Various Questions and Inventions

Having therefore supposed the above-stated suppositions, I bring forth this proposition, and I say that every balanced weight departing from the site or place of equality becomes lighter, and so much more so, the further it is from the said site of equality. And for an example of this proposition, let there be the balance a.b. (of the preceding figure) rotatable upon the said center c. with the two same (equal) bodies a. and b. hung or joined to the two extremities of both arms of the said balance, and let them stand in the same site of equality (as was supposed above). Now I say that moving one and the other of said bodies from the said site of equality (that is, lowering one and elevating the other), one and the other of those will be made lighter according to the place, and so much lighter the more they are removed from the said site of equality. And to demonstrate this, let body a. (of the said preceding figure) be lowered as far as point u. (as appears in the figure written below), and its other opposite (that is, body b.) will come to be elevated as far as point i. And let one and the other of the two arcs a.u. and i.b. be divided into as many equal parts as one wishes. Now let us place one and the other into three equal parts at the points l., n., and q., s. And from the three points n., l., i. let there be drawn the three lines n.o., l.m., and i.k. equidistant to the diameter b.a., which will cut the line e.f. of the direction at the three points z., y., x. Similarly, from the three points q., s., u. let there be drawn the three lines q.p., s.r., and u.t., also equidistant to the same line a.b., which will cut the same line of direction at the three points v., p., z. Whence, by these things so disposed, we will come to have divided the whole descent a.u. made by the said body a. in descending to point u. into three descents or equal parts, which are a.q., q.s., and s.u. And similarly, the whole descent i.b., which the said body b. would make in descending or returning to its first place (that is, to point b.), will come to be divided into three descents or three equal parts, which are i.l., l.n., and n.b. And each of these three and three parts of the descents contains a part of the line of direction; that is, the descent from a. to q. takes or contains from the line of direction the part c.v., and the descent q.s. takes or contains the part v.p., and the descent s.u. contains the part p.z. And because the part c.v. is greater than the part v.p. (as can be easily proven geometrically), therefore (by the second supposition) the descent q.s. will come to be more oblique than the descent a.q. Whence the said body a. will be lighter (by the supposition) standing at point q. than it would be standing at point a. Similarly, because the part v.p. (of the line of direction) is less than the part v.p., the descent s.u. (by the same second supposition) will be more oblique than the descent q.s. and consequently (by the first supposition) the said body a. will be lighter standing at point s. than it would be standing at point q. And all this, and by the same methods, will be demonstrated on the opposite part of body b.; that is, that its descent from point i. to point l. is more oblique than that which is from point l. to point n. (by the said second supposition) because the part x.y. which it contains of the line of direction is less than the part y.z. Whence, by the said first supposition, the said body will be lighter standing at point i. than it would be standing at point l., and by the same reasons, it will be lighter standing at point l. than it would be standing at point n., and similarly, it will be lighter standing at point n. than it would be standing at point b. (site of equality), which is what was proposed.