Let the lines through B, G, C, D and F (fig ii) cut the boundary of the fi ' ' B' ' ' D' and F, and meet the base X'X in K, L, M, N and P;

gure again in, G, C,

G' C the points A and E being at

I D » the extremities of the figure,

If; F and the lines through them

B g Z Q E meeting the base in a and e. fl' I I Then, if we take ordinates Kb, A ' - if I Lg, Mc, Nd, Pf, eilual to B'B, GG', C'C, D'D, F ', the figure

ib c, ;D' ' I abgcdfe will be the equivalent

3 Gi;; i Q trapezoid, and any ordinate

~;; 1 i ' drawn from the base to the

Y' a K LM N P ' X top of this trapezoid will be ual to the portion of this

Flo. 4. eq

ordinate (produced) which falls

within the original figure.

26. Volumes of Solids with Plane Faces(-The following are expressions for the volumes of some simple solid figures. (i) Cube: side a. Volume=a“.

(ii) Rectangular parallelepiped: sides a, b, c. Volume=abc. (iii) Right prism. Volume=length of edge X area of end. (iv) Oblique prism. Volume=height X area of end=length of edge X area of cross-section; the “ height ” being the perpendicular distance between the two ends.

The parallelepiped is a particular case. (v) yramid with rectilinear base. Volume=he1ghtX§ .area of base.

The tetrahedron is a particular case.

(vi) Wedge: parallel edges a, b, c; area of cross-section S. Volume=§ (a+b+c)S.

This formula holds for the general case in which the base is a trapezium; the wedge being thus formed by cutting a triangular prism by any two planes.

(vii) Frustum of pyramid with rectilinear base: height h; areas of ends (i.e. base and top) A and B. Volume=h ~§ (A-I-/AB-|~B). 27. The figures considered in § 26 are particulancases of the prismoid (or prismatoid), which may be defined as a solid figure with two parallel plane rectilinear ends, each of the other (i.e the lateral) faces being a triangle with an angular point in one end of the figure and its opposite side in the other. Two adjoining faces in the same plane may together make a trapezium. More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices.

If R and S are the ends of a prismoid, A and B their areas, h the perpendicular distance between them, and C the area of a section a plane parallel to R and S and midway between them, the volume %h(A+4C+B)-This

is known as the tfrzsmoidalfformula.

The formula is a eduction rom a general formula, considered later (§ 58), and may be verified in various ways. The most instructive is to regard the prismoid as built up (by addition or subtraction) of simpler figures, which are particular cases of it. (i) Let R and S be the vertex and the ase of a pyramid. Then A '= O, C = § B, and volume = § hB = § h(A +4C -|- B). The tetrahedron is a particular case.

H, .

o the prismoid is

(ii) Let R be one edge of a wedge with parallel ends, and S the face containing the other two edges. Then A=O, C=§ B, and volume = § hB = ~§ h(A+4C-l-B).

(iii) Let R and S be two opposite edges of a tetrahedron. Then the tetrahedron may be regarded as the difference of a wedge with parallel ends, one of the e ges being ~R, and a pyramid whose base LS a parallelogram, one side of the parallelogram being S (see fig. 9, § 58). Hence, by (i) and (ii), the formula holds for this figure. (iv) For the rismoid in general let ABCD be one end, and abcd the otiier. Take any point P in the latter, and form triangles by joining P to each of the sides AB, BC, . ab, bc, . . . of the ends, and also to each of the edges. Then the prismoid is divided into a pyramid with vertex P and base ABCD ., and a such as PABa or PAab By (i) and (iii), the formula holds for each of these figures; and therefore it holds for the prismoid as a whole. Another method of verifying the formula is to take a point Q in the mid-section, and divide up the prismoid into two pyramids with vertex Q and bases ABCD . and

abcd . . respectively, and a series of tetrahedral having Q as one vertex.

28. The Circle and Allied Figures.-The mensuration of the circle is founded on the property that the areas of different circles are proportional to the squares on their diameters. t ratio by fr, the area of a circle is 1ra2, where 4 *S the radii-S, and 'l|'=3'14159 approximately. The expression 21a for the length of the circumference can be deduced by considering the limit of the area cut off from a circle of radius a by a concentric circle of radius a'-a, when a becomes indefinitely small; this is an elementary case of differentiation. The lengths of arcs of the same circle being proportional to the series of tetrahedral,

B

P

A

0

c

Q o

Fig. 5.

