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268

BOOK XI

[XI. DEFF. 11-13

this definition, that there was controversy as to the correctness of describing as a solid angle the "angle" enclosed by fewer than three surfaces (including curved surfaces). Thus the scholiast says that Euclid's definition of a solid angle as made up of three or more plane angles is deficient because it does not e.g. cover the case of the angle of a "fourth part of a sphere," which is contained by more than two surfaces, though not all plane. But he declines to admit that the half-cone forms a solid angle at the vertex, for in that case the vertex of the cone would itself be an angle, and a solid angle would then be formed both by two surfaces and by one surface: "which is not true." Heron on the other hand (Def. 24) distinctly speaks of solid angles which are not contained by plane rectilineal angles, "e.g. the angles of cones." The conception of the latter "angles" as the limit of solid angles with an infinite number of infinitely small constituent plane angles does not appear in the Greek geometers so far as I know.

In modern text-books a polyhedral angle is usually spoken of as formed (or bounded) by three or more planes meeting at a point, or it is the angular opening between such planes at the point where they meet.

DEFINITION 12.

Πυραμίς ἐστι σχῆμα στερεὸν ἐπιπέδοις περιεχόμενον ἀπὸ ἑνὸς ἐπιπέδου πρὸς ἑνὶ σημείῳ συνεστώς.

This definition is by no means too clear, nor is the slightly amplified definition added to it by Heron (Def. 100). A pyramid is the figure brought together to one point, by putting together triangles, from a triangular, quadrilateral or polygonal, that is, any rectilineal, base.

As we might expect, there is great variety in the definitions given in modern text-books. Legendre says a pyramid is the solid formed when several triangular planes start from one point and are terminated at the different sides of one polygonal plane.

Mr H. M. Taylor and Smith and Bryant call it a polyhedron all but one of whose faces meet in a point.

Mehler reverses Legendre's form and gives the content of Euclid's in clearer language. "An n-sided pyramid is bounded by an n-sided polygon as base and n triangles which connect its sides with one and the same point outside it."

Rausenberger points out that a pyramid is the figure cut off from a solid angle formed of any number of plane angles by a plane which intersects the solid angle.

DEFINITION 13.

Πρίσμα ἐστὶ σχῆμα στερεὸν ἐπιπέδοις περιεχόμενον, ὧν δύο τὰ ἀπεναντίον ἴσα τε καὶ ὅμοιά ἐστι καὶ παράλληλα, τὰ δὲ λοιπὰ παραλληλόγραμμα.

Mr H. M. Taylor, followed by Smith and Bryant, defines a prism as a polyhedron all but two of the faces of which are parallel to one straight line.

Mehler calls an n-sided prism a body contained between two parallel planes and enclosed by ʼn other planes with parallel lines of intersection.

Heron's definition of a prism is much wider (Def. 105). Prisms are those figures which are connected (ovváπтоνтα) from a rectilineal base to a rectilineal area by rectilineal collocation (Kar' evlúɣpaμμov σúvbeow). By this Heron must

XI. DEFF. 13—15] NOTES ON DEFINITIONS 11-15

269

apparently mean any convex solid formed by connecting the sides and angles of two polygons in different planes, and each having any number of sides, by straight lines forming triangular faces (where of course two adjacent triangles may be in one plane and so form one quadrilateral face) in the manner shown in the annexed figure, where ABCD, EFG represent the base and opposite.

its

Heron goes on to explain that, if the face opposite to the base reduces to a straight line, and a solid is formed by connecting the base to its extremities by straight lines, as in the other case, the resulting figure is neither a pyramid nor a prism.

E

B

C

Further, he defines parallelogrammic (in the body of the definition parallelsided) prisms as being those prisms which have six faces and have their opposite planes parallel.

DEFINITION 14.

Σφαῖρά ἐστιν, ὅταν ἡμικυκλίου μενούσης τῆς διαμέτρου περιενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆμα.

