Bible Encyclopedias
Mensuration

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 0 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 xo and xo+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 xo+ z H.

The " mean ordinate " or average ordinate is an ordinate of length 1 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 0 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 xo+Oh, and to denote it by x 0, the corresponding value of u being ue. The data are then either the bounding ordinates uo, ui, ... u m _ i, u m of the strips, or their mid-ordinates 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 (xo, y o), (xo+H, y o), (x o, yo+K), and (xo-1-H, yo+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 xo+Oh and yo+Ok, and the length of the ordinate by uo, 0.

The data are usually the breadths H and K and either (i) the edges of the minor briquettes, viz. uo,o, uo,l,. u1,o, u1,i, ... or (ii) the mid-ordinates of one set of parallel faces, viz. uo ,i , uo ,i,

ui 3,. .. or ui,o, a o,

14,1, . ., or (iii) the " mid-ordinates " u1,2, u, ... u I, ... 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, for instance, they may be ... u_ 2 ,u_ l , uo, u 1

U m, u m+1 14m+2. ., where ue denotes the same function of x =xo+Bh, whether Oh lies between the limits o 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 f x o 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 Z. u. It will be found convenient to denote ? (b) - 49(a), where 4)(x) is any function of x, by [ c P(x)]; the area of the trapezette whose bounding ordinates are uo and u m may then be denoted by [Ax. u] x - xo or [A.. u 8= u , instead of by fxo udx. In the same way the volume of a briquette between the planes x = xo, y = yo, x= a, y = b may be denoted by [[Vx,y ]y=yo] u 'x' =xo.

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 crosssection at distance x from the plane A. On a straight line OX in any plane take a point N at distance x from 0, and draw an ordinate NP at right angles to OX and equal to S/1, where 1 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 0 from it is ultimately SO, when B is indefinitely small; and the area between the corresponding ordinates of the trapezette is (S/1). B =S611. 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 1, and therefore the total volume is 1 X area of trapezette.

The volume of a briquette can be found in this way if the area of the section by any principal plane can be expressed in terms of the distance of this plane from a fixed plane of the same set. The result of treating this area as if it were the ordinate of a trapezette leads to special formulae, when the data are of the kind mentioned in § 44.

49. (B) Mensuration of Graphs of Algebraical Functions. - The first class of cases to be considered comprises those cases in which u is an algebraical function (i.e. a rational integral algebraical function) of x, or of x and y, of a degree which is known.

50. The simplest case is that in which u is constant or is a linear function of x, i.e. is of the form px + q. The trapezette is then a right trapezium, and its area, if m=l, is 2h(uo + ui) or hug.

51. The next case is that in which u is a quadratic function of x, i.e. is of the form px 2 + qx -{- r. The top is then a parabola whose axis is at right angles to the base; and the area can therefore (§ 34) be expressed in terms of the two bounding ordinates and the midordinate. If we take these to be uo and u 2, and u 1 , so that m = 2, we have area = 6H(uo + 4u1 + u2) = 'h(uo + 4 u 1 + 142). This is Simpson's formula. If instead of uo, u 1, and u 2, we have four ordinates uo, ul, u2, and u 3, so that m = 3, it can be shown that area = 8h(uo + 3/41 + 3u2 - Fu3).

This is Simpson's second formula. It may be deduced from the formula given above. Denoting the areas of the three strips by A, B, and C, and introducing the middle ordinate ug, we can express A + B; B -{- C; A + B -FC; and B in terms of uo, u 1, u 2; u 1, u2, u3; uo, u, u 3; and u 1, ug, u 2 respectively. Thus we get two expressions for A + B -}- C, from which we can eliminate ug.

A trapezette of this kind will be called a parabolic trapezette.

52. Simpson's two formulae also apply if u is of the form px 3 - }- 5x 2 + rx -}- s. Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1. This will be seen by taking the mid-ordinate as the ordinate for which x = o, and noticing that the odd powers of x introduce positive and negative terms which balance one another when the whole area is taken into account.

53. When u is of degree 4 or 5 in x, we require at least five ordinates. If m = 4, and the data are ul, u2, Us, U4, we have area = h (7 u o + 3 2u 2 -112/42 + 3 2u 3 + 7u4).

For functions of higher degrees in x the formulae become more complicated.

54. The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences. The breadth of the trapezette being mh, it may be shown that its area is 2 2 N 4 4 iv I mh ug m + 24 m h u gm + 1920 m h u g m + 322560 no/tourgm -}- m3l1gu i„? +. .

9289-7280 gm where u gm, u g m, u g m ,.. . denote the values for x = xim ' of the successive differential coefficients of u with regard to x; the series continuing until the differential coefficients vanish. There are two classes of cases, according as m is even or odd; it will be convenient to consider them first for those cases in which the data are the bounding ordinates of the strips.

