SECTION II.

OF SOLID ANGLES.

420. A Solid Angle is the opening between two or more planes, each of which intersects all the others. The lines of intersection are called Edges, and the planes, or the portion of the planes between the edges, where there are more than two, are called Faces.

421. A Diedral Angle, or simply a Diedral, is the opening between two intersecting planes.

422. A Polyedral Angle, called also simply a Polyedral, is the opening between three or more planes which intersect so as to have one common point, and only one. In the case of three intersecting planes the angle is called a Triedral. The point common to all the planes is called the Vertex. The plane angles enclosing a polyedral are the Facial angles.

423. A Diedral (Angle) is measured by the plane angle included by lines drawn in its faces from any point in the edge, and perpendicular thereto. A diedral angle is called right, acute, or obtuse, according as its measure is right, acute, or obtuse. Of course the magnitude of a solid angle is independent of the distances to which the edges may chance to be produced.

ILL'S. The opening between the two planes CABF and DABE is a Diedral (angle), AB is the

the diedral. A diedral may be read by the letters on the edge, when there

would be no ambiguity, or otherwise by these letters and one in each face; thus, the diedral in (a) may be designated as AB, or as C-AB-D.

In (b) we have a Triedral (angle The edges are SA, SB, and SC; the faces ASB, BSC, and ASC: the facial angies are ASB, ASC, and BSC; and S is the vertex. Such an angle, and any polyedral (angle), may be read by naming the angle at the vertex, when there would be no ambiguity, or otherwise by naming the letter at the vertex, and then one in each edge; thus S-ABCDE designates the polyedral (c). The opening between the planes is the angle, in each case.

OF DIEDRALS.

424. A Diedral may be considered as generated by the revolution of a plane about a line of the plane, and hence we may see the propriety of measuring it by the angle included by two lines in its faces perpendicular to its edge, as stated in the preceding article.

A

ILL.-Let AB be a line of the plane CB. Conceive g perpendicular to AB. Now, let the plane revolve upon AB as an axis, whence gB describes a circle (?); and at any posi. tion of the revolving plane, as ƒBAF, since fBg measures the amount of revolution, it may be taken as the measure of the diedral f-BA-g. When gB has made of a revolution, the plane will have made of a revolution, and the diedral will be right.

425. COR.-Opposite diedrals are equal.

Thus, if C-AB-D is measured by MON, c-AB-d is measured by the equal angle nom.

O

B

426. Theorem.-Any line in one face of a right diedral, perpendicular to its edge, is perpendicular to the other face.

M

FIG. 272.

DEM.-In the face CB of the right diedral C-AB-D, let MO be perpendicular to the edge AB; then is it perpendicular to the face DB. For, draw ON in the face DB, and perpendicular to AB. Now, since the diedral is right, and MON measures its angle, MON is a right angle; whence MO is perpendicular to two lines of the plane DB, and consequently perpendicular to the plane. Q. E. D.

427. COR.-Conversely, If one plane contain a line which is perpendicular to another plane, the diedral is right.

Thus, if MO is perpendicular to the plane DB, C-AB-D is a right diedral. For MO is perpendicular to every line of DB passing through its foot (?); and hence is perpendicular to ON, drawn at right angles to AB. Whence C-AB-D is a right diedral, for it is measured by a right plane angle.

PROPOSITION II.

428. Theorem.-If two planes are perpendicular to a third, their intersection is perpendicular to the third plane.

M

DEM.-If CD and EF are perpendicular to the plane MN, then is AB perpendicular to MN. For, EF being perpendicular to MN, D-FG-E is a right diedral, and a line in EF and perpendicular to FG at B is perpendicular to MN; also a line in the plane CD, and perpendicular to DH at B, is perpendicular to MN (?). Hence, as there can be one and only one perpendicular to MN at B, and as this perpendicular is in both planes, CD and EF, it is their intersection. Q. E. D.

FIG. 273.

PROPOSITION III.

429. Theorem.-If from any point perpendiculars be drawn to the faces of a diedral angle, their included angle will be the supplement of the angle which measures the diedral, or equal to it.

DEM.-Let BD and AD be any two planes including the diedral A-SD-B, then will two lines drawn from any point, perpendicular to these planes, include an angle which is the supplement of the measure of the diedral, or equal to it.

