the portion MNKM of the circle turned about the fixed points M, N, so that it ultimately comes near to or coincides with the remaining portion MNHDM.

"Then (i) the whole diameter MAN, with all its points, clearly remains in the same position, since otherwise two straight lines would enclose a space (contrary to the first Lemma).

"(ii) Clearly no point K of the circumference NKM falls within or outside the surface enclosed by the diameter MAN and the other part, NHDM, of the circumference, since otherwise, contrary to the nature of the circle, a radius as AK would be less or greater than another radius as AH.

M

H

N

"(iii) Any radius MA can clearly be rectilineally produced only along a single other radius AN, since otherwise (contrary to the second Lemma) two lines assumed straight, e.g. MAN, MAH, would have one and the same common segment.

"(iv) All diameters of the circle obviously cut one another in the centre (Lemma 3 preceding), and they bisect one another there, by the general properties of the circle.

"From all this it is manifest that the diameter MAN divides its circle and the circumference of it just exactly into two equal parts, and the same may be generally asserted for every diameter whatsoever of the same circle; which was to be proved."

Simson observes that the property is easily deduced from III. 31 and 24; for it follows from III. 31 that the two parts of the circle are "similar segments" of a circle (segments containing equal angles, III. Def. 11), and from III. 24 that they are equal to one another.

DEFINITION 18.

Ημικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ ̓ αὐτῆς περιφερείας. κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, δ καὶ τοῦ κύκλου ἐστίν.

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

The last words, "And the centre of the semicircle is the same as that of the circle," are added from Proclus to the definition as it appears in the MSS. Scarburgh remarks that a semicircle has no centre, properly speaking, and thinks that the words are not Euclid's, but only a note by Proclus. I am however inclined to think that they are genuine, if only because of the very futility of an observation added by Proclus. He explains, namely, that the semicircle is the only plane figure that has its centre on its perimeter (!), "so that you may conclude that the centre has three positions, since it may be within the figure, as in the case of a circle, or on the perimeter, as with the semicircle, or outside, as with some conic lines (the single-branch hyperbola presumably)" !

Proclus and Simplicius point out that, in the order adopted by Euclid for these definitions of figures, the first figure taken is that bounded by one line (the circle), then follows that bounded by two lines (the semicircle), then the triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes

T

different kinds of figures bounded by two lines (pp. 159, 14—160, 9). Thus they may be formed

(1) by circumference and circumference, e.g. (a) those forming angles, as a lune (To μnvoedés) and the figure included by two arcs with convexities outward, and (b) the angle-less (dywviov), as the figure included between two concentric circles (the coronal);

(2) by circumference and straight line, e.g. the semicircle or segments of circles (avides is a name given to those less than a semicircle);

(3) by "mixed" line and "mixed" line, e.g. two ellipses cutting one another;

(4) by "mixed" line and circumference, e.g. intersecting ellipse and circle;

(5) by "mixed" line and straight line, e.g. half an ellipse.

Following Def. 18 in the MSS. is a definition of a segment of a circle which was obviously interpolated from 111. Def. 6. Proclus, Martianus Capella and Boethius do not give it in this place, and it is therefore properly omitted.

DEFINITIONS 19, 20, 21.

19. Σχήματα εὐθύγραμμά ἐστι τὰ ὑπὸ εὐθειῶν περιεχόμενα, τρίπλευρα μὲν τὰ ὑπὸ τριῶν, τετράπλευρα δὲ τὰ ὑπὸ τεσσάρων, πολύπλευρα δὲ τὰ ὑπὸ πλειόνων ἢ τεσσάρων εὐθειῶν περιεχόμενα.

20. Τῶν δὲ τριπλεύρων σχημάτων ισόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς.

21.

Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλείαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας.

19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

19.

The latter part of this definition, distinguishing three-sided, four-sided and many-sided figures, is probably due to Euclid himself, since the words τρίπλευρον, τετράπλευρον and πολύπλευρον do not appear in Plato or Aristotle (only in one passage of the Mechanics and of the Problems respectively does even Terparλeupov, quadrilateral, occur). By his use of τετράπλευρον, quadrilateral, Euclid seems practically to have put an end to any ambiguity in the use by mathematicians of the word Terpaywvov, literally "four-angled (figure)," and to have got it restricted to the square. Cf. note on Def. 22.

