cases where the orthodox geometrical term would be aπтeσ0α. Thus in Meteorologica III. 5 (376 b 9) he says a certain circle will pass through all the angles (anao@v épaveraι Twv ywvwv), and (376 a 6) M will lie on a given (circular) circumference (dedoμévns tepipepeías épáveтa Tò M). We shall find aTToba used in these senses in Book IV. Deff. 2, 6 and Deff. 1, 3 respectively. The latter of the two expressions quoted from Aristotle means that the locus of M is a given circle, just as in Pappus ἅψεται τὸ σημεῖον θέσει δεδομένης eveias means that the locus of the point is a straight line given in position.

DEFINITION 3.

Κύκλοι ἐφάπτεσθαι ἀλλήλων λέγονται οἵτινες ἁπτόμενοι ἀλλήλων οὐ τέμνουσιν ἀλλήλους.

Todhunter remarks that different opinions have been held as to what is, or should be, included in this definition, one opinion being that it only means that the circles do not cut in the neighbourhood of the point of contact, and that it must be shown that they do not cut elsewhere, while another opinion is that the definition means that the circles do not cut at all. Todhunter thinks the latter opinion correct. I do not think this is proved; and I prefer to read the definition as meaning simply that the circles meet at a point but do not cut at that point. I think this interpretation preferable for the reason that, although Euclid does practically assume in III. 11-13, without stating, the theorem that circles touching at one point do not intersect anywhere else, he has given us, before reaching that point in the Book, means for proving for ourselves the truth of that statement. In particular, he has given us the propositions 111. 7, 8 which, taken as a whole, give us more information as to the general nature of a circle than any other propositions that have preceded, and which can be used, as will be seen in the sequel, to solve any doubts arising out of Euclid's unproved assumptions. Now, as a matter of fact, the propositions are not used in any of the genuine proofs of the theorems in Book III.; 111. 8 is required for the second proof of 111. 9 which Simson selected in preference to the first proof, but the first proof only is regarded by Heiberg as genuine. Hence it would not be easy to account for the appearance of 111. 7, 8 at all unless as affording means of answering possible objections (cf. Proclus' explanation of Euclid's reason for inserting the second part of 1. 5).

External and internal contact are not distinguished in Euclid until 111. 11, 12, though the figure of 111. 6 (not the enunciation in the original text) represents the case of internal contact only. But the definition of touching circles here given must be taken to imply so much about internal and external contact respectively as that (a) a circle touching another internally must, immediately before "meeting" it, have passed through points within the circle that it touches, and (b) a circle touching another externally must, immediately before meeting it, have passed through points outside the circle which it touches. These facts must indeed be admitted if internal and external are to have any meaning at all in this connexion, and they constitute a minimum admission necessary to the proof of 111. 6.

DEFINITION 4.

Ἐν κύκλῳ ἴσον ἀπέχειν ἀπὸ τοῦ κέντρου εὐθεῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπ ̓ αὐτὰς κάθετοι ἀγόμεναι ἴσαι ὦσιν.

DEFINITION 5.

Μεῖζον δὲ ἀπέχειν λέγεται, ἐφ' ἣν ἡ μείζων κάθετος πίπτει.

DEFINITION 6.

Τμῆμα κύκλου ἐστὶ τὸ περιεχόμενον σχῆμα υπό τε εὐθείας καὶ κύκλου περιφερείας.

DEFINITION 7.

Τμήματος δὲ γωνία ἐστὶν ἡ περιεχομένη υπό τε εὐθείας καὶ κύκλου περιφερείας. This definition is only interesting historically. The angle of a segment, being the "angle" formed by a straight line and a "circumference," is of the kind described by Proclus as "mixed." A particular "angle" of this sort is the "angle of a semicircle," which we meet with again in III. 16, along with the so-called "horn-like angle" (Kepaтoeds), the supposed "angle" between a tangent to a circle and the circle itself. The "angle of a semicircle" occurs once in Pappus (VII. p. 670, 19), but it there means scarcely more than the corner of a semicircle regarded as a point to which a straight line is directed. Heron does not give the definition of the angle of a segment, and we may conclude that the mention of it and of the angle of a semicircle in Euclid is a survival from earlier text-books rather than an indication that Euclid considered either to be of importance in elementary geometry (cf. the note on III. 16 below).

