84 THE MATHEMATICAL PRINCIPLES If a body impinge upon another, and by its force change the motion of (It* other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies ; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally pro portional to the bodies. This law takes place also in attractions, as will be proved in the next scholium. COROLLARY I. A body by two forces conjoined will describe the diagonal of a parallelo gram, in the same time that it wovld describe the sides, by those forces apart. If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion / be carried from A to B ; and by the force N impressed apart in the same place, should be carried from A to c ~\) C ; complete the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the rorce N be impressed or not ; and therefore at the end of that time it will he found somewhere in the line BD. By the same argument, at the end of the same time it AY ill be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in ;i right line from A to D, by Law I. COROLLARY II. And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the re solution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from, mechanics. As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP ; and the forces of those weights to move the wheel were required. Through the rentre O draw the right line KOL, meeting the cords perpendicularly in A and L; and from the centre O, with OL the greater of the distances