Commentary on Book III Definitions¶
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DEFINITION 1.¶
Ἴσοι κύκλοι εἰσίν, ὧν αἱ διάμετροι ἴσαι εἰσίν, ἢ ὧν αἱ ἐκ τῶν κέντρων ἴσαι εἰσίν.
Many editors have held that this should not have been included among definitions. Some, e.g. Tartaglia, would call it a postulate; others, e.g. Borelli and Playfair, would call it an axiom; others again, as Billingsley and Clavius, while admitting it as a definition, add explanations based on the mode of constructing a circle; Simson and Pfleiderer hold that it is a theorem. I think however that Euclid would have maintained that it is a definition in the proper sense of the term; and certainly it satisfies Aristotle’s requirement that a “definitional statement” (ὁριστικὸς λόγος) should not only state the fact (τὸ ὅτι) but should indicate the cause as well (De anima II. 2, 413 a 13). The equality of circles with equal radii can of course be proved by superposition, but, as we have seen, Euclid avoided this method wherever he could, and there is nothing technically wrong in saying “By equal circles I mean circles with equal radii.” No flaw is thereby introduced into the system of the Elements; for the definition could only be objected to if it could be proved that the equality predicated of the two circles in the definition was not the same thing as the equality predicated of other equal figures in the Elements on the basis of the Congruence-Axiom, and, needless to say, this cannot be proved because it is not true. The existence of equal circles (in the sense of the definition) follows from the existence of equal straight lines and I. Post. 3.
The Greeks had no distinct word for radius, which is with them, as here, the (straight line drawn) from the centre ἡ ἐκ τοῦ κέντρου (εὐθεῖα); and so definitely was the expression appropriated to the radius that ἐκ τοῦ κέντρου was used without the article as a predicate, just as if it were one word. Thus, e.g., in III. 1 ἐκ κέντρου γάρ means “for they are radii”: cf. Archimedes, On the Sphere and Cylinder II. 2, ἡ ΒΕ ἐκ τοῦ κέντρου ἐστὶ τοῦ… κύκλου, BE is a radius of the circle.
DEFINITION 2.¶
Εὐθεῖα κύκλου ἐφάπτεσθαι λέγεται, ἥτις ἁπτομένη τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνει τὸν κύκλον.
Euclid’s phraseology here shows the regular distinction between ἅπτεσθαι and its compound ἐφάπτεσθαι, the former meaning “to meet” and the latter “to touch.” The distinction was generally observed by Greek geometers from Euclid onwards. There are however exceptions so far as ἅπτεσθαι is concerned; thus it means “to touch” in Eucl. IV. Def. 5 and sometimes in Archimedes. On the other hand, ἐφάπτεσθαι is used by Aristotle in certain cases where the orthodox geometrical term would be ἅπτεσθαι. Thus in Meteorologica III. 5 (376 b 9) he says a certain circle will pass through all the angles (ἁπασῶν ἐφάψεται τῶν γωνιῶν), and (376 a 6) $\text{M}$ will lie on a given (circular) circumference (δεδομένης περιφερείας ἐφάψεται τὸ Μ). We shall find ἅπτεσθαι 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 $\text{M}$ is a given circle, just as in Pappus ἅψεται τὸ σημεῖον θέσει δεδομένης εὐθείας 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 III. 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.; III. 8 is required for the second proof of III. 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 III. 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 I. 5).
External and internal contact are not distinguished in Euclid until III. 11, 12, though the figure of III. 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 III. 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 ὑπό τε εὐθείας καὶ κύκλου περιφερείας.
DEFINITION 7.¶
Τμήματος δὲ γωνία ἐστὶν ἡ περιεχομένη ὑπό 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” (κερατοειδής), 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 I. 5 above (Vol. I. 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. I. 24, 41 b 13 sqq.) that the theorem of I. 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 I. 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 τοῦ τμήματος ληφθῇ τι σημεῖον καὶ ἀπ᾽ αὐτοῦ ἐπὶ τὰ πέρατα τῆς εὐθείας, ἥ ἐστι βάσις τοῦ τμήματος, ἐπιζευχθῶσιν εὐθεῖαι, ἡ περιεχομένη γωνία ὑπὸ τῶν ἐπιζευχθεισῶν εὐθειῶν.
DEFINITION 9.¶
Ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν εὐθεῖαι ἀπολαμβάνωσί τινα περιφέρειαν, ἐπ᾽ ἐκείνης λέγεται βεβηκέναι ἡ γωνία.
DEFINITION 10.¶
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, σκυτοτομικὸς τομεύς, which suggested the name τομεύς for a sector of a circle. The derivation of the name from a resemblance of shape is parallel to the use of ἄρβηλος (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-Nairīzī (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-Nairīzī 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 (περὶ διαιρέσεων) discovered in an Arabic MS. and edited by Woepcke (cf. Vol. I. 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. $\text{I}_3$, 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 11.¶
Ὅμοια 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.