arranged in an inverse order, are termed angles equal by symmetry, or simply, symmetrical angles.

The same observation applies to solid angles formed by more than three plane angles. Thus, a solid angle formed by the plane angles A, B, C, D, E, and another solid angle formed by the same angles in an inverse order, A, E, D, C, B, may be such that the planes in which the equal angles are situated are equally inclined to each other. These two solid angles, which would in this case be equal, although not admitting of superposition, would be termed solid angles equal by symmetry, or symmetrical solid angles.

In plane figures, there is no species of equality to which this designation can belong, for all those cases to which the term might seem to apply, are cases of absolute equality, or equality of coincidence. The reason of this is, that the position of a plane figure may be altered at pleasure, and one may take the upper part for the under, and vice versa. This, however, does not hold in solids, in which the third dimension may be taken in two different directions.

SOLID GEOMETRY.

DEFINITIONS.

1. SIMILAR Solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes.

2. A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet.

3. A prism is a solid figure contained by plane figures, of which, two that are opposite are equal, similar, and parallel to each other; and the others are parallelograms.

L

4. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.

Thus, the inner side of the semicircle ABC revolving round the diameter AC, which remains fixed, generates a sphere.

A

B

5. The axis of a sphere is the fixed right line about which the semicircle revolves.

Thus AC, in the figure above, is the axis of the sphere.

6. The centre of a sphere is the same with that of the semicircle.

7. The diameter of a sphere is any right line which passes through the centre, and is terminated both ways by the superficies of the sphere.

8. A right cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.

If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuseangled; and if greater, an acute-angled cone.

Thus, the side AC, revolving round AB, one of the sides which contains the right angle and remains fixed, generates a cone.

A

C

9. The axis of a cone is the fixed right line about which the triangle revolves. In figure above, AB is the axis.

10. The base of a cone is the circle described by that side containing the right angle which revolves.

11. A cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed.

Thus, the revolution of the parallelogram AC about its side AB, which remains fixed, generates a cylinder.

12. The axis of a cylinder is the fixed right line about which the parallelogram revolves.

13. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram.

14. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

15. A cube is a solid figure contained by six equal squares.

16. A tetrahedron is a solid figure contained by four equal and equilateral triangles.

17. An octahedron is a solid figure contained by eight equal and equilateral triangles.

18. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular.

19. An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

20. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel.

PROPOSITIONS.

PROP. I.

If a prism be cut by a plane parallel to its base, the section will be equal and

like to the base.

Let AG be any prism, and IL a plane parallel to the base AC; then will the plane IL be equal and like to the base AC, or the two planes will have all their sides and all their angles equal.

I

УК

D

C

B

For, the two planes, AC, IL, being parallel, by hypothesis; and two parallel planes, cut by a third plane, having parallel sections; therefore, IK is parallel to AB, KL to BC, LM to CD, and IM to AD. But AI and BK are parallels, by Def. 3; consequently, AK is a parallelogram; and the opposite sides, AB, IK, are equal. In like manner, it is shown that KL is BC and LM = CD, and IM = AD, or the two planes, AC, IL, are mutually equilateral. But these two planes, having their corresponding sides parallel, have the angles contained by them also equal; namely, the angle A = the angle I, the angle B = the angle K, the angle C = the angle L, and the angle D = the angle M. So that the two planes, AC, IL, have all their corresponding sides and angles equal, or are equal and like, Q. E. D.

E E

PROP. II.

If a cylinder be cut by a plane parallel to its base, the section will be a circle, equal to the base.

Let AF be a cylinder, and GHI any section parallel

to the base ABC; then will GHI be a circle, equal to ABC.

For, let the planes KE, KF, pass through the axis of the cylinder MK, and meet the section GHI in the three points H, I, L; and join the points as in the figure.

M

F

E

L

H

K

C

B

Then, since KL, CI, are parallel; and the plane KI, meeting the two parallel planes ABC, GHI, makes the two sections KC, LI, parallel; the figure KLIC is therefore a parallelogram, and consequently has the opposite sides LI, KC equal, where KC is a radius of the circular base.

In like manner, it is shown that LH is equal to the radius KB; and that any other lines, drawn from the point L to the circumference of the section GHI, are all equal to radii of the base; consequently, GHI is a circle, and equal to ABC. Q. E. D.

PROP. III.

All prisms, and a cylinder, of equal bases and altitudes, are equal to

the last two propositions, the section PQ is equal to the base AB, and the sec tion RS equal the base DE.

are equal by the hypoAnd in like manner, it

But the bases AB, DE, thesis; therefore the sections PQ, RS, are also equal. may be shown, that any other corresponding sections are equal to one another. Since, then, every section in the prism AC, is equal to its corresponding section in the prism, or cylinder RS, the prisms and cylinder themselves, which are composed of those sections, must also be equal. Q. E. D.

Corol. Every prism, or cylinder, is equal to a rectangular parallelopipedon, of an equal base and altitude.

PROP. IV.

Rectangular parallelopipedons, of equal altitudes, have to each other the same proportion as their bases.

Let AC, EG, be two rectangular parallelopipedons, having the equal altitudes AD, EH; then will AC be to EG as the base AB is to the base EF.

For, let the proportion of the base AB to the base EF, be that of any one number m (3) to any other number n (2).

D

R

S с

V

G

K B

F