20.

Isosceles (ioookeλns, with equal legs) is used by Plato as well as Aristotle. Scalene (σκαληνός, with the varient σκαληνής) is used by Aristotle of a triangle with no two sides equal: cf. also Tim. Locr. 98 B. Plato, Euthyphro 12 D,

applies the term "scalene" to an odd number in contrast to "isosceles " used of an even number. Proclus (p. 168, 24) seems to connect it with σkaw, to limp; others make it akin to okoλiós, crooked, aslant. Apollonius uses the same word "scalene" of an oblique circular cone.

Triangles are classified, first with reference to their sides, and then with reference to their angles. Proclus points out that seven distinct species of triangles emerge: (1) the equilateral triangle, (2) three species of isosceles triangles, the right-angled, the obtuse-angled and the acute-angled, (3) the same three varieties of scalene triangles.

Proclus gives an odd reason for the dual classification according to sides and angles, namely that Euclid was mindful of the fact that it is not every triangle that is trilateral also. He explains this statement by reference (p. 165, 22) to a figure which some called barb-like (åkıdocidńs) while Zenodorus called it hollow-angled (Kooywvios). Proclus mentions it again in his note on 1. 22 (p. 328, 21 sqq.) as one of the paradoxes of geometry, observing that it is seen in the figure of that proposition. This "triangle" is merely a quadrilateral with a re-entrant angle; and the idea that it has only three angles is due to the non-recognition of the fourth angle (which is greater than two right angles) as being an angle at all. Since Proclus speaks of the four-sided triangle as "one of the paradoxes in geometry," it is perhaps not safe to assume that the misconception underlying the expression existed

A

in the mind of Proclus alone; but there does not seem to be any evidence that Zenodorus called the figure in question a triangle (cf. Pappus, ed. Hultsch, pp. 1154, 1206).

DEFINITION 22.

Τῶν δὲ τετραπλεύρων σχημάτων τετράγωνον μέν ἐστιν, ὁ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτερόμηκες δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλευρον δέ, ῥόμβος δέ, δ ἰσόπλευρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοειδὲς δὲ τὸ τὰς ἀπεναντίον πλευράς τε καὶ γωνίας ἴσας ἀλλήλαις ἔχον, ὃ οὔτε ἰσόπλευρόν ἐστιν οὔτε ὀρθογώνιον· τὰ δὲ παρὰ ταῦτα τετράπλευρα τραπέζια καλείσθω.

Of quadrilateral figures, a square is that which is both equilateral and rightangled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

TETράуwVOV was already a square with the Pythagoreans (cf. Aristotle, Metaph. 986 a 26), and it is so most commonly in Aristotle; but in De anima II. 3, 414 b 31 it seems to be a quadrilateral, and in Metaph. 1054 b 2, "equal and equiangular Terpάywva," it cannot be anything else but quadrilateral if "equiangular" is to have any sense. Though, by introducing TETрáπλeupov for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of TETрáywvov in much later writers. Thus Heron (Def. 104) speaks of a cube as "contained by six equilateral and equiangular Terpάywva" and Proclus (p. 166, 10) adds to his remark about the "four-sided triangle" that "you might have TeTpάywva with more than the four sides," where Terpάywva can hardly mean squares.

érepóμnkes, oblong (with sides of different length), is also a Pythagorean term. The word right-angled (öploywviov) as here applied to quadrilaterals must mean rectangular (i.e., practically, having all its angles right angles); for, although it is tempting to take the word in the same sense for a

square as for a triangle (i.e. "having one right angle"), this will not do in the case of the oblong, which, unless it were stated that three of its angles are right angles, would not be sufficiently defined.

If it be objected, as it was by Todhunter for example, that the definition of a square assumes more than is necessary, since it is sufficient that, being equilateral, it should have one right angle, the answer is that, as in other cases, the superfluity does not matter from Euclid's point of view; on the contrary, the more of the essential attributes of a thing that could be included in its definition the better, provided that the existence of the thing defined and its possession of all those attributes is proved before the definition is actually used; and Euclid does this in the case of the square by construction in 1. 46, making no use of the definition before that proposition.

