sides, the edges of one triedral may be produced, forming the symmetrical triedral, to which the other given triedral may be applied. [Let the student construct figures, and go through with the application.]

PROPOSITION XII.

447. Theorem.-Two triedrals which have two diedrals and the included facial angle equal, are equal, or symmetrical and equiva

lent.

DEM.-[Same as in the preceding. Let the student draw figures like those for the preceding, and go through with the details of the application.]

448. COR.-It will be observed that in equal or in symmetrical triedrals, the equal facial angles are opposite the equal diedrals.

PROPOSITION XIII.

449. Theorem.-Two triedrals which have two facial angles of the one equal to two facial angles of the other, each to each, and the included diedrals unequal, have the third facial angles unequal, and the greater facial angle belongs to the triedral having the greater included diedral.

DEM.-Let ASC asc, and ASB asb, while the diedral C-SA-B > c-sa-b; then CSB > csb.

For, make the diedral C-SA-oc-sa-b: and taking ASo asb, bisect the diedral o-SA-B with the plane ISA. Draw ol and oC, and conceive the planes oSI and SC. Now, the triedral S-AC8-abc, since they have two facial angles and the included diedral equal (446). For a like reason S-Alo = S-AIB, and the facial angle OSI ISB. Again, in the triedral S-loC, ØSI +

=

S

Ca

FIG. 283.

ISC > ®SC (434), and substituting ISB for øSI, we have ISB + ISC (or BSC) > oSC, or its equal bsc. Q. E. D.

450. COR.-Conversely, If the two facial angles are equal, each to each, in the two triedrals, and the third facial angles unequal, the diedral opposite the greater facial angle is the greater.

That is, if ASB = asb, and ASC = asc, while BSC > bxc, the diedral B-AS-C > b-as-c. For, if B-AS-C = b-as-c, BSC = bsc (446), and if B-AS-C < b-18-c, BSC <bsc, by the proposition. Therefore, as B-AS-C cannot be equal to nor less than b-as-c, it must be greater.

PROPOSITION XIV.

451. Theorem.-Two triedrals which have the three facial angles of the one equal to the three facial angles of the other, each to each, are equal, or symmetrical and equivalent.

AA

DEM.-Let A, B, and C represent the facial angles of one, and a, b, and c the corresponding facial angles of the other. If A = a, Bb, and C = c, the triedrals are equal. For A being equal to a, and B to b, if, of their included diedrals, SM were greater than sm, C would be greater than c; and if diedral SM were less than diedral 8m, C would be less than c, by the last corollary. Hence, as diedral SM can neither be greater nor less than diedral sm, it must be equal to it. For like reasons, diedral SN = diedral sn, and diedral SO = diedral 80. Therefore, the triedrals are equal, or symmetrical, according to the arrangement of the faces. Thus, if SN and sn are both considered as lying on the same side of the planes MSO and mso, the triedrals are equal; but, if one lies on one side and the other on the opposite side of those planes (SN in front, and sn behind, for example), the diedrals are symmetrical, and hence equivalent.

FIG. 284.

PROPOSITION XV.

452. Theorem.-Two triedrals which have the three diedrals of the one equal to the three diedrals of the other, each to each, are equal, or symmetrical and equivalent.

DEM.-In the two supplementary triedrals, the facial angles of the one will be equal to the facial angles of the other, each to each, since they are supplements of equal diedrals (438). Hence, the supplementary triedrals are equal or equivalent, by the last proposition. Now, the facial angles of the first triedrals are supplements of the diedrals of the supplementary; whence the corresponding facial angles, being the supplements of equal diedrals, are equal. Therefore, the proposed triedrals have their facial angles equal, each to each, and are consequently equal, or symmetrical and equivalent. q. E. D.

453. COR.-All trirectangular triedrals are equal.

454. SCH.-The proof that two forms are equal, includes the fact that corresponding parts are equal.

OF POLYEDRALS.

