THE MATHEMATICAL SCIENCES
WHEN we consider the questions studied by the Greek mathematicians, we are at first astonished at their great diversity. Besides 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____________________