342 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



Theon of Smyrna 2 (about 130 A.D.) called the s's and S's side and 

 diametral (diagonal) numbers and gave the above recursion f ormulffi without 

 the geometrical interpretation. 



Archimedes (third century B.C.) gave the approximations 265/153 and 

 1351/780 to V3, which can be explained in connection with x 2 3y 2 = 2, 

 z 2 - 3?/ 2 = 1. _ 



Heron of Alexandria used the approximation a + r/(2a) for Va 2 + r. 



For a more detailed account than what precedes of the connection be- 

 tween the knowledge of the early Greeks and Hindus of approximation to 

 square roots and Pell equations, see H. Konen 3 and E. E. Whitford. 4 



The history of the cattle problem of Archimedes will now be discussed 

 in detail. 



In 1773, Gotthold Ephraim Lessing 5 published a Greek epigram in 24 

 verses, from a manuscript in the Wolfenbiittel library, stating a problem 

 purporting to be one proposed by Archimedes, 6 in a letter to Eratosthenes, 

 to the mathematicians of Alexandria, as well as a scholium giving a false 

 answer, and a long mathematical discussion by Chr. Leiste. The problem 

 is to find the numbers W, X, Y, Z of the white, black (or blue), piebald (or 

 spotted) and yellow (or red) bulls, and the numbers w, x, y, z of the cows 

 of the corresponding colors, when 



(1) W = (I + |)X + Z, (2) X = (I + J)F + Z, 



(3) Y = (i + \}W + Z, (4) w = (| + J)(Z + x), 



(5) s = (i + i)(F + 2,), (6) y = (t + (Z + ), 



(7) z = (% + V(W + w), (8)W + X=n, 



(9) Y + Z = A, 



the final notations being those for a square and a triangular number. 

 Leiste found at once the integral solutions of (1), (2), (3) : 



(10) Y = 1580m, Z = 891m, W = 2226m, X = 1602m. 

 Then, by (4), m = 2p, x = 12. By (5), a = 3/3, y = 20(4/3 - 158p). 



2 Platonici . . . expositio, 1544,67. Theon Smyrnaeus, ed., E. Hiller, Leipzig, 1878, 43; 

 French transl., by J. Dupuis, Paris, 1893, 71-5. 



3 Geschichte der Gleichung t 2 Du 2 = 1, Leipzig, 1901, 2-17. Reviews by Wertheim, 

 Bibl. Math., (3), 3, 1902, 248-251; and Tannery, Bull, des Sc. Math., 27, II, 1903, 47. 



4 The Pell Equation, Columbia Univ. Diss., New York, 1912, 3-22. The following related 

 papers are not mentioned in the pages just cited: E. S. Unger, Kurzer Abriss der Gesch. 

 Z. von Pythagoras bis Diophant, Progr., Erfurt, 1843; C. Henry, Bull, des Sc. Math. 

 Astr., (2), 3, 1, 1879, 515-20; H. Weissenborn, Die irrationalen Quadratwurzeln bei 

 Archimedes und Heron, Berlin, 1884; Zeitschr. Math. Phys., Hist.-Lit. Abt., 28, 1883, 

 81; E. Mahler, ibid., 29, 1884, 41-3; W. Schoenborn, 30, 1885, 81-90; C. Demme, 31, 

 1886, 1-27; K. Hunrath, 33, 1888, 1-11; V. V. Bobynin, 41, 1896, 193-211; M. Curtze, 

 42, 1897, 113, 145; F. Hultsch, Gottingen Nadir., 1893, 367; G. Wertheim, Abh. Gesch. 

 Math., VIII, 146-160 (in Zeitschr. Math. Phys., 42, 1897); Zeitschr. Math. Naturw. 

 Unterricht, 30, 1899, 253; T. L. Heath, Euclid's Elements, 1, 1908, 398-401. 



6 Zur Geschichte der Literatur, Braunschweig, 2, 1773, No. 13, 421-446. Lessing, Sammtliche 

 Schriften, Leipzig, 22, 1802, 221; 9, 1855, 285-302; 12, 1897, 100-15; Opera, XIV, 232. 



8 Archimedes opera, ed., J. L. Heiberg, 2, 1881, 450-5; new ed., 2, 1913, 528-34. 