We have however, in the note on 1. 5 above (Vol. 1. pp. 252—3), seen evidence that the angle of a segment had played some part in geometrical proofs up to Euclid's time. It would appear from the passage of Aristotle there quoted (Anal. prior. 1. 24, 41 b 13 sqq.) that the theorem of 1. 5 was, in the text-books immediately preceding Euclid, proved by means of the equality of the two "angles of" any one segment. This latter property must therefore have been regarded as more elementary (for whatever reason) than the theorem of 1. 5; indeed the definition as given by Euclid practically implies the same thing, since it speaks of only one "angle of a segment," namely "the angle contained by a straight line and a circumference of a circle." Euclid abandoned the actual use of the "angle" in question, but no doubt thought it unnecessary to break with tradition so far as to strike the definition out also.

DEFINITION 8.

Ἐν τμήματι δὲ γωνία ἐστίν, ὅταν ἐπὶ τῆς περιφερείας του τμήματος ληφθῇ τι σημεῖον καὶ ἀπ ̓ αὐτοῦ ἐπὶ τὰ πέρατα τῆς εὐθείας, ἢ ἐστι βάσις τοῦ τμήματος, ἐπιζευχθῶσιν εὐθεῖαι, ἡ περιεχομένη γωνία ὑπὸ τῶν ἐπιζευχθεισῶν εὐθειῶν.

DEFINITION 9.

Ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν εὐθεῖαι ἀπολαμβάνωσι τινα περιφέρειαν, ἐπ' ἐκείνης λέγεται βεβηκέναι ἡ γωνία.

DEFINITION IO.

Τομεὺς δὲ κύκλου ἐστίν, ὅταν πρὸς τῷ κέντρῳ τοῦ κύκλου συσταθῇ γωνία, τὸ περιεχόμενον σχῆμα ὑπό τε τῶν τὴν γωνίαν περιεχουσῶν εὐθειῶν καὶ τῆς ἀπολαμβανομένης ὑπ' αὐτῶν περιφερείας.

A scholiast says that it was the shoemaker's knife, σкvтотоμIKÒS TOμEÚS, which suggested the name Toμeús for a sector of a circle. The derivation of the name from a resemblance of shape is parallel to the use of apßnλos (also a shoemaker's knife) to denote the well known figure of the Book of Lemmas partly attributed to Archimedes.

A wider definition of a sector than that given by Euclid is found in a Greek scholiast (Heiberg's Euclid, Vol. v. p. 260) and in an-Nairizi (ed. Curtze, p. 112). “There are two varieties of sectors; the one kind have the angular vertices at the centres, the other at the circumferences. Those others which have their vertices neither at the circumferences nor at the centres, but at some other points, are for that reason not called sectors but sector-like figures (τομοειδῆ σχήματα).” The exact agreement between the scholiast and an-Nairizi suggests that Heron was the authority for this explanation.

The sector-like figure bounded by an arc of a circle and two lines drawn from its extremities to meet at any point actually appears in Euclid's book On divisions (Tepi diapéσewv) discovered in an Arabic мs. and edited by Woepcke (cf. Vol. 1. pp. 8-10 above). This treatise, alluded to by Proclus, had for its object the division of figures such as triangles, trapezia, quadrilaterals and circles, by means of straight lines, into parts equal or in given ratios. One proposition e.g. is, To divide a triangle into two equal parts by a straight line passing through a given point on one side. The proposition (28) in which the quasi-sector occurs is, To divide such a figure by a straight line into two equal parts. The solution in this case is given by Cantor (Gesch. d. Math. 1, pp. 287-8).

