or diminish in the same proportion as its base is increased or diminished; for, in that case, it will consist of the same number of plates, all the plates being increased or diminished in the same proportion as its base is increased or diminished. If it have the same base,

its volume will increase or diminish in the same proportion as its altitude is increased or diminished; for, in that case, while the magnitude of the corresponding plates remains unaltered, their numbers will be increased or diminished in the same proportion as the altitude is increased or diminished.

(340.) The volume of a triangular prism is equal to three times the volume of a pyramid, which has the same base and altitude as the prism.

Let A B C and A'B'C' (fig. 158.) be the two bases

or ends of the prism, and let a plane be supposed to be drawn through the edge AC and the angle B'; a pyramid will thus be cut off from the prism whose base is A B C, and whose vertex is at B'. If another plane be drawn through the edge B'C' and the angle A, a second pyramid will be cut from the prism, having for its base A'B'C', and for its vertex A. The altitude of each of these two pyramids will be the same, being the distance between the bases of the prism; and their bases will be equal, being the ends of the prism. The

remainder of the prism after removing these two pyramids will be the pyramid whose base is A C C ́, and whose vertex is B'; but the volume of this pyramid will be equal to the volume of the pyramid whose base is A A'C', and whose vertex is B', because these two pyramids have the equal triangles into which the parallelogram A A' C'C is divided by its diagonal A C' for their bases, and have a common vertex B'. It follows, therefore, that the three pyramids into which the prism is divided by the planes A B'C' and A B′ C have equal volumes; and since one of these has the base of the prism for its base, and the altitude of the prism for its altitude, the volume of the prism must be equal to three times the volume of the pyramid having the same base and altitude.

(341.) The volume of any prism whatever is equal to three times the volume of a pyramid having the same altitude, and having a base of equal area; for, whatever be the form of the base of the prism, its volume will be equal to that of a triangular prism having an equal base and altitude.

(342.) A figure formed by the section of a prism by a plane not parallel to its base is called a truncated prism.

fig. 159.

B'

A

N/

M/

M

MA

B

Let A A', B B', C C' (fig. 159.), be the three parallel edges of a triangular prism, and let MNO be the section of that whatever; and let M'N'O' be plane not parallel to the former. The figure whose ends or bases are M N O and M ́N ́O' is a truncated prism.

prism by any plane its section by another

(343.) The volume of a truncated triangular prism is equal to the sum of volumes of three pyramids whose base is one of the bases of the truncated prism, and whose vertices are at the three angles of the other base.

Draw a plane through the edge MO of the base MNO, and through the angle N'; this plane will cut off from the truncated prism a pyramid having for its base the base M N O, and for its vertex the angle N'. Draw another plane through the edge M' N', and through the angle O; this will cut off another pyramid having M' N'O' for its base, and O for its vertex. The remainder of the truncated prism will be the pyramid whose base is M M' N', and whose vertex is 0. But this will be equal to the pyramid which has the same base and its vertex at O'; because O and O' are equally distant from the plane M M' N'. Hence it follows that the volume of the truncated prism is equal to the two pyramids which have M'N'O' for their common base and their vertices at M and O, together with the volume of the pyramid which has MNO for its base and its vertex at N'. But if the line N O' be drawn, the pyramids whose common base is M N N' and whose, vertices are O and O ́are equal; and if N M' be drawn, the pyramids whose common base is N N'O' and whose vertices are at M and M' will have equal volumes. It follows, therefore, that the pyramid which has M N O for its base and its vertex at N', will be equal to that which has M'N' O' for its base and its vertex at N. Hence it appears that the whole volume of the truncated triangular prism is equal to the sum of the volumes of three pyramids which have M'N'O' for their base, and their vertices at the points M, N, and O, which form the angles of the other base.

(344.) Since pyramids having equal bases and altitudes have equal volumes, it follows that the volume of a triangular truncated prism is equal to the sum of the volumes of three pyramids having one of the bases of the prism for their base, and having their altitudes

equal to perpendiculars drawn upon the one base of the prism from the three angles of the other base.

(345.) Let M" N" O" be a section of the prism by a plane perpendicular to its edges. The volume of the truncated prism whose base is M′′ N ̋ O′′, and whose superior base is M NO, will then be equal to the sum of the volumes of three pyramids upon the base M ́N ́O′′ whose vertices shall be M, N, and O; or, since the edges of the prism are perpendicular to M" N"O", it will be equal to the sum of the volumes of three pyramids upon the base M" N" O" with the altitudes M" M, N" N, and O" 0.

For the same reasons the volume of the prism on the base M"N" O", and having for its superior base M'N' O', will be equal to the sum of the volumes of three pyramids whose common base is M" N" O", and whose altitudes are respectively M" M', N" N', and O" O'. The difference between the volumes, therefore, which is in fact the volume of the truncated prism whose bases are M N O and M'N' O', is equal to the difference between the sum of the volumes of the three former pyramids having M" N" O" as their common base, and the sum of the volumes of the three latter pyramids having the same common base, which difference will be equal to the sum of the volumes of three pyramids having the same common base M" N" O", and the difference of the altitudes respectively of the two systems of pyramids as their altitudes, which differences will be M M', N N', and O O'.

It follows, therefore, that the volume of any triangular truncated prism whatever will be equal to the sum of the volumes of three pyramids whose common base is a rectangular section of the prism, and whose altitudes respectively are equal to the three edges of the truncated prism.

(346.) Since the volumes of prisms and pyramids having equal bases are proportional to their altitudes, it follows that the sum of the volumes of any number of prisms or pyramids having equal bases will be equal

to the volume of one prism or pyramid having the same base, and whose altitude shall be equal to the sum of their several altitudes.

(347.) Since the volumes of prisms and pyramids having equal altitudes are proportional to their bases, it follows that the sum of the volumes of several prisms or pyramids having equal altitudes, is equal to the volume of one prism or pyramid with the same altitude, and whose base is equal to the sum of their several bases.

(348.) Since the volume of a truncated triangular prism is equal to the sum of the volumes of three pyramids whose common base is the rectangular section of the prism, and whose altitudes respectively are its three edges, it is equal to the volume of one pyramid whose base is the same rectangular section of the prism, and whose altitude is the sum of the three edges.

A/

fig. 160.

B'

C

Di

(349.) Let A B C D and A'B'C' D ́ (fig. 160.) be the bases of a quadrangular truncated prism whose faces are perpendicular to each other, and let A"B"C"D" be a rectangular section of it; let its volume be divided by two diagonal planes, one passing through the edges A A ́,C C ́, and the other through the edges BB', DD': the volume of the truncated triangular prism whose bases are ABD and A'B'D' is equal to the volume of a pyramid whose base is A"B" D", and whose altitude is the sum of the edges AA', B B', and DD'. In like manner the volume of the truncated triangular prism whose bases are BCD and B'C'D' is equal to the volume of

B

D"

a pyramid whose base is B ́C ́ ́D", and whose altitude is the sum of the edges BB', CC', and DD'; there