PREFACE. 



Diophantine analysis was named after the Greek Diophantus, of the 

 third century, who proposed many indeterminate problems in his arithmetic. 

 For example, he desired three rational numbers, the product of any two of 

 which increased by the third shall be a square. Again, he required that 

 certain combinations of the sides, area, and perimeter of a right triangle 

 shall be squares or cubes. He was content with a single numerical rational 

 solution, although his problems usually have an infinitude of such solutions. 

 Many later writers required solutions in integers (whole numbers), so that 

 the term Diophantine analysis is used also in this altered sense. For the 

 case of homogeneous equations, the two subjects coincide. But in the 

 contrary case, the search for all integral solutions is more difficult than that 

 for all rational solutions. In his first course in the theory of numbers, a 

 student is surprised at the elaborate theory relating to the equation which 

 in analytic geometry represents a conic; but it is a real difficulty to pick 

 out those points of the conic whose coordinates are rational and a greater 

 difficulty to pick out those points whose coordinates are integral. 



Our subject has appeared not only in works on arithmetic and geometry, 

 but also in algebras; to it was devoted the larger part of Euler's famous 

 Algebra. Some of its topics, as the theory of partitions, belong equally 

 well to analysis. Although most of the problems in this domain may be 

 stated in simple language free of technical mathematics, their investigation 

 has quite often required the aid of many branches of advanced mathe- 

 matics. A mere reference to the extensive subject index will show how 

 frequently use has been made of elliptic functions and integrals, infinite 

 series and products, algebraic and complex numbers, covariants, invariants, 

 and seminvariants, Cremona and birational transformations, geometrical 

 methods, matrices, gamma and theta functions, cyclotomy, linear differ- 

 ential and difference equations, integration, approximation, limits, minima, 

 asymptotic and mean values. 



Following the plan used in Volume I, we proceed to give an account in 

 untechnical language of the main landmarks in the successive chapters. 

 If a reader will not pause to read this entire introduction, let him sample it 

 by selecting the account of the final chapter. This introduction is followed 

 by an explanation of the author's point of view in producing a work quite 

 different from conventional histories. 



The notion of triangular numbers 1, 3, 6, . . . goes back to Pythagoras, 

 who represented them by points arranged as are the shot in the base of a 

 triangular pile of shot. The number of shot in such a pile is called a tetra- 

 hedral number. In an analogous manner we may define a polygonal 

 number of m sides (m-gonal number) and a pyramidal number. Simple 

 theorems concerning these numbers occur in the Greek arithmetics of 

 Theon of Smyrna, Nicomachus (each about 100 A.D.), and Diophantus 

 (250 A.D.), who wrote also a special tract about them. They were treated 



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