no SCIENCE IN GRECO-ROMAN ANTIQUITY 



the same in what concerns the relation of law to 

 consequence. 



This being so, it is necessary, for the explanation of 

 the relations between the lines and surfaces of which 

 geometrical figures are constituted, to have recourse to 

 geometrical and numerical reasonings ; to account for 

 the phenomena of the physical world, it is necessary to 

 appeal to mechanical and physical reasonings, and, 

 finally, it is by physiological reasonings that health and 

 disease must be explained, and not by invisible powers 

 outside the body. 



By these entirely new ideas the Greeks revealed to 

 the human mind for the first time the true foundations 

 of the sciences which, from the time of the Renaissance, 

 were to blossom and give to Europe her supremacy. 

 It may be objected with truth that these foundations 

 had been laid already by the Egyptians and Chaldeans. 

 But, as we have already remarked, these peoples had 

 simply imparted to the Greeks mathematical facts and 

 empirical formulae which they had been able to estab- 

 lish through centuries of experience ; they had never 

 conceived of the possibility of creating a science worthy 

 of the name. Between the fragments of knowledge 

 which they discovered and the scientific conceptions of 

 the Greeks there is an abyss which we may fathom by 

 the following example. The Egyptians knew and made 

 use of the numerical properties of the squares con- 

 structed on the sides of a right-angled triangle. We 

 do not know how they discovered these properties, 

 but it is probable, as we have remarked before (p. 7), 

 that it was in the following manner. On the sides of a 

 right-angled triangle whose magnitudes are 5, 4, and 3, 

 let squares be described. We can divide these squares 

 into smaller squares all equal to 1 2 and easily prove the 

 equality 25 = 16 + 9. This demonstration is purely 

 empirical. It is so intuitive that a child can easily 

 understand it. As it simply states a mathematical 