(i) If m is even, ug m will be onejof the given ordinates, and we can express h 2 u, m, 4 u" m, ... in terms of ug m and its even central differences (see Differences, Calculus Of). Writing m = 2p, and grouping the coefficients of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If u is of degree 2f or 2f + i in x, we require to go up to b 2f u p, so that m must be not less than 2f. Simpson's (first) formula, for instance, holds for f = I, and is obtained by taking p = i and ignoring differences after 52up.

(ii) If m is odd, the given ordinates are uo, ... u m. We then have area =) + 24 3 + m4 50m2 + ?3 5 ?S4ugm 967680 ?S6ua „,,d -.. ., where l aug m, µ6 2 ug m,. .. denote 2 (ug 'm _ z + ug m+ g), ? (b2 u g m - s + S2u lm+ g),. 'Simpson's second formula is obtained by taking m = 3 and ignoring differences after /,u52u1m.

55. The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, where u is of degree 2f or 2f + I in x. Thus we find from (i) that Simpson's second formula, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us).

This might have been deduced directly from Simpson's first formula, by a series of eliminations.

56. Hence, for the case of a parabola, we can express the area in terms of the bounding ordinates of two strips, but, if we use mid-ordinates, we require three strips; so that, in each case, three ordinates are required. The question then arises whether, by removing the limitation as to the position of the ordinates, we can reduce their number.

Suppose that in fig. 6 (§ 34) we draw ordinates QD midway between KA and MC, and RE midway between MC and LB, meeting the top in D and E (fig. 8), and join DE, meeting KA, LB, and MC in H, J, and W. Then it may be shown that DE is parallel to AB, and that the area of the figure between chord DE and arc DE is half the sum of the areas DHA and EJB. Hence the area of the right trapezium Khjl is greater than the area of the trapezette Kacbl.

If we were to take QD and RE closer to MC, the former area would be still greater. If, on the other hand, we were to take them very close to KA and LB respectively, the area of K 0 M B Z the trapezette would be the greater. There FIG. 8. is therefore some intermediate position such that the two areas are equal; i.e. such that the area of the trapezette is represented by KL. 1-(QD + RE).

To find this position, let us write QM = MR = 8. KM. Then WC = 0 2 . VC, VW = (i - 0 2 ) VC; curved area ACB = a of parallelogram Afgb = 3kl. Vc; parallelogram Ahjb = Kl. Vw = (i - 0 2) KL. VC.

Hence the areas of the trapezette and of the trapezium will be equal if I - 2=3,0=I/%/3.

This value of 0 is the same for all parabolas which pass through D and E and have their axes at right angles to KL. It follows that, by taking two ordinates in a certain position with regard to the bounding ordinates, the area of any parabolic trapezette whose top passes through their extremities can be expressed in terms of these ordinates and of the breadth of the trapezette.

The same formula will also hold (§ 52) for any cubic trapezette through the points.

57. This is a particular case of a general theorem, due to Gauss, that, if u is an algebraical function of x of degree 2p or 2p + I, the area can be expressed in terms of p -}- i ordinates taken in suitable positions.

58. The Prismoidal Formula

It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid. If the areas of the two ends in the planes K and L are So and S2, and the area of the mid-section (i.e. the section by a plane parallel to these planes and midway between them) is S i, the volume is *H(So + 4 S 1 S 2), where H is the total breadth.

This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid. In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefore s. 2R. - 2 To show that the area of a cross-section of a - prismoid is of the form ax e -{- bx -{- c, where x is the distance of the section from one end, we may proceed as in § 27. In the case of a pyramid, of height h, the area of the section by a plane parallel to the base and at distance x from the vertex is clearly x 2 /h 2 X area of base. In the case of a wedge with parallel ends the ratio x2/h2 is replaced by x/h. For a tetrahedron, two of whose opposite edges are AB and CD, we require the area of the section by a plane parallel to AB and CD. Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC, up -f- .

3m6

147m 4 181 3 m2 - 4725 G F H A B FIG. 9.

BC, BD, AD, in P, Q, R, S (fig. 9). Then the section of the pyramid by this plane is the parallelogram Pqrs. By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x. The proposition can then be established for a prismoid generally by the method of § 27 (iv). The formula is known as the prismoidal formula. 59. Moments. - Since all points on any ordinate are at an equal distance from the axis of u, it is easily shown that the first moment (with regard to this axis) of a trapezette whose ordinate is u is equal to the area of a trapezette whose ordinate is xu; and this area can be found by the methods of the preceding sections in cases where u is an algebraical function of x. The formulae can then be applied to finding the moments of certain volumes.

In the case of the parabolic trapezette, for instance, xu is of degree 3 in x, and therefore the first moment is lh(xouo+4xlui-+x2u2). In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other. The centroid of a hemisphere of radius R, for instance, is the same as the centroid of particles of masses 0, 7rR 2, and 4. lirR 2, placed at the extremities and the middle of its axis; i.e. the centroid is at distance 8R from the plane face.

