184 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. IV 



Fermat 132 noted that d = 7 for (5, 12, 13) and (8, 15, 17) ; from these we 

 get all by his 111 rule. 



Frenicle 89 examined the 16 triangles with hypotenuses < 100 and found 

 that d = 1, 7, 7 2 , 17, 23, 31, 41, each of the form Sn 1. The triangles 

 formed from n, m and m, 2m + n have the same difference of legs. 132a 



T. T. Wilkinson 133 would start with a solution of a 2 + b 2 = c 2 and form 

 a = a + c, j3 = b -}- c, 7 = a + 6 -f c; repeat the process; we obtain 



a' = a + 7 = 2a + b + 2c, b' = + 7 = a + 26 + 2c, 

 c' = a + -f y = 2a + 26 + 3c, 



which are sides of a new triangle with a! b' = a b. H. S. Monck 

 (pp. 20-21, 76) failed in his attempt to prove that if we start with (3n, 4n, 

 5ri) and apply the process repeatedly we obtain all triangles with the same 

 difference of legs. J. W. L. Glaisher (p. 54) noted that the proof is 

 inadequate. Proof was given by S. Tebay (p. 99) and P. Mansion. 134 



T. Pepin 135 considered the problem of Fermat (Oeuvres, II, 231) to find 

 the number of right triangles the sum of whose legs is a given number A. 

 To the resulting condition x 2 2y 2 = A we may apply the theory of 

 quadratic forms and show that, if A = a a - c y , where a, , c are primes 

 SI 1, the total number of primitive triangles whose sum of legs is A is 

 i{(2a+l)...(2 T +l) -1}. 



J. H. Drummond and M. A. Gruber 136 found solutions when d is given. 



Several 137 treated the case d = 7. 



E. Bahier, 62 pp. 72-120, treated the problem at length by recurring series. 



TWO RIGHT TRIANGLES WITH EQUAL DIFFERENCES OF LEGS, AND LARGER 

 LEG OF ONE EQUAL TO THE HYPOTENUSE OF THE OTHER. 



Frenicle 138 proposed the problem to J. Wallis. Wallis (Aug., 1661) took 

 two overlapping triangles BAG and BCE with the respective hypotenuses 

 BC = 5 + x and BE. Take BA = 5 - x. Then BC 2 - BA 2 = 2Qx is a 

 square if 5x is; take 5 = ba 2 , x = be 2 . Then 



BC = ba 2 + be 2 , BA = ba 2 - be 2 , AC = 2bae. 

 On AB lay off AD = AC', on BC lay off Bd = ED. Since 



EC - CE = AB - AC = BD 



132 Oeuvres, II, 258-9; letter to St. Martin, May 31, 1643. 

 132 Oeuvres de Fermat, II, 235-7. 



133 Math. Quest. Educ. Times, 20, 1874, 20, 100. G. H. Hopkins, p. 22. On the proof sheets, 



E. B. Escott noted that " this process can be applied to other triangles than right-angled 

 triangles. Under this transformation, c 2 2ab as well as a b is invariant. Cf. 

 Dickson. 34 " 



134 Mathesis, (3), 6, 1906, 113. 



135 Mem. Pont. Accad. Nuovi Lincei, 8, 1892, 84-108; extract, Oeuvres de Fermat, 4, 1912, 



205-7; cf. 253. 



136 Amer. Math. Monthly, 9, 1902, 230, 292-3. 



137 Math. Quest. Educ. Times, (2), 7, 1905, 88-9. 



138 Cf. C. Henry, Bull. Bibl. Storia Sc. Mat. Fis., 12, 1879, 695; 13, 1880, 446; 17, 1884, 



351-2. 



