A very brief method of measuring a distance.

Let the distance be b n, in which at b stands the eye measuring that distance, and at n a mark is noted, whose distance from the eye is to be judged.

Now to make this measurement I will place myself with my feet at b, and have a staff set up between the eye and the object n, and this staff will be fixed upright. Then I will move a space from b to a, which may be much or little or as you please; but let us say that from a to b is 4 braccia; and once I have reached a, I will do the same with another staff interposed perpendicularly between the eye b and the object n. Having done this, I will observe the difference between the length of the line a b and the line c d, which we shall say that c d is 1/2 braccio shorter than a b, the base of the triangle a b n. If a b, the base of the pyramid a b n, is 30 braccia, the section of that triangle, made one braccio distant from that base, is narrower by one braccio than the base, so that 29 will remain. Now say: if the base of the triangle, which is 30, narrows by another braccio in the space of one braccio, in what distance will it narrow the whole 30 braccia? You will say: 30 times 1 makes 30. Divide that 30 by unity, and 30 units will come; and thus you will find that the base of the 30 will have its pyramid consumed in 30 braccia of distance.

Again, if a b were 4 braccia, and it narrows by 1/8 of a braccio in c d over the space of one braccio, always multiply the base by the part of the diminution, and the result will be the number of braccia in which those 4 braccia of the base will be consumed.

Therefore you will say: 4 times 8 makes 32; and 32 braccia you will say to be the length of the pyramid.

And just as you move from here to there transversally, so you may measure from up to down by altitude in the same way as I told you above; and it is briefer, because you do not have to move from your place.

Let a b c d be immovable in itself; let the angle a be a right angle; let b d be movable at d about the pole b in the manner of the leg of a compass; let c d be marked with degrees, points and minutes.

Then, to measure a distance, place the eye opposite the angle a, somewhat apart, and aim the line a c at the place whose distance you wish to know, which we shall say to be at e. Then place the eye at the angle b, and aim the side b d at the point of the distance e. Now, if you find that the width c d narrows for you by 1/100 of the width a b, you must multiply the 4 braccia of d b by a hundred, and in a hundred braccia the lines of the sides of the triangle will meet at e. But it is better to make it large on the ground and walk along its sides.