60. The method can be extended to finding the second, third,. .. moments of a trapezette with regard to the axis of u. If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+.. +Amvm, the qth moment of the trapezette is Aoxo 4 uo+Xix1 4 ui+ ... -1-Amxm'um, and the mean value of x° is (AoxoQuo + Alx1 4 u1 + ... + Amxm°um)/(Aouo + A1u1+ ... An,um). The calculation of this last expression is simplified by noticing that we are only concerned with the mutual ratios of Xo, X 1, ... and of uo, u1,.. , not with their actual values.

61. Cubature of a Briquette

To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette. This ordinate will be an algebraical function of x, and we can again apply a suitable formula. Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x=xo+h and the planes y = yo+k, y = yo+2k, and that the 12 ordinates forming the edges of these six elements are given. The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al. The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u. Hence Simpson's formula applies, and the volume is lh(A0-I-4A1+ A2).

The process is simplified by writing down the general formula first and then substituting the values of u. The formula, in the above case, is 3h{ *k(uo,o + 3 where u 0 ,0 denotes the ordinate for which x=xo+Oh, y=yo+c¢k The result is the same as if we multiplied lk(vo 3v1+3v2 + v 3) by lh(uo 4u1 +u2), and then replaced uovo, uov1, .. by uo,o, uo,i .. The multiplication is shown in the adjoining diagram; the factors s ands are kept outside, so that the sum uo,o+3uo,1+ ... +4u1,o+..

can be calculated before it is multiplied by 3h . 1k.

62. The above is a particular case of a general principle that the obtaining of an expression such as Ih(uo+4u1+u2) or lk(vo-1-3v 1 +3v 2 +v 3) is an operation performed on uo or vo, and that this operation is the suns of a number of operations such as that which obtains 3huo or 1kvo. The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order. Starting from any ordinate ue,o, the result of integrating with regard to x through a distance 2h is (in the example considered in § 61) the same as the result of the operation 3h(I + 4E + E 2), where E r denotes the operation of changing x into x+h (see Differences, Calculus oF). The integration with regard to y may similarly (in the particular example) be replaced by the operation ak(I+3E'-+3E'2+E'3), where E' denotes the change of y into y + k. The result of performipg both operations, in order to obtain the volume, is the result of the operation denoted by the product of these two expressions; and in this product the powers of E and of E' may be dealt with according to algebraical laws.

The methods of §§ 59 and 60 can similarly be extended to finding the position of the central ordinate of a briquette, or the mean q th of elements of the briquette from a principal plane.

63. (C) Mensuration of Graphs Generally. - We have next to consider the extension of the preceding methods to cases in which u is not necessarily an algebraical function of x or of x and y. The general principle is that the numerical data from which a particular result is to be deduced are in general not exact, but are given only to a certain degree of accuracy. This limits the accuracy of the result; and we can therefore replace the figure by another figure which coincides with it approximately, provided that the further inaccuracy so introduced is comparable with the original inaccuracies of measurement.

The relation between the inaccuracy of the data and the additional inaccuracy due to substitution of another figure is similar to the relation between the inaccuracies in mensuration of a figure which is supposed to be of a given form (§ 20). The volume of a frustum of a cone, for instance, can be expressed in terms of certain magnitudes by a certain formula; but not only will there be some error in the measurement of these magnitudes, but there is not any material figure which is an exact cone. The formula may, however, be used if the deviation from conical form is relatively less than the errors of measurement. The conditions are thus similar to those which arise in interpolation (q.v.). The data are the same in both cases. In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates. If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.

64. There are three classes of cases to be considered. In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact. In physical and mechanical applications, where concrete measurements are involved, there is, as pointed out in the preceding section, the additional inaccuracy due to want of exactness in the figure itself. In the case of statistical data there is the further difficulty that there is no real continuity, since we are concerned with a finite number of individuals.

The proper treatment of the deviations from mathematical accuracy, in the second and third of the above classes of cases, is a special matter. In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of u but also of some of its differential coefficients) are satisfied, subject to the small errors in the values of u actually given; the limits of these errors being known.

65. It is only necessary to consider the trapezette and the briquette, since the cases which occur in practice can be reduced to one or other of these forms. In each case the data are the values of certain equidistant ordinates, as described in §§ 43-45. The terms quadratureformula and cubature-formula are sometimes restricted to formulae for expressing the area of a trapezette, or the volume of a briquette, in terms of such data. Thus a quadrature-formula is a formula for expressing [A x .24] or fudx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[Vx, 0 . u]] or ffudxdy in terms of the values of u for certain values of x in combination with certain values of y; these values not necessarily lying within the limits of the integrations.

