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The third and fourth definitions in Euclid's Elements say:

  1. The ends of a line are points.
  2. A straight line is a line which lies evenly with the points on itself.

The fourth definition is usually regarded by modern mathematicians as meaningless or puzzling at best. A good definition we are told is that a straight line is the shortest distance between two points. However, Euclid's third, sixth and seventh definitions can provide some guidance on how to read the fourth definition.

  1. The edges of a surface are lines.
  2. A plane surface is a surface which lies evenly with the straight lines on itself.

I would argue the proper way to appreciate seventh definition is through the eye of a builder or surveyor. The straight line in this context is a builder's ideal straight edge which is used to gauge the flatness of a surface. As an ideal straight edge slides over and around the surface one looks from the edge of the surface for gaps between the edge and the surface. If no gaps are seen then the "plane surface lies evenly with straight lines on itself".

By analogy with the surface, the straight line in the fourth definition should be viewed from its end points. From this perspective if no part of the line lies away from the end points, then the line "lies evenly with the points on itself" which makes it a straight line.

Source for definitions:

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs

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  • $\begingroup$ "A good definition we are told..." requires a source: It depends on a particular axiomatization of Euclidean Geometry. For instance, in Hilbertian axiomatization, a "line" is a primitive (hence, undefined) notion. What did Euclid think about this, we can only guess. One guess (per Givental) is that Euclid struggled mightily with what to say here and decided to make an informal "definition" which conveys some intuition of the notion of a straight line. $\endgroup$ Jul 7 at 15:44
  • $\begingroup$ "A good definition we are told..." was intended to be taken ironically. Hilbert's axiomatic approach to geometry implies definitions are inconsequential to the study of geometry, so his answer to the question would be to deny the validity of question. Anyway, it was not his list of problems. ;-) $\endgroup$ Jul 7 at 16:56
  • $\begingroup$ As to whether Euclid struggled to define a straight line, that too is a guess. I think as the centuries passed, it would be more correct to say subsequent readers have struggled to understand what Euclid meant. $\endgroup$ Jul 7 at 17:09
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    $\begingroup$ Does this answer your question? What made Euclid/Heron define line as a length without breadth and point as that which has no part? $\endgroup$
    – Conifold
    Jul 8 at 4:28
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    $\begingroup$ The point I am trying to make is that just because some interpretation of a historical document is intuitively appealing to us does not mean that it is the only one, or that it reveals "the meaning" of it. Such ponderings are helpful in teaching and explaining, that is why commentators added them, but not as a guide to history. Strings, straightedges and compasses are not mentioned in the Elements because "lowly tools" detract from "eternal ideas" of geometry. Use of motions is avoided for the same reason, even though this forces proofs with cumbersome configurations of multiple triangles. $\endgroup$
    – Conifold
    Jul 10 at 6:26
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These "definitions" in Euclid are not really definitions in the modern sense. They are rather attempts to explain the intuitive meaning of the terms. And they are never used in his proofs.

What we would call axioms, is called in Euclid "Postulates" and "Common notions".

Some historians argue that "definitions" are not a part of the original text; they are later inserts. See, for example,

Lucio Russo, The Definitions of Fundamental Geometric Entities Contained in Book I of Euclids Elements, Archive for history of exact sciences, 52 (1998) 3, 195-219.

The best modern analysis of Euclid, on my opinion is

Robin Hartshorne, Geometry: Euclid and beyond, Springer 1997.

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  • $\begingroup$ From what I have read modern mathematicians aren't interested in the objects which definitions describe because they consider such objects to be irrelevant to mathematical inquiry. "Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." - Henri Poincaré $\endgroup$ Jul 9 at 15:28
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If you place two lines parallel wrt to each other then the very concept of being parallel becomes quaky. When you have two infinite and non-evenly lines they will intersect at many points. You could define a lnon-even line and say the non-evenness has to be the same for all lines. But in that case you have to put the lines next to each other in a very special way to be parallel (equal distance everywhere). I don't think that Euclid had this in mind. There are many ways to define non-even lines. There is just one line where eveness is evenly divided over the line. In contrast to non-even lines, even lines show no diversions.

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  • $\begingroup$ This is why the definition of a straight line plays a vital role in Euclid's geometry. If the definition is changed or deemed irrelevant the 5th postulate becomes murky which opens the door to speculation about alternative geometries. $\endgroup$ Jul 12 at 13:58
  • $\begingroup$ @EuclidLookedOnBeauty Precisely. It's a pity Euclid wasn't ready for this yet. Maybe relativity had been invented already in ancient Greece... $\endgroup$ Jul 12 at 14:08
  • $\begingroup$ The study of strange or fantastic geometries is welcome , but one needs a way in and out of Wonderland to be able to discern a good application from a bad application. Modern treatments of the Elements are not helpful in this regard. $\endgroup$ Jul 12 at 16:58
  • $\begingroup$ But what is a good and what is a bad application? You mention house builders but if you have seen buildings of Hundertwasser than this application is useless. Why using straight bricks if you can use uneven ones? $\endgroup$ Jul 12 at 17:05
  • $\begingroup$ The process of constructing such houses uses Euclidean geometry, but that doesn`t mean the final form has to visually represent the geometry. $\endgroup$ Jul 12 at 17:27

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