192 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v 



whence x > 2. Taking x = 5, we have 



N* + 8 2 = (8 - 5A0 2 , N =, 



and the sides are 10, 9^, 8f , while the altitude is 8. 

 If BAG is oblique, N > 32/3. He took 



Bachet's second method of solution is of greater importance, since it 

 consists in juxtaposing two rational right triangles having a common side 

 AD. Take as the latter any number, as 12. Seek two squares such that 

 the sum of each and 12 2 is a square: 35 2 + 12 2 = 37 2 , 16 2 + 12 2 = 20 2 . 

 Hence by juxtaposition, we get a rational triangle with the sides 37, 20, 

 35 + 16 = 51, and altitude 12. Using the first relation with 9 2 + 12 2 = 15 2 

 or 5 2 + 12 2 = 13 2 , we get the rational triangle (37, 15, 35 + 9) or (37, 13, 

 35 + 5). 



F. Vieta 4 started with a given right triangle with legs B, D and hypote- 

 nuse Z, and formed (Diophantus 7 of Ch. IV) a second right triangle from 

 F + D and B, having therefore the altitude A = 2B(F + D), and multi- 

 plied its sides by D, and the sides of the given triangle by A. Juxtaposing 

 the resulting two triangles with the common altitude AD, we obtain a rational 

 triangle with the sides AZ, D(F + Z)) 2 + 2 D, D(F + -D) 2 - B 2 D + BA, 

 whose angle at the vertex is acute or obtuse according as F < Z or F > Z. 



Frans van Schooten 5 used the juxtaposition of right triangles. 



The Japanese manuscript of Matsunago, 6 first half of the eighteenth 

 century, started with any two right triangles with integral sides and multi- 

 plied the sides of each by the hypotenuse of the other and then juxtaposed 

 the triangles. The sides below 1000 of the resulting oblique triangles were 

 tabulated. Removing common factors, he obtained a table of primitive 

 triangles. From Kurushima (f 1757) he quoted the result that, if 



then 



are sides of a triangle with rational area. 



Nakane Genkei 6a in 1722 considered triangles whose sides are consecutive 

 integers such that the perpendicular upon the longest side from the opposite 

 vertex shall be rational. Denote the solutions (3, 4, 5), (13, 14, 15), (51, 

 52, 53) and (193, 194, 195) by (a h b,-, c y ), j = l, 2, 3, 4. Then 



and similarly for the 6's and c's. Whether or not he made the induction 

 complete does not, however, appear. 



4 Ad Logisticem Speciosam Notae Priores, Prop. 55, Opera Math., 1646. French transl. by 



F. Hitter, Bull. Bibl. Storia Sc. Mat., 1, 1868, 274-5. 



5 Exercitationum Math., Lugd. Batav., 1657, 426^132. 



6 Y. Mikami, Abh. Gesch. Math. Wiss., 30, 1912, 230-1. 



60 D. E. Smith and Y. Mikami, A History of Japanese Mathematics, Chicago, 1914, 168. 



