134 A SHORT HISTORY OF SCIENCE 



however only in somewhat mutilated form. It is the first 

 known treatise on algebra, and is devoted to the solution of 

 equations, employing algebraic symbols and analytical methods. 

 Euclid had given the geometrical equivalent of the solution of a 

 quadratic equation, and Hero could solve the same problem alge- 

 braically but lacked a satisfactory symbolism. The algebra of 

 Diophantus was therefore not a sudden invention, but the result 

 of gradual evolution during several centuries of increasing interest 

 in arithmetical problems, and declining vogue of the abstract 

 Euclidean geometry. 



Writers on the history of algebra distinguish three classes or 

 methods of algebraic expression : 



(a) the rhetorical, where no symbols are used, but every term 

 and operation is described in full. This was the only method 

 known before Diophantus, and was later in vogue in western 

 Europe until the fifteenth century; 



(6) the syncopated, which replaces common words and operations 

 by abbreviations, but conforms to the ordinary rules of syntax. 

 This was the style of Diophantus ; 



(c) the symbolical or modern, using symbols only, without words. 



The syncopated method may be illustrated by the following 

 passage from Heath's Diophantus : 



Let it be proposed then to divide 16 into two squares. And let 

 the first be supposed to be IS ; therefore the second will be 16*7 IS. 

 Thus 16*7 IS must be equal to a square. I form the square from 

 any number of N's minus as many *7's as there are in the side of 16 IPs. 

 Suppose this to be 2N - 4*7. Thus the square itself will be 4S 16*7- 

 IQN etc. 



In his Arithmetic, which is really a treatise on algebra, Diophan- 

 tus represents the (single) unknown by the Greek sigma all the 

 other letters of the Greek alphabet standing for definite numbers 

 with successive powers to the sixth inclusive. If he requires 

 two unknowns he admits only one at a time. His originality and 

 power in the solution of problems are amply shown, though the 

 solutions are rarely complete. For quadratic equations, for ex- 



