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I decided to study Euclid for fun. I have Oliver Bryne's edition.

I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do this?

I have a compass, and a ruler. So far I've constructed the first three propositions from book one.

However, it's not clear to me how I ought to draw the fourth proposition, or whether it's only meant to be understood.

The later propositions use the earlier propositions where equal line lengths were drawn using circles. If I want to use those same deductions to construct later propositions, should I simply copy the line length with a ruler?

Surprisingly, google didn't turn up much guidance for this project. I'm assuming earlier generations of pupils would have drawn Euclid, no?

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  • $\begingroup$ I suppose, to some extent, that once you follow/believe a construction, there is no point in repeating it. A bit like using a theorem rather than proving it each time. $\endgroup$ – copper.hat Sep 7 '13 at 4:59
  • $\begingroup$ Copying with a ruler is not accurate, old-style draught-people used compasses. But accuracy in diagrams may even be undesirable. $\endgroup$ – André Nicolas Sep 7 '13 at 5:06
  • $\begingroup$ We had a special compass that worked on blackboards. $\endgroup$ – copper.hat Sep 7 '13 at 5:12
  • $\begingroup$ Andre, I've got a compass. My question is: a lot of the later diagrams depend on earlier propositions which are drawn with a series of circles. Book 1, prop 2 requires 4 circles, for example. So if I were proving later propositions manually, am I supposed to draw four circles every time I want to copy the length of a line? How did geometry students do it in earlier times? $\endgroup$ – Graeme Sep 7 '13 at 15:03
  • $\begingroup$ Yes, drawing a bunch of circles is exactly how line segments were transferred back then. $\endgroup$ – Keshav Srinivasan Sep 10 '13 at 7:33
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An actual compass and straightedge are not needed at all, but it can be fun and educational to use them. You can remember the constructions better when you actually do them. I remember when I constructed a regular pentagon on my front lawn with ropes and drew the lines with lime.

Most of Euclid's propositions aren't constructions, but use them at the beginning to add auxiliary lines needed later to prove the statement of the proposition.

An exception is proposition I.4 that you mention. There is no construction. At most there's a mental motion of one triangle to fit over another. (Euclid's proof is not convincing.)

You ask about transferring lengths with a ruler. The alternative is to use I.3 that you've already done every time you want to copy the length. That's enough justification to copy the length.

It's possible that I.3 wasn't in the original Elements. Earlier Elements might have assumed that lengths could be transferred, and Euclid, or someone else, discovered the construction in I.3 that allowed this operation. That would reduce the assumed constructions (postulates) to the simpler postulate that a circle could be drawn given a center and a point to be on the circumference of the circle.

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First, to answer the question about drawing the propositions:

Every "given" that you start with at the outset of a proposition is supposed to be guaranteed by the five postulates. The first three postulates can be oversimplified to "moves you can do with a straightedge and a compass" as you have probably noticed. So that means you will always be able to construct the given, right?

Nope! You could know how to construct the given in many trivial cases, but in more complex cases you will not always know how to perform the construction that will satisfy all of the requirements of the given. In fact, you shouldn't view yourself as responsible for creating the given at all. Pretend like the given is eternally existent or has been magiced into being by the god of euclidean geometry. Here are two examples to illustrate what this looks like in practice:

In prop 1, we are given a single finite straight line called AB. This would have been easy to construct ourselves from the first postulate (to draw a straight line from any point to any point). So there's an example of where the given is so trivial that we could have constructed it to meet all of the requirements.

In prop 4, as you have seen, we are given two triangles with certain parts of them equal: two sides are correspondingly equal and the angles contained by those sides are equal between the two triangles. Those conditions make the construction impossible for you since there is no construction (yet) to make an angle equal to another angle. You are at the mercy of the god of euclidean geometry! Pray to him to provide you with these ideal objects as described.

Later propositions will require drawing an impossible diagram! The given will start out just fine, but an assumption made for the sake of the proof will require "constructing" a straight line chord that somehow falls outside of its circle or two circles that do not share the same center, but somehow cut one another in more than two places. These examples can't be done on paper unless you fudge the diagram in one way or another.

If anything, this is the reason why you should ease up on thinking that you have to perform everything with real-world instruments. Actually drawing an object in such a way is not what guarantees its existence or its properties. The look of a drawing does not prove that a situation is possible or impossible either.

This reveals something subtle about the nature of hypothesis and conclusion of mathematics in general. We are always saying, "If such and such is the case, then so and so will necessarily follow from it." If we happen to be in bizarro land where our two hypothetical triangles from prop 4 could not exist with their properties of equal sides and angles, then the conclusion of the prop 4 would be undermined and would not be meaningfully true since it would refer to something that could not possibly exist.

Anyways, how do we know that the two triangles from prop 4 could exist even though we can't yet construct them?

Well, generic triangles are guaranteed to exist by the first postulate applied three times. Equal lines are guaranteed to exist in a subtle way by the third postulate and the definition of circle. The fact that equal lines can occur (or be constructed) anywhere in the plane is demonstrated by the third proposition building off of the first two. (That Euclid realized the necessity for props 2 and 3 is evidence of his genius and is part of what separated him from second-rate scrubs who might have attempted the same project.)

What about the equal angles then? What guarantees that they can exist? I think that's a harder question. We start with the fact that two right angles are guaranteed to be equal to one another anywhere they occur (postulate 4). By extension, this suggests that the magnitude of one angle may be equal to the magnitude of another angle anywhere on the plane, as I'll try my best to explain. Observing the common notions, the "spirit of the law" seems to be that equal multiples of equal things are equal to one another. Also, equal parts (fractions or submultiples) of equal things are equal to one another. Euclid draws on principles like this when he says that "the halves of equals are equal" in his proof of the Pythagorean Theorem. That's not explicitly a common notion, but it's not handwaving either. It feels like it's implied from the common sense we are supposed to have in using the common notions.

But smarter men than me like David Hilbert include S.A.S. congruence (prop 4) as an axiom. So what are you gonna do?

I have already rambled enough, but I wanted to also make a comment on Byrne's Elements:

I love Oliver Byrne's edition of the Elements---it's like an art piece.

Be careful though, if you want the experience of understanding Euclid's original thought and the mathematical mind of the ancient Greeks, you need to read the propositions in their rhetorical form. The version of the Elements translated by Heath is what is used in the "Great Books" colleges that still study the Elements as part of their liberal arts curriculum.

Byrne also alters the proofs of several propositions. In Book III on circles, there are many differences where Byrne tries to smooth over difficulties or break up large propositions into multiple parts. Also, his treatment of proportions as icons was really weird to me at first, but if the point is to make Book V easier on the eyes then he has succeeded in that.

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You might want to have a look at Euclidea.

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