Euclid's Elements

Euclid on Definition

Introduction, ch.9, §7

Euclid characterizes definition in Aristotelian terms. Definition does not concern the existence of a thing, but rather its possibility. Heath characterizes things this way,

It is an answer to the question what a thing is, and does not say that it is. The existence of the various things defined has to be proved, except in the case of a few primary things in each science, the existence of which is indemonstrable and must be assumed among the first principles of each science ; e.g. points and lines in geometry must be assumed to exist, but the existence of everything else must be proved.1

Here’s Euclid’s definition of a circle (definition 15):

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.2

As Heath points out, the definition is not genetic, it does not prove the existence of its figure or provide the means of constructing it. This is done only in the Third Postulate.

To describe a circle with any centre and distance.3

    Address = {Cambridge},
    Author = {Euclid},
    Publisher = {The University Press},
    Title = {Euclid's Elements: Introduction and books 1,2},
    Year = {1908}}



  1. (NO_ITEM_DATA:euclid1908), 143. ↩︎

  2. (NO_ITEM_DATA:euclid1908), 153. Definition 16 labels this point the ‘centre’. ↩︎

  3. (NO_ITEM_DATA:euclid1908), 154. The original Greek has the passive of the verb, so it literally reads, “a circle can be drawn with any centre and distance”. Proclus modifies the Greek so that it has the active form of the verb in order to match the first two postulates. ↩︎

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