Commentary on Proposition VI.33¶
Back to Proposition VI.33.
This proposition as generally given includes a second part relating to sectors of circles, corresponding to the following words added to the enunciation: “and further the sectors, as constructed at the centres” (ἔτι δὲ καὶ οἱ τομεῖς ἅτε [or οἵτε] πρὸς τοῖς κέντροις συνιστάμενοι). There is of course a corresponding addition to the “definition” or “particular statement,” “and further the sector GBOC to the sector HEQF.” These additions are clearly due to Theon, as may be gathered from his own statement in his commentary on the μαθηματικὴ σύνταξις of Ptolemy, “But that sectors in equal circles are to one another as the angles on which they stand, has been proved by me in my edition of the Elements at the end of the sixth book.” Campanus omits them, and P has them only in a later hand in the margin or between the lines. Theon’s proof scarcely needs to be given here in full, as it can easily be supplied. From the equality of the arcs BC, CK he infers [III. 29] the equality of the chords BC, CK. Hence, the radii being equal, the triangles GBC, GCK are equal in all respects [I. 8, 4]. Next, since the arcs BC, CK are equal, so are the arcs BAC, CAK. Therefore the angles at the circumference subtended by the latter, i.e. the angles in the segments BOC, CPK, are equal [III. 27], and the segments are therefore similar [III. Def. 11] and equal [III. 24].
Adding to the equal segments the equal triangles GBC, GCK respectively, we see that
the sectors GBC, GCK are equal.
Thus, in equal circles, sectors standing on equal arcs are equal; and the rest of the proof proceeds as in Euclid’s proposition.
As regards Euclid’s proposition itself, it will be noted that (1), besides quoting the theorem in III. 27 that in equal circles angles which stand on equal arcs are equal, the proof assumes that the angle standing on a greater arc is greater and that standing on a less arc is less. This is indeed a sufficiently obvious deduction from III. 27.
Any equimultiples whatever are taken of the angle BGC and the arc BC, and any equimultiples whatever of the angle EHF and the arc EF. (Accordingly the words “any equimultiples whatever” should have been used in the step immediately preceding the inference that the angles are proportional to the arcs, where the text merely states that there have been taken of the circumference BC and the angle BGC equimultiples BL and BGL.) But, if any multiple of an angle is regarded as being itself an angle, it follows that the restriction in I. Deff. 8, 10, 11, 12 of the term angle to an angle less than two right angles is implicitly given up; as De Morgan says, “the angle breaks prison.” Mr Dodgson (Euclid and his Modern Rivals, p. 193) argues that Euclid conceived of the multiple of an angle as so many separate angles not added together into one, and that, when it is inferred that, where two such multiples of an angle are equal, the arcs subtended are also equal, the argument is that the sum total of the first set of angles is equal to the sum total of the second set, and hence the second set can be broken up and put together again in such amounts as to make a set equal, each to each, to the first set, and then the sum total of the arcs will evidently be equal also. If on the other hand the multiples of the angles are regarded as single angular magnitudes, the equality of the subtending arcs is not inferrible directly from Euclid, because his proof of III. 26 only applies to cases where the angle is less than the sum of two right angles. (As a matter of fact, it is a question of inferring equality of angles or multiples of angles from equality of arcs, and not the converse, so that the reference should have been to III. 27, but this does not affect the question at issue.) Of course it is against this view of Mr Dodgson that Euclid speaks throughout of “the angle BGL” and “the angle EHN” (ἡ ὑπὸ ΒΗΛ γωνία, ἡ ὑπὸ ΕΘΝ γωνία). I think the probable explanation is that here, as in III. 20, 21, 26 and 27, Euclid deliberately took no cognisance of the case in which the multiples of the angles in question would be greater than two right angles. If his attention had been called to the fact that III. 20 takes no account of the case where the segment is less than a semicircle, so that the angle in the segment is obtuse, and therefore the “angle at the centre” in that case (if the term were still applicable) would be
greater than two right angles, Euclid would no doubt have refused to regard the latter as an angle, and would have represented it otherwise, e.g. as the sum of two angles or as what is left when an angle in the true sense is subtracted from four right angles. Here then, if Euclid had been asked what course he would take if the multiples of the angles in question should be greater than two right angles, he would probably have represented them, I think, as being equal to so many right angles plus an angle less than a right angle, or so many times two right angles plus an angle, acute or obtuse. Then the equality of the arcs would be the equality of the sums of so many circumferences, semi-circumferences or quadrants plus arcs less than a semicircle or a quadrant. Hence I agree with Mr Dodgson that VI. 33 affords no evidence of a recognition by Euclid of “angles” greater than two right angles.
Theon adds to his theorem about sectors the Porism that, As the sector is to the sector, so also is the angle to the angle. This corollary was used by Zenodorus in his tract περὶ ἰσομέτρων σχημάτων preserved by Theon in his commentary on Ptolemy’s σύνταξις, unless indeed Theon himself interpolated the words (ὡς δ᾽ ὁ τομεὺς πρὸς τὸν τομέα, ἡ ὑπὸ ΕΘΛ γωνία πρὸς τὴν ὑπὸ ΜΘΛ).