl from the oyster shell; and give out, according to the power of my intelligence, the Sārasańgraha, a small work on arithmetic, which is (however) not small in value.
20-23. Accordingly, from this ocean of Sārasańgraha, which is filled with the water of terminology and has the (eight) arithmetical operations for its bank; which (again) is full of the bold rolling fish represented by the operations relating to fractions and is characterised by the great crocodile represented by the chapter of miscellaneous examples ; which (again) is possessed of the waves represented by the chapter on the rule-of-three, and is variegated in splendour through the lustre of the gems represented by the excellent language relating to the chapter on mixed problems; and which (again) possesses the extensive bottom represented by the chapter on area-problems, and has the sands represented by the chapter on the cubic contents of excavations ; and wherein (finally) shines forth the advancing tide represented by the chapter onshadows, which is related to the department of practical calculation in astronomy--(from this ocean) arithmeticians possessing the necessary qualifications in abundance will, through the instrumentality of calculation, obtain such pure gems as they desire.
24. For the reason that it is not possible to know without (proper) terminology the import of anything, at the (very) commencement of this science the required terminology is mentioned.
Terminology relating to (the measurement of) Space.
25–27. That infinitely minute (quantity of) matter, which is not destroyed by water, by fire and by other such things, is called a paramāņu. An endless number of them makes an anu which is the first (measure) here. The trasarēņu which is derived therefrom, the ratharēņu, thence (derived), the hair-measure, the louse-measure, the sesamum-measure, which (last) is the same as the mustard-measure, then the barley-measure and (then) the ańgula are (all) -- in the case of (all) those who are born in the worlds of enjoyment and the worlds of work, which are (all) differentiated as superior, middling and inferior -- eight-fold (as measured in relation to what immediately precedes each of them), in the order (in which they are mentioned). This ańgula is known as vyavahārāńgula.
28. Those, who are acquainted with the process of measurement, say that five-hundred of this (vyavahārāńgula) constitutes (another ańgula known as) pramāņa. The finger measure of men now existing forms their own ańgula.
29. They hold that in the established usage of tho world the ańgula is of three kinds, vyavahāra and pramāņa constituting two (of them), and (then there being) one's own ańgula'; and six ańgulas make the foot-measure as moasured across.
30. Two (such) feet make a vitasi; and twice that is a hasta. Four hasta make a daņda, and two thousands of that make a krōša.
31. Those who are well versed in the measurement of space (or surface-area) say that four krōšas form a yōjana. After this, I mention in due order the terminology relating to (the measurement of) time.
Terminology relating to (the measurement of) Time
32. The time in which an atom (moving) goes beyond another atom (immediately next to it) is a samaya; innumerable samayas make an āvali.
33. A measured number of āvalis makes an ucchvāsas; seven ucchvāsas make one stōka; even stōkas make one lava, and with thirty-eight and a half of this the ghațī is formed.
34. Two ghațīs make one muhūrta ; thirty muhūrtas make one day; fifteen days make one pakşa ; and two pakşasare taken to be a month.
35. Two months make one ŗtu ; three of these are understood to make one ayana; two of these form one year. Next, I give the grain-measure.
Terminology relating to (the measurement of ) Grain.
38. Know that four șōdaśikas form here one kuḍaha ; four kuḍahas one prastha ; and four prasthas one āḍhaka.
37. Four āḍhaka make one drōņa, and four times one drōņa make one mānī ; four mānīs make one khārī; five khārīs make one pravartikā.
38. Four times that same (pravartikā) is a vāha; five pravartikās make one kumbha. After this the terminology relating to the measurement of gold is described.
Terminology relating to (the measurement of) Gold.
89. Four gaņḍakas make one guñjā ; five guñjās make one paņa, and eight of this (paņa) make one dharaņa; two dharaņas make one karșa, and four karșas make one pala.
Terminology relating to (the measurement of) Silver
40. Two grains make one guñjā ; two guñjās make one māșa ; sixteen māșas are said here to make one dharaņa.
41. Two and a half of that dharaņa make one karsa; four purānas (or karsa) make one pala--so say persons well versed in calculation in respect of the measurement of silver according to the
standard current m Magadha.Terminology relating to (the measurement of) Other Metals.
42. What is known as a kalā' consists of four pādus; six and a quarter kalās make one yava; four yavas make one aṃśa ; aṃśas make one bhāga
43. Six bhāgas make one draksūņa ; twice that (draksūņa) is one dīnāra; two dīnāras make one satēra. Thus say the learned men in regard to the (measurement of other) metals.
44. Twelve and a half palas make one prastha; two hundred palas make one tulā ; ten tulās make one bhāra. Thus say those who are clever in calculation.
45. In this (matter of measurement) twenty pairs of cloths, of jewels or of canes (are called) a kōtikā. Next I give the names of the (principal) operations (in arithmetic).
Names of the operations in Arithmetic.
46. The first among these (operations) is guņakāra (multiplication), and it is also (called) pratyutpanna; the second is what is known as bhāyahāra(division ); and kŗti (squaring) is said to be the third.
47. The fourth, as a matter of course, is varga-mūla (square root), and the fifth is said to be ghaņa(cubing) ; then gharamūla (cube root) is the sixth, and the seventh is known as citi (summation).
48. This is also spoken of as saņkalita. Then the eighth is vyutkalita (the subtraction of a part of a series, taken from the beginning, from the whole series), and this is also spoken of as śēșa. All these eight (operations) appertain to fractions also.
General rules in regard to zero and positive and negative quantities.
49. A number multiplied by zero is zero, and that (number) remains unchanged when it is divided by, *[४] combined with (or) diminished by zero. Multiplication and other operations in relation to zero (give rise to) zero and in the operation of addition,the zero becomes the same as what is added to it.
50. In multiplying as well as dividing two negative (or) two positive (quantities, one by the other), the result is a positive (quantity). But it is a negative quantity in relation to two (quantities), one (of which is) positive and the other negative. In adding a positive and a negative (quantity, the result) is (their) difference.
51. The addition of two negative (quantities or) of two positive (quantities gives rise to) a negative or positive (quantity) in order. A positive (quantity) which has to be subtracted from a (given ) number becomes negative, and a negative (quantity) which has to be (so) subtracted becomes positive.
52. The square of a positive as well as of a negative (quantity) is positive; and the square roots of those (square quantities) are positive and negative in order. As in the nature of things a negative (quantity) is not a square (quantity), it has therefore no square root.
58-62. [These stanzas give certain names of certain things, which names are frequently used to denote figures and numbers in arithmetical notation. They are not therefore translated here ; but the reader is referred to the appendix where in an alphabetical list of such of these names as occur in this work is given with their ordinary and numerical meanings.]
The names of Notational Places.
68. The first place is what is known as ēka(unit); the second place is named dașa(ten); the third they call as șata(hundred), while the fourth is sahasra(thousand)
64. The fifth is daśa-sahasra(ten-thousand) and the sixth is no other than lakșa(lakh). The seventh is daśa-lakșa (ten-lakh) and the eighth is said to be kōti(crore) 65. The ninth is daśa-kōti (ten-crore) and the tenth is śata-kōti (hundred-crore). The (place) characterised by eleven is arbuda and the twelfth (place) is nyarbuda.
66. The thirteenth place is kharva and the fourteenth is mahā-karva. Similarly the fifteenth is padma and the sixteenth mahā-padma
67. Again the seventeenth is kșōni, the eighteenth mahā-kșōni. The nineteenth place is śańkha and the twentieth is mahā-śańkha.
68. The twenty-first place is kșityā, the twenty-second mahā-kșityā. Then the twenty-third is kșōbha and tho twenty-fourth mahā-kșōbha
69. By means of the (following) eight qualifies, viz., quick method in working, forethought as to whether a desirable result may be arrived at, or as to whether an undesirable result will be produced, freedom from dullness, correct comprehension, power of retention, and the devising of now means in working, along with getting at those numbers which make (unknown) quantities known --(by means of these qualities) an arithmetician is to be known as such.
70. Great sages have briefly stated the terminology thus. What has to be further said (about it) in detail must be learnt from (a study of) the science (itself) .
Thus ends the chapter on Terminology in Sārasańgraha, which is a work on arithmetic by Mahāvirācārya.
CHAPTER II.
ARITHMETICAL OPERATIONS
The First Subject of Treatment.
Hereafter We shall expound the first subject of treatment, which is named Parikarman.
Multiplication.
The rule of work in relation to the operation of multiplication, which is the first (among the parikarman operations), is as follows:--
1. After placing (the multiplicand and the multiplier one below the other) in the manner of the hinges of a door, the multiplicand should be multiplied by the multiplier, in accordance with (either of) the two methods of normal (or) reverse working, by adopting the process of (i) dividing the multiplicand and multiplying the multiplier by a factor of the multiplicand, (ii) of dividing the multiplier and multiplying the multiplicand
1. Symbolically expressed, this rule works out thus :-- In multiplying ab by cd, the product is (i) or (ii) ; or (iii) . Obviously the object of the first two devices here is to facilitate working through the choice of suitable factors.
The anulōma or normal method of working is the one that is generally followed. The vilōma or the reverse method of working is as follows:--
- To multiply 1998 by 27 :
1998
272 x 1 == 2 x 9 == 2 x 9 == 2 x 8 == 7 x 1 == 7 x 9 == 7 x 9 == 7 x 8 == 2 1 8 1 8 1 6 7 6 3 6 3 5 6 5 3 9 4 6 by a factor of the multiplion; or (iii) of using them (in the multiplication) as they are (in themselves)
Examples in illustration hereof.
2. Lotuses were given away (in offering)-- eight of them to each Jina temple. How many (were given away) to 114 temples ?
3. Nine padmarāga gems are seen to have been offered in worship in a single Jina temple. How many will they be (at that same rate) in relation to 288 temples ?
4. One hundred and thirty-nine pușyarāga gems have to be offered in worship in a single Jina temple. Say, how many gems (have to be so offered) in 109 temples.
5. Twenty-seven lotuses have been given away in offering to a single Jina temple. Say, how many they are(which have been at that rate given away) to 1998 (temples).
6. (At the rate of) 108 golden lotuses to each temple, how many will they be in relation to 85697481 (temples) ?
7. If (the number represented by ) the group (of figures) consisting of 1, 8, 6, 4, 9, 9, 7 and 2 (in order from the units' place upwards) is written down and multiplied by 441, what is the value of the (resulting) quantity ?
8. In this (problem), write down (the number represented by) the group (of figures) consisting of 1, 4, 4. 1, 3 and 5 (in order from the units' place upwards), and multiply it by 81; and then tell me the (resulting) number.
9. In this (problem), write down the number 157683 and multiply it by 9, and then tell me, friend, the value of the (resulting) quantity.
10. In this (problem), 12345679 multiplied by 9 is to be written down; this (product) has been declared by the holy preceptor Mahāvira to constitute the necklace of Narapāla.
4. Here, 189 is mentioned in the original as 40 + 100 -- 1.
5. Here 1998 is mentioned in the original as 1068 + 900.
10. Here as well as in the following stanzas, certain numbers are said to constitute different kinds of necklaces on account of the symmetrical arrangement of similar figures which is readily noticeable in relation to them. 11. Six 3's, five 6's, and (one)7, which is at the end, are put down (in the descending order down to the units' place); and this (number) multiplied by 38 has (also) been declared to be a (kind of) necklace.12. In this (problem), write down 3, 4, 1, 7, 8, 2, 4, and 1 (in order from the units' place upwards), and multiply (the resulting number) by 7; and then say that it is the necklace of precious gems.
18. Write down (the number) 142857143, and multiply it by 7; and then say that it is the royal necklace.
14. Similarly 37087037 is multiplied by 3. Find out (the result) obtained by multiplying (this product) again to get such multiples (thereof) as have one as the first and nine as the last (of the multipliers in order) .
15. The (figures) 7, 0, 2, 2, 5 and 1 are put dowm (in order from the units' place upwards) ; and then this (number) which is to be multiplied by 73, should (also) be called a necklace (when so multiplied)
16. Write down (the number represented by) the group (of figures) consisting of 4, 4, 1, 2, 6 and 2 (in order from the units' place upwards); and when (this is) multiplied by 64, you, who know arithmetic, tell me what the (resulting) number is
17. In this (problem) put down in order (from the units' place upwards) 1, 1, 0, 1,1,, 1 and 1, which (figures so placed) give the measure of a (particular) number; and (then) if this (number) is multiplied by 91, there results that necklace which is worthy of a prince.
Thus ends multiplication, the first of the operations known as Parikarman.
11. Tho multiplicand here is 333333666667.
14. This problem reduces itself to this multiply 37037037 x 3 by 1, 2,3,4,5,6,7,8, and 9 in order.Division.
The rule of work in relation to the operation of division, which is the second (among the parikarman operations), is as follows:--
18. Put down the dividend and divide it, in accordance with the process of removing common factors, by the divisor, which is placed below that (dividend), and then give out the resulting (quotient).
Or:
19. The dividend should be divided in the reverse way (i.e., from left to right) by the divisor placed below, after performing in relation to (both of) them the operation of removing the common factors, if that be possible.Examples in illustration thereof.
20. Dinārās (amounting to) 8192 have been divided between 64 men. What is the share of one man ?
21. Tell me the share of one person when 2701 pieces of gold are divided among 37 persons.
22. Dinārās (amounting to) 10349 have been divided between 79 person. What is it that is obtained by one (person) ?
23. Gold pieces (amounting to) 14141 are given to 79 temples. What is the momy (givon} to each (temple)?
24. Jambū fruits (amounting to) 31317 have been divided between 89 persons. Tell me the share of each.
25. Jambū fruits (amounting to) 31313 have been divided between 181 persons. Give out the share of each.
26. Gems amounting to 36261 (in number) are given to 9 persons (equally). What does one man obtain here ?
27. 0 friend, gold pieces (to the value of the number wherein the figures in order from the units' place upwards are) such as
- 20. Here, 8192 is mentioned in the original as 8000+92 +100.
- 22. In the original, 10348 is given as 10000+300+ 72
- 23. Here, 14141 is given as 10000+(40+4000+1+100).
- 24. Here, 31317 is given as 17 + 300+31000.
- 25. Here, 31313 is given as 13+300+31000.
- 26. Here, 36261 is given as 30000+1+(60+200+6000)
- 27. Here, the given divided is obviously 12345654321.
begin with 1 and end with 6, and then become gradually diminished, are divided between 441 persons. What is the share of each?
28. Gems (amounting to) 28483 (in number) are given (in offering) to 13 Jina temples. Give out the share of each (temple).
Thus ends division, the second of the operations known as Parikarman.
Squaring.
The rule of work in relation to the operation of squaring, which is the third (among the Parikarman operations), is as follows:--
29. The multiplication of two equal quantities: or the multiplication of the two quantities obtained (from the given quantity) by the subtraction (therefrom), and the addition (thereunto), of any chosen quantity, together with the addition of the square of that chosen quantity (to that product): or the sum of a series in arithmetical progression, of which 1 is the first term, 2 is the common difference, and the number of terms wherein is that (of which the square is) required : gives rise to the (required) square.
30. The square of numbers consisting of two or more places is (equal to) the sum of the squares of all the numbers (in all the places) combined with twice the product of those (numbers) taken (two at a time) in order
28. Here, 28483 is given as 83 + 400 + (4000 x 7).
25. The rule given herein, expressed algebraically, comes out thus :
(i) a x a =a2; (ii) (a+x)(a-x) + x2 = a2; (īi) 1+3+5+7+ . . . up to a terms == a2.
30. The word translated by place here is स्थान; it obviously means a a place in notation. Here, as a commentary interprets it, it may also denote the component parts of a sum, as each such part has a place in the sum. According to both these interpretations the rule works out correctly.
For instance,
Similarly . 31. Get the square of the last figure (in the number, the order of counting the figures being from the right to the left,) and then multiply this last (figure), after it is doubled and pushed on (to the right by on notational place), by (the figures found in) the remaining places. Each of the remaining figures (in the number) is to be pushed on (by one place) and then dealt with similarly. This is the method of squaring.Examples in illustration thereof.
32. Give out the squares of (the numbers from) 1 to 9, of 15, 16, 25, 36 and 75.
33. What will 338, 4661 and 256 become when squared ?
34. O arithmetician, give out, if you know, the squares of 65536, 12345 and 3333.
35. (Each of the numbers) 6387, and then 7435, and (then) 1022 is squared. O clever arithmetician, tell me, after multiplying well, the value of those three (squares).
Thus ends squaring, the third of the operations known as Parikarman
31. The pushing on to she right mentioned herein will become clear from the following worked out examples:--
"To square 131. To square 132. To square 555. 12== 2 x 1 x 3== 2 x 1 x 1== 32== 2 x 3 x 1== 12== 1 6 2 9 6 1 (1) 1 7 1 6 1 12== 2 x 1 x 3== 2 x 1 x 2== 32== 2 x 3 x 2== 22== 1 6 4 9 12 4 (1) (1) 1 7 4 2 4 52== 2 x 5 x 5== 2 x 5 x 5== 52== 2 x 5 x 5== 52== 25 50 50 25 50 25 (5) (8) (5) (2) 30 8 0 2 5 33. Here, 4681 is given as 4000+61+ 600.
35. Here, 7135 is given as 185+ (1000x7).
Square Root.
The rule of work in relation to the operation of (extracting) the square root, which is the fourth (of the parikarman operations) is as follows:--
36. From the (number represented by the figures up to the) last odd place (of notation counted from the right), subtract the (highest possible) square number; then multiply the root (of this number) by two, and divide with this (product the number represented by taking into position the figure belonging to) the (next) even place; and then the square of the quotient (so obtained) is to be subtracted from the (number represented by taking into position the figure belonging to the next) odd place. (If it is so continued till the end), the half of the (last) doubled quantity (comes to be ) the resulting square root.
Examples in illustration thereof.
37. O, friend, tell me quickly the roots of the squares of the numbers from 1 to 9, and of 256 and 576.
38. Find out the square root of 6561 and of 65536.
39. What are the square roots of 4294967296 and 622521 ?
40. What are the square roots of 63664441 and 1771561 ?
41. Tell me, friend, after considering well, the square roots of 1296 and 625.
- 36. To illustrate the rule, the following example is worked out below:--
To extract the square root of 65536
- 6 | 55 | 36
- 22 = 4
- 2 x 2 = 4)25(5
- 20
- 55
- 52 = 25
- 25 x 2 = 50)303(6
- 300
- 36
- 300
- 62 = 36
- 256 x 2 = 512) 0 (0
- 0
- Square root required ==
42. Tell me, O leading arithmetician, the square roots of 110889, 12321, and 844561.
Thus ends square root, the fourth of the operations known as Parikarman.
Cubing.
Tho rule of work in relation to the operation of cubing, which is the fifth (of the parikarman operations), is as follows:--
43. The product of (any) three equal quantities: or the product obtained by the multiplication of any (given) quantity by that given quantity) as diminished by a chosen quantity and (them again) by that (given quantity) as increased by the (same) chosen quantity, when combined with the square of the chosen quantity as multiplied by the least (of the above three quantities) and (combined) also with the cube of the chosen quantity: gives rise to a cubic quantity.
44. Or, the summing up of a series in arithmetical progression, of which the first term is the quantity (the cube whereof is) required, the common difference is twice this quantity, and the number of terms is (equal to) this (same given) quantity, (gives rise to the cube of the given quantity). Or, the square of the quantity (the cube whereof is required), when combined with the product (obtained by the multiplication) of this given quantity diminished by one by the sum of a series in arithmetical progression in which the first term is one, the common difference is two and the number of terms is (equal to the given quantity, gives rise to the cube of the given quantity).
- 43. Symbolically expressed, this rule works out thus:
(i) . - 44. Algebraically, this rule means--
(i) to a terms.
(ii) to a terms).
- 43. Symbolically expressed, this rule works out thus:
45. In an arithmetically progressive series, wherein one is the first term as well as the common difference, and the number of terms is (equal to) the given number, multiply the preceding terms by the immediately following ones. The sum of the products (so obtained), when multiplied by three and combined with the last term (in the above series in arithmetical progression), becomes the cube (of the given quantity).
46. (In a given quantity), the squares of (the number represented by the figures in) the last place as also (by those in) the other (remaining places) are taken; and each of these (squares) is multiplied by the number of the other place and also by three; the sum of the two (quantities resulting thus), when combined again with the cubes of the numbers corresponding to all the (optional) places, (gives rise to) the cube (of the given quantity).
47. Or, the cube of the last figure (in the number counted from right to left is to be obtained); and thrice the square (of that last figure) is to be pushed on (to the right by one notational place) and multiplied by (the number represented by the figures found in) the remaining (places); then the square of this (number represented by the figures found in the) remaining (places) is to be pushed on (as above) and multiplied by thrice the last figure (above-mentioned). These (three quantities) are then to be placed in position (and then summed up). Such is the rule (to be carried out) here.
Examples an illustration thereof.
48. Give out the cubes of the numbers from 1 to 9 and of 15, 25, 83, 77 and 96. 49. Give out the cubes of 101, 172, 516, 717 and 1344.
- 45. .
- 46. . To make the rule general and applicable to numbers having more than two places, it is clearly implied here that ; and it is obvious that any number may be represented as the sum of two other suitable numbers.
- 47. The pushing on of a figure here referred to is similar to what is exhibited in the note under stanza 31 in this chapter.
50. The number 21B is cubed; and twice, thrice, four times and five times that (number are) also (cubed; find out the corre sponding quantitics)
51. It is seen that 168 multiplied by all the numbers from 1 to 8 is related (as base) to the required cubes. Give out those cubes quickly
52. O you, who have seen the other shore of the deep and excellent ocean of the practice of (arithmetical) operations, write down the figures 4,0,6,0,5, and 9 in order (from right to left), and work out the cube of the number (represented by those figures), and mention the result at once.
Thus ends cubing, the fifth of the operations known as Parikarman.
Cube Root
The rule of work in relation to the operation of extracting the cube root, which is the sixth (among the parikarman operations), is as follows:-
53. From (the number represented by the figures up to) the last ghana place, subtract the (highest possible) cube; then divide the (number represented by the next) bhājya place (after it is taken into position) by three times the square of the root (of that cube); then subtract from the (number represented by the next) ōdhya place (after it is taken into position) the square of the (above) quotient as multiplied by three and by the already mentioned (root of the highest possible cube); and then (subtract) from
53 and 54. The figures in any given number, the cube-root whereof is required, are conceived in these rules to be divided into groups, each of which consists as far as possible of three figures, named, in the order from right to left, as ghanā or that which is cubic, that is, from which the cube is to be subtracted, as śōdhya or that which is to be subtracted from, and as bhājya or that which is to be divided. The bhājya and śōdhya are also known as aghana or non-cubic. The last group on the left need not always consist of all these three figures ; it may the (number represented by the figure in the next) ghana place (after it is takon into position) the cube (of this same quotient).
54. One (figure in the various groups of three figures) is cubic: two are non-cubic. Divide (the non-cubic figure) by three times the square of the cube root. From the (next) non-cubic (figure) subtract the square of the quotient (obtained as above and) multiplied by three times the previously mentioned (cube-root of the highest cube that can be subtracted from the previous cubic figure) and (then subtract) tho cube of the (above, quotient (from the next cubic figure as taken into position). With the help of the cuberoot-figures (s0) obtained (and taken into position, the procedure is) as beforo.
Examples in thstation thereof:
55. What is the cube root of the numbers beginning with 1 and ending with 9, all cubed; and of 4913; and of 1860867?
56. Extract the cube root of 13824, 36926037 and 618470908.
consist of one or two or three figures, as the cause may be . The rule mentioned will be clear from the following worked out example.
- To extract the cube root of 17808776:--
- ś gh. bh. ś gh. bh. ś gh.
- 7 7 | 3 0 8 | 7 7 6
- gh. ... ... 43 == 6 4
- bh. ... ... 42 x 3 == 48)133(2
- 96
- 370
- s. ... ... 22 x 3 x 4 == 48
- 3228
- 3228
- gh. ... ... 23 == 8
- bh. ... ... 422 x 3 == 5292)32207(6
- 31752
- 4557
- 31752
- ś. ... ... 62 x 3 x 42 == 4536
- 216
- gh. ... ... 63 == 216
- Cube root == 426
The rule does not state what figures constitute the cube root; but it is meant that the cube root is the number made up of the figures which are cubed in this operation, written down in the order from above from left to right 57. Give the cube roots of 270087225844 and 76332940488. 58. Give the cube roots of 77308776 and also of 269917119. 59. Give the cube roots of 2427715584 and of 1626379776. 60. O arithmetician, who are clever in calculation, give out after examination the root of 859011369945948864, which is a cubic quantity.
Thus ends cube root, the sixth of the operations known as Parikarman.
{{rule|5em}
Summation.
The rule of work in relation to the operation of summation of series, which is the seventh (among the perikunnan operations), is as follows:-
61. The number of terms in the series is (first) diminished by one and (is then) halved and multiplied by the common difference; this when combined with the first term in the series and (then) multiplied by the number of terms (therein) becomes the sum of all (the terms in the series in arithmetical progression)
The rule for obtaining the sum of tho series in another manner:- 62. The number of terms (in the series) as diminished by one and (then) multiplied by the common difference is combined with twice the first term in the series; and when this (combined sum) is multiplied by the number of terms (in the series) and is (then) divided by two, it becomes the sum of the series in all cases,
- 61. This rule comes out thus when expressed algebraically:-
- , where a is the first term, b the common difference,n the number of terms and S the sum of the whoele series.
- 62. Similarly,
Tho rule for finding out the ādidhana, the uttaradhanaand the sarvadhana :--
63. Tho ādidhana is the first term multiplied by the number of terms (in the series). The uttaradhana is (the product of) the number of terms multiplied by the common difference (and again) multiplied by the half of the number of terms less by one. The sum of these two (gives) the sarvadhana i.e., the sum of all the terms in the series; and (this sum will be the same as that of a series which is) characterised by a negative common difference, when (the order of the terms in the series is reversed, so that) the last term is made to be the first term.
The rule for finding the 'antyadhana, the madhyadhana and the sarvadhana:-
64. The number of terms (in the series) lessened by one and multiplied by the common differece and (then) combined with the first term (gives) the antyodhana. Half of the sum of
63-34. In these rules, each of the terms in an arithmetically progressive series is supposed to be obtained by adding to the first term thereof a multiple of the common difference, the nature of this multiple being determined by the position which any specified term holds in the series. According to this conception we have to find in every term of the series the first term along with a multiple of the common difference. The sum of all such first terms so found is what is here called the ādidhana; the sum of all such multiples of the common difference constitutes the uttaradhana; and the sarvadhana which is obtained by adding these two sums is of course the sum of the whole series. The expression antyadhana denotes the value of the last term in an arithmetically progressive series. And madhyadhana means the value of the middle term which value, however corresponds to the arithmetical mean of the first and the last terms in the series, so that when there are 2n + 1 terms in the series, the value of the (n + 1)th term is the madhyadhana, but when there are 2n terms in the series the arithmetical mean of the value of the nth term and of that of the (n+1)th term becomes the madhyadhana. Accordingly we have
- (1) Ādidhana .
- (2) Uttaradhana == .
- (3) Antyadhana.
- (4) Madhyadhana ==
- (5) Sarvadhana === ;
- or
this (antyadhana) and the first term (gives) the madhyadhana. The product of this (madhyadhana) and the number of terms (in the series gives) the desired sum of all the terms therein.
Examples in illustration thereof
65. (Each of) ten merchants gives away money (in an arithmetically progressive series) as a religious offering, the first terms of the (ten) series being from 1 to 10, the common difference (in each of these series ) being of the same value (as the first terms thereof), and the number of forms being 10 (in every one of the series). Calculate the sums of those (series).
66. A certain excellent śrāvaka gave gems in offering to 5 temples (one after another) commencing (the offering ) with 2 (gems), and then increasing (it successively) by 3(gems). O you who know how to calculate, mention what their (total) number is.
67. The first term is 3; the common difference is 8; and the number of terms is 12 . All these three (quantities) are (gradually) increased by 1, until (there are) 7 (series). O arithmetician, give out the sums of all (those series) 68. O you who possess enough strength of arms to cross tho ocean of arithmetic, give out the total value of the offerings made in relation to 1000 cities, commencing (the offering) with 4 and increasing it successively by 8.
The rule for finding out the number of terms (in a series in arithmetical progression) :-- 69. When, to the square root of the quantity obtained by the addition of the square of the difference between twice the first
It is quite obvious that an arithmetically progressive series having a negative common difference becomes changed into one with a positive common difference when the order of the terms is reversed throughout so as to make the last of them become the first.
66. A śrāvaka is a lay follower of tho Jaina religion, who merely hears, i.e. listens to and learns the dharmas Or duties, as opposed to the ascetics who are entitled to teach those religious duties.
69. Algebraically this rule works out thus:--
term and the common difference to 8 times the common difference multiplied by the sum of the series, the common difference is added, and the resulting quantity is halved; and when (again) this is diminished by the first term and then divided by the common difference, we gat the number of terms in the series.
The rule for finding out the number of terms (stated) in another manner:--
70. When, from the square root of (the quantity obtained by) the addition of the square of the difference between twice the frst term and the common difference to 8 times the common difference multiplied by the sum of the series, the kșēpapada is subtracted, and (the resulting quantity) is halved; and (when again this is) divided by the common difference, (we get) the number of terms in the series.
Examples in illustration thereof.
71. The first term is 2, the common difference 8; these two are increased successively by 1 till three (series are so made up). The sums of the three series are 90, 276 and 110 in order. What is the number of terms in each Series ?
72. The first term is 5, the common difference 8, and the sum of the series 333. What is the number of terms ?
The first term (of another series) is 6, the common difference 8, and the sum 420. What is the number of terms?
The rule for finding out the common difference as well as the first term :--
73. The sum (of the series) diminished by the ādidhana,and (then) divided by half (the quantity represented by) the square
70. Kșēpapada is half of the difference between twice the first term and the the common difference. i.e. . It is obvious that this stanza varies the rule mentioned in the previous stanza only to the extent necessitated by the introduction of this kșēpapada therein.
73. For ādidhana and uttaradhana, see note under stanzas 63 and 64 in this chapter. Symbolically expressed this stanza works out thus:--
and
of the number of terms as lessened by the number of terms, (gives) the common difference. The sum (of the series) diminished by uttaradhana and (then) divided by the number of terms, (gives) the first term of the series.
The rule for finding out the first ten as well as the common difference :--
74. The sum of the semics divided by the number of terms (therein), when diminished by the product of the common difference multiplied by the half of the number of terms less by one gives the first term of the series. The common difference is (obtained when) the sum, divided by the number of terms and then diminished by the first term, is divided by the half of the number of terms less by one.
Two rules for finding out, in another way, the common difference and the first term :--
75. Understand that the common difference is (obtained, when) the sum of the series, multiplied by two and divided by the number of terms (therein), is diminished by twice the first term, and is (then) divided by the number of terms lossened by one.
76. Twice the sum of the series divided by the number of terms therein, and (then) diminished by the number of terms as lessened by one and multiplied by the common difference, when divided by two, (gives) the first term of the series.
Examples in the illustration thereof.
77. The first term is 9; the number of terms is 7; and the sum of the series is 105. Of what value is the common difference?
- 74. Algebraically, ; and
- 75. Symbolically,
- 76. Algebraically,
The common difference (in respect of another series) is 5, the number of terms is 8, and the sum is 156. Tell me the first term.
The rule for finding out how (when the sum is given) the first term, the common difference, and the number of terms may, as desired, be arrived at:--
78. When the sum is divided by any chosen number, the divisor becomes the number of terms (in the series) ; when the quotient here is diminished by any number chosen (again), this subtracted number becomes the first term (in the series); and the remainder (got after this subtraction) when divided by the half of the number of terms lessened by one becomes the common difference
Example in illustration thereof.
79. The sum given in this problem is 540. O crest-jewel of arithmeticians, tell me the number of terms, the common difference, and the first term.
Three rule-giving stanzas for splitting up (into the component elements) such a sum of a series (in arithmetical progression) as is combined with the first term, or with the common difference, or with the number of terms, or with all these.
80. O crest-jewel of calculators, understand that the miśradhana diminished by the uttaradhana, and (then) divided by the number of terms to which one has been added, gives rise to the first term.
81. The miśradhana , diminished by the ādidhana, and (then) divided by the (quantity obtained by the) addition of one to the (product of the) number of terms multiplied by the half of the number of terms lessened by one, (gives rise to) the common
78. Symbolically, the problem herein is to find out b, when S is given, and a and m are allowed to be chosen at option. Naturally, there may be in relation to any given value of S any values of b, which depend upon the chosen values of a and n. When the values of a and n are definitely chosen, the rule herein given for finding out b turns out to be the same as that given in stanza 74 above.
80-82. The expression miśradhanameans a mixed sum. It is used here to denote the quantity which may be obtained by adding the first term or the common difference or the number of terms or all three of these to the sum of a difference. (In splitting up the number of terms from the miśradhana), the (required) number of terms (is obtained) in accordance with the rule for obtaining the number of terms, provided that the first term is taken to be increased by one (so as to cause a corresponding increase in all the terms).
82. The miśradhana is diminished by the first term and the number of terms, both (of these) being optionally chosen: (then) that quantity, which is obtained (from this difference) by applying the rule for (splitting up) the uttara-miśradhana happens to be the common difference (required here ). This is the method of work in (splitting up) the all-combined (miśradhana).
Examples in illustration thereof.
83. Forty exceeded by 2, 3, 5 and 10, represents (in order) the ādi-miśradhana and the other (miśradhana). Tell me what (respectively) happens in these cases to be the first term, the common difference, the number of terms and all (these three).
series in arithmetical progression. There are accordingly four different kinds of miśradhana mentioned here; and they are respectively ādi-miśradhana' and uttara-miśradhanas', gaccha-miśradhana and sarva-miśradhana. For ādidhana and uttaradhana sē note under stanzas 63 and 64 in this chapter.
Algebraically, stanza 80 works out thus: , where Sa is the ādi-miśradhana, i.e.,.
- And stanza 81 gives where is the uttara-miśradhana,
- i.e., ; and further points out that the vlaue of n may be found out, when the value of Sn, which, being the gaccha-miśradhana, is equal to , is given, from the fact that, when upto n terms.
- Since, in stanza 82, the choice of a and n are left to our option, the problem of finding out a,n, and b from the given value of S a n b, which, being the sarva-miśradhana, is equal to , resolves itself easily to the finding out of b from any given value of Sb in the manner above explained.
83. The problem expressed in plainer terms is:-- (1) Find out a when , and . (2) Find out b, when and </math>n==5</math>. (3) Find out n when and . And (4) find out a,b, and n when
The rule for finding out, from the known sum, first term, and common difference (of a given series in arithmetical progression), the first term and the common difference (of another series), the optionally chosen sum (whereof) is twice, three times, half, one third, or some such (multiple or fraction of the known sum of the given Ser1es):--
84. Put down in two places (for facility of working) the chosen sum as divided by the known (i.e., the given) sum ; this (quotient) when multiplied by the (known) common difference gives the (required) common difference ; and that (same) quotient when multiplied by the (known) first term gives the (required) first term of (the series of which) the sum is either a multiple or a fraction (of the known sum of the given series).
Eachamples in illustration thereof.
85. Sixty is the (known) first term, and the (known) common difference is twice that, and the number of terms is the same. i.e., 4 (in the given series as well as in all the required series ). Give out the first terms and the common differences of these required (series, the sums whereof are ) represented by that (known sum) as multiplied or divided by the (numbers) beginning with 2.
The rule for finding out, in relation to two (series), the number of terms wherein are optionally chosen, their mutually interchanged first term and common difference, as also their sums which may be equal, or (one of which may be) twice, thrice, half, or one third, or any such (multiple or fraction of the other):-
86. The number of terms (in one series, multiplied by itself as lessened by one and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by
84. Symbolically, , where S1, a1 and b1, are the sum, the first term and the common difference, in order of the series whose sum is chosen. Given the sums of two series, the ratio between the two first terms and that between the two common differences need not always be . The solution here given is hence applicable only to certain particular cases.
86. Algebraically, and , where a, b and n are the first term, the common difference and the number of twice the number of terms in the other series (gives rise to the interchangeable) first term of one (of the series). The square of the (number of terms in the other (series), diminished by that (number of terms) itself, and (then) diminished (again) by the product of two (times the) chosen (ratio) and the number of terms (in the first series gives rise to the interchangeable) difference (of that series).
Examples in illustration thereof.
87. In relation to two men, (whose wealth is measured respectively by the sums of two series in arithmetical progression) having 5 and 8 for the number of terms, the first term and the common difference of both these series being interchangeable (in relation to each other); the sums (of the series) being equal or the sum (of one of them) being twice, thrice, or any such (multiple of that of the other)--O arithmetician give out (the value of these) sums and the interchangeable first term and common difference after calculating (them all) well.
88. In relation to two series (in arithmetical progression), having 12 and 16 for their number of terms, the first term and the common difference are interchangeable. The sums (of the series) are equal, or the sum (of one of them) is twice or any such multiple, or half or any such fraction (of that of the other). You, who are versed in the science of calculation, give out (the value of these sums and the interchangeable first term and common difference).
The rule for finding out the first terms in relation to such (series in arithmetical progression) as are characterised by varying common differences, equal numbers of terms and equal sums :-
89. Of that (series) which has the largest common difference, one is (taken to be) the first term. The difference between this
terms in the first series, n1 the number of terms in the second series, and p the ratio between the two sums: a and b being thus found out, the first term and the common difference of the second series are a and b respectively in value.
89. The solution herein given is only a particular case of the general rule , where a and a1 are the first terms of the series, and largest common difference and (any other) remaining common difference is multiplied by the half of the number of terms lessened by one ; and when this (product) is combined with one (we get,) O friend, the first terms of (the various series having) the remaining (smaller) common differences.
Examples in illustration thereof.
90. Give out quickly, O friend, the first terms of (all the series found in two sets of) such (series) as have equal sums (in relation to each set) and are characterised by 9 as the number of terms in each (series), when those (series belonging to the first and second sets) have (respectively) common differences beginning with 1 and ending with 6 (in one case) and have 1, 3, 5 and 7 as the common differences (in the other case).
The rule for finding ont the common difference in relation to such (series in arithmetical progression) as are characterised by varying first terms, equal numbers of terms and equal sums:--
91. Of that (series) which has the largest first term, one is taken to be the common difference. The difference between this largest first term and (each of the) remaining (smaller) first terms is divided by the half of the number of terms lessened by one; and when this (quotient in each case) is combined with one, (we get) the common differences of (the various series having) the remaining (smaller) first terms.
An example illustration thereof.
92. O arithmetician, who have been the other shore of calculation, give out the common differences of (all) those (series) which are characterised by equal sums and have 1, 3, 5, 7, 9 and 11 for their first terms and 5 for the number of terms in each.
b and b1, their corresponding common differences. It is obvious that in this formula, when b,b1 and n are given, a1 is determined by choosing any value for a; and one is chosen as the value of a in the rule here.
91. The general formula in this case is:-
, wherein also the value of b is taken to be one in the rule given above. The rule for finding out the guņadhana and the sum of a series in geometrical progression :--93. The first term (of a series in geometrical progression), when multiplied by that self-multiplied product of the common ratio in which (product the frequency of the occurrence of the common ratio is) measured by the number of terms (in the series), gives rise to the guņadhana. And it has to be understood that this guņadhana, when diminished by the first term, and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression.
Another rule also for finding out the sum of a series in geometrical progression:--
94. The number of terms in the series is caused to be marked (in a separate column) by zero and by one (respectively) corresponding to the even (value) which is halved and to the uneven (value from which one is subtracted till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again ) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (whereever zero happens to be the denoting item). When (the result
93. The guņadhana of a series of n terms in geometrical progression corresponds in value to the (n + 1)th term thereof, when the series is continued. The value of this guņadhana algebraically stated is r x r x r ..... up to n such factors x a i.e., arn. Compare this with the uttaradhana.
This rule for finding out the sum may be algebraically expressed thus:-- {{space|em|7}, where a is the first term. r is the common ration, and n is the number of terms.
94. This rule differs from the previous one in so far as it gives a new method for finding out rn by using the processes of squaring and ordinary multiplication; and this method will become clear from the following example:--
Let rn be equal to 12 .
is even; it has therefore to be divided by 2, and to be denoted by 0;
is even; it has therefore to be divided by 2, and to be denoted by 0;
is odd; 1 is therefore to be subtracted from it, and it is denoted by 1;
is even; it has therefore to be divided by 2, and to be denoted by 0;
is odd; 1 is therefore to be subtracted from it, and it is denoted by 1;
, which concludes this part of the operation. of) this (operation) is diminished by one, and (is then) multiplied by the first term, and (is then) divided by the common ratio lessened by one. it becomes the sum (of the series).The rule for finding out the last term in a geometrically progressive series as also the sum of that (series):--
95. The antyadhana or the last term of a series in geometrical progression is the guņadhana (of another series) wherein the number of arms is less by one. This (antyadhana), when multiplied by the common ratio, and (then) diminished by the first term, and (then) divided by the common ratio lessened by one, gives rise to the sum (of the series)
An example illustration thereof.
96. Having (first) obtained 2 golden coins (in some city), a man goes on from city to city, earning (everywhere) three times (of what he earned immediately before ). Say how much he will make in the eighth city.
Now, in the representative column of figures so derived and given in the margin--
The table's caption 0 the lowest 1 is multiplied by r, which gives r: since this lowest 1 has 0 0 above it, the r obtained as before is squared, which gives r2 ; since this 0 1 has 1 above it, the r2 now obtained is multiplied by r,which gives r2; 0 since this 1 has 0 above it, this r2 is squared, which gives r2 1 again this 0 has another 0 above it, this r6 is squared, which gives r12
Thus the value of r may be arrived at by using as few times as possible the processes of squaring and simple multiplication. The object of the method is to facilitate the determination of the value of rn; and it is easily seen that the method holds true for all positive and integral values of n.
95. Expressed algebraically, , The antyadhana is the value of the last term in a series in geometrical progression; for the meaning and value of guņadhana, see stanza 93 above in this chapter. The antyadhana of a geometrically progressive series of n terms is arn-1, while the guņadhana of the same series is arn. Similarly the antyadhana of a geometrically progressive serires of n-1 terms is arn-2, while the guņadhana thereof is arn-1. Here it is evident that the antyadhana of the series of n terms is the same as the guņadhana of the series of n-1 terms.
