DISCOVERY OF THE LAWS OF MOTION. 831 



from this theoretical result. For it may be observed, that the curvilinear 

 paths ascribed to military projectiles by Rivius and Tartalea, and bj 

 other writers who followed them, as Digges and Norton in our own 

 country, though utterly different from the theoretical form, the parab- 

 ola, do, in fact, approach nearer the true paths of a cannon or inuskei 

 ball than a parabola would do ; and this approximation more especially 

 exists in that which at first sight appears most absurd in the old 

 theory ; namely, the assertion that the ball, which ascends in a sloping 

 direction, finally descends vertically. In consequence of the resistance 

 of the air, this is really the path of a projectile ; and when the velocity 

 is very great, as in military projectiles, the deviation from the parabolic 

 form is very manifest. This cause of discrepancy between the theory, 

 which does not take resistance into the account, and the fact, Galileo 

 perceived; and accordingly he says, 12 that the velocities of the projec- 

 tiles, in such cases, may be considered as excessive and supernatural. 

 With the due allowance to such causes, he maintained that his theory 

 was verified, and might be applied in practice. Such practical appli- 

 cations of the doctrine of projectiles no doubt had a share in estab- 

 lishing the truth of Galileo's views. We must not forget, however, 

 that the full establishment of this second law of motion was the result 

 of the theoretical and experimental discussions concerning the motion 

 of the earth : its fortunes were involved in those of the Copernican 

 system ; and it shared the triumph of that doctrine. This triumph 

 was already decisive, indeed, in the time of Galileo, but not complete 

 till the time of Newton. 



Sect. 4. Generalization of the Laws of Equilibrium. Principle of 



Virtual Velocities. 



IT was known, even as early as Aristotle, that the two weights 

 which balance each other on the lever, if they move at all, move with 

 velocities which are in the inverse proportions of the weights. The 

 peculiar resources of the Greek language, which could state this rela- 

 tion of inverse proportionality in a single word (avrnrtTrovOev), fixed 

 it in men's minds, and prompted them to generalize from this property. 

 Such attempts were at first made with indistinct ideas, and on conjec- 

 ture only, and had, therefore, no scientific value. This is the judg- 

 ment which we must pass on the book of Jordanus Nemorarius, which 



11 Op. vol. iii. p. 147. 