66. There are two principal methods. The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an algebraical function of x or of x and y, and expressing the area or volume of this latter figure (exactly or approximately) in terms of the given ordinates. The second consists in taking a comparatively simple expression obtained in this way, and introducing corrections which involve the values of ordinates at or near the boundaries of the figure. The various methods will be considered first for the trapezette, the extensions to the briquette being only treated briefly.

67. The Trapezoidal Rule

The simplest method is to replace the trapezette by a series of trapezia. If the data are uo, u 1,.

um, the figure formed by joining the tops of these ordinates is a trapezoid whose area is h(Iuo -}- ui+u2 -I-

+ um-1 + Ium>. This is called the trapezoidal or chordal area, and will be denoted by C1. If the data are u;, U I, ... u m _ 4 , we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u 4 .+u 2 +... +um_4). This is called the tangential area, and will be denoted by T1. The - - 4X + g tangential area may be expressed in terms of chordal areas. If we write CI for the chordal area obtained by taking ordinates at intervals Zh, then T i =2CI-C I. If the trapezette, as seen from above, is everywhere convex or everywhere concave, the true area lies between C 1 and T1.

68. Other Rules for Trapezettes. - The extension of this method consists in dividing the trapezette into minor trapezettes, each consisting of two or more strips, and replacing each of these minor trapezettes by a new figure, whose ordinate v is an algebraical function of x; this function being c h osen so that the new figure shall coincide with the original figure so far as the given ordinates are concerned. This means that, if the minor trapezette consists of k strips, v will be of degree k or k - I in x, according as the data are the bounding ordinates or the mid-ordinates. If A denotes the true area of the original trapezette, and B the aggregate area of the substituted figures, we have A B, where 41-denotes approximate equality. The value of B is found by the methods of §§ 49-55. The following are some examples.

(i) Suppose that the bounding ordinates are given, and that m is a multiple of 2. Then we can take the strips in pairs, and treat each pair as a parabolic trapezette. Applying Simpson's formula to each of these, we have A -__9= *h(uo + 4 14 1 + u2) + 3 h (14 2 + 4u3 + u4) + .. _f)_ 111040+ 4u1 + 2u2 + 4 u 3 + 2464 +. .. + 214m-2 + 4 u m-i + um)This is Simpson's rule. (ii) Similarly, if m is a multiple of 3, the repeated application of Simpson's second formula gives Simpson's second rule A 1? -gh(uo+3ui+3u2+ 2 143+3 14 4+... +314m_4+2um-3+ 314m_2 + 314m_1 + um).

(iii) If mid-ordinates are given, and m is a multiple of 3, the repeated application of the formula of § 55 will give A -??- ah(3u + 2u 3 + 3u; + 3u; +.. + 2um-? + Su m _). 69. The formulae become complicated when the number of strips in each of the minor trapezettes is large. The method is then modified by replacing B by an expression which gives the areas of the substituted figures approximately. This introduces a further inaccuracy; but this latter may be negligible in comparison with the main inaccuracies already involved (cf. § 20 (iii)).

Suppose, for instance, that m=6, and that we consider the trapezette as a whole; the data being the bounding ordinates. Since there are seven of these, v will be of degree 6 in x; and we shall have (§ 54 (i)) B =6h(v 3 + 6 2 v3 + 20 14v 3 + i 4 1 1 b 6v 3) = 6h(u3+2.6 2 u3 + 2 OS 4 u3 +? O16u3)If we replace 440136u3 in this expression by g405 6 u 3, the method of § 68 gives A -Q AIL h (uo + 5 u 1 + u2 + 6u3 + u4 + 5 14 6 + us); the expression on the right-hand side being an approximate expression for B, and differing from it only by s1eH5 6 u 3. This is Weddle's rule. If m is a multiple of 6, we can obtain an expression for A by applying the rule to each group of six strips.

70. Some of the formulae obtained by the above methods can be expressed more simply in terms of chordal or tangential areas taken in various ways. Consider, for example, Simpson's rule (§ 68 (i)). The expression for A can be written in the form h(2 uo + ul+ u2 + u3 +

+ um-2+ um -1 + 2 u m) 3 h (2140+ u2 + u. 2um) Now, if p is any factor of m, there is a series of equidistant ordinates uo, up, 142p, um - p, um; and the chordal area as determined by these ordinates is ph (2uo + up + u2p +. ... + u m-p + zum), which may be denoted by Cp. With this notation, the area as given by Simpson's rule may be written in the form sC l - 3 C2 or CI+ 1 3 `-(C1C2). The following are some examples of formulae of this kind, in terms of chordal areas.

(i) m a multiple of 2 (Simpson's rule).

A -ns (4 C i - C2 C1 + 1(C1 - C2).

(ii) m a multiple of 3 (Simpson's second rule).

A

' 1(9 C 1 - C 3) -` i - C1 + -s(C 1 - C3).

