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According to most discussions of Euclid's Elements, this work - and indeed, much of Ancient Greek geometry - should be seen as engaged in the game of figuring out what can be done with straightedge and (collapsible) compass.

Although Euclid's Elements can certainly be (mathematically) interpreted in this way, Euclid never says anything to explicitly support this. Nor does one find an explicit discussion of the idea that this is how we should view the ancient practice of geometry (with respect to Euclid or the Greek geometrical tradition in general) in Plato, Aristotle, Proclus and so on - as far as I can tell.

Is there any ancient source who specifically talks about straightedges and compasses and the idea that much of geometry revolves around questions of what can and can't be done with such tools? I am aware that much discussion of Greek mathematics simply assumes this, so I'm not looking for references to modern people who repeat this piece of folklore - there are examples of that everywhere. I'm rather looking for actual extant ancient texts that support this piece of folklore.

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    $\begingroup$ The three construction postulates of Euclid's Elements are:"To draw a straight line from any point to any point", "To produce a finite straight line continuously in a straight line", and "To describe a circle with any center and radius". "Straightedge and compass" is simply a convenient shorthand. If you are asking about physical straightedges and compasses the Platonist tradition opposed acknowledgement of such lowly sensible tools when describing the pure intelligibles of geometry. $\endgroup$
    – Conifold
    Commented Apr 4, 2018 at 2:13
  • $\begingroup$ "Straightedge and compass is simply a convenient shorthand" is still a claim that needs to be justified. $\endgroup$
    – provocateur
    Commented Apr 4, 2018 at 2:21
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    $\begingroup$ You'll have to be more specific as to what you mean by "justified". Straightedges draw straight lines and compasses draw circles according to their dictionary definitions. $\endgroup$
    – Conifold
    Commented Apr 4, 2018 at 2:26
  • $\begingroup$ I'm talking about the ordinary sense of justification. Just because someone is talking about circles, doesn't mean they are talking about compasses. $\endgroup$
    – provocateur
    Commented Apr 4, 2018 at 3:39
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    $\begingroup$ I do not follow. Euclid does not use the straightedge and compass terminology, we do, to describe what he is doing. And we do it because it is short, because teachers did use literal straightedges and compasses in the recent centuries, etc. Egyptian land measurers, from whom Pythagoreans imported geometry, used stretched ropes for both purposes instead, hence the nickname rope stretchers. $\endgroup$
    – Conifold
    Commented Apr 4, 2018 at 4:25

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“Geometry … teaches us the use of the rule and compasses.” Vitruvius, De architectura I.1.4.

“I describe a circle with the compass. … With the straight ruler I set to work to inscribe a square within this circle.” Aristophanes, Birds, 1000.

“A circle drawn with geometrical compasses.” Plutarch, Maxime 1.

As for the importance of knowing what can and cannot be done by these tools, Pappus (Collection IV) says it is “no small error for geometers” to use more advanced means when a problem “can be solved by means of a straight line and a circumference of a circle,” that is, by ruler and compass.

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  • $\begingroup$ Thanks. The Vitriuvius comment, when looked at in full, also endorses the use of 'the square, the level, and the plummet' in the same sentence, and so seems to leave my question open. The Aristophanes and Plutarch passages just point out the circles can be drawn with compasses, which just seems like a triviality. The Pappus quote is perhaps the most compelling here, and I would like to think about it further. $\endgroup$
    – provocateur
    Commented Apr 4, 2018 at 0:03
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    $\begingroup$ @provocateur Vitruvius and Plutarch specifically associate compasses with geometry, not just mention it as a practical tool. Aristophanes’s character is inscribing a square within a circle, which is exactly the kind of thing Euclid is doing. $\endgroup$
    – Viktor Blasjo
    Commented Apr 4, 2018 at 7:21

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