The rule for finding out the first term and the common ratio in relation to a (given) guņadhana:--
97. The guņadhana when divided by the first term becomes equal to the (self-multiplied) product of a certain quantity in which (product) that (quantity) occurs as often as the number of terms (in the series); and this (quantity) is the (required) common ratio. The guņadhana, when divided by that (self-multiplied) product of the common ratio in which (product the frequency of the occurrence of this common ratio) is measured by the number of terms (in the series), gives rise to the first term.
The rule for finding out in relation to a given guņadhana the number of terms (in the corresponding geometrically progressive series):--
98. Divide the guņadhana (of the series) by the first term (thereof). Then divide this (quotient) by the common ratio (time after time) so that there is nothing left (to carry out such a division any further) whatever happens (here) to be the number of vertical strokes. (each representing a single such division), so much is (the value of) the number of terms in relation to the (given) guņadhana.
Examples in illustration thereof.
99. A certain man (in going from city to city) earned money (in a geometrically progressive series) having 5 dīnāras for the first term (thereof) and 2 for the common ratio. He (thus) entered 8 cities. How many are the dīnāras (in) his (possession) ?
100. What is (the value of) tho wealth owned by a merchant (when it is measured by the sum of a geometrically progressive series), the first term whereof is 7, the common ratio 3, and the number of terms (wherein) is 9: and again (when it is measured by the sum of another geometrically progressive series), the first
97 and 98. It is clear that arn, when divided by a gives rn ; and this is divisible by r as many times as n, which is accordingly the measure of the number of terms in the series. Similarly r * r * r ... up to n times gives rn; and the guņadhana i.e., arn divided by rn gives a, which is the required first term of the series. term, the common ratio and the number of terms thereof being 3, 5 and 15 (respectively) ?
The rule for finding out the common ratio and the first term in relation to the (given) sum of a series in geometrical progression :--
101. That (quantity) by which the sum of the series divided by the first term and (then) lessened by one is divisible throughout (when this process of division after the subtraction of one is carried on in relation to all the successive quotients) time after time--(that quantity) is the common ratio. The sum, multiplied by tho common ratio lessened by one, and (then) divided by that self-multiplied product of the common ratio in which (product) that (common ratio) occurs as frequently as the number of terms (in the series), after this (same self-multiplied product of the common ratio) is diminished by one, gives rise to the first term.
102. When the first term is 3, the number of terms is 6, and the sum is 4095 (in relation to a series in geometrical progression), what is the value of the common ratio ? The common ratio is 6, the number of terms is 5, and the sum is 3110 (in relation to another series in geometrical progression). What is the first term here ?
101. The first part of the rule will become clear from the following example:-
The sum of the series is 4095, the first term 3, and the number of terms 6. Here, dividing 4095 by 3 we get 1365. Now, . Choosing by trial 4, ; ; ; ; ; ; ; . Hence 4 is the common ratio. The principle on which this method is based will be clear from the following:--
- ; and , which is obviously divisible by r
- The second part expressed algebraically is .
The rule for finding out the number of terms in a geometrically progressive series:--
103. Multiply the sum (of the given series in geometrical progression) by the common ratio lessened by one: (then) divide this (product) by the first term and (then) add one to this (quotient). The number of times that this (resulting quantity) is (successively) divisible by the common ratio--that gives the measure of the number of terms (in the series ).
Examples in illustration thereof.
104. O my excellently able mathematical friend, tell me of what value the number of terms is in relation to (a series, whereof) the first term is 3, the common ratio is 6, and the sum is 777.
105. What is the value of the number of terms in those (series) which (respectively) have a for the first term, 2 for the common ratio, 1275 for the sum : 7 for the first term, 3 for the common ratio, 68887 for the sum : and 3 for the first term, 5 for the common ratio and 22888183593 for the sum ?
Thus ends summation, tho seventh of the operations known as Parikarman.
Vyutkalita.
Tho rule of work in relation to the operation of Vyutkalita,* which is the eigth (of tho Pirikarman operations), is as follows:--
106. (Take) tho chosen-off number of terms as combined with the total number of terms (in the series), and (take) also your own chosen-off number of terms (simply); diminish (each of)
<nowiki>*<nowiki> In a given series, any portion chosen of from the beginning is called ișța or the chosen-of part; and the rest of the series is called śēșa, and it contains the remaining terms and forms the remainder-series. It is the sum of these śēșa terms which is called vyutkalita.
106. Algebraically, vyutkalita or and the sum of the ișța or ; where d is the number of terms in the chosen-off part of the series.
these (quantities) by one and (then) halve it and multiply it by the common difference; and (then) add the first term to (each of) these (resulting products). And these (resulting quantities), when multiplied by the remaining number of terms and the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series (in order).
The rule for obtaining in another manner the sum of the remainder-series and also the sum of the chosen-of part of the given series :--
107. (Take) the chosen-off number of terms as combined with the total number of terms (in the series), and (take) also the chosen-off number of terms (simply); diminish (each of) these by one, and (then) multiply by the common difference, and (then) add to (each of) these (resulting products) twice the first term. These (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series (in order).
The rule for finding out the sum of the remainder-series in respect of an arithmetically progressive as well as a geometrically progressive series, as also for finding out the remaining number of terms (belonging to the remainder-series) :--
108. The sum (of the given series) diminished by the sum of the chosen-off part (of the series) gives rise to the sum of the remainder-series in respect of the arithmetically progressive as well as the geometrically progressive series; and when the difference between the total number of terms and the chosen-off number of terms (in the series) is obtained, it becomes the remaining number of terms belonging to that (remainder-series) .
107. Again, , and . The rule for finding out the first term in relation to the remaining number of terms (belonging to the remainder-series):--
109. The chosen-off number of terms multiplied by the common difference and (then) combined with the first term (of the given series) gives rise to the first term in relation to the remaining terms (belonging to the remainder-series) The already mentioned common difference is the common difference in relation to these (remaining terms also) ; and in relation to the chosen-off number of terms (also both the first term and the common difference) are exactly those (which are found in the given series).
The rule for finding out the first term in relation to the remaining number of terms belonging to the remainder-series in a geometrically progressive series:--
110. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike (in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of (the first term in relation to) the remaining number of terms (in the remainder-Series) viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term (of the remainder-series).
Examples in illustration thereof.
111. Calculate what the sums of the remainder-series are in respect of a series in arithmetical progression, the first term of which is 2, the common difference is 3, and the number of terms is 14, when the chosen-off numbers of the terms are 7, 8, 9, 6 and 5 (respectively).
112. (In connection with a series in arithmetical progression) here (given), the first term is 6, the common difference is 8, the number of terms is 36, and the chosen-of numbers of terms are 10,
109. The first term of the remainder series . The series dealt with in this rule is obviously in arithmetical progression.
10. The first term of remainder series is . 12 and 16 (respectively). In connection with another (similar series), the first term and the other things are 5, 5, 200 and 100 (in order). Say what the sums are of the (corresponding) remainder-series.
118. The number of terms (in a series in arithmetical progression) is 216; the common difference is 8; the first term is 14; 37 is the chosen-off number of terms (to be removed). Find the sums both of the remainder-series and of the chosen-off part(of the given series).
114. The first term (in a given series in arithmetical progression) is, in this (problem), 64; the common difference is minus 4; the number of terms is 16. What are the sums of the remainder-series when the chosen-off numbers of terms are 7, 9, 11 and 12?
Examples on vyutkalita in respect of a geometrically progressive series.
115. Where (in the process of reckoning of the fruits on trees in serial bunches), 4 happens to be the first term, 2 the common ratio, and 16 the number of terms, while the chose-off number of terms (removed) are 10, 9, 8, 7, 6, 5 and 4 (respectively)-- there. say, O you who know arithmetic and have penetrated into the interior of the forest of practical mathematical operations, (the interior) wherein wild elephants sport--(there say) what the total of the remaining fruits is on the top of the various good trees (dealt with therein).
Thus ends vyutkalita, the eighth of the operations known as Parikarman
Thus ends the first subject of treatment known as Parikarman in Sārasańgraha, which is a work on arithmetic by Mahavirācārya.
115. In this problem, there are given 7 different fruit trees, each of which has 16 bunches of fruits. The lowest bunch on each tree has 4 fruits; the fruits in the higher bunches are geometrically progressive in number, the common ratio being 2; and 10, 9, 8, 7, 6, 5 and 4 represent the numbers of the bunches removed from below in order from 7 the trees. We have to find out "the total of the remaining fruits on the tops of the various good trees". Mattēbhavikrīḍita, as it occurs in this stanza is the name of the metre in which it is composed, at the same time that it means the sporting of wild elephants.CHAPTER III.
FRACTIONS
The Second Subject of Treatment.
1. Unto that excellent Lord of the Jinas, by whom the tree of karman has been completely uprooted, and whose lotus-like feet are enveloped in the halo of splendor proceeding from the top of the crowns belonging to the chief sovereigns in all the three worlds --(unto that Lord of the Jinas), I bow in devotion.
Hereafter, we shall expound the second subject of treatment known as Kalāsavarņa* (i.e., fractions).
Multiplication of Fractions
The rule of work here, in relation to the multiplication of fractions, is as follows:--
2. In the multiplication of fractions, the numerators are to be multiplied by the numerators and the denominators by the denominators, after carrying out the process of cross-reduction, if that be possible in relation to them.
3. Tell me, friend, what a person will get for of a pala of dried ginger, if he gets of a paņa for 1 pala of such ginger.
4. Where he price of 1 pala of pepper is of a paņa, there, say, what the price will be of of a pala.
5. A person gets of a pala, of long pepper for 1 paņa. O arithmetician, mention. after multiplying, what (he gets) for paņas.
6. Where a merchant buys of a pala of cumin seeds for 1 paņa, there, O you who possess complete knowledge, mention what {he buys) for paņas. 7. The numerators of the given fractions begin with 2 and go on increasing gradually by 2; again their denominators begin
*Kalāsavarņa literally means parts resembling , since kalā denotes the sixteenth part, Hence the term Kalāsavarņa has come to signify fractions in general.
2. When is reduced as , the process of cross-reduction is applied.
7. The fractions herein mentioned are: .
with 3 and go on increasing by 2; those (numerators and (denominators) are, in both (the cases), 10 in number. Mention, of what value the products here will be, when those (fractions) are multiplied, they being taken two by two.
Thus ends multiplication of fractions.
Division of Fractions.
The rule of work, in relation to the division of fractions, is as follows:--
8. After making the denominator of the divisor its numerator (and vice versa), the operation to be conducted then is as in the multiplication (of fractions). Or, when (the fractions constituting) the divisor and the dividend are multiplied by the denominators of each other and (these two products) are (thus reduced so as to be) without denominators, (the operation to be conducted) is as in the division of whole numbers
Examples in illustration thereof.
9. When the cost of half a pala of asafoetida is of a paņa, what does a person get if he sells 1 pala at that (same) rate ?
10. In case a person gets of a paņa, for of a pala, of red sandalwood, what will he get for 1 pala (of the same wood) ?
11. When palas, of the perfume nakha is obtainable for of a paņa, what (will be obtainable) for 1 paņa at that (same rate) ?
12. The numerators (of the given fractions) begin with 3 and go on increasing gradually by 1, till they are 8 in number; the denominators begin with 2 and are (throughout) less by one (than the corresponding numerators). Tell me what the result is when the succeeding (fractions here) are divided (in order by the preceding ones)
Thus ends the division of fractions.
8. (i) ; (ii).
The Squaring, Square-Root, Cubing, and Cube-Root of Fractions.
In connection with the squaring, the square root, the cubing, and the cube root of fractions the rule of operation is as follows:--
18. If, after getting the square, the square root, the cube (or) the cube root of the (simplified) denominator and numerator (of the given fraction), the (new) numerator (so obtained) is divided by the (similarly new) denominator, there arises the result of the operation of squaring or of any of the other above-mentioned (operations as the case may be) in relation to fractions.
Examples in illustration thereof.
14. O arithmetician, tell me the squares of and
15. The numerators (of the given fractions) begin with 3 and (gradually) rise by 2; the denominators begin with 2 and (gradually) rise by 1; the number of these (fractions) is known to be 12. Tell me quickly their squares, you who are foremost among arithmeticians.16. Tell me quickly, O arithmetician, the square roots of and
17. O clever man, tell me what the square roots are of the squared quantities which are found in the (examples bearing on the) squaring of fractions and also of .
18. The following quantities, namely, and are given. Tell me their cubes separately.
19. The numerators (of the given fractions) begin with 3, and (gradually) rise by 4; the denominators begin with 2 and (gradually) rise by 2; the number of such (fractional) terms is 10. Tell me their cubes quickly, O friend who are possessed of keen intelligence in calculation.
- 17. Here is given in the original as
20. Give the cube roots of and
21. O friend of prominent intelligence, give the cube roots of the cubed quantities found in (the examples on) the cubing of fractions and (give also the cube root) of .
Thus end the squaring, square-root, cubing and cube-root of fractions.
Summation of fractional series in progression.
In regard to the summation of fractional series, the rule of work is as follows:--
22. The optional number of terms (making up the fractional series in arithmetical progression) is multiplied by the common difference, and (then it) is combined with twice the first term and diminished by the common difference. And when this (resulting quantity) is multiplied by the half of the number of terms, it gives rise to the sum in relation to a fractional series (in arithmetical progression).
Examples in illustration thereof.
23. Tell me what the sum is (in relation to a series) of which and are the first term, the common difference and the number of terms (in order); as also in relation to another of which and (constitute these elements).
24. The first term, the common difference and the number of terms are and (in order in relation to a given series in arithmetical progression). The numerators and denominators of all (these fractional quantities) are (successively) increased by 2 and 3 (respectively) until seven ( series are so made up). What is the sum (of each of these) ?
22. Algebraically . Cf. note under 62, Chap. II
23. Whenever the number of terms in a series is given as a fraction, as here, it is evident that such a series cannot generally be formed actually number of terms. But the intention seems to be to show that the rule holds good even in such cases. The rule for arriving, in relation to (a series made up of any) optional number of term, at the first term, the common difference and the (related) sum, which is equivalent firstly to the square and secondly to the cube (of the number of terms) :--
25. Whatever is (so) chosen is the number of terms, and one is the first term. The number of terms diminished by the first term, and (then) divided by the half of the number of terms diminished by one, becomes the common difference. The sum (of the series) in relation to these is the square of the number of terms. This multiplied by the number of terms becomes the cube thereof.
Examples in illustration thereof.
26. The optional number of terms (in a given series) is (taken to be) ; and the numerator as well as the denominator (of this fraction) is (successively) increased by one till ten (such different fractional terms) are obtained. In relation to these (fractions taken as the number of terms of corresponding arithmetically progressive series), give out the first term, the common difference and the square and the cube (values of the sums in the manner explained above).
The rule for finding out the first term, the common difference and the number of terms, in relation to the sum (of a series in arithmetical progression ) which (sum) happens to be the cube of (any) chosen quantity:--
27. One-fourth of the chosen quantity is the first term; and from this first term, when it is multiplied by two, results the
25. It is obvious that in the formula , the value of S becomes equivalent to when , and . In the multiplication of this sum by n, there is necessarily involved the multiplication of a as well as of b by n, so that, when and . A little consideration will show the value of b as makes it possible to arrive at as the value of S whatever may be the value of a, whether fractional or integral.
27. This rule gives only a particular case of what may be generally applied. The rule as given here works out thus: up to 2x terms.
common difference. The common difference multiplied by four is the number of terms (in the required series ). The sum as related to these is the cube (of the chosen quantity)
Examples in illustration thereof.
28. The numerators begin with 2 and are successively increased by 1; the denominators begin with 3 and are (also) successively increased by 1; and both these kinds of terms (namely, the numerators and the denominators) are (severally) five (in number). In relation to these (chosen fractional quantities), give out, O friend, the cubic sum and the (corresponding) first term, common difference, and number of terms.
The rule for finding out, from the known sum, first term and common difference (of a given series in arithmetical progression), the first term and the common difference (of a series), the optionally chosen sum (whereof) is twice, three times, half, one-third, or some such (multiple or fraction of the known sum of the given series):--
29. Put down in two places (for facility of working) the chosen sum as divided by the known sum. This (quotient), when multiplied by the (known) common difference, gives the (required) common difference -- and that (same quotient), when multiplied by the (known) first term, gives the (required) first term--of (the series of which) the sum is either a multiple or a fraction (of the known sum of the given series).
Examples in illustration thereof.
30. The first term (of a series) is , the common difference is 1, and the number of terms common (to the given as well as the
. The general applicability of this process can be at once made out from the equality, , so that in all such cases the number of terms in the series is obtained by multiplying by the first term, which is representable as and the common difference is of course taken to be twice this first term in every case.
29. See note under 84, Chap. II. required series) is (taken to be) . The sum of the required series is of the same value ). Find out, O friend, the first term and the common difference (of the required series).
31. The first term is twice the common difference (which is taken to be 1); the number of terms is (taken to be). The sum of tho required series is . Find out the first term and the common difference.
82. The first term is 1, the common difference is and the number of terms common (to both the given as well as the required series) is (taken to be) . The sum of the required series is . Give out the first term and the common difference (of the required series).
The rule for finding out the number of terms (in a series in arithmetical progression):-
38. When, to the square root of (the quantity obtained by) the addition, of the square of the difference between the half of the common difference and the first term, to twice the common difference multiplied by the sum of the series, half the common difference is added, and when (this sum is) diminished by the first term, and (then) divided by the common difference, (we get) the number of terms in the series.
He (the author) states in another way (the rule for finding out) the same (number of terms):--
34. When, from the square root of (the quantity obtained by) the addition, of the square of the difference between the half of the common difference and the first term, to twice the common difference multiplied by the sum of the series, the kşēpapada is subtracted, and when (this resulting quantity is) divided by the common difference. (we get) the number of terms in the series.
33. Symbolically expressed, . Cf. note under 69, in Chap. II
34. For kşēpapada, see note under 70 in Chap. II
Examples in illustration thereof.
35. In relation to this (given) series, the first term is , the common difference is , and the sum given is ; again (in relation to another series), the common difference is , the value of the first term is , and the sum is . In respect of these two (series), O friend, give out the number of terms quickly.
The rule for finding out the first term as well as the common difference :--
36. The sum (of the series) divided by the number of terms (therein) when diminished by (the product of) the common difference multiplied by the half of the number of terms less by one, (gives) the first term (of the series). 'I'he common difforence is (obtained when) the sum, divided by the number of terms and (then) diminished by the first term, is divided by the half of the number of terms less by one.
Examples in illustration thereof.
37. Give out the first term and the common difference (respectively) in relation to (the two series characterised by) as the sum. and having (in one case) as the common difference and as the number of terms, and (in the other case) as the first term and as the number of terms.
The rule for finding out in relation to two (series), the number of terms wherein is optionally chosen, their mutually interchanged first term and the common difference, as also their sums which may be equal, or (one of which may be) twice, thrice, half or one- third (of the other):-
38. The number of terms (in one series) multiplied by itself as lessened by one, and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by twice the number of terms in the other (series, gives rise to the interchangeable) first term (of one of the series). The square of the
36. See note under 74, Chap. II.
38. See note under 86, Chap. II.
(number of terms in the) other (series), diminished by that (number of terms) itself, and (then) diminished (again) by the product of two (times) the chosen (ratio) and the number of terms (in the first series, gives rise to the interchangeable) common difference (of that series).
Examples in illustration thereof.
39. In relation to two series, having and to (respectively) represent their number of terms, the first term and the common difference are interchangeable, the sum of one (of the series) is either a multiple or a fraction (of that of the other, this multiple or fraction being the result of the multiplication or the division as the case may be) by means of (the natural numbers) commencing with 1. O friend, give out (these) suns, the first terms and the common differences.
The rule for finding out the guņadhana, and the sum of a series in geometrical progression :--
40. The first term (of a series in geometrical progression), when multiplied by that Self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the number of terms (in the series), gives rise to the guņadhana. And it has to be understood that this ( guņadhana), when diminished by the first term and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression.
The rule for finding out the last term in a geometrically progressive series as well as the sum of that (series):--
41. The antyadhana or the last term of a series in geometrical progression is the guņadhana of (another series) wherein the number of terms is less by one. This (antyadhana), when multiplied by the common ratio and (then) diminished by the first term and (then)
40. See note under 93, Chap. II.
41. See note under 95, Chap. II.
divided by the common ratio lessened by onegives rise to the sum (of the series).
An example in illustration thereof.
42. In relation to a series in geometrical progression, the first term is , the common ratio is and the number of terms is here 5. Tell me quickly bhe sum and the last term of that (series).
The first term, the common ratio and the number of terms, in relation to the guņadhana and the sum of a series in geometrical progression, should also be found out by means of the rules stated already (in the last chapter).*
The rule for finding out the (common) first term of two series having the same sum, one of them being in arithmetical progression and the other in geometrical progression, their optionally chosen number of terms being equal and the similarly chosen common difference and common ratio also being equal in value.
43. One is (taken as) the first term, the number of terms and the common ratio as well as the common difference (which is equal to it) are optionally chosen. he uttaradhana (here), divided by the sum of this geometrically progressive series as diminished by the ādidhana (thereof), and (then) multiplied by whatever is taken as the first term, gives rise to the (required common) first term in relation to the two series, (one of which is in geometrical progression and the other in arithmetical progression, and both of) which are characterised by sum of the same value.
- *For these rules, see 97, 98, 101 and 103, Chap. II.
43. For ādidhana and uttaradhana, see note under 68 and 64, Chap II. This rule, symbolically expressed, works out thus where b==r. For facility of working, 1 is chosen as the provisional first term, but it is obvions that any quantity may be so provisionally chosen. The use of the provisional first term is seen in facilitating the statement of the rule by means of the expressions ādidhana and uttaradhana. The formula here given is obtained by equatingtbhe formulae giving the sums of the geometrical and the arithmetical series. It is worth noting that the word caya is used here to denote both the common difference in an arithmetical and the common ratio in a geometrical series.
Examples an illustration thereof.
44. T'he number of terms are 5, 4 and 3 (respectively) and the common ratios as well as the (equal) common differences are and (in order). What is the value of the (corresponding) first terms in relation to these (sets of two series, one in geometrical progression and the other in arithmetical progression), which are oharacterised by sums of the same value ?
Thus ends the summation of fractions in series,
Vyutkalita of fractions in series.
The rule for performing the operation of youtkrc:lita is as follows :--
45. (Take) the chosen-of number of torms as combined with the total number of terms (in the series), and (take) also your chosen-off number of terms (separately). Multiply each of theso quantities by the common difference and diminish (the products ) by the common difference; (then) multiply by two; and these (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the (given) series (in order).
The rule for inding out the first term in relation to the remaining number of terms (making up the remainder-series) :--
46. The first term (of the series), diminished by the half of the common difference, and combined with the chosen-off number of terms as multiplied by the common difference, as also with the half of the common difference, (gives) the first term of the remaining number of terms (making up the remainder-series). And the common difference (of the remainder-series) is the same as what is found in the given series.
45. Cf. note under 106, Chap. II.
47. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike (in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of (the first term among the remaining number of terms (constituting the remaindor-series), viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term (of the remainder-series).46. Cf. note under 109, Chap. II.
Examples in illustration thereof.
48. Calculate what the sum of the remainder-series is in relation to that (series) of which is the common difference, the first term, and is (taken to be) the number of terms, when the chosen-off number of terms (to be removed) is (taken as) .
49. In relation to a series in arithmetical progression, the first term is , the common difference is , and the number of terms is (taken to be) . When the chosen-off number of terms (to be removed) is (taken as) , give out, O you who know calculation, the sum of the remainder-series.
50. What is the value of the sum of the remainder-series in relation to a series of which the first term is , the common difference is , and the number of terms is (taken to be) , when the chosen-off number of terms is ?
51. The first term is , the common difference is , and the number of tenums is (talken as) , and the chosen-off number of terms is taken to be or . O you, who, being the abode of kalās * , are the moon shining with the moon-light of wisdom, tell me the sum of the remaining number of terns .
52. Calculate the sum of the remaining number of terms in relation to a series of which the number of terms is 12, the common difference is minus , and the first term is , the chosen-off number of terms being 3, 4, 5 or 8.
7. See note under 110, Chap. II.
* Kāla is here used in the double sense of 'learning' and 'the digits of the moon'Example in illustration of vyutkalita in relation to a series in geometrical progression.
53. The first term is , the common ratio is , and the number of torms is 8; and the chosen-off number of torms is 3, 4 or 5. What are the first term, the sum and the number of terms in relation to the (rospective) remainder-series ?
Thus ends the vyutkalita of fractions.
The six varieties of fractions.
Hereafter we shall expound the six varieties of fractions.
54. Bhāga (or simple fractions), Prabhāga (or factions of fractions), then Bhāgābhāga (or complex fractions), then Bhāgānubandha or (or fractions in association), Bhāgāpavāha(or fractions in dissociation), together with Bhāgamātŗ (or fractions consisting of two or more of the above-mentiomed fractions)--these are here said to be the six varieties of fractions.
Simple fractions: (addition and subtraction).
The rule of operation in connection with simple fractions therein :--
55. If, in the operations relating to sinmple fractions, the numerator and the denominator (of each of two given simple fractions) are multiplied in alternation by the quotients obtained
55. The method of reducing fractions to common denomenators described in this rule applies only to paris of fractions. This rule will be clear from the following worked out example:--To simplify . Here, a and xy are to be multiplid by z which is the quotient obtained by dividing yz, the donominator of the other fraction, by y which is the common factor of the denomenators. Thus we get .
Similarly in the second fraction, by multiplying b and yz by x which is the quotient obtained by dividing the first denominator xy by y the common factor. we get . Now . by dividing the denominations by means of a common factor thereof, (the quotient derived from the denominator of either of the fractions being used in the multiplication of the numerator and the denominator of the other fraction), those (fractions) become so reduced as to have equal denominators. (Then) removing one of these (equal) denominators, the numerators are to be added (to one another) or to be subtracted (from one another, so that the result may be the numerator in relation to the other equal denominator).
Another rule for arriving at the common denominator in another manner:--
6. The niruddha (or the least common multiple) is obtained by means of the continued multiplication of (all) the (possible) common factors of the denominators and (all) their (ultimate) quotients. In the case of (all) such multiples of the denominators and the numerators (of the given fractions), are obtained by multiplying those (denominators and numerators) by means of the quotients derived from the division of the niruddha by the (respective) denominators, the denominators become equal (in value).
Examples in illustration thereof.
57 and 58. A śrāvaka purchased, for the worship of Jina, jambu fruits, limes, oranges, cocoanuts, plantains, mangoes and pomegranates for and of the golden coin in order ; tell (me) what the result is when these (fractions) are added together.
59. Add together and
60. (There are 3 sets of fractions), tho denominators whereof begin with 1, 2 and 3, (respectively) and go on increasing gradually by one till the last (of such denominators) becomes 9, 10 and
60. The resulting problems are to find the values of--
- (i)
- (ii)
- (iii)
16 (in order in the respective sets); the numerators (of these sets of fractions) are of the same value as the first number (in those sets of denominators), and every one of these (above-mentioned denominators in each set) is multiplied by the next one, (the last denominator, however, remaining in each case unchanged or want of a further multiplying denominator). What is the sum of (each of) these (finally resulting sets of fractions)?
61 and 62. (There are 4 sets of fractions), the denominators whereof begin with 1, 2, 3 and 4 (respectively) and rise successively in value by 1 until 20, 42, 25 and 36 become the last (denominators in the several sets) in order; the numerators of these (sets of fractions) are of the same value as the first number (in these sets of denominators). And every one of these (denominators in each set) is multiplied in order by the next one(the last denominator, however, remaining unchanged in each case). What is the sum on adding these (finally resulting sets of fractions)?
68. A man purchased on account of a Jinn-festival sandalwood, camphor, agaru and Saffron for and of a golden coin. What is the remainder (left thereof) ?
64. A worthy śrāvaka gave me two golden coins and told me that I should bring, for the purpose of worshipping in the temple of Jina, blossomed white lotuses, thick curds, ghee, milk and sandal-wood for and of a golden coin,(respectively, out of the given amount). Now tell me, O arithmetician, what remains after subtracting the (various) parts (so spent).
65 and 66. (There are two sets of fractions) the denominators whereof begin with 8 and 5 (respectively) and rise in both cases successively in value by 1, until 30 becomes (in both cases) the last (denominator). The numerators of these (sets of fractions) are of the same value as the first term in each (of these sets of denominators). And every one of the denominators (in each set) is multiplied by the next one, the last (denominator) being (in each case) multiplied by 4. After subtracting from 1, (each of) these two (sums obtained by the addition of the sets of fractions finally resulting as above), tell me O friend, who have gone over to the other shore of the ocean of simple fraction, what it is that remains. 67 to 71. The denominators (of certain given fractions) are stated to be 19, 23, 62, 29, 123, 35, 188, 87, 98, 47, 140, 141, 116, 31, 92, 57, 73, 55, 110, 49, 74, 219, (in order); and the numerators begin with 1 and rise successively in value by 1 (in order). Add (all) these (fractions) and give the result. O you who have reached the other shore of the ocean of simple fractions.
Here, the rule for arriving at the numerators, (when the denominators and the sum of a number of fractions are given, is as follows):--
72. Make one the numerator (in relation to all the given denominators); then, multiply by means of such (numbers) as are optionally chosen, those numerators which (are derived from these fractions so as to) have a common denominator. (Here), those (numbers) turn out to be the required numerators, the sum of the products whereof, obtained by multiplying them with the numerators (derived as above), is equal to the numerator of) the given sum(of the fractions concerned).
The rule for arriving at the numerators, (the denominators and the sum being given as before), in relation to such (fractional) quantities as have their numerators (successively) rising in value by one, when, in the (given) sum (of these fractions), the denominator is higher in value than the numerator:--
73. The quotient obtained by dividing the (given) sum (of the fractions concerned) by the sum of those (tentative fractions)
72. This rule will become clear from the working of the example in stana No. 74, wherein we assume 1 to be the provisional numerator in relation to each of the given denominators; thus we get and which, being reduced so as to have a common denominator, become and . When the numerators are multiplied by 2, 3 and 4 in order, the sum of the products thus obtained becomes equal to the numerator of the given sum, namely, 877. Hence, 2, 3, and 4 are the required numerators. Here it may be pointed out that this given sum also must be understood to have the same denominators as the common denominator of the fractions
73. To work out the sum given under 74 below, according to this rule:--
Reducing to the same denominator the fractions formed by assuming 1 to be the numerator in relation to each of the given denominators, we get, and . Dividing the given sum by the sum of these fractions , we get the quotient 2, which is the numerator in relation to the first denominator. The remainder 279 which, (while having the given denominators), have one for the numerators and (are then reduced so as to) have a common denominator becomes the (first required) numerator among those which (successively) rise in value by one (and are to be found out). On the remainder (obtained in this division) being divided by the sum of the other numerators (having the common denominator as above), it, (i.e., the resulting quotient), becomes another (viz., tho second required ) numerator (if added to the first one already obtained). In this manner (the problem has to be worked out) to the end.An example in illustration thereof.
7. The Sum of (certain numbers which are divided (respectively) by 9, 10 and 11 is 877 as divided by 990. Give out what the numerators are (in this operation of adding fractions).
The rule for arriving at the (required) denominators (is as follows):--
75. When the sum of the (different fractional) quantities having one for their numerators is one the (required) denominators are such as, beginning with one, are in order multiplied (successively) by
obtained in this division is then divided by the sum of the remaining provisional numerators, i.e., 189, giving the quotient 1, which, combined with the numerator of the first fraction, namely 2, becomes the numerator in relation to the second denominator. The remainder in this second division, viz, 50, is divided by the provisional numerator 90 of the last fraction, and the quotient 1, when combined with the numerator of the previous fraction, namely 3, gives rise to the numerator in relation the last enumerator. Hence the fractions, of which the sum, are and .
It is noticeable here that the numerators successively found out thus become the required numerators in relation to the given denominators in the order in which they are given.
Algebraically also, given the denominators a, b & c, in respect of 3 fractions whose sum is , the numerators x,x + 1 and x + 2 are easily found out by the method as given above.
75. In working out an example according to the method stated herein, it will be found that when there are n fractions, there are, after leaving out the first term and the last fraction n - 2 terms in geometrical progression with as the first term and as the common ration. The sum of theese n - 2 terms is , which when reduced becomes , which is the same
three, the first and the last (denominators so obtained) being (however) multiplied (again) by 2 and (respectively).
Examples in illustration thereof.
76. The sum of five or six or seven (different fractional) quantities, having 1 for (each of) their numerators, is 1 (in each case). O you, who know arithmetic, say what the (required) denominators are.
The rule for finding out the denominators in the case of an even number (of fractions):--
77. When the sum of the (different fractional}quantities, having one for each of their numerators, is one, the (required) denominators are such as, beginning with two, go on (successively) rising in value by one, each (such denominator) being (further) multiplied by that
as . From this it is clear that, when the first fraction and last fraction are added to this result, the sum becomes 1.
In this connection it may be noted that, in a series in goemetrical progression consisting of n terms, having as the first term and as the common ratio, the sum is, for all positive integral values of a, less than by the (n+1)th term in the series. Therefore, if we add to the sum of the series in goemetrical progression the (n + 1)th term which is the last fraction according to the rule stated in this stanza, we get . To this , we have to add in order to get 1 as the sum. This is mentioned in the rule as the first fraction, and so 3 is the value chosen for a, since the numerator of all the fractions has to be 1.
77. Here note
- ==
- ==
- ==
(number) which is (immediately) next to it (in value) and then halved.
The rule for arriving at the (required) denominators (in the case of certain intended fractions), when their numerators are (each) one or other than one, and when the (fraction constituting their) sum has one for its numerator:--
78. When the sum (of certain intended fractions) has one for its numerator, then (their required denominators are arrived at by taking) the denominator of the sum to be that of the first (quantity), and (by taking) t!his (denominator) combined with its own (related) numerator to be (the denominator) of the next (quantity) and so on, and then by multiplying (further each such denominator in order) by that which is (immediately) nex to it, the last (denominator) being (however multiplied) by its own (related) numerator.
Examples in illustration thereof
79. The sums (of certain intended fractions) having for their numerators 7, 9, 3 and 18 (respectively) are (firstly) 1, (secondly) and (thirdly) . Say what the denominators (of those fractional quantities) are.
The rule for arriving at the denominators (of certain intended fractions) having one for their numerators, when tho sum (of those fractions) has one or (any quantity) other than one for its numerator:--
78. Algebraically, if the sum is and a, b, c and d are the given numerators, the fractions summed up are as below:--
- ==
- ==
- ==
- ==
80. The denominator (of the given sum , when combined with an optionally chosen quantity and then divided by the numerator of that sum so as to leave no remainder, becomes the denominator related to the first numerator (in the intended series of fractions); and the (above) optionally chosen quantity, when divided by this (denominator of the first fraction) and by the denominator of the (given) sum, gives rise to (the sum of) the remaining (fractions in the series). From this (known sum of the remaining fractions in the series, the determination) of the other (denominators is to be carried out) in this very manner.
Examples in illustration thereof.
81. Of three (different) fractional quantities having 1 for each of their numerators, the sum is ; and of 4 (such other quantities, the sum is) . Say what the denominators are.
The rule for arriving at the denominators (of certain intended fractions) having either one or (any number) other than one for their numerators, when the sum (of those fractions) has a numerator other than one:--
82. When the known numerators are multiplied by (certain) chosen quantities, so that the sum of these (products) is equal to the numerator of the (given) sum (of the intended fractions), then, if the denominator of the sum (of the intended fractions ) is divided by the multiplier (with which a given numerator has) itself (been multiplied as above), it gives rise to the required denominator in relation to that (numerator).
80. Algebraically, if is the sum, the first fraction is ; and the sum of the remaining fractions is mentioned in the role to be , where p is the optionally chosen quantity . This is obtained obviously by simplifying
We must here give such a value to p that n + p becomes exactly divisible by a.Examples an illustration thereof.
88. Say what the denominators are of three (different fractional) quantities each of which has 1 for its numerator, when the sun (of those quantities) is .
84. Say what the denominators are of three (fractional quantities) which have 3, 7 and 9 (respectively) for their numerators, when the sum (of those quantities) is .
The rule for arriving at the denominators of two (fractional) quantities which have one for each of their numerators, when the sum (of those quantities) has one for its numerator:--
85. The denominator of the (given) sum multiplied by any chosen number is the denominator (of one of the intended fractional quantities); and this (denominator) divided by the (previously) chosen (number) as lessened by one gives rise to the other (required denominator). Or, when in relation to the denominator of the (given) sum (any chosen) divisor (thereof) and the quotient (obtained therewith) are (each) multiplied by their sum, they give rise to the two (required) denominators,
Examples in illustration thereof.
86. Tell me, O you who know the principles of arithmetic, what the denominators of the two (intended fractional) quantities are when their sum is either or .
The first rule for arriving at the denominators of two (intended fractions) which have either one or (any number) other
85. Algebraically, when is the sum of two intended fractions, the fractions according to this rule are - and , where p is any clnosen quantity. It will be seen at once that the sum of these two fractions is .
Or, when the sum is , the fractions may be taken to be and . than one for their numerators, when the sum (of those fractions) has either one or (any number) other than one for its numerator:--
87. (Either) numerator multiplied by a chosen (number), then combined with the other numerator, then divided by the numerator of the (given) sum (of the intended fractions) so as to leave no remainder, and then divided by the (above) chosen number and multiplied by the denominator of the (above) sum (of the intended fractions), gives rise to a (required) denominator. The denominator of the other (fraction), however, is this (denominator) multiplied by the (above) chosen (quantity).
Examples in illustration thereof.
88. Say what the denominators are of two (intended fractional) quantities which have 1 for each of their numerators, when the sum (of those fractional quantities) is either or ; as also of two (other fractional quantities) which have 7 and 9 (respectively) for (their) numerators.
The second rule (is as follows):--
89. The numerator (of one of the intended fractions) as multiplied by the denominator of the sum (of the intended fractions), when combined with the other numerator and then divided by the numerator of the sum (of the intended fractions), gives rise to the denominator of one (of the fractions). This (denominator), when multiplied by the denominator of the sum (of the intended fractions), becomes tho denominator of the other (fraction).
67. Algebraically, if is the sum of two intended fractions with a and b as their numerators, then the fractions are and where p is any number so chosen that ap + b is divisible by m. The sum of these fractions it will be found, is .
89. This rule is only a particular case of the rule given in stanza No. 87, as the denominator of the sum of the intended fractions is itself substituted in this rule for the quantity to be chosen in the previous rule.
Examples in illustration thereof.
90. O friend, tell me what the denominators are of two (fractional) quantities which have 1 for each of their numerators, when the sum (of those intended fractions) is ; as also of (two other intended fractions) which have 6 and 8 (respectively) for (their) numerators.
91. The sum of and is 1. When is left out here, what two (fractions) having 1 (for each of their numerators) have to be added (instead so as to give the same total)?
92. and is 1. If and is 1. If is left out here, what two (fractions) having 7 and 11 for their numerators should be added (instead so as to give the same total) ?
The rule for arriving at the denominators (of a number of intended fractions) by taking (them) in pairs:--
98. After splitting up the sum (of all the intended fractions) into as many parts having one for each of their numerators as here are (numbers of) pairs (among the given numerators), these (parts) are taken (severally) as the sums of the pairs ; (and from them) the (required) denominators are to be found out in accordance with the rule relating to two (such component fractional quantities).
Examples in illustration thereof.
94. What are the denominators of (those intended) fractions whose numerators are 3, 5, 13, 7, 9 and 11, when tho sum of (those fractional) quantities is 1 or ?
The rule for arriving at (a number of) denominators, with the help of the denominators that have one as (their corresponding) numerators and are arrived at according to one of the (already given) rules (for finding out the denominators), as also with the help of the denominators that have one as (their corresponding) numerators and are arrived at according to any other of those
98. The rules relating to two fractional qantities have been given in stanza 85, 87 and 89. rules, when the sum (of all the intended fractions) is one; and also (the rule) for getting at (the value of) the part that is left out:--
95. The denominators derived in accordance with (any) chosen rule, when (severally) multiplied by the denominators derived in accordance with another rule, become the (required) denominators. The sum (of all the fractions), diminished by the sum of the specified part (thereof), gives the measure of the optionally left-out part.
Examples in illustration hereof.
96. The number of fractions (obtained) by rule No. 77 is 13, and 4 (is obtained) by rule No. 78. When the sum (of the fractions arrived at with the help of these rules) is 1, how many are the (component) fractions ?
97. The number of fractions (obtained) by rule No. 78 is 7, and 3 (is obtained) by rule No. 77. When the sum (of the fractions arrived at) with the help of these (rules) is 1, how many are the (component) fractions?
98. Certain fractions are given with 1 for each of their numerators, and 2, 6, 12 and 29 for their respective denominators. The (fifth fractional) quantity is here left out. The sum of all (these five) being 1, what is that {fractional) quantity (which is leftout)?
Here end Simple Fractions,
Compound and Complex Fractions.
The rule for (simplifying) compound and complex fractions:--
99. In (simplifying) compound fractions, the multiplication of the numerators (among themselves) as well as of the denominators (among themselves) shall be (tho operation). In the operation (of simplification) relating to complex fractions, the denominator of (the fraction forming) the denominator (becomes) the multiplier of the number forming the numerator (of the given fraction).
98. The complex fraction pro dealt with is of the sort which has an integer for the numerator and a fraction for the denominator
Examples in compund fractions.
100 to 102. To offer in worship at the fact of Jina, lotuses, jasamines, kēlakīs and lilies were purchased in return for the payment of , , , , , , , , , and , of a paņa. Sum of these (paid quantities) and give out the result.
103 and 104. A certain person gave (to a vendor) , , , , and , , (of a paņa) out of the 2 paņas (in his possession), and brought fresh ghee for (lighting) the lamps in a Jina temple. O friend, give out what he remaining balance is.
105 and 106. If you have taken pains in connection with compound fractions, give out (the resulting sum) after adding these (following fractions):- , , , , and
The rule for finding out the one unknown (element common to each of a set of compound fractions whose sum is given):--107. The given sum, when divided by whatever happens to be the sum arrived at in accordance with the rule (mentioned) before by putting down one in the place of the unknown (element in the compound fractions), gives rise to the (required) unknown (element) in (the summing up of) compound fractions.
An example in illustration thereof.
108. The sum of , , , , of a certain quantity is . What is this unknown (quantity) ?
The rule for finding out more than one unknown (element, one such occurring in each of a set of compound fractions whose sum is given):--
109. Make the unknown (values of the various partially known compound fractions) to be (equivalent to) such optionally chosen
109. This rule will be clear from the following working of the problem given in stanza No. 110:--
Splitting up , the sum of the intended fractions, into 3 fractions according to rule No. 78, we get . Making these the values of the three
quantities, as, (being equal in number to the given compound fractions), have their sum equal to the given sum (of the partially given compound fractions) : then, divide these (optionally chosen) values of the unknown (compound-fractional) quantities by (their) known (elements) respectively
An example in illustration thereof.
