A very short way to measure a distance. Let the distance be b n, in which at b stands the eye that measures the distance, and at n a mark is noted whose distance from the eye is to be judged. Now, to make such a 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 stand upright. Then I will move a space from b to a, which may be much or little, as you please; but let us say that from a to b is 4 braccia; and once I am at a, I will do the same with another staff set perpendicular 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, and let us 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 away from the base, is narrower than the base by one braccio, so that 29 remains. Now say: if the base of the triangle, which is 30, is narrowed by another braccio in the space of one braccio, at what distance will it narrow away the whole 30 braccia? You will say: 30 times 1 makes 30. Divide that 30 into units, and 30 units will result; and thus you will find that the base of 30, at 30 braccia of distance, will have its pyramid consumed. Again, if a b were 4 braccia and it narrows by 1/8 of a braccio in the space of one braccio at c d, 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. So you will say: 4 times 8 makes 32; and you will say the length of the pyramid to be 32 braccia. And just as you move crosswise from here to there, so you can measure from top to bottom by altitude in the same way I told you above; and it is shorter, because you need not shift your position.

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 arm of a compass; let c d be marked in degrees, points and minutes. Then, to measure a distance, place the eye opposite the angle a, somewhat back, and aim the line a c at the place whose distance you wish to know, which we will say is at e. Then put 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 have to multiply the 4 braccia of d b by a hundred, and at 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.