GREEK SCIENCE IN ALEXANDRIA 89 



finality. It consisted of thirteen books, of which only the 

 first six are ordinarily included in modern editions. The whole 

 is essentially a systematic introduction to Greek mathematics, 

 consisting mainly of a comparative study of the properties and 

 relations of those geometrical figures, both plane and solid, which 

 can be constructed with ruler and compass. The comparison of 

 unequal figures leads to arithmetical discussion, including the 

 consideration of irrational numbers corresponding to incommensu- 

 rable lines. The contents may be briefly summarized as follows : 

 Book I deals with triangles and the theory of parallels : Book II 

 with applications of the Pythagorean theorem, many of the prop- 

 ositions being equivalent to algebraic identities, or solutions of 

 quadratic equations, which seem to us more simple and obvious 

 than to the Greeks. It should be noted however that the geomet- 

 rical treatment is relatively advantageous for oral presentation. 

 Book III deals with the circle, Book IV with inscribed and cir- 

 cumscribed polygons. These first four books thus contain a 

 general treatment of the simpler geometrical figures, together 

 with an elementary arithmetic and algebra of geometrical magni- 

 tudes. In Book V, for lack of an independent Greek arithmetical 

 analysis, a theory of proportion (which has thus far been 

 avoided) is worked out, with the various possible forms of the 



equation - = - The results are applied in Book VI to the com- 

 b a 



parison of similar figures. This contains the first known problem 

 in maxima and minima, the square is the greatest rectangle 

 of given perimeter, also geometrical equivalents of the solution 

 of quadratic equations. The next three Books are devoted to the 

 theory of numbers, including for example the study of prime and 

 composite numbers, of numbers in proportion, and the determina- 

 tion of the greatest common divisor. He shows how to find the 

 sum of a geometrical progression, and proves that the number of 

 prime numbers is infinite. 



If there were a largest prime number n then the product 1 X 2 X 

 3 ... X n increased by 1 would always leave a remainder 1 when di- 

 vided by n or by any smaller number. It would thus either be prune 



