Here it is shown how the proportion of the upright with its hypotenuse is the same as that which the weight of the upright has with the weight that upright imparts to its two hypotenuses.

I say that the upright h c is 3/4 of one of its hypotenuses c f. Therefore the weight m is 3/4 of the weight the two uprights feel.

What is said is proved. Because from the line b c a quarter is cut off in b h, and 3/4 remain in h c. It follows that these 3/4 stand in the same proportion to their whole b c as the upright stood to its hypotenuse. Therefore the hypotenuse is like the whole perpendicular b c. But what has been said so far does not conclude. It is proved by the balance f h and f e, which has 2 of counter-lever in f h and 3 of lever in f e: it follows that 3 at h can withstand 2 at e, and so the weights are in the same proportion as were the arms of the balance between themselves, that is 2/3. Now because at m there is 4, and in this second demonstration we have set it as 3 against the 2 of the pendant c i; now because at m there is 3 and at i 2, and that 2 is 2/3 of 3, and answers to the proportion of the arms of the said balance; therefore at i there is 2/3 of the weight m, which is 3. In this proof the weight that falls to a single hypotenuse has been made, that is 2, and as much is loaded on the opposite hypotenuse, making 4; and so we have made the weight m, which is 3, be 3/4 of the weight the two hypotenuses feel; and it agrees with the first, that the upright was 3/4 of one of its hypotenuses.

The semicircle f h c is drawn upon the hypotenuse f c only to find within it the right angle f e c, which is the rectangular junction the lever f e makes with its pendant e i.

I suspend the weight b, which is 2, attached to the cord h K. What counterweight must there be at a, c to sustain it in the position of equilibrium, the balances being equal and their arms equal to one another?

The second figure here is of the same substance as the first one above.

Here the weight 4 is examined, in what way it varies in its supports f, g, h placed in various positions, their junctions with the rod being always right-angled.

I wish to move a heavy body suspended in the air along different lines, and I seek to know what power its mover must use for each of the given lines.

As many as are the varieties of weight that one same heavy body gives of itself to its mover, so many are the varieties of obliquity from which the mover moves that weight.

This suspended heavy body is so conditioned that it is all in b by similitude of power, so that the side of the rectangle f a feels no resistance at the beginning of the motion it makes to move a from its position. The mover e feels such proportion with its remainder as the lever g b has with the counter-lever b a; it is 13/14, that is, the lesser lever being 13 and the greater 14, a being 13 libre and g requiring 14 of power. Next the mover d n, at right angles to the lever n b, which lever is 10/14 of the counter-lever b a, so that 14 at d resists 10 at a; then the mover c m of the lever m b, which is 5/14 of the counter-lever b c, so that 14 at c is against 5 at b.

This calculated figure is like heavy bodies dragged over different obliquities, and nothing else varies except the power of the friction of the weight, because in the air no friction is given.