BOSCOVICH'S THEORY. 



761 



h'i mined; lie points out and supplies the defect of proof 

 ' in the common way of finding the centre of seve- 

 ral bodies ; illustrating the subject by the multiplica- 

 phy. tien ' numbers, and the composition or forces ; and 



- he demonstrates the celebrated theorem of Newton, 



that the centre of gravity is undisturbed by mutual 

 internal forces ; consequently, that the quantity of 

 motion in the universe is preserved always the same, 

 when computed in the same direction, and therefore 

 that action and reaction are always equal and con- 

 trary. 

 Collision. From this law of the equality of action and re- 



action, readily flow the laws of collision, discovered 

 at the same time by Wren, Huygens, and Wallis, 

 as is mentioned by Newton, when treating of this very 

 law. (Prin. lib. i. Cor. 4. Ax.) Boscovich derives 

 them in this way. Suppose a soft globe or ball goes 

 forward with a less velocity, and followed by another 

 soft globe with a greater velocity, so that their centres 

 be always carried in the line which joins them, and 

 that the one at length hits the other, which is called a 

 direct collision ; this hitting, according to our au- 

 thor, is not done by an immediate contact, but be- 

 fore they come in contact, the after parts of the first 

 and the fore parts of the last are compressed by the 

 mutual repulsive force ; and this compression goes 

 on increasing until they come to have equal veloci- 

 ties, then all further access ceases, and, consequently, 

 all further compression ; and since the bodies are 

 soft, they exert no mutual force after compression, 

 but continue to go on with equal velocity. And 

 since the quantity of motion will be the same in the 

 same direction, we must, in order to find the com- 

 mon velocity after collision, multiply each mass into 

 its velocity, and divide the sum of these products by 

 the sum of the masses. If one of the globes were at 

 rest, its velocity might be made = 0, and, if moving 

 in the opposite direction, it might be taken with a 

 negative value. 



From soft bodies, the transition is easy to those 

 which are elastic. In these, after the greatest com- 

 pression and change of figure, the two globes con- 

 tinue to act on each other, until they recover their 

 first shape, and this action doubles the effect of the 

 former. If the elasticity be imperfect, and the force 

 in losing shape be to the force in recovering it in any 

 given ratio, the effect of the former to that of the 

 latter will also be in a given ratio, ( See Collision) ; 

 the deductions of Boscovich being no way different 

 from those given in other elementary treatises. 



Proceeding now to oblique concourse, let the two 

 globes A and C in Fig. 8. come in a given time, by 

 the right lines AB, CD, which measure their velo- 

 cities into physical contact at B and D. By the 

 common mode, the effect of the contact is thus ex- 

 plained : Join the centres by the straight line BD, 

 to which, produced if necessary, draw the perpen- 

 diculars AF, CH ; and completing the rectangles, 

 AFBE, CHDG, each of the motions AB, CD is 

 resolved into two, the one into AF, AE, or BE, BF, 

 the other into CH, CG, or GD, DH. The first 

 of these on tach side remains entire ; the second, FB 

 and HD, make a direct collision. We must there- 

 fore find, by the law of direct collision, the velocities 

 DK, DI, which, according to that law, will be dif- 



VOL. III. PART IV. 



I.XV. 

 F* 8. 



ferent for different sorts of bodies ; and we must Boscovicl-.'i 

 compound these with the forces or velocities expres- Iv 601 ^ . ot 

 sed by the straight lines BL, DQ lying in the same j>(,iloso- 

 straight lines with BE and GD, and equal to them ; pi,y. 

 therefore BM and DP will express the velocities and -. t . . i 

 direction of the motions after collision. The reso- 

 lution of motions in this way is considered as a real 

 and actual resolution, the one of which continues un- 

 altered, the other undergoes a change ; and in the 

 case which this figure expresses, is altogether ex- 

 tinguished, and then another is produced again. But 

 the thing takes place, in fact, without any real resolu- 

 tion, in the following manner : The mutual force No real re- 

 which acts upon the balls B, D, gives to them, du- solution ot 

 ring the whole time of the collision, the contrary ve- forces, 

 locities BN, DS, equal, in this case, to those two, of 

 which the one is commonly supposed destroyed and 

 the other reproduced ; these forces, compounded with 

 BO and DR., equal, and in the same direction with 

 AB, CD, and therefore expressing the entire effect 

 of the preceding velocities, exhibit the very same re- 

 sulting velocities BM, DP. For it is evident, that 

 LO will be equal to AE or BF, and therefore MO 

 rzBN, and BMNO a parallelogram. In like manner, 

 DM PS is a parallelogram. Wherefore there is no 

 real resolution in this case, but merely a composition 

 of motions ; namely, the former velocity persevering 

 by the vis inertia, and that compounded with the 

 new velocity which the forces produce that act in the 

 collision. 



In the same manner, when a ball strikes obliquely 

 on a plane, when a heavy body descends on an in- 

 clined plane, or is constrained to move in the arch of 

 a circle, by being suspended by a thread, the case 

 may be always explained without having recourse to 

 the resolution of forces or motions, and all the pheno- 

 mena shown to depend only on the composition of 

 forces: thus the procedure of nature is always simple 

 and uniform. And, indeed, that this is general, appears 

 evident from the theory j since no motion can be par- 

 tially obstructed, when there is no such thing as ab- 

 solute contact ; and that any point is freely moved 

 in empty space, and at liberty to obey, at the same 

 time, the velocity it had previously acquired, and the 

 forces which arise from all the other points of mat- 

 ter. Accordingly Boscovich can see no necessity for 

 introducing the principle of the vires tiw, which 

 Leibnitz and others have brought forward to ex- 

 plain the common doctrine of the resolution of forces, 

 since those very instances employed to demonstrate 

 their existence may be equally well explained with- 

 out them. One instance may be given in the oblique 

 collision of elastic bodies. Let (Fig. 0.) the triangles p LATE 

 ADB, BHG, GML, be right angled at D, H, M, l,xv. 

 so that the sides BD, GH, LM, are each equal to k~'g- 0. 

 half the base AB ; and let BG, GL, LQ be paral- 

 lel to AD, BH, GM, the ball A, with the velocity 

 AB=2, hits at B the equal ball C, lying in DB 

 produced from the oblique impact, it communicates 

 to it the velocity CE=1 = DB, which it loses itself, 

 and then goes on in BG with the velocity =.y/3=r AD. 

 In like manner, if it meets the ball I, it communi- 

 cates to it the velocity IK=1, while it loses 1H ; and 

 its velocity in GL 16=^/2; then communicating to 

 L the velocity OP=l, it goes on with the velocity 

 5c 



