A scale in this sense is a series of evenly spaced graduations like the marks on a ruler:


Today, we use an End Mill or a Dividing Engine to manufacture the scales.

Scales were used frequently on astrolabes (curved scales) and rulers (straight scales). I have not been able to find out how they did it.

I am looking for first hand accounts or instructions for manufacturing these scales in medieval times. Does anyone know of any such documents?

Note: I asked about medieval technology, because I am most familiar with the history of European manufacturing processes. However, I will happily accept documentation from any preindustrial cultures that did not have the benefit of calibrated lead screws.

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    the ancient greeks used a compass and geometry for making gears with a certain number of teeth in them, this would be your scale on a curved surface. i recall seeing a documentary where they recreated the antikythera machine using ancient techniques and they showed how to make a gear of any specific number of teeth using this method, i am trying to find the documentary now for you but google is being a pain. here is a good primer on gear technology and the math involved though - geartechnology.com/issues/0584x/geardesign.pdf
    – ed.hank
    Oct 6, 2021 at 14:40
  • Making a linear scale is a trivial application of Euclid's Book VI, Proposition 10: "To cut a given uncut straight line similarly to a given cut straight line." This is the entire point of proving Book VI Prop. 10. Oct 6, 2021 at 14:45
  • @ed.hank; Not quite "any specific number of teeth": as Gauss proved in 1796 that seven teeth is impossible with compass and straight edge and was also the first to find a technique for seventeen teeth (the heptadecagon). Certainly many possible number of teeth involving powers of 1/2 and working from angle measures of (at least) 45, 30, and 18 degrees. Oct 6, 2021 at 14:49
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    @PieterGeerkens That is one of those theory vs application issues. It doesn't take into account the error in the compass marks, for example. I only mention this, because I don't want your use of the word trivial to deter potential respondents.
    – Craeft
    Oct 6, 2021 at 14:55
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    @Craeft - its not just theory vs application, the greeks actually built machines using these techniques, the antikythera mechanism for one, plus tons of other less complicated examples which are extremely accurate for being over 2000 years old. it shows the greeks understood both the theory and how to apply it to actual production devices.
    – ed.hank
    Oct 6, 2021 at 15:06

1 Answer 1


Note - at time of writing this answer, the question was phrased:

  • am looking for first hand accounts or instructions for laying out these scales in medieval times.*

"Laying out these scales" is exactly what I have described below. That is a key, and perhaps the most difficult, part of "manufacture". It is OP's responsibility to ask the desired question in clear and unambiguous terms; and to not then fiddle with the question asked so as to invalidate existing answers.

The laying out of linear scales predates Euclid by centuries, if not millennia. Euclid's Prop. IV.10 - "To cut a given uncut straight line similarly to a given cut straight line." - proves the theoretical soundness of this technique. That is to say, that it is theoretically sound and as accurate as the tools available and care taken will allow.

The laying out of angular scales is simply the process of inscribing regular polygons to a circle. Euclid in Book IV gives specific direction with compasses and straight edge) for a regular triangle, square, pentagon, hexagon, and 15-gon. This translates into access to angles of 30 degrees (hexagon), 90 degrees (square), 18 degrees (pentagon), and 24 degrees (15-gon); plus any multiple or power-of-two fraction of these; plus any sum of these. That's a lot of angles, but not even the complete set of all angles accessible with compasses and straightedge - not until 1796 did Gauss derive a method of inscribing a heptdecagon (17-gon) and prove the impossibility of constructing a heptagon (7-gon) by such methods. Note of course that Viete had constructed a heptagon in 1593 using, in addition to compasses and straight edge, the neusis:

..., let it be allowed from any point to any two straight lines, to draw a straight line cutting off between them any segment fixed in advance.

Gears, from a theoretical perspective, require that the angles be exact. However some tolerance is always available, and the Greeks were well aware that, having obtained an angle sufficiently close by theoretic means, a little manual effort could tune the mechanism and fit.

Additionally, Greek craftsmen had access to tools and techniques beyond those proven theoretically by Euclid, including for example the neusis construction described above.

  • Geometry alone is not sufficient for manufacturing a product any more than knowledge that shovels and picks exist is sufficient to describe the information in De Re Metallica. I asked for primary sources on manufacturing processes, which The Elements is not.
    – Craeft
    Oct 6, 2021 at 16:13
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    @Craeft: Unfortunately I am forbidden by site regulations from responding properly to your comment. The Elements is an original source. It is complete, a rarity amongst sources so old and particularly one of its size. Further, it encompasses a huge body of knowledge then known to have been, for the most part, known to the Greeks for centuries previous. Oct 6, 2021 at 16:24
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    Ah, that is a fine criticism, which I accept. "Laying out" does indeed have diverse meanings. I was worried that phrase would lead to confusion when I wrote it, but it the phrase that is commonly used for the whole process in several sources I have. What would you suggest to make it clearer?
    – Craeft
    Oct 6, 2021 at 16:54
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    @Craeft, I'm not sure why you think a scale needs to be heat-treated, or even need to be made from metal.
    – Mark
    Oct 7, 2021 at 1:14
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    i would also note for measurements not as precise but that will work for many applications all you need is a string and a peg board with two nails, with this you can very closely approximate any number you want, you wrap a string between two nails however many times you want, mark each part of the string at the nail, then you place the string on your curved surface and mark the surface at each location the string was marked.
    – ed.hank
    Oct 7, 2021 at 17:33

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