If the point from which the lines are drawn is not in the edge SD, we may conceive two lines drawn through any point, as S, in this edge, which shall be parallel to the two proposed, and hence include an equal angle, and have their plane parallel to the plane of the proposed angle (416). Let the latter lines be SO and SP. We are to show that OSP is supplemental to the measure of A-SD-B. A plane passed through S, perpendicular to the edge SD, will contain the lines SO and SP (388); and its intersections with the faces, as SB and SA, will form an angle (ASB) which is the measure of the diedral (423). Now, PSA = a right angle (?), and OSB = a a right angle (?). Hence, PSO and ASB are either equal or supplemental (283). Q. E. D.

D

P B

M

FIG. 274.

N

OF TRIEDRALS.

430. Triedrals are Rectangular, Birectangular, or Trirectangular, according as they have one, two, or three, right diedral angles.

FIG. 275.

ILL'S. The corner of a cube is a Trirectangular triedral, as S-ADC. Conceive the upper portion of the cube removed by the plane ASEF; then the angle at S, i. e., S-AEC is a Birectangular triedral, A-SC-E and A-SE-C being right diedrals.

431. An Isosceles Triedral is one that has two of its facial angles equal. An Equilateral Triedral is one that has all three of its facial angles equal.

432. Two Symmetrical Triedrals are such as have the facial angles of the one equal to the facial angles of the other, each to each; but in which the equal facial angles are not similarly situated, and hence the triedrals are not necessarily capable of superposition.

A

S

FIG. 276.

a

B

ILL'S.-Let the edges of the triedral S-ABC, be produced beyond the vertex, forming a second triedral S-abc; then are the two triedrals symmetrical, i. e., the faces are equal plane angles, but disposed in a different order. Thus, ASB = aSb, ASC = aSc, and BSC = bSc; but the triedrals cannot be made to coincide. To show this fact, conceive the upper triedral detached, and the face aSc placed in its equal face ASC, Sa in SA, and Sc in SC. Now, the edge Sb, instead of falling in SB will fall on the left of the plane ASC.

Symmetrical solids are of frequent occurrence: the two hands form an illustration; for, though the parts may be exactly alike, the hands cannot be placed so that their like parts will be similarly situated; in short, the left glove will not fit the right hand.

433. Two triedrals are Supplementary when the edges of one are perpendicular to the faces of the other. (See 438a.)

PROPOSITION IV.

434. Theorem.-The sum of any two facial angles of a triedral is greater than the third.

DEM.-This proposition needs demonstration only in case of the sum of the two smaller facial angles as compared with the greatest (?). Let ASB and BSC each be less than ASC; then is ASB + BSC > ASC. For, make the angle ASb' = ASB, and Sʊ' = Sb, and pass a plane through b and b', cutting SA and SC in a and c. The two triangles aSb and aSb' are equal (?), whence ab' = ab. Now, ab + bc> ac (?), and subtracting ab from the first member, and its equal ab' from the second, we have be> b'c. Whence the two triangles ¿Sc and b'Sc have two sides equal, but the third side be > than the third side b'c, and consequently angle bSc > 'Sc. Adding ASB to the former, and its equal ASʊ' to the latter, we have ASB + BSC > ASC. Q. E. D.

a

S

FIG. 277.

435. COR.-The difference between any two facial angles of a triedral is less than the third facial angle (?).

PROPOSITION V.

436. Theorem.-The sum of the facial angles of a triedral may be anything between 0 and four right angles.

DEM.-Let ASB, BSC, and ASC be the facial angles enclosing a triedral; then, as each must have some value, the sum is greater than 0, and we have only to show that ASB + ASC + BSC < 4 right angles. Produce either edge, as AS, to D. Now, in the triedral S-BCD, BSC < BSD + CSD. To each member of this inequality add ASB ASC, and we have

ASB+ ASC + BSC < ASB + ASC + BSD + CSD. But, ASB + BSD = 2 right angles (?), and ASC + CSD = 2 right angles; whence ASB + ASC + BSD + CSD = 4 right angles; and consequently, ASB + ASC + BSC < 4 right angles. Q. E. D.

B

FIG. 278.

PROPOSITION VI.

437. Theorem.-Two triedrals having the facial angles of the one equal to the facial angles of the other, each to each, and similarly arranged, are equal.