F

G

through the common angular point are in one straight line. If ABCF, CDEG be similar and similarly situated parallelograms, so that BCG, DCF are straight lines, and if the diagonals AC, CE be drawn, the triangles ABC, CDE are similar and are placed exactly as described in VI. 32, so that AC, CE are in a straight line. Hence Simson suggests that there may have been, in addition to the indirect demonstration in vi. 26, a direct proof covering the case just given which may have used the result of vi. 32. I think however

that the place given to the latter proposition in Book vi. is against this view.

E

PROPOSITION 33.

In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.

Let ABC, DEF be equal circles, and let the angles BGC, EHF be angles at their centres G, H, and the angles BAC, EDF angles at the circumferences;

I say that, as the circumference BC is to the circumference EF, so is the angle BGC to the angle EHF, and the angle BAC to the angle EDF.

For let any number of consecutive circumferences CK, KL be made equal to the circumference BC,

and any number of consecutive circumferences FM, MN equal to the circumference EF;

and let GK, GL, HM, HN be joined.

Then, since the circumferences BC, CK, KL are equal to one another,

the angles BGC, CGK, KGL are also equal to one another;

H. E. II.

[111. 27] 18

therefore, whatever multiple the circumference BL is of BC, that multiple also is the angle BGL of the angle BGC.

For the same reason also,

whatever multiple the circumference NE is of EF, that multiple also is the angle NHE of the angle EHF.

If then the circumference BL is equal to the circumference EN, the angle BGL is also equal to the angle EHN; [111.27] if the circumference BL is greater than the circumference EN, the angle BGL is also greater than the angle EHN; and, if less, less.

There being then four magnitudes, two circumferences BC, EF, and two angles BGC, EHF,

there have been taken, of the circumference BC and the angle BGC equimultiples, namely the circumference BL and the angle BGL,

and of the circumference EF and the angle EHF equimultiples, namely the circumference EN and the angle EHN. And it has been proved that,

if the circumference BL is in excess of the circumference EN, the angle BGL is also in excess of the angle EHN; if equal, equal;

and if less, less.

Therefore, as the circumference BC is to EF, so is the angle BGC to the angle EHF. [v. Def. 5] But, as the angle BGC is to the angle EHF, so is the angle BAC to the angle EDF; for they are doubles respectively.

Therefore also, as the circumference BC is to the circumference EF, so is the angle BGC to the angle EHF, and the angle BAC to the angle EDF.

Therefore etc.

Q. E. D.

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" (erɩ dè kai oi toμeîs åte [or οἶτε] 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 μabnμatiky σúvraέis 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 [1. 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 [111. 27], and the segments are therefore similar [111. Def. 11] and equal [111. 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 111. 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.

(2) 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 111. 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 ΕΗΝ” (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 ouvrages, unless indeed Theon himself interpolated the words (ὡς δ' ὁ τομεὺς πρὸς τὸν τομέα, ἡ ὑπὸ ΕΘΛ γωνία πρὸς τὴν ὑπὸ ΜΘΛ).

BOOK VII.

DEFINITIONS.

I. An unit is that by virtue of which each of the things that exist is called one.

2. A number is a multitude composed of units.

3. A number is a part of a number, the less of the greater, when it measures the greater;

4.

but parts when it does not measure it.

5. The greater number is a multiple of the less when it is measured by the less.

6. An even number is that which is divisible into two equal parts.

7. An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

8. An even-times even number is that which is measured by an even number according to an even number.

9. An even-times odd number is that which is measured by an even number according to an odd number.

IO. An odd-times odd number is that which is measured by an odd number according to an odd number.