110. (The following partially known compound fractions, viz., ) of a certain quantity, of another (quantity), and of (yet) another (quantity give rise to) as (their) sum. What are the unknown (elements here in respect of these compound fractions)?
Examples in complex fractions
111. (Given) and ; say what the sum is when these are added.
112. After subtracting , and also and , from 9, give out the remainder.
Thus end Compound and Complex Fractions.
Bhāgānubandha Fractions.
The rule in respect of the (simplification of Bhāgānubandha or associated fractions :--
113. In the operation concerning (the simplification of) the Bhāgānubandha class (of fractions), add the numerator to the
partially known compound fractions, we divide them in order by , and respectively. The fractions thus obtained, viz. and , are the quantities to be found out.
118. Bhāgānubandha literally means an associated fraction. This rule contemplates two kinds of associated fractions. The first is what is known as a mixed number, i.e., a fraction associated with an integer. The second kind consists of fractions associated with fractions, e.g., associated with , associated with its own , and with of this associated quantity. The expression " associated with " means . The meaning of the other example here of . This kind of relationship is what is denoted by association in additive consecntion.
(product of the associated) whole number multiplied by the denominator. (When, however, the associated quantity is not integral, but is fractional), multiply (respectively) the numerator and denominator of the first (fraction, to which the other fraction is attached) by the denominator combined with the numerator, and by the denominator (itself, of this other fraction).
Examples on Bhāgānubandha fractions containing associated integers
114. Nișkas 2, 3, 6 and 8 in number are (respectively) associated with and . O friend, subtract(the sum of these) from 20
115. Lotuses were purchased for , camphor for and Saffron for (nișkas). What is (their total) value when added ?
116. O friend, subtract from 20 (the following):-- and .
117. A person, after paying and māșas offered in worship in a Jina temple, garlands of blooming kuravaka, kunda, jāti and mallī' flowers. O arithmetician, tell me quickly (the sum of those māșas') after adding them.
Examples on Bhāgānubandha fractions containing associated fractions.
118. (Here) is associated with its own and with (of this associated quantity); and also (is similarly associated ); is associated with its own and with (of this associated quantity). What is the value when these are (all) added ?
119. For the purpose of worshipping the exalted Jinas a certain person brings--flowers (purchased) for (nișka) associated (in additive consecution) with fractions (thereof) commencing with and ending with (in order); all scents (purchased) for (nișka) associated (similarly) with and (thereof); and incense (purchased) for (nișka) associated (similarly) with and (thereof): what is the sum when these (nișkas) are added ? 120. O friend, subtract (the following) from 3: associated with of itself and with of this (assicuated quantity), associated with and of itself (in additive consecution), (similarly) associated with (fractions thereof) commencing with and ending with , and associated with of itself.
121. O friend, you, who have a thorough knowledge of Bhāgānubandha, give out (the result) after adding associated with of itself, associated with of itself, associated with of itself, associated with of itself, and associated with of itself.
Now the rule for finding out the one unknown (element) at the beginning (in each of a number of associated fractions, their sum being, given):--
122.[५] The optionally split up parts of the (given) sum, which are equal (in number) to the (intended) component elements (thereof), when divided in order by the resulting quantities arrived at by taking one to be the associated quantity (in relation to these component elements), give rise to the value of the (required) unknown (quantities in association).
Examples in illustration thereof.
123. A certain fraction is associated with and of itself (in additive consecution); another (is similarly associated with and of itself; and another (again is similarly associated) with and of itself; the sum of these (three fractions so associated) is 1: what are these fractions ?
124. A certain fraction, when associated (as above) with and of itself, becomes . Tell me, friend, quickly the measure of this unknown (fraction),
The rule for finding out any unknown fraction in other required places (than the beginning) :-125. The optionally split up parts of the (given) sum when divided in order by the simplified known quantities (in the intended Bhāgānubandha (fractions), and (then) diminished by one become the unknown (fractional quantities) in the required places of our choice.
Thus ends the Bhāgānubandha class (of fractions).
Bhāgāpavāha Fractions.
Then (comes) the rule for the (simplification of) Bhāgāpavāha (or the dissociated) variety (in fractions):--126. In the operation concerning (the simplification of) the Bhāgāpavāha class (of fractions), subtract the numerator from the (product of the dissociated) whole number as multiplied by the denominator. (When, however, the dissociated quantity is not integral, but is fractional,) multiply (respectively) the numerator and the denominator of the first (fraction to which the other fraction is negatively attached) by the denominator diminished by the numerator, and by the denominator (itself, of this other fraction).
Examples of Bhāgāpavāha fractions containing dissociated integers.
127. Karșās 3, 8, 4 and 10, diminished by and of a karșā, are offered by certain men for the worship of tīrthańkaras. What is (the sum) when they are added ?
125. The method given in this rule is similar to what is explained under stanza No. 122: only the results thus obtained have to be, in this case, each diminished by one.
126. Bhāgāpavāha literally means fractional dissociation. As in Bhāgānubandha, there are two varieties here also. When an integer and a fraction are in Bhāgāpavāha relation, the fraction is simply subtracted from the integer.
Two or more fractions may also be in such relation, as for example, dissociated from of itself or dissociated from and of itself. It is meant here that is to be subtracted from in the first example; and the second example comes to of .
128. Tell me friend, quickly the amount of the money remaining after subtracting from 6 x 4 of it, (the quantities) 9, 7 and 9 as diminished in order by and
Examples on Bhāgāpavāha containing dissociated fractions
129. Add and which are (respectively) diminished by and of themselves in order; and (then) give out (the result)
130. (Given) of a paņa diminished by and of itself (in consecution); (similarly) diminished by and of itself; (similarly) diminished by and of itself: and another (quantity). viz., diminished by of itself--when these are (all) added, what is the result ?
131. If you have taken pains, O friend, in relation to Bhāvāpavāha fractions, give out the remainder after subtracting from (the following quantities): diminished (in consecution) by and of itself; also (similarly) diminished by and of itself; and (also)(similarly) diminished by and of itself.
Here, the rule for finding out the (one) unknown element at the beginning (in each of a number of dissociated fractions, their sum being given):--
132. The optionally split up parts of the (given) sum which are equal (in number) to the (intended) component elements (thereof), when divided in order by the resulting quantities arrived at by taking one to be the dissociated quantity (in relation to these component elements), give rise to the value of the (required) unknown (quantities in dissociation) .
Examples in illustration thereof.
188. A certain fraction is diminished (in consecution) by and of itself; another fraction is (similarly) diminished by and of itself; and (yet) another is (similarly) diminished by ,
132. The working is similar to what has been explained under stanza No. 122. and of itself. The sum of these (quantities so diminished) is . What are the unknown fractions here?
134. A certain fraction, diminished (in consecution) by and of itself, become . O you, who know the principles of arithmetic, what is that (unknown) fraction?
The rule for finding out any unknown fraction in other required places (than the beginning) :--
135. The optionally split up parts derived from the (given) sum, when divided in order by the simplified known quantities (in the intended Bhāgāpavādha fractions), and (then) subtracted from one (severally), become the unknown (fractional quantities) in the (required) places of our choice.
Thus ends the Bhāgāpavāha variety of fractions.
The rule for finding out the unknown fractions in all the places in relation to a Bhāgānubandha or Bhāgāpavāha variety of fractions (when their ultimate value is known):--
136. Optionally choose your own desired fractions in relation to all unknown places, excepting (any) one. Then by means of the rules mentioned before, arrive at that (one unknown) fraction with the help of these (optionally chosen fractional quantities).
Examples in illustration thereof.
187. A certain fraction combined with five other fractions of itself (in additive consecution) becomes ; and a certain (other) fraction diminished (by five other fractions of itself in consecution) becomes , O friend, give out (all) those fractions.
135. This rule is similar to the rule already given in stanza No. 125.
136. The previous rules here intended are those given in stanzas 122, 125, 132 and 135.
137. In working out the first case in this example, choose the fractions and in places other than the beginning; and then find out, by the rule given in stanza 122, the first fraction which comes to be . Or choosing and , find out the fraction left out in a place other than the beginning in accordance with the rule given in stanza 125; the result arrived at is . Similarly, the second case which involves fractions in dissociation can be worked out with the help of the rules given in stanzas 132 and 135.
Bhāgamātŗ Fractions.
Tho rule for (the simplification of) that class of fractions which contains all the foregoing varieties of fractions:--
138. In the case of the Bjāgamātŗ class of fractions (or that class of fractions which contains all the foregoing varieties), the respective rules pertaining to the (different) varieties beginning with simple fractions (hold good). It, i.e., Bhāgamātŗ, is of twenty-six kinds.
One is (taken to be) the denominator (in the case ) of a quantity which has no denominator.
Examples in illustration thereof
139 and 140. (Given) associated with of itself; then associated with of itself; 1 diminished by ; 1 diminished by . diminished by of itself; and diminished by of itself : after adding these according to the rules which are strung together in the manner of a garland of blue lotuses made up of fractions, give out, O friend, (what the result is).
Thus ends the Bhāgamātŗ variety of fractions.
Thus ends the second subject of treatment known as Fractions in Sārasańgraha which is a work on arithmetic by Mahāvīrācārya
18. The twenty-six varieties here mentioned are Bhāga, Prabhāga, Bhāgabhaga, Bhāgānubandha, and Bhāgāpavāha, in combinations of two, three, for or five of these at a time; such as, the variety in which Bhāga and Prabhāga are mixed, or Bhāga and Bhāgabhāga are mixed, and so on. The number of varieties obtained by mixing two of them at a time is 10, by mixing three of them at a time is 10, and by mixing four of them at a time is 5, and by mixing all of them at a time is 1; so there are 26 varieties. The example given in stanza 139 belongs to this last-mentioned variety of Bhāgamātŗ in which all the five simple varieties are found.
139. The word utpalamālikā occurs in this stanza, means a garland of blue lotuses, at the same time that it happens to be the name of the metre in which the stanza is composed!
CHAPTER IV.
MISCELLANEOUS PROBLEMS (ON FRACTIONS).
The Third Subject of Treatment.
1. After Saluting the Lord Jina, Mahavira, whose collection of infinite attributes is highly praiseworthy, and who vouchsafes boons to (all) the three worlds that worship (him), I shall treat of miscellaneous problems (on fractions)
2. May Jina, who has destroyed the darkness of unrighteousness, and is tho authoritative exponent of the syādvāda, and is the joy of learning, and is the great disputant and the best of sages, be (ever) victorious. Hereafter we shall expound the third subject of treatment, viz., miscellaneous problems (on fractions).
3. There are these ten (varieties in miscellaneous problems on fractions, namely), Bhāga, Śēșa, Mūla, Śēșamūla, the two varieties
3. The Bhāga variety consists of problems wherein is given the numerical value of the portion remaining after removing certain specified fractional parts of the total quantity to be found out. The fractional parts removed are each of them called a bhāga, and the numerical value of the known remainder is termed dŗśya.
The Śēșa variety consists of problems wherein the numerical value is given of the portion remaining after removing a known fractional part of the total quantity to be found out as also after removing certain known fractional parts of the successive sēșas or remainders.
The Mūla variety consists of problems wherein the numerical value is given of the portion remaining after subtracting from the total quantity certain fractional parts thereof as also a multiple of the square root of that total quantity.
The Śēșamūla variety is the same as the mūla variety with this difference, viz., the square root here is of the remainder after subtracting the given fractional parts, instead of being of the whole.
The Dvirogra-śēşamūla variety consists of problems wherein a known number of things is first removed, then some fractional parts of the successive remainders and then some multiple of the square root of the further remainder are removed, and lastly the numerical value of the remaining portion is given. The known number first removed is called pūrvāgra'
In the Amśamūla variety, a multiple of the square root of a fractional part of the total number is supposed to be first removed, and then the numerical value of the remaining portion is given. Dviragraśēşamūla and Amśmūla, and then Bhāgābhyāsa', then Amśavarga, Mūlamiśra and Bhinnadŗśya.
The rule relating to the Bhāga and the Śēșa varieties therein, (i.e., in miscellaneous problems on fractions).
4. In the operation relating to the Bhāga' variety, the (required) result is obtained by dividing the given quantity by one as diminished by the (known) fractions. In the operation relating to the śēșa variety, (the required result) is the given quantity divided by the product of (the quantities obtained respectively by) subtracting the (known) fractions from one.
Examples in the Bhāga variety.
5. Of a pillar, part was seen by me to be (buried) under the ground, in water. in moss, and 7 hastas (thereof was free) in the air. What is the length of the) pillar?
In the Bhāgābhyāsa or Bhāgasamvarga variety, the numerical value is given of the portion remaining after removing from the whole the product or products of certain fractional parts of the whole taken two by two.
The Amśvarga variety consists of problems wherein the numerical value is given of the remainder after removing from the whole the square of a fractional part thereof, this fractional part being at the same time increased or decreased by a given number.
The Mūlamiśra variety consists of problems wherein is given the numerical value of the sum of the square root of the whole when added to the square root of the whole as increased or diminished by a given number of things.
In the Bhinnadŗśya variety, a fractional part of the whole as multiplied by another fractional part thereof is removed from it, and the remaining portion is expressed as a fraction of the whole . Here it will be seen that unlike in the other varieties the numerical value of the last remaining portion is not actually given, but is expressed as a fraction of the whole.
4. Algebraically, the rule relating to the Bhāga' variety is where x is the unknown collective quantity to be found out, a is the dŗśya or agra, and b is the bhāga or the fractional part or the sum of the fractional parts given.
It is obvious that this is derivable from the equation .
The rule relating to the Śeşa variety, when algebraically expressed, comes to where b1,b2,b3,&c are fractional parts of the successive remainders. This formula also is derivable from the equation 6. Out of a collection of excellent bees, took delight in pālalī' trees, in kadamba tree, in mango threes, in a campaka tree with blossoms fully opened; in a collection of full-blown lotuses, opened by the rays of the sun and (finaly), a single intoxicated bee has been circling in the sky. What is the number (of bees) in that collection ?
7. A certain śrāvaka, having gathered lotus, and loudly uttering hundreds of prayers, offered those (lotuses) in worship, of those lotuses and and of this respectively to four tīrthaňkaras commencing with the excellent Jina Vŗșabha; then to Sumati as well as of this (same of the lotuses); (thereafter) he offered in worship to the remaining (19) tīrthaňkaras 2 lotuses each with a mind well-pleased. What is the numerical value of (all) those (lotuses) ?
8 to 11. There was seen a collection of pious men, who had brought their senses under control, who had driven away the poison-like sin of karma, who were adorned with righteous conduct and virtuous qualities and whose bodies had been embraced by the Lady Mercy. Of that (collection). was made up of logicians; this diminished by of itself was made up of the teachers of the true religion; the difference between these two (, namely, and of ) was made up of those that knew the Vedas ; this (last proportional quantity) multiplied by 6 was made up of the preacher of the rules of conduct, and this very same (quantity) diminished by of itself was made up of astrologers; the difference between these two (last mentioned quantities) was made up of controversialists; this (quantity) multiplied by 6 was made up of penitent ascetics ; and 9X8 leading ascetics were (further) seen by me near the top of a mountain with their shining bodies highly heated by the rays of the sun. Tell me quickly (the measure of this) collection of prominent sages.
12 to 16. (A number of) parrots descended on paddy-field beautiful with (the crops) bent down through the weight of the ripe corn. Being scared away. by men, all of them suddenly flew up. One-half of them went to the east, and went to the south-east; the difference between those that went to the east and those that went to the south-east, diminished by half of itself and (again) diminished by the half of this (resulting difference), went to the south : the difference between those that went to the south and those that went to the south-east diminished by of itself, went to the south-west; the difference between those that went to the south and those that went to the south-west, went to the west; the difference between those that went to the south-west and those that went to the west, together with of itself, went to the north-west; the difference between those that went to the north-west and those that went to the west, together with of itself, went to the north ; the sum of those that went to the north-west and those that went to the north, diminished by of itself, went to the north-east; and 280 parrots were found to remain in the sky (above). How many were the parrots (in all) ?
17 to 22. One night, in a month of the spring season, a certain young lady . . . . was lovingly happy along with her husband on . . . . the foor of a big mansion. white like the moon, and situated in a pleasure-garden with trees bent down with the load of the bunches of flowers and fruits, and resonant with the sweet sounds of parrots, cuckoos and bees which were all intoxicated with the honey obtained from the flowers therein. Then on a love-quarrel arising between the husband and the wife, that lady's necklace made up of pearls became sundered and fell on the floor. One-third of that necklace of pearls reached the maid-servant there; fell on the bed: then of what remained (and one-half of what remained thereafter and again of what remained thereafter) and so on, counting six times (in all), fell all of them everywhere; and there were found to remain (unscattered) 1,161 pearls; and if you know (how to work) miscellaneous problems (on fractions), give out the (numerical) measure of the pearls (in that necklace).
23 to 27. A collection of bees characterized by the blue color of the shining indranīla gem was seen in a flowering pleasure-
17. Certain epithets here have not been considered fit for translation. garden. One-eighth of that (collection) became hidden in aśōka trees, in kuțaja trees. The difference between those that hid themselves in the kuțaja trees and the aśōka trees, respectively, multiplied by 6, became hidden in a crowd of big pātalī' trees. The difference between those that hid themselves in the pātalī trees and the aśōka trees, diminished by of itself became hidden in an extensive forest of sāla trees. The same difference, together with of itself, became hidden in a forest of madhuka trees ; of that whole collection of bees was seen hidden in the vakula trees with well-blossomed flower-buds; and that same part was found hidden in tilaka, kuravaka, sarala and mango trees, and on collections of lotuses, and at the base of the temples of forest elephants; and 33(remaining) bees were seen in a crowd of lotuses, that were variegated in color on account of the large quantity of (their) filaments. Give out, O you arithmetician, the (numerical) measure of that collection of bees.
28. Of a herd of cattle, is on a mountain; of that is at the base of the mountain; and 6 more parts, each being in value half of what precedes it, are found together in an extensive forest, and there are (the remaining) 82 cows seen in the neighbourhood of a city. Tell me, O you my friend, the (numerical) measure of that herd of cattle.
Here end the examples in the Bhāga variety.
Examples in the Śēșa variety.
29-30. Of a collection of mango fruits, the king (took) ; the queen (took) of the remainder, and three chief princes took and (of that same remainder) ; and the youngest child took the remaining three mangoes. O you, who are clever in (working) miscellaneous problems on fractions give out the measure of that (collection of mangoes).
31. One-seventh of (a herd of) elephants is moving on a mountain ; portions of the herd, measuring from in order up to in the end, of every successive remainder, wander about in a forest; and the remaining 6 (of them) are seen near a lake . How many are those elephants ? 32. Of (the contents of) a treasury, one man obtained part; others obtained from in order to , in the end, of the successive remainders ; and (at last) 12 purāņas were seen by me (to remain). What is the (numerical) measure (of the purāņas contained in the treasury)?
Here end examples in the Śēșa variety.
The rule relating to the Mūla variety (of miscellaneous problems on fractions) :--
33. Half of (the coefficient of) the square root (of the unknown quantity) and (then) the known remainder should be (each) divided by one as diminished by the fractional (coefficient of the unknown) quantity . The square root of the (sum of the) known remainder (so treated), as combined with the square (of the coefficient) of the square root (of the unknown quantity dealt with as above), and (then) associated with (the similarly treated coefficient of) the square root (of the unknown quantity), and (thereafter) squared (as a whole), gives rise to the (required unknown) quantity in this mūla variety (of miscellaneous problems on fractions).
Examples in illustration thereof.
34. One-fourth of a herd of camels was seen in the forest; twice the square root (of that herd) had gone on to mountain slopes; And 3 times 5 Camels (were), however, (found) to remain on the bank of a river. What is the (numerical) measure of that herd of camels?
35. After listening to the distinct sound caused by the drum made up of the series of clouds in the rainy season, and (of a collection) of peacocks, together with of the remainder and of the remainder (thereafter), gladdened with joy, kopt on dancing on
33. Algebraically expressed, this rule comes to ; this is easily obtained from the equation . This equation is the algebraical expression of problems of this variety. Here c stands for the coeffcient of the square rōt of the unknown quantity to be found out. the big theatre of the mountain top; and 5 times tho square root (of that collection) stayed in an excellent forest of vakula trees; and (the remaining) 25 were seen on a punnāga tree. O arithmetician, give out after calculation (the numerical measure of) the collection of peacocks.
86. One-fourth (of an unknown number) of sārasa birds is moving in the midst of a cluster of lotuses; and parts (thereof) as well as 7 times the square root (thereof) move on a mountain; (then) in tho midst of (some) blossomed vakula trees (the remainder) is (found to be) 56 in number. O you clever friend, tell me exactly how many birds there are altogether.
37. No fractional part of a collection of monkeys (is distributed anywhere); three times its square root are on a mountain; and 40 (remaining) monkeys are seen in a forest. What is the measure of that collection (of monkeys) ?
38. Half (the number) of cuckoos were found on the blossomed branch of a mango tree; and 18 (were found) on a tilaka tree. No (multiple of the) square root (of their number was to be found anywhere). Give out (the numerical value of) the collection of cuckoos.
39. Half of a collection of swans was found in tho midst of vakula trees ; five times the square root (of that collection was found) on the top of tamāla' trees ; and here nothing was seen (to romaim therafter). O friend, give out quickly the numerical measure of that (collection).
Here ends the Mūla variety (of miscellaneous problems on fractions).
The rule relating to the Śēșamūla variety (of miscellaneous problems on fractions).
40. (Take) the square of half (the coefficient) of the square root (of the remaining part of the unknown collective quantity), and
40. Algebraically, . From this the value of x is to be found out according to rule 4 given in this chapter. This value of x-bx is obtained easily from the equation . combined it with the known number remaining, and (then extract) the square root (of this sum, and make that square root become) combined with half of the previously mentioned (coefficient of the) square root (of the remaining part of the unknown collective quantity). The square of this (last sum) will here be the required result, when the remaining part (of tho unknown collective quantity) is taken as the original (collective quantity itself). But when that remaining part (of the unknown collective quantity) is treated merely as a part, the rule relating to the bhāga variety (of miscellaneous problems on fractions) is to be applied.
Examples in illustratino thereof.
41. One-third of a herd of elephants and three times tho square root of the remaining part (of the herd) were seen on a mountain slope; and in a lake was seen a male elephant along with female elephants (constituting the ultimate remainder). How many were the elephants here ?
42 to 45. In a garden beautified by groves of various kinds of trees, in a place free from all living animals, many ascetics were seated. Of them the number equivalent to the square root of the whole collection were practising yōga at the foot of the trees. One-tenth of the remainder, the square root (of the remainder after deducting this), (of the remainder after deducting this), then the square root (of the remainder after deducting this), (of the remainder after deducting this, the square root (of the remainder after deducting this), (of the remainder after deducting this), the square root (of the remainder after deducting this), ( of the remainder after deducting this), the square root (of the remainder after deducting this),(of the remainder after deducting this), the square root (of the remainder after deducting this)—these parts consisted of those who were learned in the teaching of literature, in religious law, in logic, and in politics, as also of those who were versed in controversy, prosody, astronomy, magic, rhetoric and grammar and of those who possessed the power derived from the 12 kinds of austerities, as well as of those who possessed an intelligent knowledge of the twelve varieties of the ańga-śāstra; and at last 12 ascetics were seen (to remain without being included among those mentioned before). O(you) excellent ascetic, of what numerical value was (this) collection of ascetics ?
46. Five and one-fourth times the square root (of a herd) of elephants are sporting on a mountain slope; of the remainder sport on the top of the mountain; five times the square root of the remainder (after deducting this) sport in a forest of lotuses; and there are 6 elephants then (left) on the bank of a river. How many are (all) the elephants here ?
Here ends the Śēșamūla variety (of miscellaneous problems on fractions).
The rule relating to the Śēșamūla variety involving two known (quantities constituting the) remainders:--
The (coefficient of the) square root (of the unknown collective quantity), and the (final) quantity known (to remain), should (both) be divided by the product of the fractional (proportional) quantities, as subtracted from one (in each case); then the first known quantity should be added to the (other) known quantity (treated as above). Thereafter the operation relating to the Śēșamūla variety ( of miscellaneous problems on fractions is to be adopted).
47. Algebraically, this rule enables us to arrive at the expressions and , which are required to be substituted for c and a respectively in the formula for Śēșamūla, which is . In applying the value of b becomes zero, as the mūla square root involved in the dviragra-śēşamūla is that of the total collective quantity and not of a fractional part of that quantity. Substituting a desired, we get . This result may easily be obtained from the equation where b1,b2, &c are, the various fractional parts of the successive remainders; and 11 and a2 are the first known quantity and the final known quantity respectively.
Examples in illustration thereof.
48. A single bee (out of a swarm of bees) was seen in the sky; of the remainder (of the swarm), and of the remainder (left thereafter), and (again), of the remainder (left thereafter) and (a number of bees equal to the square root (of the numericial value of the swarm, were seen) in lotuses and two (bees remaining at last were seen) on a mango tree. How many are those (bees in the swarm)?
49. Four (out of a collection of) lions were seen on a mountain; and fractional parts commencing with and ending with of the successive remainders (of the collection) , and (lions equivalent in number to) twice the square root (of the numerical value of the collection),as also (the finally remaining) four (lions, were seen a forest. How many are those (lions in the collection )?
50. (Out of a herd of deer) two pairs of young female deer were seen in a forest; fractional parts commencing with and ending with of the (successive) remainders (of the herd were seen) near a mountain; (a number) of them (equivalent to) 3 times the square root (of the numerical value of the herd) were seen in an extensive paddy field; and (ultimately) only ten remained on the bank of a lotus-lake. What is the (numerical) measure of the herd १
Thus ends the Śēșamūla variety involving two known quantities.
The rule relating to the Amsamūla variety (of miscellaneous problems on fractions).
51. Write down (the coefficient of) the square root (of the given fraction of the unknown collective quantity) and the known quantity (ultimately remaining, both of these) having been
50. The word hariņī occurring in this stanza not only means a 'female deer' but is also the name of the metre in which the stanza is composed.
51, Algebraically stated, this rule helps us to arrive at cb and ab, wich are required to be substituted for c and b respectively in the formula , as in the Śēșamūla variety. As pointed out in the note multiplied by the (given proportional) fraction; then that result which is arrived at by means of tho operation of finding out (the unknown quantity) in the Śēșamūla variety (of miscellaneous problems on fractions), when divided by the (given proportional) fraction, becomes the required quantity in the Amsamūla variety (of miscellaneous problems on fractions).
Another rule relating to the Amsamūla variety.
52. The known quantity given as the (ultimate) remainder is divided by the (given proportional) fraction and multiplied by four; to this the square (of the coefficient) of the square root (of the given fraction of the unknown collective quantity) is added ; then the square root (of this sum), combined with (the above mentioned coefficient of) the square root (of the fractional unknown quantity), and (then) halved, and (then) squared, and (then) multiplied by the (given proportional) fraction, becomes the required result.
Examples in illustration thereof.
58. Eight times the square root of part of the stalk of a lotus is within water, and 16 ańgulas (thereof are) in the air (above water); give out the height of the water (above the bed) as well as of the stalk (of the lotus).
54–55. (Out of a herd of elephants), nine times the square root of part of their number, and six times the square root of of the remainder (left thereafter), and (finally) 24 (remaining) elephants with their broad temples wetted with the stream of the exuding ichor, were seen by me in a forest. How many are (all) the elephants ?
under stanza 47, x-bx becomes x here also. After substituting as desired, and dividing the result by b, we get . This value of x may be easily arrived at from the equation .
52. Algebraically stated, . This is obvious from the equation given in the note under the previous stanza. 56. Four times the square root of the number of a collection of boars went to a forest wherein tigers were at play; 8 times the square root of , of the remainder (of the collection) went to a mountain; and 9 times the square root of of the (further) remainder (left thereafter) went to the bank of a rivor; and bears equivalent in (numerical) measure to 56 were seen (ultimately) to remain (where they were) in the forest. (Give out the (numerical) measure of (all) those (boars).
Thus ends the Amśamūla variety.
The rule relating to the Bhāgasamvarya variety (of miscellaneous problems on fractions):--
57. From the (simplified) denominator (of the specified compound fractional part of the unknown collective quantity), divided by its own (related) numerator, (also simplified), subtract four times the given known part (of the quantity), then multiply this (resulting difference) by that same (simplified) denominator (dealt with the above). The square root (of this product) is to be added to as well as subtracted from that (same) denominator (so dealt with); (then) the half (of either) of these (two quantities resulting as sum or difference is the unknown) collective quantity (required to be found out).
Examples in illustration thereof.
58. A cultivator obtained (first) of a heap of paddy as multiplied by (of that same heap); and (then) he had 24 vāhas (left in addition). Give out what the measure of the heap is.
59. One-sixteenth part of a collection of peacocks as multiplied by itself, (i.e., by the same a part of the collection), was found
56. The word śārdūlavikrīḍita in this stanza means 'tigers at play', and at the same time happens to be the name of the metre in which the stanza is composed
57. Algebraically stated ; and this value of x may easily be obtained from the equation , where and are the fractions contemplated in the rule. In a mango tree; the remainder as multiplied by that same ( part of that same remainder), as also (the remaining) fourteen (peacocks) were found in a grove of tamāla trees. How many are they (in all) ?
60. One-twelfth part of a pillar, as multiplied by part thereof, was to be found underwater; of the remainder, as multiplied by thereof was found (buried) in the mire (below); and 20 hastas of the pillar were found in the air (above the water). O friend, you give out the measure of the length of the pillar.
Here ends the Bhāgassamvarga vareity.
The rule relating to the Aṃśavarga variety (of miscellaneous problems on fractions), characterised by the subtraction or addition (of known quantities):--
61. (Take) the half of the denominator (of tho specified fractional part of the unknown collective quantity), as divided by its own (related) numerator, and as increased or diminished by the (given) known quantity which is subtracted from or added to (the specified fractional part of the unknown collective quantity). The square root of the square of this (resulting quantity), as diminished by the square of (the above known) quantity to be subtracted or to be added and (also) by tho known remainder (of the collective quantity), when added to or subtracted from the square root (of the square quantity mentioned above) and then divided by the (specified) fractional part (of the unknown collective quantity), gives the (required) value (of the unknown collective quantity).
Examples of the minus variety.
62.[62] (A number) of buffaloes (equivalent to) the square of (of the whole herd) minus 1 is sporting in the forest. The
62.^ Algebraically, . This value is obtained from the equation , where d is the given known quantity. (remaining) 15 (of them) are seen grazing grass on a mountain. How many are they (in all) ?
63. (A number) of elephants (equivalent to) of the herd minus 2, as multiplied by that same ( a of the herd minus 2), is found playing in a forest of sallakā trees. The (remaining) elephants of the herd measurable in number by the square of 6 are moving on a mountain. How many(together) are (all) these elephants here?
An example of the plus variety.
64.[64] (A number of peacocks equivalent to) of their whole collection plus 2, multiplied by that same ( of the collection plus 2), are playing on a jambā' tree. The other (remaining) proud peacocks (of the collection), numbering 22 x 5, are playing om a mango tree. O friend, give out the numerical measure of (all) these (peacocks in the collection).
Here ends tho Aṃśavarya variety charecterised by plus or minus quantities.
The rule relating to the Mūlamiśravariety (of miscellaneous problems on fractions).
65.[65]To the square of the (known) combined sum (of the square roots of the specified unknown quantities), the (given) minus quantity is added, or the (given) plus quantity is subtracted (therefrom); (then) the quantity (thus resulting) is divided by twice the combined sum (referred to above); (this) when squared gives rise to the required value (of the unknown collection). In relation to the working out of the Mūlamiśra variety of problems, this is the rule of operation.
64.^ The word meta mattamayūra occurring in the stanza means a proud peacock and is also the name of the metre in which the stanza is composed.
65.^ Algebraically . This is easily derived from the equation . The quantity m is here the known combined sum mentioned in the rule.
Examples of the minus variety.
66. On adding together (a number of pigeons equivalent to) the square root of the (whole) collection of pigeons and (another number equivalent to the square root of the (whole) collection as diminished by 12, (exactly) 6 pigeons are seen (to be the result). What is (the numerical value of) that collection (of pigeons)?
67. The sum of two (quantities, which are respectively equivalent to the) square roots of the (whole) collection of pigeons and of (that same) collection as diminished by the cube of 4 , amounts to 16. How many are the birds in that collection ?
An example of the plus variety
68. The sum of the two (quantities, which are respectively equivalent to the ) square root (of the numerical value) of a collection of superior swans and (the square root of that same collection) a combined with 68, amounts to (62 - 2. Give out how many swans there are in that collection.
Here ends the Mālamiśra variety.
The rule relating to the Bhinnadŗśya variety (of miscellaneous problems on fractions)
69.[69] When one , diminished by the (given) fractional remainder (related to the unknown quantity), is divided by the product of the (specified) fractional parts (unrelated thereto), the result which is (thus) arrived at becomes the (required) answer in working out the Bhinnadŗśya variety (of problems on fractions).
Examples in illustration thereof.
70. One-eighth part of a pillar, as multiplied by the part (of that same pillar), was found (to be buried) in the sands ; of the pillar was visible (above). Say how much the (vertically measured) length of the pillar is.
69.^ Algebraically stated, . This is obvious from the equation . 71.[71] (Elephants equivalent in number to) part of the whole herd of elephants, as multiplied by (of that same herd ) as divided by 2, are in a happy condition on a plain. The remaining (ones forming) (of the herd), resembling exceedingly dark masses of clouds in form are playing on a mountain. O friend, you tell me how the numerical measure of the heard of elephants.
72. (Ascetics equivalent in number to ) part of a collection of ascetics, as divided by 3 and as multiplied by that same ( part divided by 3), are living in the interior of a forest ; (the remaining ones forming) part (of that collection) are living on a mountain. O you, who have crossed over to the other shore of the ocean-like miscellaneous problems on fractions, tell me quickly the (numerical) value of that (collection of ascetics).
Here ends the Bhinnadŗśya variety.
Thus ends the third subject of treatment known as Prakārņaka in Sārasangraha , which is a work on arithmetic by Mahāvērācārya.
71.^ The word pŗthvi occurring in this stanza means 'the earth', and is also the name of the metre in which the stanza is composed.CHAPTER V.
RULE-OF-THREE
The fourth subject of treatment.
1. Salutation to that blessed Vardhamāna, who is like a (helpful) relation to (all) the three worlds, and is (resplendent) like the sun in the matter of absolute knowledge and has cast off (the taint of) all the karmas
Next we shall expound the fourth subject of treatment, viz., rule-of-three.
The rule of operation in respect thereof is as follows:--
2.[2] Here, in the rule-of-throe, Phala multiplied by Ichhā and divided by Pramāņā, becomes the (required) answer, when the Icchā and the Pramāņā are similar, (i.e., in direct proportion); and in the case of this (proportion) being inverse, this operation (involving multiplication and division) is reversed, (so as to have division in the place of multiplication and multiplication in the place of division}.
Examples relating to the former half of the above rule, i.e.,
on the direct rule-of-three3.[3] The man who in days goes over yōjanas-give out what (distance) he (goes over) in a year and a day.
4. A lame man walks over of a krōśa together with (thereof) in 7 days. Say what (distance) he (goes over) in 3 years (at this rate ).
5. A worm goes in of a day over of an ańgula. In how many days will it reach the top of the Mēru moumtain from its bottom?
6. The man who in 3 days uses up 1 kārşāpaņas--in what time (will) he (use up) 100 purāņas along with one paņa ?
2.^ Pramāņa and Phala together give the rate, in which Phala is a quantity of the same kind as the required answer and 'Pramāņa is of the same kind as Icchā. This Icchā is the quantity about wich something is required to be found out at the given rate. For instance the problem stanza 3 here, days is the Pramāņa, yōjanas is the Phala, and 1 year and 1 day is the Icchā.
3.^ The height of the Mēru mountain is supposed to be 99,000 yōjanas or 76,032,000,000 ańgulas. 7.[7]A good piece kṛṣṇāgaru[1], 12 hastas in length and 3 hastas in diameter, is consumed (at the rate of) 1 cubic angula a day. What is the time required for the (complete) consumption of this cylinder ?
8. (If) a vāha of very superior black gram, along with 1 drōņa,1 āḍhaka and 1 kuḍava (thereof), has been purchased by means of svarņas, what measure (may we purchase of it) by means of svarņas?
9. Where 1 pala of kuńkuṃa is obtainable by means of purāņas, what measure (of it) may (we obtain) there by means of 100 purāņas ?
10. By means of palas of ginger, paņas were obtained; say, O friend, what (may be obtained) in return for palas of ginger?
11. By means of kārṣāpaṇas, a man obtains palas of silver; what (weight does he obtain thereof) by means of 10,000 karṣas
12. By means of palas of camphor, a man obtains 5 dīnāras along with 1 bhāga, 1 aṃśa and 1 kalā'. What (does he obtain) here by means of 1,000 palas (thereof) ?
13. The man who purchases here palas of ghee by means of paṇas-what (measure of it does the purchase) by means of karṣas?
14. By means of purāṇas, pains of cloths were obtained. O friend, say what (number of them may be obtained) by means of 61 karṣas?
15-16.[15-16]There is a square well without water, (cubically) measuring 512 hastas. A hill rises on its bank; from the top
7.^ Here the process of finding out, from the given diameter, the area of the cross-section of a cylinder is supposed to be known. This is given in the sixth Vyavahāra, in the 19th stanza, where the area of a circle is said to be approximately equal to the diameter squared and then divided by 4 and multiplied by 3.
^ Kṛṣṇāgaru is a kind of fragrant wood burnt in fire as intense.
15-16.^ In this problem, the stream of water is as long as the mountain is high, so that as soon as it reaches the bottom of the mountain, it is supposed to cease to flow at the summit. For finding out the quantity of water in Vāhas, etc., the relation between cubical measure and liquid measure should have been given. The Sanskrit commentary in P and the Kanarese ṭīkā in B state that 1 cubic aṅgula of water is equal to 1 kaṛṣā thereof in liquid measure. thereof hows down, (to the bottom) a crystal-clear stream of water having 1 aṅgula for the diameter of its circular section, and the well becomes quite filled with water within. What is the height of the hill, and (what) the numerical value (of the liquid-measure) of water ?
17.[17]A king gave, on (the occasion of) the saṅkrāti, to 6 Brahmins, 2 daōṇas of kidney-bean, 9 kuḍabas of ghee, 6 drōṇas if ruce, 8 pairs of cloths, 6 cows with calves and 8 svarṇas. Give out quickly, O friend, what (the measure) is (of) the kidney-bean and the other things given by him (at that rate) to 336 Brahmins.
Here ends the (direct) rule-of-three .
Examples on rule-of-three as explained in the fourth pāda[*]
(of the rule given above)18.[18]How much is the gold of 9 varṇas for 90 of pure gold, as also for 100 gold (Dharaṇas) along with a guṅjā thereof made up of gold of varṇas?
19. There are 300 pieces of China silk of 6 hastas in breadth as well as in length; give out, O you who know the method of inverse proportion, how many pieces (of that same silk) there are (in them, each) measuring 5 by 3 hastas.
Here ends the inverse rule-of-three.
An example on inverse double rule-of-three.
20. Say how many pieces of that famous clothing, each measuring 2 hastas in breadth and 3 hastas in length, are to be found in 70(pieces) of China Silk, (each) measuring 5 hastas in breadth and 9 hastas in length.
An example on inverse treble rule-of-three.
21.Say how many images of Tirthaṅkaras, (each) measuring 2 by 6 by 1 hastas, there may be in a big gem, which is 4 hastas in breadth, 9 hastas in length and 8 hastas in height.
17.^ Saṇkranti is the passage of the sun from one zodiacal sign to another.
18.^ Pure gold is here taken to be of 16 varṇas,
^* The reference here is to the fourth quarter of the second stanya in this clhapter
An example on verse quadruple rule-of-three.
22. There is a block of stone (suited for building purposes), which measures 6 hastas in breadth, 30 hastas in length and 8 hastas in height, and (it is) 9 in worth. By means of this (given in exchange), how many (blocks) of such stone, fit to be used in building a Jina temple, (may be obtained, each) measuring 2 by 6 by 1 (hastas), and being 5 in worth?
Thus ends the inverse double, treble and quadruple rule-of-three.
The rule in regard to (problems bearing on associated) forward and backward movement.
28. Write down the net daily movement, derived from the difference of (the given rates of) forward and backward movements, each (of these rates) being (first) divided by its own (specified) time ; and then in relation to this (net daily movement), carry out the operation of the rule-of-three.
Examples in illustration thereof.
24-25. In the course of of a day, a ship goes over of a krōśa in the ocean; being opposed by the wind she goes back (during the same time) — of a krōśa. Give out, O you who have powerful arms in crossing over the ocean of numbers well, in what time that (ship) will have gone over yōjanas.
26. A man earning (at the rate of) 1 of a gold coin in days, spends in days of the gold coin as also of that itself; by what time will he own 70 (of those gold coins as his not earnings)?
27. That excellent elephant, which, with temples that are attacked by the feet of bees greedy of the (flowing) ichor, goes over as well as of a yōjana in days, and moves back in days over of a krōśa: say in what time he will have gone over (a net distance of) I00 yōjanas less by krōśa.
28-30.[28-30]A well completely filled with water is 10 daṇḍas depth a lotus sprouting up therein grows from the bottom
28-80.^ The 'depth' of the well is mentioned in the original as 'height' measured from the bottom of it. (at the rate of) aṅgulas in a day and half; the water (thereof) flows out through a pump (at the rate of) aṅgulas (of the well in depth) in days; aṅgulas of water (in depth) are lost in a day by evaporation owing to the (heating) rays of the sun; a tortoise below pulls down aṅgula of the stalk of the lotus plant in days. By what time will the lotus be on the same level with the water (in the well) ?
31. A powerful unvanquished excellent black snake, which is 32 hastas in length, enters into a hole (at the rate of) aṅgulas in of a day; and in the course of of a day its tail grows by of an aṅgu la. O ornament of arithmeticians, tell me by what time this same (serpent) enters fully into the hole. [*]
Thus end the (problems bearing on associated) forward and backward movements.
The rule of operation relating to double, treble and quadruple rule-of-thrē.
