In Book 1 of Euclid’s Elements, Definition 7 reads (Fitzpatrick 6):
|ζʹ. ̓Επίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ’ ἑαυτῆς εὐθείαις κεῖται.||7. A plane surface is (any) one which lies evenly with the straight-lines on itself.|
The Greek word ἐπίπεδος [epipedos] is the nominative masculine singular of the adjective ἐπίπεδος [epipedos], which means flat, smooth. It can also be used as a noun, meaning plane surface, though Euclid only uses it in this acceptation in other definitions. For example, Book 11, Definition 4 (Fitzpatrick 424):
|δʹ. ̓Επίπεδον πρὸς ἐπίπεδον ὀρθόν ἐστιν, ὅταν αἱ τῇ κοινῇ τομῇ τῶν ἐπιπέδων πρὸς ὀρθὰς ἀγόμεναι εὐθεῖαι ἐν ἑνὶ τῶν ἐπιπέδων τῷ λοιπῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν.||4. A plane is at right-angles to a(nother) plane when (all of) the straight-lines drawn in one of the planes, at right-angles to the common section of the planes, are at right-angles to the remaining plane.|
Euclid’s technical term for a surface, ἐπιφάνειά [epiphaneia], is feminine. So why does Euclid write ἐπίπεδος ἐπιφάνειά rather than επίπεδη ἐπιφάνειά ? Because ἐπίπεδος is one of those adjectives that use the same form for both the masculine and the feminine (J B Calvert). These are the so-called two-termination adjectives, which are compounds of simpler elements: ἐπί + πέδον = on + ground.
In Definition 5, a general surface was defined as that which has length and breadth only. Euclid now defines a special type of surface: a plane surface. The surface of a sphere is a surface according to Definition 5, but it is not a plane surface according to Definition 7. The relationship between these two definitions is akin to that between Definitions 2 and 4, which defined a line and a straight line respectively. As Thomas Heath comments:
The Greek follows exactly the definition of a straight line mutatis mutandis ... Proclus remarks that, in general, all the definitions of a straight line can be adapted to the plane surface by merely changing the genus. Thus, for instance, a plane surface is “a surface the middle of which covers the ends” (this being the adaptation of Plato’s definition of a straight line). Whether Plato actually gave this as the definition of a plane surface or not, I believe that Euclid’s definition of a plane surface as lying evenly with the straight lines on itself was intended simply to express the same idea without any implied appeal to vision (just as in the corresponding case of the definition of a straight line). (Heath 171)
Euclid’s definition of a plane surface seems to be another of those cases in which he assumes the existence of the geometric object he is defining, assumes that the reader is familiar with this object, and is simply letting the reader known what technical term he will be using for objects of this sort. Heath again:
Gauss observed in a letter to Bessel that the definition of a plane surface as a surface such that, if any two points in it be taken, the straight line joining them lies wholly in the surface (which, for short, we will call “Simson’s” definition) contains more than is necessary, in that a plane can be obtained by simply projecting a straight line lying in it from a point outside the line but also lying on the plane; in fact the definition includes a theorem, or postulate, as well. The same is true of Euclid’s definition of a plane as the surface which “lies evenly with (all) the straight lines on itself,” because it is sufficient for a definition of a plane if the surface “lies evenly” with those lines only which pass through a fixed point on it and each of the several points of a straight line also lying in it but not passing through the point. But from Euclid’s point of view it is immaterial whether a definition contains more than the necessary minimum provided that the existence of a thing possessing all the attributes contained in the definition is afterwards proved. This however is not done in regard to the plane. No proposition about the nature of a plane as such appears before Book XI, although its existence is presupposed in all the geometrical Books I-IV and VI; nor in Book XI is there any attempt to prove, e.g. by construction, the existence of a surface conforming to the definition. The explanation may be that the existence of the plane as defined was deliberately assumed from the beginning like that of points and lines, the existence of which, according to Aristotle, must be assumed as principles unproved, while the existence of everything else must be proved; and it may well be that Aristotle would have included plane surfaces with points and lines in this statement had it not been that he generally took his illustrations from plane geometry (excluding solid). (Heath 172)
The phrase which lies evenly with the straight-lines on itself echoes the corresponding part of Definition 4: which lies evenly with points on itself. As we saw in the article on that definition, the latter was possibly intended to capture the visual appearance of a straight line segment: if you look along the segment from one end, you cannot see the point at the other end. How does this concept translate to two dimensions? Presumably, if you take any two lines (not necessarily straight) that lie on the plane and try to look from one of them towards the other, you will not be able to see the other line because the plane will get in the way.
Proclus’ interpretation of these phrases, however, is quite different:
But Euclid and his successors make the surface the genus and the plane a species of it, as the straight line is a species of line. That is why, by analogy with the straight line, he defines the plane separately from the surface. For the straight line, he says, is equal to the interval that lies between its points, and the plane likewise occupies a place equal to that between two straight lines lying on it. This is what is meant by “lying evenly with the straight lines on itself.” (Morrow 94-95)
And that’s a good place to stop.
- 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 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)