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 ; 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 vivce, 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. 9.) the triangles ADB, BHG, GML, be right angled at D, H, M, so that the sides BD, GH, LM, are each equal to 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 DE 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=0=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 is =V2; then communicating to L the velocity it goes on with the velocity LQ=1, and which it soinmunicates, by direct con course, to the ball R. Wherefore, say they, with that force which it had with the velocity 2, it has communicated to four balls equal to it, forces which being each =1, make a total of 4; and since the ori ginal velocity was 2, the forces are not as the simple velocities into the masses, but as the squares of the velocities. But in the theory of Boscovich this ar gument has no force. The ball A does not commu nicate a part of its velocity AB resolved into DB, TB, to the ball C, and with it a part of its force. There, acts upon the balls a new and mutual force in opposite directions, which impresses upon the one the velocity. CE, and BD on the other. The velo city of the former ball, expressed by BF, equal and in the same direction with AB, is compounded with the new velocity BD, and there arises the velocity BG, less than BF from,tbe obliquity of the composition. In like manner, a new mutual force acts on the balls at G and I, L and 0, Q and R, and the new velocities of the first ball GL, LQ, zero, compose the velocities GH and GN, LM and LS, ,J.,Q and QL, without either any actual resolution or translation of vis viva.
In the collision of bodies and reflected motion, it may be observed, that since, by this theory, there are no continuous globes, no continuous planes ; the most part of the phenomena above mentioned take place only perceptibly, and not with a strict ac curacy. The change of direction in impact is not made in one point, but by a continued curve, since the forces act at a distance, something in the way of AB and DM, Fig. 10. if the forces act only by re pulsion. If there be alternate attractions and repul sions, the body will proceed by a winding course. But it is still evident, that if the forces be equal, at equal distances, the two halves ABQ and QDM are equal and similar. If the plane. CO be rough, as must be the case in nature, and as we have exhibited in the Figure, this equality of forces will not take place ; but if the inequalities be very small in respect of the distance, the irregularity, this cause, will also be small; and it must be observed, that all the points within the segment RTS will be in action, which will render the inequality so much the more imperceptible.
In this manner one may observe, that light will be reflected at equal angles, from glass sufficiently polished, although the polishing matter has left some small inequalities. But from surfaces, which are sen sibly rough, it must be dispersed irregularly and in all directions.
To apply the theory to the refraction of light, let there be two parallel surfices AB, CD, Fig. 11, and
a moveable point without them. At some distance it is not acted on by any force, but, within that., is urged by forces which, however, are always perpen dicular to the, plane. Let it approach either of them in the direction GE, with the velocity HE. Let this be expressed, or, as it is usually called, resolved into the two HS, and SE. After ingress, between the planes, its motion will be ineurvatd by these forces, in such a manner, however, as not to alter its velocity parallel to the, planes ; but its perpendicular velocity will be materially changed. There are three cases • 1st, The velocity ES may be extinguished somewhere in X, arid then the body being reflected 1 back by the same forces, will pass off in XIMK ; and we have the same phenomena as in Fig. 10. 2d, The body may pass on to CD, with a diminished velocity as at 0, where taking PN=HS, but OP less than SE, the angle DON is less than the angle GEA of incidence. 3d, If the velocity be increa ped, then op being greater than SE, the angle D o n will be greater. And it will be easy to demonstrate, that the sine of the angle NES of incidence, is in a constant ratio to the sine of the angle of refrac tion PON.
We shall now consider the mutual action of three masses, being a more general application of the sys tem of three points.. Let the three masses, of which the centres of gravity are A, B, and C, act on each other, with forces directed towards the centres of gravity ; and first let us consider the directions of the forces. The force of the point C, when attrac tive on either side, as CV, Cd, will be C e; if repul. sive, as CY, C a, it willke CZ ; and the direction, in either case, will pass through the triangle, at least when produced to the opposite parts, bitting in the ' one case the interior angle ACB, and in the other, the one vertically opposite. With the attractive force CV towards B, and repulsive CY from A, the resulting Force is CX. The opposite supposition gives C b, each of which keeps without the triangle, and cuts the external angle. To the first, Cc, the attractions BP and AG correspond, and these, with the attrac tions AE and BN, would produce the forces AF and BO ; but with the repulsions AI and BR, they would. produce A H and BQ. In either case the forces lie to wards the same side of the line AB, and either both enter the triangle tending towards it, or both of them go away from it,.and tend in a direction opposite to that of the force C c in respect of AB. To the se cond, CZ must correspond the repulsions BT and AL, which, with the repulsions AI, BR, constitute AK, BS ; but with the attractions AE, BN, they form AD and BM. Of these, the first pair, as well as the last, lie towards the same side of AB, and the directions of both, when produced backwards, enter the triangle, but with contrary directions to CZ ; or they go away without the triangle in opposite direc tions from CZ. Thirdly, if CX be got, which would. be produced by CV, CY, then BP and AL correspond to it, and, if the first be conioined with BN, we have BO entering the triangle ; but if with BR, then indeed BQ falls without the triangle as well as CX ; but the corresponding forces AL and AI produce AK, which, at least, enters the triangle when back : wherefore there is, in every case, some one of the directions which passes through the triangle; and. then what was said in the cases of C e and. CZ, re turns respecting the other two.. We have therefore the following theorem : If three masses act on other, with forces directed to their centres of gravity, the compound force, acting on one at least, has a di rection, which, at. least, when produced towards the opposite parts, will cut the internal angle of the tri angle, and enter it : The remaining two both enter,. or they both, avoid the triangle, and always proceed, towards the same parts, in respect of the line joining the centres of the masses : And, in the first case, all the three forces tend towards the interior of the tri angle, lying in the internal angles ; or all tend away from the triangle lying in the vertical opposite an gles : But in The second case, with respect to the line joining the two masses, they tend towards the opposite parts from that towards which the force of the first mass is directed.