94 THE MATHEMATICAL PRINCIPLES And in like manner in the pully, or in a combination of pullies, the force of a hand drawing the rope directly, which is to the weight, whethel ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other. The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body. The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the propress of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines. The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary : from whence in all sorts of proper machines, we have the solution of this problem ; 7 move a given weight with a given power, or with a given force to over come any other given resistance. For if machines are so contrived that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant, but with a greater disparity of ve locity will overcome it. So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those re sistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of {he machine as in the resisting body. But to treat of mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law ot motion. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjuncth from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts, the action and reaction 'YL the use of all sorts of machines will b" found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon tic resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction.