The Operations of the Geometric and Military Compass

number, one took the third, and then in place of the third, one took the second; that is, the same will give us the second number taken directly and applied transversally to the first, taking then the third transversally and measuring it directly, which would give us the third taken directly and applied transversally to the first, then taking the second transversally and measuring it directly. In one way and the other, we will find 150. It is good to have noted this because, according to different occasions, this or that way of operating will be more convenient for us.

Regarding the operation of this Rule of Three, some cases may occur that could produce some difficulty if one is not careful. We demonstrate next how one must proceed in them. And first, it could sometimes happen that of the three proposed numbers, neither the second nor the third taken directly could be applied transversally to the first. As if one were to say: 25 gives me 60, what will 75 give? Here, both 60 and 75 exceed the double of the first, that is, of 25, so that neither one nor the other can be taken directly and applied transversally to that 25. Therefore, to achieve our intent, we will take either the second or the third directly and we will apply it transversally to the double of the first, that is, to 50 (and when the double is not enough, we will apply it to the triple, to the quadruple, etc.). Then, taking the other transversally, we will affirm that that which it shows us when measured directly will be the half (or the third, or fourth part) of that which we seek. And so, in the proposed example, 60 taken directly appli-