XI. DEFF. I—

-5]

DEFINITIONS And notes

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(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. 13.)

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

Στερεοῦ δὲ πέρας ἐπιφάνεια.

DEFINITION 2.

In like manner Aristotle says (Metaph. 1066 b 23) that the notion (Móyos) of body is "that which is bounded by surfaces" (éédois in this case) and (Metaph. 1060 b 15) "surfaces (émipávetai) are divisions of bodies."

So Heron (Def. 13): "Every solid is bounded (πeраτоvται) 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 οὖσας 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 1. Def. 7, Vol. 1. 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.

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[XI. DEFF. 5, 6

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, 1. 19 and 18.

DEFINITION 6.

Ἐπιπέδου πρὸς ἐπίπεδον κλίσις ἐστὶν ἡ περιεχομένη ὀξεία 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.

A

M

E

B

IN

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

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

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

Legendre then extends the proof to the case where the plane angle incommensurable by reference to an exactly similar extension in his propo 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 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 word which is translated "which do not meet" is ȧovμπтшта, the term has been adopted for the asymptotes of a curve.

DEFINITION 9.

Ομοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων ἐπιπέδων περιεχόμενα ἴσ πλῆθος.

DEFINITION IO.

Ισα δὲ καὶ ὅμοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων ἐπιπέδων περιεχ ἴσων τῷ πλήθει καὶ τῷ μεγέθει.

These definitions, the second of which practically only substitute words "equal and similar" for the word "similar" in the first, have bee mark of much criticism.

Simson holds that the equality of solid figures is a thing which ought proved, by the method of superposition, or otherwise, and hence that D is not a definition but a theorem which ought not to have been placed a the definitions. Secondly, he gives an example to show that the definiti theorem is not universally true. He takes a pyramid and then erects base, on opposite sides of it, two equal pyramids smaller than the first. addition and subtraction of these pyramids respectively from the first giv

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[XI. DEF. 10 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

XI. DEFF. 10, 11] NOTES ON DEFINITIONS 10, 11

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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 II.

Στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐν τῇ αὐτῇ ἐπιφανείᾳ οὐσῶν πρὸς πάσαις ταῖς γραμμαῖς κλίσις. Αλλως· 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 (vviorάueva) 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. 24) even omits the word "broken" and says that A solid angle is in general (Koivŵs) 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

I