CHAPTER I 

 THE MATHEMATICAL SCIENCES 



1. THE PURPOSE AND SCOPE OF GREEK 



MATHEMATICS 



WHEN we consider the questions studied by 

 the Greek mathematicians, we are at first 

 astonished at their great diversity. Be- 

 sides completed works, we find in the compendium 

 of Diophantus the principles of a theory of numbers, 

 in Apollonius the first idea of an analytical geometry, 

 in Archimedes the clear conception of the infinitesimal 

 calculus, and in Euclid the almost perfect application 

 of a method of exposition which has remained the 

 basis of more modern works. 1 



Important as they are, these discoveries only 

 embrace a portion of the vast field of mathematics. 

 The relations of numbers and figures constitute a world 

 so extraordinarily complex, that much of it is still 

 unexplored by modern science. And amongst all the 

 aspects of this world of relations, the Greek scientists 

 have been obliged to make a choice. What have been 

 the reasons and circumstances which determined their 

 choice ? It is on this question that we must attempt 

 to shed some light. 



On the nature of the mathematical fact there is 

 unanimous agreement. The Greek mathematician 

 admits implicitly or explicitly that the science of 

 number and space deals with ideal objects, changeless 

 and incorruptible. Plato has powerfully expounded 

 1 4 Boutroux, Ideal, p. 31. 

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