CHAP. IV] DlVISOKS OF SlDES OF A RlGHT TRIANGLE. 171 



PAPERS WITHOUT NOVELTY. 



G. Oughtred, Opuscula Math., Oxonii, 1677, 130-8. 



A. Thacker, A Miscellany of Math. Problems, Birmingham, 1, 1743, 171-8 [Proof of (1)]. 



Anonymous, Ladies' Diary, 1752, 39, Quest. 344 [Proof of (I)]- 



A. D. Wheeler, Amer. Jour. Arts. Sc. (ed., Silliman), 20, 1831, 295 [Plato's rule]. 



J. A. Grunert, Kliigel's Math. Worterbuch, 5, 1831, 1141-3 [Euler 20 ]. 



C. M. Ingleby and S. Bills, Math. Quest. Educ. Times, 6, 1866, 39-40 [Proof of (1)]. 



M. A. Gruber, Amer. Math. Monthly, 4, 1897, 106-8. 



H. Schubert, Niedere Analysis, 1, 1902, 159-162 [Proof of (1)]. 



F. Thaarup, Nyt Tidsskrift for Mat., 15, A, 1904, 33 [Proof of (1)]. 



A. Aubry, Mathesis, 5, 1905, 6-13 [historical]. 



A. Holm, Math. Quest. Educ. Times, (2), 9, 1906, 92; 10, 1906, 56 [Proof of (1)]. 



V. Varali-Thevenet, Rivista Fis. Mat. Sc. Nat., 8, I, 1906, 422-3. 



C. Botto, Giornale di Mat., 46, 1908, 297-8 [Poinsot 23 ]. 



P. Richert, Unterrichtsblatter Math., 14, 1908, 55-7, 87. 



C. Botto, Suppl. al Periodico di Mat., 12, 1908-9, 68-74. 



T. S. Rao, Jour. Indian Math. Club, Madras, 1, 1909, 130-4. 



School Sc. and Math., 10, 1910, 683; 11, 1911, 293^; 13, 1913, 320-2. 



SlDES OF A RIGHT TRIANGLE DIVISIBLE BY 3, 4, OR 5. 



Frenicle de Bessy 52 (f 1675) noted that if the g.c.d. of the integral 

 sides of a right triangle is a square or the double of a square, the sides are 

 of the form (1), and that one of the sides is divisible by 5, one of the legs 

 by 3 and one by 4. If the sides are relatively prime, the sum and difference 

 of the legs are of the forms 8k 1. 



P. Lentheric 53 noted that the product xyz of the numbers (1) is divisible 

 by 60, since mn(m 2 n 2 ) is divisible by 6 and if no one of m, n, m n is 

 divisible by 5, m 2 + n 2 is. F. Paulet added (p. 382) the remark that 

 m 4 n 4 is divisible by 5 if neither m nor n is, since m 4 = 10& + 1 or 10k + 6. 



L. Poinsot 23 stated, as if new, that if x, y, z are relatively prime solutions 

 of x 2 + y 2 = 2 2 , 3 is a factor of x or y, 4 a factor of x or y, and 5 a factor of 

 x, y or z. This was proved by E. R. Grenoble 54 by considering the residues 

 modulo 3, 4 or 5, and by J. Binet (pp. 686-7, 755) by use of Fermat's 

 theorem. J. Liouville remarked (p. 687) that x, y, x + y or x y is 

 divisible by 7. Bourdat 55 stated that he had found these facts in 1839 and 

 added that, if x z + y 2 = z 4 , 5 is a factor of x, y or z, likewise 7 and 24. 

 If z 2 + y 2 = z 8 , one of the numbers has the factor 2 4 3 7. 



A. Vermehren 56 proved that xyz is divisible by 60. 



A. Levy 57 noted that in a 2 + fc 2 = c 2 , 7 divides a -f b or a - b if 7 is 

 prime to a, 6, c; 11 divides one of 5a &, 56 a if 11 is prime to a, b, c. 



62 TraitS des triangles rectangles en nombres, I, Paris, 1676, 24-25, pp. 59-61. Re- 



printed with part II in 1677 at end of Problemes d'Architecture de Blondel. Both parts 

 in Mem. Acad. R. Sc. Paris, 5, 1666-99; e"d. Paris, 1729, pp. 146-7. C. Henry, Bull. 

 Bibl. Storia Sc. Mat. Fis., 12, 1879, 691-2, gave a list of Frenicle'a writings; cf. Nouv. 

 Ann. Math., 8, 1849, 364-5. 



63 Annales de Math, (ed., Gergonne), 20, 1829-30, 376-382; 21, 1830-1, 96-98. Cf. Jour, fiir 



Math., 5, 1830, 386; Jour, de math. e"le"m. sp6c., 1880, 261. 

 M Comptes Rendus Paris, 28, 1849, 665-6. 

 66 Bull, de 1'Acad. Delphinale, Grenoble, 3, 1850, 37-43. 

 86 Die Pythagoraischen Zahlen, Progr. Domschule, Glistrow, 1863. 

 " Bull, de math, elem., 15, 1909-10, 277. 