[32]. Transpose the Phala from its own place to the other place (wherein a similar concrete quantity would occur); (then, for the purpose of arriving at the required result), the row consisting of the larger number (of different quantities) should be, (after they are all multiplied together, divided by the row consisting of the
32.^ The transference of the Phala and the other operations herein mentioned will be clear from the following worked out example.
The data in the problem in stanza No.36 are to be first represented thus:-
9Mānīs 1Vāha + 1 Kumbha. 3 Yōjanas. 10 Yōjanas. 60 Paṇas
When the Phala here, viz., 60 pounds, is transferred to the other row we have--9 Mānīs. 1 Vāha + 1 Kumbha = Vāha. 3 Yōjanas. 10 Yōjanas 60 Paṇas.
Now the right hand row, consisting of a larger number of different quantities, should be, after they are all multiplied together, divided by the smaller left hand row similarly dealt with.
Then we have
The result here gives the number of paṇas to be found out.
smaller number (of different quantities, after these are also similarly thrown together and multiplied); but in the matter of the buying and selling of living animals (the transposition is to take place) only (in relation to the numbers representing) them.
Examples in illustration thereof.
33. At the rate of 2, 3 and 4 per cent. (per month), 50, 60 and 70 Purāṇas (respectively) put to interest by a person desiring profit. How much interest does he obtain in ten months ?
34. The interest on gold coins for of a month is . How much (will it be) on gold coins for months ?
35. He who obtains 20 gems in return for 100 gold piccas of 16 varṇas—what (will he obtain) in return for 288 gold pieces of 10 varṇas ?
36. A man, by carrying 9 mānīs of wheat over 3 yōjanas,obtained 60 paṇas. How much (would he obtain) by carrying one kumbha along with one vāha (thereof) over 10 yōjanas ?
Examples on barter
87. A man obtains 3 karṣas of musk for 10 gold coins and 2 karṣas of camphor for 8 gold coins. How many (karṣas of camphor does he obtain) in return for 800 karṣas of musk ?
88. In return for 8 (māṣas in weight of silver), a man obtains 60 jack fruits; and in return for 10 māṣas (in weight of silver he obtains) 80 pomegranates. How many pomegranates (does he obtain) in return for 900 jack fruits ?
Examples of(problems bearing on) the buying and selling
of animals39. Ṭwenty horses, (each) of 16 years (of age), are worth 100,000 gold coins. O leading arithmetician, say how much 70 horses, (each) of 10 years (of age), will be (worth) at this (rate).
40. Three hundred gold coins form the price of 9 damsels, (each) of 10 years (of age). What is the price of 36 damsels, (each) of 16 years (of ago) ? 41. What is the interest for 10 months on 90, invested at the rate of 6 per 100 (per month)? O you, who are a mirror to the face of arithmeticians, say, with the aid of the two (other requisite) known quantities, what the time in relation to that (interest) is, and what the capital is (in relation to that interest and time).
An example on treble rule-of-three.
42. Two pieces of sandal-wood, measuring 3 and 4 hastas in diameter and length respectively, are worth 8 gold coins. At this (rate) how much will be the worth of 14 (pieces of sandal-wood, each) measuring 6 and 9 hastas in diameter and length (respectively) ?
Thus ends treble rule-of-three.
An example on quadraple rule-of-thrē
[43]. A household well, measuring 5, 8 and 8 hastas in breadth, length and height (from the bottom, respectively), contain 6 vāhas of water; O you, who are learned, give out how much (water) 9 wells, (each being) 7 hastas in breadth, 60 in length and 5 in height (from the bottom, will contain ).
Thus ends the fourth subject of treatment known as Rule-of-three in Sārasaṅgraha which is a work on arithmetic by Mahāvirācārya.
43.^ The word śālinī occurring in this stanza indicates the name of the metre in which the stanza is composed, at the same time that it means 'bolonging to a house'. CHAPTER VI--MIXED PROBLEMS 9B
CHAPTER VI.
MIXED PROBLEMS
The Fifth Subject of Treatment
1. For attaining the supreme good, we worshipfully salute the holy Jinas, who are in possession of the fourfold infinite attributes, who are the makers of tīrthas, who have attained self-conquest, are pure, are honoured in all the three worlds and are also excellent preceptors—the Jinas who have gone over to the (other) shore of the ocean of the Jaina doctrines, and are the guides and teachers of (all) born beings, and who, being the abode of all good qualities, are good in themselves and do good to others.
Hereafter We shall expound the fifth subject of treatment known as mixed problems. It is as follows:--
Statement of the meaning of the technical terms saṅkramaṇa and vișama-saṅkramaṇa:
2. Those who have gone to the end of the ocean of calculation say that the halving of the sum and of the difference (of any two quantities) is (known as) saṅkramaṇa, and that the saṅkramaṇa of two quantities which are (respectively) the divisor and the quotient is that which is vișama(i.e., vișama-saṅkramaṇa).
Examples in illustration thereof.
3. What is the saṅkramaṇa where the number 12 (is associated) with 2; and what is the divisional vișama-saṅkramaṇa of that (same) number (12 in relation to 2) ?
1.^ Tirtha is interpreted to mean a ford intended to cross the river of mundane existence which is subject to karma and reincarnation. The Jinas are conceived to be capable of enabling the souls of men to get out of the stream of samsāra or the recurring cycle of embodied existence. The Jinas are therefore called tīrthaṅkaras.
2{{note|2}. Algebraically the saṅkramaṇa of any two quantities a and b is finding out and their vișama-saṅkramaṇa is ariving at and .Double Rule-of-three.
The rule for arriving at (the value of) the interest which (operation) is of the nature of double rule-of-three :--
4. The number representing the Icchē, i.e., the amount the interest whereon is desired to be found out, is multiplied by the time connected with itself and is then multiplied by (the number representing ) the (given ) rate of interest for the given capital ; (then the resulting product) is divided by the time and the capital quantity (connected with the rate of interest); this (quotient) is, in arithmetic, the interest of the desired amount.
Examples in illustration thereof.
5.Purānas, 50, 60, and 70 (in amount) were lent out on interest at the rate of 3, 5 and 6 per cent (per mensem respectively); what is the interest for 6 months ?
6.(A sum of) 30 kārșāpaṇas and 8 paṇas were lent out on interest at the rate of per cent (per month) ; what is the interest produced in exactly months ?
7.The interest on 60 for 2 months is seen to be 5 purāṇas with 3 paṇas; what would be the interest on 100 for 1 year?
8.The interest for 1 months and a half on leading out 150 is 15. What would be the interest obtained at this rate on 300 for 10 months ?
9.A merchant lent out 63 kārșāpaṇas at the rate of 8 for 108 (per month). What (is the interest) for months ?
The rule for finding out the capital lent out:--
10.The capital quantity (involved in the rate of interest) is multiplied by the time connected with itself and is then divided
4. Symbolically , where T, C and I are respectively the time, capital and interest of the pramāṇa or the rate, and t, c and i are respectively the time, capital and interest of the iccha. For an explanation of pramāṇa, iccha, &c,see note under Ch. V. 2.
5. Unless otherwise mentioned, the rate of interest is for 1 month.
10. Symbolically by the interest connected with itself. (Then) this (quotient) has to be divided by the time connected with the capital lent out; (this last) quotient when multiplied by the interest (that has accrued) becomes the capital giving rise to that (interest).
Examples in illustration thereof.
11. In lending out at the rate of per cent (per mensem), a month and a half (is the time for which interest has accrued), and a certain person thus obtains 5 purāaṇas as the interest. Tell me what the capital is in relation to that (interest).
12. IThe interest on 70 for months is exactly . When the interest is for months what is the capital lent out?
13. In lending out at the rates of 3, 5 and 6 per cent (per mensem), the interest has so accrued in 6 months as to be 9, 18 and ; (respectively); what are the capital amounts lent out?
The rule for finding out the time (during which interest has accrued):-
[14] Take the capital amount (involved in the given rate of interest) as multiplied by the time (connected therewith); then cause this to be divided by its own (connected) rate-interest and by the capital lent out; then this (quotient) here is multiplied by the interest that has accrued on the capital lent out. Wise men say that the resulting (product) is the time (for which the interest has accrued)
Examples in illustration thereof.
15. O friend, mention, after calculating the time, by what time 28 will be obtained as interest on 80, lent out at the rate of per cent (per mensem).
16. The capital amount lent out at the rate of 20 per 600 (per mensem) is 420. The interest also is 84. O, friend, you tell me quickly the time (for which the interest has accrued)
14.^ Symbolically,
17. It is 96 that is lent out at the rate of 6 per cent (per mensem): the interest thereon is seen to be . What is the time (for which interest has accrued) ?
The rule regarding barter or exchange of commodities:-
18. The quantity of the commodity taken in exchange is divided by its own price as well as by the quantity of the commodity given in exchange. (It is then) multiplied by the price of the commodity given in exchange, and hereafter multiplied by the quantity of the commodity intended to be exchanged. This (resulting) product is the required quantity corresponding to the prices of the commodity given in exchange as well as of the commodity taken in exchange.
An example an illustration thereof.
19 and 20. Palas 8 of dried ginger were purchased for paṇas and palas 5 of long pepper for paṇas. Think out and tell me quickly, O you who know arithmetic, how many palas of long pepper have been purchased by one (at the above rate) by means of 80 palas of dried ginger.
Thus end the problems on double rule-of-three in this chapter on mixed problems .
Problems bearing on interest.
Next, in the chapter on mixed problems, we shall expound problems bearing on interest.
The rule for the separation of the capital and interest from their mixed sum:-
[21]. The result arrived at by carrying out the operation of division in relation to the given mixed sum of capital and interest
21.^ Symbolically, . where m = c + i; hence i = m - c. by means of one, to which the interest thereon for the (given) time is added, (happens to be the required) capital; and the interest required is the combined sum nuing this capital.
An example at illustration thereof.
22. If one lends out money at the rate of 5 per cent (per month), the combined sum of interest and capital becomes 48 in 12 months. What are the capital and the interest therein ?
Again another rule for the separation of the capital and the interest from their combined sum:-
23. The product of the given, time and the rate-interest, divided by the rate-time and the rate-capital and then combined with one, is the divisor of the combined sum of the capital and interest ; the resulting quotient has to be understood as the (required) capital
An example in illustration thereof.
24. Having given out on interest some money at the rate of per cent (per mensem), one obtains 33 In 4 months as the combined sum (of the capital and the interest). What may be the capital (therein) ?
The rule for the separation of the time and the interest from their combined sum:-
25. Take the rate-capital multiplied by the rate-time and divided by the rate-interest and by the given capital, and then combine this (resulting quantity) with one; then the quotient obtained by dividing the combined sum (of the time and interest) by this (resulting sum) indeed becomes the (required) interest.
Examples in illustration thereof.
26. Money amounting to 60 exactly was lent out at the rate of 5 per cent (per month) by one desirous of obtaining interest.
23. Symbolically . It is evident that this is very much the same as the formula given under 21.
25. Symbolically, , where m=i+t. The time (for which the interest has accrued) combined with the interest therefor is 20. What is the time here ?
27. The capital put to interest at the rate of per (per mensem) is 705. The mixed sum of its time and interest is 80. (What is the value of the time and of the interest ?)
28. The capital put to interest at the rate of per 80 for months is 400, and the mixed sum of time and interest is 36. (What is the time and what the interest ?)
The rule for arriving at the separation of the capital and the time of interest from their mixed sum:--
[29]. From the square of the given mixed sum (of the capital and the time), the rate-capital divided by its rate-interest and multiplied by the rate-time and by four times the given interest is to be subtracted. The square root of this (resulting remainder) is then used in relation to the given mixed sum so as to carry out the process of saṅkramaṇa.
Examples in illustration thereof.
30. This, viz., 4 Purāṇas is the interest on 70 (per month). The interest (obtained on the whole) is 25. The mixed sum (of the capital used and the time of interest) is . What is the capital lent out ?
31. By landing out what capital for what time at the rate of 3 per 60 (per mensem) would a man obtain 18 as interest, 66 being the mixed sum of that time and that capital ?
32. It has been ascertained that the interest for months on 60 is only . The interest here (in the given instance) is 24, and
29.^ Symbolically, or tas the case may be, where m=c+t.
The value of the quantity under the rōt, as given in the rule, in (c-t)2; and the square rōt of this and the miśra hae the operation of saṅkramaṇa performed in relation to them.
For explanation of the saṅkramaṇa see Ch. VI.2 60 is (the value of) the time combined with the capital lent out, (What is the time and what the capital ? )
The rule for arriving at the separation of the rate-interest and the required time from their sum :--
33. The rate-capital is multiplied by its own rate-time, by the given interest and by four, and is then divided by the other (that is, the given) capital. The square root of the remainder (obtained by subtracting this resulting quotient) from the square of the given mixed sum is then used in relation to the mixed sum so as to carry out the process of saṅkramaṇa.
An example in illustration thereof.
34. The mixed sum of the rate-interest and of the time (for which interest has accrued) at the rate of the quantity to be found out per 100 per month and a half is , the capital lent out being 30 and the interest accruing thereon being 5. (What is the rate of interest and what the time for which it has accrued ?)
The rule for arriving separately at the capital, time, and the interest from their mixed sum:-
35. Any (optionally chosen) quantity subtracted from the given mixed sum may happen to be the time required. By means of the interest on one for that same time, to which interest one is added, (the quantity remaining after the optionally chosen time is subtracted from the given mixed sum) is to be divided. (The resulting quotient) is the required capital. The mixed sum diminished by its own corresponding time and capital becomes the (required) interest.
An example in illustration thereof.
36. In a loan transaction at the rate of 5 percent (per mensem), the quantities representing the time, the capital and the interest
33. Symbolioally, is used with m in carrying out the required saṅkramaṇa, m being equal to I+t.
35. Here, of the three unknown quantities, the value of the time is to be optionally chosen, and the other two quantities are arrived at in accordance with rule in Ch. VI. 21. (connected with the loan) are not known. Their sum however is 82. What is the capital, what is the time, and what the interest ?
The rule for arriving separately at the various amounts of interest accruing on various capitals for various periods of time from the mixed sum of (those) amounts of interest:-
37. Let each capital amount, multiplied by the (corresponding) time and multiplied (also) by the (given) total (of the various amounts) of interest, be separately divided by the sum of the products obtained by multiplying each of the capital amounts by its corresponding time, and let the interest (of the capital so dealt with) be (thus) declared.
An example in illustration thereof.
88. In this (problem), the (given) capitals are 40, 30, 20 and 50; and the months are 5, 4, 8 and 6 (respectively). The sum of the amounts of interest is 34. (Find out each of these amounts.)
The rule for separating the various capital amounts from their mixed sum:-
30. Let the quantity representing the mixed sum of the various capitals lent out be divided by the sum of those (quotients) which are obtained by dividing the various amounts of interest by their corresponding periods of time and let the (resulting) quotient be multiplied (respectively) by (the various) quotients obtainod by
37. Symbolically, ;
- and ; where and etc., are the various capitals, and etc., are the various periods of time.
39. Symbolically, ;
- and
dividing the various amounts of interest by their corresponding period of time. Thus the various capital amounts happen to be found out.
Examples in illustration thereof.
49. (Sums represented by) 10, 6, 3 and 15 are the (various given) amounts of interest, and 5, 4, 3 and 6 are the (corresponding) months (for which those amounts of interest have accrued); the mixed sum of the (corresponding) capital amounts is seen to be . (Find out these capital amounts.)
41. The (various) amounts of interest are , 6, , 16 and 30; (the corresponding periods of time are) 5, 6, 7, 8 and 10 months; 80 is the mixed sum (of the various capital amounts lent out. What are these amounts respectively ?)
The rule for arriving separately at the various periods of time from their given mixed sum:-
[42]. Let the quantity representing the mixed sum of the (various) periods of time be divided by the sum of those (various quotients) obtained by dividing the various amounts of interest by their corresponding capital amounts; and (then) let the (resulting) quotient be multiplied (separately by each of the above mentioned quotients). (Thus) the (various) periods of time happen to be found out.
An example in illustration thereof
43. Here, (in this problem, the (given) capital amounts are 40, 30, 20 and 50; and 10, 6, 3 and 15 are the (corresponding) amounts of interest; 18 is the quantity representing the mixed sum of the (respective) periods of time (for which interest has accrued. Find out those periods of time separately).
42.^ Symbolically, where . Similarly etc., may be found out The rule for arriving separately at the rate-interest of the rate-capital from the quantity representing the mixed sum obtained by adding together the capital amount lent out, which is itself equal to the rate-interest, and the interest on such capital lent out :-
[44]. The rate-capital as multiplied by the rate-time is divided by the other time (for which interest has accrued); the square root of this (resulting quotient) as multiplied by the (given) mixed sum once, and (then) as combined with the square of half of that (above-mentioned) quotient, when diminished by the half of this (same) quotient, becomes the (required) rate-interest (which is also equal to the capital lent out).
Examples in illustration thereof.
45.The rate-interest per 100 per 4 months is unknown. That (unknown quantity) is the capital lent out; this, when combined with its own interest, happens to be 12; and 25 months is the time for (which) this (interest has accrued. Find out the rate-interest equal to the capital lent out).
46. The rate-interest per 80 per 3 months is unknown ; is the mixed sum of that (unknown quantity taken as the) capital lent out and of the interest thereon for 1 year. What is the capital here and what the interest ?
The rule for separating the capital, which is of the same value in all cases, and the interest (thereon for varying periods of time), from their mixed sum :-
[47].Know that, when the difference between (any two of) the (given) mixed sums as multiplied by each other's period [*] (of
44.^ , Symbolically, which is equal to c
47.^ Symbolically, .[*]By "the period of interest" here is meant the time for which interest has accrued in connection with any of the given mixed sums of capital and interest. interest) is divided by the difference between those periods, what happens to be the quotient is the required capital in relation to (all) those (given mixed sums).
Examples in illustration thereof
43. The mixed sums are 50, 58 and 66, and the months (during which interest has accrued respectively) are 5, 7 and 9. Find out what the interest is (in each case)
49 and 50. O arithmetician, a certain man paid out to 4 persons 30, , and 35, (these) being the mixed sums (of the same capital and the interest due thereon) at the end of 3, 4, 5 and 6 months (respectively). Tell me quickly, what may be the capital here ?
The rule for separating the capital, which is of the same value in all cases, and the time (during which interest has accrued}, from their mixed sum :-
51. Wise men say that that is the (required) capital, which is obtained as the quotient of the difference between (any two of) the (given ) mixed sums as multiplied by each other's interest, when this (difference) is divided by the difference between the (two chosen) amounts of interest.
Examples in illustration thereof.
52. The (given) mixed sums of the capital and the periods of interest are 21, 23 and 25; here, (in this problem,) the amounts of interest are 6, 10 and 14. What may be the capital of equal value here ?
53. The (given) mixed suns are 35, 37 and 39; and the amounts of interest are 20, 28 and 36. (What is the common capital ?)
51.^ Symbolically, , where etc., are the various miśras or mixed sums. The rule for arriving at the capital dealt out at two different rates of interest:-
54.[54] Let the balance quantity (i.e., the difference between the two amounts of interest,) be divided by the difference between those (two quantities) which form the interest on one for the given periods of time; (this quotient) becomes the capital thought of by one's self before.
Examples in illustration thereof.
55. Borrowing at the rate of 6 percent, and then lending out at the rate of 9 per cent, one obtains in the way of the differential gain 81 duly at the end of 3 months. What is the capital (utilized here) ?
56. Borrowed at the rate of 3 per cent per mensem, a certain capital amount is put out to interest at the rate of 8 per cent per mensem. The differential gain is 80 at the end of 2 months. How much is the capital (so used)?
The rule for arriving at the time when both capital and interest will become paid up (by installments):-
57.[57] The capital lent out is multiplied by its time (of installment) and is again multiplied by the rate-interest; this product, when divided by the rate-capital and the rate-time, becomes the interest in relation to the installment. The capital (in the installment) and the time (of discharge of the debt are to be made out) as before from (this) interest.
Examples in illustration thereof.
58. The rate of interest is 5 for 70 per mensem; the (amount of the) installment to be paid is 18 in (every) 2 months; the capital lent out is 84. What is the time of discharge?
54.^ Symbolically, .
57.^ Symbolically, interest in the installment, where p is the time of each installment. 3. The monthly interest on 60 is exactly 5. The capital lent out is 35; the (amount of the) installment (to be paid) is 15 in (every) 3 months. What is the time (of discharge) of that (debt) ?
The rule for separating various capital amounts, on which the sume interest has accrued, from their mixed sum:-
[60]. Let the (given) nixed sum multiplied by the time (give) in relation to it be divided by the sum of that quantity, wherein are combined the various rate-capitals as multiplied by their respective rate-times and as divided by their respective rate-interests. Ihe interest (is thus arrived at); and (from this) the capital amounts are arrived at as before.
Examples in illustration thereof.
61. The mixed sum (of the capital amounts lent out) at the rates of 2, 6 and 4 per cent per mensem is 4,400. Here the capital amounts are such as have equal amounts of interest accruing after 2 months. What (and the capital amounts lent, and what is the equal interest)?
62. An amount represented (on the whole) by 1,900 was lent out at the rates of 3 per cent, 5 per 70, and per 60 (per mensem); the interest (accrued) in 3 months (on the various lent parts of this capital amount) is the same (in each case). (What are these amounts lent out and what is the interest?)
The rule for arriving at the lent out capital in relation to the known time of discharge by instalments:-
[63]. Lat the amount of the installment as divided by the time thereof and as multiplied by the time of discharge be divided by
60.^ Symbolically, , from this, the capitals are found out by the rule in Ch. VI.10.
63.^ Symbolically, , where s=amount of installment, p=the time of an installment, and t=the time of discharge. that interest on one for the time of discharge to which one is added; the capital lent out is (thus arrived at).
Examples in illustration thereof.
64. In accordance with the rate of 5 per cent (per mensem), 2 months is the time for each installment; and paying the installment of 8 (on each occasion), a man here became free (from debt) in 60 months. What is the capital (borrowed by him) ?
65. A certain person gives once in 12 days an installment of the late of interest being 3 percent (per mensem). What is the capital amount of the debt discharged in 10 months?
The rule for arriving separately at the various capital amounts which, when combined with or diminished by their respective interests, are equal to one another, from their mixed sum, (the interests being either added to the capital amounts in all the given cases or subtracted from them similarly in all the given cases):-
[66]. One is to be either combined with or diminished by the interest (accruing) thereon for the (given) period of time (in each case in accordance with the respectively given rate of interest; then again in each case,) one is divided respectively by these (combined or diminished quantities arrived at as before). Thereafter the (given) mixed sum (of the various capital amounts lent out) is divided by the sum of these (resulting quotients), and in relation to the mixed sum (so treated) the process of multiplication is to be conducted (separately in each case by multiplying it) by (the corresponding) proportionate part (of the above mentioned sum of the quotients). This gives rise to the capital
66.^ Symbolically, ,
- Similarly, do. do. .
amounts lent out, which on being combined with or diminished by their respective amounts of interest are equal (in value).
Examples in illustration thereof
67. The total capital represented by 8,520 is invested (in parts) at the (respective) ratios of 3, 5 and 8 per cent (per month). Then, in this investment, in 5 months the capital amounts lent out are, on being diminished by the (respective) amounts of interest, (seen to be) equal in value. (What are the respective amounts invested thus?)
68. The total capital represented by 4,250 is invested (in parts) at the (respective) rates of 3, 6 and 8 for 60 for 2 months; then, in this investment, in 8 months the capital amounts lent out are, on being diminished by the (respective) amounts of interest, (seen to be) equal in value. (What are the respective amounts invested thus ?)
69. The total capital represented by 13,740 is invested (in parts) at the (respective) rates of 2, 5 and 9 per cent (per month) ; then, in this investment, in 4 months the capital amounts lent out are, on being combined with the (respective) amounts of interest, (seem to be) equal in value. What are the respective amounts invested thus?)
70. The total capital represented by 3,646 is invested (in parts) at the (respective) rates of , and for 80 (per month); then, in this investment, in 8 months (the capital amounts lent out are, on being combined with the respective amounts of interest, seen to be equal in value. What are the respective amounts invested thus?)
The rule for arriving at the capital, the interest, and the time of discharge (of the debt) in relation to the debt-amount (paid up) in installments in arithmetical progression:-
[71]. (The required capital amount in the due debt) is that capital amount (which results) by adding the product of the
71.^ The rule is very elliptical and will become clear from the following working of the example contained in stanzas 72–:-
Here the mūla or the maximum available amount of an installment is 60; this, when divided by 7, the al mount of the first installment, gives or or , of which optionally chosen (maximum available amount of an installment) by (whatever happens to be) the outstanding (fractional part of the number of terms in the scrics), to the amount of the (first installment as multiplied by the sum of that series in arithmetical progression, which has (one for the first term, one for the common difference, and has one for the number of terms the integral value of) the quotient obtained by dividing (the above optionally chosen maximum) amount of debt (discharged at an installment) by the (above amount of the first) installment. The interest thoreon is that which accrues for the period of an installment. The time (of an installment) divided by the amount of the (first) installment and multiplied by the (optionally chosen maximum) amount of debt (discharged at an installment) gives rise to the time (which is the time of the discharge of the whole debt).
Examples in illustration thereof.
72 and . A certain man utilised, (for the discharge of a debt) bearing interest at 5 per cent (per month), 60 (as the available maximum amount) with 7 as the first installment amount, increasing it by 7 in successive installments due every of a month. He thus gave in discharge of the debt tho sum of a series in arithmetical progression consisting of terms, and gave also the interest accruing on those multiples of 7. What is the debt amount corresponding to the sum of the series, what is that interest (which he paid), and (what is) the time of discharge of that debt ?
to 76. A certain man utilised for the discharge of a debt, bearing interest at 5 per cent (per mensem), 80(as the available maximum amount) with 8 as tho first installment amount, increasing it by 8 in successive installments due every of a month. He thus
8 represents the number of terms of the series in arithmetical progression, which has 1 for the first term and 1 for the common difference; and is the agra or the outstanding fractional part. The sum of the above-mentioned series, viz, 36, multiplied by 7, the amount of the first installment, is added to the product of , and 60, which latter is the maximum available amount of an installment. Thus, we get , which is the required capital amount in the due debt. The interest on for of a month at the rate of 5 per cent per mensem will be the interest paid on the whole. The time of discharge will be months . gave in discharge of the debt the sum of a series in arithmetical progression consisting of terms and gave also the interest accruing on those multiples of 8. The debt amount (corresponding to the sum of the series), the interest (which he paid), and the time of discharge (of that debt)-tell me, friend, after calculating, what the (respective) value of those quantities is.
The rule for arriving at the average common interest:-
77 and .[*] Divide the sum of the (various accruing) interests by the sum of the (various corresponding) interests due for a month; the resulting quotient is the required time. The product of the (assumed) rate-time and the rate-capital is divided by this required time then multiplied by the sum of the (various accruing) interests and then divided again by the sum of the (various given) capital amounts. This gives rise to the (required) rate-interest.
An example in illustration thereof.
. In this problem, four hundreds were (separately) invested at the (respective) rates of 2, 3, 5 and 4 per cent (per mensem) for 5, 4, 2 and 3 months (respectively). What is the average common time of investment, and what the average common rate of interest?
Thus end the problems bearing on interest in this chapter on mixed problems.
77 and ^ The various accruing interests are the various amounts of interests accruing on the several amounts at the various rates for their respective periods.
Symbolically, or average time;
and or average interest.
Proportionate Division.
Hereafter we shall expound in (this) chapter on mixed problems the working of proportionate division:-
[*]. The operation of proportionate division is that wherein the (given) collective quantity (to be divided) is first divided by the sum of the numerators of the common-denominator-fractions (representing the various proportionate parts), donominators of which fractions are struck off out of consideration; and (then it) has to be multiplied (respectively in each case) by (these) proportional numerators. This is called kuṭṭīkāra by the learned.
Examples in illustration thereof.
. Here, (in this problem,) 120 gold pieces are divided among 4 servants in the (respective) proportional parts of and . O arithmetician, tell me quickly what they obtained.
. (The sum of) 363 dīmāras was divided among five, the first one(among them) getting 3 parts, and 3 being the common ratio successively (in relation to the shares of the others). What was the state of each ?
to . A certain faithful śrāvaka took a number of lotus flowers, and going into the Jina temple conducted (therein) with devotion the worship of the chief Jinas that were worthy of worship. He offered part to Vṛṣabha, to worthy Pārśva, and to Jinapati, and to sage Suvrata.; he devotedly gave to Aristanémi who dostroyed all the eight kinds of karmas' and to Jinaśānti. 480 lotuos were brought (for this purpose. ) By adopting the operation known
.^ In working the example in stanza according to this rule we get . After removing the denominators here, we have 6, 4, 3 and 2. These are also called prakṣēpas or proportional numerators. The sum of these is 15, by which the amount to be distributed, viz., 120 is divided; and the resulting quotient 8 is separately multiplied by the proportional numerators,6,4,3 and 2. Then the amounts thus obtained are 6 x 8 or 48, 4 x 8 or 32, 3 x 8 or 24, 2 x 8 or 16. It is worthy of note that prakṣēpas means both the operation of proportionate division and a proportional numerator . as prakṣēpaka, give out the proportionate distribution of the flowers.
86.(A sum of) 480 was divided among five men in the proportion of 2, 3, 4, 5 and 6; O friend, give out (the share of each).
The rule for arriving at (certain) results in required proportions:-
87.[*] The (number representing the) rate-price is divided by (the number representing) the thing purchasable therewith; (it) is (then) multiplied by the (given) proportional number; by means of this, (we get at) the sum of the proportionate parts, (through) the process of addition. Then the given amount multiplied by the (respective) proportionate parts and then divided by (this sum of) the proportionate parts gives rise to the value (of the various things in the required proportion).
Another rule for this (same) purpose:-
88.[*] Multiply the numbers representing the rate-prices (respectively) by the numbers representing the (given) proportions of the (various) things (to be purchased); then divide (the result) by the (respective) numbers measuring the things purchasable for the rate-price; the resulting quantities happen to be the (requisite) multipliers in the operation of prakṣēpaka. The intelligent man may (then) give out the required answer by adopting the rule-of-three.
Again a rule for this (same) purpose:-
89.[*] The (numbers representing the various) rate-prices are respectively divided by their own related (numbers representing the) things purchasable therefor and are (then) multiplied by their related proportional numbers. With the help of these, the remainder (of the operation should be carried out) as before.
87 to 89.^ In working the example in stanza 90 and 91 according to these rules 2, 3 and 5 are divided by 3,5 and 7 respectively and are similarly multiplied by 6,3 and 1. Thus we have . These are the proportional parts. The rules in stanzas 88 and 89 require thereafter the operation of prakṣēpaka to be applied in relation to these proportional parts; but the rule in stanza 87 expressly describes this operation. The required result is well arrived at by going through the process of the rule-of-three.
Examples in illustration thereof.
90 and 91. Pomegranates, mangoes and woodapples are obtainable at the (respective) rates of 3 for 2, 5 for 3, and 7 for 5 paṇas. O you friend, who know the principles of calculation, come quickly having purchased fruits for 76 paṇas, so that tho mangoes may be thrē times as the woodapples, and the pomegranates six times as much.
99 to 94. A follower of Jina had the image of Jina bathed in potfuls of curds, ghee and milk. Three pots became filled with 72 palas(of those); 32 palas were found in the first pot and 24 in the second pot and 16 in the third pot. From those (potfuls of mixed-up) curd, ghee and milk, find out each of those (ingredients) separately and give them out, there being altogether 24 palas of ghee, 16 palas of milk and 32 palas of curds.
95 and 96. Three puraṇas formed the pay of one man who is a mounted soldier; and at that rate there were 65 men in all. Some (among them) broke down, and the amount of their pay was given to those that remained in the field. Of this, each man obtained 10 purāṇas. You tell me, after thinking well, how many remained in the field and how many broke down.
The rule for the operation of proportionate division, wherein there is the addition or the subtraction of certain optionally chosen integral quantities:-
97[*]. The given total quantity is diminished by the integral quantities that are to be added, or is combined with the positive integral quantities that are to be subtracted; then with the help of this resulting quantity the operation of proportionate division is to be conducted, and the resulting proportionate parts are respectively combined with those (integral quantities that are to be added to them), or they are diminished (respectively) by those (integral quantities that are to be subtracted).
97^ . The operation of proportionate division to the conducted here is according to any of the rules in stanzas 87 to 89.
Examples in illustration thereof
98.[1] Four men obtained their shares in successively doubled proportions and with successively doubled differences in addition, the first man obtaining, one share: 67 (is the quantity so to be distributed) here. What is the share of each ?
99. (A sum of) 78 is divided by these four (among themselves) in proportions which are successively from the first 1 times (what precedes) and with differences (in addition, which,) commencing with 1, (go on) increasing three-fold. Give out the (value of the) parts obtained (by each.)
100. (The shares of) five (persons) are (successively) from the first 1 times (what goes before), and the differences in addition are quantities which are (successively) 2 times(the preceding difference) 51 is (the total quantity) to be divided. (Find out the values of the portions obtained by each.)
101. (A sum of) 400 minus 15 is divided by four men (among themselves) in proportions which from the first are 2 times (what precedes), and which (besides) are loss by differences which are (successively) 4 times (the preceding difference) . (Find out the values of the various portions obtained.)
The rule for arriving at the value of tho prices producing equal sale-proceeds and at the value of the highest capital (invested in the transactions concerned):--
102.[2] The largest capital (invested) combined with one becomes the vending rate of the commodity (to be sold). That (same vending rate), multiplied by the (given) price at which the remnant is to be sold, and diminished by one, gives rise to the
98.^ The difference quantity to be added to the shares here is 1 in the case of the second man, and twice the preceding difference in the case of each of the remaining two men; and this difference in the case of the second man is not expressly mentioned as in this example as well as in the example in stanza 101)
102.^ The examples bearing on this rule contemplate the purchase of a commodity at a certain common rate for various capital amounts; then the commodity so purchased is to be sold at a certain other common rate. That quantity of the commodity which is left over, owing to its not being enough to be sold for a unit of bhe kind of money employed in the transaction, is here purchasing rate. By revorsing the processes, one may arrive at the valuation of the highest capital (invested in the transaction).
Examples in illustration thereof.
103. The capital amounts invested by (three) men are (respectively) 2, 8 and 36; 6 is the price at which the remnants of the commodity are to be sold. Having purchased and sold at the same rates, they became possessors of equal wealth. (Find out the buying and selling prices.)
104. Those three persons took up 1, and 2 (as their respective capital amounts) and conducted the operations of buying and selling (in relation to the same commmodity at the same rates of price); by selling the remnant (in the end) at a price presented by 6, they become possessors of equal wealth. (Find out their buying and selling prices,
105.[*] The quantity measuring the equal wealth is 41, and the price at which the remnants of the commodity are bold is 6. O arithmetician, tell me quickly what the highest capital (invested) is, and what the (various) capitals are.
106. In the case where 35 dīnāras give the numerical measure of the equal wealth, and 4 is the price at which the remnant is to be sold, you tell me, O arithmetician, what the highest capital (invested) is.
spoken of as the remnant, and the price at which this remnant is sold is the remnant-price.
Symbolically, let a, a + b and a + b + c be the capitals, where the last is the ज्येष्ठधन or the largest capital, and let 2 be the चरमार्घ or the remnant price; then, according to the rule, a + b + c + 1 = the vending rate ; and (a + b + c + 1)p - 1= the purchasing rate.
From these, it can be easily shown that the sum of the amounts realised by selling the commodity at the vending rate and the remnant at the remaining price turns out to be the same in each case.
It may be noted that the purchasing rate happens in problems bearing on this rule to be the same in value as the समधन or the equal sale-proceeds.
105. ^ It may be noted here that, according to the rule, it is only the largest sapital that is found out ; while the other capitals required in the problem are optionally chosen, so as to be less than the largest capital. The rule for arriving at the value of the prices producing equal sale-proceeds when the price of the remnant is fractional in character:--
107. When the remnant price is fractional in character, the selling and the buying rates are to be derived as before with (the data consisting of) the (invested) capitals and the remnant-price reduced to the same denominator, which is (however) ignored (for the time being); these selling and buying rates are (then respectively) to be multiplied by (this) denominator and the square of (this) denominator (for arriving at the required selling and buying rates). The value of the equal sale-proceeds is (then obtained) by means of the rule-of-three.
An example in illustration thereof.
108. (In a transaction) are the capital amounts (invested respectively by three porsons); the remnant-price is . By purchasing and selling at the same prices, they became possessed of equal sale-proceeds. (What is the buying price, what the selling price, and what the equal sale-amount ?)
Again, another rule for arriving at the value of the equal sale-proceeds, when the remnant-price is fractional:-
109. The continued product of the highest numerator, of two, and of (all) the denominators (to be found in the values of the capital amounts invested), when combined with the (last) denominator belonging to the value of the remnant-price, gives rise to the selling rate. This multiplied by the remnant-price, and then diminished by one, and then multiplied (successively) by two and all the denominators, becomes the purchasing rate. Then the rule-of-three (is to be used for arriving at the common value of the sale-amounts).
{{c|An example in illustration thereof
110. Having invested (respectively), and having bought and sold (the same commodity), and with as the remnant price, three merchants became possessors of equal sale-proceeds (in the end. What is the buying price, what is the selling price, and what the equal sale-amount?)
The rule for arriving at (the solution of a problem wherein) optionally chosen quantities (are) bestowed in optionally chosen multiples for an optionally chosen number of times :-
111. Let the penultimate quantity be added to the ultimate quantity as divided by its own corresponding multiple number, and let the result of this operation be divided by that (multiple number which is associated with this) penultimate quantity (given in the problem). What results (from carrying out this operation throughout in relation to all the various quantities bestowed) happens to be the (required) original quantity.
Examples in illustration thereof.
112 and 118. A certain lay follower of Jainism went to a Jina temple with four gate-ways, and having taken (with him) fragrant flowers offered them (thus) in worship with devotion:- At the four gate-ways, they became doubled, then trebled, then quadrupled and then quintupled (respectively in order.) The number of flowers offered by him was five at every (gate-way). How many were the lotuses (originally taken by him)?
114. Flowers were obtained and offered in worship by devotees with devotion, the flowers (so offered) being (successively) 3, 5. 7 and 8; (their corresponding) multiple quantities being and (in order. Find out tho original number of flowers).
Thus ends proportionate division in this chapter on mixed problems.
Vallikā-kuṭṭīkāra.
Hereafter we shall explain the process of calculation known as Vallikā-kuṭṭīkāra[*]:-The rule unederlying the process of calculation which is a special kind of division or distribution):-
115. Divide the (given) group-number by the (given) divisor; discard the first quotient; then put down one below the other the (various) quotients obtained by the successive division (of the various resulting divisors by the various resulting remainders; again), put down below this the optionally chosen number,
^ It is so called because the method of kuṭṭīkāra. explained in the rule is based upon a creeper-like chain of figures.
115. The rule will become clear from the following working of the problem in stanza No. 117.
Here it is stated that 68 heaps of plantains together with 7 separate fruits are exactly divisible among 23 persons; it is required to find out the number of fruits in a heap. Here the 63 is called the ‘group-number' and the numerical value of the fruits contained each heap is called the 'group-value' and it is this latter which has to be found out.
Now, according to the rule, we divide first the rāśi, or group-number 63, by the chēda or the divisor 23; and then we continue the process of division as in finding out H.C.F. of two numbers:-
- 23)63(2
- 46
- ----
- 17)23(1
- 17
- ----
- 6)17(2
- 12
- ----
- 5)6(1
- 5
- ---
- 1)5(4
- 4
- ----
- 1
1
2
1
4
Here, the first quotient 2 is discarded; the other quotients are written down in a line one below the other as in the margin ; then we have to choose such a number as, when multiplied by the last remainder 1, and then combined with 7, (the number of separate fruits given in the problem) will be divisible by the last divisor 1. We accordingly choose 1, which is written down below the last figure in the chain ; and below this chosen number, again, is written down the quotient obtained in the above division with the help of the chosen number. Here we stop the division with the fifth remainder as it is the least remainder in the odd position of order in the series of divisions carried out here. with which the least remainder in the odd position of order (in the above-mentioned process of successive division ) is to be multiplied; and (then put down) below (this again) this product increased or decreased (as the case may be by the given known number) and then divided (by the last divisor in the above mentioned process of successive division. Thus the Vallikā or the creeper-like chain of figures is obtained. In this) the sum obtained by adding (the lowermost number in the chain) to the product obtained by multiplying the number above it with the number (immediately) above (this upper number, this process of addition being in the same way continued till the whole chain is exhausted, this sum, is to be divided by the (originally
1--51
2--38
1--13
4--12
1
8
Thus we get the chain or Vallikā noted in the first column of figures in the margin. Then we multiply the penultimate figure below in the chain, viz., 1, by 4, which is above it, and add 8, the last number in the chain; the resulting 12 is written down so as to be in the place corresponding to 4; then multiplying this 12 by 1 which is the figure above it in the creeper chain, and adding 1, the figure similarly below it, we get 13 in the place of 1; proceeding in the same manner 38 and 51 are obtained in the places of 2 and 1 respectively. This 51 is divided by 28, the divisor in the problem ; and the remainder 5 is seen to be the length number of fruits in a bunch.
The rationale of the rule will be clear from the following algebraical representation:-.
, where
Thus we have,
given) divisor. (The remainder in this last division becomes the multiplier with which the originally given group-number is to be multiplied for the purpose of arriving at the quantity which is to be divided or distributed in the manner indicated in the problem. Where, however, the given group-numbers, increased or decreased in more than one way, are to be divided or distributed in more than one proportion,) the divisor related to the larger group-value,(arrived at as explained above in relation to either of two specified distributions), is to be divided (as above) by the divisor (related to
By choosing a value for p4, such that , which is, as shown above, the value of p4,becomes an integer, and by arranging in a chain , we get at the value of x by proceeding as stated in the rule, that is, by the processes of multiplication by the upper quantity and the addition of the lower quantity in the chain, which are carried up to the topmost quantity. The value of x so obtained is divided by A, and the remainder represents the least value of x; for the values of x which satisfy the equation, an integer, are all in an arithmetical progression wherein the common difference is A.
This same rule contemplates problems where two or more conditions are given, such as the problems given in stanzas 121 to 129. he problem in 121 may be thus worked out according to the rule:- It is given that a heap of fruits when diminished by 7 is exactly divisible among 8 men, and the same heap when diminished by 3 is exactly divisible among 13 men.
Now, according to the method already given, find out first the least number of fruits that will satisfy the first condition, and then find out the number of fruits that will satisfy the second condition. Thus we get the group-values 15 and 16 respectively. Now, the divisor related to the larger group-value is divided as before by that related to the smaller group-value to obtain a fresh vallikā chain. , Thus dividing 13 by 8 and continuing the division, we have:-
8)13(1
8
----
5)8(1
5
----
3)5(1
3
----
2)3(1
2
----
1)2(1
1
--
1
1
1
1
1
1
2
From this the Vallikā chain comes out thus:- Choosing 1 as the mati, and adding the difference between the two group-values already arrived at, that is, 16–15, or 1, to the product of the mati and the last divisor, and dividing this sum by the last divisor, we have 2, which is to be written down below the mati in the Vallikā chain. Then proceeding as before with the vallikā, we get 11, which, when divided by the first divisor 8, leaves the remainder 3. This is multiplied by the divisor related to the larger group-value, viz., 13, and then is combined with the larger group-value. Thus 65 is the number of fruits in the heap. the smaller group-value, so that a creeper-like chain of successive quotients may be obtained in this case also as before. Below the lowermost quotient in this chain, the optionally chosen multiplier of the last remainder in the odd position of order in this last successive division is to be put down; and below this again is to be put down the number which is obtained by adding the difference between the two group-values (already referred to) to the product (of the least remainder in the last odd position of order multiplied by the above optionally chosen multiplier thereof, and then by dividing the resulting eum by the last divisor in the
The rationale of this process will be clear from the following considerations:-
(iii) is an integer. In (i) Let the lowest value of- In (ii) ,, ,, ,, .
- In (iii) ,, ,, ,, .
(iv) When both (i) and (ii) are to be satisfied, has to be euqal to , so that . That is .
From (iv), which is an indeterminate equation with the values o d and k unknown, we arrive, according to what has been already proved, at the lowest positive integral value of d/ This value of d multiplied by A1, and then increased by s1, gives the value of x which will satisfy (i) an (ii).
Let this be t1; and let the next higher value of x which will satisfy both these equations be t2.
- (v) Now, ;
- (vi) and .
- .
- .
- Substituting in (v) and (vi), we have
- .
From this it is obvious that the next higher value of x satisfying the two equations is obtained by adding the least common multiple of A1 and A2 to the lower value.
Now again, let v be the value of x which satisfies all the three equations.
Then , (where r is a positive integer) =(say) ; and .
- .
above division chain. Thus the creeper-like chain of figures required for the solution of this latter combined problem is obtained. This chain is to be dealt with as before from below upwards, and the resulting number is to be divided as before by the first divisor in this last division chain. The remainder obtained in this operation is then) to be multiplied by the divisor (related to the larger group-value, and to the resulting product, this) larger group-value is to be added. (Thus the value of the required multiplier of the given group-number is obtained; and this will satisfy both the specified distributions taken together into consideration).
Examples in illustration thereof.
116. Into the bright and refreshing outskirts of a forest, which were full of numerous trees with their branches bent down with the weight of flowers and fruits, trees such as jambū trees, lime trees, plantains, areca palms, jack trees, date-palms, hintāla trees, palmyras, punnāga trees and mango trees--(into the outskirts, the various quarters whereof were filled with the many sounds of crowds of parrots and cuckoos found near springs containing lotuses with bees roaming about them (into such forest outskirts) a number of weary travellers entered with joy.
117. (There were) 63 (numerically equal) heaps of plantain fruits put together and combined with (more) of those same fruits and these were (equally) distributed among 23 travellers so as to leave no remainder. You tell (me now) the (numerical) measure of a heap (of plantains.)
118. Again, in relation to 12 (numerically equal) heaps of pomegranates, which. after having been put together and
By applying the principle of vallikā-kuṭṭīkāra in the last equation, the value of c is obtained, and thence the value of v can be easily arrived at.
It is seen from this that, when, in order to find out v, we deal with t1 and s3 in accordance with the kuṭṭīkāra method, the chēda or the divisor to be taken in relation to , or the least common multiple of the divisor in the first two equations combined with 5 of those (same fruits) , were distributed similarly among 19 travellers. Give out the (numerical) measure of (any) one (heap).
119. A traveller sees heaps of mangoes (equal in numerical value), and makes 31 heaps less by 3 (fruits) and when the remainder (of these 31 heaps) is (equally) divided among 73 men, there is no remainder. Give out the numerical value of one (of these heaps).
120. In the forest 37 heaps of wood-apples were seen by the travellers. After 17 fruits were removed (therefrom the remainder) was (equally) divided among 79 persons (so as to leave no remainder). What is the share obtained by each ?
121. When, after seeing a heap of mangoes in the forest and removing 7 fruits (therefrom, it was divided equally among 8 of the travellers and when again after removing 3 (fruits) from that (same) heap it was (equally) divided among 13 of them left no remainder (in both cases). O arithmetician, tell me (the numerical measure of this) single heap.
122. A single heap of wood-apples divided among 2, 3, 4, or 5 (persons) leaves 1 as remainder (in each case). O you who know arithmetic, tell me the (numerical) measure of that (heap).
123. When (divided) by 2, the remainder is 1 when by 3, it is 2; when by 4, it is 3; when by 5, it is 4. 'Tell me, O friend, what this heap is.
124. When (divided) by 2, the remainder is 1; when by 3, there is no remainder; when by 4, it is 3; when by 5, it is 4. Tell me, 0 friend, what the heap is (in numerical value).
125. When divided by 2, there is no remainder; when by 3, there is 1 as remainder ; when by 4, there is no remainder and when by 5, there is one as remainder. What is this quantity ?
126. When divided by 2 (the remainder is) 1; when by 3, there is no remainder; when by 4, (the remainder is) 3; and when divided by 5, there is no remainder. Tell me how what (this) quantity is.
127. The travellers saw on the way certain (equal) heaps of jambū fruits. Of then, 2(heaps) were equally divided among 9 ascetics and left 3(fruits) as remainder. Again 3 (heaps) were (similarly) divided among 11 persons, and the remainder was 5 fruits; then again 5 of those heaps were similarly divided among 7, and there were 4 more fruits (left out) of them. O you arithmetician who know tho meaning of the kuṭṭīkāra process of distribution, tell me after thinking out well the numerical measure of a heap (here).
128. In the interior of the forest, 3 heaps (equal in value) of pomegranates were divided (equally) among 7 travellers, leaving 1 (fruit) as remainder; 7 (of such helps) were divided (similarly) among 9, leaving a remainder of 3(fruits; again) 5 (of such heaps) were (similarly) divided among 8, leaving 2 fruits as remainder, O arithmetician, what is the numerical value of a heap here).
129. There were 5 (heaps of fruits equal in numerical value), which after being combined with 2 (fruits of the same kind) were (equally) divided among 9 travellers (and left no remainder); 6 (heaps) combined with 4 (fruits) were (similarly) divided among 8 of them; and 4 (heaps) combined with 1 (fruit) were (also similarity) divided among 7 of them. Give out the numerical measure (of a heap here).
The rule for arriving at the original quantity distributed (as desired), after obtaining the remainder due to (the removal of certain specified) known quantities:-
130[*]. (Obtain) the product of the (given) known quantity (to be removed), as multiplied by the fractional proportion of what is 1eft (after a specified fractional part of what remains on the removal of the given known quantity has been given away). The next quantity is (obtained by means of) this (product), to whicn
130.^ Here the known quantity to be removed is called the agra. What remains after the removal of the agra is the remainder. That fraction of this remainder which is given or taken away is the agrāṅsa, and what is left of the remainder after the agrāṁśa is given or taken away is the śēsāṁśa or the remaining fractional proportion of the remainder. For example, where x is the quantity to be found out, and a is the agra in relation to the first distribution with as the fractional proportion distributed, happens to be the agrāṁsa, and to be the śēșāṁśa. the specified known quantity which is to be taken away (from the previous remainder) is added: (and this resulting sum) is multiplied by that (same kind of) remaining fractional proportion (of the remainder as has been mentioned above). This is to be done as many times as there are distributions to be made. Then these quantities so obtained should be deprived of their denominators; and these denominator-less quantities (and the successive products of the above-mentioned remaining fractional proportions of the remainder) are (to be used as) the known quantity and the (other elements, viz., the coefficient) multiple (of the unknown quantity and the divisor, required in relation to a problem on Vallikā-kuṭṭikāra).
Examples in illustration thereof.
131. On a certain man bringing mango fruits (home, his) elder son took one fruit first and then half of what remained (On the elder son going away after doing this), the younger (son) did similarly (with what was left there. He further took half
The rule will be clear from the following working of the problem in 132-133:-
Here 1 is the first agra, and is the first agrāṁśa; therofore or is the śēșāṁśa. Now, obtain the product of agra and śēșāṁśa or or . Write it down in 2 places.I
Repeat the quantities ; add the second agra 1(to one of the quantities) Then we have;
multiply both by the next śēșāṁśa or , so that you getTake these figures and add the third agra 1 as before; and you have ; multiply by the next śēșāṁśa or and by the last aṁsa or and you haveII
III
The denominators of the first fractions in these three sets of fractions marked I, II, III, are dropped, and the numerators represent negative agra in a problem on Vallikā-kuṭṭīkāra, wherein the numerator and the denominator of each of the second fractions in those sets represent respectively the dividend coefficients and the divisor. Thus we have
is an integer; is an integer; and is an integer.
The value of a satisfying these three conditions gives the number of flowers. of what was thereafter left) ; and the other (son) took the other half. (Find the number of fruits brought by the father.)
132 and 133. A certain person went (with flowers) into a Jina temple which was (in height) three times the height of a man. At first he offered one (out of those flowers) in worship at the foot of the Jina and then (offered in worship) one-third of the remaining number (of flowers) to the first height-measure (of the Jina). Out of the remaining two-thirds (of the number of flowers, he conducted worship) in the same manner in relation to the second height-measure; and (then he did) the same thing in relation to the third height-measure also. The two-thirds which remained at last were also made into 3 equal parts (by him) ; and having worshipped the 24 tīrthaṅkaras (with those parts at the rate of eight tīrthaṅkaras for each part), he went away with no (flower) on hand. (Find out the number of flowers taken by him.)
Thus ends simple Kuṭṭīkāra in this chapter on mixed problems.
[*] Vișama-kuṭṭīkāra
Hereafter we shall expound complex kuṭṭīkāra
Tho rule relating to complex kuṭṭīkāra:-
134. The (given) divisor, (written down) in two (places), is to be multiplied (in each place) by an optionally chosen number; and the (known) quantity given (in the problem) for the purpose of being added is to be subtracted (from the product in one of these places); and the quantity given (in the problem) for the purpose of being subtracted is to be added (to the product noted down in the other place. The two quantities thus obtained are) to be divided by the known (coefficient) multiplier (of the unknown
*^ The words Vișama and Bhinna here used in relation to kuṭṭīkāra have obviously the same meaning and refer to the fractional character of the dividend quantities occurring in the problems contemplate by the rule. quantities to be distributed in accordance with the problem). Each (of the quotients so obtained) happens to be the required (quantity which is to be multiplied by the given) multiplier in the process of Bhinnakuṭṭikāra.[*]
An example in illustration thereof.
135. A certain quantity multiplied by 6, (then) increased by 10 and (then) divided by 9 leaves no remainder. Similarly, (a certain other quantity multiplied by 6, then) diminished by 10 (and then divided by 9 leaves no remainder). Tell me quickly what those two quantities are (which are thus multiplied by the given multiplier here).
Sakala-kuṭṭīkāra
The rule in relation to sakala-kuṭṭīkāra.
136[*]. The quotient in the first among the divisions, carried on by means of the dividend-coefficient (of the unknown quantity to be distributed), as well as by means of the divisor and the (successively) resultiug remainders, is to be discarded. The other quotients obtained by means of this mutual division ( carried on till the divisor and the remainder become equal) are to be written down (in a vertical chain along with the ultimately equal remainder and divisor); to the lowermost figure (in this chain), the remainder (obtained by dividing the given known quantity in the problem by the divisor therein), is to be added. (Then by means of those numbers in the chain), the sum(which has to be) obtained by adding (successively to the lowermost number) the product of the two
136.^ This rule will become clear from the following working of the problem given in 137:-
The problem is, when is an integer, to find out the value of x. Removing the common factors, we have is an integer. numbers immediately above it, (till the topmost figure in the chain becomes included in the operation), is to be arrived at. (Thereafter) this resulting sum and the divisor in the problem (give rise), in the shape of two remainders, (to the two values of) the unknown quantity (which is to be multiplied by the given dividend-coefficient in the problem),which (values)are related either to the known given quantity that is to be added or to the known given quantity that is to be subtracted, according as the number of figure-links in the above-mentioned chain of quotients is even or odd. (Where, however, the given groups, increased or decreased in more than one way, are to be divided or distributed in more than one proportion), the divisor related to the larger group-value, (arrived at as explained above in relation to either of two specified distributions, is to be divided over and over (as above by the divisor
Carry out, the required process of continued division:--
67)59(0
0
----
59)67(1
59
----
8)59(7
56
-----
3)8(2
6
-----
2)3(1
2
----
1)2(1
1
----
1
After discarding the first quotient, the others are written down in a chain thus:--1
7
2
1
1
1
1+13=14
Below this are next written down 1 and 1, the last equal divisor and remainder. Here also, as in Vallikā-kuṭṭīkāra is worthy of note that in the last division there can be really no remainder, as 2 is fully divisible by 1 . But since the last remainder is wanted for the chain, it is allowed to occur by making the last quotient smaller than possible. And to the last number 1 here, add 13, which is the remainder obtained by dividing 80 by 67; the 14 so obtained is also written down at the bottom of the chain,which now becomes complete.
1-392
7-345
2-47
1-16
1-15
1
14Now, by the continued process of multiplying and adding the figures in this chain, as already explained in the note under stanza No. 115. This is then divided by 67; and the remainder 57 is one of the values of x, when 80 is taken as negative owing to the number of figures in the chain being odd. When 80 is taken as positive, the value of x is 67 - 57 or 10. If the number of figures in the chain happen to be even, then the value of x first arrived at is in relation to the positive agra; if this value be subtracted from the divisor, the value of x in relation to a negative agra is arrived at. related to the smaller group-value obtained as above so that a creeper-like chain of successive quotients may be obtained in this case also. Below the lowermost quotient in this chain the optionally chosen multiplier of the least remainder in the odd position of order in this last successive division is to be put down
The principle underlying the process given in the rule is the same as that explained in the rule regarding Vallikā-kuṭṭīkāra—but with this difference namely, that the last two figures in the chain here are obtained in a different way.
Again from the rationale given in the footnote to role in 115, Ch. VI, it will be seen that the agra, b, associated with the remainder in the odd position of order, has the same algebraical sign as is given to it in the problem; while the sign of the agra, b, associated with the remainder in the even position of order is opposite to its sign as given in the problem. Hence, when the continued division is carried up to a remainder in the odd position of order, the value of x arrived at therefrom is in relation to, such an agra as has its sign unchanged; on the other hand, when the continued division is carried up to a remainder in the even position of order, the value of x arrived at therefrom is in relation to an agra that has its sign changed. When the number of remainders obtained is odd the number of quotients in the chain is even; and when the remainders are even, the quotients are odd in number. As the agra associated with the last remainder is in this rule always taken to be positive, the value of x arrived at is in relation to the positive agra, if the last remainder happens to be in the odd position of order. And it is in relation to the negative agra, if the last remainder happens to be in the even position of order. In other words, if the number of quotients be even, the value is in relation to the positive agra; and if the number of quotients be odd. it is in relation to the negative agra.
The value of x in relation to the positive or the negative agra being thus found out, the other value is arrived at by subtracting this value from the divisor in the problem. How this turns out will be clear from the following representation:-
an integer. Here let an integer. We know that is also an integer. Hence or is an integer.
It has to be noted here that the common factor, if any, of the three given numerical quantities is to be removed before the operation of continued division is begun. The last divisor and the last remainder being required to be equal it will invariably happen that these come to be 1.
The mati, required to be chosen in the rule relating to the Vallikā-kuṭṭīkāra and required to be written below tho chain of quotients, is in this rule always 1, the last divisor being 1. Therefore the last divisor here takes the place of the mati in the Vallikā-kuṭṭīkāra. It will be seen further that the last figure of the chain obtained according to this rule, i.e., 1 + agra is the same as the last figure in the chain obtained in the Vallikā-kuṭṭīkāra by dividing by the last divisor by the last sum of agra and the product of the mati as multiplied by the last remainder as before; and below this again is to be put down) the number which is obtained by adding the difference between the two group-values, (already referred to, to the product of the least remainder in the odd position of order multiplied by the above optionally chosen multiplier thereof, and then by dividing this resulting sun by the last divisor in the above division chain.
Thus the creeper-like chain of figures required for the solution of this latter kind of problem is obtained. This chain is to be dealt with as before from below upwards, and the resulting number is to be divided as before by the first divisor in this last division chain. he remainder obtained in this operation is then to be) multiplied by the divisor (related to the larger group-value); and to the resulting product this larger group-value is to be added.
(Thus the value of the required multiplier of the given group number is obtained so as to satisfy the two specified distributions taken into consideration.)
Examples in illustration thereof.
137. One hundred and seventy-seven (is the dividend-coefficient of the unknown factor), 240 is the known quantity associated (with the product so as to be added to or subtracted from it); the whole is divided by 201 (and leaves no remainder What is the (unknown) factor here (with which the given dividend coefficient is to be multiplied) ?
138. Thirty-five and other quantities, 16 in number, rising (thence successively in value) by 3, (are the given dividend-coefficients). The given divisors are 32 (and others) as successively increased by 2. And 1 successively increased by 3 gives rise to the associated known (positive and negative) quantities. What are the values of the (unknown) factors (of the known dividend coefficients), according as they are additively associated with positive or negative (known) numbers ? The rule for separating the prices of (an interchangeable) larger and (a similar) smaller number of two different things from the given mixed sums of the prices of these things:-
139. From the higher price-sum, as multiplied by the corresponding larger number of one of the two kinds of things, subtract the lower price-number as multiplied by the smaller number relating to the other of the two kinds of thing. Then divide the result by the difference between the squares of the numbers relating to these things. This gives rise to the period of the thing which is larger in number. The other, that is, the price of the thing which is smaller in number, is obtained by interchanging the multipliers.
An example in illustration thereof.
140 to 142. The mixed price of 9 citrous and 7 fragrant wood-apples is 107; again the mixed price of 7 citrons and 9 fragrant wood-apples is 101. O you arithmetician, tell me quickly the price of a citron and of a wood-apple here, having distinctly separated those prices well.
The rule for separating the prices and the numbers of different mixed quantities of different kinds of things from their given mixed price and given mixed values:-
143. The (different) given (mixed) quantities (of the different things) are to be multiplied by an optionally chosen number; the given (mixed) price (of these mixed quantities) is to be diminished (by the value of these products separately). The resulting quantities
139. Algebraically, if
- and ,
- then
- and
- .
- .
143. The rule will become clear by the following working of the problem in stanzas 144 and 145:-
The total number of fruits in the first heap is 21.
Do. do. second do. 22.
Do. do. third do. 23.
are to be divided (one after another) by an optionally chosen number (and the remainders again are to be divided by an optionally chosen number, this process being repeated) over and over again. The given (mixed) quantities of the different things are to be (successively) diminished by the corresponding quotients in the above process. (In this manner the numerical values of the various things in the mixed sums are arrived at). The optionally chosen divisors (in the above processes of continued division ) combined with the optionally chosen multiplier as also that multiplier constitute (respectively) the prices (of a single thing in each of the varieties,of the given different things).
Choose any optional number, say 2, and multiply with it these total numbers; we get 42, 44, 46. Subtract these from 73, the price of the respective heaps. The remainders are 31, 29, and 27. These are to be divided by another optionally chosen number, say 8. The quotients are 3, 3, 3, and the remainders are 7, 5 and 3. These remainders are again divided by a third optionally chosen number say 2. The quotients are 3, 2, 1, and the remainders are 1, 1, 1. These last remainders are in their turn divided by a fourth optionally chosen number which is 1 here. The quotients are 1, 1, 1 with no remainders. The quotients derived in relation to the first total number are to be subtracted from it. Thus we get ; this number and the quotients 3, 3, 1 represent the number of fruits of the different sorts in the first heap. Similarly we get in the second group 16, 3, 2, 1, and in the third group 18, 3, 1, 1, as the number of the different sorts of fruits.
The prices are the first chosen multiplier, viz., 2, and its sums with the other optionally chosen multipliers. Thus we get 2, 2 + 8 or 10, 2 + 2 or 4, and 2 + 1 or 3, as the price of each of the four different kinds of fruits in order.
The principle underlying this method will be clear from the following algebraical representation:-
I II Let w = s. Multiplying II by s, we haveIII Subtracting III from I, we getIV Dividing IV by , we get a as the quotient, and as the remainder, where is a suitable integer.
Similarly we proceed till the end.
Thus it will be seen that the successively chosen divisors , and , when combined with s,give the value of the various prices, s by itself being the price of the first thing; and that the successive quotients a, b, c, along with are the numbers measuring the various kinds of things.
It may be noted that, in this rule, the number of divisions to be carried out is one less than the number of the kinds of things given, and that there should be no remainder left in the last division.
An example illustration thereof.
144 and 145. There are here fragrant citrons, plantains, wood-apples and pomegranates mixed up (in three heaps). The number of fruits in the first (heap) is 21, in the second 22, and in the third 23. The combined price of each of these (leaps) is 73. What is the number of tho (various) fruits (in each of the heaps), and what the price (of the different varieties of fruits) ?
The rule for arriving at the numerical value of the prices of dearer and cheaper things (respectively) from the given mixed value (of their total price):-
146.[*] Divide (the rate-quantities of the given things) by their rate-prices. Diminish (these resulting quantities separately) by the least among them. Then multiply by the least (of the above mentioned quotient-quantities) the given mixed price of all the things and subtract (this product) from the given (total number of the various) things. Then split up (this remainder optionally) into as many (bits as there are remainders of the above quotient quantities left after subtraction); and then divide (these bits by those remainders of the quotient-quantities. Thus the prices of the various cheaper things are arrived at). These, separated from the total price, give rise to the price of the dearest article of purchase.
Examples in illustration thereof.
147 to 149. "In accordance with the rates of 3 peacocks for 2 paņas 4 pigeons for 3 paņas,5 swans for 4 paņas, and 6 sārasa
146.^ The rule will be clear from the following working of the problem given in 147-149:-
Divide the ratio-quantities 3, 4, 5, 6 by the respective rate-prices 2, 3, 4, 5; thus we have . Subtract the least of these from each of the other three. we get . By multiplying the given mixed price, 56 by the above mentioned least quantity , we have . Subtract this from the total number of birds, 72. Split up the remainder into any three parts, say . Dividing these respectively by we get the prices of the first three kinds of birds,. The price of the fourth variety of birds can be found out by subtracting all these three prices from the total 56. birds for 5 paņas, purchase, O friend, for 56 paņas 72 birds and bring them (to me)". So saying a man gave over the purchase-money (to his friend). Calculate quickly and find out how many birds (of each variety he,bought) for how many paņas.
150. For 3 paņas, 5 palas of ginger are obtained; for 4 paņas, 11 palas of long pepper; and for 8 paņas, 1 pala of pepper is obtained. By means of the purchase-money of 60 paņas,quickly obtain 68 palas (of these drugs).
The rule for arriving at tho desired numerical value of certain specified objects purchased at desired rates for desired sums of money as their total price:-
151. The rate-values (of the various things purchased are each separately) multiplied by the total value (of the purchase-money), and the various values of the rate-money are (alike separately)
151. The following working of the problem given in 152-153 will illustrate the rule :-
5 7 9 3 3 5 7 9 500 700 900 300 300 500 700 900 0 0 0 600 200 200 200 0 0 0 0 6 2 2 2 0 0 0 0 36 6 8 10 0 6 4 4 6 6 6 6 4 6 6 6 4 18 30 42 36 30 42 54 12 3 5 7 6 5 7 9 2 9 20 35 36 16 28 45 12
Write down the rate-things and the rate-prices in two rows, one below the other . Multiply by the total price and by the total number of things respectively. Then subtract. Remove the common factor 100. Multiply by the chosen numbers 3, 4, 5, 6. Add the numbers in each horizontal row and remove the common factor 6. Change the position of these figures, and write down in two rows each figure as many times as there are component elements in the corresponding sum changed in position. Multiply the two rows by the rate-prices and the rate-things respectively. Then remove the common factor 6. Multiply by the already chosen numbers 3, 4, 5, 6. The numbers in the two rows represent the proportions according to which the total price and the total number of things become distributed.This rule relates to a problem indeterminate equations, and as such, there may be many sets of answers, these answers obviously depending upon the quantities chosen optionally as multipliers.
It can be easily seen that, only when certain sets of numbers are chosen as optional multiplier, integral answers are obtained; in other cases, fractional answers are obtained, which are of course not wanted. For an explanation of the rationale of the process, see the note given at the end of the chapter multiplied by the total number of things purchased; (the latter products are subtracted in order from the former products ; the positive remainders are all written down in a line below, the negative remainders in a line above and all these are reduced to their lowest terms by the removal of the factors which are common to all of them. Then each of these reduced) differences is multiplied by (a separate) optionally chosen quantity; (then those products which are in a line below as well as those which are so above are separately added together); and the sums are written upside down, (the sum of the lower row of numbers being written above and the sum of the upper row being written below. These sums are also reduced to the lowest terms by means of the removal of common factors, if any; and the resulting quantities) are each of them written down twice, (so as to make one be below the other, as often as there are component elements in the corresponding alternate sum. These numbers thus arranged in two rows) are multiplied by their respective rate-prices and rate-values of things, (the rate-price multiplication being conducted with one row of figures and the rate-number multiplication being in relation to the other row of figures. The products so obtained are again reduced to their lowest terms by the removal of such factors as are common to all of them. The resulting figures in each vertical row are separately) multiplied (each) by (means of its corresponding originally chosen) optional multiplier. (And the products should be written down as before in two horizontal rows. The numbers in the upper row of products give the proportion in which the purchase money is distributed ; those in the lower row of products give the proportion in which the corresponding things purchased are distributed. Therefore) what remains thereafter is only the operation of prakșēpaka-karņa (proportionate distribution in accordance with rule-of-three).
An example in illustration thereof.
152 and 153. Pigeons are sold at the rate of 5 for 3 (paņas), sārasa birds at the rate of 7 for 5 (paņas), swans at the rate of 9 for 7 (paņas), and peacocks at the rate of 3 for 9 (paņas). A certain man was told to bring at these rates 100 birds for 100 paņas for the amusement of the king's son, and was sent to do so. What (amount) does he give for each (of the various kinds of birds that he buys) ?
The rule for arriving at the measure of two given commodities whose prices are interchanged:-
154. Let (the numerical value of) the sum of the (total selling and buying) money-prices (of the two given commodities) be divided by (the numerical measure of) the sum (of the commodities put together); then let the difference (between the above-mentioned buying and selling prices) be divided by the (numerical measure of any such) difference as may be obtained by subtracting any optionally chosen commodity-quantity from the given measure of the sum of the given commodities. If the operation of saṅkramaņa is conducted in relation to these, (viz., the quotient obtained in the first operation above and any one of the many quotients that may be obtained in the second operation), the rates at which those commodities are purchased is obtained. Then if the same operation of saṅkramaņa as relating to the sum of the commodities and to their difference is carried out, it of course gives rise to (the numerical measure of) the commodities (in question). The alternation (of these above-mentioned purchase-rates) gives rise to the sale-rates. This is the solution of (this kind of) problems as propounded by the learned; and the rule (itself) has been declared by the great Jina.
154. The algebraical representation of the method described in the rule may be given thus in relation to the problem proposed in stanzas 155 and 156--
LetI II III Adding I and II, we haveIV V Again subtracting I from II, we get
Now 2b is optionally chosen to be equal to 6.
VI VII Carry out the operation of saṅkramaņa with reference to VII and V and VI and III; and the values of x,y,a and b are all made out.
An example in illustration thereof.
155 and 156. The original price of one piece of sandalwood and one piece of agaru wood, they being together 20 palas (in weight), is 104 paņas; when after a time they were sold with their prices mutually altered, 116 paņas were obtained. You give out their buying and selling rates and the numerical measure of the commodities, taking 6 and 8 separately as the optional (number) needed by the rule.
The rule for arriving at the distance in yōjanas travelled by the horses of the sun's chariot when yoked as desired:-
157. The number representing the total yōjanas, divided by the total number of horses, gives the yōjanas (which each has at a stage to travel) in turn. These yōjanas multiplied by the optionally chosen number of horses to be yoked, give the measure of the distance to be travelled over by each horse.
An example in illustration thereof.
158. It is well-known that the horses belonging to the sun's chariot are 7. For horses (have to) drag it along, being harnessed to the yoke. They have to do a journey of 70 yōjanas. How many times are they unyoked and how many times yoked (again) in for four?
The rule for arriving at the value of the commodity to be found in the hands of each (of a body of joint proprietors}, from the conjoint remainder left after subtracting whatever is desired from the total value of all the commodities:--
159. Let the sum (of the values of the conjoint remainders) of the commodities be divided by the number of men lessened by one; the quotient will be the total value of all the commodities (owned in common). This total value as diminished by the specified values gives (in the corresponding cases) the value of commodity in the hands (of each of the proprietors in turn).
An example an illustration thereof.
160 to 162. Four merchants who had invested their money in common were asked each separately by the customs officer what the value of the commodity (they were dealing in) was; and one eminent merchant (among them), deducting his own investment, said that that (value) was in fact 22. Then another said that it was 23; then another said 24; and the fourth said that it was 7 (in saying so) each of them deducted his own invested amount (from the total value of the commodity for sale). 0 friend, tell me separately the value of the (share in the) commodity owned by each.
The rule for arriving at equal amounts of wealth, (as owned in precious gems,) after mutually exchanging any desired number of gems:--
163.[*] The number of gems to be given away is multiplied by the total number of men (taking part in the exchange). This product is (separately) subtracted from the number (of the gems) for sale (owned by each); the continued product of the remainders (so obtained) gives rise to the value of the gem (in each case), provided the remainder relating to it is given up (in obtaining such a product).
Examples in illustration thereof.
164. Tho first man had 6 azurE-blue gEms (of equal value), the second man had 7 (similar) emeralds, the other the third man--had 8(similar) diamonds. Each (of them), on giving to each (of the others) the value of a single gem (owned by himself), became equal (in wealth-value to the others . What is the value of a gem of each variety ?)
165 and 166. The first man has 16 azure-blue gems, the second has 10 emeralds, and the third mam has 8 diamonds. Each among them gives to each of the others 2 gems of the kind owned by himself; and then all three men come to be possessed of equal
163.^ Let m,n,p, be respectively the numbers of the three kinds of gems owned by three different persons, and a the number of gems mutually exchanged; and let x,y,z be the value in order of a single gen in the three varieties concrened.
wealth. Of what nature are the prices of those azure-blue gems, emeralds, and diamonds ?
The rule or arriving at the (value of the) invested capital by means of the rate of purchase, the rate of sale, and the profit obtained:-
167.[*] The buying and the selling rate-measures of the commodity are each multiplied alternately by the rate-prices (the product obtained with the help of) the buying rate-measure is divided by (the other product obtained with the aid of) the selling rate-measure. The profit, divided by the resulting quotient as diminished by one gives rise to the originally employed capital amount
An example in illustration thereof.
168. A merchant buys at the rate of 7 prasthas of grain for 3 paņas, and sells it at the rate of 9 prasthas for 5 paņas, and makes a profit of 72 paņas. What is the capital employed in this transaction?
Thus ends Sakala-kuțțīkāra in the chapter on mixed problems.
Suvarna-kuțțīkāra
Hereafter we shall explain that kuțțīkāra which consists of calculations relating to gold.
The rule for arriving at the top of the resulting mixed gold obtained by putting together (different component varieties of) gold of (various) desired varņas:-
169. It has to be known that the (sum of the various) products of (the various component quantities of gold as multiplied by (their respective) varņas, when divided by (the total quantity of)
167.^ If the buying rate is a things for b and the selling rate is c things for d, and if it is the gain by the transaction, then the capital invested is-
the mixed gold gives rise to the (resulting) varņa. (The original varņa of any component part thereof), when divided by the latter resulting varņa (of the mixed up whole), and multiplied by the (given) quantity of gold (in that component part), gives rise to (that) corresponding quantity of (the mixed) gold (which is equal in value to that same component part thereof).
An example in illustration thereof.
170 to 171. There are 1 part (of gold) of 1 varņa, 1 part of 2 varņas, 1 part of 3 varņas, 2 parts of 4 varņas, 4 parts of 6 varņas,7 parts of 14 varņas, and 8 parts of 15 varņas. Throwing these into the fire; make them all into one (mass), and then (say) what the varņa of the mixed gold is. This mixed gold is distributed among the owners of the foregoing parts. What does each of them get ?
The rule for arriving at the required weight of gold (of any desired varņa equivalent in value to given quantities of gold) of given varņas:-
172. The given quantities of gold are all (separately) multiplied by their respective varņas, and the products are added. The resulting sum is divided by the total weight of the mixed gold; the quotient is to be understood as the resulting average varņa. This (above-mentioned sum of the products) is separately divided by the desired varņas (to arrive at the required equivalent weight of this gold).
Examples in illustration thereof.
173. Twenty paņas (in weight of gold) of 16 varņas have been exchanged for (gold of) 10 varņas in quality; you give out how many purāņas (in weight) they become now.
174. One hundred and eight (in weight of) gold of varņas is exchanged for (gold of) 14 varņas. What is the (equivalent quantity of this new) gold ?
The rule for finding out the unknown varņa :-
175. From the product obtained by multiplying the total quantity of gold by the resulting varņa of the mixture, the sum of the products obtained by multiplying the (several component) quantities of gold by (their respective varņas) is to be subtracted. The remainder, when divided by the known component quantity of gold, (the varņa of which is to be found out), gives rise to the required varņa; and when divided by the difference between the resulting varņa and the known varņa (of an unknown component quantity of gold) gives rise to the (required weight of that) gold.
Another rule in relation to the unknown varņa :-
176. The sum of the products of the (various component quantities of) gold as multiplied by their respective varņa is to be subtracted from the product of the total quantity of gold as multiplied by the resulting varņa. Wise people say that this remainder when divided by the weight of the gold of the unknown varņa gives rise to the required varņa.
Examples in illustration thereof.
177 and 178. With gold of 6, 4 and 3 (in weight), characterised respectively by 13,8 and 6 as their varņas in weight of gold of an unknown varņa is mixed. The resulting varņa of the mixed gold is 11. O you, friend, who know the secrets of calculation, tell me the numerical value of this unknown varņa,
179. Seven in weight (of a given specimen) of gold has exactly 14 as the measure of its varņa; then 4 in weight (of another specimen of gold) is added to it. The resulting varņa is 10. Give out the unknown varņa (of this second specimen of gold).
The rule for arriving at the unknown weight of gold:-
180. Subtract the sum, obtained by adding together the products of the (various component quantities of) gold as multiplied by their respective varņas, from the product of the sum (of the known weights) of gold as multiplied by the now durable resulting varņa; the renmainder divided by the difference between the (known) varņa of the unknown quantity of gold and the resulting durable varņa (of the mixed gold) gives rise to the (weight of) gold.An example in distribution thereof.
181. Three pieces of gold, of 3 each in weight, and of 2, 3, and 4 varņas (respectivel ), are added to (an unknown weight of) gold of 18 varņas. The, resulting varņa comes to be 10. Tell me, O friend, the measure (of the unknown weight) of gold.
Tho rule for arriving at (the weights of gold (corresponding to two given varņas) from (the known weight and varņa of) the mixture of two (given specimens of) gold of (given) varņas:-
182. Obtain the differences between the resulting varņa (of the mixture on the one hand) and the known higher and lower varņas (of the unknown component quantities of gold on the other hand); divide one by these differences (in order); then carry out as before the operation of prakșēpaka (or proportionate distribution with the aid of these various quotients). In this manner it is possible to arrive even at the value of many component quantities of gold also.
Again, the rule for arriving at (the weights of) gold (corresponding to two given varņas) from (the known weight and varņa of) the mixture of two (given specimons of gold of (given) varņas:-
183. Write down in inverse order the difference between the resulting varņa and the higher (of the two given varņas of the two component quantities of gold), and also the difference between the resulting varņa and the lower (of the two given varņas). The result arrived at by means of the operation of proportionate distribution (carried out with the aid of these inversely arranged differences), that (result) gives the required (weights of the component quantities of) gold.
An example in illustration thereof.
184. If gold of 10 varņas, on being combined with gold of 16 varņas produces as result 100 in weight of gold of 12 varņas, give out separately (the measures in weight of) the two different varieties of gold. The rule for arriving at the (weights of) many (component quantities of) gold (of known varņa in a mixture of known your and weight):-
185. (In relation to all the known component varņas) excepting one of them, optionally chosen weights may be adopted. Then what remains should be worked out as in relation to the previously given cases by means of the rule bearing upon the (determination of an) unknown weight of gold.
An example an illustration thereof.
186. The (given) varņas (of the component quantities of gold) are 5, 6, 7, 8, 11, and 13 (respectively); and the resulting zero is in fact 9; and if (the total) weight (of all the component quantities) of gold be 60, what may be the several measures (in weight of the various component quantities) of gold?
The rule for arriving at the unknown varņas of two (known quantities of gold when the resulting part of the mixture is known):-
187. Divide one (separately) by the two (given weights of) gold; multiply (separately each of the quotients thus obtained) by (the weight of) the (corresponding quantity of) gold and (also) by the (resulting) varņa; write down (both the products so obtained) in two (different) places; (each of these in each of the two sets,) if diminished and increased alternately by one as divided by (the
185. The rule preferred to here is found in stanza 180 above. 187. The rule will become clear by the following working of the problem in stanza 188:-
- are written down in two places
- thus:
- Then and are added and subtracted alternately in each of the two sets thus:
- These give the two sets of answers.
known weight of) the corresponding (variety of gold, gives rise as a matter of course, to the required varņas.
An example in illustration thereof.
188. If, the (component) varņas not being known, the resulting varņa obtained by means of two (different kinds of) gold weighing 16 and 10 (respectively) happens to be 11, what would be the (respective) varņas of those two (different kinds of gold ?
Again, the rule for arriving at the unknown varņas of two (known quantities of gold, when tho resulting varņa of the mixture is known):-
189. Choose an optional varņa in relation to one (of the two given quantities of gold); what remains (to be found out) may then be arrived at as before. In relation to (the known quantities of all) the numerous varieties of gold excepting one, the varņas are optional; then (proceed) as before.
An example in illustration thereof.
190. On fusing together (two different kinds of gold which are 12 and 14 (respectively in weight), the resulting varņa is made out to be 10. Think out and say (what) the varņas of those two (kinds of gold are).
An example to illustrate the latter half of the rule.'
191. On fusing together 7, 9, 3, and 10 (in weight respectively of four different kinds) of gold, the resulting mixture turns out to be (gold of) 12 varņas. Give out the varņas (of the various component kinds of gold) separately.
The rule regarding how to arrive at (an estimate of the value of) the test sticks (of gold):-
192. The varņa of every stick is to be separately divided by the (given) maximum varņa, and (the quotients so obtained) are (all) to be added together. The resulting sum gives (the measure of) the required quantity of (pure) gold. From the summed up (weight of all the) sticks, this is to be subtracted. What remains is (the quantity of) the prapūraņikā (that is, the quantity of the baser metal mixed).
An example in illustration thereof.
198-196. (Three) merchants, well acquainted with the varņa of gold, were desirous of making test sticks of gold, and produced (such) golden sticks. The gold of the first (merchant) was of 12 varņas; (that of tho second was of) 14 varņas; and that of the third was of 16 varņas. The (various specimens of the test sticks of) gold in the case of the first (merchant) were (regularly) less by 1 (in varņa); those of the second were less by and ; and those of tho third were (in regular order) lass by . (The specimens of test gold) possessed by the first (merchant) began with that of (his) maximum varņa, and ended with that of 1 varņa; (similarly, those of the second began with that of his maximum varņa and) ended with that of 2 varņas; and those of the third merchant (began with that of his maximum varņa and) ended with that of 3 varņas. Every test stick is 1 māșa in weight. O mathematician, if you indeed know gold calculation, tell me separately and soon what the measure of pure gold here is, and what that of the baser metal mixed.
The rule for arriving at (the different weights of) gold obtained in exchange and characterised by (two given) varņas:--
197.[*] The two differences between, (firstly,) the product of the (given weight of) gold to be exchanged as multiplied by the (given) varņa (thereof) and the product of the weight of gold obtained in exchange as multiplied by the (first of the two specified) varņas (of the exchanged gold)-(and, secondly, between the first product above-mentioned and the product of the weight of
197.^ This rule will be clear from the following working of the problem given in stanza 198:-
, and are altered in position and written down as 896 and 1120; and these, when divided by 12-10 or 2, give rise to the answers, namely, 448 and 560 in weight of gold of 10 and 12 varņas respectively. gold obtained in exchange as multiplied by the second of the specified varņas out of the exchanged gold-these two differences) have to be written down. If then, they are altered in position and divided by the difference between the (two specified) varņas (of the two varieties) of the exchanged gold, the result happens to be the (two required) quantities (of the two kinds) of gold (obtained in exchange).
An example in illustration, thereof.
198. Seven hundred in weight of gold characterised by 16 varņas produces, on being exchanged, 1,008 (in weight) of two kinds of gold characterised (respectively) by 12 and 10 varņas. Now, what is the weight (of each of these two varieties) of gold?
The rule for finding out the (various weights of) gold obtained as the result of many (specified) kinds of exchange :-
199. If the (given) weight of gold (to be exchanged) as multiplied by the varņa (thereof) is divided by (the quantity of) the desired gold (obtained in exchange), there arises the uniform average varņa. On carrying out (further) operations as mentioned before, the result arrived at gives the required weights of the various kinds of gold obtained in exchange.
An example in illustration thereof.
200-201. In the case of a man exchanging 800 in weight of gold characterised by 14 varņas; the gold (obtained in exchange) is seen to be altogether 500 in weight, (the various parts whereof are respectively) characterised by 12, 10, 8 and 7 varņas. What is the weight of gold separately corresponding to each of these (different) varņas?
