The Operations of the Geometric and Military Compass

and having found it to be, for example, 89, one will take the distance 89 directly with the compass, of which, having fixed one leg in G, one will note with the other the intersection of the arc R I Q with the first arc O I N, made at the point I, to which the line H I must be directed; and it will without doubt make the angle H equal to the angle B, and the line H I proportional to the B C. And with such an order, the other points K L M, corresponding to the angle D E F, will be found.

The Rule of Three Solved with the means of the Compass, and of the same Arithmetical Lines. Oper. IV.

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 the Rule of Three, which practitioners ask for, is nothing other than finding the fourth number proportional to the three proposed. Therefore, demonstrating the whole with an example for a clearer understanding, let us say:

If 80 gives us 120, what will 100 give us? You have therefore three numbers placed in this order:

And to find the fourth number that we seek, take the second number of those proposed, that is 120, directly upon the Instrument, and apply it transversally to the first, that is to the 80. Then take the third number, that is 100, transversally, and measure it directly upon the scale; and that which you find, that is 150, will be the fourth number sought. And note that the same would occur if, instead of taking the second