Denoting the constan

angles subtended by them at the centre, we get the idea of circular measure.

Let O be the common centre of two circles, of radii a and b, and let radii enclosing an angge 0 (circular measure) cut their circumferences in A, B and C, respectively (fig. 5). Then the area of ABDC is

éb20- saw = (b -a) -§ (b +a)0.

If we bisect AB and CD in P and Q respectively, and describe the arc PQ of a circle with centre O, the length of this arc is %(b-l-a)0; and b—a=AB. Hence area ABDC =AB>< arc PQ. The figure ABDC is a sector of an annulus, which is the portion of a circle left after cutting out a concentric circle. 29. By considering the circle as the limit of a polygon, it follows that the formulae (iii) and (v) of § 26 hold fora right circular cylinder and a right circular cone; i.e.

volume of right circular cylinder=length X area of base; volume of right circular cone =height X é area of base. These formulae also hold for any right cylinder and any cone. 30. The curved surfaces of the cylinder and of the cone are develop able surfaces; i.e. they can be unrolled on a plane. The curved surface of any right cylinder (whether circular or not) becomes a rectangle, and therefore its area=length .X perimeter of base. The curved surface of a right circular cone becomes a sector of a circle, and its area=%-slant height X perimeter of base. 31. If a is the radius of a sphere, then (i) volume of sphere=§ 1ra3;

(ii) surface of sphere=41ra2=curved surface of circumscribing cylinder.

The first of these is a particular case of the prismoidal formula (§ 58). To obtain (i) and (ii) together, we show that the volume of a sphere is proportional to the volume of the cube whose edge is the diameter; denoting.the constant ratio by ik, the volume of the sphere is M3, and thence, by taking two concentric spheres (cf. § 28), the area of the surface is 3)a2. This surface may be split up into elements, each of which is equal to a corresponding element of the curved surface of the circumscribing c linder, so that 3>»a'=curved surface of cylinder=2a. 21ra=41ra2. Hibnce)=§ 1r The total surface of the cylinder is 41ra2+1ra2+1ra2=61ra', and its volume is 2a.1ra”=21ra3. Hence

volume of sphere = § volume of circumscribing cylinder; surface of sphere = 3; surface of circumscribing cylinder. These latter formulae are due to Archimedes.-32. Moments and Centroidsf-For every material body there is a point, fixed with regard to the body, such that the moment of the body with regard to any plane is the same as if the whole mass were collected at that point; the moment being the sum of the products of each element of mass of the body by its distance from the plane. This point is the centroid of the body. The ideas of moment and of centroid are extended to geometrical figures, whether solid, superficial, or linear. The moment of a figure with regard to a plane is found by dividing the figure into elements of volume, area or length, multiplying each element by its distance from the plane, and adding the products. In the case of aplane area or a plane continuous line the moment with regard to a straight line in the plane is the same as the moment with regard to a perpendicular plane through this line; i e. it is the sum of the products of each element of area or length by its distance from the straight line. The centroid of a figure is a point fixed with regard to the figure, and such that its moment with regard to any plane (or, in the case of a plane area or line, with regard to any line in the plane) is the same as if the whole volume, area or length were concentrated at this point. The centroid is sometimes called the centre of volume, centre of area, or centre of arc. The proof of the existence of the centroid of a figure is the same as the proof of the existence of the centre of ravity of a body. (See MECHANICS.) The moment as descried above is sometimes called the jirst moment. The second moment, third moment, . . of a plane or solid figure are found in the same way by multiplying each element by the uare, cube, . of its distance from the line or plane with regafd to which the moments are being taken. If we divide the first, second, third, . moments by the total volume, area or length of the figure, we get the mean distance, mean square of distance, mean cube of distance . .. of the figure from the line or plane. The mean distance of a plane figure from a line in its plane, or of any figure from a plane, is therefore the same as the distance of the centroid of the figure from the line or plane. We sometimes require the moments with regard to a line or plane through the centroid. If N0 is the area of a plane figure, and Ni, Ng, . .are its moments with regard to a line in its plane, the moments Ml, Mg, . . with regard to a parallel line through the centroid are given byM1=N1';xN0=0, = N2 2xN1+ x2N0 = N2 ' x2N0,

M, = N, - qxN, , 1 -l-i@'7i2x”N, , 2. +<-yi-lgxf-1N, + 2. .

()"xNo;