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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

Euclid Looked On Beauty Bare
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  • "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. – Moishe Kohan Jul 07 '21 at 15:44
  • "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. ;-) – Euclid Looked On Beauty Bare Jul 07 '21 at 16:56
  • 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. – Euclid Looked On Beauty Bare Jul 07 '21 at 17:09
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    Does this answer your question? What made Euclid/Heron define line as a length without breadth and point as that which has no part? – Conifold Jul 08 '21 at 04:28
  • No, but the discussion and the links are interesting to read. Absent from the analysis is a view of the Elements as a theory of the art and science of surveying the Earth. – Euclid Looked On Beauty Bare Jul 08 '21 at 14:25
  • Surveying the Earth, and applications generally, were derided by Plato and avoided by mathematicians of Platonic persuasion, like Euclid, see When were the concepts of pure and applied Mathematics introduced? However, Aristotle, Archimedes and Apollonius were more sympathetic, and possibly influenced "definitions" inserted into Elements, see Acerbi, Two Approaches to Foundations in Greek Mathematics, p.160ff. – Conifold Jul 08 '21 at 23:29
  • According to Proclus, Apollonius's explanation of the notion of line was:"we have a notion when we ask only to measure the lengths, either of a road or of a wall: for we do not take into account the breadth, but reckon the distance in one [direction] only, in the same way as, when we measure domains, we look at the surface, when [we measure] wells, at the solid". But these later explanations have little to do with Euclid or what goes on in the Elements. – Conifold Jul 08 '21 at 23:36
  • I believe Euclid's definitions should be interpreted as idealizations which emerged from the technics of surveying and building. Since it involves idealizations I don't think this is incompatible with Platonism. – Euclid Looked On Beauty Bare Jul 09 '21 at 15:37
  • The definitions are likely not Euclid's, and theorizing about geometry by Pythagoreans, who are the Euclid's source, was very far removed from surveying and building, especially after Plato. Idealizations from practical activities are the opposite of what Plato had in mind by "ideas" and the "ideal" nature of geometry, he even criticized the abstract talk of "constructions" as "corrupting the good of geometry". So such reinterpretation is far from the spirit of Elements. But it was offered later by Apollonius, Heron and others. – Conifold Jul 09 '21 at 23:29
  • String lines were used for thousands of year prior to Euclid in surveying and construction and they are still used today. The point I am trying to make is that the meaning of Euclid's definitions becomes evident when a string line is used to intuitively model a Euclidean line instead of a draftsman`s line. Although the word "draw" is used many times in the Elements the terms compass and straight edge do not appear anywhere in the books .This suggests a Euclidean line is quite unlike a line drawn on surface , and instead exists independently of a surface the way a string line does. – Euclid Looked On Beauty Bare Jul 10 '21 at 05:20
  • The exception is a circle line which Euclid states is a plane figure. – Euclid Looked On Beauty Bare Jul 10 '21 at 05:23
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    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. – Conifold Jul 10 '21 at 06:26
  • The intuitions of subsequent commentators have made understanding Euclid difficult. However, we can reverse the trend and let the text speak for itself which means avoiding the temptation to explain away the parts which don`t conform to modern conceptions of geometry. It is also true that calling the Elements a platonic endeavor does not help in this regard. – Euclid Looked On Beauty Bare Jul 11 '21 at 03:46
  • Texts do not speak for themselves, ever. Reading them that way is exactly the recipe for reading in modern meanings and connotations, and often also personal tastes and biases. This is as intended for literary works, but not for historical documents. Mathematical objects as idealizations of worldly things is a modernized conception, popular in teaching. Documents should be read in the context of their time, informed by contemporaneous documents and artifacts, not by "rational reconstruction" of the "text itself" from intuitions and vague generalities about historical trajectories. – Conifold Jul 14 '21 at 20:26
  • @Conifold, I agree texts don't literally speak for themselves, but they don`t get to speak at all if they are dismissed as gibberish according to the precepts of modern mathematics. – Euclid Looked On Beauty Bare Jul 14 '21 at 21:20

2 Answers2

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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.

Alexandre Eremenko
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  • 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

     – Euclid Looked On Beauty Bare Jul 09 '21 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.

