THE MATHEMATICAL SCIENCES 115 



of a circle may therefore take a concrete form in 

 sensible representations, but it is not exhausted by any 

 of them, and it is not these representations which 

 justify its existence, for they are never anything but 

 an imperfect image. As will be seen, the definition 

 sheds light upon the structure of mathematical 

 principles and shows them to be distinct from the data 

 furnished by sensible perception. This distinction 

 impresses itself on the geometer apart from the meta- 

 physical reasons, always debatable, by which it may be 

 justified. What is certain, is that the principles of 

 mathematics, thanks to their definition, can serve as 

 a basis for strict reasoning, which can never be con- 

 tradicted by any sensible experience. If we take at 

 random two points on a circumference and if with these 

 two points and the centre of the circle as vertex, we 

 construct a triangle, we can affirm that this triangle 

 is isosceles and has two equal angles. This affirmation 

 is directly derived from the definitions which have been 

 given of the isosceles triangle and of the circle. Thus to 

 the Greeks belongs the great merit of having demon- 

 strated that numerical expressions and geometrical 

 figures possess peculiar properties of their own, judged by 

 other criteria, and dependent on other methods of investi- 

 gation than the phenomena of the sensible world. But 

 this does not enable us to understand what has guided 

 them in their choice of the innumerable problems pre- 

 sented by arithmetic and geometry. Doubtless it is 

 very important to recognize the quality of the materials 

 and the way to utilize them for the construction of a 

 building, but it is also necessary to sort them according 

 to the plan of the building. Now, the regular combina- 

 tions of numbers or figures are unlimited in number. 

 Analytical geometry has revealed to us several curves 

 (the curve called by French mathematicians la coitrbe 

 dn diable, for example) of which the Greek scientists 

 had not the slightest idea. Why did they stop at a 



