Commentary on Book XI Definitions

Back to Book XI Definitions.

DEFINITION 1.

Στερεόν ἐστι τὸ μῆκος καὶ πλάτος καὶ βάθος ἔχον.

This definition was evidently traditional, as may be inferred from a number of passages in Plato and Aristotle. Thus Plato speaks (Sophist, 235 D) of making an imitation of a model (παράδειγμα) “in length and breadth and depth” and (Laws, 817 E) of “the art of measuring length, surface and depth” as one of three μαθήματα. Depth, the third dimension, is used alone as a description of “body” by Aristotle, the term being regarded as connoting the other two dimensions; thus (Metaph. 1020 a 13, 11) “length is a line, breadth a surface, and depth body”; “that which is continuous in one direction is length, in two directions breadth, and in three depth.” Similarly Plato (Rep. 528 B, D), when reconsidering his classification of astronomy as next to (plane) geometry: “although the science dealing with the additional dimension of depth is next in order, yet, owing to the fact that it is studied absurdly, I passed it over and put next to geometry astronomy, the motion of (bodies having) depth.” In Aristotle (Topics VI. 5, 142 b 24) we find “the definition of body, that which has three dimensions (διαστάσεις)”; elsewhere he speaks of it as “that which has all the dimensions” (De caelo I. 1, 268 b 6), “that which has dimension every way” (τὸ πάντῃ διάστασιν ἔχον, Metaph. 1066 b 32) etc. In the Physics (IV. 1, 208 b 13 sqq.) he speaks of the “dimensions” as six, dividing each of the three into two opposites, “up and down, before and behind, right and left,” though of course, as he explains, these terms are relative.

Heron, as might be expected, combines the two forms of the definition. “A solid body is that which has length, breadth, and depth: or that which possesses the three dimensions.” (Def. 11.)

Similarly Theon of Smyrna (p. 111, 19, ed. Hiller): “that which is extended (διαστατόν) and divisible in three directions is solid, having length, breadth and depth.”

DEFINITION 2.

Στερεοῦ δὲ πέρας ἐπιφάνεια.

In like manner Aristotle says (Metaph. 1066 b 23) that the notion (λόγος) of body is “that which is bounded by surfaces” (ἐπιπέδοις in this case) and (Metaph. 1060 b 15) “surfaces (ἐπιφάνειαι) are divisions of bodies.”

So Heron (Def. 11): “Every solid is bounded (περατοῦται) by surfaces, and is produced when a surface is moved from a forward position in a backward direction.”

DEFINITION 3.

Εὐθεῖα πρὸς ἐπίπεδον ὀρθή ἐστιν, ὅταν πρὸς πάσας τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ἐπιπέδῳ ὀρθὰς ποιῇ γωνίας.

This definition and the next are given almost word for word by Heron (Def. 115).

That a straight line can be so related to a plane as described in Def. 3 is established in XI. 4. The fact has been made the basis of a definition of a plane which is attributed by Crelle to Fourier, and is as follows. “A plane is formed by the totality of all the straight lines which, passing through one and the same point of a straight line in space, stand perpendicular to it.” Stated in this form, the definition is open to the objection that the conception of a right angle, involving the measurement of angles, presupposes a plane, inasmuch as the measurement of angles depends ultimately upon the superposition of two planes and their coincidence throughout when two lines in one coincide with two lines in the other respectively. Cf. my note on I. Def. 7, Vol. I. pp. 173—5.

DEFINITION 4.

Ἐπίπεδον πρὸς ἐπίπεδον ὀρθόν ἐστιν, ὅταν αἱ τῇ κοινῇ τομῇ τῶν ἐπιπέδων πρὸς ὀρθὰς ἀγόμεναι εὐθεῖαι ἐν ἑνὶ τῶν ἐπιπέδων τῷ λοιπῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν.

Both this definition and Def. 6 use the common section of two planes, though it is not till XI. 3 that this common section is proved to be a straight line. The definition however, just like Def. 3, is legitimate, because the object is to explain the meaning of terms, not to prove anything.

The definition of perpendicular planes is made by Legendre a particular case of Def. 6, the limiting case, namely, where the angle representing the “inclination of a plane to a plane” is a right angle.

DEFINITION 5.

Εὐθείας πρὸς ἐπίπεδον κλίσις ἐστίν, ὅταν ἀπὸ τοῦ μετεώρου πέρατος τῆς εὐθείας ἐπὶ τὸ ἐπίπεδον κάθετος ἀχθῇ, καὶ ἀπὸ τοῦ γενομένου σημείου ἐπὶ τὸ ἐν τῷ ἐπιπέδῳ πέρας τῆς εὐθείας εὐθεῖα ἐπιζευχθῇ, ἡ περιεχομένη γωνία ὑπὸ τῆς ἀχθείσης καὶ τῆς ἐφεστώσης.

