plane of the base: hence these pyramids are equivalent. But the pyramid K-FƒH may be regarded as having its vertex in f, and thus its altitude will be the same as that of the frustum: as to its base FKH, we are now to show that this is a mean proportional between the bases FGH and fgh. Now, the triangles FHK, fgh, have each an equal angle F=ƒ; hence

FHK: fgh: : FK × FH: fg× fh (Book IV. Prop. XXIV.) ; but because of the parallels, FK=fg, hence

We have also,

FHK fgh: FH : fh.

FHG : FHK :: FG: FK or fg.

But the similar triangles FGH, fgh give

hence,

FG: fg: FH : fh;

FGH FHK:: FHK : fgh;

or the base FHK is a mean proportional between the two bases FGH, fgh. Hence the frustum of a triangular pyramid is equivalent to three pyramids whose common altitude is that of the frustum and whose bases are the lower base of the frustum, the upper base, and a mean proportional between the two bases.

PROPOSITION XIX. THEOREM.

Similar triangular prisms are to each other as the cubes of their homologous sides.

Let CBD-P, cbd-p, be two similar triangular prisms, of which BC, bc, are homologous sides: then will the prism CBD-P be to the prism cbd-p, as BC3 to bc33.

P

a

a

D

d

H

B

C с

For, since the prisms are similar, the planes which contain the homologous solid angles B and b, are similar, like placed, and equally inclined to each other (Def. 17.): hence the solid angles B and b, are equal (Book VI. Prop. XXI. Sch.). If these solid angles be applied to each other, the angle cbd will coincide with CBD, the side ba with BA, and the prism cbd-p will take the position Bcd-p. From A draw AH perpendicular to the common base of the prisms: then will the plane BAH be perpendicular to the plane of the com

mon base (Book VI. Prop. XVI.). Through a, in the plane BAH, draw ah perpendicular to P BH: then will ah also be perpendicular to the base BDC (Book VI. Prop. XVII.); and AH, ah will be the altitudes of the two prisms.

Now, because of the similar triangles ABH,aBh, and of the similar parallelograms AC, ac, we have

D

租

a

AH: ah :: AB : ab : : BC : bc.

But since the bases are similar, we have

base BCD base bcd :: BC2: bc2 (Book IV. Prop. XXV.) ; hence,

:

base BCD base bcd :: AH2 : ah2. Multiplying the antecedents by AH, and the consequents by ah, and we have

base BCD × AH : base bcd× ah :: AH3 ah3.

But the solidity of a prism is equal to the base multiplied by the altitude (Prop. XIV.); hence, the

prism BCD-P: prism bcd-p :: AH3 : ah3 :: BC3 : bc3, or as the cubes of any other of their homologous sides.

Cor. Whatever be the bases of similar prisms, the prisms will be to each other as the cubes of their homologous sides. For, since the prisms are similar, their bases will be similar polygons (Def. 17.); and these similar polygons may be di. vided into an equal number of similar triangles, similarly placed (Book IV. Prop. XXVI.): therefore the two prisms may be divided into an equal number of triangular prisms, having their faces similar and like placed; and therefore, equally inclined (Book VI. Prop. XXI.); hence the prisms will be similar. But these triangular prisms will be to each other as the cubes of their homologous sides, which sides being proportional, the sums of the triangular prisms, that is, the polygonal prisms, will be to each other as the cubes of their homologous sides.

PROPOSITION XX. THEOREM.

Two similar pyramids are to each other as the cubes of their homologous sides.

For, since the pyramids are similar, the solid angles at the vertices will be contained by the same number of similar planes, like placed, and equally inclined to each other (Def. 17.). Hence, the solid angles at the vertices may be made to coincide, or the two pyramids may be so placed as to have the solid angle S common.

B

C

In that position, the bases ABCDE, abcde, will be parallel; because, since the homologous faces are similar, the angle Sab is equal to SAB, and Sbc to SBC; hence the plane ABC is parallel to the plane abc (Book VI. Prop. XIII.). This being proved, let SO be the perpendicular drawn from the vertex S to the plane ABC, and o the point where this perpendicular meets the plane abc: from what has already been shown, we shall have

SO: So: SA: Sa: AB: ab (Prop. III.); and consequently,

SO So: AB: ab.

