Where the real appendage a b cuts, with its real or potential alignment, the beam of the balance between right angles, there is generated a space between that cut and the centre of the balance, which is the true length of that arm of such a balance. Therefore the semi-real appendage c b, cutting the arm of the balance d f at the point c, makes that arm f d remain at the length c d, and thus I intend to have defined the other arms of such a balance.

a e, e f are real together with the appendage e c; the arm of resistance is potential, namely a b, and the arm of power a d is also potential. But the arm of resistance is fourfold the arm of power a d; for which reason one pound placed at b, the end of the arm a b, resists 4 pounds placed at the end of the arm a d moved by the appendage d f, etc.

Here, as has been said, one calculates with mathematical lines and not with the weight of the members of the instrument.

Here all the lines are potential, and the 2 beams n a, n m with the appendage n c are real, along with the weight c.

Wishing to know what gravity the weight c, joined to the cord n c, gives of itself to that cord, and moreover with what gravity that weight loads itself upon the bases of those beams n a, n m, we will have recourse to the right-angle joining created by the arms of the balance with their appendages, and with the help of such an arm we will give true knowledge of these powers.

Here we have true notice how the weight c, which is 8 pounds, loads itself with 4 pounds upon the base of the beam n m, and with another 4 pounds upon the base of the beam n a, which are 8 pounds in all. And this is natural weight; and besides this, those bases receive a quantity of accidental weight, which weight by another name is called force, and which gains so much the more power as the upper joining of the beams is made with a more obtuse angle.

The balance a b, a o of the weight that the base of the beam n a receives has its arms in double proportion, of which arms a o is the lever and a b is its counter-lever. Whence 8 pounds, placed at the appendage o p of the half-length arm a o, resist the descent of the double-length arm a b with half of the weight 8, that is with 4 pounds; at the end of the double arm one resists the power of eight pounds placed on the half-length arm a o. It follows that here there arise 8 pounds of weight and 8 pounds of force, besides the natural weight of the instrument.

Four are the appendages that support the weight m, namely those fixed to the arms of the balance, of which f g, f b are on one side, d n, d c on the other, and these are real; and of these there are 2 semi-real, namely d n c, n c; and the potential part e d is the real part of d c; o b is the potential part of f b. But of the arms of the balance there are 2 real and 3 potential: the real ones are a g, a b; the potential ones are a b, a c, a e; and this is the character of this machine, of which in its place its power and strength will be fully defined.

The power of the real appendages joined to the opposite arms of the balance by an oblique position are not the true arms of such a balance, because they are of a simple denomination, that is a real and not potential position. But the true denomination is composed of real and potential, which happens when the joining of the real appendages with the arm of the balance is at right angles; and this not being at right angles, it is necessary to carry out the calculation of the weights placed on the arms of such balances with the potential right angles, there being no real right angles.

Therefore, from what has been declared above, we find, in the balance figured here, that the joining of the five appendages is not at right angles with the arms of its balance, that is it is an acute joining, as the appendages e m, f m, n g, h i, K i show. For which reason, wishing the true calculation made with the true appendages, it is necessary to make use of the potential appendages joined at right angles to the potential arms of the aforesaid balance, which are a, b above the balance, c, d below that balance, which are right angles composed of the true potential arms and the true potential appendages. There remains for us one right angle beneath the pole of the balance, which is called a semi-real angle because it is composed of a real appendage and a potential arm.

Having given the definition and names of the arms and appendages of the balance, it is necessary to give their calculation to conclude their true powers; for which reason we will place the right and left potential arms around the pole of the balance by means of the rotation made above, and in the same proportions we will divide the weight h and give to each appendage its part.

a b, b c, b f are real powers; a e is the potential arm to which the right-angle joining is made at the point e with the real appendage e f; therefore whatever proportion a e has with a d, such does the motive power d c have with the power that resists e f.

a d, the lever, stands as 4 against 3 of a e the counter-lever. Therefore the appendage d c lifts 4 at the appendage e f.