ii2 SCIENCE IN GRECO-ROMAN ANTIQUITY 



so, it is necessary to prove first that the triangles of 

 each pair are equal and then that one of them is equal 

 in area to the half of one of the squares, etc. The 

 demonstration in this form is quite general, independent 

 of particular cases, but it supposes a whole series of 

 propositions previously demonstrated and which are 

 rigorously linked together ; for example, all triangles 

 which have the same base and the same height as a 

 rectangle have equal areas, which are equivalent to 

 half that of the rectangle. 1 To establish all these pro- 

 positions, they must be based on the general properties 

 of the straight line and the angle, in other words, on 

 axioms and definitions. These axioms or definitions 

 must be logical and in no way obscure to the mind, 

 otherwise the deduction would remain doubtful and 

 would lack exactitude. 



Thus, the ideal which the Greeks have more and more 

 conscientiously pursued is the following : to place at 

 the basis of all science a number of principles which 

 guarantee a strict logical reasoning, and then by their 

 means to construct an edifice of consequences the value 

 of which is assured by a rational deduction. Without 

 insisting further it can be seen how much the Greek 

 ideal of knowledge differed from that of primitive 

 peoples or even of the peoples of the East. 



1 In this demonstration the investigation of the congruency 

 plays a preponderant part as M. E. Meyerson rightly remarks : 

 De I 'explication dans les sciences, vol. I, p. 137 et seq., Payot, 

 Paris, 192 1). 



