328 FIRST EXPLANATION 



of direction in ivhatever line [de quelque cote] 29 we take it in 

 the ivorld. That is to say, drawing any straight line you 

 please, and taking also such bodies and so many of them 

 as you please, you will find that, considering all these 

 bodies together, without omitting any of those which act 

 upon any one of those which you have taken, there 

 will always be the same amount [quantite] of progression 

 in the same direction [du meme cote] in all lines parallel 

 to the straight line you have taken observing that the 

 total amount of progression is to be calculated by sub- 

 tracting from the amount of progression of the bodies 

 which go in the given direction, the amount of progression 

 of those which go in the opposite direction 30 . This law, 

 being as good and as general as the other, deserved as 



29 E. reads vers quelque cote. 



30 See Principles of Nature and of Grace, n, note 48. Cf. Epistola ad 

 Bernoullium (1696) (G. Math. iii. 243; E. 108 note): 'In the next 

 place it is to be observed that I make a distinction between absolute 

 force and directing force, although I can deduce and demonstrate 

 directing force from the sole consideration of absolute power. And 

 indeed I prove that there is conservation not only of the same 

 absolute force or quantity of action in the world, but also of the 

 same directing force and the same quantity of direction in the 

 same lines [ad easdem paries], i.e. the same quantity of progression, 

 its direction being taken into account and the quantity of pro- 

 gression being counted equal to the mass multiplied by the 

 velocity, and not by the square of the velocity' [mv, not ?nt/ 2 ]. 

 1 Nevertheless this quantity of progression differs from quantity 

 of motion in this way, that when two bodies are moving in 

 opposite directions their total quantity of motion (in the Cartesian 

 sense) is to be got by adding together the quantity of motion of 

 each (calculated as the mass into the velocity) ; but the quantity 

 of progression is to be got by subtracting the one from the other ; 

 for in such a case the difference between the quantities of motion 

 will be the quantity of progression. Therefore when Descartes 

 thought that he could safeguard the soul's power of acting on the 

 body in this way, that while the soul cannot increase or diminish 

 the quantity of motion in the world, it can nevertheless increase 

 or diminish the quantity of direction of the [animal] spirits, he 

 erred through not knowing this new law of ours regarding the 

 conservation of the quantity of direction, which is no less beautiful 

 and inviolable than the law of the conservation of absolute force 

 [virtus'] or power of action.' The ' quantity of progression ' would 

 now be called a projection of the quantity of motion. A full 

 explanation, with diagrams, will be found in the appendix to 

 Boutroux's edition of the Monadologie. 



