Operations of the Geometric and Military Compass

and having found it is, for example, 89, take the distance 89 directly with the compass, from which, having fixed one leg in G, you will mark with the other the intersection of the arc R I Q with the first arc O I N, made at point I, to which the line H I must be directed. And angle H will undoubtedly be equal to angle B, and the line H I proportional to B C. With this order, the other points K L M corresponding to angle D E F will be found.

The Rule of Three solved by means of the Compass and the same Arithmetic Lines. Operation IV.

A large decorative initial 'S' begins the text.

The present lines serve us not only for the resolution of various linear problems, but also for some rules of Arithmetic, among which we will place this one, which corresponds to that in which Euclid teaches us: given three numbers, find the fourth proportional. For the Golden Rule, which practitioners call the rule of three, is nothing other than finding the fourth number proportional to the three proposed. Therefore, demonstrating the whole with an example for clearer understanding, let us say:

If 80 gives us 120, what will 100 give us? You have, therefore, three numbers placed in this order: 80, 120, 100. And to find the fourth number that we are looking for, take the second of the proposed numbers directly upon the Instrument—that is, 120—and apply it transversally to the first—that is, to 80. Then, take the third number—that is, 100—transversally, and measure it directly upon the scale; and that which you will find, which is 150, will be the fourth number sought. And note that the same would happen if, instead of taking the second