common vertex is 0, the sum of all the angles is equal to the four right angles about 0, together with the interior angles of the polygon, and this sum is equal to twice as many right angles as there are triangles. Since the number of triangles, in each set, is the same, it fol lows that these sums are equal. But in the triedral angle whose vertex is B, ABS+SBC>ABC (P. 19), and the like may be shown at each of the other vertices, C, D, E, A: hence, the sum of the angles at the bases, in the triangles whose common vertex is S, is greater than the sum of the angles at the bases, in the set whose common vertex is 0: therefore, the sum of the vertical angles about S is less than the sum of the angles about 0: that is, less than four right angles.

Scholium. This demonstration is founded on the supposition that the polyedral angle is convex, or that the plane of no one face produced can ever meet the polyedral angle; if it were otherwise, the sum of the plane angles would no longer be limited, and might be of any magnitude.

PROPOSITION XXI. THEOREM.

If two triedral angles are included by plane angles which are equal each to each, the planes of the equal angles are equally inclined to each other.

Let S and T be the vertices of two triedral angles, and let the angle ASC=DTF, the angle ASB=DTE, and the angle BSC ETF'; then will the inclination of the planes ASC, ASB, be equal to that of the planes DTF, DTE. For, having taken SB at

S

Τ

TE=SB; draw EP perpendicular to the plane DTF; from the point P draw PD, PF, perpendicular respectively to TD, TF; lastly, draw DE and EF.

P

equal to the triangle TDE; therefore, SA=TD, and AB=DE

In like manner, it may be shown, that SC=TF, and BC=EF. That proved, the quadrilateral ASCO is equal to the quadrilateral DTFP: for, place the angle ASC upon its equal DTF; because SA=TD, and SC=TF, the point A will fall on D, and the point C on F; and, at the same time, AO, which is perpendicular to SA, will fall on DP, which is perpendicular to TD, and, in like manner, OC on PF; wherefore, the point O will fall on the point P, and hence, AO is equal to DP.

But the triangles AOB, DPE, are right-angled at O and P; the hypothenuse AB=DE, and the side A0=DP: hence, those triangles are equal (B. I., P. 17); and, conse quently, the angle OAB=PDE. But the angle OAB measures the inclination of the two faces ASB, ASC; and the angle PDE measures that of the two faces DTE, DTF ; hence, those two inclinations are equal to each other.

Scholium 1. It must, however, be observed, that the angle A of the right-angled triangle AOB is properly the inclination of the two planes ASB, ASC, only when the perpendicular BO falls on the same side of SA, with SC; for, if it fell on the other side, the angle of the two planes would be obtuse, and the obtuse angle together with the angle A of the triangle OAB would make two right angles. But in the same case, the angle of the two planes TDE, TDF, would also be obtuse, and the obtuse angle together with the angle D of the triangle DPE, would make two right angles; and the angle A being thus always equal to the angle D, it would follow that the inclination of the two planes ASB, ASC, must be equal to that of the two planes TDE, TDF.

Scholium 2. If two triedral angles are included by three

plane angles, respectively equal to each other, and if, at the same time, the equal or homologous angles are disposed in the same order, the two triedral angles will coincide when applied the one to the other, and consequently, are equal (A. 14).

For, we have already seen that the quadrilateral SAOC may be placed upon its equal TDPF; thus, placing SA upon TD, SC falls upon TF, and the point O upon the point P. But because the triangles AOB, DPE, are equal, OB, perpendicular to the plane ASC, is equal to PE, perpendicu lar to the plane TDF; besides, these perpendiculars lie in the same direction; therefore, the point B will fall upon the point E, the line SB upon TE, and the two angles will wholly coincide.

Scholium 3. The equality of the triedral angles does not exist, unless the equal faces are arranged in the same manner. For, if they were arranged in an inverse order, or, what is the same, if the perpendiculars OB, PE, instead of lying in the same direction with regard to the planes ASC, DTF, lay in opposite directions, then it would be impossible to make these triedral angles coincide the one with the other. The theorem would not, however, on this account, be less true, viz.: that the faces containing the equal angles must be equally inclined to each other; so that the two triedral angles would be equal in all their constituent parts, without, however, admitting of superposi tion. This sort of equality, which is not absolute, or such as admits of superposition, ought to be distinguished by a particular name: we shall call it, equality by symmetry.

Thus, those two triedral angles, which are formed by faces respectively equal to each other, but disposed in an inverse order, will be called triedral angles equal by symme try, or simply symmetrical angles.

BOOK VII.

POLYEDRONS.

DEFINITIONS.

1. POLYEDRON is a name given to any solid bcunded by polygons. The bounding polygons are called faces of the polyedron; and the straight line in which any two adjacent faces meet each other, is called an edge of the polyedron.

2. A PRISM is a polyedron in which two of the faces are equal polygons with their planes and homologous sides parallel, and all the other faces parallelograms.

3. The equal and parallel polygons are called bases of the prism-the one the lower, the other, the upper baseand the parallelograms taken together, make up the lateral or convex surface of the prism.

4. The ALTITUDE of a prism is the distance between its two bases, and is measured by a line drawn fro: a point in one base, perpendicular to the plane of the ott.

5. A right prism is one whose edges, formed by the intersection of the lateral faces, are perpendicular to the planes of the bases. Each edge is then equal to the altitude of the prism. In every other case, the prism is oblique, and each edge is then greater than the altitude.

6. A TRIANGULAR PRISM is one whose bases are triangles: a quadrangular prism is one whose bases are quadrilaterals: a pentangular prism is one whose bases are pentagons: a hexangular prism is one whose bases are hexagons, &c.

7. A PARALLELOPIPEDON is a prism whose bases are parallelograms.

8. A RECTANGULAR PARALLELOPIPEDON is one whose faces are all rectangles. When the faces are squares, it is called a cube, or regular hexaedron.

9. A PYRAMID is a solid bounded by a polygon, and by triangles meeting at a common point, called the vertex. The polygon is called the base of the pyramid, and the triangles, taken together, the convex, or lateral surface. The pyramid, like the prism, takes different names, according to the form of its base: thus,

it

may be triangular, quadrangular, pentangular, &c.

10. The ALTITUDE of a pyramid is the perpendicular let fall from the vertex on the plane of the base.

11. A RIGHT PYRAMID is one whose base is a regular polygon, and in which the perpendicular let fall from the vertex upon the base passes through the centre of the base. This perpendicular is then called the axis of the pyramid.

12. The SLANT HEIGHT of a right pyramid, is the per pendicular let fall from the vertex to either side of the polygon which forms the base.

13. If a pyramid is cut by a plane parallel to its base, forming a second base, the part lying between the bases, is called a truncated pyramid, or frustum of a pyramid.