William Fleetwood Sheppard

“ If the area of the cross-section, in every position, is known in terms of its distance from one of the bounding planes, or from a fixed plane. A parallel to them, the volume of the solid can be expressed in terms of the area of a trapezette. Let S be the area of the cross-section at distance x from the plane A. On a straight line OX in any plane take a point N at distance x from O, and draw an ordinate NP at right angles to OX and equal to S/l, where l is some fixed length (e.g. the unit of measurement) . ”

“ If any line in a plane revolves about an external axis in the plane, the area of the curved surface generated by the revolution is equal to the product of the length of the line and the distance travelled by the centroid of the line. ”

“ Representation of Volume by Area.—An important plane graph is that which represents the volume of a solid figure.
Suppose that we take a pair of parallel planes, such that the solid extends from one to the other of these planes. The section by any intermediate parallel plane will be called a “cross-section.” The solid may then be regarded as generated by the cross-section moving parallel to itself and changing its shape, or its position with regard to a fixed axis to which it is always perpendicular, as it moves. ”

“ This use of formulae for dealing with numbers, which express magnitudes in terms of units, constitutes the broad difference between mensuration and ordinary geometry, which knows nothing of units. Mensuration involves the use of geometrical theorems, but it is not concerned with problems of geometrical construction. The area of a rectangle, for instance, is found by calculation from the lengths of the sides, not by construction of a square of equal area. ”

“ Such a body of formulae cannot, of course, be regarded as constituting a science; it has no power of development from within, and can only grow by accretion. It may be of extreme importance for practical purposes; but its educational value, if it is studied apart from the methods by which the formulae are obtained, is slight. Vitality can only be retained by close association with more abstract branches of mathematics.
10. On the other hand, mensuration, in its practical aspect, is of importance for giving reality to the formulae themselves and to the principles on which they are based. ”

“ It must, however, be remembered that in all practical applications of formulae the data have first to be ascertained by direct or indirect measurement; and this measurement involves a certain margin of error.
The two sources of error mentioned under (a) and (b) above are closely related. Suppose, for instance, that we require the area of a circular grass-plot of measured diameter. As a matter of fact, no grass-plot is truly circular; and it might be found that if the breadth in various directions were measured more accurately the want of circularity would reveal itself. ”

“ This is the simplest case of generation of a plane figure by a moving ordinate; the corresponding figure for generation by rotation of a radius vector is a circle.
To regard a figure as being generated in a particular way is essentially the same as to regard it as being made up of a number of successive elements, so that the analytical treatment involves the ideas and the methods of the infinitesimal calculus. It is not, however, necessary that the notation of the calculus should be employed throughout. ”

“ The second group comprises the mechanic arts, in which the bodies to be measured are solid bodies which can be handled; in these cases plane figures appear mainly as sections of a solid. In developing a system of mensuration-formulae the importance of this latter group of cases must not be overlooked. A third group, of increasing importance, comprises cases in which curves or surfaces arise out of the application of graphic methods in engineering, physics and statistics. The general formulae applicable to these cases are largely approximative. ”

“ While, for instance, it may be difficult to realize the equality of area of two plots of ground of different shapes, it may be easy to realize the equality of the amounts of a given material that would be required to cover them to a particular depth. ”

“ Mensuration, then, is mainly concerned with quadrature-formulae and cubature-formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume. ”

“ The use of the square bricks familiarizes the scholar with the ideas of parallel lines, of equality of lengths, and of right angles. The conception of the right angle is strengthened, by contrast, by the use of bricks in the form of a rhombus. ”

“ 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. ”

“ The data of a trapezette are usually its breadth and either the bounding ordinates or the mid-ordinates of a series of minor trapezettes or strips into which it is divided by ordinates at equal distances. If there are m of these strips, and if the breadth of each is h, so that H=mh, it is convenient to write x in the form x0+θh, and to denote it by xθ, the corresponding value of u being uθ. The data are then either the bounding ordinates u0, u1, . ”

“ 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. ”