First.

The joining of the lever with its counter-lever will always be at right angles.

In the calculations of mechanical powers no account is taken of the weight or the position of their material members, but only of the potential, that is mathematical, lines.

I wish to know what weight the heavy body gives of itself to its two supports, namely m q and n b. I will draw out the two potential levers b c, q r, which will be at a right-angle joining with b n, m q at the points b, q. Then I will draw the straight line b q from b to q; and in the middle of these the perpendicular f n will descend beneath the centre of the heavy body a, and thus I will have made the right angle b f n at the point f. Besides this I will draw the appendage c d at a right-angle joining to the lever b c, and thus I will have made the lever b c with the appendage c d, against the potential counter-lever b f with its appendage f n. And if the adversary should say that this counter-lever is not potential but real, that is the counter-lever b n, here it is answered by the 10th of this [book], where it says that all appendages directed toward the centre of the world are at a right-angle joining with the arms of balances set in the position of equality; and by the first above, every joining of an appendage to the arm of the lever is at right angles, which cannot be, for the adversary, in the joining of b n with the appendage f n. Therefore the right angle b f n serves our purpose, etc.

Here one calculates the mathematical powers, and not the real weights, that is the weights of the members of the instruments.

Calculation.

Whatever proportion the potential lever b c has to the real counter-lever b f, such will the power of the angle c have to the resistance of the angle f. Therefore, the powers being inverse, b c, which is half of b n, will require double power to support b n.

The power figured here below is similar to the one above. But this one gives account of the weight that the cord a b c of the heavy body s sustains, that is of the natural weight, called gravity, and of the accidental weight, called force; and it teaches how to divide the gravity from the force. And this operation is carried out by those cords a b c below the line of equality a d c. But above, account is given of the weight and force that the heavy body a, or if you will n, gives of itself, above the line of equality b f q, by the two beams b n, m q that support the said heavy bodies a, n.

The lever and the potential appendage, a e and e b, will always be found in a right-angle joining at the point e, in the same alignment as the real appendage b c; that is, continuing the real cord c b in a straight line, it will meet the potential lever a e at the point e in a right-angle joining, and the potential counter-lever of the lever a e will be found on the line a d, terminated by the potential appendage d b above the real appendage b s.

Calculation.

Whatever proportion the lever a e has with the counter-lever a d, such here does the force have with the weight; but it is inverse, because the counter-lever is double the lever, etc. ...

Therefore, the calculation of half of the weight s will be found in the potential lever and counter-lever a e and a d, of which the appendages e c and d s are semi-real, as is seen here.

Here 4 of counterweight upon the pulley n will move 4 of weight on the pulley m, the axles of such pulleys being of equal thickness and the wheels of various diameters but of equal thickness.

Here the real cord m e ends in the potential cord b c, which terminates in the potential arm a b, and the real arm a n is transformed into the potential arm a d.

Let there be given the centre of gravity that the weight c has, which is 1 upon the rod a b and is suspended by the cords a d on one side, and f e, c b on the other.