The scholiast observes that this definition is not properly a definition of a sphere but a description of the mode of generating it. But it will be seen, in the last propositions of Book XIII., why Euclid put the definition in this form. It is because it is this particular view of a sphere which he uses to prove that the vertices of the regular solids which he wishes to "comprehend" in certain spheres do lie on the surfaces of those spheres. He proves in fact that the said vertices lie on semicircles described on certain diameters of the spheres. For the real definition the scholiast refers to Theodosius' Sphaerica. But of course the proper definition was given much earlier. In Aristotle the characteristic of a sphere is that its extremity is equally distant from its centre (rò lσov åπéxei Tov μéσov тò čσxarov, De caelo II. 14, 297 a 24). Heron (Def. 77) uses the same form as that in which Euclid defines the circle: A sphere is a solid figure bounded by one surface, such that all the straight lines falling on it from one point of those which lie within the figure are equal to one another. So the usual definition in the text-books: A sphere is a closed surface such that all points of it are equidistant from a fixed point within it.

DEFINITION 15.

*Αξων δὲ τῆς σφαίρας ἐστὶν ἡ μένουσα εὐθεῖα, περὶ ἣν τὸ ἡμικύκλιον στρέφεται.

That any diameter of a sphere may be called an axis is made clear by Heron (Def. 79). The diameter of the sphere is called an axis, and is any straight line drawn through the centre and bounded in both directions by the sphere, immovable, about which the sphere is moved and turned. Cf. Euclid's Def. 17.

270

BOOK XI

[XI. DEFF. 16-18

DEFINITION 16.

Κέντρον δὲ τῆς σφαίρας ἐστὶ τὸ αὐτό, δ καὶ τοῦ ἡμικυκλίου.

Heron, Def. 78. The middle (point) of the sphere is called its centre; and this same point is also the centre of the hemisphere.

DEFINITION 17.

Διάμετρος δὲ τῆς σφαίρας ἐστὶν εὐθεῖα τις διὰ τοῦ κέντρου ἠγμένη καὶ περατούμενη ἐφ͵ ἑκάτερα τὰ μέρη ὑπὸ τῆς ἐπιφανείας τῆς σφαίρας.

DEFINITION 18.

Κωνός ἐστιν, ὅταν ὀρθογωνίου τριγώνου μενούσης μιᾶς πλευρᾶς τῶν περὶ τὴν ὀρθὴν γωνίαν περιενεχθὲν τὸ τρίγωνον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆμα. κἂν μὲν ἡ μένουσα εὐθεῖα ἴση ᾖ τῇ λοιπῇ [τῇ] περὶ τὴν ὀρθὴν περιφερομένῃ, ὀρθογώνιος ἔσται ὁ κώνος, ἐὰν δὲ ἐλάττων, ἀμβλυγώνιος, ἐὰν δὲ μείζων, ὀξυγώνιος.

This definition, or rather description of the genesis, of a (right) cone is interesting on account of the second sentence distinguishing between rightangled, obtuse-angled and acute-angled cones. This distinction is quite unnecessary for Euclid's purpose and is not used by him in Book XII.; it is no doubt a relic of the method, still in use in Euclid's time, by which the earlier Greek geometers produced conic sections, namely, by cutting right cones only by sections always perpendicular to an edge. With this system the parabola was a section of a right-angled cone, the hyperbola a section of an obtuse-angled cone, and the ellipse a section of an acute-angled cone. The conic sections were so called by Archimedes, and generally until Apollonius, who was the first to give the complete theory of their generation by means of sections not perpendicular to an edge, and from cones which are in general oblique circular cones. Thus Apollonius begins his Conics with the more scientific definition of a cone. If, he says, a straight line infinite in length, and passing always through a fixed point, be made to move round the circumference of a circle which is not in the same plane with the point, so as to pass successively through every point of that circumference, the moving straight line will trace out the surface of a double cone, or two similar cones lying in opposite directions and meeting in the fixed point, which is the apex of each cone. The circle about which the straight line moves is called the base of the cone lying between the said circle and the fixed point, and the axis is defined as the straight line drawn from the fixed point, or the apex, to the centre of the circle forming the base. Apollonius goes on to say that the cone is a scalene or oblique cone except in the particular case where the axis is perpendicular to the base. In this latter case it is a right

cone.

