Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

1 It is a fact that Euclid's enunciations not infrequently leave something to be desired in point of clearness and precision. Here he speaks of the triangles having “the angle equal to the angle, namely the angle contained by the equal straight lines” (τὴν γωνίαν τῇ γωνίᾳ ἴσην ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην), only one of the two angles being described in the latter expression (in the accusative), and a similar expression in the dative being left to be understood of the other angle. It is curious too that, after mentioning two “sides,” he speaks of the angles contained by the equal “straight lines,” not “sides.” It may be that he wished to adhere scrupulously, at the outset, to the phraseology of the definitions, where the angle is the inclination to one another of two lines or straight lines. Similarly in the enunciation of I. 5 he speaks of producing the equal “straight lines” as if to keep strictly to the wording of Postulate 2.

2 I agree with Mr H. M. Taylor (Euclid, p. ix) that it is best to abandon the traditional translation of “each to each,” which would naturally seem to imply that all the four magnitudes are equal rather than (as the Greek ὲκατέρα ὲκατέρᾳ does) that one is equal to one and the other to the other.

3 Here we have the word base used for the first time in the Elements. Proclus explains it (p. 236, 12-15) as meaning (1), when no side of a triangle has been mentioned before, the side “which is on a level with the sight” (τὴν πρὸς τῇ ὄψει κειμένην), and (2), when two sides have already been mentioned, the third side. Proclus thus avoids the mistake made by some modern editors who explain the term exclusively with reference to the case where two sides have been mentioned before. That this is an error is proved (1) by the occurrence of the term in the enunciations of I. 37 etc. about triangles on the same base and equal bases, (2) by the application of the same term to the bases of parallelograms in I. 35 etc. The truth is that the use of the term must have been suggested by the practice of drawing the particular side horizontally, as it were, and the rest of the figure above it. The base of a figure was therefore spoken of, primarily, in the same sense as the base of anything else, e.g. of a pedestal or column; but when, as in I. 5, two triangles were compared occupying other than the normal positions which gave rise to the name, and when two sides had been previously mentioned, the base was, as Proclus says, necessarily the third side.

4 ὑποτείνειν ὑπό, “to stretch under,” with accusative.

5 The full Greek expression would be ἡ ὑπὸ τῶν ΒΑ, ΑΓ περιεχομένη γωνία, “the angle contained by the (straight lines) BA, AC.” But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books X.—XIII.), to use the abbreviation αἱ ΒΑΓ for αἱ ΒΑ, ΑΓ, “the (straight lines) BA, AC.” Thus, on περιεχομένη being dropped, the expression would become first ἡ ὑπὸ τῶν ΒΑΓ γωνία, then ἡ ὑπὸ ΒΑΓ γωνία, and finally ἡ ὑπὸ ΒΑΓ, without γωνία, as we regularly find it in Euclid.

6 The difference between the technical use of the passive ἐφαρμόζεσθαι “to be applied (to),” and of the active ἐφαρμόζειν “to coincide (with)” has been noticed above (note on Common Notion 4, pp. 224-5).

7 Heiberg (Paralipomena su Euklid in Hermes, XXXVIII., 1903, p. 56) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairīzī (Codex Leidensis 399, 1, ed. Besthorn-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.E.D., which shows that they constitute a scholium only. They were doubtless added by some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F, the straight line BC coincides with the straight line EF, an inference which really follows from the definition of a straight line and Post. 1; and no doubt the Postulate that “Two straight lines cannot enclose a space” (afterwards placed among the Common Notions) was interpolated at the same time.

8 Where (as here) Euclid's conclusion merely repeats the enunciation word for word, I shall avoid the repetition and write “Therefore etc.” simply.