Page:EB1911 - Volume 18.djvu/157

Let the lines through B, G, C, D and F (fig 4.) cut the boundary of the figure again in B′, G′, D′ and F′, and meet the base X′X in K, L, M, N and P; the points A and E being at the extremities of the figure, and the lines through them meeting the base in a and e. Then, if we take ordinates Kb,
Fig. 4. Lg, Mc, Nd, Pf, equal to B′B, GG′, C′C, D′D, FF′, the figure abgcdfe will be the equivalent trapezoid, and any ordinate drawn from the base to the top of this trapezoid will be equal to the portion of this 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＝a³.

⁠(ii) Rectangular parallelepiped: sides a, b, c. Volume＝abc.

(iii) Right prism. Volume＝length of edge × area of end.

(iv) Oblique prism. Volume＝height × area of end＝length of edge × area of cross-section; the “height” being the perpendicular distance between the two ends.

The parallelepiped is a particular case.

⁠(v) Pyramid with rectilinear base. Volume＝height × 1/3.area of base.

The tetrahedron is a particular case.

(vi) Wedge: parallel edges a, b, c; area of cross-section S. Volume＝1/3 (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.1/3(A+√AB+B).

27. The figures considered in § 26 are particular cases 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 of the prismoid is

1/6h(A+4C+B).

This is known as the prismoidal formula.

The formula is a deduction from 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 base of a pyramid. Then A＝O, C＝1/4B, and volume＝1/3hB＝1/6h(A +4C + B). The tetrahedron is a particular case.

(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＝1/2B, and volume＝1/2hB＝1/2h(A+4C+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 edges being R, and a pyramid whose base is 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 prismoid in general let ABCD . . . be one end, and abcd . . . the other. 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 series of tetrahedra, 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 tetrahedra having Q as one vertex.

Fig. 5.

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. Denoting the constant ratio by 1/4π, the area of a circle is πa², where a is the radius, and π＝3·14159 approximately. The expression 2πa 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−α, when α becomes indefinitely small; this is an elementary case of differentiation.

The lengths of arcs of the same circle being proportional to the 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 angle θ (circular measure) cut their circumferences in A, B and C, respectively (fig. 5). Then the area of ABDC is

1/2b²θ−1/2a²θ＝(b−a)·1/2(b+a)θ.

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 1/2(b+a)θ; 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 for a right circular cylinder and a right circular cone; i.e.

volume of right circular cylinder＝length × area of base;

volume of right circular cone＝height × 1/3 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 developable 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 × perimeter of base. The curved surface of a right circular cone becomes a sector of a circle, and its area＝1/2·slant height × perimeter of base.

31. If a is the radius of a sphere, then

(i) volume of sphere＝4/3πa³;

(ii) surface of sphere＝4π²＝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 1/8λ, the volume of the sphere is λa³, and thence, by taking two concentric spheres (cf. § 28), the area of the surface is 3λa². This surface may be split up into elements, each of which is equal to a corresponding element of the curved surface of the circumscribing cylinder, so that 3λa²＝curved surface of cylinder＝2a. 2πa＝4πa². Hence λ＝4/3π.

The total surface of the cylinder is 4πa2+πa²+πa²＝6πa², and its volume is 2a.πa³＝2πa³. Hence

volume of sphere＝2/3 volume of circumscribing cylinder;

surface of sphere＝2/3 surface of circumscribing cylinder.

These latter formulae are due to Archimedes.

32. Moments and Centroids.—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 a plane 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 gravity of a body. (See Mechanics.)

The moment as described above is sometimes called the first moment. The second moment, third moment, . . . of a plane or solid figure are found in the same way by multiplying each element by the square, cube, . . . of its distance from the line or plane with regard 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 N₁, N₂, . . . are its moments with regard to a line in its plane, the moments M₁, M₂, . . . with regard to a parallel line through the centroid are given by

M₁＝N₁ − xN₀＝0,

M₂＝N₂ − 2xN₁+ x²N₀＝N₂ − x²N₀,

  :

  :

M_q＝N_q − qxN_q−1 +q(q − 1)/2!x²N_q−2 . . . + ( − )^q−1qx^q−1N₁ + ( − )^qxN₀;