ON THE PRINCIPLES OF FORTIFICATION.

Figure: A

FIRST PROPOSITION.

Let there be given the length of line AB, of three score and twelve feet original: "trois vingts & douze pieds." This equals 72 feet. Early modern French often used a base-20 counting system (vigesimal), much like the English "three score.", draw a sufficient line CD, and divide it in such a manner: first, take with the compass a very small part, and place it ten times upon the line, proceeding from D toward C; then take those ten parts together, and place such a distance as many times as you see fit, approaching nearer and nearer to the extremity C; this being done, you must count as many parts as the line AB ought to have, that is to say, seventy-two, which make EF; upon the Base EF, a Triangle shall be made having its sides all entirely equal to the Base An equilateral triangle.; such a triangle shall be EGF; close the compass to the length of AB, and with such a distance mark GH and GI; draw a line through H and I; and from each point previously marked on CD, let a line be drawn, aimed toward point G; thus the line drawn through H and I will be divided by these latter lines, and KL will be the scale made from the length AB.

Three geometrical diagrams labeled No. 1, No. 2, and No. 3. No. 1 shows a large triangle EGF with a horizontal line HI intersecting it, and a series of converging lines from a base line CD meeting at apex G, used to create a scale KL. No. 2 shows a similar construction with multiple parallel horizontal lines and converging lines to a point. No. 3 shows a complex diagram with multiple horizontal lines labeled with letters A, B, C, and D, and vertical/diagonal lines intersecting them to create a grid-like scale. These diagrams illustrate the "Sector" principle—a way to use proportional triangles to shrink or expand measurements accurately.

In the same manner, the scale shall be made, given AB of seven hundred and twenty feet. But then the small parts of KL will mark ten feet, the others one hundred feet; always taking good care that the parts of line CD be larger than those of KL. Goldman is explaining that the "source" line CD must be larger than the "target" scale KL for the geometry to work correctly, as the lines must converge toward the apex G.