CHAP. IV] TABLES OF RIGHT TRIANGLES. 189 



J. Kersey 66 (p. 143) took the right triangle with the rational sides 



AC = p(p z + & 2 ), AB = X? 2 ~ & 2 ), BC = p(2bp). 

 The bisector AD of angle A divides the base into two segments 



CD = b(p* + 6 2 ), BD = b(p 2 - b 2 ) 



proportional to AC and AB. Since AB : BD = p : b, we have 



AD = h(p* - 6 2 ), 



if h, p, b are sides of any rational right triangle. 



Several 160 writers found a right triangle with a rational bisector of one 

 acute angle. 



E. Turriere 161 found a rational right triangle with rational interior and 

 exterior bisectors of an acute angle. 



TABLES OF RIGHT TRIANGLES WITH INTEGRAL SIDES. 



The tables are usually arranged according to the magnitude of the 

 hypotenuse h or the area A. 



An Arab manuscript 9 of 972 gave a brief table (see Ch. XVI). 



J. Kersey, Elements of Algebra, Books 3, 4, 1674, 8, h ^ 265. 



J. C. Schulze, Sammlung Log., Trig. Tafeln, Berlin, II, 1778, 308, 

 gave the decimal values of tan /2 = mfn for 200 pairs of relatively prime 

 integers m, n each ^ 25, m < n; also right triangles with an angle w. 



A. Aida 17 (1747-1817) listed the 292 primitive triangles with h < 2000. 



Le pere Saorgio, Mem. Acad. Sc. Turin, 6, annees 1792-1800, 1801, 

 239-252, quoted a table of primitive right triangles from Schulze. 



C. A. Bretschneider, Archiv Math. Phys., 1, 1841, 96, h ^ 1201. 



Du Hays, Jour, de Math., 7, 1842, 331-4, gave four tables each with 

 32 entries to illustrate the systematic tabulation of primitive right triangles, 

 using (1) with m, n relatively prime, m > n. First, give to m the values 

 2, 3, and to n the values < m and prime to m, such that one of m, n is 

 even. Second, take 1, 3, 5, as the odd side and factor each into two 

 factors m =b n. Third, begin with the even side 2mn. Fourth, take a sum 

 of two squares as the hypotenuse. 



A. Wiegand, Sammlung Trig. Aufgaben, Leipzig, 1852, 131 triangles 

 and their angles. 



D. W. Hoyt, Math. Monthly (ed., Runkle), Cambridge, Mass., 2, 

 1860, 264-5, h < 100. 



E. Sang, Trans. Roy. Soc. Edinburgh, 23, III, 1864, 757, h ^ 1105. 



S. Tebay, Elements of Mensuration, London and Cambridge, 1868, 

 111-2, gave an incomplete table arranged according to area A, the largest 

 A, 863550, being an error for 934800. Reprinted by G. B. Halsted, Metrical 

 Geometry, 1881, 147-9. 



H. Rath, Archiv Math. Phys., 56, 1874, 188-224, used formulas [due to 

 de Lagny 18 ] to form a double-entry table, and noted an error by Berkhan. 



160 Amer. Math. Monthly, 7, 1900, 83-5. 



161 L'enseignement math., 18, 1916, 407-8. 



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