Rule of the angular balance.

Tested.

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

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

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

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

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

As may be said of the angular balance c e f, whose pole is at the angle e: the opposite ends f and c have, in their distances from the central line a b, such proportion as that of the length of their arms e c and e f; but it is inverse: that is, the smaller arm has its end as much further from the central line as it is smaller than its larger one. And so the space which the larger arm has from that central line is as much smaller as its arm is larger than its smaller one.

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

Rule for proceeding in the way of placing the distances of the ends of the balances, of whatever kind of angle they may be, from their central line. And the distinction of the balances is described here below: and first the balances of obtuse angle, beginning with the rectilinear balance, where the larger arm o d is shown to be double the smaller 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 balance written above have from the central line of the pole of the balance will be the same as the distance of the ends of their arms and that of the lengths of these arms.

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

It is not to be blamed to graft into the order of the progress of the science some 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 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, and their thirds and their fourths, and so by every other corresponding part.

If the pyramid f n e is one third of its cylinder, and that cylinder is 18, the pyramid comes to 6, and the cylinder a f is half of the said cylinder, which is 9; and its "seratile" is 3, which multiplied by 3 remakes the weight of its cylinder, which makes 9. Therefore we have beyond the pole 6 and on this side 9, that is in the cylinder. Hence the balances are not equal.

Therefore it is necessary that the cylinder a f be 6, like the pyramid set against it, which was born of a cylinder that was 18; of which 18, 6 is the third. So you will make the cylinder a f the third of the cylinder f e in length; and thus you will have made the sums equal about the pole, because a f is 1/3 of the greater cylinder, and the pyramid f n e likewise is 1/3 of it. And so they remain equal.

a is the centre of natural gravity.

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

Of the whole pyramid a b suspended at the fourth, 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 pole are.

The arms of the balance that of themselves resist the descent of one another, even though they be various in weight, length, shape and thickness — those of their parts that are equally distant from its pole will equally resist the equal weights attached to them.

Bread 2 s. Wine 6

Meat 5

Fruit 2 s.

Cloth 10 s.

Debt 16.