Of the screw and friction.

I have shown in the next-to-last [proposition] of the third [book] how all the various amounts of friction that one and the same weight can make upon one and the same nature of resistance or support are all of equal labor to its mover. And certainly it does seem that nature cannot by any cunning ever be deceived; for if a man wishes to make easy the motion of dragging over one same plane, this will be impossible. For if you give it great contact, the parts of the support on which it rests are many in number, and each is little charged; and if the contact of that weight is reduced to a smaller number of supports, each part is the more burdened. For example, I have a heavy body which is 4 parts on one side and 8 on the other and weighs 8 pounds. I say that when it rests on its 4 parts it touches 4 parts of its support, and, that weight being, as said, 8 pounds, each of the 4 parts of the support is burdened with 2 pounds of weight. And if that weight is set upon its 8 parts, it occupies 8 parts of the support too; whence of the eight pounds of the said weight one will fall to each part of the support. So we shall say that the labor of dragging 8 upon 8 of resistance is as much as of 8 upon 8; for if the second was of 4 parts of contact, and each was well burdened with 2 pounds, that comes again to 8, as the first.

Of the screw.

Now I find the weight moved by force of the screw; that screw does not move it save by way of friction. And, drawing one same weight, it is as much to the mover that the screw touch 8 threads of its nut as that it touch 4, by the reason stated above, and as much a hundred as one, provided the screw stays loose, so that the friction of itself does not render its motion laborious. Again, the pivots that turn charged with the weight of themselves will do the same office named. But when the cause of its revolution is more remote from it, that revolution will be easier, and conversely, being nearer. The power of the screw is measured by proportioning together the lever with its counter-lever. And the lever is understood from the center of the hand, or force that moves the screw, to the center of that screw; and the counter-lever extends from that center of the screw to the middle of its opposite thread, at which end of the counter-lever you will reckon, against its counter-lever, not the weight drawn by the screw but the friction that the threads make which bear that weight. And if the weight is dragged, you will have two frictions, that is, of teeth and of weight.

Of the power of friction.

m c r n h, S t, f

I wish to push the weight m out of its place with the instrument n, by means of the weight f, which is the cause of the motion. I say then that the motion of m will be made by double or even triple friction. And the cause is that the weight of itself will not move without friction made of itself together with the plane that supports it. The second friction will be that which that weight makes with the instrument that pushes it, which follows the motion of its counterweight. Yet another friction I will add, which will be that which the pivot of the pushing instrument makes with its support. It being needful to know the nature of the motion, and how one must measure and understand the qualities and quantities of the powers that overcome and of those overcome by them, it is necessary to know the power in itself of the weight, or quantity of the weight. Besides this, by how much the power of its friction is less than that of the weight. Third, you must understand the qualities of the surfaces that touch one another, that is whether they be rough or smooth, rare or dense, evenly or not, and what difference there is from one to the other. And I make you understand that if they be equally rough, the weight cannot be pushed or dragged without great labor. But if the one be rough and the other dense and smooth, their motion will be as if both were smooth. The hard and the soft will be of a middling difficulty between the two named. Turn the page.