19 line which remains, when from a right line is taken a right line incommensurable to it in power, the squares of the whole line and of the part taken together make a medial superficies, and twice he rectangle contained by them makes up a medial superficies incomnmensurable to the first medial superficies. The Tenth Book is the longest of the elements and contains in all 109 propositions. Thus in the first ten books is taught whatever is requisite and necessary to the knowledge of all superficial figures of any Bort whatever. The remaining books are concerned with solid figures (घनक्षेत्र), such as cubes. cones ( शकुछ ), Pyramids सूचीफलकघनक्षे ), cylinders (समतलमस्तकपरिधिरूपशकुघन समतलमस्तकशङ्कुक्षेत्र ), prisms (छेदितघनक्षेत्र ), spheres (गोलक्षेत्र ) and parallelepipeds ( समानान्तरधरातलघनक्षेत्र or घनहस्तक्षेत्र ). The eleventh book contains 41 propositions and propositions, 24th to bhe end, threat of the properties of parallelepipeds. or The Twelfth Book sets forth the properties of pyramids prisms, cones, cylinders and spheres, and compares pyramids to pyramids and prisms Likewise are compared cones, cy- linders and sphere8; and to prove the properties of these bodies it is first established that like polygons inscribed in circles and bhe circles themselves are to one another as the squares of their diameters. The enunciations of a few propositions will clearly shew bhe nature of the book.* Every pyramid having a briangle as its base may be divided into four parts, of which two are equal and similar pyramids and the other two are equal prisms greater than half the whole pyramid' Pyramids having tri angles as their bases and of the same altitude are to one an other as their bases.' + Every prism can be divided into three equal pyramids having briangles as their bases.’ ‘ If two py ramids having triangles as their bases be similar, they shall be in the treble ratio of that which their like sides have. A cone is a third part of a cylinder having the same base and altitude with it.' ‘ Like cylinders and cones are in bhe treble ratio of bhat in which the diameters of their circles (bases) are.' 'Spheres • Props. 3, 5, 6, 8, 9, 10, and 15, Book XII.