(iii) m a multiple of 4.

A -n- (64C1 - 20C2+C4) ' R C 1+9 (C 1 - C2) ?1(C l - C4).

(iv) m a multiple of 6 (Weddle's rule, or its repeated application).

A ?- *(1 5 C 16C 2+ C 3) C1 +$(C i - C2) -lO(C 2 - C3)(v) m a multiple of 12.

A 41311 (56C 1 - 28C 2 + 8C 3 - C4) C1+2(Ci - C2) - (C2 - C3) + s i s(C3 - C4).

There are similar formulae in terms of the tangential areas T1, Thus (iii) of § 68 may be written A -a -1-(9T 1 - T3).

71. The general method of constructing the formulae of § 7 0 for chordal areas is that, if p, q, r, ... are k of the factors (including 1) of m, we take A -IPCp+ Q C 4+ RC ,. +.. ., where P, Q, R, ... satisfy the k equations P+Q+R+...

Pp2 + Q q2 + Rr 2 + ... Pp4-I-Qg4+Rr4+....

- + Q g2k-2 + 2k The last k - I of these equations give I /P: I /Q 1/R. .. = p2 (p2 - q2) (p 2 - r2) q2 (q2 - p2) (q2 r 2). ... 7.2(y2 - p 2) (r 2 - q 2). ... .

Combining this with the first equation, we obtain the values of P, Q, R, .. .

The same method applies for tangential areas, by taking A -2 PTp -}- QT 4 + RT r + .. .

provided that p, q, r, ... are odd numbers.

72. The justification of the above methods lies in certain properties of the series of successive differences of u. The fundamental assumption is that each group of strips of the trapezette may be replaced by a figure for which differences of u, above those of a certain order, vanish (§ 54). The legitimacy of this assumption, and of the further assumption which enables the area of the new figure to be expressed by an approximate formula instead of by an exact formula, must be verified in every case by reference to the actual differences.

73. Correction by means of Extreme Ordinates

The preceding methods, though apparently simple, are open to various objections in practice, such as the following: (i) The assignment of different coefficients of different ordinates, and even the selection of ordinates for the purpose of finding C3, &c. (§ 70), is troublesome. (ii) This assignment of different coefficients means that different weights are given to different ordinates; and the relative weights may not agree with the relative accuracies of measurement. (iii) Different formulae have to be adopted for different values of m; the method is therefore unsuitable for the construction of a table giving successive values of the area up to successive ordinates. (iv) In order to find what formula may be applied, it is necessary to take the successive differences of u; and it is then just as easy, in most cases, to use a formula which directly involves these differences and therefore shows the degree of accuracy of the approximation.

The alternative method, therefore, consists in taking a simple formula, such as the trapezoidal rule, and correcting it to suit the mutual relations of the differences.

74. To illustrate the method, suppose that we use the chordal area C1, and that the trapezette is in fact parabolic. The difference between C 1 and the true area is made up of a series of areas bounded by chords and arcs; this difference becoming less as we subdivide the figure into a greater number of strips.

The fact that C 1 does not give the true area is due to the fact that in passing from one extremity of the top of any strip to the other extremity the tangent to the trapezette E, _- changes its direction. We have therefore B in the first place to see whether the difference can be expressed in terms of the directions of the tangents.

Let Kabl (fig. to) be one of the strips, of breadth h. Draw the tangents at A and B, meeting at T; and through T draw a line parallel to KA and LB, meeting the arc AB in C and the chord AB in V. Draw AD and BE perpendicular to this line, and DF and TG perpendicular to LB. Then AD =EB = 2h, and the triangles AVD and BVE are equal.

The area of the trapezette is less (in fig. to) than the area of the trapezium Kabl by two-thirds of the area of the triangle ATB (§ 34). This latter area is Obte -Datd = Abtg-Datd = 8h2 tan GTB - $h2 tan DAT. Hence, if the angle which the tangent at the extremity of the ordinate u 0 makes with the axis of x is denoted by fie, we have area from uo to u1= 2h(uo + ui) - -- i i h 2 (tan y l - tan t u 2 = Wu ' + u2) - 1 Tih 2 (tan 4,2 - tan um-1 t0 26 m, - 2 h(um-1 + um) i h (tan 4, m - and thence, by summation, A =C I - i i h 2 (tan - tan 1,1/o).

This, in the notation of §§ 46 and 54, may be written ? Since h = H/m, the inaccuracy in taking C I as the area varies as I /m2.

It might be shown in the same way that A=Ti+ ih 2 (tan Yamtan 4) = T1 + [h2u]::.

75. The above formulae apply only to a parabolic trapezette. Their generalization is given by the Euler-Maclaurin formula = I, = 0, = 0, = 0 .

M FIG. IO.