The word rhombus (póμßos) is apparently derived from péußw, to turn round and round, and meant among other things a spinning-top. Archimedes uses the term solid rhombus to denote a solid figure made up of two right cones with a common circular base and vertices turned in opposite directions. We can of course easily imagine this solid generated by spinning; and, if the cones were equal, the section through the common axis would be a plane rhombus, which would also be the apparent form of the spinning solid to the

The difficulty in the way of supposing the plane figure to have been named after the solid figure is that in Archimedes the cones forming the solid are not necessarily equal. It is however possible that the solid to which the name was originally given was made up of two equal cones, that the plane rhombus then received its name from that solid, and that Archimedes, in taking up the old name again, extended its signification (cf. J. H. T. Müller, Beiträge zur Terminologie der griechischen Mathematiker, 1860, p. 20). Proclus, while he speaks of a rhombus as being like a shaken, i.e. deformed, square, and of a rhomboid as an oblong that has been moved, tries to explain the rhombus by reference to the appearance of a spinning square (TETρáywvov ῥομβούμενον).

It is true that the definition of a rhomboid says more than is necessary in describing it as having its opposite sides and angles equal to one another. The answer to the objection is the same as the answer to the similar objection to the definition of a square.

Euclid makes no use in the Elements of the oblong, the rhombus, the rhomboid, and the trapezium. The explanation of his inclusion of definitions of the first three is no doubt that they were taken from earlier text-books. From the words "let quadrilaterals other than these be called trapezia," we may perhaps infer that this was a new name or a new application of an old

name.

As Euclid has not yet defined parallel lines and does not anywhere define a parallelogram, he is not in a position to make the more elaborate classification of quadrilaterals attributed by Proclus to Posidonius and appearing also in Heron's Definitions. It may be shown by the following diagram, distinguishing seven species of quadrilaterals.

It will be observed that, while Euclid in the above definition classes as trapezia all quadrilaterals other than squares, oblongs, rhombi, and rhomboids, the word is in this classification restricted to quadrilaterals having two sides (only) parallel, and trapezoid is used to denote the rest. Euclid appears to have used trapezium in the restricted sense of a quadrilateral with two sides parallel in his book repì diapéo ewv (on divisions of figures). Archimedes uses it in the same sense, but in one place describes it more precisely as a trapezium with its two sides parallel.

DEFINITION 23.

Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ' ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Пapános (alongside one another) written in one word does not appear in Plato; but with Aristotle it was already a familiar term.

eis ameipov cannot be translated "to infinity" because these words might seem to suggest a region or place infinitely distant, whereas eis amepov, which seems to be used indifferently with en' amepov, is adverbial, meaning "without limit," i.e. "indefinitely." Thus the expression is used of a magnitude being "infinitely divisible," or of a series of terms extending without limit.

In both directions, ¿p' ékátepa тà μépη, literally "towards both the parts" where "parts" must be used in the sense of "regions" (cf. Thuc. 11. 96).

It is clear that with Aristotle the general notion of parallels was that of straight lines which do not meet, as in Euclid: thus Aristotle discusses the question whether to think that parallels do meet should be called a geometrical or an ungeometrical error (Anal. post. 1. 12, 77 b 22), and (more interesting still in relation to Euclid) he observes that there is nothing surprising in different hypotheses leading to the same error, as one might conclude that parallels meet by starting from the assumption, either (a) that the interior (angle) is greater than the exterior, or (b) that the angles of a triangle make up more than two right angles (Anal. prior. II. 17, 66 a 11).

Another definition is attributed by Proclus to Posidonius, who said that "parallel lines are those which, (being) in one plane, neither converge nor diverge, but have all the perpendiculars equal which are drawn from the points of one line to the other, while such (straight lines) as make the perpendiculars less and less continually do converge to one another; for the perpendicular is enough to define (opičev dúvarai) the heights of areas and the distances between lines. For this reason, when the perpendiculars are equal, the distances between the straight lines are equal, but when they become greater and less, the interval is lessened, and the straight lines converge to one another in the direction in which the less perpendiculars are" (Proclus, p. 176, 6—17).

Posidonius' definition, with the explanation as to distances between straight lines, their convergence and divergence, amounts to the definition quoted by Simplicius (an-Nairīzi, p. 25, ed. Curtze) which described straight lines as parallel if, when they are produced indefinitely both ways, the distance between them, or the perpendicular drawn from either of them to the other, is always equal and not different. To the objection that it should be proved that the distance between two parallel lines is the perpendicular to them Simplicius.