455. A Convex Polyedral is a polyedral in which none of the faces, when produced, can enter the solid angle. A section of such a polyedral made by a plane cutting all its edges is a convex polygon. [See Fig. 285.]

PROPOSITION XVI.

456. Theorem.-The sum of the facial angles of any convex polyedral is less than four right angles.

A

DEM.-Let S be the vertex of any convex polyedral. Let the edges of this polyedral be cut by any plane, as ABCDE, which section will be a convex polygon, since the polyedral is convex. From any point within this polygon, as O, draw lines to its vertices, as OA, OB, OC, etc. There will thus be formed two sets of triangles, one with their vertices at S, and the other with their vertices at O; and there will be an equal number in each set, for the sides of the polygon form the bases of both sets. Now, the sum of the angles of these two sets of triangles is equal. But the sum of the angles at the bases of the triangles having their vertices at S is greater than the sum of the angles at the bases of the triangles having their vertices at O, since SBA + SBC > ABC, SCB + SCD > BCD, etc. (434). Therefore the sum of the angles at S is less than the sum of the angles at O, i. e, less than 4 right angles. Q. E. D.

B

FIG. 285.

2. Fig. 286 represents the appearance of a rectangular parallelopiped, as seen from a certain position. Now, all the angles of such a solid are right angles: why is it that they nearly all appear oblique? Can you see a right parallelopiped from such a position that all the angles seen shall appear as right angles?

3. The diedral angles of crystals are measured with great care, in order to determine the substance of which the crystals consist. How must the measure be taken? If we measure obliquely around the edge, shall we get the true value of the angle?

4. Cut out any triedral from a block of wood (or a potato), and stick three pins into it, as near the vertex as you can, one in each face, and perpendicular to that face. What figure do the three pins form? What relation does the angle included between any two adjacent pins sustain to one of the diedrals of the block? Which ones are they that sustain this relation?

5. Can three planes intersect each other and yet not form a triedral angle? In how many ways? Can they all three have a common point, and yet not form a triedral?

B

6. From a piece of pasteboard cut two figures of the same size, like ABCDS and abcds. Then drawing SB and SC so as to make 1 the largest angle and 3 the smallest, cut the pasteboard almost through in these lines, so that it will readily bend in them. Now fold the edges AS and DS together, and a triedral will be formed. From the piece beads form a triedral in like manner, only let the lines sc and sa be drawn so as to make the angles 1, 2, and 3 of the same size as before, while they occur in the order given in beads. Now, see if you can slip one triedral into the other, so that they will fit. What is the difficulty?

a

FIG. 287.

=

7. In the last case, if 1 equals of a right angle, 2 = of a right angle, and 3 of a right angle, can you form the triedral? Why? If you keep increasing the size of 1, 2, and 3, until the sum becomes equal to 4 right angles, will it always be possible to form a triedral ? How is it when the sum equals 4 right angles?

SECTION III.

OF PRISMS AND CYLINDERS.

457. A Prism is a solid, two of whose faces are equal, parallel polygons, while the other faces are parallelograms. The equal parallel polygons are the Bases, and the parallelograms make up the Lateral or Convex Surface. Prisms are triangular, quadrangular, pentagonal, etc., according to the number of sides of the polygon forming a base.

458. A Right Prism is a prism whose lateral edges are perpendicular to its bases. An Oblique prism is a prism whose lateral edges are oblique to its bases.

459. A Regular Prism is a right prism whose bases are regular polygons; whence its faces are equal rectangles.

460. The Altitude of a prism is the perpendicular distance between its bases: the altitude of a right prism is equal to any one of its lateral edges.

461. A Truncated Prism is a portion of a prism cut off by a plane not parallel to its base. A section of a prism made by a plane perpendicular to its lateral

462. A Parallelopiped is a prism whose bases are parallelograms: its faces, inclusive of the bases, are consequently all parallel