On the proof, whether it is well calculated.

4, b, a, 15 — c, 1 and 1/15, d — 4/15, f. It amounts to letting stand the one that you took from the 16 of the a and placed in d. Divide the friction by the number into which the arm of the counterweight is divided, and if the result is equal in power to the counterweight of that friction, the operation has been good. Divide the friction of the whole weight by that number into which the arm of the counterweight was divided, and if the result stands in the same proportion to the said friction as the arm of the friction has to the arm of the counterweight, that operation stands well. Let the said friction of the weight be f d e, which is 16/15, divided by the arm of the counterweight a b, which is 4; and if the result, which is 4/15, is subquadruple to the 16/15 of the d e, as the arm of the friction b c e is to the arm of the counterweight b a, which is subquadruple to the greater arm, then such calculation stands well. Again, that friction always has to be the quarter of the rubbed weight, as you see the friction d e, which is 16/15, and is a quarter of the 64/15 at b f. Here the 15 of the a, the arm of the counterweight, is not established as being 15, for it could be 15 whole, or 15 halves or thirds, or 15 times 2/3, or another fraction; but as soon as it is made divisor of the friction of the pole, its units immediately acquire their denomination, that is 15/15, and this number 15 serves for nothing else than to make the aforesaid denomination. The one that you took from the 16, which remained 15 in a, is placed in d and there takes the same denomination of fifteenths that the 15 took, the remainder of that unit; whence that one is now named 1/15. The one that you took from the 15, which, after it divided the friction of the pole, was named 15/15, this one is named 1/15 and is the friction of the counterweight. And this 1/15, joined with the friction of the pole, has the name 16/15, as when it was joined with 15/15 whence it was taken. Always the unit taken from the 15 will unite with another 15 of friction; and so, departing from the 16, it left 15, and joining to the 15 of the friction it made it become 16. And this rule of undoing the number of the divisions of the counterweight's arm, whence one departs, and remaking a like number to that whence one departs by joining to the friction of the pole, this rule will hold in all weights, fractions as well as whole numbers.

12; 3, a — 3, b; 1 — 4. If here the one that was joined to the 3 of the a joins to the 3 of the b, it will make the 4 that was in a transmute and come into b; and so it does in all the numbers.