J xo u dx = C I + [_h 2 u' + th h4u'" - s? Ohsuv x= J x m. x=xo, and an analogous formula (which may be obtained by substituting 2h and C, for h and C 1 in the above and then expressing T 1 as 2C -CO) A =J xo udx=T1+ [4h2u'_+ 15487$5 h8uvii ? . = x0 To apply these, the differential coefficients have to be expressed in terms of differences.

76 If we know not only the ordinates uo, ul,. .. or u I, U I,

, but also a sufficient number of the ordinates obtained by continuing the series outside the trapezette, at both extremities, we can use central-difference formulae, which are by far the most convenient. The formulae of § 75 give 11?

A=C I + h [-121? 6u + 7 1 20 / 1 ' 63u - O $Oµ65u+3.522489L-011671/ - ... xxm MU - RV - 0 7u g xxo A=T 1 + h [4 8u - rh 13u + g07 80 65u 4-`644K640oa7u+... ]m. x xo 77. If we do not know values of u outside the figure, we must use advancing or receding differences. The formulae usually employed are A = C1+h) d 3 2 0u0- 02u0+.1.186.z3u0-160p4uo+.. .

(((+i,'um-84a'2um+W125A'3um ifiO O ' 4u m +

. A =T1+h - 84 0u 1+2 1 4 A2u i - .3.272831743211z8°8364uI - .. .

- 24 m -2 l 80y4um-3 80. Moments of a Trapezette. - The above methods can be applied, as in §§ 59 and 60, to finding the moments of a trapezette, when the data are a series of ordinates. To find the pth moment, when uo, u l, u 2, ... are given, we have only to find the area of a trapezette whose ordinates are xo P uo, x 1 'u 1, x2 P u2,

81. There is, however, a certain set of cases, occurring in statistics, in which the data are not a series of ordinates, but the areas A I, A I,. .. A m _ I of the strips bounded by the consecutive ordinates uo, 72 1, ... um. The determination of the moments in these cases involves special methods, which are considered in the next two sections.

82. The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients u', u", u"',. are practically negligible for = and for x =x m . The method adopted in these cases is to treat the areas A I, A I,. .. as if they were ordinates placed at the points for which i x=x l, ... , to calculate the moments on this assumption, and then to apply certain corrections. If the first, second,. .. moments, so calculated, before correction are denoted by p 1, P21

, we have pl =xIA'I+xIA3?-...

P2 = x2 I A I + x 2 I A I +. .. + -IAm p p - x 2 1AI +x P IAg -F. .. -[- x P „,IA r _ These are called the raw moments. Then, if the true moments are denoted by Pi, v 2, ..., their values are given by vi ?Pl P2 - 11. p2 - ] 5h2po v3?P3-1h2pi, + .5hh4po v5Pp5 - h2p3+48h4p1 where po (or po) is the total area A; -}- Ag -}- ... -FA r _i; the general expression being pf I !?12! (p -2) ! h2 PP -2 + X 4! ( p 4) h 4PP-4 - where X1 - 12, X2= 240, X 3 = Tit/, X 4 - S840, X 5 - $37 g 2,

The establishment of these formulae involves the use of the integral calculus.

The position of the central ordinate is given by x = v 1 /po, and therefore is given approximately by pl/po. To find the moments with regard to the central ordinate, we must use this approximate value, and transform by means of the formulae given in § 32. This can be done either before or after the above corrections are made. If the transformation is made first, and if the resulting raw moments with regard to the (approximate) central ordinate are o, 72, 71-3, ..., the true moments µ1, u2, /13, ...with regard to the central ordinate are given by Lo=o 1 i / h2,3 83. These results may be extended to the calculation of an expression of the form fxo u4(x)dx, where 0(x) is a definite function of x, and the conditions with regard to u are the same as in § 82.

(i) If cp(x) is an explicit function of x, we have (o ' n ud(x)dx?A3 I P(x 1)+ Ap P(x 3)+ ... Am-I?(xm- where (x)=Ih(x) 4h 2 0" (x) -i-ih 4 4, iv (x) - ..., the coefficients X i, X2 i .... having the values given in § 82.

(ii) If 4)(x) is not given explicitly, but is tabulated for the values.. x, x,. .. of x, the formula of (i) applies, provided we take (x)=(I - 24 62 +04?1 s4 -1-h-06 +

) Y' (x).

The formulae can be adapted to the case in which cp(x) is tabulated for xl,. .. 84. In cases other than those described in § 82, the pth moment with regard to the axis of u is given by Pp = XPrA where A is the total area of the original trapezette, and S 2 _ 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are o, x1P -1A, x2 P-1 (AI+AI),. xn= i(AI -?' AI {- ... + A m -), xm lA. The value of S 13 _ 1 has to be found by a quadrature-formula. The generalized formula is fxo u¢ (x)dx = Aq(xm) - where 0, A 2, ... have the usual meaning (Duo = u l - uo, L 2 uo = Du 1 - Auo, . .), and O', O' 2, ... denote differences read backwards, so that 0'um = um-1 -um, A' 2 um = um-2 -!2um_1+ um,

The calculation of the expressions in brackets may be simplified by taking the pairs in terms from the outside; by finding the successive differences of uo + um, ill + um_l, ..., or of uI u i +umi, ..

