Elements of machines.

Conclusion concerning the weight attached at the angle of the cord, and how it distributes itself to each side of that angle.

It is impossible that any power could ever straighten a cord stretched crosswise, and it will be all the more impossible when it has some weight attached at the middle of its length.

It is proved. Let d n b be the cord stretched crosswise, and e the weight attached to it. Here, by the 7th of the equilibrium, I will make the potential balance, of which b a and b c will be its arms, and c d and a e are its semi-real pendants. I say therefore that, wishing to straighten that cord d n b, its angle n always grows, the lever b c diminishes, and the counter-lever b a becomes greater. And because that lever b c is divisible to infinity, it would therefore be necessary to increase the weight infinitely in order to diminish it. Which weight would have to be increased infinitely, if the cord and its supports did not break.

The potential lever and counter-lever (calculator of the weight sustained by the angle of the cord) will never be double one another, because when they are at the position of their doubling they are immediately consumed. And here ends the company of natural weight with accidental, together with the smallest of all acute angles.

If it were possible to straighten the cord stretched crosswise and to undo the last, greatest of the obtuse angles created by the weight that is sustained at the middle of that cord, then the power of that cord would be infinite in sustaining a small weight, and the lever and counter-lever would be consumed. Which is impossible, because in such a disposition of the cord no power is without a lever.

No movement made by rod or cord, wheel or screw, will be generated without the power of a lever and counter-lever, real or potential.

Here, the lever a c being double the counter-lever c b, the cords n a and m c sustain the whole natural weight e f by means of the potential pendant b e, which falls upon the centre e, the centre of that weight.

Here it is proved why the weight felt by the 2 hypotenuses, from the weight attached at their angle, has such proportion as one of those hypotenuses has with its upright.

The cause why the hypotenuses make their upright account for the weight attached at the angle is that the line f c is equal to that hypotenuse by the definition of the circle; and one could say that the same proportion the part b c has with the whole f c, the weight felt by the hypotenuses has with the natural weight attached at the angle where those hypotenuses join.

But because at the upper junction of the 2 hypotenuses at the point f the accidental weight is finished, only the natural weight remains; and when the ends of those hypotenuses begin to separate and move by circular motion, then they begin to lower, and then the accidental weight is born, which is called force, and which, although it does not weigh, does the office of weight, and gains as much magnitude as the ends of the hypotenuses descend; and the measure of the descent is found in the perpendicular line f c; and therefore this line is measured, imagined to be cut by a straight line extending from one end of the upright to the other end of the other upright. And thus with such a line a triangle is finished, which with its base cuts the line f g; and what remains within the body of that line is called the upright of that triangle, and is the measure of how much the fronts of the two hypotenuses have been lowered. And therefore there the weight remaining to the said hypotenuses is measured, which has such proportion with the natural weight of c as that upright has with the whole upright f c, or, if you will, one of the hypotenuses, which is the same.