Operations of the Geometric and Military Compass

catchword: cato

take the third number, and then in place of the third, take the second; that is, the same result will be given to us by the second number taken directly and applied to the first transversally, then taking 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, such that in either way we will find 150. And it is well to have noted this, because according to the different occasions, this or that mode of operating will be more convenient for us.

Regarding the operation of this rule of three, some cases may occur that could cause some difficulty if one were not aware, and we shall demonstrate next how one should proceed in them. And first, it could sometimes occur 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 us 60, what will 75 give? Where both 60 and 75 exceed double the first, that is 25, so that neither one nor the other can be applied transversally to that 25 when taken directly. Therefore, to achieve our intent, we will take either the second or the third directly and apply it transversally to the double of the first, that is, to 50 (and when it is not enough to take the double, we will apply it to the triple, to the quadruple, etc.). Then, taking the other transversally, we will affirm that what it shows us when measured directly will be the half (or the third, or the fourth part) of that which we seek. And thus, in the proposed example, 60 taken directly and applied

to the double of 25, that is to 50 transversally, and immediately taking the 75, also transversally, and measuring this directly, we will find that it gives us 90, whose double, that is 180, is the fourth number that was sought.

It could furthermore occur that if the second or the third of the proposed numbers could not be applied to the first because this first is too large, so that it exceeds the number marked upon the lines, which is 250, as if we were to say: 280 gives 130, what will 195 give me? In such a case, having taken 130 directly, it will be placed transversally to the half of 280, which is 140. Then one will take transversally the half of the third number, that is of 195, which is 97 and a half, and this distance measured directly will give us 90 and a half, which is that which was sought.

Another precaution will be useful to keep in mind for use when the second or third of the proposed numbers are very large, while the other two are mediocre, as when one might say: if 60 gives 390, what will 45 give me? Having therefore taken 45 directly, it will be applied transversally to 60, and not being able to take the 390 in its entirety, we will take it in pieces, as it pleases us; for example, I will take 90 transversally, which, measured directly, will give me 67 and a half, which I will note aside; then I will take 100 transversally, which measured directly will give me 75, and because within 390 there is one time 90 and three times 100, I will take three times the 75 found, and in addition the 67 and a half that was found by virtue of the 90, and this whole sum makes 292 and a half, for the fourth number that is sought.

Lastly, we will not refrain from saying how one can operate