If ABCD be the given figure, E the middle point of BD and EC at right angles to BD,

the broken line AEC clearly divides the figure into two equal parts.

Join AC, and draw EF parallel to it meeting AB in F.

Join CF, when it is seen that CF divides the figure into two equal parts.

DEFINITION II.

B

D

E

Όμοια τμήματα κύκλων ἐστὶ τὰ δεχόμενα γωνίας ἴσας, ἢ ἐν οἷς αἱ γωνίαι ἴσαι ἀλλήλαις εἰσίν.

De Morgan remarks that the use of the word similar in "similar segments" is an anticipation, and that similarity of form is meant. He adds that the definition is a theorem, or would be if "similar" had taken its final meaning.

BOOK III. PROPOSITIONS.

PROPOSITION I.

To find the centre of a given circle.

Let ABC be the given circle;

thus it is required to find the centre of the circle ABC. Let a straight line AB be drawn

5 through it at random, and let it be bisected

at the point D;

from D let DC be drawn at right angles to AB and let it be drawn through to E; let CE be bisected at F;

10 I say that F is the centre of the circle ABC.

15

For suppose it is not, but, if possible,

let G be the centre,

and let GA, GD, GB be joined.

Then, since AD is equal to DB,

and DG is common,

G

B

E

the two sides AD, DG are equal to the two sides BD, DG respectively;

and the base GA is equal to the base GB, for they are 20 radii ;

therefore the angle ADG is equal to the angle GDB. [1. 8] But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right;

25 therefore the angle GDB is right.

[1. Def. 10]

30

But the angle FDB is also right;

therefore the angle FDB is equal to the angle GDB, the greater to the less: which is impossible.

Therefore G is not the centre of the circle ABC.

Similarly we can prove that neither is any other point except F.

Therefore the point F is the centre of the circle ABC. PORISM. From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at 35 right angles, the centre of the circle is on the cutting straight line.

Q. E. F.

12. For suppose it is not. This is expressed in the Greek by the two words Mǹ yáp, but such an elliptical phrase is impossible in English.

17. the two sides AD, DĜ are equal to the two sides BD, DG respectively. As before observed, Euclid is not always careful to put the equals in corresponding order. The text here has "GD, DB."

Todhunter observes that, when, in the construction, DC is said to be produced to E, it is assumed that D is within the circle, a fact which Euclid first demonstrates in III. 2. This is no doubt true, although the word dixow, "let it be drawn through," is used instead of exßeßλnow, "let it be produced." And, although it is not necessary to assume that D is within the circle, it is necessary for the success of the construction that the straight line drawn through D at right angles to AB shall meet the circle in two points (and no more): an assumption which we are not entitled to make on the basis of what has gone before only.

Hence there is much to be said for the alternative procedure recommended by De Morgan as preferable to that of Euclid. De Morgan would first prove the fundamental theorem that "the line which bisects a chord perpendicularly must contain the centre," and then make III. I, III. 25 and iv. 5 immediate corollaries of it. The fundamental theorem is a direct consequence of the theorem that, if P is any point equidistant from A and B, then P lies on the straight line bisecting AB perpendicularly. We then take any two chords AB, AC of the given circle and draw DO, EO bisecting them perpendicularly. Unless BA, AC are in one straight line, the straight lines DO, EO must meet in some point 0 (see note on iv. 5 for possible methods of proving this). And, since both DO, EO must contain the centre, O must be the centre.

B

A

This method, which seems now to be generally preferred to Euclid's, has the advantage of showing that, in order to find the centre of a circle, it is sufficient to know three points on the circumference. If therefore two circles have three points in common, they must have the same centre and radius, so that two circles cannot have three points in common without coinciding entirely. Also, as indicated by De Morgan, the same construction enables us (1) to draw the complete circle of which a segment or arc only is given (111. 25), and (2) to circumscribe a circle to any triangle (IV. 5).