The rule for arriving at (the various weights of) gold obtained in exchange which are characterised by known varņas and are (definite) multiples in proportion:-
202-203. The sum of the (given) proportional multiple numbers is to be divided by the sum of the products (obtained) by
199. The operation which is stated here as having been mentioned before is what it given in stanza 185 above. multiplying the (given proportional quantities of the various kinds of the exchanged) gold by (their respective specified) varņas. (The resulting quotient) is to be multiplied by the original varņa (of the gold to be exchanged). If by this product as diminished by one, tho increase (in the weight of gold due to exchange) is divided, and the quotient (so obtained) is subtracted from the original wealth of gold, the remaining (weight of unexchanged) gold is arrived at. This (weight of the unexchanged gold) is then to be subtracted from the sum (of the weight) of the original gold and the increase (in weight due to exchange). Then if the resulting remainder (here) is divided by the sum of the proportional multiple numbers connected with the exchange, and is then multiplied by (each of those) proportional numbers (separately), the (various weights of) gold obtained in exchange and characterised by the specified varņas and the specified proportions are arrived at.
An example in illustration thereof.
204-205. There is a certain merchant desires of obtaining profit; and the gold (in his possession) is of 16 varņas and 200 in weight. A portion of it is exchanged in return for (four different kinds of gold characterised respectively by 12, 8, 9 and 10 varņas, (so that those varieties of gold are by weight) in proportions which begin with 1 and are then (regularly) multiplied by 2. The gain (in the weight of gold resulting out of this exchange transaction) is 102. What is the remaining (weight of the unexchanged) gold? Tell me also the weights of gold obtained in exchange corresponding to those (above-mentioned varņas ).
The rule for arriving at (the weight of the original (quantity of) gold with the aid of the gold exchanged (in part), and with the aid (of the weight) of gold seen to be in excess (in consequence of the exchange )
206. Each specified part of (the original) gold (to be exchanged) is divided by the varņa corresponding to its exchange (The resulting quotient is in each case to be) multiplied by the optionally chosen varņa (of the originally given gold; and then all these products are to be added). From this sum, the sum of the (various) fractional (exchanged) parts (of the original gold) is to be subtracted. (If now) the observed excess (in the weight of gold due to the exchange) is divided by this resulting remainder, what comes out here happens to be the original wealth of gold.
An example an illustration thereof.
207-208. A certain small ball of gold of 16 varņas belonging to a merchant is taken; and , and parts thereof are in order exchanged for (different kinds of) gold characterised (respectively) by 12, 10 and 9 varņas. (The weights of these exchanged varieties of gold are) added to what remains (unexchanged) of the original gold. Then 1,000 is observed to be in excess on removing from the account the weight of the original gold. What then is (the weight of this) original gold ?
The rule for arriving at the desired varņa with the help of the (mutual) gift of a desired fractional part of the gold (owned by the other), and also for arriving at the (weights of) gold (respectively) corresponding to those optionally gifted parts:-
209 to 212.[**] One divided by (the numerical measure of each of two specifically gifted)|parts is to be noted down in reverse order; and (if each of the quotients so obtained is) multiplied by an
209-212.^ The rule will be clear from the following working of the problem in 213-215:-
Dividing 1 by and , we get respectively 2, 3; altering their position and multiplying them by any optionally chosen number, say 1, we get 3, 2. These two numbers represent the quantities of gold owned respectively by the two merchants.
Choosing 9 as the varņa of the gold owned by the first merchant, we can easily arrive, from the exchange proposed by him, at 13 as the varņa of the gold owned by the second merchant. These varņas 9 and 13, give, in the exchange proposed by the second merchant, the average varņa of , while the average varņa as given in the sm ha8 to be 12 or
Therefore the varņa 9 and 13 have to be altered. If 8 is chosen instead of 9, 13 has to be increased to 16 in the first exchange. Using these two varņas, 8 and 16, in the second exchange, we obtain as the average varņa, instead of optionally chosen quantity, (it) gives rise to (the weights of each of the two small) balls of gold. The varņa (of each) of these (little balls of gold) as also that of the gold gifted by the other person (in the transaction) has to be arrived at as before with the aid of the (given) final average varņa (in each case). It in this manner both sets of answers (arrived at) happen to tally (with the requirements of the problem, the two varņa arrived at in accordance with the previously adopted option become the verified varņas mentioned in relation to the two (given) little balls of gold. If, (however, these answers do) not (tally), the varņas belonging to the first set (of answers) have to be made (as the case may be) a little less or a little more; (then the average varņa corresponding to these modified component varņa has to be further obtained). Thereafter, the difference between this (average) varņa and the previously obtained (untallying average) varņa is written down; (and the required proportionate quantities) are (therefrom) derived by means of the operation of the Rule of Three: and the varņas (arrived at according to the option chosen before, when respectively) diminished by one of these two quantities and increased by the other, turn out to be evidently the required varņas (here).
An example is illustration thereof.
213-215. Two merchants well versed in estimating the value of gold asked each other (for an exchange of gold). Then the first (of them) said to the other—If you give me half (of your gold), I shall combine that small pellet of gold with my own gold and make (the whole become gold of) 10 varņas” Then this other said--"If I only obtain your gold by one-third (thereof), I shall likewise make the whole (gold in my possession become
Thus, in the second exchange, we see an increase of 40-35 or 5 in the sun of tho products of weight and varņa, while the decrease and the increase in relation to the originally chosen varņas are 9-8 or 1 and 16-13 or 3.
But the required increase in sum of the products of weight and varņa in the second exchange is 36–35 or 1. Applying the Rule of Three, we get the corresponding decrease and increase in the varņas to be and .
Therefore, the varņa are 9-- or and or gold of) 12 varņas with the aid of the two pellets.” O you, who know the secret of calculation, if you possess cleverness in relation to calculations bearing upon gold, tell me quickly, after thinking out well, the measures of the quantities of gold possessed by both of them, and also of the varņas (of those quantities of gold).
Thus ends Suvarna-kuțțīkāra in the chapter on mixed problems
Vicitra-kuțțīkāra
Hereafter we shall expound the Vicitra-kuțțīkāra in the chapter on mixed problems.
The rule in regard to (the ascertaining of) the number of truthful and untruthful statements (in a situation like the one given below wherein both are simultaneously possible):-
216. The number of men, multiplied by the number of those liked (among them) as increased by one, and (then) diminished by twice the number of men liked, gives rise to the number of untruthful statements. The square of the number representing all the men, diminished by the number of those (untruthful statements), gives rise to the statements that are truthful.
216. The rationale of this rule will be clear from the following algebraical representation of the problem given in stanza 2]7 below:-
Let a be the total number of persons of whomb are liked. The number of utterances is a, and each statement refers to a persons. Hence the total number of statements is .
Now, of these a persons, b are liked, and are not liked. When each of the b number of persons is told — "You alone are liked," the number of untruthful statements in each case is . Therefore, the total number of untruthful statements in b statements is . . . . . . . . . . I
When, again, the same statement is made to each of the persons, the number of untruthful statements in each case is . Therefore, the total number of untruthful statements in utterances is . . . . . . . . II
Adding I and II, we get .
This represents the total of untruthful statements; and on subtracting it from , which is the measure of all the statements, truthful and untruthful, we arrive obviously at the measure of the truthful statements.
An example in illustration thereof.
217. There are five lustful men. Among them three are in fact liked by a public woman. She says (separately) to each (of them) "I like you (alone)". How many (of her statements, explicit as well as implicit) are true ones ?
The rule regarding the (possible) varieties of combinations (among given things):-
218. Beginning with one and increasing by one, let the numbers going up to the given number of thingy be written down in regular order and in the inverse order (respectively) in an upper and a lower (horizontal) row. (If) the product (of one, two, three, or more of the numbers in the upper row) taken from right to left (be) divided by the (corresponding) product (of one, two, three, or more of the numbers in the lower row) also taken from right to left, (the quantity required in each such case of combination) is (obtained as) the result.
Examples in illustration thereof.
219. 'Tell (me) now, O mathematician, the combination varieties as also the combination quantities of the tastes, viz., the astringent, the bitter, the sour, the pungent, and the saline, together with the sweet taste (as the sixth).
220. O friend, you (tell me quickly how many varieties there may be, owing to variation in combination, of a (single string) necklace made up of diamonds, Sapphires, emeralds, corals, and pearls.
221. O (my) friend, who know the principles of calculation, tell (me) how many varieties there may be, owing to variation in combination, of a garland made up of tho (following) flowers-- kētakī, aśōka, campaka, and nīlōtpala.
218. This rule relates to a problem in combination The formula given here is ; and this is obviously equal to The rule to arrive at (the unknown) capital with the aid of certain known and unknown profits (in a given transaction):-
222. By means of the operation of proportionate distribution, the (unknown) profits age to be determined from the mixed sum (of all the profits) minus the (known) profit. Then the capital of the person whose investment is unknown results from dividing his profit by that (same common factor which has been used in the process of proportionate distribution above).
An example in illustration thereof.
223-225. According to agreement some three merchants carried put (the operation of) buying and selling. The capital of the first (of them) consisted of six purāņas., that of the second of eight purāņas, but that of the third was not known. The profit obtained by all those (three) men was 96 purāņas. In fact the profit obtained by him (this third person) on the unknown capital happened to be 40 purāņas. What is the amount thrown by him (into the transaction), and what is the profit (of each) of the other two merchants? O friend, if you know the operation of proportionate distribution, tell (me this) after making the (necessary) calculation.
The rule for arriving at the wages (due in kind for having carried certain given things over a part of the stipulated distance according to a given rate):-
226.[*] From the square of the product (of the numerical value) of the weight to be carried and half of the (stipulated distance
226.^ Algebraically, the formula given in the rule is :
, where x=wages to be found out, a = the total weight to be carried, D= the total distance, d is the distance gone over, and b=the total wages promised. it may be noted here that the rate of the wages for the two stages of the journey is the same, although the amount paid for each stage of the journey is not in accordance with the promised rate for the whole journey.
The formula is easily derived from the following equation containing the data in the problem:-
measured in) yōjana, subtract the (continued) product of (the numerical value of) the weight to be carried, (that of the stipulated) wages, the distance already gone over, and the distance still to be gone over. Then, if the fraction ( viz., half) of the weight to be carried over, as multiplied by the (whole of the stipulated) distance, and then as diminished by the square root of this (difference above mentioned), be divided by the distance still to be gone over, the required answer is arrived at.
An example in illustration thereof.
227. Here is a man who is to receive, by carryiug 32 jackfruits over 1 yōjana, 7 of them as wages. He breaks down at half the distance. What (amount within the stipulated wages) is (then) due to him ?
The rule for arriving at the distances in yōjanas (to be travelled over) by the second or the third weight-carrier (after the first or the second of them breaks down):-
228. From the product of the (whole) weight to be carried as multiplied by the (value of the stipulated) wages, subtract the square of the wages given to the first carrier. This (difference has to be used as the) divisor in relation to the (continued) product of the difference between the (stipulated) wages (and the wages already given away), the (whole) weight to be carried, and the (whole) distance (over which the weight has to be carried. The resulting quotient gives rise to the distance to be travelled over by the second (person).
An example in illustration thereof.
229. A man by carrying 24 jack-fruits over (a distance of) five yōjanas has to obtain 9(of them) as wages therofor. When 6 of these have been given away as wages (to the first carrier), what is the distance the second carrier has to travel over (to obtain tho remainder of the stipulated wages) ?
228. Algebraically , which can be easily found out from the equation in the last note. The rule for arriving at (the value of) the wages corresponding to the various stages (over which varying numbers of persons carry a given weight):-
280. The distance (travelled over by the various numbers of men), are (respectively) to be divided by the numbers of the men that are (doing the work of carrying) there. The quotients (so obtained) have to be combined so that the first (of them is taken at first separately and then) has (1, 2, and 3, etc., of) the following (quotients) added to it. (These quantities so resulting are to be respectively) multiplied by the numbers of the men that turn away (from the journey at the various stages. Then) by adopting (in relation to these resulting products) the process of proportionate distribution (prakșēpaka), the wages (due to the men leaving at the different stages) may be found out.
An example in illustration thereof.
231-232. Twenty men have to carry a palanquin over (a distance of) 2 yōjanas, and 720 dīnāras form their wages. Two men stop away after going over two krōśas; after going over two (more) krōśas, three others (stop away): after going over half of the remaining distance, five men stop away . What wages do they (the various bearers) obtain?
The rule for arriving at (the value of the money contents of) a purse which (when added to what is on hand with each of certain persons) becomes a specified multiple (of the sum of what is on hand with the others):-
233-235.[*] The quantities obtained by adding one to (each of the specified) multiple numbers (in the problem, and then)
233–235.^ In the problem given in 236-237, let x,y,z represent the money on hand with the three merchants, and y the money in the purse.
- where a,b,c represent the multiples given in the problem.
multiplying these sums with each other, giving up in each case the sum relating to the particular specified multiple, are to be reduced to their lowest terms by the removal of common factors. (These reduced quantities are then) to be added. (Thereafter) the square root (of this resulting sum) is to be obtained, from which one is (to be subsequently) subtracted. Then the reduced quantities referred to above are to be multiplied by (this) square root as diminished by one. Then these are to be separately subtracted from the sum of those same reduced quantities. Thus the moneys on hand with each (of the several persons) are arrived at . These (quantities measuring the moneys on hand) have to be added to one another, excluding from the addition in each case the value of the money on the hand of one of the persons; and the several sums so obtained are to be written down separately. These are (then
- Then, . . . . .
I - where
- Similarly, . . . . .
II - and . . . . .
III - Adding, I, II, III,
- . . . . .
IV - Subtracting separately I, II, III,
By removing the common factors, if any, in the right-hand side of the proportion, we get at the smallest integral values of x,y,z.
This proportion is given in the rule as the formula.
It may be noted that the square root mentioned in the rule has reference only to the problem given in the stanzas 236-287. Correctly speaking, instead of "square root", we must have '3'.
It can be seen easily that this problem is possible only when the sum of any two of greater than the third. to be respectively) multiplied by (the specified) multiple quantities (mentioned above) ; from the several products so obtained the (already found out) values of the moneys on hand are (to be separately subtracted). Then the (same) value of the money in the purse is obtained (separately in relation to each of the several moneys on hand).
An example in illustration thereof.
236-237. Three merchants saw (dropped) on the way a purse (containing money). One (of them) said (to the others), "If I secure this purse, I shall become twice as rich as both of you with your moneys on hand.” Then the second (of them) said, "I shall become three times as rich.” Then the other, (the third), said, "I shall become five times as rich.” What is the value of the money in the purses also the money on hand (with each of the three merchants) ?
The rule to arrive at the value of the moneys on hand as also the money in the purse (when particular specified fractions of this latter, added respectively to the moneys on hand with each of a given number of persons, make their wealth become in each case) the same multiple (of the sum of what is on hand) with all (the others )
238.[*] The sum of (all the specified) fractions (in the problem the denominator being ignored is multiplied by the (specified common) multiple number. From this product, the products obtained by multiplying (each of the above-mentioned) fractional parts (as reduced to a common denominator, which is then ignored), by the product of the number of cases of persons minus one and the specified multiple number this last product being diminished
238.^ The formula given in the rule is--
where x,y,z are the moneys on hand,m the common multiplier, and a,b,c, the specified fractional parts given.
These values can be easily found out from the following equations:-
- where P is the money in the purse.
by one, are (severally) subtracted. The resulting remainders constitute the several values of the moneys on hand. The value of the money in the purse is obtained by carrying out operations as before and then by dividing by any particular specified fractional part (mentioned in the problem).
An example in illustration thereof.
239–240. Five merchants saw a purse of money. They said one after another that by obtaining (respectively) of the contents of the purse, they would each become with what he had on hand three times as wealthy as all the remaining others with what they had on hand together. O arithmetician, (you tell) me quickly what moneys these had on hand (respectively), and what the value of the money in the purse was.
The rule for arriving at the measure of the money contents of a purse, when specified fractional parts (thereof added to what may be on hand with one among a number of persons) makes him a specified number of times (as rich as all the others with what they together have on hand)
241. The specified fractional parts relating to all others (than the person in view) are (reduced to a common denominator, which is ignored for practical purposes. These are severally) multiplied by the specified multiple number (relating to the person in view). To these products, the fractional part (relating to the person) in view (and treated like other fractional parts) is added. The resulting sums are (severally) divided each by its (corresponding specified) multiple quantity as increased by one. Then these quotients are also added. The several sums (so obtained in relation
241. The formula given in the rule is:-
and so on; where x,y, . . . . are moneys on hand; a,b,c,d, . . . . fractional parts; m,n,q, r,. . . . . various multiple numbers; and s the number of persons concerned in the transaction.
to the several cases) are diminished by the product of the particular specified fractional part as multiplied by the number of cases less by two. The difference is divided by the particular specified multiple quantity as increased by one. The result is the money on hand(in The particular case).
Examples in illustration thereof
242-243. Two travellers saw a purse containing money (dropped) on the way. One of them said (to the other), "By securing half of this money (in the purse), I shall become twice as rich (as you).", The other said, "By securing two-thirds (of the money in the purse), I shall, with the money I have on hand, have three times as much money as what you have on hand." What are the moneys on hand, and what the money in the purse?
244-244. Two travellers saw on the way a purse containing money; and the first of them took it up and (said, that) that money along with the money that he had on band became twice the money of the other (traveller. This) other (said that that money in the purse with the aid of what he had on hand would be) three times (the money in the hand of the first traveller). What is the money on hand (in the case of each of them). and what the money in the purse ?
245-247. Four men saw on the way a purse containing money. The first among them said, "If I secure this purse, I shall with the money already on hand with me become (possessed of money which will be) eight times (the money on hand with the remaining travellers). Another (said, that the money in the purse with what he had on band) would be nine times the money on hand with the rest (among them). Another (said that similarly he) would be possessed of ten times the money, and another (that he) would be possessed of eleven times the money. Tell me quickly, O mathematician, what the money in the purse was and how much the money in the hand of each of them was.
248. Four men saw on the way a purse containing money. (Then), with what each of them had on hand, the , , , and parts (respectively) of this (money in the purse) became twice, thrice, five times and four times (that money which the others together had on hand. What is the money in the purse, and what the money on hand with each of them ?)
249-250. Three merchants saw on the way a purse containing money. The first among them said, "If I get of this money in the purse, I shall (with what I have on hand) become (possessed of) twice the (money on hand with) both of you." Another said that, if he secured part of the money in the purse, ho would with the money on hand with him (become possessed of) thrice (tho money on hand with the others). The third man said, " If I obtain of this (money in the purse), I shall become possessed of four times the money (on hand with both of you)." Tell me quickly, O mathematician, what the money on hand, with each of them was, and what was the money in the purse.
The rule for arriving at the money on hand, which, with the moneys begged (of others), becomes a specified multiple (of the money on hand with the others)
251-252.[*] The sums of the moneys begged are multiplied each by its own corresponding multiple quantity as increased by one. With the aid of these (products) the moneys on hand are arrived at according to the rule given in stanza 241. These quantities ( so obtained) are reduced so as to have a common denominator. Then they are (severally) divided by the sum as diminished by unity of the specified multiple quantities (respectively) divided by (those same) multiple quantities as increased by one. (The resulting quotients) themselves should be understood to be the moneys on hand , with the various persons) .
251-252. ^ Algebraically,
Similarly for y,s, etc. Here a,b,c,d,f,g, are sums of money begged of each other.
Examples in illustration thereof.
253-255. Three merchants begged money from the hands of each other. The first begged 4 from the second and 5 from the third man, and became possessed of twice the money (then on hand with both the others). The second (merchant) begged 4 from the first and 6 from the third, and (thus) got three times the money (held on hand at the time by both the others together. The third man begged 5 from the first and 6 from the second, and (thus) became 5 times (as rich as the other two). O mathematician, if you know the mathematical process known as citra-kuṭṭīkāra-miśra, tell me quickly what may be the moneys they respectively had on hand.
256-258. There were three very clever persons. They begged money of each other. The first of them begged 12 from the second and 13 from the third, and became thus 3 times as rich as these two were then. The second of them begged 10 from the first and 13 from the third, and thus became 5 times as rich (as the other two at the time). The third man begged 12 from the second and 10 from the first, and became (similarly) 7 times as rich. Their intentions were fulfilled. Tell me, O friend, after calculating, what might be the moneys on hand with them.
The rule for arriving at equal capital amounts, on the last man giving (from his own money) to the penultimate man an amount equal to his own, (and again on this man doing the same in relation to the man who comes behind him, and so on) :-
259.[*] One divided by the optionally chosen multiple quantity (in respect of the amount of money to be given by the one to the other) becomes the multiple in relation to the penultimate man's amount. This (multiplier) increased by one becomes the multiplier of the amounts (in the hands) of the others. The
259. ^ The rule will be clear from the following working of the problem given in st. 263.
or 2 is the multiple with regard to the penaltimate man's amount; this 2 combined with 1, i.e., 3 becomes the multiple in relation to the amounts of the others. amount of the last person (so arrived at) is to have one added to it. This is the process to be adopted.
Examples in illustration thereof.
260-261. Three sons of a merchant, the eldest, the middle, and the youngest, were going out along a road. The eldest son gave out of his capital amount to the middle on exactly as much as the capital amount of (that same) middle son. This middle son gave (out of his amount) to the last son just as much as he had. (In the end), they all became possessed of equal amounts of money. O mathematician, think out and say what amounts they (respectively) had (with them) on hand (to start with).
262. There were five sons of a merchant. From the eldest (of them) the one next to him obtained as much money as he himself had on hand . All others also did accordingly (each one giving to the brother next to him as much as he had on hand. In the end) they all became possessed of equal amounts of money. What were the amounts of money they (respectively) had on hand (to start with) ?
268. Fire merchants became possessed of equal amounts of money after each of them gave out of his own property to the one who went before him half of what he possesed. Think out and
- Now
1,1 Multiplying the penultimate 1 and 2 by the other by 3, we get2,3 Adding 1 to the last2,4 Write down2,4,4 Multiply the penultimate 4 by 2, and the others by 3, and add 1 to the last6,8,13 Again6,8,13,13 Repeating the same operations as above we get18,24,26,40,
54,72,78,80,121The figures in the last row represent the amounts in the hands of the 5 merchants,
Algebraically-
- ; where a,b,c,d are the amounts on hand with the fice merchants
say what amounts of money they (respectively) had on hand (to start with)
264. There were six merchants. The elder ones among them gave in order, out of what they respectively had on hand, to those who were next younger to them exactly two-thirds (of what they respectively had on hand. Afterwards, they all became possessed of equal amounts of money. What were the amounts of money they (severally) had on hand (to start with) ?
The rule for arriving at equal amounts of money on hand, after a number of persons give each to the others among them as much as they (respectively) have (then) on hand :-
265.[*] One is divided by the optionally chosen multiple quantity(in the problem). (To this), the number corresponding to the men (taking part in the transaction) is added. The first (man's) amount (on hand so start with is thus arrived at). This (and the results thereafter arrived at) are written down (in order), and each of them is multiplied by the optional multiple number as increased by one; and the result is then diminished by one. (Thus the money on hand with each) of the others (to start with is arrived at).
Examples in illustration thereof.
266. Each of three merchants gave to the others what each of these had on hand (at the time). Then they all became possessed of equal amounts of money. What are the amounts of money which they (respectively) had on hand (to start with)?
265.^ The rule will be clear from the following working of the problem given in st. 266:-
1, divided by the optionally chosen multiple 1, and increased by the number of person, 3, gives 4; this is the money in the hand of the first man.
This 4, multiplied by the optionally chosen multiple, 1, as increased by 1, becomes 8; when 1 is subtracted from this, we get 7, which is the money on hand with the second person.
This 7, again, treated as above, i.e., multiplied by 2 and then diminished by 1, gives 13, the money on hand with the third man.
This solution can be easily arrived at from the following equations:-
-
- 267. There were four merchants. Each of them obtained from the others half of what he had on hand (at the time of the
respective transfer of money). Then they all became possessed of equal amounts of money. What is the measure of the money (they respectively had) on hand (to start with) ?
The rule for arriving at the gain derived (equally) from success and failure (in a gambling operation):-
268-269. The two sums of the numerators and denominators of the (two fractional multiple) quantities (given in the problem) have to be written down one below the other in the regular order,and (then) in the inverse order. The (summed up) quantities (in the first of these sets of two sums) are to be multiplied according to the vajrāpavartana process by the denominator, and (those in the second set) by the numerator, (of the fractional quantity) corresponding to the other (summed up quantity). The results (arrived at in relation to the first set) are written down in the form of denominators, and (those arrived at in relation to the second set are written down) in the form of numerators : (and the difference between the denominator and numerator in each set is noted down). Then by means of these differences the products obtained by multiplying (the sum of) the numerator and the denominator (of each of the given multiple fractions in the problem) with the denominator of the other are (respectively) divided. These resulting quantities, multiplied by the value of the desired gain, give in the inverse order the measure of the moneys on hand (with the gamblers to stake) .
An example in illustration thereof.
270–272. A great man possessing power of magical charm and medicine saw a cock-fight going on, and spoke separately in
268-269. Algebraically,
, where x and y are the moneys on hand with the gamblers, and , the fractional parts taken from them, and p the gain. This follows from . confidential language to both the owners of the cocks. He said to one: "If your bird wins, then you give the stake-money to me If, however, you prove unvictorious, I shall give you two-thirds of that stake-money then. He went to (the owner of) the other (cock) and promised to give three-fourths (of his stake-money on similar conditions). From both of them the gain to him could be only 12 (gold-pieces in each case). You tell me, O ornament on the forehead of mathematicians, the (values of the) stake-money which (each of ) the cock-owners had on hand.
The rule for separating the (unknown) dividend number, the quotient, and the divisor from their combined sum :-
273. Any (suitable optionally chosen) number (which has to be) subtracted from the (given) combined sum happens to be the divisor (in question). On dividing, by this (divisor) as increased by one, the remainder (left after subtracting the optionally chosen number from the given combined sum), the (required) quotient is arrived at. The very same remainder (above mentioned), as diminished by (this) quotient becomes the (required dividend) number.
An example in illustration thereof.
274. A certain unknown quantity is divided by a certain (other) unknown quantity. The quotient here as combined with the divisor and the dividend number is 53. What is that divisor and what (that) quotient ?
The rule for arriving at that number, which becomes a square either on adding a known number (to the original number), or on subtracting (another) given number (from that same original number) :-
275.[*] The sum of the quantity to be added and the quantity to be subtracted is multiplied by one as associated with whatever may happen to be the excess above the even number (nearest to
275.^ Algebraically, let x be the quantity to be found out, and a,b, the respective quantities to be added to or subtracted from it; then, the formula to represent the rule will be that sum). The resulting product is (then) halved and (then) squared. (From this squared quantity), the (above-referred-to possible) excess quantity is subtracted. The result is divided by four, and then combined with one. Then the resulting quantity is either added to or subtracted from (respectively) by the half of the difference between the two given quantities as diminished or increased by the odd-making excess quantity (above referred to) according as the original given quantity to be subtracted is greater or less than the original given quantity to be added. The result arrived at in this manner happens to be the (required) number, which (when associated as desired with the (given) quantities) surely yields the square root (exactly).
Examples in illustration thereof.
276. A certain number when increased by 10 or decreased by 17 yields an exact square root. If possible, O arithmetician, tell me quickly that number
277. A certain quantity either as diminished by 7, or as added to by 18, yields the square root exactly. O arithmetician, give it out after calculation.
278[*]. A certain quantity diminished by , or again that same (quantity) increased by , yields the square root (exactly). Tell me that quantity quickly, O arithmetician, after thinking out what it may be.
The rationale of this may be made out thus:-
, an odd number; and an even number; where n is any integer.
From , the rule shows how we may arrive at when we know , to be equal to .
278.^ Since the quantities represented by b and a in the note on stanza 275 are seen to be fractional in this problem, being actually and , it is necessary to have these fractional quantities removed from the process of working out the problem in accordance with the given rule. For this purpose they are first reduced to the same denominator, and come to be represented by respectively: then these quantities are multiplied by , so as to yield 294 and 189, which are assumed to be the b and the a in the problem. The result arrived at with these assumed values of b and a is divided by , and the quotient is taken to be the answer of the problem. The rule for arriving at the square root of (an unknown) number as increased or diminished by a known number:-
279.[1] The known quantity which is given is first halved and (then) squared and then one is added (to it). The resulting quantity either when increased by the desired given quantity or when diminished by the (same) quantity yields the square root (exactly).
An example an illustration thereof.
280. Here is a number which, when increased by 10 or diminished by the same 10, yields an exact square root. Think out and tell me that number, O mathematician
The rule for arriving at the two required square quantities, with the aid of those required quantities as multiplied by a known number, and also with the aid of (the same known number as forming the value of) the square root of the difference (between these products):-
281.[2] The given number is increased by one; and the given number is also diminished by one. The resulting quantities when halved and then squared give rise to the two (required) quantities. Then if these be (separately) multiplied by the given quantity, the squrare root of the difference between these (products) becomes the given quantity.
An example in illustration thereof.
282-283. Two unknown squared quantities are multiplied by 71. The square root of the difference between these (two resulting products is also 71. O mathematician, if you know the process of calculation known as citra-kuṭṭīkāra, calculate and tell me what (those two unknown) quantities are.
279.^ This is merely a particular case of the rule given in stanza 275 wherein a is taken to be equal to b.
281.^ Algebraically, when the given number is d, are the required square quantities The rule for arriving at the required increase or decrease in relation to a given multiplicand and a given multiplier (so as to arrive at a given product):-
284.[1] The difference between the required product and the resulting product (of the given multiplicand and the multiplier) is written down in two places. To (one of the factors (of the resulting product) one is added, and (to the other) the required product is added. That (difference written above in two positions as desired) is (severally) divided in the inverse order by the sums (resulting thus). These give rise to the quantities that are to be added (respectively to the given multiplicand and the, multiplier) or (to the quantities that are) to be (respectively) subtracted (from them).
Examples in illustration thereof.
285. The product of 3 and 5 is 15; and the required product is 18; and it is also 14. What are the quantities to be added (respectively to the multiplicand and the multiplier) here, or what to be subtracted (from then)?
The rule for arriving at (the required result by the process of working backwards:-
286. To divide where there has been a multiplication, to multiply where there has been a division, to subtract where there has been an addition, to get at the square root where there has been a squaring, to get at the squaring where the root has been given—this is the process of working backwards.
An example in illustration thereof.
287. What is that quantity which when divided by 7, (then) multiplied by 3, (them) squared, (then) increased by 5, (then)
284.^ The quantities o be added or subtracted are:- .
For , where a and b are the given factors, and d the required multiple. divided by , (then) halved, and then reduced to its square root, happens to be the number 5 ?
The rule for arriving at (the number of arrows in a bundle with the aid of the even number of) arrows constituting the common circumferential layer (of the bundle) :-
288.[1] Add three to the number of arrows forming the circumferential layer; then square this (resulting sum) and add again three (to this square quantity). If this be further divided by 12, the quotient becomes the number of arrows to be found in the bundle.
An example in illustration thereof.
289. The circumferential arrows are 18 in number. How many (in all) are the arrows to be found (in the bundle) within tho quiver? O mathematician, give this out if you have taken pains in relation to the process of calculation known as vicitra-kuṭṭīkāra.
Thus ends vicitra-kuṭṭīkāra in the chapter on mixed problems.
288.^ The formula here given to find out the total number of arrows is where n is the number of circumferential arrows. This formula can be arrived at from the following considerations. It can be proved geometrically that only six circles can be described round another circle, all of them being equal and each of them touching its two neighbouring circles as well as the central circle; that, round these circles again, only twelve circles of the same dimension can be described similarly; and that round these again, only 18 such circles are possible, and so on . Thus, the first round has 6 circles, the second 12, the third 18, and so on. So that the number of circles in any round, say p, is equal to 6 p.
Now the total number of circles in the given number of rounds p, calculated from the central circle, is . If the value of 6p is given, say, as n, the total number of circles in . which is easily reducible to the formula given at the beginning of this note,
Summation of Series.
Hereafter we shall expound in (this) chapter on mixed problems the summation of quantities in progressive series.
The rule for arriving at the sum of a series in arithmetical progression, of which the common difference is either positive or negative:-
290. The first term is either decreased or increased by the product of the negative or the positive common difference and the quantity obtained by having the number of terms in the series as diminished by one. (Then, this is (further) multiplied by the number of terms in the series. (Thus, the sum of a series of terms in arithmetical progression with positive or negative common difference is obtained.
Examples in illustration thereof.
290.[1] The first term is 14; the negative common difference is 3; the number of terms is 5. The first term is 2; the positive common difference is 6; and the number of terms is 8. What is the sum of the series in (each of these cases?
The rule for arriving at the first term and the common difference in relation to the sum of a series in arithmetical progression, the common difference whereof is positive or negative :-
292.[2] Divide the (given) sum of the series by the number of terms (therein) and subtract (from the resulting quotient) the product obtained by multiplying the common difference by the half of the number of terms in the series as diminished by one. (Thus) the first term (in the series) is arrived at. The sum of the series is divided by the number of terms (therein). The first term is subtracted (from the resulting quotient); the remainder when divided by half of the number of terms in the series as diminished by one becomes the common difference.
290.^ Algebraioally, , where n is the number of terms, a the first term, b the common difference, and s the sum of the series.
292.^ Algebraically, and .
Examples in illustration thereof.
293. The sum of the series is 40; the number of terms is 5; and the common difference is 3; the first term is not known now. (Find it out.) When the first term is 2, find out the common difference.
The rule for arriving at the sum and the number of terms in a series in arithmetical progression (with the aid of the known lābha, which is the same as the quotient obtained by dividing the sum by the unknown number of terms therein):-
294.[1] The lābha is diminished by the first term, and (then) divided by the half of the common difference and on adding one to this same (resulting quantity), the number of terms in the series (is obtained). The number of terms in the series multiplied by the lābha becomes the sum of the series.
An example in illustration thereof.
295.[2] (There were a number of utpala flowers, representable as the sum of a series in arithmetical progression, whereof) 2 is the first term, and 3 the common difference. A number of women divided (these) utpala flowers (equally among them). Each woman had 8 for her share. How many were the women, and how many the flowers ?
The rule for arriving at the sum of the squares (of a given number of natural numbers beginning with one):-
296.[3] The given number is increased by one, and (then) squared; (this squared quantity is) multiplied by two, and (then) diminished by the given quantity as increased by one. (The remainder thus
294.^ Algebraically, , which is the lābha.
295.^ The number of women in this problem is conceived to be equal to the number of terms in the series.
296.^ Algebraically,, which is the sum of the squares of the natural numbers up to n. arrived at is) multiplied by the half of the given number. This gives rise to the combined sum of the square (of the given number), the cube (of the given number), and the sum of the natural numbers (up to the given number). This combined sum, divided by three, gives rise to the sum of the squares (of the given number of natural numbers).
Examples in illustration thereof.
297. (In a number of series of natural numbers), the number of natural numbers is (in order) 8, 18, 20, 60, 81, and 36. Tell me quickly (in each case) the combined sum of the square (of the given number), the cube (of the given number), and the sum of the given number of natural numbers. (Tell me) also the sum of the squares of the natural numbers (up to the given number).
The rule for arriving at the sum of the squares of a number of terms in arithmetical progression, whereof the first term, the common difference, and the number of terms are given:-
298.[1] Twice the number of terms is diminished by one, and (then) multiplied by the square of the common difference, and is (then) divided by six. (To this), the product of the first term and the common difference is added. The resulting sum is multiplied by the number of terms as diminished by one(To the product so arrived at), the square of the first term is added. This sum multiplied by the number of terms becomes the sum of the squares of the terms in the given series.
Again, another rule for arriving at the sum of the squares of a number of terms in arithmetical progression, whereof the first term, the common difference, and the number of terms are given:-
299. Twice the number of forms (in the series) is diminished by one, and (then) multiplied by the square of the common difference, and (also) by the number of terms as diminished by one. This
298.^ sum of the squares of the terms in a series in arithmetical progression. product is divided by six. (To this resulting quotient), the square of the first term and the (continued product) of the number of terms as diminished by one, the first term, and the common difference, are added. The whole (of this) multiplied by the number of terms becomes the required result.
Examples in illustration thereof.
300. In a series in arithmetical progression), the first term is 3, the common difference is 5. the number of terms is 5. Give out the sum of the squares (of the terms) in the series. (Similarly, in another series), 5 is the first term, B the common difference, and 7 the number of terms. What is the sum of the squares (of the terms) in this series ?
The rule for arriving at the sum of the cubes (of a given number of natural numbers):-
301.[1] The quantity represented by the square of half the (given) number of terms is multiplied by the square of the sum of one and the number of terms. In this (science of) arithmetic, this result is said to be the sum of the cubes (of the given number of natural numbers) by those who know the secret of calculation.
Examples in illustration thereof.
302. Give out (in each case) the sum of the cubes of (the natural numbers up to) 6, 8, 7, 25 and 256.
The rule for arriving at the sum of the cubes (of the terms in a series in arithmetical progression), the first term, the common difference, and the number of terms whereof are optionally chosen:-
303.[2] The sum (of the simple terms in the given series), as multiplied by the first term (therein), is (further) multiplied by the
301.^ Algebraically , which sum the cubes of the natural numbers up to n.
303.^ Algebraically, the sum of the cubes of the terms in a series in arithmetical progression, where the sum of the simple terms of the series. The sign of the first term in the formula is according as difference between the first term and the common difference (in the series). (Then) the square of the sum (of the series) is multiplied by the common difference. If the first term is smaller than the common difference, then (the first of the products obtained above is) subtracted (from the second product). If, however, (the first term is) greater (than the common difference), then (the first product above-mentioned is added (to the second product). (Thus) the (required) sum of the cubes is obtained.
Examples in illustration thereof.
304. What may be the sum of the cubes when the first term is 3, the common difference 2, and the number of terms 5; or, when the first term is 5, the common difference 7, and the number of terms 6 ?
The rule for arriving at the sum of (a number of terms in a series wherein the terms themselves are successively) the sums of the natural numbers (from 1 up to a specified limit, these limiting numbers being the terms in the given series in arithmetical progression):-
305–305.[1] Twice the number of terms (in the given series in arithmetical progression) is diminished by one and (then) multiplied by the square of the common difference. This product is divided by six and increased by half of the common difference and (also) by the product of the first term and the common difference. The sum (so obtained) is multiplied by the number of terms as diminished by one and then increased by the product obtained by multiplying the first term as increased by one by the first term itself. The quantity (so resulting) when multiplied by half the number of terms (in the given series) gives rise to the required sum of the series wherein the terms themselves are sums (of specified series)
305-305.^ Algebraically, is the sum of the series in arithmetical progression, wherein each term represents the sum of a series of natural numbers up to a limiting number, which is itself a member in a series in arithmetical progression.
Examples in illustration thereof.
306. It is seen that (in a given series) the first term is 6, the common difference 5, and the number of terms 18. In relation to (these) 18 terms, what is the sum of the sums of (the various series having 1 for the first term and 1 for the common difference.
The rule for arriving at the sum of the four quantities (specified below and represented by a certain given number):-
307.[1] The given number is increased by one and (then) halved. This is multiplied by the given number and (then) by seven. From the (resulting) product, the given number is subtracted; and the (resulting) remainder is divided by three. The quotient (thus obtained),when multiplied by the given number is increased by one, gives rise to the (required) sum of (the four specified quantities, namely,) the sum of the natural numbers (up to the given number), the sum of the sums of the natural numbers (up to the given number), the square (of the given number), and the cube (of the given number)
'Examples in illustration thereof.
308. The given numbers are 7, 8, 9, 10, 16, 50 and 61. Taking into consideration the required rules, separately give out in the case of each of them the sum of the four (specified) quantities.
The rule for arriving at the collective sum (of the four different kinds of series already dealt with):-
309. [2] The number of terms is combined with three; it is (then) multiplied by the fourth part of the number of terms ; (then) one
307.^ .^ Algebraically, is the sum of the four quantities specified in the rule. These are (i) the sum of the natural numbers up to n; (ii) the sum of the sums of the various series of natural numbers respectively limited by the various natural numbers up to n (iii)the square of n; and (iv) the cube of n.
309. ^ Algebraically, is the collective sum of the sums, namely, of the sums of the different series dealt with in rules 296, 301, 305 to 305 above, and also of the sum of the series of natural numbers up to n. is added (thereunto). The (resulting) quantity when multiplied by the square of the number of terms as increased by the number of terms gives rise to the (required) collective sum.
Examples in illustration thereof.
310. What would be the (required) collective sum in relation to the (various) series represented by (each of) 49, 66, 13, 14, and 25?
The rule for arriving at the sum of a series of fractions in geometrical progression:-
311.[1] The number of terms (in the series) is caused to be marked (in a separate column) by zero and by one (respectively), corresponding to the even (value) which is halved, and to the uneven (value from which one is subtracted, till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (wherever zero happens to be the denoting item). The result (of this operation) is written down in two positions. (In one of them, what happens to be) the numerator in the result (thus obtained) is divided (by the result itself; then) one is subtracted (from it); the (resulting) quantity is multiplied by the first term (in the series) and (then) by (the quantity placed in) the other (of the two positions noted above). The product (so obtained), when divided by one as diminished by the common ratio, gives rise to the required sum of the series.
Examples in illustration thereof.
312-313. In relation to 5 cities, (the first term is) dīnāra and the common ratio is . (Find out the sum of tho dīnāras obtained in all of them.) The first term is , the common ratio is
311.^ In this rule, the numerator of the fractional common ratio is taken to be always 1. See stanza 94, Ch. II and the note thereunder. , and 7 is the number of terms. If you are acquainted with calculation, then tell me quickly what the sum of the series of fractions in geometrical progression here is.
The rule for arriving at the sum of a series in geometrical progression wherein the terms are either increased or decreased (in a specified manner by a given known quantity):-
314. The sum of the series in (pure) geometrical progression (with the given first term, given common ratio, and the given number of terms, is written down in two positions) ; one (of these sums so written down) is divided by the (given) first term. From the (resulting) quotient, the (given) number of terms is subtracted. The (resulting) remainder is (then) multiplied by the (given) quantity which is to be added to or to be subtracted (from the terms in the proposed series ). The quantity (so arrived at) is (then) divided by the common ratio as diminished by one. (The sum of the series in pure geometrical progression written down in) the other (position) has to be diminished by the (last) resulting quotient quantity, if the given quantity is to be subtracted (from the terms in the series). If, however, it is to be added, (then the sum of the series in geometrical progression written down in the other position) has to be increased by the resulting quotient (already referred to. The result in either case gives the required sum of the specified series)
Examples in illustration thereof.
315. The common ratio is 5, the first term is 2, and the quantity to be added (to the various terms) is 3, and the number of terms is 4. O you who know the secret of calculation, think out and tell me quickly the sum of the series in geometrical progression, wherein the terms are increased (by the specified quantity in the specified manner).
