8G THE MATHEMATICAL PRINCIPLES because the force with which the weight p presses the plane pQi is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line joH towards both the planes, as joN to pH ; and to the force with which it presses the other plane pG, as joN to NH. And thus the force of the screw may be deduced from a like resolution of forces ; it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther con firmed. For on what has been said depends the whole doctrine of mechan ics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullics, levers, cords, and weights, ascending directly or obliquely, and other mechan ical powers ; as also the force of the tendons to move the bones of animals. COROLLARY III. The (/uaittity of motion, which is collected by taking the sum of the mo tions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action oj bodies among themselves. For action and its opposite re-action are equal, by Law III, and there fore, by Law II, they produce in the motions equal changes towards oppo site parts. Therefore if the motions are directed towards the same parts. whatever is added to the motion of the preceding body will be subducted from the motion of that which follows ; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both ; and therefore the difference of the motions directed towards opposite parts will remain the same. Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, the motion of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many ; and therefore after reflexion A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts ; or it will go back with 1 part, having not only lost its whole motion, but (if 1 may so say) one part more; or it will go back with 2 parts, because a progressive mo tion of 12 parts is taken off. And so the sums of the Conspiring motions 15 ,1, or 16-1-0, and the differences of the contrary i otions 17 — 1 and