But the bases ABCDE, abcde, being similar figures, we have ABCDE: abcde: AB2: ab2 (Book IV. Prop. XXVII.). Multiply the corresponding terms of these two proportions; there results the proportion,

ABCDESO: abcdex So :: AB3: ab3.

Now ABCDEX SO is the solidity of the pyramid S-ABCDE, and abcdex So is that of the pyramid S-abcde (Prop. XVII.); hence two similar pyramids are to each other as the cubes of their homologous sides.

General Scholium.

The chief propositions of this Book relating to the solidity of polyedrons, may be exhibited in algebraical terms, and so recapitulated in the briefest manner possible.

Let B represent the base of a prism; H its altitude: the solidity of the prism will be Bx H, or BH.

Let B represent the base of a pyramid; H its altitude: the solidity of the pyramid will be BxH, or H×B, or BH.

Let H represent the altitude of the frustum of a pyramid, having parallel bases A and B ; AB will be the mean proportional between those bases; and the solidity of the frustum will be Hx (A+B+ √AB).

In fine, let P and p represent the solidities of two similar prisms or pyramids; A and a, two homologous edges: then we shall have

P: P :: A3: a3.

BOOK VIII.

THE THREE ROUND BODIES.

Definitions.

1. A cylinder is the solid generated by the revolution of a rectangle ABCD, conceived to turn about the immoveable. side AB.

In this movement, the sides AD, BC, continuing always perpendicular to AB, describe equal circles DHP, CGQ, which are called E the bases of the cylinder, the side CD at the same time describing the convex surface.

The immoveable line AB is called the axis of the cylinder.

M

Every section KLM, made in the cylinder, at right angles to the axis, is a circle equal to F either of the bases; for, whilst the rectangle ABCD turns about AB, the line KI, perpen

P

G

dicular to AB, describes a circle, equal to the base, and this circle is nothing else than the section made perpendicular to the axis at the point I.

Every section PQG, made through the axis, is a rectangle double of the generating rectangle ABCD.

2. A cone is the solid generated by the revolution of a rightangled triangle SAB, conceived to turn about the immoveable side SA.

In this movement, the side AB describes a circle BDCE, named the buse of the cone; the hypothenuse SB describes the convex surface of the cone.

The point S is named the vertex of the cone, SA the axis or the altitude, and SB the side or the apothem.

Every section HKFI, at right angles to the axis, is a circle; every section SDE, through the axis, is an isosceles triangle, double of the generating triangle SAB.

S

FH

E

B

D

3. If from the cone S-CDB, the cone S-FKH be cut off by a plane parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone.

We may conceive it to be generated by the revolution of a trapezoid ABHG, whose angles A and G are right angles, about the side AG. The immoveable line AG is called the axis or altitude of the frustum, the circles BDC, HEK, are its bases, and BH is its side.

4. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases.

5. If in the circle ACD, which forms the base of a cylinder, a polygon ABCDE be inscribed, a right prism, constructed on this F base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism.

The edges AF, BG, CH, &c. of the prism, being perpendicular to the plane of the base, are evidently included in the convex sur- . face of the cylinder; hence the prism and the cylinder touch one another along these edges.

6. In like manner, if ABCD is a polygon, circumscribed about the base of a cylinder, a right prism, constructed on this base ABCD, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, or the cylinder to be inscribed in the prism.

N

M

W

B

R

K

Let M, N, &c. be the points of contact in the sides AB, BC, &c.; and through the points M, N, &c. let MX, NY, &c. be drawn A perpendicular to the plane of the base: these perpendiculars will evidently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of

contact.

7. If in the circle ABCDE, which forms the base of a cone, any polygon ABCDE be inscribed, and from the vertices A, B, C, D, E, lines be drawn to S, the vertex of the cone, these lines may be regarded as the sides of a pyramid whose base is the polygon ABCDE and vertex S. The sides of this pyramid are in the convex A surface of the cone, and the pyramid is said to be inscribed in the cone.

M

E

N

B

S

B