Page:EB1911 - Volume 18.djvu/159

of elements of the graph from an axis through O at right angles to OX. Its calculation therefore involves the calculation of the area and the first moment of the graph.

40. The processes which have to be performed in the mensuration of figures of this kind are in effect processes of integration; the distinction between mensuration and integration lies in the different natures of the data. If, for instance, the graph were a trapezium, the calculation of the area would be equivalent to finding the integral, from x＝a to x＝b, of an expression of the form px+q. This would involve p and q; but, for our purposes, the data are the sides pa+q and pb+q and the base b−a, and the expression of the integral in terms of these data would require certain eliminations. The province of mensuration is to express the final result of such an elimination in terms of the data, without the necessity of going through the intermediate processes.

41. Trapezettes and Briquettes.—A figure of the kind described in § 39 is called a trapezette. A trapezette may therefore be defined as a plane figure bounded by two straight lines, a base at right angles to them, and a top which may be of any shape but is such that every ordinate from the base cuts it in one point and one point only; or, alternatively, it may be defined as the figure generated by an ordinate which moves in a plane so that its foot is always on a straight base to which the ordinate is at right angles, the length of the ordinate varying in any manner as it moves. The distance between the two straight sides, i.e. between the initial and the final position of the ordinate, is the breadth of the trapezette. Any line drawn from the base, at right angles to it, and terminated by the top of the trapezette, is an ordinate of the figure. The trapezium is a particular case.

Either or both of the bounding ordinates may be zero; the top, in that case, meets the base at that extremity. Any plane figure might be converted into an equivalent trapezette by an extension of the method of § 25 (iv).

42. The corresponding solid figure, in its most general form, is such as would be constructed to represent the relation of a magnitude E to two magnitudes F and G of which it is a function; it would stand on a plane base, and be comprised within a cylindrical boundary whose cross-section might be of any shape. We are not concerned with figures of this general kind, but only with cases in which the base is a rectangle. The figure is such as would be produced by removing a piece of a rectangular prism, and is called a briquette. A briquette may therefore be defined as a solid figure bounded by a pair of parallel planes, another pair of parallel planes at right angles to these, a base at right angles to these four planes (and therefore rectangular), and a top which is a surface of any form, but such that every ordinate from the base cuts it in one point and one point only. It may be regarded as generated either by a trapezette moving in a direction at right angles to itself and changing its top but keeping its breadth unaltered, or by an ordinate moving so that its foot has every possible position within a rectangular base.

43. Notation and Definitions.—The ordinate of the trapezette will be denoted by u, and the abscissa of this ordinate, i.e. the distance of its foot from a certain fixed point or origin O on the base (or the base produced), will be denoted by x, so that u is some function of x. The sides of the trapezette are the “bounding ordinates”; their abscissae being x₀ and x₀+H, where H is the breadth of the trapezette.

The “mid-ordinate” is the ordinate from the middle point of the base, i.e. the ordinate whose abscissa is x₀+1/2H.

The “mean ordinate” or average ordinate is an ordinate of length l such that Hl is equal to the area of the trapezette. It therefore appears as a calculated length rather than as a definite line in the figure; except that, if there is only one ordinate of this length, a line drawn through its extremity is so placed that the area of the trapezette lying above it is equal to a corresponding area below it and outside the trapezette. Formulae giving the area of a trapezette should in general also be expressed so as to state the value of the mean ordinate (§§ 12 (v), 15, 19).

The “median ordinate” is the ordinate which divides the area of the trapezette into two equal portions. It arises mainly in statistics, when the ordinate of the trapezette represents the relative frequency of occurrence of the magnitude represented by the abscissa x; the magnitude of the abscissa corresponding to the median ordinate is then the “median value of x.”

The “central ordinate” is the ordinate through the centroid of the trapezette (§ 32). The distance of this ordinate from the axis of u (i.e. from a line drawn through O parallel to the ordinates) is equal to the mean distance (§ 32) of the trapezette from this axis; moments with regard to the central ordinate are therefore sometimes described in statistics as “moments about the mean.”

