226 THEORIES AND MODELS 



modern geometers concern themselves, because his postulates did 

 lead to the relations already known to him. 



Galileo's original conception of free fall, as uniformly accelerated 

 with increasing distance traveled, is formally unassailable. But only 

 his second conception, of a motion uniformly accelerated with in- 

 creasing time traveled, proves compatible with what his experi- 

 ments showed to be the case in nature. Newton might perfectly well 

 have postulated an inverse first power law of gravitation. The com- 

 bination of this with the other Newtonian postulates might have led 

 to a system formally wholly unobjectionable, but certainly not to an 

 acceptable theory of celestial dynamics: this system does not yield 

 theorems identifiable with Kepler's laws, which were already known. 

 The choice of the familiar inverse square law, on the other hand, at 

 once yields a convincing derivation of just those laws. Thus "intui- 

 tion" comes under the control of experience through the mediation 

 of logic and mathematics— in very much the fashion Newton himself 

 suggests. 



hi mathematics we are to investigate the quantities of [hypothetical] 

 forces with their proportions consequent upon any conditions sup- 

 posed; then, when we enter upon physics, we compare those propor- 

 tions with the phenomena of nature, that we may know what condi- 

 tions of those forces answer to the several kinds of attractive bodies. 



"Induction" represents the co-ordinated function of imaginative, 

 formal, and empirical components prettily displayed in Rheticus' ac- 

 count of the procedures of his revered master, Copernicus. 



... he assumes new hypotheses, not indeed without divine inspira- 

 tion and the favor of the gods; by applying mathematics, he geometri- 

 cally establishes the conclusions which can be drawn from them by 

 correct inference; . . . 



These conclusions were then compared with the data, Rheticus tell- 

 ing us that: 



. . . my teacher always has before his eyes the observations of all 

 ages together with his own, assembled in order as in catalogs; . . . 



DEDUCTION 



The whole conception of scientific theories as postulational sys- 

 tems assumes the availability of e£Eective machinery of formal de- 