34. Algebraically, is the sum of the series of the following form: and so on, 316. The first term is 3, the common ratio is 8, the quantity to be subtracted (from the terms) is 2, and the number of terms is 10. O you mathematician, think out and tell me quickly what happens to be here the sum of the series in geometrical progression, whereof the terms are diminished (by the specified quantity in the specified manner).
The rule for arriving at the first term, the common difference and the number of terms, from the mixed sum of the first term, the common difference, the number of terms, and the sum (of a given series in arithmetical progression):-
317.[1] (An optionally chosen number representing) tho number of terms (in the series) is subtracted from the (given) mixed sum. (Then) the sum of the natural numbers (beginning with one and going up to) one less than this optionally chosen number is combined with one. By means of this as the divisor (the remainder from the mixed sum as above obtained is divided). The quotient here happens to be the (required) common difference; and the remainder (in this operation of division ) when divided by the (above optionally chosen) number of terms as increased by one gives rise to the (required) first term.
An example in illustration thereof.
318. It is seen here that the sum (of a series in arithmetical progression) as combined with the first term, the common difference, and the number of terms (therein) is 50. O you who know calculation, give out quickly the first term, the common difference, the number of terms, and the sum of the series (in this case).
The rule for arriving at the common limit of time when one, who is moving (with successive velocities representable) as the terms in an arithmetical progression, and, another moving with steady unchanging velocity, may meet together again (after starting at the same instant of time):-
317.^ See stanzas 80-82 in Ch. II and the note relating to them. 319.[1] The unchanging velocity is diminished by the first term (of the velocities in series in arithmetical progression), and is (then) divided by the half of the common difference. On adding one (to the resulting quantity), the (required) time (of meeting) is arrive at. (Where two persons travel in opposite directions, each with a definite velocity), twice (the average distance to be covered by either of them) is the (whole) way (to be travelled). This when divided by the sum of their velocities gives rise to the time of (their) meeting.
An example in illustration thereof.
320. A certain person goes with a velocity of 3 in the beginning increased (regularly) by 8 as the (successive) common difference. The steady unchanging velocity (of another person) is 21. What may be the time of their meeting (again, if they start from the same place at the same time, and move in the same direction)?
An example in illustration of the latter half of the rule given in the steps above
321-321. One man travels at the rate of 6 yōjanas and another at the rate of 3 yōjanas. The (average) distance to be covered by either of them moving in opposite directions is 108 yōjanas. O arithmetician, tell me quickly what the time of their meeting together is.
The rule for arriving at the time and distance of meeting together, (when two persons start from the same place at the same time and travel) with (varying) velocities in arithmetical progression.
322.[2] The difference between the two first terms divided by the difference between the two common differences, when multiplied by two and increased by one, gives rise to the time of coming together on the way by the two persons travelling simultaneously (with two series of velocities varying in arithmetical progression).
319.^ Algebraically, , where v is the unchanging velocity, and t the time.
322.^ Algebraically, .
An example in illustration thereof.
323. A person travels with velocities beginning with 4, and Increasing (successively) by the common difference of 8. Again, a second person travels with velocities, beginning with 10, and increasing (excessively) by the common difference of 2. What is the time of their meeting?
The rule for arriving at the time of meeting of two persons (starting at the same time and travelling in tho same direction with varying velocities in arithmetical progression), the common difference (in the one case) being positive, and (in the other) negative
324. The difference between the two first terms is divided by half of the sum of the numbers representing the two (given) common differences, and (then) one is added (to the resulting quantity). This becomes the time of meeting on the way by the two persons (starting at the same time and travelling simultaneously (with velocities in arithmetical progression, the common difference in the one case being positive and in the other negative).
An example in illustration thereof.
325. The first man travels with velocities beginning with 5, and increased (successively) by 8 as the common difference. In the case of the second person, the commencing velocity is 45, and the common difference is minus 8. What is the time of meeting?
The rule for arriving at the time of meeting of two persons, (starting at different times and) travelling (respectively) with a quicker and a less quick velocity (in the same direction):-
326. He who travels less quickly and he who travels more quickly-both move in the same direction. What happens to be the distance to be overtaken here is divided by the difference between those (two) velocities. In the course of the number of days represented by the quotient (here), the more quickly moving person goes to the less quickly moving one.
324. Compare this with the rule given in 322; above
An example in illustration thereof.
327. A certain person travels at the rate of 9 yōjanas (a day); and 100 yōjanas have already been gone over by him. Now, a messenger sent after him goes at the rate of 18 yōjanas(a day). In how many days will this (messenger) meet him ?
The rule for working out the circumferential number of arrows in the quiver with the aid of the (given) uneven number of arrows (contained in the quiver ; and vice versa):-
328. The number of the circumferential arrows is increased by three and (then) halved. This is squared and (then) divided by three. On adding one (to the resulting quantity), the number of arrows (in the quiver) is obtained. When, however, the number of the circumferential arrows has to be arrived at, the reverse process is (to be adopted in relation to those operations).
Examples in illustration thereof:
329. The circumferential number of the arrows is 9. Their total number, however, is not known. (What is that ?). The total number of arrows (in the quiver) is 18. Tell me, O arithmetician, the number of the circumferential arrows also in this case.
The rule for arriving at the number of bricks to be found in structures made up of layers (of bricks one over another)
330. The square of the number of layers is diminished by one, divided by three, and (then), multiplied by the number of layers. On adding (to the quantity so obtained) the product, obtained by multiplying the optionally chosen number (representing the bricks in the topmost layer) by the sum of the (natural numbers beginning with one and going up to the given) number of layers, the required answer is obtained.
330. Algebraically, is the total number of bricks in the structure, where n is the number of layers, and a the optionally chosen number of bricks in the topmost layer. The number of bricks along the length or breadth of any layer is one less than the same in the immediately lower layer.
Examples in illustration thereof.
331. There is constructed an equilateral quadrilateral structure consisting of 5 layers. The topmost layer is made up of 1 brick. O you who know the calculation of mixed problems, tell me how many bricks there are (here in all).
332. There is a structure built up of successive layers of bricks, which is in the form of the nandyāvarta. There are 4 layers built symmetrically with 60 (as the numerical measure of the top‑bricks in single row). Tell me how many are all the bricks (here).
Rules regarding the six things to be known in the science of prosody:-
333-336. (The number of syllables in a given syllabic metre or chandas is caused to be markel in a separate column) by zero and
332. The nandyāvarta figure referred to in the stanza is 卐;
333-336. As each syllable found in a line forming a quarter of a stanza may be short or long, there arises a number of varieties corresponding to the different arrangements of long and short syllables. In arranging these varieties, a certain order is followed. The rules given here enable us to find out (1) the number of varieties possible in a metre consisting of a specified number of syllables, (2) the number of arrangement of the syllables in these varieties, (3) the arrangement of the syllables in a variety specified by its ordinal position (4) the or ordinal position of a specified arrangement of syllables, (5) the number of varieties containing a specified number of long or short syllables, and (6) the amount of vertical space required for exhibiting the varieties of a particular metre.
The rules will become clear from the following working of the problems given in stanza 337.:-
(1) There are 3 syllable in a metre; now we proceed thus:
3 - 1 1 2 2 0 1 - 1 1 0
Now, multiplying by 2 the figures in the right-hand chain, we obtain . By the process of multiplication and squaring, as explained in the note to stanza 94, Ch. II, we get 8; and this is the number of varieties.
(2) The manner of arrangement of the syllable each variety is arrived at thus:-
1st variety: 1. being odd, denotes a long syllable; so the first syllable is long. Add 1 to this 1, and divide the sum by 2; the quotient is old. and denotes another long syllable. Again, 1 is added to this quotient 1, and divided by 2; the result.
by one (respectively), corresponding to the even (value) which is halved, and the uneven (value from which one is subtracted, till by continuing these processes zero is ultimately reached. The number in the chain of figures so obtained are) all doubled, (and then in the process of continued multiplication from the bottom to the top of the chain, those figures which come to have a zero above them) are squared. The (resulting) product (of this continued multiplication) gives the number (of the varieties of stanzas possible in that syllabic metre or chandas).
The arrangement (of short and long syllables in all the varieties of stanzas so obtained) is shown to be arrived at thus:-
(The natural numbers commencing with one and ending with the measure of the maximum number of possible stanzas in the given metre being noted down), every odd number (therein) has one added to it, and is (then) halved. (Whenever this process is gone through), a long syllable is decidedly indicated. Where
1st variety: 1, again odd, denotes a third long syllable. Thus the first variety consists of three long syllables, and is indicated thus ʃ ʃ ʃ
2nd variety: 2, being even. indicates a short syllable; when this 2 is divided by 2, the quotient is 1, which being odd indicates a long syllable. Add 1 to this 1, and divide the sum by 2; the quotient being odd indicates a long syllable; thus we get | ʃ ʃ
Similarly the other six varieties are to be found out. (3) The fifth variety, for instance may be found out as above. (4) To find out, for instance, the ordinal position of the variety,| > | we proceed thus:-
| ? |
1 2 4Below these syllables, write down the terms of a series in geometrical progression, having 1 as the first term and 2 as the common ratio. Add the figures 4 and 1 under the the short syllables, and increase the sum by 1; we get 6: and we, therefore, say that this is the sixth variety in the tri-syllabic metre.
(5) Suppose the problem is: How many varieties contain 2 short syllables? Write down the natural numbers in the regular and in the inverse order thus: . Taking two terms from right to left, both from above and from below, we divide the product of the former by the product of the latter . And the quotient 3 is the answer required.
(6) It is prescribed that the symbols representing the long and short syllables of any variety of metre should occupy an aṅgula of vertical space, and that the intervening space between any two varieties should also be an aṅgula. The amount, therefore of vertical space required for the 8 varieties of this metre is aṅgula. the number is even, it is (immediately) halved and this indicates a short syllable. In this manner, the process (of halving with or without the addition of one as tho onse may be, noting down at the same time the corresponding long and short syllables as indicated), is to be regularly carried on (till the actual number of syllables in the metre is arrived at in each case).
(If the number representing in the natural order any given variety of a stanza), the arrangement of the syllables wherein has to be found out, (happens to be even, it) has to be halved, and indicated a short syllable. (If it happens to be however odd), one has to be added to it, and (then) it is to be halved: and this indicates a long syllable. Thus (the long and short syllables have to be put down over and over again (in their respective positions) till the maximum number of syllables in the stanza is arrived at. This gives the arrangement (of long and short syllables in the required variety of the stanza).
Where (a stanza of a particular variety is given, and) its ordinal position (among the varioties of stanzans possible in the metre) is to be found out, the terms (of a series in geometrical progression) commencing with one and having two as the as the conmmon ratio are written dowm, (the number of terms in the series being equal to the number of syllables in the given metre. Above those terms, the corresponding long or short syllables are noted down). Then the terms (immediately) below the position of short syllables are all added; the sum (so obtained) is increased by one. (This gives the required ordinal number.)
Natural numbers commencing with one, and going up to the number (of syllables in the given metre), are written down in the regular and in the inverse (order in two rows) one below the other When the numbers in the row are multiplied (1, 2, 3 or more at a time) from the right to the left, and the products (so obtained in relation to the upper row) are divided by the (corresponding) products (in relation to tho lower row), the quotient represents the result of the operation intended to arrive at (the number of varieties of stanzas in the given metre, with 1, 2, 3 or more) short or long syllables (in the verse) The possible number (of the varieties of stanzas in the given metre) is multiplied by two and (then} diminished by one. This result gives (the measure of what is called) adhvan, (wherein an interval equivalent to a stanza is conceived to exists between every two successive varieties in the metre).
Examples in illustration thereof.
337. In relation to the metre made up of 3 syllables, tell me quickly the six thingष्ठ to be known--viz., (1) the (maximum) number (of possible stanzas in the metre), (2) the manner of arrangement (of the syllables in those stanzas), (3) the arrangement of the syllables (in a given variety of the stanza, the ordinal position whereof among the possible varieties in the metre is known), (4) tho ordinal position (of a given stanza), (5) the number (of stanzas in the given metro containing any given number of short or long syllables, and (6) the (quantity known as) adhvan.
Thus ends the process of summation of series in the chapter on mixed problems.
Thus ends the fifth subject of treatment, known as Mixed Problems, in Sārasaṅgraha, which is a work on arithmetic by Mahāvērācārya.
CHAPTER VII.
CALCULATION RELATING TO THE MEASUREMENT
OF AREAS.1. For the accomplishment of the object held in view, I how again and again with true earnestness to the most excellent Siddhas who have realized the knowledge of all things.
Hereafter we shall expound the sixth variety of calculation forming the subject known by the name of the Measurement of Areas. And that is as follows:
2. (The measurement of) area has been talking to be of two kinds by Jina in accordance with (tho nature of the result, namely, that which is approximate) for practical purposes and that which is minutely accurate. Taking this into consideration, I shall clearly explain this subject.
3. (Mathematical) teachers, who have reached the other shore of the ocean of calculation, have given out well (the various kinds of) areas as consisting of those that are trilateral, quadrilateral and curvi-linear, being differentiated into their respective varieties.
4. A trilateral area is differentiated in three waye; a quadrilateral one in five ways; and a curvi-linear one in eightways. All the remaining (kinds of) areas are indeed variations of the varieties of these (different kinds of areas).
5. Learned men say that the trilateral area may be equilateral, isosceles or scalene, and that the quadrilateral area also may be
5 and 6. The various kinds of enclosed areas mentioned in these stanzas are illustrated below :
1 2
3
Samatribhuja = Equilateral
trilateral figureVisāmatribhuja = Isoceles
trilateral figureViṣamatribhuja = Scalene
trilateral figureequi-lateral, equi-dichastic, equi-bilateral, equi-trilateral and in equi-lateral.
4
5
Samacaturaśra = Equi-lateral quadrilateral.
Dvidvisamacaturaśra=Equi-dichastic quadrilateral.
6
7
Dvisamacaturaśra = Equi-bilateral quadrilateral.
Trisamacaturaśra = Equi-trilateral quadrilateral.
8
9
Viṣamacaturaśra = Inequi-lateral quadrilateral.
Samavṛtta = Circle
10
11
Ardhavṛitta = Semicircle.
Āyatavṛtta = Ellipse.
6. (The curvi-linear area may be) a circle, a semicircle, an ellipse, a conchiform area, a concave circular area, a convex circular area, an out-lying annulus or an in-reaching annulus.
12
13
14
Kambukāvṛtta = conchiform area.
Nimmavṛtta = concave circular area.
Unnaiavṛtta = convex circular area.
15
16
Bahiścakravālavṛtta = Out-lying annulus. Antaścakravālavṛita = In-reaching annulus
From a consideration of the rules given for the measurement of the dimensions and areas of quadrilateral figures, it has to be concluded that all the quadrilateral figures mentioned in this chapter are cyclic. Hence an equilateral quadrilateral is a square, an equidichastic quadrilateral is an oblong; and equiilateral and equi-trilateral quadrilaterals have their topside parallel to the baseCalculation relating to approximate measurement
(of areas).The rule for arriving at the (approximate) measure of the areas of trilateral and quadrilateral fields:-
7.[1] The product of the halves of the sums of the opposite sides becomes the (quantitative) measurement (of the area) of trilateral and quadrilateral figures. In the case of (a figure constituting a circular annulus like) the rim of a wheel, half of the sum of the (inner and outer) circumferences multiplied by (the measure of) the breadth (of the annulus gives the quantitative measure of the area thereof). Half of this result happens to be here the area of (a figure resembling) the crescent moon.
Examples in illustration thereof.
8. In the case of a trilateral figure, 8 daṇdas happen to be the measure of the side, the opposite side and the base; tell me quickly, after calculating, the practically approximate value (of the area) thereof.
9. In the case of a trilateral figure with two equal sides, the length (represented by the two sides) is 77 daṇdas and the breadth (measured by the base) is 22 daṇdas associated with 2 hastas. (Find out the area)
7. A trilateral figure is here conceived to be formed by making the topside, i.e., the side opposite to the base, of a quadrilateral so small as to be neglected. Then the two lateral sides of the trilateral figure become the opposite sides, the topside being taken to be nil in value. Hence it is that the rule speaks of opposite sides even in the case of a trilateral figure.
As half the sum of the two sides of a triangle is, in all cases, bigger than the altitude, the value of the area arrived at according to this rule cannot be accurate in any instance.
In regard to quadrilateral figures the value of the area arrived at according to this rule can be accurate in the case of a square and an oblong, but only approximate in other cases.
Nēmi is the area enclosed between the circumferences of two concentric circles; and the rule here stated for finding out the approximate measure of the area of a Nēmikṣētra happens to give the accurate measures thereof. In the case of a figure resembling the crescent moon, it is evident that the result arrived at according to the rule gives only an approximate measure of the area.
10. In the case of a scalene trilateral figure, one sida is 18 daṇdas, the opposite side is 15 daṇdas and the base is 14 daṇdas. So what is the quantitative measure (of the area) of this (figure) ?
11.[1] In the case of a figure resembling (the medial longitudinal section of) the tusk of an elephant, the length of the outer curve is seen to be 88 daṇdas; that of the inner curve is (seen to be) 72 daṇdas; the measure of (the thickness at) the root of the tusk is 80 daṇdas. (What is the measure of the area ?)
12. In the case of an equilateral quadrilateral figure, the sides and the opposite sides (whereof) are each 60 daṇdas in measure, you tell me quickly, O friend, the resulting (quantitative) measure (of the area thereof).
13. In the case of a longish quadrilateral figure here, the length is 61 daṇdas breadth is 32. Give out the practically approximate measure (of the area thereof).
14. In the case of a quadrilateral with two equal sides, the length (as measured along either of the equal sides) is 67 daṇdas, the breadth of this figure is 88 daṇdas (at the base) and 33 daṇdas (at the top. What is the measure of the area of the figure?)
15. In the case of a quadrilateral figure with three equal sides, (each of these) three sides measures 108 daṇdas (remain ing side here called) mukha, or top-side measure 8 daṇdas and 3 hastas. Accordingly, tell me, O mathematician (the measure of the area of this figure).
16. In the case of a quadrilateral the sides of which are all unequal, the side forming the base measures 33 daṇdas, the side forming the top is 82 daṇdas; one of the lateral sides is 50 daṇdas and the other is 60 daṇdas. What is (the area) of this (figure) ?
17. In an annulus, the inner circular boundary neasure 30 daṇdas; the outer circular boundary is seen to be 300. The breadth
11.^ The shape of the figure mentioned in this stanza seems to be what is given here in the margin: it is intended that this should be treated as a trilateral figure, and that the area thereof should bo found out in accordance with the rule given in relation to trilateral figures of the annulus is 45. What is the calculated measure of the area of (this) annulus?
18. In the case of a figure resembling the crescent moon, the breadth is seen to be 2 hastas, the outer curve 68 hastas and the inner curve 82 hastas. Say what the (resulting) area is.
The rule for arriving at the (practically approximate value of the) area of the circle :-
19.[1] The (measure of the) diameter multiplied by three is the measure of the circumference ; and the number representing the square of half the dianmeter, if multiplied by three, gives the (resulting) area in the case of a complete circle. Teachers say that, in the case of a semicircle, half (of these) give (respectively) the measure (of the circumference and of the area).
Examples an illustration thereof.
20. In the case of a circle, the diameter is 18. What is the circumference, and what the (resulting) area (thereof)? In the case of a semicircle, the diameter is 18: tell me quickly what the calculated measure is (of the area as well as of the circumference).
The rule for arriving at (the value of) the area of an elliptical figure:-
21.[2] The longer diameter, increased by half of the (shorter) diameter and multiplied by two, gives the measure of the circumference of the elliptical figure. One-fourth of the (shorter) diameter, multiplied by the circumference, gives rise to the (measure of the) area (thereof).
19.^ The approximate character of the measure of the circumference as well as of the area as given here is due to the value of a being taken as 3.
21.^ The formula given for the circumference of an ellipse is evidently an approximation of a different kind. The area of an ellipse is , where a and b are the semi-axes. If is taken to be equal to 3, then . But the formula given in the stanza makes the area equal to .
An example an illustration thereof.
22. In the case of an elliptical figure the (shorter) diameter is 12, and the longer diameter is 36. What is the circumference and what is the (resulting) area (thereof)?
The rule for arriving at the (resulting) area of a conshiform curvilinear figure:-
23. In the case of a conchiform curvilinear figure, the measure of the (greatest) breadth diminished by half the measure of the mouth and multiplied by three gives the measure of the perimeter. One third of the square of half (this) perimeter, increased by three-fourths of tho square of half the measure of the mouth, (gives the area)
An example in Illustration thereof.
24. In the case of a conchi-form figure the breadth is 18 hastas, and the measure of the month thereof is 4 (hastas). You tell me what the perimeter is and what the calculated area is.
The rule for arriving at the (resulting) area of the concave and convex circular surfaces:-
25. Understand that one-fourth of the circumference multiplied by the diameter gives rise to the calculated (resulting) area. Thence, in the case of concave and convex areas like that of a
23. If a is the diameter and m is the measure of the month, then is the measure of the circumference; and is the measure of the area. The exact shape of the figure is not clear from the description given; but from the values given for the circumference and the area, it may be conceived to consist of 2 unequal semicircles placed so that their diameters coincide in position as shown in figure 12, given in the footnote to stanza 6, in this chapter.
25. The area here specified seems to be that of the surface of the segment of a sphere; and the measure of the area is stated to be, when symbolically represented, equal to , where c is the circumference of the sectional circle, and d is the diameter thereof. But the area of the surface of a spherical segment of this kind is equal to , where r is the radius of the sectional circle and h is the height of the spherical segment.
sacrificial fire-pit and like that of (the back of) the tortoise, (the required result is to be arrived at).
An example in illustration thereof.
26. In the case of the area of a sacrificial fire-pit the measure of the diameter is 27, and the measure of the circumnference is seen to be 56. What is the calculated measure of the area of that same (pit) ?
An example about a convex circular surface resembling (the back) of a tortoise.
27. The diameter is 15, and the circumference is seen to be 36. In the case of this area resembling the (back of a) tortoise, what is the practically approximate measure as calculated ?
The rule for arriving at the practically approximate value of the area of an in-lying annular figure as well as of an out-reaching annular figure:-
28.[1] The (inner) diameter increased by the breadth (of the annular area) when multiplied by three and by the breadth (of the annular area) gives the calculated measure of the area of the out-reaching annular figure. (Similarly the measure of the calculated area) of the in-lying annular figure (is to be obtained) from the diameter as diminished by the breadth (of the annular area).
Examples in illustration thereof.
29. The diameter is 18 hastas, and the breadth of the out reaching annular area is 3 in this case: the diameter is 18 hastas and again the breadth of the in-lying annular area is 3 hastas. What may be (the area of the annular figure in each case) ?
28.^ The shape of the अन्तश्चक्रवालवृत्तक्षेत्र I as well as of the बहिश्चक्रवालवृत्तक्षेत्र is identical with the shape of the नेमिक्षेत्र mentioned in the note to stanza 7 in this chapter. Hence the rule given for arriving at the area of all these figures works out to be the same practically.
The rule for arriving separately at the numerical measures of the circumference, of the diameter, and of the area of a circular figure, from the combined sum obtained by adding together the approximate measure of its area, the measure of its circumference and the measure of its diameter:-
30. In relation to the combined sum (of the three quantities) as multiplied by 12, the quantity thrown in so as to be added is 64. of this (second) sum the square root diminished by the square root of the quantity thrown in gives rise to the measure of the circumference.
An example an illustration thereof.
31. The combined sum of the measures of the circumference, of the diameter and of the area (of a circle) is 1116. Tell me what the (measure of the) circumference is, what (that of) the calculated area and what (of) the diameter is.
The rule for arriving at the practically approximate value of surface-areas resembling (the longitudinal sections of) the yava grain, (of) the mardala, (of) the paṇava, and (of) the vajra, the
32. In the case of areas shaped in the form of the yava grain, of the muraja, of the page, and of the vajra, the
30. This rule will be clear from the following algebraical representation:- Let c be the circumference of the circle as is taken to be equal to 3, is the diameter and is the area of the circle. If m stands for the combined sum of the circumference, the diameter and the area of the circle, then the rule given in the stanza to the effect that may be easily arrived at from the quadratic equation containing the data in the problem:-
32. Muraja means the same thing as mardala and mṛdaṅga. The shape of the various figures mentioned in this stanza is as follows:
Yavākārakṣētra.
Murajākārak,sētra
Paṇavākārakṣētra.
Vajrākārakṣētra.(require dmeasurement of) area is that which results by multiplying half the sum of the end measure and the middle measure by the length.
Examples in illustration thereof.
33. In the case of an are resembling the configuration of a yuva grain, the length is 80 and the breadth in the middle is 40. Tell me, what may be the calculated measure of that area ?
34. Tell (me what may be the calculated measure of the area) in relation to a field which has the outline configuration of the mṛdaṅga, and of which the length is 80 daṇdas, the end measure is 20 and the middle measure is 40 daṇdas.
35. In the case of a field having the outline of the paṇava, the length is 77 daṇdas, the measure of each of the two ends is 8 daṇdas, and the measure in the middle is 4 daṇdas. (What is the measure of the area ?)
36. Similarly in the case of a field having the outline of the vajra, the length i8 96 daṇdas, in the middle there is the middle point ; and at the ends the measure is daṇdas. (What is the measure of the area ?)
The rule for arriving at the measure of areas such as the ubhaya-niṣēdha or de-deficient area :
37. On subtracting the product of the length into half the breadth from the product of the length into the breadth, you
The measures of the area arrived at according to the rule given in this stanza are approximately correct in the case of all the figures, as the rule is based on the assumption that each of the bounding curved lines may be taken to be equal to the sum of two straight lines formed by joining the end of the curves with the middle point thereof.
37. The figures mentioned in this stanza are those given below:-
These are looked upon as being derived from a quadrilateral figure which is divided into four triangles by means of its diagonals crossing each other. The declare the measure of the di-deficient area. That which is less (than the latter product here) by half of this (above-mentioned quantity to be subtracted) is the measure of the area of the uni-definient figure.
An example in illustration thereof.
38. The length is 36 and the breadth is only 18 daṇdas. What is the resulting measure of the area in the case of a di-deficient area, and what in the case of the uni-deficient area ?
The rule for arriving at the practically approximate measure of the area of fields resembling the outline of a multiplex vajra:-
39.[1] One-third of the square of half the perimeter, divided by the number of sides and (then) multiplied by the number of sides as diminished by one, gives indeed in the result the value of the area of all figures made up of sides. In the case of the area
di-deficient figure is that in which any two of the opposite triangles out of the four making up the quadrilateral are left out of consideration, the uni-deficient figure being that in which only one out of the four triangles is neglected.
39.^ The rule stated in this stanza gives the area of figures made up of any number of sides. If s is half the sum of the measures of the sides, and n the number of sides, the area is said to be equal to . This formula is found to give the approximate value of the area in the case of a triangle, a quadrilateral, a hexagon and a circle conceived as a figure of infinite number of sides. The other part of the rule deals with the interspace bounded by parts of circles in contact, and the value of the area arrived at according to the rule here given is also approximate. The figure below shows an interspace so bounded by four touching circles.
included between circles (in contact), one-fourth of the result thus arrived at gives the required measure).
Examples in instruction thereof.
40. In the case of a six-sided figure the measure of a side is 5, and in the case of another figure of 16 sides the measure of a side is 3. Give out (the measure of the area in each case).
41. In the case of a trilateral figure one of the sides is 5, the opposite (i.e., the other) side is 7, and the base is 6. In the case of another hexalateral figure the sides are in measure from 1 to 6 in order. (Find out the value of the area in each case).
42. (Give out) the value of the interspace included inside four (equal) circles (in contact) having a diameter which is 9 in measure; and (give out) the value of the area of the interspace included inside three circles having diameters measuring 6, 5 and 4 (respectively).
The rule for arriving at the practically approximate area of a field resembling a bow in outline:-
43.[1] In the case of a bow-shaped field the calculated measure (of the area) is obtained by adding together (the measure of) the arrow and (that of) the string and multiplying the sum by half (the measure of) the arrow. The square root of the square of the (measure of the) arrow as multiplied by 5 and (then) as combined with the square of the (measure of the) string gives the (measure of the bent) stick (of the bow).
43.^ The field resembling a bow in outline is in fact the segment of a circle, the bow forming the arc, the bow-string forming the chord, and the arrow measuring the greatest perpendicular distance between the arc and the chord. If a, c and p represent the length of these three lines, then, according to the rules given in stanzas 43 and 45-
For accurate value see stanzas in this chapter.
An example in illustration thereof.
44. A bow-shaped field is seen whereof the string-measure is 26, and the arrow-measure is 13. Tell me quickly, O mathematician, what the calculated measure of this (area) is, and what the measure of this (bent) stick (curve).
The rule for arriving at the arrow-measure as well as the string-measure (in relation to a bow-shaped field):-
45. The difference between the squares of the string and of the bent bow is divided by 5. The square root (of the resulting quotient) gives the intended measure of the arrow. The square of the arrow is multiplied by 5; and (this product) is subtracted from the square (of the arc ) of the bow. The square root (of the resulting quantity) gives the measure corresponding to the string.
Examples in illustration thereof.
46. In the case of this (already given bow-shaped) field the measure of the arrow is not known; and in the case of another (similar field) the measure of the string is not known. O you who know calculation, give out both these measures.
The rule for arriving at the practically approximate value of the area of the circle which is circumscribed about or inscribed within a four-sided figure :-
47.[1] Half of three times (the measure of the area of the inscribed quadrilateral figure) gives the measure of the area of the circle in the case in which it is circumscribed outside. In the case where it is inscribed within and the quadrilateral is the other way (i.e., escribed), half of the above measure (is the required quantity).
47.^ The formula here given may be seen to be accurate in the case of a square, but only approximate in the case of other quadrilateral, if 3 be taken to be the correct value of
An example in illustration thereof.
48. In relation to a quadrilateral figure, each of whose sides is 15 (in measure), tell me the practically approximate value of the inscribed and the escribed circles.
Thus ends the calculation of practically approximate value in relation to areas.
The Minutely Accurate calculation of the
Measure of Areas.Hereafter in the calculation regarding the measurement of areas we shall expound the subject of treatment known as minutely accurate calculation. And that is as follows:-
The rule for arriving at the measure of the perpendicular (from the vertex to the base of a given triangle) and (also) of the segments into which the base is thereby divided):-
49. The process of saṅkramaṇa carried out between the base and the difference between the squares of the sides as divided by the base gives rise to the values of the two segments (of the base) of the triangle. Learned teachers say that the square root of the difference between the square root of the difference between these (segments) and of the (corresponding adjacent) side gives rise to the measure of the perpendicular.
49. Algebraically represented
- . Here a,b,c represent the meaesures of the sides of a triangle the measures of the segments of the base whose total length is c; and p represents the length of the perpendicular.
The rule for arriving at the minutely accurate measurement of the area (of trilateral and quadrilateral figures):-
50.[1] Four quantities represented (respectively) by half the sum of the sides as diminished by (each of) the sides (taken in order) are multiplied together; and the square root (of the product so obtained) gives the minutely accurate measure (of the area of the figure). Or the measure of the areas may be arrived at by multiplying by the perpendicular (from the top to the base) half the sum of the top measure and the base measure. (The latter rule does) not (hold good) in the case of an inequi-lateral quadrilateral figure.
Examples in illustration thereof.
51. In the case of an equilateral triangle, 8 daṇdas and give the measure of the base as also of each of the two sides. You, who know calculation, tell me the accurate value of the area (thereof) and also of the perpendicular (to the base) as well as of the segments (of the base caused thereby).
52. In the case of an isosceles triangle (each of the two (equal) sides measures 13 daṇdas, and the base measures 10. (What is) the accurate measure of the area thereof, and of the perpendi.
50.^ Algebraically represented:-
- Area of the trilateral figure = ; where s is half the sum of the sides, a,b,c, the respective measures of the sides of the trilateral figure;
- or , where p is the perpendicular distance of the vertex from the base.
- Area of a quadrilateral figure where s is half the sum of the sides, and a,b,c,d the measures of the respective sides of the quadrilateral figure;
- or (except in the case of an iequilateral quadrilateral) where p is the measure of either of the perpendiculars drawn to the base from the extremities of the top side.
The formulas here given for trilateral figures are correct; but those given for quadrilateral figures hold good only in the case of cyclic quadrilaterals, as in these formulas sight is lost of the fact that for the same measure of the sides the value of the area as well as of the perpendicular may vary. cular (to the base) as also of the segments (of the baso caused thereby) ?
53. In the case of a scalene triangle one of the sides is 13 (in measure), the opposite side is 15, and the base is 14. What indeed is the calculated measure (of the area of this figure), and what of the perpendicular (to the base) and of the basal segments ?
Hereafter (we give) the rule for arriving at the value of the diagonal of the five varieties of quadrilateral figures.
54.[1] The two quantities obtained by multiplying the basal side by the (larger and the smaller of the right and the left sides are (respectively) combined with the two (other) quantities obtained by multiplying the top side by the smaller and the larger of the right and the left sides. The (resulting) two sums constitute the multiplier and the divisor as also the divisor and the multiplier in relation to the sum of the products of the opposite sides. The square roots (of the quantities so obtained) give the required measures of the diagonals.
Examples in illustration thereof.
55. In the case of an equilateral quadrilateral which has all around a side measure of 5, tell me quickly,O friend who know the secret of calculation, the value of the diagonal and also the accurate value of the area.
54.^ Algebraically represented bhe measure of the diagonal of a quadrilateral figure as given here is:–
These formulas also are correct only for cyclic quadrilaterals. Bhāskarācārya is aware of the futility of attempting to give the measure of the area of a quadrilateral without previously knowing the values of the perpendicular or of |he diagonals. Vide the following stanza from his Līlāvatī:-
- लम्बयोः कर्णयोर्वेकमनिर्दिश्यापरान् कथम्।
पृच्छत्यनियतत्वेऽपि नियतं चापि तत्फलम् ।।
स पृच्छकः पिशाचो वा वक्ता वा नितरां ततः ।
यो न वेत्ति चतुबाहुक्षेत्रस्यानियतां स्थितिम् ।
- लम्बयोः कर्णयोर्वेकमनिर्दिश्यापरान् कथम्।
56. In the case of a longish quadrilateral, the (horizontal) side is 12 in measure and the perpendicular side is 3 in measure. Tell me quickly what the measure of the diagonal is and what the accurate measure of the area.
57. The basal side of an equi-bilateral quadrilateral is 36. One of the sides is 61 and the other also is the same. The top side is 14. What is the diagonal and what the accurate measure of the area?
58. In the case of an equi-trilateral quadrilateral, the square of 13 (gives the measure of an equal side); the base, however, is 407 in measure. What is the value of the diagonal, of the basal segments, of the perpendicular and of the area ?
59. The (right and the left sides of an inequilateral quadrilateral are 13 x 15 and 13 x 20 (respectively in measure) : the top side is , and the side below is 300. What are all the values here beginning with that of the diagonal ?
Hereafter (are given) the rules for arriving at the minutely accurate values relating to curvilinear figures. Among them the rule for arriving at the minutely accurate values relating to a circular figure is as follows:-
60.[1] The diameter of the circular figure multiplied by the square root of 10 becomes the circumference (in measure). The circumference multiplied by one-fourth of the diameter gives the area. In the case of a semicircle this happens to be half (of what it is in the case of the circle).
Examples in illustration thereof.
61. In the case of one (circular) field the diameter of the circle is 18; in the case of another it is 60 ; in the case of yet another it is 22. What are the circumferences and the areas ?
60. ^ The value of a given in this stanza is , which is equal to 3.16....... Compare this with the more approximate value given by Āryabhaṭa. Bhāskarācārya also gives to it the same value, and represents it in reduced terms as 62. In the case of a semicircular field of a diameter measuring 12, and of (another) field having a diameter of 36 in measure what is the circumference and what the area ?
The rule for arriving at the minutely accurate values relating to an elliptical figure:-
63.[1] The square of the (shorter) diameter is multiplied 6 by and the square of twice the length (as measured by the longer diameter) is added square root sum ) the to this. (The of this gives measure of the circumference. This measure of the circumference multiplied by one-fourth of the (shorter) diameter gives the minutely accurate measure of the area of an elliptical figure.
An example in illustration thereof.
64. In the case of an elliptical figure, the length (as measured by the longer diameter) is 36, and the breadth (as measured by the shorter diameter) is 12. Tell me, after calculation. what the measure of the circumference is, and what the minutely accurate measure of the area.
The rule for arriving at the minutely accurate values in relation to a conchiform figure:-
65.[2] The (maximum measure of the) breadth (of the figure), diminished by half (the measure of the breadth) of the mouth, and (then) multiplied by the square root of 10, gives rise to the measure of the perimeter. The square of half the (maximum)
63.^ If a represents the measure of the longer diameter and b that of the shorter diameter of an ellipse, then, according to the rule given here the circumference is and the area is . It may be noted that this stanza, as found in the MSS., omits to mention that the square root of the quantity is to be taken for arriving at the value of the circumference. The formula for the area given here is only an approximation, and seems to be based on the analogy of the area of a circle as represented by where d is the diameter and is the circumference.
65.^ Algebraically, circumference
; where a is the measure of the breadth (of the figure) as diminished by half the (breadth of the) mouth, and the square of one-fourth of the (breadth of the mouth are added together; and the resulting sun is multiplied by the square root of 10. This gives rise to the minutely accurate measure of the area in the case of the conchiform figure.
An example in illustration thereof.
66. In the case of a conchiform curvilinear figure the (maximum breadth is 18 daṇdas, and the breadth of the mouth is 4 (daṇdas). What is the measure of the perimeter and what the minutely accurate measure of the area as calculated ?
The rule for arriving at the minutely accurate measures in relation to outreaching and inlying annular figures:-
67. The (inner) diameter, to which the breadth (of the annulus) is added, is multiplied by the square root of 10 and by the breadth (of the annulus). This gives rise to the value of the area of the out-reaching annulus. The (outer) diameter as diminished by the breadth (of the annulus) gives rise (on being treated in the same manner as above) to the value of the area of the inlying annular figure.
Examples an illustration thereof.
68. Eighteen daṇdas measure the (inner or the outer) diameter of the annulus (as the case may be) ; the breadth of the annulus is, however, 3 (daṇdas). You give out the minutely accurate value of the area of the outreaching as well as the inlying annular figure.
69. The (outer) diameter is 18 daṇdas, and the breadth of the inlying annulus is 4 daṇdas. You give out the minutely accurate value of the area of the inlying annular figure.
maximum breadth, and in the measure of the mouth of a conchiform figure. As observed in the note relating to stanza 23 of this chapter, the figure intended is obviously made up of two unequal semicircles The rule for arriving at the minutely accurate values relating to a figure resembling (the longitudinal section of) the yava grain, and also to a figure having the outline of a bow :-
70.[1] It should be known that the measure of the string (chord) multiplied by one-fourth of the measure of the arrow, and then multiplied by the square root of 10, gives rise to the (accurate) value of the area in the case of a figure having the outline of a bow as also in the case of a figure resembling the (longitudinal) section of a yava grain.
Examples in illustration thereof.
71. In the case of a figure resembling (the longitudinal section of the yava grain, the (maximum) length is 12 daṇdas. the two ends are needle points, and the breadth in the middle is 4 daṇdas. What is the area ?
72. In the case of a figure having the outline of a bow, the string is 24 in measure; and its arrow is taken to be 4 in measure. What may be the minutely accurate value of the area ?
The rule for arriving at the measure of the (bent) stick of the bow as well as of the arrow in the case of a figure having the outline of a bow :-
73.[2] The square of the arrow measure is multiplied by 6. To this is added the square of the string measure. The square
70.^ The figure resembling a bow is obviously the segment of a circle. The area of the segment as given here . This formula is not accurate. It seems to be based on the analogy of the rule for obtaining the area of a semi-circle, which area is evidently equal to the product of , the diameter and one-fourth of the radius, i.e.,
The figure resembling the longitudinal section of a good grain may be easily seen to be made up of two similar and equal segments of a circle applied to each other so as to have a common chord. It is evident that in this case the value of the arrow-line becomes doubled. Thus the same formula is made to hold good here also
73 & 74.^ Algebraically,
root of that (which happens to be the resulting sum here) gives rise to the measure of the (bent) bow-stick. In the case of finding out the measure of the string and the measure of the arrow, a course converse to this is adopted.
The rule relating to the process according to the converse (here mentioned):-
74. The measure of the arrow is taken to be the square root of one-sixth of the difference between the square of the string and the square of the (bent stick of the) bow. And the square root of the remainder, after subtracting six times the square of the arrow from the square of the (bent stick of the) bow, gives rise to the measure of the string.
An example in illustration thereof.
75. In the case of a figure having the outline of a bow, the string-measure is 12, and the arrow-measure is 6. The measure of the bent stick is not known. You (find it out), O friend. (In the case of the same figure) what will be the string-measure (when the other quantities are known), and what its arrow-measure (when similarly the other requisite quantities are known) ?
The rule for arriving at the minutely accurate result in relation to figures resembling a Mṛudaṅga, and having the outline of a Paṇava, and of a Vajra-
76.[1] To he resulting area, obtained by multiplying the (maximum) length with (the measure of the breadth of the mouth, the value of the areas of its associated bow-shaped figures is added. The resulting sum gives the value of the area of a figure resembling (the longitudinal section of) a Mṛdaṅga. In the case
In giving the rule for the measure of the arc in terms of the chord and the largest perpendicular distance between the arc and the chord, the arc forming a semicircle is taken as the basis, and the formula obtained for it is utilized for arriving at the value of the arc of any segment. The semicircular arc : based on this is the formula for any arc where p = the largest perpendicular distance between the arc and the chord, and c = the chord .
76.^ The rationale of the rule here given will be clear from the figures given in the note under stanza 32 above. of those two (other) figures which resemble (the longitudinal section of) the Paṇava, and (of) the Vajra, that (same resulting area, which is obtained by multiplying the maximum length with the measure of the breadth of the mouth)is diminished by the measure of the areas of the associated bow-shaped figures. (The remainder gives the required measure of the area concerned.)
Examples in illustration thereof.
77. In the case of a figure having the outline configuration of a Mṛdaṅga, the (maximum) length is 24; the breadth of (each of) the two mouths is 8; and the ( maximum) breadth in the middle is 16. What is the area ?
78. In the case of a figure having the outline of a Paṇava, the (maximum) length is 24; similarly the measure (of the breadth of either) of the two mouths is 8; and the central breadth is 4. What is the area ?
79. In the case of a figure having the outline of a Vajra, the (maximum) length is 24; the measure (of the breadth of either) of the two mouths is 8; and the centre is a point. Give out as before what the area is.
