plane AB; but HM and GL are parallel, therefore the planes in which they are situated are not inclined to each other in that direction; for the same reason they are not inclined to each other in the direction of the parallels HS and GR. The same may be shown with respect to lines in the two planes in every direction from the points G and H; consequently the two planes are not inclined to each other in any direction; and we say that-Two planes are parallel when they are not inclined to each other, in any direction. We say, therefore, that -Two planes perpendicular to the same straight line, are parallel. And-Two parallel planes have common perpendiculars.

212. Remark 1st. As two parallel planes are not inclined to each other, thy must be throughout at the same distance from each other; and, however far produced, can never meet; therefore-All the perpendiculars between parallel planes, are equal.

213. Remark 2d. If two straight lines which cut each other, are parallel respectively to two other straight lines which cut each other, the plane determined by the first two lines, will be parallel to the plane determined by the other two (211).

214. Let the two parallel planes AB and CD (fig. 116) be cut by a third plane FH. The two intersec- Fig.116. tions EF and GH, being in the same plane FH, will meet unless they are parallel (12); and if these lines will meet, the planes AB and CD in which they are situated, will also meet; but these planes being parallel to each other, can never meet; consequently the two intersections EF and GH cannot meet; they are then parallel; and we say-When two parallel planes are cut by a third plane, the two intersections are parallel.

Remark. If the lines FG and EH are parallel, being between the parallels EF and GH they must be equal (60); we say therefore-Parallel lines between parallel planes, are equal.

215. Let the two straight lines GH and IK (fig. 117) be cut by the three parallel planes, AB, CD, EF. Fig.117. Draw HI piercing the plane CD in M; draw also GIÍ and LM, MN and HK. Then G1 and LM are the intersections of the plane HGI with the parallel planes AB and CD, they are therefore parallel; for a similar reason MN is parallel to HK. In the triangle GHI, LM being par

allel to the base GI, we have

GL

IM

; and in the

LH MH

triangle IHK, MN being parallel to HK, we have

that is,

; but these two equations having the com

IM
MH'

GL IN

the other two members are equal ;

; and we say—If two straight lines LH NK

meet three-parallel planes, they will be cut into proportional parts by these planes.

*

216. When several planes, as ASB, BSC, CSD, DSE, Fig 118. ESF, FSG, and GSA, (fig. 118) all passing through the point S, are joined, two and two, by their sides diverging from S, the space which they comprehend, unlimited in the direction opposite to S, is called a polyedral angle.* Angles of this kind are distinguished by the number of planes which meet at the summit or vertex of the angle. A triedral angle is an angle of three faces, as the angle SABC (fig. 119); a tetraedral angle is one which has four faces. The angle SABCDEFG (fig. 118) has seven faces and is called a heptaedral angle, &c.

The things to be considered in a polyedral angle, are the plane angles formed by the edges of the polyedral angle; and the respective inclinations of these planes, or the diedral angles formed by contiguous faces.

217. Problem. Three plain angles being given to construct a triedral angle. Upon the side AS of one of the Fig.119. angles, ASB (fig. 119), place the side AS of another of them, so that their vertices may coincide at S; then raising the part C of the second, make it turn about the intersection AS; it will be perceived that, as the inclination of the plane CAS to the plane BAS increases, the angular space CSB will increase; there is therefore a certain inclination of the first two planes, which allows the third angle to be introduced to complete the triedral angle; therefore the magnitude of each of the plane angles, determines the inclination of the other two. We say therefore-If two triedral angles are formed of plane angles respectively equal, the inclination of any two planes in the one, will be equal to the inclination of the homologous planes in the other.

* This was formerly called a solid angle.

218. Remark 1. If the plane angles be unequal, but are placed in the same order in both, the two triedral angles will coincide, when placed together; in this case they are called equal. If the order of the planes be different (fig. 120), the two triedral angles cannot coincide; for, placing two equal faces together, so that they may coincide, two other equal faces will be inclined to this plane on opposite sides. In this case they are said to be symmetrical.

219. Remark 2d. If the three plane angles which form a triedral angle are all equal, the three inclinations, that is, the three diedral angles will be equal. If each of the plane angles is a right-angle, the diedral angles will be right-angles; and the triedral angle may be called right-angled.

220. If the magnitude of the third angle is just equal to the difference between the other two, this third plane can be introduced without inclining the other two, that is, the diedral angle will be nothing; the three planes will then be in the same plane, and will therefore comprehend no space. If the third angle be less than the difference between the other two, it will not fill the remaining space, the inclination of the two being nothing. We say, therefore, that―The sum of any two of the plane angles which form a triedral angle, is always greater than the third.

