In Book 1 of Euclid’s Elements, Definition 4 reads (Fitzpatrick 6):
| Greek | English |
|---|---|
| δʹ. Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ ̓ ἑαυτῆς σημείοις κεῖται. | 4. A straight-line is (any) one which lies evenly with points on itself. |
Definition 2 has already defined a line [γραμμή, grammē] as length without breadth. We saw how Euclid uses the technical term line to refer not only to what modern mathematicians call lines and line segments (or, more simply, segments) but also to curves. In Definition 4, Euclid tells us how a straight line or straight line segment differs from a line as previously defined.
The Greek word εὐθεῖα [eutheia] is the nominative feminine singular of the adjective εὐθύς [euthus], which means straight, direct. It can also be used as a noun, meaning straight line:
Euclid himself uses εὐθεῖα as a noun many times throughout The Elements (eg Book 1, Proposition 1).
Euclid’s definition sounds strange to modern ears. What exactly does he mean by lies evenly with points on itself? We are all familiar with pictorial representations of straight lines, and Euclid’s definition may be an attempt to put into words what it is we see when we look at such a line. It sounds more like a description than a definition. He might just as easily as said: When I use the expression straight line, I am using it in the usual sense with which we are all already familiar.
Before Euclid, Plato had defined a straight line segment quite differently:
Furthermore, straight is that whose middle stands in the way of the two extremities. (Cooper 372, Parmenides 137 e)
In Thomas Heath’s opinion, this means that to an eye placed at either end of the line and looking along it, the middle blocks the eye’s view of the other end (Heath 1956:165). Note that Plato’s definition does not apply to a continuous line, which has no ends. Heath took this interpretation, no doubt, from Proclus, who quotes Plato’s definition in A Commentary on the First Book of Euclid’s Elements:
Plato, however, defines the straight line as that whose middle intercepts the view of the extremes. This is a necessary property of things lying on a straight line but need not be true of things on a circle or any other extension. (Morrow 89)
Aristotle quotes Plato’s definition in terms that are more-or-less equivalent to Plato’s, along with another definition of a straight line—one which we will meet shortly, when we discuss Euclid’s Definition 6:
Suppose, e.g., that some one has defined a finite straight line as the limit of a finite plane, such that its centre is in a line with its extremes; if now the account of a finite line is the limit of a finite plane, the rest (viz. ‘such that its centre is in a line with its extremes’) ought to be an account of straight. But an infinite straight line has neither centre nor extremes and yet is straight, so that the remainder is not an account of the remainder. (Barnes 556, Topics 148 b 27)
According to Thomas Heath, the definition in The Elements was a novelty of Euclid’s own devising:
As Aristotle mentions no definition of a straight line resembling Euclid’s, but gives only Plato’s definition and the other explaining it as the “extremity of a surface,” the latter being evidently the current definition in contemporary textbooks, we may safely infer that Euclid’s definition was a new departure of his own. (Heath 1956:166)
Did Euclid concoct his definition to cover both finite and infinite lines? Or did he devise it, as Glen Morrow surmises, to eliminate any implied appeal to vision (Morrow 89 fn 46)?
In his translation of The Elements, Heath devotes almost four whole pages to this definition in an attempt to get at the Euclid’s meaning, but he concedes:
While the language is thus seen to be hopelessly obscure, we can safely say that the sort of idea which Euclid wished to express was that of a line which presents the same shape at and relatively to all points on it, without any irregular or unsymmetrical feature distinguishing one part or side of it from another. (Heath 1956:167)
His conclusion is:
The question arises, what was the origin of Euclid’s definition, or, how was it suggested to him? It seems to me that the basis of it was really Plato’s definition of a straight line as “that line the middle of which covers the ends.” Euclid was a Platonist, and what more natural than that he should have adopted Plato’s definition in substance, while regarding it as essential to change the form of words in order to make it independent of any implied appeal to vision, which, as a physical fact, could not properly find a place in a purely geometrical definition? I believe therefore that Euclid’s definition is simply an attempt (albeit unsuccessful, from the nature of the case) to express, in terms to which a geometer could not object as not being part of geometrical subject-matter, the same thing as the Platonic definition. (Heath 1956:168)
Archimedes
Neither Euclid’s nor Plato’s definition caught on with mathematicians. Archimedes, who flourished about sixty years after Euclid, assumed that the shortest “line” between two given extremities was a straight line segment:
Of all lines which have the same extremities the straight line is the least. (Heath 1897:3 , On the Sphere and Cylinder, Assumption 1 )
This definition anticipates the calculus of variations, which can be used to prove that the shortest path between two points is a straight line segment.
And that’s a good place to stop.
References
- Jonathan Barnes (editor), The Complete Works of Aristotle, Volumes 1 & 2, Princeton University Press, Princeton, NJ (1984)
- John M Cooper, Plato: Complete Works, Hackett Publishing Company, Indianapolis (1997)
- Richard Fitzpatrick (translator), Euclid’s Elements of Geometry, University of Texas at Austin, Austin, TX (2008)
- Thomas Little Heath (translator & editor), The Works of Archimedes, Cambridge University Press, Cambridge (1897)
- 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)
- Glenn R Morrow (translator), Proclus: A Commentary on the First Book of Euclid’s Elements, Princeton University Press, Princeton, NJ (1970)
Image Credits
- Relief of Archimedes: Commonist (photographer), Russian State Library, Moscow, Public Domain
- The School of Aristotle: Gustav Adolph Spangenberg (artist), Martin Luther University of Halle-Wittenberg, Public Domain
Online Resources