In other words, the inclination of a straight line to a plane is the angle between the straight line and its projection on the plane. This angle is of course less than the angle between the straight line and any other straight line in the plane through the intersection of the straight line and plane; and the fact is sometimes made the subject of a proposition in modern text-books. It is easily proved by means of the propositions XI. 4, I. 19 and 18.

DEFINITION 6.

Ἐπιπέδου πρὸς ἐπίπεδον κλίσις ἐστὶν ἡ περιεχομένη ὀξεῖα γωνία ὑπὸ τῶν πρὸς ὀρθὰς τῇ κοινῇ τομῇ ἀγομένων πρὸς τῷ αὐτῷ σημείῳ ἐν ἑκατέρῳ τῶν ἐπιπέδων.

When two planes meet in a straight line, they form what is called in modern text-books a dihedral angle, which is defined as the opening or angular opening between the two planes. This dihedral angle is an “angle” altogether different in kind from a plane angle, as again it is different from a solid angle as defined by Euclid (i.e. a trihedral, tetrahedral, etc. angle). Adopting for the moment Apollonius’ conception of an angle as the “bringing together of a surface or solid towards one point under a broken line or surface” (Proclus, p. 123, 16), we may regard a dihedral angle as the bringing together of the broken surface formed by two intersecting planes not to a point but to a straight line, namely the intersection of the planes. Legendre, in a proposition on the subject, applied provisionally the term corner to describe the dihedral angle between two planes; and this would be a better word, I think, than opening to use in the definition.

The distinct species of “angle” which we call dihedral is, however, measured by a certain plane angle, namely that which Euclid describes in the present definition and calls the inclination of a plane to a plane, and which in some modern text-books is called the plane angle of the dihedral angle.

It is necessary to show that this plane angle is a proper measure of the dihedral angle, and accordingly Legendre has a proposition to this effect. In order to prove it, it is necessary to show that, given two planes meeting in a straight line,

  1. the plane angle in question is the same at all points of the straight line forming the common section;

  2. if the dihedral angle between two planes increases or diminishes in a certain ratio, the plane angle in question will increase or diminish in the same ratio.

  1. If MAN, MAP be two planes intersecting in MA, and if AN, AP be drawn in the planes respectively and at right angles to MA, the angle NAP is the inclination of the plane to the plane or the plane angle of the dihedral angle.

Let MC, MB be also drawn in the respective planes at right angles to MA.

Then since, in the plane MAN, MC and AN are drawn at right angles to the same straight line MA,

MC, AN are parallel.

For the same reason, MB, AP are parallel.

Therefore [XI. 10] the angle BMC is equal to the angle PAN.

And M may be any point on MA. Therefore the plane angle described in the definition is the same at all points of AM.

M C E B A N D P

  1. In the plane NAP draw the arc NDP of any circle with centre A, and draw the radius AD.

Now the planes NAP, CMB, being both at right angles to the straight line MA, are parallel; [XI. 14] therefore the intersections AD, ME of these planes with the plane MAD are parallel, [XI. 16] and consequently the angles BME, PAD are equal. [XI. 10]

If now the plane angle NAD were equal to the plane angle DAP, the dihedral angle NAMD would be equal to the dihedral angle DAMP; for, if the angle PAD were applied to the angle DAN, AM remaining the same, the corresponding dihedral angles would coincide.

Successive applications of this result show that, if the angles NAD, DAP each contain a certain angle a certain number of times, the dihedral angles NAMD, DAMP will contain the corresponding dihedral angle the same number of times respectively.

Hence, where the angles NAD, DAP are commensurable, the dihedral angles corresponding to them are in the same ratio.

Legendre then extends the proof to the case where the plane angles are incommensurable by reference to an exactly similar extension in his proposition corresponding to Euclid VI. 1, for which see the note on that proposition.

Modern text-books make the extension by an appeal to limits.

DEFINITION 7.

Ἐπίπεδον πρὸς ἐπίπεδον ὁμοίως κεκλίσθαι λέγεται καὶ ἕτερον πρὸς ἕτερον, ὅταν αἱ εἰρημέναι τῶν κλίσεων γωνίαι ἴσαι ἀλλήλαις ὦσιν.

DEFINITION 8.

Παράλληλα ἐπίπεδά ἐστι τὰ ἀσύμπτωτα.

