a, 3 12; b, 6 3, 4, 3; c, 3 1, 3; d, 2 3; e, 2, 2, 3

General rule. If you know the weight first placed on one of the arms of the balance, and the degrees of the arm that sustains it, and likewise you know the second arm, and you want, from these three pieces of knowledge, the knowledge of the second weight, do thus: multiply the degrees of the first weight by those of the arm that sustains it, and divide by the degrees of the second arm. And what comes of it will be the number of the degrees of the weight that was unknown. And if you know the first weight and likewise the arm that sustains it, and also the degrees of the third weight, and you want the knowledge of the degrees of the arm that sustains it, do thus: as before, multiply the degrees of the first weight by those of the arm that sustains it, and divide by the degrees of the third weight. And what comes of it will be the degrees of the second arm, which was unknown to you; and do likewise for the other arm.

If you have hung a weight on one of the arms of the balance, and you wish to place on the opposite arm, at whatever point you please, enough weight to make resistance to the descent of the first weight, do thus: multiply the degrees of the first weight by the degrees of the arm that sustains it, and divide the resulting sum by the degrees of the opposite arm. And what comes of it will be the number of the second degrees of the weight that will resist the aforesaid first weight. And know that 'arm' is understood from the centre of the pole of the balance to the middle of the contact that the cord which sustains the weight makes with the balance. But if, against the weight placed on one of the arms of the balance, you wished to make resistance on the opposite arm with 2 different weights, placed as you please, and to know what these weights must be, do thus: divide the weight that stands alone on the first arm; divide it into 2 parts in your mind, and by the rule given above give it such weight at one of the given sites on the opposite arm that it makes resistance. And do the like with the 2 second weights that remain, and it will be a good rule. This second rule gives you, against a known weight, two different unknown weights in two known sites. This third will give you two known weights, against the first (also known), in two different unknown sites. And do thus: divide likewise the first weight, which stands by itself on one of the arms, in the way you did before, and go proportioning them together with the weights placed on the opposite arm. That is, if you have, for the first weight divided into 3 and 2, placed at three degrees of its arm, you will give to the 3... I hang a weight of 2 degrees of gravity on an arm of the balance which is 2 degrees of length and stands on this side of the pole. And I also have 3 degrees of weight, which weight I should like to attach to the other arm of the balance, on the far side of the pole. But I do not know, at present, how long that arm must be so that precisely the weight of 3 degrees, attached at its end, makes resistance to the descent of the two degrees of weight. But to find such length, I shall make that arm have such proportion to the first known arm as the first weight has to the second, which stands in subsesquialtera proportion, and it will make the said resistance. And to do this, I shall divide the said first arm by the 3 of the second weight, and I shall say: 2 times 3 makes 6, which 6 are all thirds. Then I shall put the two and the 3 of the degrees of weight one above the other, and it will stand thus 2/3, which means the 2/3 of 6, which is 4/3. And thus 4 thirds of degrees of arm length will hold the 3 of the degrees of weight.