Archimedes called the right cone an isosceles cone. This fact, coupled with the appearance in his treatise On Conoids and Spheroids (7, 8, 9) of sections of acute-angled cones (ellipses) as sections of conical surfaces which are proved to be oblique circular cones by finding their circular sections, makes it sufficiently clear that Archimedes, if he had defined a cone, would have defined it in the same way as Apollonius does.

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DEFINITION 21.

Κύλινδρός ἐστιν, ὅταν ὀρθογωνίου παραλληλογράμμου μενούσης μιᾶς πλ τῶν περὶ τὴν ὀρθὴν γωνίαν περιενεχθὲν τὸ παραλληλόγραμμον εἰς τὸ αὐτὸ 1 ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆμα.

DEFINITION 22.

*Αξων δὲ τοῦ κυλίνδρου ἐστὶν ἡ μένουσα εὐθεῖα, περὶ ἣν τὸ παραλληλόγρα στρέφεται.

DEFINITION 23.

Βάσεις δὲ οἱ κύκλοι οἱ ὑπὸ τῶν ἀπεναντίον περιαγομένων δύο πλε γραφόμενοι.

DEFINITION 24.

Ὅμοιοι κώνοι καὶ κύλινδροί εἰσιν, ὧν οἵ τε ἄξονες καὶ αἱ διάμετροι τῶν βά ἀνάλογόν εἰσιν.

DEFINITION 25.

Κύβος ἐστὶ σχῆμα στερεὸν ὑπὸ ἓξ τετραγώνων ἴσων περιεχόμενον.

DEFINITION 26.

Οκτάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ ὀκτὼ τριγώνων ἴσων καὶ ἰσοπλε περιεχόμενον.

DEFINITION 27.

Εἰκοσάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ εἴκοσι τριγώνων ἴσων καὶ ἰσοπλε περιεχόμενον.

DEFINITION 28.

Δωδεκάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ δώδεκα πενταγώνων ἴσων καὶ ἰσοπλο καὶ ἰσογωνίων περιεχόμενον.

BOOK XI. PROPOSITIONS.

PROPOSITION I.

A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.

For, if possible, let a part AB of the straight line ABC be in the plane of reference, and a part

BC in a plane more elevated.

There will then be in the plane of reference some straight line continuous with AB in a straight line.

Let it be BD;

therefore AB is a common segment of the two straight lines ABC, ABD:

which is impossible, inasmuch as, if we describe a circle with centre B and distance

A

B

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AB, the diameters will cut off unequal circumferences of the circle.

Therefore a part of a straight line cannot be in the plane of reference, and a part in a plane more elevated.

Q. E. D.

I. the plane of reference, Tò &Toкeiμevov éπíπedov, the plane laid down or assumed. more elevated, μετεωροτέρῳ.

2.

There is no doubt that the proofs of the first three propositions are unsatisfactory owing to the fact that Euclid is not able to make any use of his definition of a plane for the purpose of these proofs, and they really depend upon truths which can only be assumed as axiomatic. The definition of a plane as that surface which lies evenly with the straight lines on itself, whatever its exact meaning may be, is nowhere appealed to as a criterion to show whether a particular surface is or is not a plane. If the meaning of it is what I conjecture in the note on Book I., Def. 7 (Vol. 1. p. 171), if, namely, it only tries to express without an appeal to sight what Plato meant by the "middle covering the extremities" (i.e. apparently, in the case of a plane, the fact that a plane looked at edgewise takes the form of a straight line), then it is perhaps possible to connect the definition with a method of generating a plane which

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