The First Six Books of the Elements of Euclid, with a Commentary and Geometrical Exercises

Let the base abcde be placed upon the base A B C D E, so that the several homologous sides shall coincide. Let the side bg be inclined to ba and bc, at the same angles as B G is inclined to BA and BC. Hence the side bg coincides with BG. Since the point a coincides with A and the lines BAF and baf are in the same plane, and the angles B A F and b af are equal, the line af must coincide with AF; and in the same manner it may be proved that the several sides of the prism whose base is a b c d e will coincide with the corresponding sides of the other prism, and therefore the angles under these sides and those of the base are respectively equal to each.

PROPOSITION II.

(73) If two prisms have equal bases and one pair of corresponding sides equal and similarly inclined

to the sides of the bases with which they form solid angles, the prisms will be equal in every respect.

For by the demonstration of (72) it appears, that the base of one may be so applied to the base of the other that the several sides of the one will respectively coincide with the sides of the other; and since these sides are equal the opposite bases must coincide, and therefore the several vertices of the one prism will coincide with those of the other, and the two solids will fill exactly the same spaces and be bounded by the same lines and planes.

(74) COR.-Hence it obviously follows, that right prisms, which have equal and similar bases and equal altitudes, are equal in all respects.

PROPOSITION III.

(75) If two parallelopipeds have three conterminous edges in the one equal to three conterminous

edges in the other and including angles which are equal each to each, the parallelopipeds are equal in all respects.

For if two conterminous edges in one be equal to two in the other, the faces of which these edges are sides will be equal, and thus the proposition becomes a particular case of (73).

(76) COR.-If the vertex of a solid angle of a parallelopiped

be given in position, and the three edges terminated at that vertex be given in magnitude and position, the parallelopiped is given.

PROPOSITION IV.

(77) Every prism may be divided into as many triangular prisms as there are triangles into which

its base may be resolved by diagonals drawn from the vertex of any of its angles.

Since each pair of sides are equal and parallel, it follows that the diagonals of the bases which connect the extremities of the sides are equal and parallel, and the figure formed by the sides and diagonal is therefore a parallelogram. There are as many of these parallelograms, which we shall call diagonal planes, as there are different diagonals of the bases of the prism; and it is evident that the prism may be resolved into triangular prisms by diagonal planes, all of which pass through any one side and severally through the other sides, except those which are adjacent to that side which is their common intersection. This will be evident on inspecting the figure.

(78) COR.-It is evident that each diagonal plane is parallel to the sides of the prism, and also that the intersection of any two such planes is parallel to the sides.

PROPOSITION V.

(79) The sections of a prism by parallel planes are similar and equal rectilinear figures.

Let A B C D E and a b c d e be two parallel sections. Since A B and a b are the intersections of parallel planes with the same plane they are parallel, therefore Aab B is a parallelogram, and therefore A B and a bare equal. In the same manner it may be proved that BC is equal to bc, CD to cd, and so on.

Since the sides of the angle ABC are parallel to those of a b c, and in the same direction, the angle A B C is equal to a b c; and in like manner it may be proved that the angle B C D is equal to b c d, and so on. Hence the two sections A B C D E and abc de are equal and similar.

(80) COR. 1.-Hence all sections of a prism parallel to its bases are equal and similar to its bases.

(81) Cǝr. 2.-All sections of a parallelopiped parallel to any face are parallelograms equal and similar to that face.

PROPOSITION VI.

(82) In every parallelopiped the opposite angles are included by plane angles which are equal each to each, but not similarly placed, so that the solid angles do not admit of being placed with their edges mutually coincident.

Since A a and a b are parallel to D d and d c, and in the same direction the angles A a band D d c are equal. But

the angle D d c is equal to the opposite angle D C c,
therefore A a b is equal to DC c.
In the same
manner it may be proved that the angle A ad is
equal to c C B and bad to B C D.

But if the point C be conceived to be placed at a, and the edge C c upon a A, and C B upon a d, the

edge CD will not coincide with a b, but will extend in the opposite direction from the vertex a; and in the same manner whatever pair of edges of the angle D be placed in coincidence with the pair of corresponding edges of B, the remaining edges will be found not to coincide.

(83) DEF. Two solid angles contained by plane angles which are equal each to each, but which do not admit of coincidence, are said to be symmetrically equal.

PROPOSITION VII.

(84) Of four parallel edges of a parallelopiped the diagonal planes which pass through each pair

of them intersect, each dividing the other into two equal parallelograms, and the diagonals of each bisecting and being bisected by their common intersection.

Let A a c C be the diagonal plane through the opposite edges A a and C c, and D db B that through the opposite

edges D d and Bb, and let m M be the intersection

of these planes which will be equal and parallel to the edges A a, Bb, &c.

The diagonals a c and b d of the face a b c d bisect each other at m, and in like manner AC and

BD bisect each other at M. Hence it is evident that the parallelogram A m is equal to the parallelogram C m, and in like manner B m is equal to D m.

Also since a c is bisected at m, and m O is parallel to a A, A è must be bisected at O. And since m O is half of a A, it is also half of m M which is equal to a A, and therefore m M is bisected at O. By similar reasoning it may be pròved that b D is bisected at O.

In this way it may be proved that every diagonal of the parallelopiped passes through Ò and is bisected at that point. (85) COR.-Every right line whatever drawn through the point Ò, and terminated in the faces of the parallelopiped, is bisected at O. For let p q be such a line terminated in the faces Ab and Dc; and draw q D and p b. The lines p b and q D are parallel, being the intersections of the plane of the lines p q and Db with the parallel planes D c and A b. Hence the triangles qOD and b Op are mutually equiangular, and since b O is equal to O D, they are mutually equilateral. Therefore p O is equal

(86) DEF.-Hence the point O is called the centre of the parallelopiped.

(87) DEF.-The quantity of space included within the surface or surfaces of a solid figure is called its volume.

Thus when we speak of the volume of a solid we do not take into account its figure. If any solid figure were divided in any manner into parts, and these parts changed in their arrangement so as to form another and different solid figure, still the volume would remain the same although the shape be altered.

PROPOSITION VIII.

(88) The two prisms into which a diagonal plane divides a parallelopiped are equal in volume.

Let A B CD, abcd, be the bases of the parallelopiped, and A c a diagonal plane. Through a and A draw planes at right angles, to the sides of the given parallelopiped so as to form a rectangular parallelopiped, of which the bases are A B'C' D', a b' c' d', and the sides of which coincide with those of the given

one.

Since Cc and C'c' are each equal to A a they are equal to each other. Taking away the common part C c' we have c c'