The rule for arriving at the minutely accurate value of the areas of figures resembling (the annulus making up) the rim of a wheel, (resembling) the crescent moon and the (longitudinal) section of the tusk of an elephant:-
80. In the case of (a circular annulus resembling ) the rim of a wheel, the sum of the measures of the inner and the outer curves is divided by 6, multiplied by the measure of the breadth
80. The rule here given for the area of an annulus, if expressed algebraically, comes to be , are the measures of the two circumferences, and p is the measure of the breadth of the annulus. On a comparison of this value of the area of the annulus with the approximate value of the same as given in stanza 7 above (vide note thereunder), it will be evident that the formula here does not give the accurate value, the value mentioned in the rule in stanza 7 being itself the accurate value. The mistake seems to have arisen from a wrong notion that in the determination of the value of this area, is involved even otherwise than in the values of . of the annulus, and again multiplied by the square root of 10. (The result gives the value of the required area.) Half of this is the (required) value of the area in the case of figures resembling the crescent moon or (the longitudinal section of) the tusk of an elephant.
Examples in illustration thereof.
81. In the case of a field resembling (the circular annulus forming) the rim of a wheel, the outer curve is 14 in measure and the inner 8; and the (breadth in the) middle is 4. (What is the area?) What is it in the case of a figure resembling the crescent moon, and in the case of a figure resembling (the longitudinal section of the tusk of an elephant (the measures requisite for calculation being the same as above) ?
The rule for arriving at the minutely accurate value of the area , of a figure forming the interspace included inside four (equal) circles (touching each other):-
82. [1] If the minutely accurate measure of the area of any one circle is subtracted from the quantity which forms the square of the diameter (of the circle), there results the value of the area of the interspace included within four equal circles (touching each other):-
An example in illustration thereof.
83. What is the minutely accurate measure of the area of the interspace included within four mutually touching (equal) circles whose diameter is 4 (in value)?
82.^ The rationale of the rule will be clear from the figure below:-
The rule for arriving at the minutely accurate value of the figure formed in the interspace caused by three (equal) circular figures touching each other:-
84.[1] The minutely accurate measure of the area of an equilateral triangle, each side of which is equal in measure to the diameter (of the circles) is diminished by half the area of any of the (three equal) circles. The remainder happens to be the measure of the interspace area caused by three (mutually touching equal circles).
An example in illustration thereof.
85. What is the minutely accurate calculated value of a figure forming the interspace enclosed by three mutually touching (equal) circles the diameter (of each) of which is 4 in measure ?
The rule for arriving at the minutely accurate values of the diagonal, the perpendicular and the area in the case of a (regular) six-sided figure:-
86.[2] In the case of a (regular) six-sided figure, the measure of the side, the square of the side, the square of the square of the side multiplied respectively by 2, 3 and 3 give rise, in that same order, to the values of the diagonal, of the square of the perpendicular, and of the square of the measure of the area.
84. ^ Similarly the figure here elucidates at once the reason of the rule:-
86. ^ The rule seems to contemplate a regular hexagon. The formula given for the value of the area of the hexagon is , where a is the length of a side. The correct formula, however is .
An example in illustration thereof.
87. In the case of a (regular) six-sided figure each side is 2 daṇdas in measure. In relation to it, what are the squares of the measures of the diagonal, of the perpendicular and of the minutely accurate area of the figure ?
The rule for arriving at the numerical measure of the sum of a number of square root quantities as well as of the remainder left after subtracting a number of square root quantities one from another in the natural order :-
88. (The square root quantities are all) divided by (such) a (common) factor (as will give rise to quotients which are square quantities). The square roots (of the square quantities so obtained) are added together, or they are subtracted (one from another in the natural order). The sun and remainder (so obtained) are (both) squared and (then) multiplied (separately) by the divisor factor (originally used). The square roots (of these resulting products) give rise to the sum and the (ultimate) difference of the quantities (given in the problem). Know this to be the process of calculation in regard to (all kinds of) square root quantities.
An example in illustration thereof.
89. O my friend who know the result of calculations, tell me the sum of the square roots of the quantities consisting of 16, 36 and 100; and then (tell me) also the (ultimate) remainder in relation to the square roots (of the same quantities). Thus ends the minutely accurate calculation (of the measure of areas).
88. The word karaṇi occurring here denotes any quantity the square root of which is to be found out, the root itself being rational or irrational as the case may be. The rule will be clear from the following working of the problem given in stanza 89:-
- To find the value of
Subject of treatment known as the Janya operation.
Hereafter we shall give out the janya operation in calculations relating to measurement of areas. The rule for arriving at longish quadrilateral figure with optionally chosen numbers as bījas:-
90.[1] In the case of the optionally derived longish quadrilateral figure the difference between the squares (of the bīja numbers) constitutes the measure of the perpendicular-side, the product (of the bīja numbers) multiplied by two becomes the (other) side, and the sum of the squares (of the bīja numbers) becomes the hypotenuse.
Examples in illustration thereof.
91. In relation to the geometrical figure to be derived optionally, 1 and 2 are the bījas to be noted down. Tell (me) quickly after calculation the measurements of the perpendicular-side, the other side and the hypotenuse.
92. Having noted down, O friend, 2 and 3 as the bījas in relation to a figure to be optionally derived, give out quickly, after calculating, the measurements of the perpendicular-side, the other side and the hypotenuse.
Again another rule for constructing a longish quadrilateral figure with the aid of numbers denoted by the name of bījas:-
93. The product of the sum and the difference of the bījas forms the measure of the perpendicular-side. The saṅkramaṇa of
90.^ Janya literally means “arising from” or “apt to be derived” ; hence it refers here to trilateral and quadrilateral figures that may be derived out of certain given data. The operation known as janya relates to the finding out of the length of the sides of trilateral and quadrilateral figures to be so derived.
Bīja, as given here, generally happens to be a positive integer. Two such are invariably given for the derivation of trilateral and quadrilateral figures dependent on them.
The rationale of the rule will be clear from the following algebraical representation :-
If a and b are the bīja numbers, then is the measure of the perpendicular, 2 ab that of the other side, and that of the hypotenuse, of an oblong. From this it is evident that the bījas are numbers with the aid of the product and the squares whereof, as forming the measures of the sides, a right-angled triangle may be constructed. the squares of that (sum and the difference of the bījas) gives rise (respectively) to the measures of the (other) side and of the hypotenuse. This also is a process in the operation of (constructing a geometrical) figure to be derived (from given bījas).
An example in illustration thereof.
94. O friend, who know the secret of calculation, construct a derived figure with the aid of 3 and 5 as bījas, and then think out and mention quickly the numbers measuring the perpendicular-side, the other side and the hypotenuse (thereof).
The rule for arriving at the bīja numbers relating to a given figure capable of being derived (from bījas).
95.[2] The operation of saṅkramaṇa between (an optionally chosen exact) divisor of the measure of the perpendicular-side and the resulting quotient gives rise to the (required) bījas. (An optionally chosen exact) divisor of half the measure of the (other) side and the resulting quotient (also) form the bījas (required). Those bījas are, (respectively), the square roots of half the sum and of half the difference of the measure of the hypotenuse and the square of a (suitably) chosen optional number.
An example in illustration thereof.
96. In relation to a certain geometrical figure, the perpendicular is 16: what are the bījas? Or the other side is 30: what are the bījas? The hypotenuse is 34: what are they(the bījas) ?
The rule for arriving at the numerical measures of the other side and of the hypotenuse, when the numerical measure of the perpendicular-side is known; for arriving at the numerical measures of the perpendicular-side and of the hypotenuse, when the numerical measure of the other side is known; and for arriving
93.^ In the rule given here, are represented as .
95.^ The processes mentioned in this rule may be seen to be converse to the operations mentioned in stanza 90.
at the numerical measure of the perpendicular-side and of the other side, when the numerical measure of the hypotenuse is known:-
97.[1] The operation of saṅkramaṇa, conducted between (an optionally chosen exact) divisor of the square of the measure of the perpendicular-side and the resulting quotient, gives rise to the measures of the hypotenuse and of the other side (respectively). Similarly (the same operation of saṅkramaṇa) in relation to the square of the measure of the other side (gives rise to the measures of the perpendicular-side and of the hypotenuse). Or, the square root of the difference between the squares of the hypotenuse and of a (suitably chosen) optional number forms, along with that chosen number, the perpendicular-side and the other side respectively.
An example in illustration thereof.
98. In the case of a certain (geometrical) figure, the perpendicular-side is 11 in measure; in the case of another figure, the (other) side is 60; and in the case of (still) another figure the hypotenuse is 61. Tell me in these cases the measures of the unmentioned elements.
The rule regarding the manner of arriving at a quadrilateral figure having two equal sides (with the aid of the given bījas):-
99.[2] The perpendicular-side of the primary figure derived (with the aid of the given bījas), on being added to the perpendicular-side (in another figure) derived with the aid of the two optionally chosen) factors of half the base of (this original) derived
97.^ This rule depends on the following identities :-
99.^ The problem solved in the role stated in this stanza is to construct with the aid of two given bījas a quadrilateral having two equal sides. The lengths of the sides, of the diagonals, of the perpendicular from the end-points of the top-side to the base, and of the segments thereof caused by the perpendicular are all derived from two rectangles constructed with the aid of the given bījas. The first of these rectangles is formed according to the rule given in stanza 90 above. The second rectangle is formed according to the same rule from two optionally chosen factors of half the length of the base of the first rectangle, figure (taken as the bījas), gives rise to the measure of the base of the (required) quadrilateral with two equal sides. The difference (between the measures of these two perpendiculars) gives the top-measure (of the quadrilateral). The smaller of the diagonals (relating to the two derived figures already mentioned) gives the measure of (either of the two equal) sides. The smaller of the (two) perpendicular-sides (in relation to the two derived figures under reference) gives the measure of the (smaller) segment (of the base formed by the perpendicular dropped thereunto from either of the end-points of the top-side). The larger of the (two) diagonals (in relation to the two derived figures of reference) gives the measure of the (required) diagonal. The area of the larger (of two derived figures of reference) is the area of the (required)
taken as bījas. Hence the first rectangle is called the primary figure in the translation to distinguish it from the second rectangle.
The rationale of the rule will be clear from the following diagrams illustrating the problem given for solution in stanza 100. Here 5 and 6 are the bījas given; and the first rectangle or the primary figure derived from the bījas is ABCD :-
Half the length of the base in this figure,is 30; and two factors of this, namely, 3 and 10 may be chosen. The rectangle constructed with the aid of these numbers as bījas is EFGH:-
To construct the required quadrilateral with two equal sides, one of the two triangles into which the first rectangle is divided by its diagonal is applied to the second rectangle on one side, and a portion equal to the same triangle is removed from the same second rectangle on the other side, as shown in the figure H A'FC'. figure; and the measure of the base (of either of the derived figures of reference) happens to be the measure of the perpendicular (dropped to the base from either of the old-points of the topside in the required figure).
An example in Illustration thereof.
100. In relation to a quadrilateral with two equal sides constructed with the aid of 5 and 6 as bījas give out the measures of the top side, of the base, of (either of the two equal) sides, of the perpendicular (from the top to the base), of the diagonal, of the (lesser) segment (of the base), and of the area.
The rule for arriving at the measures of the top-side, of the base, of (any one of ) the (equal) sides, of the perpendicular (from the top to the base), of the diagonal, of the (lesser) segment (of the base) and of the area, in relation to a quadrilateral having three equal sides (with the aid of given bījas) :
The process will be clear from a comparison of the diagrams:
Area of the required quadrilateral, HA'FC' = area of the second rectangle EFGH.
Base A'F = perpendicular-side of the first rectangle plus perpendicular-side of the second rectangle. i.e., AB + EF.
Top side HC = perpendicular-side of the second rectangle minus perpendicular-side of the first rectangle i.e., GH - CD
Diagonal HF = diagonal of the second rectangle.
Smaller segment of the base, i.e., A'E= perpendicular-side of the first rectangle, i.e.,AB.
Perpendicular HE = base of the first or of the second rectangle, i.e.,BC or FG.
Each of the lateral equal sides A'H and FC' = diagonal of the first rectangle i.e., AC.
101. The difference between the (given) bījas is multiplied by the square root of the base (of the quadrilateral immediately derived with the aid of those bījas). The area of (this immediately) derived (primary) quadrilateral is divided (by the product so obtained). Then, with the aid of the resulting quotient and the divisor (in the operation utilized as bījas, a second derived quadrilateral of reference is constructed. A third quadrilateral of
101. If a and b represent the given bījas, the measures of the sides of the immediately derived quadrilateral are :-
- Perpendicular-side
- Base
- Diagonal
- Area
As in the case of the construction of the quadrilateral with two equal sides (vide stanza 99 ante), this rule proceeds to construct the required quadrilateral with three equal sides with the aid of two derived rectangles. The bījas in relation to the first of these rectangles are :-
Applying the rule given in stanza 90 above, we have for the first rectangle :
- Perpendicular-side
- Base
- Diagonal
The bījas in the case of the second rectangle are:
The various elements of this rectangle are :
- Perpendicular-side
- Base
- Diagonal
With the help of these two rectangles, the measures of the sides, diagonals, etc., of the required quadrilateral are ascertained as in the rule given in stanza 99 above. They are :
- Base =sum of the perpendicular-sides
- Top-side = greater perpendicular-side minus smaller perpendicular-side
- Either of the lateral sides = smaller diagonal
- Lesser segment of the base = smaller perpendicular-side
- Perpendicular = base of either rectangle
- Diagonal = the greater of the two diagonals
- Area = area of the larger rectangle
It may be noted here that the measure of either of the two lateral sides is equal to the measure of the top-side. Thus is obtained the required quadrilateral with three equal sides.
reference is further constructed) with the aid of the measurements of the base and the perpendicular-side (of the immediately derived quadrilateral, above referred to, used as bījas. Then, with the aid of these two last derived secondary quadrilaterals, all the required) quantities appertaining to the quadrilateral with three equal sides are (to be obtained) as in the case of the quadrilateral with two equal sides.
An example in illustration thereof.
102. In relation to a quadrilateral with three equal sides and having 2 and 3 as its bījas, give out the measures of the top-side, of the base, of (any one of) the (equal) sides, of the perpendicular (from the top to the base), of the diagonal, of the (lesser) segment (of the base) and of the area.
The rule for arriving at the measures of the top-side, of the base, of the (lateral) sides, of the perpendiculars (from the ends of the top-side to the base), of the diagonals, of the segments (of the base) and of the area, in relation to a quadrilateral the sides of which are (all) unequal :-
103. [1] With the longer and the shorter diagonals (of the two derived rectangular quadrilateral figures related to the two sets
103.^ The rule will be clear from the following algebraical representation. Let a, b, and c, d, be two sets of given bījas. Then the various required elements are as follow:-
- Lateral sides
- Base
- Top-side
- Diagonals
- Perpendiculars
- Segments
of given bījas), the base and the perpendicular-side (of the smaller and the larger derived figures of reference) are respectively multiplied. The products (so obtained) are (separately) multiplied (again) by the shorter diagonal. The resulting products give the measures of the two (unequal) sides, of the base and of the top-side (in relation to the required quadrilateral). The perpendicular-sides (of the derived figures of reference) are multiplied by each other's bases; and the two products (so obtained) are added together. Then to the product of the (two) perpendicular-sides (relating to the two figures of reference), the product of the bases (of those same figures of reference) is added. The (two) sums (so obtained), when multiplied by the shorter of the two) diagonals (of the two figures of reference), give rise to the measures of the (required) diagonals. (Those same) bums, when multiplied by the base and the perpendicular-side (respectively) of the smaller figure (of reference), give rise to the measures of the perpendiculars (dropped from the ends of the diagonals); and when multiplied (respectively) by the perpendicular-side and the base (of the same figure of reference), give rise to the measures of the segments of the base (caused by the perpendiculars). The measures of these segments, when subtracted from the measure of the base, give the values of the (other) segments (thereof). Half of the product of the diagonals (of the required figure arrived at as above) gives the measure of the area (of the required figure).
An example in illustration thereof.
104. After forming two derived figures (of reference) with 1 and 2, and 2 and 3 as the requisite bījas give out, in relation to a quadrilateral figure the sides whereof are all unequal, the values of the top-side, of the base, of the (lateral) sides, of the perpendiculars, of the diagonals, of the segments (of the base), and of the area.
Again another rule for arriving at (the measures of the sides, etc., in relation to) a quadrilateral, the sides of which are all unequal :105–107.[1] The square of the diagonal of the smaller (of the two derived oblongs of reference), as multiplied (separately) by the base and also by the perpendicular-side of the larger (oblong of reference), gives rise to the measures (respectively) of the base and of the top-side (of the required quadrilateral having unequal sides). The base and the perpendicular-side of the smaller (oblong of reference, each) multiplied successively by the two diagonals (one of each of the oblongs of reference), give rise to the measures respectively) of the two (lateral) sides (of the required quadrilateral). The difference between the base and the perpendicular-side of the larger (oblong of reference) is in two positions (separately) multiplied by the base and by the perpendicular-side of the smaller (oblong of reference). The two (resulting) products (of this operation) are added (separately) to the product obtained by multiplying the sum of the base and the perpendicular-side of the smaller (oblong of reference) with the perpendicular-side of the larger (oblong of reference). The two sums (so obtained), when multiplied by the diagonal of the smaller (oblong of reference), give rise to the values of the two diagonals (of the required quadrilateral). The diagonals (of the required quadrilateral) are (separately) divided by the diagonal of the smaller (oblong of
105-107.^ The same values as are mentioned in the footnote to stanza 103 above are given here for the measures of the sides, etc.; only they are stated in a slightly different way. Adopting the same symbols as in the note to stanza 103, we have:-
Diagonals =
and
Perpendiculars= ;
and
The above four expressions can be reduced to the form in which the measures of the diagonals and the perpendiculars are given in stanza No. 103. The measures of the segments of the base are here derived by extracting the square root of the difference between the squares of the side and of the perpendicular corresponding to the segment. reference). The quotients (so obtained) are multiplied respectively by the perpendicular-side and the base of the smaller (oblong of reference). The (resulting) products give rise to the measures of the perpendiculars (in relation to the required quadrilateral). To these (two perpendiculars), the above values of the two sides (other than the base and the top-side) are (separately) added, (the larger side being added to the larger perpendicular and the smaller side to the smaller perpendicular). The differences between these perpendiculars and sides are also obtained (in the same order). The sums (above noted) are multiplied (respectively) by (these) differences. The square roots (of the products so obtained) give rise to the values of the segments (of the base in relation to the required quadrilateral). Half of the product of the diagonals (of the required quadrilateral) gives the value of (its) area.
The rule for arriving at an isosceles triangle with the aid of a single derived oblong (of reference).
108.[1] The two diagonals (of the oblong of reference constructed with the aid of the given bījas) become the two (equal) sides of the (required) isosceles triangle. The base (of the oblong of reference), multiplied by two, becomes the base (of the required triangle). The perpendicular-side (of the oblong of reference) is the perpendicular (of the required triangle from the apex to the base thereof). The area (of the required triangle) is the area (of the oblong of reference).
108.^ The rationale of the rule may be made out thus:- Let ABCD be an oblong and let AD be produced to E so that AD=DE. Join EC. It will be seen that ACE is an isosceles triangle whose equal sides are equal to the diagonals of the oblong and whose area is equal to that of the oblong.
An example in illustration thereof.
109. O mathematician, calculate and tell me quickly the measures of the two (equal) sides, of the base and of the perpendicular in relation to an isosceles triangle derived with the aid of 3 and 5 as bījas.
The rule regarding the manner of constructing a trilateral figure of unequal sides :-
110.[1] Half of the base of the (oblong of reference) derived (with the aid of the given bījas) is divided by an optionally chosen factor. With the aid of the divisor and the quotient (in this operation as bījas), another (oblong of reference) is derived. The sum of the perpendicular-sides belonging to these two (oblongs of reference) gives the measure of the base of the (required) trilateral figure having unequal sides. The two diagonals (related to the two oblongs of reference) give the two sides (of the required triangle). The base (of either of the two oblongs of reference) gives the measure of the perpendicular (in the case of the required triangle).
An example in illustration thereof.
111. After constructing a second (derived oblong of reference) with the aid of half the base of the (original) figure (i.e. oblong of reference) derived with the aid of 2 and 3 as bījas, you tell (me) by means of this (operation) the values of the sides, of the base and of the perpendicular in a trilateral figure of unequal sides.
Thus ends the subject of treatment known as the Janya operation.
{{rule}
110. ^ The rule will be clear from the following construction:- Let ABCD and EFGH be the two derived oblongs, such that the base AD = the base EH. Produce BA to K so that AK=EF. It can be easily shown that DK = EG and that the triangle BDK has its base BK=BA + EF, called the perpendiculars of the oblongs, and has its sides equal to the diagonals of the same oblongs.
Subject of treatment known as Paiśācika or
devilishly difficult problems.Hereafter we shall expound the subject of treatment known as Paiśācika.
The rule for arriving, in relation to the equilateral quadrilateral or longish quadrilateral figures, at the numerical measure of the base and the perpendicular-side, when, out of the perpendicular-side,the base, the diagonal, the area and the perimeter, any two are optionally taken to be equal, or when the area of the figure happens to be the product obtained by multiplying respectively by optionally chosen multipliers any two desired quantities (out of the elements mentioned above): that is-(the rule for arriving at the numerical values of the base and the perpendicular-side in relation to an equilateral quadrilateral or a longish quadrilateral figure,) when the area of the figure is (numerically) equal to the measure of the perimeter (thereof); or, when the area of the figure is of numerically equal to the measure the base (thereof); or, when the area of the figure is numerically equal to the measure of the diagonal (thereof); or, when the area of the figure is numerically equal to half the measure of the perimeter; or, when the area of the figure is numerically equal to one-third of the base or, when the area of the figure is numerically equal to one-fourth of the measure of the diagonal; or, when the area of the figure is numerically equal to that doubled quantity which is obtained by doubling the quantity which is the result of adding together wide the diagonal, three times the base, four times the perpendicular side and the perimeter and so on :-
112.[1] The measure of the base (of an optionally chosen figure of the required bype), on being divided by the (resulting) optional factor in relation thereto, (by multiplying with which the area
112.^ The rule will be clear from the following working of the first example given in stanza 113:- Here the problem is to find out the measure of the side of an equilateral quadrilateral, the numerical value of the area where, of is equal to the numerical value of the perimeter. Taking an equilateral quadrilateral of any dimension, say, with 5 as the measure of its side, we have the perimeter equal to 20, and the area equal to 25. The factor with which of the said optionally chosen 'figure happens to be arrived at); or the base (of such an optionally chosen figure of the requisite type), on being multiplied by the factor with which the area (of the said figure) has to be multiplied (to give the required kind of result); gives rise to the measures of the bases of the (required) equilateral quadrilateral and other kinds of derived figures.
Examples in illustration thereof.
113. In the case of an equilateral quadrilateral figure, the (numerical measure of the) perimeter is equal to (that of) the area. What then is the numerical measure of (its) base ? In the case of another (similar) figure, the (numerical) measure of the area is equal to (that of) the base. Tell me in relation to that (figure) also (the numerical measure of the base).
114. In the case of an equilateral quadrilateral figure, the (numerical) measure of the diagonal is equal to (that of) the area. What may be the measure of (its) base ? And in the case of another (similar) figure, the (numerical) measure of the perimeter is twice that of the area. Tell me (what may be the measure of its base).
115. Here in the case of a longish quadrilateral figure, the (numerical) measure of the area is equal to that of the perimeter; and in the case of another (similar) figure, the (numerical) measure of the area is equal to that of the diagonal. What is the measure of the base (in each of these cases)?
116. In the case of a certain equilateral quadrilateral figure, the (numerical) measure of the base is three times that of the area. (In the case of) another equilateral quadrilateral figure, the (numerical) measure of the diagonal is four times that of the area. What is the measure of the base (in each of these cases) ?
the measure of the perimeter, viz. 20, has to be multiplied in order to make it equal to the measure of the area, viz., 25, is . If 5, the measure of a side of the optionally chosen quadrilateral is divided by this factor 4, the measure of the side of the required quadrilateral is arrived at.
The rule gives also in another manner what is practically the same process thus: The factor with which the measure of the area, viz. 25 has to be multiplied in order to make it equal to the measure of the perimeter, viz. 20, is . lf 5, the measure of a side of the optionally chosen figure is multiplied by this factor , the measure of the side of the required figure is arrived at. 117. In the case of a longish quadrilateral figure, (the numerical measures of) twice the diagonal, three times the base and four times the perpendicular-side being taken, the measure of the perimeter is added to them. Twice (this sum) is the (numerical) measure of the area. (Find out the measure of the base.)
118. In the case of a longish quadrilateral figure, the (numerical) measure of the perimeter is 1. Tell me quickly, after calculating, what the measure of its perpendicular side is, and what that of the base.
119. In the case of a longish quadrilateral figure, the (numerical measures of twice the diagonal, three times the base, and fours times the perpendicular, on being added to the (numerical) measure of the perimeter, become equal to 1. (Find out the measure of the base.)
Another rule regarding the process of arriving at the number the bījas in relation to the derived longish quadrilateral figure :-
120.[1] The operation to arrive at the generating (bījas) in relation to a longish quadrilateral figure consists in getting at the square roots of the two qnantities represented by (1) half of the diagonal as diminished by the perpendicular-side and (2) the difference between this quantity and the diagonal.
An example in illustration thereof.
121. In the case of a longish quadrilateral figure, the perpendicular-side is 55, the base is 48, and then the diagonal is 73, What are the bījas here ?
120.^ The rule in stanza 95 of this chapter relates to the method of arriving at the bījas from the base or the perpendicular or the diagonal of a longish quadrilateral. But the role in this stanza gives a method for finding out the bījas from the perpendicular and the diagonal of a longish quadrilateral. The process described is based on the following identities:-
,
where is the measure of the diagonal, and is the measure of the perpendicular-side of a longish quadrilateral, a and b being the required bījas. The rule for arriving at the (longish quadrilateral) figure associated with a diagonal having a numerical value optionally determined:-
122.[1] Each of the various figures that are derived with the aid of the, given (bījas) is written down; and by means (of the measure) of its diagonal the (measure of the) given diagonal is divided. The perpendicular-side, the base, and the diagonal (of this figure) as multiplied by the quotient (here) obtained, give rise to the perpendicular-side, the base and the diagonal (of the required figure).
An example in illustration thereof.
123-124.O mathematician, quickly bring out with the aid of the given (bījas) the (value of the) perpendicular-sides and the bases of the four longish quadrilateral figures that have respectively 1 and 2, 2 and 3, 4 and 7, and 1 and 8, for their bījas, and are also characterised by different bases. And, (in the problem) here, the diagonal is (in value) 65. Give out (the measures of) what may be the (required) geometrical figures (in that case).
The rule for arriving at the numerical values of the base and the perpendicular side of that derived longish quadrilateral figure, the numerical measures of the perimeter as also of the diagonal whereof are known:-
125.[2] Multiply the square of the diagonal by two; (from the resulting product), subtract the square of half the perimeter; (then) get at the square root (of the resulting difference). If (this square root be thereafter) utilized in the performance of the
122.^ The rule is based on the principle that the sides of a right angled triangle vary as the hypotenuse, although for the same measure of the hypotenuse there may be different sets of values for the sides.
125.^ If a and b represent the sides of a rectangle, then is the measure of the diagonal, and is the measure of the perimeter. It can be seen easily that
- ; and
- .
These two formulas represent algebraically the method described in the rule here. operation of saṅkramaṇa along with half the perimeter, the (required) base and also the perpendicular-side are arrived at.
An example in illustration thereof.
126. The perimeter in this case is 34; and the diagonal is seen to be 13. Give out, after calculating, the measures of the perpendicular-side and the base in relation to this derived figure.
The rule for arriving at the numerical values of the base and the perpendicular-side when the area of the figure and the value of the diagonal are known :-
127. Twice the measure of the area is subtracted from the square of the diagonal. It is also added to the square of the diagonal. The square roots (of the difference and of the sum so obtained) give rise to the measures of the (required) perpendicular-side and the base, if the larger (of the square roots) is made to undergo the process of saṅkramaṇa in relation to the smaller (square root).
An example in illustration thereof.
128 In the case of a longish quadrilateral figure, the measure of the area is 60, and the measure of its diagonal is 13. I wish to hear (from you) the measures of the perpendicular-side and the base.
The rule for arriving at the numerical values of the base and the perpendicular-side in relation to a longish quadrilateral figure, when the numerical value of the area of the figure and the numerical value of the perimeter (thereof) are known:-
129. From the quantity representing the square of half the perimeter, the measure of the area as multiplied by four is to be
127. Adopting the same symbols as in the note to stanza 126, we have the following formula to represent the rule here given :-
- , as the case may be.
129. Here we have , as the case may be.
subtracted. Then, on carrying out the process of saṅkramaṇa with the square root (of this resulting difference) in relation to half the measure of the perimeter, the values of the (required) base and the perpendicular-side are indeed obtained.
130. In a derived longish quadrilateral figure, the measure of the perimeter is 170; the measure of the given area is 1,500. Tell me the values of the perpendicular-side and the base (thereof).
The rule for arriving at the respective pairs of (required) longish quadrilateral figures, (1) when the numerical measures of the perimeter are equal, and the area of the first figure is double that of the second; or, (2) when the areas of both the figures are equal, and the numerical measure of the perimeter of the second figure is twice the numerical measure of that of the first figure; or, (3) (again) when, in relation to the two required figures, the numerical measure of the perimeter of the second figure is twice the numerical measure of the perimeter of the first figure, and the area of the first figure is twice the area of the second figure:
131-133. (The larger numbers in the give ratios of) the perimeters as also (of) the areas (relating to the two required longish quadrilateral figures,) are divided by the smaller (numbers) corresponding to them. (The resulting quotients) are multiplied (between themselves) and (then) squared. (This same quantity,)
131 to 133. If x and y represent the two adjacent sides of the first rectangle, and a and b the two adjacent sides of the second rectangle, the conditions mentioned in the three kinds of problems proposed to be solved by this rule may be represented thus:-
The solution given in the rule seems to be correct only for the particular cases given in the problems in stanzas 134 to 136 on being multiplied by the given optional multiplier, gives rise to the value of the perpendicular-side. And in the case in which the areas (of the two required figures) are (held to be) equal; (this measure of) the perpendicular-side as diminished by one becomes the measure of the base. But, in the other case (wherein the areas of the required figures are not held to be equal), the larger (ratio number) relating to the areas is multiplied by the given optional multiplier, and (the resulting product is) diminished by one. The measure of the perpendicular-side (arrived at as above) is diminished by the quantity (thus resulting) and is (then) multiplied by three: thus the measure of the base (is arrived at ). Then, in respect of arriving at the other (of the two required quadrilateral figures), its base and perpendicular are to be brought out with the aid of the (now knowable) measure of its area and perimeter in accordance with the rule already given (in stanza 129).Examples in illustration thereof.
134. There are two (quadrilateral) figures, each of which is characterised by unequal length and breadth; and the given multiplier is 2. The measure of the area of the first (figure) is twice (that of the second), and the two perimeters are equal. What are the perpendicular-sides and the bases here (in this problem) ?
135. There are two longish quadrilateral figures; and the (given) multiplier is also 2. (Their) areas are equal, (but) the perimeter of the second (figure) is twice that of the first. (Find out their perpendicular-sides and bases.)
136. There are two longish quadrilateral figures. The area of the first (figure) here is twice (that of the second figure). The perimeter of the second (figure) is twice (that of the first). Give out the values of their bases and their perpendicular-sides.
The rule for arriving at a pair of isosceles triangles, so that the two isosceles triangles are characterised either by the values of their perimeters and of their areas being equal to each other, or by the values of their perimeters and of their areas forming multiples of each other:187. The squares (of the ratio-values) of the perimeters (of he required isosceles triangles) are multiplied by (the ratio-values of) the areas (of those triangles) in alternation. (Of the two products so obtained), (the larger one is) divided by the smaller; and (the resulting quotient) is multiplied by six and (is also separately multiplied) by two. The smaller (of the two products so obtained) is diminished by one. The larger product and the diminished smaller product constitute the two bījas (in relation to the longish quardrilateral figure) from which one (of the required triangles) is to be obtained. The difference between these (two bījas above, noted) and twice the smaller one (of those bījas constitute the ins (in relation to the longish quadrilateral figure) from which the other (required triangle) is to be obtained. (From the two longish quadrilateral figures formed with the aid of their respective bījas)the sides and the other things (relating to the required triangles) are to be arrived at as (explained) before.
187. When a : b is the ratio of the perimeters of the two isosceles triangles, and c : d the ratio of their areas, then, according to the rule, and and and are the two sets of bījas, with the help of which the values of the various required elements of the two isosceles triangles may be arrived at. The measures of the sides and the altitudes, calculated from these bījas according to stanza 108 in this chapter, when multiplied respectively by a and b, (the quantities occurring in the ratio of the perimeters), give the required measures of the sides and the altitudes of the two isosceles triangles. They are as follow:-
- I
- Equal side
- Base
- Altitude
- II
- Equal side
- Base
- Altitude
Now it may be easily proved from these values that the ratio of the perimeters is a:b, and that of the areas is c:d, as taken for granted at the beginning.
Examples in illustration thereof.
138. There are two isosceles triangles. Their area , is the same. The perimeters are (also) equal in value. What are the values of their sides, and what of their bases ?
139. There are two isosceles triangles. The area of the first one is twice (that of the second). The perimeter of both (of them) is the same. What are the values of (their) sides, and what of (their) bases ?
140. There are two isosceles triangles. The perimeter of the second (triangle) is twice (that of the first). The areas of the two (triangles) are equal. What are the values of (their) sides, and what of (their) bases?
141. There are two isosceles triangles. The area of the first (triangle) is twice (that of the second); and the perimeter of the second (triangle) is twice (that of the first). What are the values of (their) sides, and what of (their) bases?
The rule for arriving at an equilateral quadrilateral figure, or for arriving at a regular circular figure, or for arriving at an equilateral triangular figure, or for arriving at a longish quadrilateral figure, with the aid of the numerical value of the proportionate part of a given suitable thing (from among these), when any optionally chosen number from among the (natural) numbers, starting with one, two, &c, and going beyond calculation, is made to give the numerical measure of that proportionate part of that given suitable thing:-
142. The (given measure of the ) area (of the proportionate part) is divided by the (appropriately) similarised measure of the part held (in the hand). The quotient (so obtained), if multiplied by four gives rise to the measure of the breadth of the circle and
142. In problems of the kind given under this rule, a circle, or a square, or an equilateral triangle, or an obalong is divided into a desired number of equal parts, each part being bounded on one side by a portion of the perimeter and bearing the same proportion to the total area of the figure as the portion of the perimeter bears to the perimeter as a whole. It will be seen that in the case of a circle each part is a sector, in the case of a square and an oblong it is a rectangle, and in the case of an equilateral triangle it is a triangle. The area of each part and the length of the original perimeter contained in each part are both of given (also) of the square. (That same) quotient, if multiplied by six, gives rise to the required measure of the base of the (equilateral) triangle as also of the longish quadrilateral figure. Half (of this) is the measure of the perpendicular-side (in the case of the longish quadrilateral figure).
An example in illustration thereof.
143-145. A king caused to be dropped an excellent carpet on the floor of (his) palace in the inner apartments of his zenana amidst the ladies of his harem. That (carpet) was (in shape) a regular circle. It was held (in hand) by those ladies. The fist fuls of, both their arms made each (of them) acquire 15 (daṇdas out of the total area of the carpet). How many are the ladies, and what is the diameter (of the circle) here ? What are the sides of the square (if that same carpet be square in shape) ? and what the
magnitude. The stanza states a rule for finding out the measure of the diameter of the circle, or of the sides of the square, or the equilateral triangle or the oblong. If m represents the area of each part and n the length of a part of the total perimeter, the formulas given in the rule are--
- diameter of the cirole, or side of the square;
- and side of the equilateral triangle or of the oblong;
- and half of the length of the perpendicular-side in the case of the oblong.
The rationale will be clear from the following equation, where x represents the number of parts into which each figure is divided, a is the length of the radius in the case of the circle, or the length of a side in the case of the other figures; and b is the vertioal side of the oblong:
- In the case of the Circle ;
- In the case of the Square ;
- In the case of the Oblong ; here b is taken to be equal to half of a.
It has to be noted that only the approximate value of the area of the equilateral triangle, as given in stanza 7 of this chapter, is adopted here. Otherwise the formula given in the rule will not hold good.
143-145. What is called fistful in this problem is equivalent to four aṅgulas in measure. sides of the equilateral triangle (if it be equilaterally triangular in shape)? Tell (me, O friend, the measures of the perpendicular side and the base, in (case the carpet happens to be) a longish quadrilateral figure (in shape).
The rule for arriving at an equilaterally quadrilateral figure or at a longish quadrilateral figure when the numerical value of the area of the figure is known:-
146. The square root of the accurate measure of the (given) area gives rise to the value of the side of the (required) equilateral quadrilateral figure. On dividing the (given) area with an optionally chosen quantity (other than the square root of the value of the given area, this) optionally chosen quantity and the resulting quotient constitute the values of the perpendicular-side and the base in relation to the (required) longish quadrilateral figure.
An example in illustration thereof.
147. What indeed is that equilateral quadrilateral figure, the area whereof is 64 ? The accurate value of the area of the longish (quadrilateral) figure is 60. What are the value of the perpendicular-side and the base here?
The rule for arriving at a quadrilateral figure with two equal sides having the given area of such a quadrilateral figure with two equal sides after getting at a derived longish quadrilateral figure with the aid of the given numerical bījas and also after utilizing a given number as the required multiplier, when the numerical value of the accurate measure of the area of the required quadrilateral figure with two equal sides is known:-
148. The square of the given (multiplier) is multiplied by the that (given) area. The (resulting) product is diminished by the value of the area (of the longish quadrilateral figure) derived (from the given bījas). The remainder, when divided by the base
148. The problem here is to construct a quadrilateral figure of given area and with two equal sides. For this purpose an optionally chosen number and a set of two bījas are given. The process described in the rule will become clear by applying it to the problem given in the next stanzna. The bījas mentioned therein are 2 and 3; and the given area is 7, the given optional number being 3. (of this derived longish quadrilateral figure), gives rise to the measure of the top-side. The value of the perpendicular-side (of the derived longish quadrilateral figure) on being multiplied by two and increased by the value of the top-side (already arrived at, gives rise to the value of the base. The value of the base (of the derived longish quadrilateral figure) is (the same as that
The first thing we have to do is to construct a rectangle with the aid of the given bījas in accordance with the rule laid down in stanza 90; in this chapter. That rectangle comes to have 5 for the measure of its smaller side, 12 for the measure of its larger side, and 13 for the measure of its diagonal; and its area is 60 in value. Now the area given in the problem is to be multiplied by the square of the given optional number in the problem, so that we obtain . From this 63, we have to subtract 60, which is the measure of the area of the rectangle constructed on the basis of the given bījas: and this gives 3 as the remainder. Then the thing to be done is to construct a rectangle, the area whereof is equal to this 3, and one of the sides is equal to the longer side of the rectangle derived from the same bījas. Since this longer side is equal to 12 in value, the smaller side of the required rectangle has to be in value as shown in the figure here. Then the two triangles, into which the rectangle derived from the bījas, may be split up by its diagonal are added one on each side to this last rectangle, so that the sides measuring 12 in the case of these triangles coincide with the sides of the rectangle having 12 as their measure. The figure here exhibits the operation.
Thus in the end we get the quadrilateral figure having two equal sides, each of which measures 13, the value of the other two sides being and respectively. From this the values of the sides of the quadrilateral required in the problem may be obtained by dividing by the given optional number namely 3, the values of its sides represented by and . of) the perpendicular dropped (from the end of the top-side); and the diagonals (of the derived longish quadrilateral figure) are (equal in value to the sides. These (elements of the quadrilateral figure with two equal sides arrived at in this manner) have to be divided by the given multiplier (noted above to arrive at the required quadrilateral figure with two equal sides)
An example an illustration thereof.
149. The accurate value of the (given) area is 7; the optional given multiplier is 3; and the bījas are seen to be 2 and 3. Give out the values of the two sides of a quadrilateral figure with two equal sides and of its top-side, base, and perpendicular.
The rule for arriving at a quadrilateral figure with three equal sides, having an accurately measured given area, (with the aid of a given multiplier)
150. The square of the value of the (given) area is divided by the cube of the given (multiplier). (Then) the given (multiplier) is added (to the resulting quotient. Half (of the sum so obtained) gives the measure (of one ) of the (equal) sides. The given
150. It is stated in the rule here that the given area when divided by the given optional number gives rise to the value of the perpendicular in relation to the required figure. As the area is equal to the product of the perpendicular and half the sum of the base and the top-side, the given optional number represents the measure of half the sum of the base and the top-side. If ABCD be a quadrilateral with three equal sides, and CE the perpendicular from C on AD, then AD is half the sum of AD and BC, and is equal to the be given optional number. It can easily shown that 2 AD.
Here the given area of the quadrilateral. This last formula happens to be what is given in the rule for finding out any of the three equal sides of the quadrilateral contemplated in the problem.
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- ↑ These four attributes of Jina Mahavira are said to be his faith, understanding, blissfulness and power.
- ↑ The syādvāda is a process of reasoning adopted by the Jainas in relation to the question of the reality or otherwise of the totality of the perceptible objects found in the phenomenal universe. The word is translatable as the may-be-argument: and this may-be-argument declares that the phenomenal universe (1) may be real, (2) may not be real, (3) may and may not be real, (4) may be indescribable, (5) may be real and indescribable, (6) may be unreal and indescribable, and (7) may be real and unreal and indescribable. The position presented by this argument is not, therefore, one of a single conclusion.
- ↑ The tripraśna is the name of a chapter in Sanskrit astronomical works and the fact that it deals with three questions is responsible for that name. The questions dealt with are Dik (direction), Dīśa ( position) and Kāla (time) as appertaining to the planets and other heavenly bodies.
- ↑ It can be easily seen here that a number when divided by zero does not really remain unchanged. Bhāskara calls the quotient of such zero-divisions khahara and rightly assigns to it, the value of infinity. Mahāvīrācārya obviously thinks that a division by zero is no division at all.
- ↑ 122. This rule will be clear from the working of example No. 123 as explained below:--
There are three sets of fractions given and splitting up the sum, 1, into three fractions according to rule No. 75, we get and . By dividing these fractions by the quantities obtained by simplifying the three given sets of fractions wherein 1 is assumed as the unknown quantity, we obtain and and which are the required quantities.