Heron has the same definition of parallel planes (Def. 115). The Greek word which is translated “which do not meet” is ἀσύμπτωτα, the term which has been adopted for the asymptotes of a curve.

DEFINITION 9.

Ὅμοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων ἐπιπέδων περιεχόμενα ἴσων τὸ πλῆθος.

DEFINITION 10.

Ἴσα δὲ καὶ ὅμοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων ἐπιπέδων περιεχόμενα ἴσων τῷ πλήθει καὶ τῷ μεγέθει.

These definitions, the second of which practically only substitutes the words “equal and similar” for the word “similar” in the first, have been the mark of much criticism.

Simson holds that the equality of solid figures is a thing which ought to be proved, by the method of superposition, or otherwise, and hence that Def. 10 is not a definition but a theorem which ought not to have been placed among the definitions. Secondly, he gives an example to show that the definition or theorem is not universally true. He takes a pyramid and then erects on the base, on opposite sides of it, two equal pyramids smaller than the first. The addition and subtraction of these pyramids respectively from the first give two solid figures which satisfy the definition but are clearly not equal (the smaller having a re-entrant angle); whence it also appears that two unequal solid angles may be contained by the same number of equal plane angles.

Maintaining then that Def. 10 is an interpolation by “an unskilful hand,” Simson transfers to a place before Def. 9 the definition of a solid angle, and then defines similar solid figures as follows:

Similar solid figures are such as have all their solid angles equal, each to each, and which are contained by the same number of similar planes.

Legendre has an invaluable discussion of the whole subject of these definitions (Note XII., pp. 323—336, of the 14th edition of his Éléments de Géométrie). He remarks in the first place that, as Simson said, Def. 10 is not properly a definition, but a theorem which it is necessary to prove; for it is not evident that two solids are equal for the sole reason that they have an equal number of equal faces, and, if true, the fact should be proved by superposition or otherwise. The fault of Def. 10 is also common to Def. 9. For, if Def. 10 is not proved, one might suppose that there exist two unequal and dissimilar solids with equal faces; but, in that case, according to Definition 9, a solid having faces similar to those of the two first would be similar to both of them, i.e. to two solids of different form: a conclusion implying a contradiction or at least not according with the natural meaning of the word “similar.”

What then is to be said in defence of the two definitions as given by Euclid? It is to be observed that the figures which Euclid actually proves equal or similar by reference to Deff. 9, 10 are such that their solid angles do not consist of more than three plane angles; and he proves sufficiently clearly that, if three plane angles forming one solid angle be respectively equal to three plane angles forming another solid angle, the two solid angles are equal. If now two polyhedra have their faces equal respectively, the corresponding solid angles will be made up of the same number of plane angles, and the plane angles forming each solid angle in one polyhedron will be respectively equal to the plane angles forming the corresponding solid angle in the other. Therefore, if the plane angles in each solid angle are not more than three in number, the corresponding solid angles will be equal. But if the corresponding faces are equal, and the corresponding solid angles equal, the solids must be equal; for they can be superposed, or at least they will be symmetrical with one another. Hence the statement of Deff. 9, 10 is true and admissible at all events in the case of figures with trihedral angles, which is the only case taken by Euclid.

Again, the example given by Simson to prove the incorrectness of Def. 10 introduces a solid with a re-entrant angle. But it is more than probable that Euclid deliberately intended to exclude such solids and to take cognizance of convex polyhedra only; hence Simson’s example is not conclusive against the definition.

Legendre observes that Simson’s own definition, though true, has the disadvantage that it contains a number of superfluous conditions. To get over the difficulties, Legendre himself divides the definition of similar solids into two, the first of which defines similar triangular pyramids only, and the second (which defines similar polyhedra in general) is based on the first.

Two triangular pyramids are similar when they have pairs of faces respectively similar, similarly placed and equally inclined to one another.

Then, having formed a triangle with the vertices of three angles taken on the same face or base of a polyhedron, we may imagine the vertices of the different solid angles of the polyhedron situated outside of the plane of this base to be the vertices of as many triangular pyramids which have the triangle for common base, and each of these pyramids will determine the position of one solid angle of the polyhedron. This being so,

Two polyhedra are similar when they have similar bases, and the vertices of their corresponding solid angles outside the bases are determined by triangular pyramids similar each to each.

