viii PREFACE. 



for which there exists an additional partition into an even or odd number 

 of parts according as h is even or odd. This implies the corollary of Legendre 

 mentioned above. Analogous theorems were obtained by von Sterneck in 

 1897 and 1900. 



Mention should be made of the various papers by Glaisher of 1875-76 

 and 1909-10, that of Csorba of 1914, and the asymptotic formulas obtained 

 by Hardy and Ramanujan jointly in 1917-18. 



Chapter IV reports on the extensive, mostly old, literature on rational 

 right triangles, a subject which was the source of various problems treated 

 in later chapters. Diophantus knew that if the sides of a right triangle are 

 expressed by rational numbers they are proportional to 2mn, nfi n 2 , 

 w 2 +n 2 , and referred to the right triangle having the latter sides as that 

 "formed from the two numbers m and n." Pythagoras and Plato had 

 given special cases. Among the many problems on rational right triangles 

 treated by Diophantus, Vieta, Bachet, Girard, Fermat, Frenicle, De Billy, 

 Ozanam, Euler, and others, are the following: Find n(n^3) rational right 

 triangles of equal areas; two whose areas have a given ratio; one whose 

 area is given or becomes a square on adding a given number or a certain 

 function of the sides; one whose legs exceed the area by squares; one whose 

 legs differ by unity or by a given number; right triangles the sum of whose 

 legs is given; or with a rational angle-bisector. 



Chapter V deals with rational triangles, whose sides and area are ra- 

 tional, and rational quadrilaterals, having also rational diagonals. By the 

 juxtaposition of two rational right triangles with a common leg, we obtain 

 a rational triangle. During 1773-82, Euler wrote a series of four papers 

 on triangles whose sides and medians are all rational, while Bachet in 1621 

 had been content when a single median or single angle-bisector is rational. 

 The Hindus Brahmegupta and Bhascara showed how to form a rational 

 quadrilateral by juxtaposing four right triangles with pairs of equal legs 

 such that the right angles have a common vertex and do not overlap. In 

 1848 Kummer showed how to obtain all rational quadrilaterals. Euler 

 gave (p. 221) a construction for a polygon of n sides inscribed in a circle of 

 radius unity such that the sides, diagonals, and the area are all rational. 

 No mention will be made of the 160 further papers reported on in this 

 chapter, which closes with the papers on rational pyramids, trihedral 

 angles, and spherical triangles. 



Chapters VI-IX deal with the specially interesting literature on the 

 representation of numbers as sums of 2, 3, 4, n squares. Diophantus knew 

 how to express the product of two sums of two squares as a sum of two 

 squares in two ways : 



He knew that no number of the form 4n 1 is a sum of two squares. But 

 Girard in 1625 and Fermat a few years later were the first to recognize 

 that a number is a sum of two squares if, and only if, its quotient by the 

 largest square dividing it is a product of primes of the form 4n+l or the 



