22 6. Most of the propositions have one or nmore alternative proofs given for them. The following propositions have alter native proofs: Book I. 5, 9, 11, 1218, 20, 21, 24, 25, 26, 32, 83, 84, 47. Of these 18th and 20th Propositions have two alternative proofs: and Prop. 47th is proved in seventeen ways by describ ing squares in different waysIn each of these seventeen alter native proofs there are three diagrams caused by the equality and the inequality of the sides that contain the right angle. Book I 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14 Of these Propositions 9 and 10 have two alternative proofs. Book III. 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 30, 31, 83, 86. Prop. 30 has bhree alternative proofs. Book IV. 1, 2, 3, 6, 10, 11, 12, 13, 14. Propositions 10th and 13th have two alternative proofs. Book V. 14, 1, 18, 19, 20, 22. Proposition 22nd has two alternative proofs. Book VIL 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 15, 18, 29, 81, B2. Propositions 9 and 10 have two alternative proofs. Book VII. 7, 13, 28. Book X. 1, 1264, 65. Book XII. 9, 18.