Deschele Schilder
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  • 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. – Euclid Looked On Beauty Bare Jul 12 '21 at 13:58
  • @EuclidLookedOnBeauty Precisely. It's a pity Euclid wasn't ready for this yet. Maybe relativity had been invented already in ancient Greece... – Deschele Schilder Jul 12 '21 at 14:08
  • 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. – Euclid Looked On Beauty Bare Jul 12 '21 at 16:58
  • 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? – Deschele Schilder Jul 12 '21 at 17:05
  • The process of constructing such houses uses Euclidean geometry, but that doesn`t mean the final form has to visually represent the geometry. – Euclid Looked On Beauty Bare Jul 12 '21 at 17:27
  • Do you mean constructing houses with curved bricks uses Euclidean geometry? – Deschele Schilder Jul 12 '21 at 17:30
  • Yes, if you don`t want it to fall down. – Euclid Looked On Beauty Bare Jul 12 '21 at 17:59
  • :D But I can imagine that the bricks are all evenly curved and fit. The walls made out of these bricks can be curved too. Like an iglo. – Deschele Schilder Jul 12 '21 at 18:07
  • Thanks for introducing me to the buildings of Hundertwasser. I had not heard of him. Anyway I am not sure why Euclidean geometry is considered restrictive or limiting. It is not a cage like Cartesian geometry. It doesn't tell the artist what sort of lines or shapes are desirable, but because it clarifies what is straight and flat it accentuates the beauty of "non-Euclidean" lines and forms. – Euclid Looked On Beauty Bare Jul 13 '21 at 14:27
  • What is the difference between Euclic and Cartesian geometry? Both are in flat space. Is Cartesian geometry more rigid? Only straight lines and coordinates? – Deschele Schilder Jul 13 '21 at 17:35
  • Cartesian geometry and so called non-Euclidean geometries consist of a pre-existing orderly array of points. That kind of pre-existing order is not present in Euclidean geometry. Freedom in cartesian space requires bending the rules, euclid postulates – Euclid Looked On Beauty Bare Jul 14 '21 at 00:17
  • You are right about the coordinates. Also lines and circles have no meaning beyond simply being equations. – Euclid Looked On Beauty Bare Jul 14 '21 at 00:27
  • This 7 minute instructional video explains how to use a string line. How to use a string line. Both a Euclidean line and a Euclidean straight line (which is like a taut string) connect two points. A Euclidean line does not already contain points as it does in Cartesian geometry. Euclidean lines are similar to lines that are used in graph theory. – Euclid Looked On Beauty Bare Jul 20 '21 at 13:34
  • @EuclidLookedOnBeautyBare Ah! Now I see the difference. The Euclidean plane is an affine plane? – Deschele Schilder Jul 20 '21 at 13:39
  • Yes it is more like a affine plane, but there is still no preferred system of parallel lines on which a system of coordinates can be built. It is possible to construct an affine coordinate system by "tilting" the y-axis in a cartesian coordinate system. – Euclid Looked On Beauty Bare Jul 20 '21 at 13:50
  • @EuclidLookedOnBeautyBare so the Euclidean plane cannot be coordinated? – Deschele Schilder Jul 20 '21 at 13:58
  • Yes it can be coordinated. Something is gained by coordinating plane, but has something also been lost or forgotten? – Euclid Looked On Beauty Bare Jul 20 '21 at 15:04
  • @EuclidLookedOnBeautyBare What could have been lost? Freedom? – Deschele Schilder Jul 20 '21 at 15:08
  • @EuclidLookedOnBeautyBare Coordinates impose points. With coordinates. What is lost by this? Maybe connectivity. No two points can touch. However close. Every in between contains points. Can a line be seen as a collection of points? – Deschele Schilder Jul 20 '21 at 16:11
  • Yes a line can be seen as collection points and this supports a local or mechanistic philosophy of nature. However, if a line is allowed to exist independently of points, then a line can be a non-local connector between separate points. – Euclid Looked On Beauty Bare Jul 20 '21 at 16:31
  • @EuclidLookedOnBeautyBare Thats the most beautifull description of a line I have ever heard! What makes a straight line different from a curved one? Does it make sense to make a distiction at all? Definitions can destroy. – Deschele Schilder Jul 20 '21 at 16:40
  • "What makes a straight line different from a curved one?" I tried to answer this in the body of my question, but I will rephrase my answer. If you move around a line with two end points until both end points seem to coincide in one point then the line is a straight line if the line appears to fit within this one point. – Euclid Looked On Beauty Bare Jul 20 '21 at 18:09
  • A definition can be destructive or creative. – Euclid Looked On Beauty Bare Jul 20 '21 at 18:14
  • I don`t see the value in eliminating the difference between a straight line and a curved line. Some differences should be valued. On the other hand, topologists begin with the assumption that there is no effective difference between a straight line and a circle. Physicists have also collapsed differences by reducing everything to a form of energy. – Euclid Looked On Beauty Bare Jul 20 '21 at 18:36
  • But does it make sense to try to explain all of nature and our humanity in a way that would please a modern topologist or a modern physicist? – Euclid Looked On Beauty Bare Jul 20 '21 at 18:52
  • @EuclidLookedOnBeautyBare A curved line would be nothing with a straight one. Lines would not exists wïthout points, though you coulď argue that it is us who chop them up. A line would be nothing witjout a plane. A plane would be nothing without a volume. – Deschele Schilder Jul 20 '21 at 19:04