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I propose an alternative: set your compass to BC, draw a circle around A, draw a line from A to any point on the circumference, basta.

Why is his better? We're both using postulate 3. We both end up with the new line at a random angle. But I don't need prop 1 which he uses to draw an equilateral triangle at the outset.

Adrian May
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    For clarity, you should include the statement in question. Otherwise, "transporting" a length as you propose is really justified by this proposition. If you prefer this argument, you'd need to justify it by other means. – lulu Jun 13 '23 at 16:07
  • Thanks for linking the text. I'm not sure of your point tho. I think mine is better cos it doesn't rely on proposition 1. And it's much simpler. Otherwise they do the same thing. I don't deny that a method of transporting a length is needed. – Adrian May Jun 13 '23 at 16:18
  • Again, "transporting" the length is exactly what this proposition does. One could add it as an axiom, but it's not needed. – lulu Jun 13 '23 at 16:19
  • Still confused. The original and mine both transport lengths. I'm asking why he does it the hard way. I wasn't proposing to add it as an axiom. What do you think I need as an axiom that he doesn't? – Adrian May Jun 13 '23 at 16:20
  • There's no transportation involved in Euclid's argument. Each time a circle is constructed we have the center and a point on the circumference. – lulu Jun 13 '23 at 16:24
  • So you mean the bit I use that he doesn't is that I select a random point on the circumference? But I could select a definite point more easily just by drawing an infinite line from A to B or C. The angle of the line he ends up with is not exactly by design either. – Adrian May Jun 13 '23 at 16:25
  • You can't construct the circle in the first place! You have the center, but you don't know any point on the circumference. Again, this is the entire point of the proposition. It tells us that given the length, even if it does not involve the desired center, we can still construct the circle with the desired center and radius equal to the given length. – lulu Jun 13 '23 at 16:28
  • I don't think postulate 3 specifies that restriction. "To draw a circle with any centre and distance" – Adrian May Jun 13 '23 at 16:29
  • here is postulate $3$. The given data is the center and a radius. You haven't got a radius, all you have is a line segment that you want to have the same length as the radius. But, to Euclid, a "radius" is a line segment, it isn't a number. Another way to read postulate $3$ is to say that given a point $C$ and a different point $P$ we can construct the circle with center $C$ which passes through $P$ (and which therefore has $CP$ as a radius). – lulu Jun 13 '23 at 16:31
  • I'd rather have a quote from the original than a website written 2300 years later. Or did he just forget to say it? In definitions 15-17 this rule is clearly not there. But I can see that if I'm not allowed to move the point of the compass even tho I'm allowed to move it's legs apart, then that would invalidate my proposal. – Adrian May Jun 13 '23 at 16:32
  • Please feel free to consult the original. And, no, he didn't forget...the entire point of the proposition in question is to show that you can use any line segment as a radius (in the sense of length). Hence showing that the (apparently stronger) version of the postulate follows from the (apparently weaker) version. – lulu Jun 13 '23 at 16:35
  • I am consulting the text. The definition of a circle, and how to draw one, makes no such restriction. – Adrian May Jun 13 '23 at 16:36
  • But I suppose we can infer from the existence of prop 2 that he meant to say it. Thanks! Alles klar now. Pop in an official answer and I'll approve it. – Adrian May Jun 13 '23 at 16:37
  • What is the meaning of "faff" ? Please don't use slang : most people here know only classical english. – Jean Marie Jun 13 '23 at 19:21

1 Answers1

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To summarize the discussion in the comments:

Postulate $3$, concerning the construction of circles, is somewhat ambiguous. Modern language agrees with classical in the notion of a center. But modern language would interpret "radius" as a number while, classically, the radius was a line segment connecting the center to any point on the circle.

The proposition in question here resolves the ambiguity. Specifically, it shows that we can use any line segment, to construct a circle with a fixed center. It doesn't matter whether the line segment has the desired center as an endpoint or not.

In the language of construction, the issue is whether or not we work a "collapsing" compass. This was the notion preferred by Plato, and it specifically precludes the ability to "lift the compass off the page" while retaining the measured length. Again, the given proposition shows that whether you prefer rigid compasses or the collapsing sort doesn't change the things you can construct. It just takes more steps to work with the collapsing sort.

this article goes into these (and related) issues with greater depth.

lulu
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