Sheet 40 verso.

The difficulty of moving this wheel while the axle stays fixed will be the same as that of turning the axle within the wheel while the wheel stays still.

I find 2 pounds at a from the weight of the roller a, and I find another 2 pounds at b from weight b, which make 4; and by the 5th proposition of the 1st book these have as friction one quarter of that weight, that is 1 pound, which alone bears upon the friction n.

Since I have one pound of frictional force at r and 2 of weight at s, which joined together make 3 pounds, it is necessary that on an equal arm of the balance they be resisted at t by another 3 pounds.

Having given the counterweight to c r, which is 3 pounds, with another 3 pounds at an equal distance from their centre, it is necessary that these 3 pounds yield one quarter of their weight on the side of the friction, which will be 3/4.

Now I find I have added 3/4 of resistance in the roller, arising from the 3 pounds of the added counterweight; and for these 3/4 to be matched by an equal force of weight, you must take a weight whose quarter is equal, that is, whose friction, added to those 3/4 of other friction, makes the force of that friction equal to the force of that weight.

Sheet 41 recto.

The friction of 2 will be 1/4; divide this by 3, which gives 1/12, and clothe this twelfth with its body, that is multiply it by 4, giving 4/12 against 1/4, that is 3/12. To this 3/12 of friction add the quarter of 4/12 for its friction, which is 1/12, and you will have 4/12 of friction against 4/12 of weight; and thus the force is balanced with the resistance.

The frictions of weights are of as many various resistances as there are various bodies that rub together; and if the surfaces of the bodies that rub together are dense and of resistant flatness, then a large contact will be of resistance equal to that of a small one.

Circular friction will be equal to straight-line friction.