same manner, at the point C we have BCO+OCD< BCS+SCD; and so with all the angles of the polygon ABCDE. Whence it follows that the sum of all the angles at the bases of the triangles whose common vertex is in O, is less than the sum of all the angles at the bases of the triangles whose common vertex is in S: hence, to make up the deficiency, the sum of the angles formed about the point O is greater than the sum of the angles about the point S. But the sum of the angles about the point O is equal to four right-angles, (B. I, Prop. 1, Cor. 3;) therefore the sum of the plane angles, which form the solid angle S, is less than four rightangles.

Schol. This demonstration is founded on the supposition that the solid angle is convex, or that the plane of no one surface produced can ever meet the solid 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 solid angles are composed of three plane angles respectively equal to each other, the planes which contain the equal angles will be equally inclined to each other.

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.

F

P

E

T

Having taken SB at pleasure, draw BO perpendicular to the plane ASC; from the point O at which that perpendicular meets the plane, draw OA, OC perpendicular to SA, SC; join AB, BC; next take TE=SB; draw EP perpendicular to the plane DTF; from the point P draw PD, PF perpendicular to TD, TF; lastly, join DE, EF.

The triangle SAB is right-angled at A, and the triangle TDE is right-angled at D, (B. VI, Prop. vI ;) and since the angle ASB-DTE, we have SBA=TED. Likewise SB TE; therefore the triangle SAB is 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 granted, the quadrilateral SAOC is equal to the quadrilateral TDPF; for, place the angle ASC upon its equal DTF; because SA=TD, and SC =TF, the point A will coincide with D, and the point C with F; and at the same time AO, which is perpendicular to SA, will coincide with PD which is perpendicular to TD, and in like manner CO with FP; wherefore the point O will coincide with the point P, and AO will be equal to DP. But the triangles AOB, DPE are right-angled at O and P; the hypothenuse AB=DE, and the side AO=DP: hence those triangles are equal; therefore the angle OAB=PDE. The angle OAB is the inclination of the two planes ASB, ASC, and the angle PDE is that of the two planes DTE, DTF: hence these two inclinations are equal to each other.

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 as SC falls; for if it

fell on the other side, the angle of the two planes would be obtuse, and, added to the angle A of the triangle OAB, it would make two right-angles. But, in the same case, the angle of the two planes TDE, TDF would also be obtuse, and, added to the angle D of the triangle PDE, it would make two right-angles; and the angle A being thus always equal to the angle at D, it would follow, in the same manner, that the inclination of the two planes ASB, ASC must be equal to that of the two planes DTE, DTF.

BOOK SEVENTH.

DEFINITIONS.

1. A prism is a solid contained by plane figures, of which two that are opposite are.equal, similar, and parallel to one another; and the others are parallelograms. To construct this solid, let ABCDE be any rectilineal figure. In a plane parallel

K

to ABC, draw the lines FG, GH,
HI, &c., parallel to the sides AB,
BC, CD, &c.; thus there will be
formed a figure FGHIK, similar
to ABCDE. Now let the vertices
of the corresponding angles be
joined by the lines AF, BG, CH,
&c.; the faces ABGF, BCHG,
&c., will evidently be parallelo-
grams, and the solid thus formed will be a prism.

A

2. The equal and parallel plane figures ABCDE, FGHIK are called the bases of the prism. The other planes or parallelograms, taken together, constitute the lateral or convex surface of the prism.

3. The altitude of a prism is the perpendicular distance between its bases; and its length is a line equal to any one of its lateral edges, as AF or BG, &c.

4. A right prism is one in which the lateral edges AF, BG, &c., are perpendicular to the planes of its

bases; then each of them is equal to the altitude of the prism in every other case, the prism is oblique.

5. A prism is triangular, quadrangular, pentagonal, etc., according as the base is a triangle, a quadrilateral, a pentagon, etc.

6. A prism which has a parallelogram for its base, has all its faces parallelograms, and is called a par- G allelopipedon. A parallelopipedon is rectangular, when all its faces are rectangles.

F

7. When the faces of a rectangular parallelopipedon are square, it is called a cube.

8. A pyramid is a solid formed by several triangular planes which meet in a point, as S, and terminate in the same plane rectilineal figure ABCDE.

The plane figure ABCDE is called the base of the pyramid; the point S is its vertex; and the triangles ASB, BSC,

D

B

&c., taken together, form the convex or lateral surface of the pyramid.

9. The altitude of a pyramid is the perpendicular drawn from the vertex to the plane of its base, produced if necessary.

10. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, a quadrangle, etc.

11. A pyramid is regular, when its base is a regular figure, and the perpendicular from its vertex passes through the centre of its base; that is, through the centre of a circle which may be conceived to circumscribe its base.