An alternative method, which is in some ways preferable, is to complete the table of differences by repeating the differences of the highest order that will be taken into account (see Interpolation), and then to use central-difference formulae.

78. In order to find the corrections in respect of the terms shown in square brackets in the formulae of § 75, certain ordinates other than those used for C 1 or T I are sometimes found specially. Parinentier's rule, for instance, assumes that in addition to u I; u I.. u m _, we know uo and um; and u I - uo and u m - u m _ i are taken to be equal to zhu'o and Zhu'„, respectively. These methods are not to be recommended except in special cases.

79. By replacing h in § 75 by 2h, 3h,. .. and eliminating h2u', Ott"',. .. , we obtain exact formulae corresponding to the approximate formulae of § 70. The following are the results (for the formulae involving chordal areas), given in terms of differential coefficients and of central differences. They are not so convenient as the formulae of § 76, but they serve to indicate the degree of accuracy of the approximate formulae. The expressions in square brackets are in each case to be taken as relating to the extreme values x =xo and x=xm, as in §§ 75 and 76.

(i) A=3(4C1-C2)+[- ii-uh4um+ iuiT h6uv -14406 h8uvii + ...

= (4C1- C2)+h[ ih/ 153u +T - 05 u 00 2()O/167u+... ].

A= ? C1-C3 + [-Z5h4u"' hsu .thanh8uvii = (9 C 1C 3)+ h [ 85µ 63u +6720 415u -4M 7 b0/i 17u '+.. .].

(iii) A= g(64C1-20C2+C4) - ... ] = 45(64C1-20C2+C4)+h[- $41;µ65utas - 0-07u- ... ].

A=io(15C1-6C2+C3)+[ - a4 h suv rx41ff-uhsuvii- ... 1 = 1 O(i 5 C 10 C2C 3) + h [- 84T1 455 u. .. ].

A= (56C1-28C2+8C3-C4) +[- 20Oh8uvii J..

= x 1 (56C1-28C2+8C3-C4)+h[- 210Oµ 67 u+ ... ].

The general expression, if p, q, r,.. . are le of the factors of m, is A = PC P + QC / + r + ... + [ (-) kb k h2k d x 2k l 1)k+lbk+1h2k+2 + J x = xo, where P, Q, R,. .. have the values given by the equations in § 71, and the coefficients b k, bk+1, . are found from the corresponding coefficients in the; uler-Maclaurin formula (§ 75) by multiplying them by Pp2k+(1g2k+Rr2k.. ., Pp2k+2+.Qg2k-1-2 Rr2k+2 ?.. .

i g h oo hsuvii where T is the area of a trapezette whose ordinates at successive distances h are o, Ali' (x i), (Al +A4) ,g 5' (x2), (Az +A 2 + ... - A„L_i)4'(x„1_i), Ag5'(xm); the accents denoting the first differential coefficient.

85. Volume and Moments of a Briquette

The application of the methods of §§ 75-79 to calculation of the volume of a briquette leads to complicated formulae. If the conditions are such that the methods of § 61 cannot be used, or are undesirable as giving too much weight to particular ordinates, it is best to proceed in the manner indicated at the end of § 48; i.e. to find the areas of one set of parallel sections, and treat these as the ordinates of a trapezette whose area will be the volume of the briquette.

86. The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x =x0, x = x i,

and y = yo, y = Yi, The method of constructing the formulae is explained in § 62. If we write -fxo f yox s yiu dx dy, we first calculate the raw values coo., ai,o, 0.1,1,

of So,1, Si,o, S1,1,. on the assumption that the volume of each minor briquette is concentrated along its mid-ordinate (§ 44), and we then obtain the formulae of correction by multiplying the formulae of § 82 in pairs. Thus we find (e.g.) -2x1,1 52,1 2 x2,1 - i h2a o, .

S 1,2 2a 1,2 - 1 1 k20.10 52,2 ?? x 2,2 - k 2 a2,o - h 2 a6,2 +ia4h2k2ao,0 5 2x - 1100-1,1 5 2x h2x k2x3,o+41-gh2k°0-1,o where ao,o is the total volume of the briquette.

