Take away the lesser arm and as much of the greater arm as touches that lesser one, and at the centre of what remains you will place the counterweight at the end of the lesser arm. Let the lesser arm be c b, from which you remove the counterweight c n. There remains n a, whose centre is m, which is the centre of two pounds. Now find a weight that is in proportion with the said two pounds of m. Then set m o with o b, and put the greater weight at the end of the lesser arm b, that is, attach the greater arm m o to the end of the lesser arm b; and the heaviness of the arms will be made equal.

When one makes the balance of a simple rod, attach the greater arm to the end of the lesser, and the powers of the weights will be made equal.

Battista Alberti says, in a work of his sent to the lord Malatesta of Rimini, that when the balance a b c has the arms b a and b c in double proportion, then the weights attached to it, which dispose it in such proportion, are also in the same proportion as the arms themselves; but it is inverse, that is, the greater weight on the lesser arm. To which experience answers him that it is false; but his proposition will succeed when to the lesser arm is added the length of the greater arm, as is shown above.

And if you wish to throw away the whole of b d, there remains to you S, the centre of gravity of the remainder, which here is 2, and it is 2 distant from the pole, while b is 1 distant. Now put 4 at b against 2 at S.

This is not the centre of accidental gravity, but of natural gravity, and therefore it is not a valid proof. Now you have that the centre of gravity of a c is three times more remote from the central line than the centre of c d.

If you wish to know how much n m weighs, put the point of the compass at c, and open the compass to the width of the space c d, and make the semicircle b c d; and the remainder a b will be equal to the weight n m. It is proved by the central line o p, which lies midway between equal weights, since the space of the rod a c is divided into 3 equal parts and lies beyond the central line; and on this side of that central line there are 3 other similar parts of rod, equal to those beyond the said central line: namely c d and n m. And so the weights on this side of the pole are equal to those beyond the pole. Which is what we wished to prove. There remains for us the fallacy of the centres of gravity, which are one more distant than the other from that central line. In c d is the lesser arm of the balance on this side of the central line, which resists with the centre of its gravity by 1/3 of a pound against the centre of gravity of the greater arm at g. There remains g, centre of gravity of 3, remaining at 2 1/2. Now I must hang at d such a weight as resists 2 1/2, and I have 3 in the greater arm of space and 2 in the lesser; and say thus: if in 3 of space I am to have 2 of weight against 3, in 2 of space beyond the pole and in place of the 2 ... Extract from the weight of the balance a b the weight of the lesser arm c d, which has all its weight at its centre; which weight, being 1, resists 1/3 at the centre of the greater arm, which weighed 3. There remains to it a power of 2 2/3. Now I have to find a weight which, hung at the end of the lesser arm, resists 2 2/3 at the centre of that greater arm; which will be 4. And I prove it thus, reducing all the weights to the fraction of the said third: of 2 2/3 they make 8/3, which is the weight left to the greater arm; and of the weight of the rod hung on the lesser arm, which was 4 whole, they make 12/3. Now you see that the proportion which 8 has to 12 is the same that 2 has to 3, for each lesser number is 2/3 of the greater. So that from c to d is 2/3 of the space from c to g. And thus the inverse proportion of the weights with the arms of their balance holds good.

Always throw away the weight of the lesser arm, and throw away as much weight from the centre of gravity of the greater arm; then put the weight you wish to weigh at the end of the lesser arm, which you counter against the weight left at the centre of the greater arm. And the balance will stand level.

A re-proof against Battista Alberti.