Rule of the angular balance.

Tested (proven by experiment).

If the arms of the balance are unequal, and their junction at the fulcrum is angular, then if their ends are equally distant from the central line of the fulcrum, the weights hung on them, being equal, will weigh equally.

By 'arm' is meant the place where the centre of the weight hung on it falls.

Here the portions of the circles are not equal in the motion of the arms, but they are equal in the distances from the central line.

The angular balance is that of which the junction of its straight arms is angular; in which angle its fulcrum is placed.

Always the distances that the opposite ends of the angular balance have from the central line of its fulcrum will be in the same proportion as that which the lengths of the arms of the balance have among themselves; but it will be the inverse proportion.

As, for example, of the angular balance c e f, of which the fulcrum is at the angle e: the opposite ends f and c have, in their distances from the central line a b, such a proportion as that of the length of their arms e c and e f; but it is inverse: that is, that the lesser arm has its end so much farther from the central line as it is smaller than the greater. And likewise the space that the greater arm has from that central line is so much smaller as its arm is greater than its lesser.

The line a b does not tilt when equal weights are hung on its arms c, m.

Rule for proceeding in the manner of placing the distances of the ends of balances, of whatever kind of angle it may be, from its central line. And the distinction of the balances is described here below: and first of the balances of obtuse angle, beginning with the rectilinear balance, where the greater arm o d is shown to be double the lesser arm o e.

d o e a o i

c o f b o k

b o g obtuse angles c o L acute angles

a o h d o m

d o n

The proportions that the distances of the centres of gravity of the arms of the above-written balance have from the central line of the fulcrum will be the same as the distance of the ends of their arms and that of the lengths of those arms.

See the end i of the arm o i, which is twice as far from the central line e d as is the end a of the arm o a; and this happens because o a is double o i: and so the arm o K follows against the arm b d, as is shown at S, where the perpendicular h K is cut. And so follows q o, the space arising from the perpendicular L, which is double b d. And so it follows step by step to infinity, always with the same rule.

It is not to be blamed to graft into the ordered process of the science any general rule born of the aforesaid conclusion.

n, the centre of gravity of the arm e f, is twice as far from the central line a b as is g, the centre of gravity of the arm e c. And what the centre of gravity of these arms does is done by their ends, their thirds and their quarters, and thus by every other corresponding part.

If the pyramid f n e is a third of its cylinder, and that cylinder is 18, that pyramid comes to 6, and the cylinder a f is half of the said cylinder, which is 9; and its closing part is 3, which multiplied by 3 remakes the weight of its cylinder, which makes 9. Therefore we have 6 beyond the fulcrum and 9 on this side, that is, in the cylinder. Whence the scales are not even.

Therefore the cylinder a f must be 6, like the pyramid set against it, which arose from a cylinder that was 18; of which 18, 6 is the third. Whence you will make the cylinder a f one third of the cylinder f e in length; and thus you will have made the sums equal about the fulcrum, because a f is 1/3 of the greater cylinder, and the pyramid f n e is also itself 1/3. And so they remain even.

a is the centre of the natural gravity.

n is the centre of the accidental gravity, because you have 5 against 3.

Of the whole pyramid a b, suspended at the quarter, m, of its length toward the base, one may use the space of the half a n as a perfect balance.

That this is so is proved. m is the centre of the accidental gravity of the whole pyramid, whence the weight hung at n will be equal to the weight hung at a, just as their distances from the fulcrum are.

The arms of a balance that by themselves resist the descent of one another, even though they be various in weight, length, figure and thickness, resist equally the equal weights attached to them in those parts that are equally distant from its fulcrum.

bread 2 s. wine 6

meat 5

fruit 2 s.

cloth 10 s.

debt 16.