As a matter of fact, Cauchy proved that two convex solid figures are equal if they are contained by equal plane figures similarly arranged. Legendre gives a proof which, he says, is nearly the same as Cauchy’s, depending on two lemmas which lead to the theorem that, Given a convex polyhedron in which all the solid angles are made up of more than three plane angles, it is impossible to vary the inclinations of the planes of this solid so as to produce a second polyhedron formed by the same planes arranged in the same manner as in the given polyhedron. The convex polyhedron in which all the solid angles are made up of more than three plane angles is obtained by cutting off from any given polyhedron all the triangular pyramids forming trihedral angles (if one and the same edge is common to two trihedral angles, only one of these angles is suppressed in the first operation). This is legitimate because trihedral angles are invariable from their nature.

Hence it would appear that Heron’s definition of equal solid figures, which adds “similarly situated” to Euclid’s “similar” is correct, if it be understood to apply to convex polyhedra only: Equal solid figures are those which are contained by equal and similarly situated planes, equal in number and magnitude: where, however, the words “equal and” before “similarly situated” might be dispensed with.

Heron (Def. 118) defines similar solid figures as those which are contained by planes similar and similarly situated. If understood of convex polyhedra, there would not appear to be any objection to this, in view of the truth of Cauchy’s proposition about equal solid figures.

DEFINITION 11.

Στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐν τῇ αὐτῇ ἐπιφανείᾳ οὐσῶν πρὸς πάσαις ταῖς γραμμαῖς κλίσις. Ἄλλως· στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γωνιῶν ἐπιπέδων περιεχομένη μὴ οὐσῶν ἐν τῷ αὐτῷ ἐπιπέδῳ, πρὸς ἑνὶ σημείῳ συνισταμένων.

Heiberg conjectures that the first of these two definitions, which is not in Euclid’s manner, was perhaps taken by him from some earlier Elements.

The phraseology of the second definition is exactly that of Plato when he is speaking of solid angles in the Timaeus (p. 55). Thus he speaks (1) of four equilateral triangles so put together (ξυνισταμένα) that each set of three plane angles makes one solid angle, (2) of eight equilateral triangles put together so that each set of four plane angles makes one solid angle, and (3) of six squares making eight solid angles, each composed of three plane right angles.

As we know, Apollonius defined an angle as the “bringing together of a surface or solid to one point under a broken line or surface.” Heron (Def. 22) even omits the word “broken” and says that A solid angle is in general (κοινῶς) the bringing together of a surface which has its concavity in one and the same direction to one point. It is clear from an allusion in Proclus (p. 123, 1—6) to the half of a cone cut off by a triangle through the axis, and from a scholium to 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. 22) 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. 99), 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 cal! 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.

Πρῖσμα 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 n 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 (συνάπτοντα) from a rectilineal base to a rectilineal area by rectilineal collocation (κατ’ εὐθύγραμμον σύνθεσιν). By this Heron must 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 its opposite.

F G E D A B C

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.

Further, he defines parallelogrammic (in the body of the definition parallel-sided) 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 (τὸ ἴσον ἀπέχειν τοῦ μέσου τὸ ἔσχατον, De caelo II. 14, 297 a 24). Heron (Def. 76) 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.

Ἄξων 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. 78). 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.

DEFINITION 16.

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

Heron, Def. 77. 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 εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ περατουμένη ἐφ’ ἑκάτερα τὰ μέρη ὑπὸ τῆς ἐπιφανείας τῆς σφαίρας.

DEFINITION 18.

Κῶνος 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 κῶνος, 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 right-angled, 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.

DEFINITION 19.

Ἄξων δὲ τοῦ κώνου ἐστὶν ἡ μένουσα εὐθεῖα, περὶ ἣν τὸ τρίγωνον στρέφεται.

DEFINITION 20.

Βάσις δὲ ὁ κύκλος ὁ ὑπὸ τῆς περιφερομένης εὐθείας γραφόμενος.

DEFINITION 21.

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

DEFINITION 22.

Ἄξων δὲ τοῦ κυλίνδρου ἐστὶν ἡ μένουσα εὐθεῖα, περὶ ἣν τὸ παραλληλόγραμμον στρέφεται.

DEFINITION 23.

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

DEFINITION 24.

Ὅμοιοι κῶνοι καὶ κύλινδροί εἰσιν, ὧν οἵ τε ἄξονες καὶ αἱ διάμετροι τῶν βάσεων ἀνάλογόν εἰσιν.

DEFINITION 25.

Κύβος ἐστὶ σχῆμα στερεὸν ὑπὸ ἓξ τετραγώνων ἴσων περιεχόμενον.

DEFINITION 26.

Ὀκτάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ ὀκτὼ τριγώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.

DEFINITION 27.

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

DEFINITION 28.

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