drawn equal and parallel to the sides AB, BC, CD, &c. thus forming the polygon FGHIK equal to ABCDE; if in the next place, the vertices of the angles in the one plane be joined with the homologous vertices in the other, by straight lines AF, BG, CH, &c., the faces ABGF, BCHG, &c. will be parallelograms, and ABCDEFGHIK, the solid so formed, will be a prism.

V. The equal and parallel polygons ABCDE, FGHIK_are called the bases of the prism; the plane parallelograms taken together constitute the lateral or convex surface of the prism; the equal straight lines AF, BG, CH, &c. are called the sides of the prism.

VI. The altitude of a prism is the distance of its two bases, or the perpendicular drawn from a point in the upper base to the plane of the lower base.

VII. A prism is right when the sides AF, BG, CH, &c. are perpendicular to the planes of the bases; and then each of them is equal to the altitude of the prism. In any other case the prism is oblique, and the altitude less than the side.

VIII. A prism is triangular, quadrangular, pentagonal, hexagonal, &c. when the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c.

IX. A prism whose base is a parallelogram has all its faces parallelograms; it is named a parallelepipedon. (See the diagram of Prop. 4. VI.)

The parallelepipedon is rectangular when all its faces are rectangles.

X. Among rectangular parallelepipedons, we distinguish the cube, or regular hexaedron, bounded by six equal squares.

XI. A pyramid (see the diagram of Prop. 22. V.) is the solid formed by several triangular planes proceeding from the same point S, and terminating in the different sides of the same polygonal plane ABCDE.

The polygon ABCDE is called the base of the pyramid, the point S its vertex; and the whole of the triangles ASB, BSC, &c. form its convex or lateral surface.

XII. The altitude of a pyramid is the perpendicular let fall from the vertex upon the plane of the base, produced if necessary.

XIII. A pyramid is triangular, quadrangular, &c. according as its base is a triangle, a quadrilateral, &c.

XIV. A pyramid is regular, when its base is a regular polygon, and when, at the same time, the perpendicular let fall from the vertex to the plane of the base passes through the centre of this base. That perpendicular is then called the axis of the pyramid.

XV. The diagonal is the straight line joining the vertices of two solid angles which are not adjacent to each other.

XVI. We shall give the name, symmetrical polyedrons, to any two polyedrons which having a common base, are constructed similarly, the one above this base, the other beneath it, and so that the vertices of their corresponding solid angles are situated at equal distances from the plane of the base, and in the same straight line perpendicular to that plane.

If the straight line ST, for example, is perpendicular to the plane ABC, and also bisected at the point O, where it meets this plane, the two pyra- s

mids SABC, TABC,

T

which have the common base ABC, will be two symmetrical polyedrons.

XVII. Two triangular pyramids are similar, when two faces in each are respectively similarly placed, and equally inclined to

each other.

XVIII. If a triangle is formed by joining the vertices of three angles taken on one face or on the base of a polyedron, then the vertices of the different solid angles of the polyedron, which are situated without this base may be conceived as being the ver

tices of so many triangular pyramids having the triangle just described for a common base; and each of those pyramids will determine the position of a solid angle of the polyedron with reference to the base. Now,

Two polyedrons are similar, when having similar bases, the vertices of their corresponding solid angles lying without those bases are determined by triangular pyramids which are similar each to each.

XIX. By the vertices of a polyedron, we mean the points situated at the vertices of its different solid angles.

Note. The only polyedrons we intend at present to treat of, are polyedrons with salient angles, or convex polyedrons. They are such that their surface cannot be intersected by a straight line in more than two points. In polyedrons of this kind, the plane of any face, when produced, can in no case cut the solid; the polyedron therefore cannot be in part above the plane of any face, and in part below it; it must lie wholly on the same side of this plane.

PROPOSITION I. THEOREM.

Two polyedrons cannot have the same vertices and in the same number without coinciding.

For, suppose one polyedron to be already constructed: if a second is to be formed, having the same vertices and in the same number, the planes of the latter must either not all pass through the same points with the planes of the former, or the two polyedrons will not differ from each other. But if those planes of the latter do not all pass through the same points with the planes of the former, some of them must cut the first polyedron; one or more of whose vertices must therefore lie above them, one or more below; which cannot be the case with a convex polyedron: hence if two polyedrons have the same vertices and in the same number, they must necessarily coincide with each other.

Scholium. The points A, B, C, K, &c. which are to be the vertices of a polyedron, being given, it is easy to describe the polyedron.

