=

C; oa = a, ob = b, oc = c.

OA = A, OB = B, OC Now, if we increase a, b, c, by x, y, z, so as to become commensurable with A, B, C, and construct the parallelopipedon on (a + x), (b+ y), (c+ z), from what has already been proved, there

results

=

ABC

parallelopipedon [(a + x), (b + y), (c + z)] _ (a + x)(b+y) (c + z); parallelopipedon [A, B, C] par'dn [a, b, c] solid [x, y, z] abc par’dn [A, B, C]' par'dn [A, B,

or

+

=

[C] ABC

...

+ABC

.. (63), pard'n [A, B, C] : par'dn [a, b, c] :: ABC: abc.

Q. E. D. Cor. 1. The rectangular parallelopipedon is measured (499) by the product of its three dimensions, provided the cube, whose edge is the linear unit, be assumed as the unit of solidity.

For, from

we have

par'dn [a, b, c] = 1, a = b = c = 1,
par'dn [A, B, C]= ABC.

Cor. 2. The right prism with a right angled triangular (500) base, is measured by its base multiplied into its altitude.

For the diagonal plane divides the rectangular parallelopipedon into two rectangular prisms, capable of superposition.

Fig. 1152.

Cor. 3. Any right prism with a triangular base is equal (501) to its base multiplied into its altitude.

For the prism may be split into two, having right

angled triangles for bases.

Fig. 1153.

Cor. 4. Any right prism is measured by the product of (502)

its base and altitude.

For the solid may be divided into prisms, having triangular bases.

Fig. 1154.

Cor. 5. The right cylinder is measured by the product of (503) its base and altitude.

For (502) is obviously independent of the number and magnitude of the sides.

The pyramid is equal to one-third of the product of its (504) base and altitude, and the cone has a like measure.

In the first place, suppose the pyramid, y, to have a triangular base, z, to which one of the edges, x, is perpendicular; and, for the purpose of finding the func- yx tion y = fx, give to y, z, z, the corresponding increments k, h, i. Since the prisms constructed with the kh altitude h, and upon the bases z, z+i, will be inscribed in and circumscribe the solid, k, we have

Fig. 116.

where no constant is to be added, because yo=0, and the proposition is proved for this particular case.

Next, let a right angled triangle, revolving about its perpendicular, p, and its base, varying in any way whatever, describe, the one a cone or pyramid, y, the other its base, x; giving to y and x the vanishing increments, k and h, from what has just been proved, we find

[k]=\p[h], _.._y' =‡p;
y = 3px;

x h

Fig. 1162.

and the proposition is demonstrated for all cones and pyramids, in which the perpendicular falls within the base.

Lastly, let the base, u, be any whatever, and the perpendicular, p, fall upon its production; then, joining the foot of p, with two points of a contiguous portion of the perimeter so as to form the base, v, of a second pyramid or cone (p, v), we have

and

cone (p,u+v) = ‡p(u+v),

cone (p,v) = pv ;

cone (p,u) pu. Q. E. D..

=

PROPOSITION III

v

Fig. 1163.

The Frustrum of a cone or pyramid is measured by (505) one-third of the product of its altitude, multiplied into the sum of its bases augmented by a mean proportional between them.

frustrum [A,B] = f(x + a)A − ‡xB = [aA + x(A−B)];

(505)

frustrum [A,B.a] = ‡a[A+ (AB)*+B].

Cor. The prism or cylinder, whether right or oblique, is (506) measured by the product of its base and altitude.

For

BA, gives a[A + (AB) + B] = Aa.

PROPOSITION IV.

V

If a solid, V, be generated by the motion of a plane, U, (507) varying according to the law of continuity, and remaining constantly similar to itself, and perpendicular to the axis of x: then will the derivative of V, regarding V as a function of x, be equal to the generating plane, U, also regarded as a function of x, or

h

Fig. 117

For, from a little reflection, it will be evident that the incremental solid [U,U2,h] must be measured as in (505), or that

=

· [ƒx + {(ƒx • f(x + h)} * + f(x + h)]

= [ƒx+fx+fx] = fx = U. Q. E. D.

Cor. 1. For any solid, generated by the revolution of a (508) curve about the axis of x, the ordinate, y, describing the plane, U, we have

ELLIPSOIDS.

V'v_F = U=πy2 = π( ƒx)2.

257

Cor. 2. For any volume embraced between the surfaces (509) described by the revolution of two curves, or the two branches of the same curve, U being described by the difference of the corresponding ordinates, y, Y2, we have

29

V'1 = Tу2 — TY22 = π(y2 — Y22) = π(Y + Y2)(Y — Y2). · Cor. 3. For the ellipsoidal frustrum, estimated from its equator, or from x = 0, we find

DC

Fig. 1172.

Cor. 4. The corresponding frustrum of the circumscribing sphere, is

since (510) becomes (511) when b= a.

Cor. 5. Prolate Ellipsoid=2V... = ‡πab2
παδ
= f. 2a. πb2 = f(circumscribed cylinder)
2(inscribed double cone).

=

Cor. 6. Sphere radius = a = ‡πα3 = † • 2a • πa2

=

(circumscribed cylinder)

= 2(inscribed double cone)

(511)

(512)

Fig. 1174.

(513)

=‡а• 4πа2 = a(surface of sphere).

Scholium. The last relation might have been found by imagining the sphere filled with pyramids, having their common vertex at its centre, and their bases resting on its surface.

Cor. 7. The prolate ellipsoid and its circumscribing (514) sphere, and their frustrums corresponding to the

same abscissa, are to each other as the square of

the minor to the square of the major axis.

=

Prolate ellipsoid : sphere, V.: V1 = b2 : a2.

e

Fig. 1175.

Cor. 8. Analogous relations may be found for the Ob- (515)

late Ellipsoid, described by the revolution of the ellipse about its minor axis by changing a into b, and b into a.

= 2 • † • 2b • πa2 = 2(inscribed double cone);

=

= :

Fig. 1176.

Sphere radius-b .. Oblate Ellipsoid: Inscribed Sphere V2, V2,c=a2: b2. Cor. 9. Common Paraboloid V =прг2 = ‡x • пy2

(516)

Similar solids are to each other as the cubes of their (517) like dimensions.

We understand by similar solids those in which all like dimensions are proportional; and, consequently, the sections through such dimensions similar.

The proposition becomes evident for the solids already investigated by making A and B vary as a2 in (505), and b as a in (512) and (515), or by putting Aca3, B = c,a2, b = c,a, the cs being constant, whence

Similar Cones, Pyramids, and their similar Frustrums, consequently similar Parallelopipedons, also similar Ellipsoids and Spheres

vary as

[c + (cc2) * + c2] • a3, ‡πc2 • a3, or as a3, or as the cubes of any like dimensions.

Next let any two similar perimeters, whether rectilin- (fig. 99.) ear or curvilinear, be similarly placed and revolve about any line OaA of corresponding radii vectores as an axis of x, the origin being at O; the ordinates y, y, of the extremities P, p, of any other corresponding radii vectores, will describe planes U, U,, terminating