B

E

C

Let the three angles, BAD, DAC, BAC, form the solid angle A. The sum of any two of these is greater than the third. When these angles are all equal, it is evident that the sum of any two is greater than the third, and the proposition needs demonstration only when one of them, as BAC, is greater than either of the others; we are then to prove that it is less than their sum.

On the line AB, take any point, B, and draw any line, as BD. From the same point, B, make the angle ABC-ABD, and join DC. From the point A, and on the plane BAC, draw the angle BAE=BAD. Now the two plane triangles BAD and BAE, have a common side, AB, and the angles adjacent equal (th. 14, b. 1); therefore, the two As are, in all respects, equal; and AD AE, and BD=BE. In the triangle BDC, But,

By subtraction, .

BC BD+DC
BE-BD

EC DC

In the two triangles, DAC and EAC, DA-AE, and AC is common, but EC is less than CD; therefore, the angle DAC, opposite DC, is greater than the angle EAC, opposite EC. (Converse of th. A, b. 1).

That is,

But,

By addition,

DAC+DAB>BAC. (Ax. 2). Q. E. D.

PROPOSITION

11. THEOREM.

The sum of any plane angles forming any solid angle, is always

less than four right angles.

Let the planes which form the solid angle at A, be cut by another plane, which we may call the plane of the base, BCDE. Take any point, a, in this plane, and join aB, aC, aD, aE, &c., thus making as many triangles on the plane of the base, as there are triangular planes forming the solid angle A. But as the sum of the angles of every ▲ is two

right angles, the sum of all the angles of the As which have their vertex in A, is equal to the sum of all angles of the As which have their vertex in a. But the angles BCA+ACD, are, together, greater than the angles BCa+aCD, or BCD, by the last proposition. That is, the sum of all the angles at the bases of the As which have their vertex in A, is greater than the sum of all the angles at the bases of the As which have their vertex in a. Therefore, the sum of all the angles at a, is greater than the sum of all the angles at A, but the sum of all the angles at a, is equal to four right angles; therefore, the sum of all the angles at A, is less than four right angles. Q. E. D.

PROPOSITION 12. THEOREM.

If two solid angles are formed by three plane angles respectively equal to each other, the planes which contain the equal angles will be equally inclined to each other.

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, at D; and since the angle ASB=DTE, we have SBA=TED. Likewise, SB TE; therefore, the triangle SAB is equal to the triangle TDE; hence, SA=TD, and 4B-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 fall on D, and the point C on F;

and, at the same time, AO, which is perpendicular to SA, will fall on PD, which is perpendicular to TD, and, in like manner, OC on PF; wherefore, the point O will fall on the point P, and AO will be equal to DP. But the triangles AOB, DPE, are right angled at O and P; the hypotenuse AB-DE, and the side A0=DP; hence, those triangles are equal; hence, the angle OAB=PDE. The angle OAB is the inclination of the two planes ASB, ASC; the angle PDE, is that of the two planes DTE, DTF; consequently, those two inclinations are equal to each other. Hence, If two solid angles are formed, &c.

Scholium. The angles which form the solid angles at S and T, may be of such relative magnitudes, that the perpendiculars, BO and EP, may not fall within the bases, ASC and DTF; but they will always either fall on the bases or on the planes of the bases produced, and O will have the same relative situation to A, S, and C, as P has to D, T, and F. But, in case that O and P fall on the planes of the bases produced, the angles BCO and EFP, would be obtuse angles; but the demonstration of the problem would not be varied in the least.

BOOK VII.

воок

SOLID GEOMETRY.

THE object of Solid Geometry is to estimate and compare the surfaces and magnitudes of solid bodies; and, like Plane Geometry, it must rest on definitions and axioms.

To the definitions already given, we add the following, as being exclusively applicable to Solid Geometry.

Surfaces are measured by square units; so solids are measured by cube units.

1. A Cube is a solid, bounded by six equal square sur

faces, forming eight equal solid angles.

All other solids are referred to a unit of this figure for measurement.

2. A Prism is a solid, whose ends are parallel, equal, and form equiangular plane figures; and its sides, connecting these ends, are parallelograms.

3. A prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c.

4. A right or upright prism, is that which has the planes of the sides perpendicular to the planes of the ends or base. 5. A Parallelopipedon is a prism bounded by six parallelograms, every opposite two of which are equal, alike, and parallel.

6. A rectangular parallelopipedon, is that whose bounding planes are all rectangles, which are perpendicular to each other. A rectangular parallelopipedon becomes a cube when all its planes are equal.

7. A Cylinder is a round prism, having circles for its ends; and is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis.

8. The axis of a cylinder, is the right line joining the

centers of the two parallel circles, about which the figure is described.

9. A Pyramid is a solid, whose base is any right lined plane figure, and its sides triangles, having all their vertices meeting together in a point above the base, called the vertex of the pyramid.

10. A pyramid, like the prism, takes particular names from the figure of the base.

11. A Cone is a convex pyramid, having a circular base, and is conceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle.

12. The axis of a cone is the right line joining the vertex, or fixed point, and the center of the circle about which the figure is described.

13. Similar cones and cylinders, are such as have their altitudes and the diameters of their bases proportional.

14. A Sphere is a solid, having but one surface, which is in every part equally convex; and every point on such a surface is equally distant from a certain point within, called the center.

15. A sphere may be conceived as having been generated by the revolution of a semicircle about its axis.

The diameter of such a semicircle is the diameter of the sphere; and the center of the semicircle is the center of the sphere.

16. The altitude of any solid is the perpendicular distance between the parallel planes, one of which is the base of the solid, and the other is a plane, parallel with the plane of the base, passing through the vertex of the solid.

17. The area of the surface is measured by the product of its length and breadth (as explained by scholium on page 32); and these dimensions are always conceived to be exactly at right angles with each other.

18. In a similar manner, solids are measured by the product of their length, breadth, and hight, when all their dimensions are at right angles with each other.

The product of the length and breadth of a solid, is the measure of the surface of its base.