First choose three adjacent points D, E, H, such that the plane DEH shall pass, if need be, through the new points K, C, but leaving all the rest on the same side, all above the plane or all below it; the plane DEH or DEHKC, thus determined, will be one face of the solid. Along EH one of its sides, extend a plane, which turn round upon EH as an axis till it meet a new

D

vertex F, or several at once as F, I; it will give a second face FEH or FEHI. Continue the same process, making planes to pass through the sides successively determined, till the solid is bounded on all quarters: this solid will be the polyedron required, since there cannot be two which have the same vertices.

PROPOSITION II. THEOREM.

In two symmetrical polygons, the homologous faces are respectively equal, and the inclination of two adjacent faces in one of those solids is equal to the inclination of the two homologous faces in the other.

Let ABCDE be the common base of the two polyedrons; M and N the vertices of any two solid angles in the one, M' and N' the homologous vertices of the other; then (Def. 16.) the straight lines MM', NN', must be perpendicular to the A plane ABC, and be divided into two equal parts at the points m and n, where they meet it. Now we are to shew that MN is equal to M'N'.

For, if the trapezium m'M'N'n be made to revolve about mn till the plane of it falls upon the plane

m

M

R

E

D

B

-R

mMNn; by reason of the right angles at m and n, the side mM' will fall on its equal mM, and nN' upon nN; hence the trapeziums will coincide, and we shall have MN=M'N'.

Let P be a third vertex of the upper polyedron, and P^ its homologous vertex in the other; we shall, as before, have MP=MP', and NP-NP'; hence the triangle MNP, which joins any three vertices of the upper polyedron, is equal to the triangle M'N'P', which joins the three corresponding vertices of the other polyedron.

If, among those triangles, we confine our attention to such as are formed at the surface of the polyedrons, we may already conclude that the surfaces of the two polyedrons are each composed of the same number of triangles respectively equal in both.

It is now to be shewn, that, if any of those triangles lie on the same plane in the upper surface, and form one and the same polygonal face, the corresponding triangles will lie on the same plane in the under surface, and there form one equal polygonal face.

To prove this, let MPN, NPQ, be two adjacent triangles supposed to lie on the same plane; and let M'P'N', N'P'Q', be their corresponding triangles. The angle MNPM'N'P', the angle PNQ=P'N'Q'; and if MQ and M'Q' were joined, the triangle MNQ would be equal to M'N'Q'; hence we should have the angle MNQ=M'N'Q'. But since MPNQ is one single plane, the angle MNQ=MNP+PNQ; hence we shall likewise have M'N'Q-M'N'P'+P'N'Q. Now, if the three planes M'N'P', P'N'Q', M'N'Q' were not all in one plane, those three planes would form a solid angle, and (20. V.) we should have the angle M'N'Q' M'N'P'+P'N'Q'; which conclusion not being true, the two triangles M'N'P', P'N'Q' are in one and the same plane.

Hence each face, whether triangular or polygonal, in the one polyedron, corresponds to an equal face in the other polyedron, and thus the two polyedrons are each included under the same number of planes respectively equal in both.

We have still to shew, that the inclination of any two adjacent faces in the one polyedron is equal to the inclination of the two corresponding faces in the other.

Let MPN, NPQ be two triangles formed on the common edge NP, in the planes of two adjacent faces; let M'P'N', N'P'Q' correspond to them: conceive a solid angle to be formed at N, by the three plane angles MNQ, MNP, PNQ; and another at N', by the three M'NQ', M'N'P', P'N'Q'. Now it has been shewn already, that those plane angles are respectively equal; hence the inclination of the two planes MNP, PNQ is equal (22. V.) to that of their corresponding planes M'N'P', P'N'Q'.

Therefore, in symmetrical polyedrons, the faces are equal each to each; and the planes of any two adjacent faces, in the one solid have the same inclination as the planes of the two corresponding faces in the other solid.

Scholium. It may be observed, that the solid angles of the one polyedron are symmetrical with the solid angles of the other; for as the solid angle N is formed by the planes MNP, PNQ, QNR, &c., so its corresponding angle N is formed by the planes M'N'P', P'N'Q', Q'N'R', &c. The latter appear to be arranged in the same order as the former; but since the two solid angles are in an inverse position with regard to each other, the real arrangement of the planes which form the solid angle N', must be the reverse of the arrangement which occurs in the corresponding angle N. Farther, in both solids, the inclinations of the consecutive planes are respectively the same; hence those solid angles are symmetrical each with the other. (See the scholium of Prop. 23. V.)