In Book 1 of Euclid’s Elements, Definition 23—the last definition in this book—reads (Fitzpatrick 6):
| Greek | English |
|---|---|
| κγʹ. Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾿ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις. | 23. Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direction, meet with one another in neither (of these directions). |
Definition 1:23 defines the geometric property of parallelism, which Euclid applies only to straight lines in the same plane.
As Thomas Heath points out, the Greek word παράλληλος [parallēlos] means literally alongside one another. It does not occur in Plato, but Aristotle uses it several times with the modern geometric sense. For example:
This is what those persons do who suppose that they are constructing parallel lines; for they fail to see that they are assuming facts which it is impossible to demonstrate unless the parallels exist. (Barnes 238 : Prior Analytics 65 a 5 ff)
According to Liddell and Scott’s Greek-English Lexicon, this form was first introduced by Aristotle (Liddell & Scott 1135).
Fitzpatrick translates Euclid’s εἰς ἄπειρον [eis apeiron] as to infinity, but Heath objects to this on grounds that seem frivolous to me:
εἰς ἄπειρον cannot be translated “to infinity” because these words might seem to suggest a region or place infinitely distant, whereas εἰς ἄπειρον, which seems to be used indifferently with επ’ ἄπειρον, is adverbial, meaning “without limit,” i.e. “indefinitely.” Thus the expression is used of a magnitude being “infinitely divisible,” or of a series of terms extending without limit. (Heath 190)
In his French translation of the Elements (Paris 1814), François Peyrard translates the phrase as à l’infini (Peyrard 5). As Peyrard also notes, the form επ’ ἄπειρον [ep’ apeiron] is found in David Gregory’s Edition of the Elements (Oxford 1703). This was the first complete edition of the Elements in Greek.
Definition 23 tells us the meaning of the technical term parallel, but it does not tell us whether it is actually possible to construct parallel lines. As David Joyce points out, this will be the subject of one of the propositions (theorems) of Book 1:
This definition only defines what it means for straight lines to be parallel; it does not say that there are any parallel lines. Proposition I.31 gives a construction for a line parallel to a given line through a given point. (Joyce)
In his Commentary on the First Book of Euclid’s Elements, the 5th-century philosopher Proclus informs us that another scholar—Posidondius of Apameia—defined parallel lines in a slightly different manner:
This is the way Euclid defines parallel straight lines. But Posidonius says that parallel lines are lines in a single plane which neither converge nor diverge but have all the perpendiculars equal that are drawn to one of them from points on the other. Those lines between which the perpendiculars become progressively [longer or] shorter [intersect somewhere because they] converge upon one another; for a perpendicular can determine both the heights of figures and the distances between lines. Hence when the perpendiculars are equal, the distances between straight lines are equal, but when they become greater or less, the distance increases or decreases and the lines converge on the side on which the perpendiculars are shorter. (Morrow 138. The words in brackets are translated from Barocius’s Latin translation of 1560)
Proclus also discusses other “lines” such as circles and curves, which may also never intersect when extended indefinitiely, but which are not said to be parallel: Euclid reserves this term strictly for straight lines in the same plane (Morrow 138-139). For example, the circumferences of concentric circles lie in the same plane, do not intersect, and maintain the same distance from each other : but they are not said to be parallel.
Another interesting case raised by Proclus is that of a curve and its asymptote (ie a straight line which the curve approaches arbitrarily closely as it is extended indefinitely but which it never intersects). It was possibly to eliminate such cases that Posidonius reworded Euclid’s definition. But Euclid’s definition specifies εὐθεῖαι [eutheiai], or straight lines, and explicitly requires that they be ἐν τῷ αὐτῷ ἐπιπέδῳ [en tōi autōi epipedōi], or in the same plane.
Thomas Heath devotes more than four pages of closely-reasoned text to the history of parallelism and how it has been defined by mathematicians throughout the ages, but while his discussion is of historical interest, it has little to do with Euclid’s Elements.
And that’s a good place to stop.