221. Remark. Though the magnitude of the plane angles determines their inclination, in a triedral angle; this is not the case with any other polyedral angle. If we have four plane angles given, we can form an infinite variety of tetraedral angles with them, as they will admit of an infinite variety of inclinations; but if, in a tetraedral angle, one of the inclinations be fixed (the plane angles and their order of arrangement being given) this fact will determine all the others.

Fig.120.

222. If the sum of the three plane angles given to form a triedral angle (fig. 121) were equal to four right Fig.121. angles, the angular space left for the introduction of the third plane, would not be sufficient to receive it until the second plane was turned quite over into the same plane with the first (15). In this case the three plane angles would be in the same plane. The same would be true, of any number of plane angles given to form a polyedral angle; if their sum were equal to four right-angles

their sides would coincide when they were all in the same plane, their vertices being placed together. And it is evident that, if the sum of three angles were greater than Fig.122. four right angles (fig. 122) they could not be brought together at their vertices, their sides coinciding two and two; they could not therefore form a triedral angle. Whence we say-The sum of all the plane angles which form a convex polyedral angle, is always less than four right-angles.

223. Remark. By a convex polyedral angle, we mean one, all of whose diedral angles are salient, that is, have their edges standing out, as in figure 118. When there are, in the polyedral angles, re-entering diedral angles, Fig.123. (as the diedral angle DESF (fig. 123), there is no general limit to the sum of the plane angles which may form a polyedral angle.

It is evident that the sum of the diedral angles in triedral angles, varies with the plane angles of which the triedral angle is composed. Are there any limits to this variety and if so, what are these limits?

Of Polyedral Bodies.

224. From the discussion of polyedral angles, we proceed naturally to that of polyedral bodies, or polyedrons,* that is-bodies whose surfaces are composed entirely of planes. The word polyedron significs a body of many faces. When we speak of geometrical bodies we do not include necessarily the idea of matter or substance which resists the approach of other bodies or excludes them from the places which these occupy, as a block of wood or of marble. Geometry, as we said at first, is the science of magnitude and form, and takes no cognizance. of even the existence of matter. It discusses, it is true, the form and extent of material substances; but it is the form and extent merely; or rather, it is the form and meas

* These bodies have been called solid polyedrons ; but it has been very properly objected to the term solid, that it has in common language an appropriate use, totally different from that which has been assigned to it in geometry. This new application of a familiar term, is calculated to mislead the learner, and should never be adopted.

ure of those portions of space in which these bodies are extended, that makes the subject of geometry. When we speak of a geometrical body, therefore, we mean a portion of space of that particular form and extent; and when we speak of the face of a polyedral body, we mean that ideal division between the portion of space occupied by the body, and the portion without.

225. In a polyedral angle it is observed, that the figure is open and unlimited in the direction opposite to the summit (216); it follows, therefore, that with only one polyedral angle, of how many planes soever it be composed, no portion of space can be entirely enclosed. In order then to form a polyedral body, it will be necessary to have more than one polyedral angle.

The simplest polyedral angle we have seen (fig. 124 ) Fig.124. is one of three faces (216). If these three planes were cut by a fourth, a certain portion of space would be enclosed; and if the contiguous sides of the plane faces, forming a triedral angle, are equal, the sides opposite to the common vertex S in these triangles, as AB, BC, and CA, will be in the same plane (197); the plane triangle ABC may therefore be considered as one face of the polyedron SABC. Without this last plane ABC, we had but one polyedral angle, that whose vertex is at S; we have now four polyedral angles; the summits of which are S, A, B and C. Each of these is a triedral angle, the most simple of polyedral angles; and each of the four faces is a triangle, the simplest of rectilinear figures. It seems quite evident, therefore, that if merely these circumstances are considered, this is the simplest of polyedral bodies. It is called a tetraedron, a body having four faces.

226. We consider this body as contained by the three triangles ASB, BSC, CSA, (having a common vertex in S,) and the triangle ABC; which last we will call the base of the polyedron.

If we had four triangles, having the vertices in S (fig. 125) and their bases in the same plane ABCD, the body Fig.125. contained by these five planes we should call a pentaedron, and in general, a body contained by six plane faces is called a hexaedron; of eight, an octaedron, &c. But all bodies of which one of the faces is a polygon, and all the others triangles having their summit in the same point, are called pyramids. The body SABCDE (fig. 126) is Fig.126.