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 x_θ, the corresponding value of u being u_θ. The data are then either the bounding ordinates u₀, u₁, . . . u_m−1, u_m, of the strips, or their mid-ordinates u1/2, u3/2, . . . u_m−1/2.

44. In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette. If the lengths of these sides are H and K, the coordinates of the angles of the base—i.e. the co-ordinates of the edges of the briquette—are (x₀, y₀), (x₀+H, y₀), (x₀, y₀+K), and (x₀+H, y₀+K).

The briquette may usually be regarded as divided into a series of minor briquettes by two sets of parallel planes, the planes of each set being at successively equal distances. If the planes of one set divide it into m slabs of thickness h, and those of the other into n slabs of thickness k, so that H＝mh, K＝nk, then the values of x and of y for any ordinate may be denoted by x₀+θh and y₀+φk, and the length of the ordinate by uθ, φ.

The data are usually the breadths H and K and either (i) the edges of the minor briquettes, viz. u_0,0, u_0,1, . . . u_1,0, u_1,1, . . . or (ii), the mid-ordinates of one set of parallel faces, viz. u_0,1/2, u_0,3/2, . . ., u_1,1/2, . . . or u_1/2,0, u_3/2,0, . . . u_1/2,1, . . ., or (iii) the “mid-ordinates” u_1,1/2, u_1/2,3/2 . . . u_3/2,1/2 . . . of the minor briquettes, i.e. the ordinates from the centres of their bases.

A plane parallel to either pair of sides of the briquette is a “principal plane.” The ordinate through the centroid of the figure is the “central ordinate.”

45. In some cases the data for a trapezette or a briquette are not only certain ordinates within or on the boundary of the figure, but also others forming the continuation of the series outside the figure. For a trapezette, or instance, they may be . . . u₋₂, u₋₁, u₀, u₁ . . . u_m, u_m+1, u_m+2 . . ., where u_θ denotes the same function of x≡x₀+θh, whether θh lies between the limits 0 and H or not. These cases are important as enabling simpler formulae, involving central differences, to be used (§ 76).

46. The area of the trapezette, measured from the lower bounding ordinate up to the ordinate corresponding to any value of x, is some function of x. In the notation of the integral calculus, this area is equal to $\int _{x_{0}}^{x}udx$; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area. It is therefore better to use some independent notation, such as A_x . u. It will be found convenient to denote φ(b)−φ(a), where φ(x) is any function of x, by [φ(x)]x＝b
x＝a; the area of the trapezette whose bounding ordinates are u₀ and u_m may then be denoted by [A_x . u]x＝x_m
x＝x₀ or [A_x . u]θ＝m
θ＝0, instead of by $\int _{x_{0}}^{x_{m}}udx$.

In the same way the volume of a briquette between the planes x＝x₀, y＝y₀, x＝a, y＝b may be denoted by

[[V_{x, y} . u] y＝b 
y＝y₀] x＝a 
x＝x₀.

47. The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u＝f(x), must be distinguished from u ＝f(x) as the equation to the top of the trapezette.

In elementary geometry we deal with lines and curves, while in mensuration we deal with areas bounded by these lines or curves. The circle, for instance, is regarded geometrically as a line described in a particular way, while from the point of view of mensuration it is a figure of a particular shape. Similarly, analytical plane geometry deals with the curve described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.

In the same way, in the case of a figure in three dimensions, analytical geometry is concerned with the form of the surface, while analytical mensuration is concerned with the figure as a whole.

48. 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.

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 this is done for every possible value of x, there will be a series of ordinates tracing out a trapezette with base, along OX. The volume comprised between the cross-section whose area is S and a consecutive cross-section at distance θ from it is ultimately Sθ, when θ is indefinitely small; and the area between the corresponding ordinates of the trapezette is (S/l) . θ＝Sθ/l. Hence the volume of each element of the solid figure is to be found by multiplying the area of the corresponding element of the trapezette by l, and therefore the total volume is l × area of trapezette.