Apeiron 54 (2):233-259 (2021)

In this paper, I attempt a reconstruction of the theory of intersections in the geometry of Euclid. It has been well known, at least since the time of Pasch onward, that in the Elements there are no explicit principles governing the existence of the points of intersections between lines, so that in several propositions of Euclid the simple crossing of two lines is regarded as the actual meeting of such lines, it being simply assumed that the point of their intersection exists. Such assumptions are labelled, today, as implicit claims about the continuity of the lines, or about the continuity of the underlying space. Euclid’s proofs, therefore, would seem to have some demonstrative gaps that need to be filled by a set of continuity axioms.I show that Euclid’s theory of intersections was not in fact based on any notion of continuity at all. This is not only because Greek concepts of continuity were largely insufficient to ground a geometrical theory of intersections, but also, at a deeper level, because continuity was simply not regarded as a notion that had any role to play in the latter theory. Had Euclid been asked to explain why the points of intersections of lines and circles should exist, it would have never occurred to him to mention continuity in this connection. Ancient geometry was very different from ours, and it is only our modern views on continuity that tend to give rise to the expectation that this latter must be included in the foundations of elementary mathematics.We may hope to reconstruct Euclid’s views on intersections if we undertake a critical examination both of the extension and of the limits of ancient diagrammatic practices. This is tantamount to attempting to understand to what extent the existence of an intersection point may be inferred just from the inspection of a diagram, and in which cases we need rather to supplement such inference by propositional rules. This leads us to discuss Euclid’s definition of a point, and to work out the details of the complex interaction between diagrams and text in ancient geometry. We will see, then, that Euclid may have possessed a theory of intersections that was sufficiently rigorous to dispel the main objections that could be advanced in antiquity.
Keywords No keywords specified (fix it)
Categories No categories specified
(categorize this paper)
DOI 10.1515/apeiron-2019-0012
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 61,064
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

Euclid’s Common Notions and the Theory of Equivalence.Vincenzo De Risi - 2021 - Foundations of Science 26 (2):301-324.

Add more citations

Similar books and articles

Proofs, Pictures, and Euclid.John Mumma - 2010 - Synthese 175 (2):255 - 287.
Euclid’s Common Notions and the Theory of Equivalence.Vincenzo De Risi - 2021 - Foundations of Science 26 (2):301-324.


Added to PP index

Total views
3 ( #1,305,924 of 2,439,687 )

Recent downloads (6 months)
3 ( #209,095 of 2,439,687 )

How can I increase my downloads?


My notes