If I had to find the answer to this problem, I would start with the open opening, which will be circular as the blades of the iris are retracted.
It would be the big open. I would take the diameter of this opening and divide the focal length by this one to obtain the first "stop". Let's say it's a f / 4 goal. Scratch marks the opposite spot to "the index".
Close the aperture until the intensity of light passing through the lens is half of it (using a light intensity meter). The mark engraved on the aperture of the lens would be f / 5.6 because each stop is equal to a multiple of 2 (half it's halted and twice it's open).
Note: The light meter can be calibrated using the inverse squares law: the intensity varies with the square of the distance, in another experiment prior to this one .
I would continue this way until I have the full stops, half and even the third if I wanted it and if the space on the lens barrel allowed it.
In short, the mathematical relationship between the intensity of light and f / stops is "the tool" that I would use for the task.