An f-number is determined as the ratio of the focal length to the diameter of the entrance pupil. But the actual aperture diaphragm is usually not a perfect circle. Its shape is determined by the number of diaphragm blades and their edge (curved or straight). Depending on those the aperture might be a pentagon, hexagon, octagon... and with straight or curved edges. So how is the diameter actually determined? Is it for example twice the distance between the center and the widest point at an intersection of the blades? Twice the distance between the center and the nearest point on an edge?
If I had to figure out the answer to this problem, I would start with the open aperture which will be circular as the blades of the iris are retracted.
That would be the f/wide-open. I'd take the diameter of that aperture and divide the focal length by it to get the first "stop." Let's say it's an f/4 lens. Scratch marks the spot opposite the focussing "index."
Close the aperture until the light intensity that passes through the lens is half of that (using a light intensity measuring device of some sort). The engraved mark on the lens aperture would be f/5.6 as each stop is equal to a multiple of 2 (half if stopped down and twice if opened.)
Note: The light measuring device can be calibrated using the inverse-square law: Intensity varies with the square of the distance, in another experiment before this one.
I would continue in this manner until I had full, half, and even third stops if I wanted and if space on the lens barrel permitted.
In short, the mathematical relationship of light intensity to f/ stops is the "tool" I'd use for the task.