VOL. XrV.] PHILOSOPHICAL TRANSACTIONS. 65 



are the 20th, 22d, and 31st, of the 3d book, which contain the properties of 

 circles. 



To demonstrate the 47th, I suppose the meaning of the terms made use of 

 to be known, and that two angles or superficies are equal when one being put 

 on the other, it neither exceeds nor is exceeded; this being allowed, I say the 

 sides about the right angle are either equal or unequal; if equal, as in fig. 8, 

 let all the squares be described ; the whole figure exceeds the square of the hy- 

 pothenuse BC by the two triangles M, V, and exceeds also the squares of the 

 other two sides A B, AC, by the two triangles ABC and S ; which excesses are 

 equal, for M is equal to ABC, the two sides about the right angle being two 

 sides of a square upon AB, by supposition equal to AC, and the third side equal 

 to BC, therefore the whole triangles are equal ; after the same manner S and V 

 are proved to be equal, therefore the square of CB is equal to the square of the 

 two other sides. Q. E. D. 



But if the sides be unequal, as in fig. 7, let the squares be described, and the 

 parallelogram LQ completed, the whole figure exceeds the square upon BC, by 

 three triangles X, R, Z, and exceeds also the squares LA, AD, by the triangle 

 ABC, and the parallelogram PQ, which excesses are equal; for Z is equal to 

 ABC, the side OC = BC, AC = CD, the angle D = A, andOCD = ABC, 

 which is manifest by taking the common angle ACO out of the two right angles 

 BCO, ACD; therefore, by superimposition, the whole triangles are equal. In 

 like manner X is proved equal to ABC, as also R; and the parallelogram PQ 

 double of the triangle ABC; thus the excesses being proved equal, the re- 

 mainders also will be equal, viz. the square of BC to the squares of AB, AC, 

 Q. E. D. Manifest corollaries from hence are the 35th and 36th of the 3d 

 book, also the 12th and 13th of the 2d. 



The first ten propositions of the 2d book are evidently demonstrated, only 

 by substituting species or letters instead of lines, and multiplying them accord- 

 ing to the tenor of the proposition ; thus to instance in one or two, call the 

 whole line a, and its parts b and c, therefore a-^ b -\- c, and consequently 

 aaz=bb -f cc -|- 2 be, which is the very sense of the 4th of the 2d book. Thus 

 also, let a line be cut into equal parts e, e, and let another line d be added 

 thereto, it is manifest that 4ee -\- Ade -f- idd = lee -{■ lee -f- idd -j- Ade, 

 which is the 10th proposition of the same book. 



Almost the whole doctrine of proportionals, viz. permutation, inversion, 

 conversion, composition, division of ratios, and proportion ex aequo, and con- 

 sequently the most useful propositions of the 5th book are clearly demonstrated 

 by one definition, and that is of similar or like parts, which are said to be such 

 as are after the same manner or equally contained in their wholes; thus the 



VOL. III. K 



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