In the first book of The Elements, Euclid defines a line in the following terms (Fitzpatrick 6):
| Greek | English |
|---|---|
| βʹ. Γραμμὴ δὲ μῆκος ἀπλατές. | 2. And a line is a length without breadth. |
As we saw in the preceding article in this series, Euclid defines a point as that of which there is no part. I compared this to the traditional definition of a point in plane geometry as an object which has neither length nor breadth. For Euclid, then, a line has length but not breadth. In our modern terminology: a point is a non-dimensional object, while a line is a one-dimensional object.
Note, however, that Euclid does not define points and lines in terms of any properties that they may or may not have. We are accustomed to say that a line has length but not breadth, while a point has neither length nor breadth. But Euclid says that a line is a length without breadth, while a point is that of which there is no part. As geometric objects are wholly abstract, there is nothing concrete in them to which one might attach the property or accident of length.
According to Heath, Euclid borrowed this definition from the Platonists—if not from Plato himself (Heath 158). Aristotle discusses this definition in the Topics, though his objections are of more interest to the philosopher than to the mathematician:
Moreover, see if he divides the genus by a negation, as those do who define a line as length without breadth; for this means simply that it has not any breadth. The genus will then be found to partake of its own species; for, since of everything either the affirmation or the negation is true, length must always either lack breadth or possess it, so that length as well, i.e. the genus of line, will be either with or without breadth. But length without breadth is the account of a species, as also is length with breadth; for without breadth and with breadth are differentiae, and the genus and differentia constitute the account of the species. Hence the genus will admit of the account of its species. Likewise, also, it will admit of the account of the differentia, seeing that one or the other of the aforesaid differentiae is of necessity predicated of the genus. This principle is useful against those who posit Ideas; for if length itself exists, how will it be predicable of the genus that it has breadth or that it lacks it? For one assertion or the other will have to be true of length universally, if it is to be true of the genus; and this is contrary to the fact; for there exist both lengths which have, and lengths which have not, breadth. Hence the only people against whom the rule can be employed are those who assert that every genus is numerically one; and this is what is done by those who posit the Ideas [ie the Platonists]; for they allege that length itself and animal itself are the genus. It may be that in some cases the definer is obliged to employ a negation, e.g. in defining privations. For a thing is blind which cannot see when its nature is to see. (Barnes 94, Topics 143 b 11 ff)
Terminology
The Greek word that Euclid uses for line is γραμμη (grammē), which is derived from the verb γράφω, I scratch, I graze, I write. It literally means a stroke made by a pen:
This is the word that was generally used by the ancient Greeks to describe a geometric line—and, indeed, it is still the technical term for this object in modern Greek.
For Euclid, however, γραμμη does not just refer to straight lines and line segments: he also uses this word where modern mathematicians would use the term curve. In Euclid, a parabola is a line, as is the circumference of a circle. Other mathematicians, including some of Euclid’s predecessors, distinguish straight lines from curved lines, but in The Elements this distinction is never made explicit and is not required (Heath 159). It should, however, be kept in mind.
And that’s a good place to stop.
References
- Jonathan Barnes (editor), The Complete Works of Aristotle, Volume 1, Princeton University press, Princeton, NJ (1991)
- Richard Fitzpatrick (translator), Euclid’s Elements of Geometry, University of Texas at Austin, Austin, TX (2008)
- Thomas Little Heath (translator & editor), The Thirteen Books of Euclid’s Elements, Second Edition, Dover Publications, New York (1956)
- Johan Ludvig Heiberg, Heinrich Menge, Euclidis Elementa edidit et Latine interpretatus est I. L. Heiberg, Volumes 1-5, B G Teubner Verlag, Leipzig (1883-1888)
- Henry George Liddell, Robert Scott, A Greek-English Lexicon, Eighth Edition, American Book Company, New York (1901)
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