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



W. A. Whitworth, Proc. Lit. Phil. Soc. Liverpool, 29, 1875, 237, h < 2500. 

 Whitworth and G. H. Hopkins, Math. Quest. Educ. Times, 31, 1879, 

 67-70; D. S. Hart, Math. Visitor, 1, 1880, 99, forty triangles with 



h = 5-13-17-29. 



N. Fitz, Math. Magazine, 1, 1884, 163, primitive with h < 500. 



G. B. Airy, Nature, 33, 1886, 532, h < 100. 



A. Tiebe, Zeitschr. Math. Naturw. Unterricht, 18, 1887, 178, 420, 

 solved a 2 + x 2 = h 2 by setting h = x + y, whence 2x = a-/y y, so that y 

 is to be chosen as a divisor of a 2 (a > 2) such that the difference is even. 

 Whence he constructed a table with h < 100. Cf. T. Meyer, ibid., 36, 

 1905, 339. 



H. Lieber and F. von Liihmann, Trig. Aufgaben, ed. 3, Berlin, 1889, 

 287-9, gave the 131 primitive triangles with h < 999. 



P. G. Egidi, Atti Accad. Pont. Nuovi Lincei, 50, 1897, 126-7, h ^ 320. 



J. Sachs, Tafeln zum Math. Unterricht, Wiss. Beilage zum Jahres- 

 bericht Gym. Baden-Baden, 1905, h < 2000; 2000 < h < 5000, h a product 

 of primes 4n + 1 ; one side < 500. 



J. Gediking, 42 h < 1000. 



A. Martin, Math. Mag., 2, 1910, 301-324 (preface, 2, 1904, 297-300), 

 tabulated the values of p 2 g 2 , 2pq and area A = pq(p 2 q 2 ) for p =i 65, 

 q < p, q prime to p, q even if p is odd. Omitting the entries p = 33, 

 q = 22, and p = 35, q = 14, we have 862 triangles of which 443 have 

 A ^ 934800 (the largest A of the 178 triangles in Tebay's table). There 

 is a table of the sides p 2 d= g 2 , 2pq of triangles for which p = q -{- 1 =i 157 

 and those with p Si 312, q = 1, whence h exceeds a leg by 1 or 2 respectively. 

 . P. Barbarin, I'intermediaire des math., 18, 1911, 117-120, gave the 

 35 pairs of primitive triangles with the same h < 1000. A. Martin, ibid., 

 19, 1912, 41, 134, noted the omission of one pair and stated that there are 

 41 pairs with 1000 < h < 2000. 



A. Martin, Proc. Fifth Internat. Congress Math., 2, 1912, 40-58, gave 

 the primitive triangles with h < 3000, noting two omissions by Sang. 

 He listed many sets of A; (k ?== 15) triangles whose h's are consecutive 

 integers; also sets of three triangles whose h's are sides of a right or scalene 

 triangle. A product of n distinct primes 4m + 1 is the hypotenuse of 

 (3 n l)/2 different right triangles, only 2 n ~ 1 of which are primitive. 



W. Konnemann, Rationale Losungen Aufgaben, Berlin, 1915, h < 1000 

 (adverse review, Zeitschrift Math. Naturw. Unterricht, 46, 1915, 390). 



E. Bahier, 62 pp. 255-9, tabulated the primitive triangles with a leg ^ 300. 



On systems of equations including x 2 + if- = z 2 see papers 76, 77, 80, 

 46, 84, 89, 139, 140 of Ch. XVI; 5 of Ch. XVII; 51, 146 of Ch. XIX; 

 354, 357, 360, 362, 366, 369-71, 436 of Ch. XXI; 109, 113, 313 of Ch. 

 XXII; 207 of Ch. XXIII. 



PAPERS NOT AVAILABLE FOR REPORT. 



G. M. Pagnini, Collczione d' Opuscoli Sc., Firenze, 3, 1807, 3-24; Giornale di Fisica, Chimica e 

 Storia Nat., Pavia, 3, 1810, 193-207. [Series of rational right triangles.] 



Gruhl, Die Aufstellung Pythagoreischer Zahlen, Blatter Fortbildung d. Lehrcrs u. d. Lehrerin, 

 Berlin, 4, 1911, 998-1000. 



