DECLINE OF ALEXANDRIAN SCIENCE 137 



2x is the original number and x the number added, (an arbitrary 

 and presumably erroneous assumption), 8x* + x = 27x 3 , giving 

 19z 2 = 1. The coefficient 19 not being a square, he now seeks to 

 find two cubes whose difference is a square. If (x + I) 3 x 3 is 

 equated to (2x I) 2 the special solution x = 7 is easily obtained. 

 Returning to the original problem, the new assumption is made : 

 let x = number to be added, 7x original number. 



(7z) 3 + x = (8z) 3 whence x = ^ 



In another type to find a square between 10 and 11, he multi- 

 plies both by successive squares of integers until between the prod- 

 ucts (by 16) he finds a square, 169. The number required is 

 10 A- Such processes naturally give particular, not general, 

 solutions. 



His lost Porisms are believed to have " contained propositions 

 in the theory of numbers most wonderful for the time." Sum- 

 marizing his methods of dealing with equations we may say 

 that : 



(1) he solves completely equations of the first degree having 

 positive roots, showing remarkable skill in reducing simultaneous 

 equations to a single equation in one unknown ; 



(2) he has a general method for equations of the second degree 

 but employs it only to find one positive root ; 



(3) more remarkable than his actual solutions of equations are 

 his ingenious methods of avoiding equations which he cannot^ solve. 



How far his work was original, how far like Euclid in his Elements 

 it was the result of compilation, cannot be definitely ascertained. 

 As a whole it is somewhat uneven and makes rather the impression 

 of great learning than of exceptional originality. He seems in- 

 debted in part to predecessors unknown to us. For him the 

 earlier Greek distinction between computation and arithmetic has 

 lost its force. 



In reviewing the work of Pappus and Diophantus Gow says : 



the Collections of Pappus can hardly be deemed really important. . . . 

 But among his contemporaries, Pappus is like the peak of Teneriffe in 

 the Atlantic. He looks back from a distance of 500 years, to find 



