The Operations of the Geometric and Military Compass

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

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

Another precaution will be good for us to use when the second or third of the proposed numbers are very large, while the other two are moderate, as when one says: if 60 gives me 390, what will 45 give? Having taken 45 directly, it will be applied transversally to 60. And since one cannot take the 390 in full, 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 on the side. I will then take 100 transversally, which, measured directly, will give me 75. And because in 390 there is one 90 and three 100s, I will take three times the 75 found, and additionally the 67 and a half which 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 fail to say how one can oper-