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The mathematical concept of distance is fundamental in all mathematics and, since Bernard Riemann’s definition of manifolds, is also foundational in geometry and geometry of physics.

Contrary to a widespread belief, probably encouraged by that particular distance known as Euclidean Distance,

  1. I could not find a mathematical definition of distance in Euclid's Elements and
  2. I could not find any significant use of the word or of the concept except for Postulate 3, Book 1 ("To produce any circle with any centre and distance") – postulate in which the word is used just in the common sense of “separation” (notes on postulate 3, p.199, “Euclid’s the Elements”, Sir T. L. Heath, Dover)

Euclid, instead, speaks and deals with mathematical length – a particular magnitude.

I am not interested in the empirical and practical equivalence of length and distance. I am concerned with their significant difference with respect to the foundations of geometry.

I would appreciate some help on the following questions:

  1. Are there specific references to the concept of mathematical distance in the Elements that I have been missing – may be because they are disguised in other wordings?

  2. Are there any explicit discussions, in the Elements, that help to understand Euclid’s point of view on mathematical distance?

  3. Is there any scholarly work in which the difference between distance and length has been analyzed with respect to the foundations of modern geometry – including historical analysis? (I am aware some geometers work on “Length Geometry” but I am not aware if and where they have done any analysis of the kind above)

Thanks

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  • $\begingroup$ @JoséCarlosSantos ...and how is the history of science and mathematics not a part of mathematics? $\endgroup$
    – Circulwyrd
    Commented Mar 3, 2021 at 19:08
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    $\begingroup$ You can find in Euclid the fact that in a non-flat triangle the sum of the lengths of two sides is less than the length of the third (see here Proposition 20 Book I, which is the third and most important property of a distance. $\endgroup$ Commented Mar 3, 2021 at 19:45
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    $\begingroup$ In Euclid's Elements there are some basic different "objects", including numbers and magnitudes. See e.g. Ivor GRATTAN-GUINNESS, Numbers Magnitudes Ratios and Proportions in Euclid’s Elements (1996) $\endgroup$ Commented Mar 4, 2021 at 8:32
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    $\begingroup$ We can use numbers to "measure" magnitudes; see Book V: "A magnitude is a part of a magnitude, the less of the greater, when it measures the greater." There is no def of "to measure"; the implicit meaning is: we choose whatever "unit" we want for a specific kind of magnitude (a "unit line", a "unit square") and we use it ("the less") to measure "the greater". $\endgroup$ Commented Mar 4, 2021 at 8:34
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    $\begingroup$ In conclusion: there is no Euclidean metric in Euclid's Elements. $\endgroup$ Commented Mar 4, 2021 at 10:29

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The concept of distance is far from being fundamental to mathematics, in the sense that fundamental is foundational. After all, nothing in set theory presupposes distance, and topology dispenses with the notion altogether.

Let us say that the notion of distance has been important in mathematics.

In Euclid, the theory of measurement, ratio and proportion are in the fifth, sixth, seventh and tenth books. The fifth book, in particular, is devoted to theory of ratio and proportion of magnitude in general and the sixth book applies this theory to plane geometry.

According to the commentator Proclus, the theory of ratio and proportion was actually pioneered by Eudoxus a century before Euclid.

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  • $\begingroup$ Apart from disagreeing with me about distance being fundamental, not foundational, in mathematics, what would your answer to my questions be? $\endgroup$
    – massimo
    Commented Mar 15, 2021 at 22:22
  • $\begingroup$ @Massimo: Did you read the third paragraph? $\endgroup$ Commented Mar 17, 2021 at 12:31
  • $\begingroup$ @dear Ullah: this is the point: the things you talk about are not "distance"! $\endgroup$
    – massimo
    Commented Mar 18, 2021 at 13:05
  • $\begingroup$ @massimo: They are distance to Euclid in the same way we have displacement vector and metric to explain distance today. $\endgroup$ Commented Mar 19, 2021 at 21:42
  • $\begingroup$ There is also quite a lot of additional theorems regarding ratios and magnitude in Euclid's less popular book, "Data" $\endgroup$
    – Slereah
    Commented Jan 3 at 12:55

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