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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 perpendiculvr 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. Itemark 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 F.H. The two intersections 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 GHI 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. Draw HI piercing the plane CD in M ; draw also GI 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, I.M. being par7%

Fig.116.

Fig.117. Fig 118.

Fig.119.

GL IM allel to the base GI, we have LH F MH

triangle IHK, MN being parallel to HK, we have
IM IN .
MHT NK'
IM
mon member MFI’ the other two members are equal ;
that is, #. - * ; and we say—If two straight lines
meet three-parallel planes, they will be cut into propor-
tional parts by these planes.
216. When several planes, as ASB, BSC, CSD, DSE,
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 con-
struct a triedral angle. Upon the side AS of one of the
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 in-
creases, 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 com-
plete 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 in-
clination of any two planes in the one, will be equal to the
inclination of the homologous planes in the other.

and in the

but these two equations having the com

* 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. 222. If the sum of the three plane angles given to

Fig.120.

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

Fig.122.

Fig.123.

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
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,
(as the diedral angle DESF (fig. 123), there is no gen-
eral 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 trie-
dral angles, varies with the plane angles of which the
triedral angle is composed. Are there any limits to this
variety 1 and if so, what are these limits 7

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 meats 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 po!yedron 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.

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