87. If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have y "`x P u dx dy = K + L + R - X111010-0,0 f xo yo i'? y where K-=4, X qth moment with regard to plane y =o, Lm yn X pth moment with regard to plane x =o, and R is the volume of a briquette whose ordinate at (x,.,y s ) is found by multiplying by pq x r P - 1 ys 4-1 the volume of that portion of the original briquette which lies between the planes x =xo, y =yo, y = ys. The ordinates of this new briquette at the points of intersection of x =x 0, x = xi,. .. with y =yo, y =y 1,. .. are obtained from the data by summation and multiplication; and the ordinary methods then apply for calculation of its volume. Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both expressions, we have a formula which may be written in a more general form f f 4 u4(x, y) dx dy = u dx dy. q) a J O l x f o udxdy (1619(X q) dx 4 P u dx dy d 4)(b, y) dy dy +. f b f 4 f x f P u dx dy d x dy) dx dy. d The second and third expressions on the right-hand side represent areas of trapezettes, which can be calculated from the data; and the fourth expression represents the volume of a briquette, to be calculated in the same way as R above.

88. Cases of Failure

When the sequence of differences is not such as to enable any of the foregoing methods to be applied, it is sometimes possible to amplify the data by measurement of intermediate ordinates, and then apply a suitable method to the amplified series. There is, however, a certain class of cases in which no subdivision of intervals will produce a good result; viz. cases in which the top of the figure is, at one extremity (or one part of its boundary), at right angles to the base. The Euler-Maclaurin formula (§ 75) assumes that the bounding values of u', u"',.. . are not infinite; this condition is not satisfied in the cases here considered. It is also clearly impossible to express u as an algebraical function of x and y if some value of du/dx or duldy is to be infinite.

No completely satisfactory methods have been devised for dealing with these cases. One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied. Suppose, for instance, that we require the area of the trapezette ABL in fig. II; the curve being at right angles to the base AL at A. If QD is the bounding ordinate of one of the component strips, we can calculate the area of Qdbl in the ordinary way. The data for the area ADQ are a series of values of u corresponding to equidifferent values of x; if we denote by y the distance of a point on the arc AD from QD, we can from the series of values of u construct a series of values of y corresponding to equidifferent values of u, and thus find the area of ADQ, treating QD as the base. The process, however, is troublesome.

89. Examples of Applications

The following are some examples of cases in which the above methods may be applied to the calculation of areas and integrals.

(i) Construction of Mathematical Tables

Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae. The formula of § 76 may (see Differences, Calculus Of) be written f x udx = h .µxu + h(- 2 µ6u + 2 0 = (hu-- f 2 6 hu +.; 2 0 6 hu - udx = h. au h (1146 u - = a (hu + 5 6 2 hu - 6 +,-6 4 hu +. ..).

The second of these is usually the more convenient. Thus, to construct a table of values of f x udx by intervals of h in x, we first form a table of values of hu for the intermediate values of x, from this obtain a table of values of 62 - - ) hu for these values of x, and then construct the table of f x udx by successive additions. Attention must be given to the possible accumulation of errors due to the small errors in the values of u. Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from 'a known value of X udx. The process may be repeated. Thus we have x f x udx dx = (x + 2 4 5 - ri o 6 3)2h2u = a2 62 ,A 1 80 64 rs2 RI-vs 66 + ...)h2u = a2(h2u - 40 6 4 h 2 u). Here there are two arbitrary constants, which may be adjusted in various ways.

The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hdv/dx or h 2 d 2 v /dx 2 can be conveniently expressed in terms of v and x to a greater degree of accuracy than it could be found from the table. The process practically consists in using the table as it stands for improving the first or second differences of v and then building up the table afresh.

(ii) Life Insurance

The use of quadrature-formulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of the tabulated values and their differences.

90. The following are instances of the application of approximative formulae to the calculation of the volumes of solids.

(i) Timber Measure

To find the quantity of timber in a trunk with parallel ends, the areas of a few sections must be calculated as accurately as possible, and a formula applied. As the measurements can only be rough, the trapezoidal rule is the most appropriate in ordinary cases.

(ii) Gauging

To measure the volume of a cask, it may be assumed that the interior is approximately a portion of a spheroidal figure. The formula applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56). In the latter case the two sections are taken at distances t 2H/ A l 3 = = 2887H from the middle section, where H is the total internal length; and their arithmetic mean is taken to be the mean section of the cask. Allowance must of course be made for the thickness of the wood. 91. Certain approximate formulae for the length of an arc of a circle are obtained by methods similar to those of §§ 71 and 79. Let a be the radius of a circle, and 0 (circular measure) the unknown angle subtended by an arc. Then, if we divide 0 into m equal parts, and L 1 denotes the sum of the corresponding chords, so that L i =2ma sin (0/2m), the true length of the arc is L1 +a9 3 - 5 + ..., where cp. =B/2m. Similarly, if L2 repre sents the sum of the chords when m (assumed even) is replaced by 2m, we have an expression involving L2 and 20. The method of § 71 then shows that, by taking a (4L 0 - L 2) as the value of the arc, we get rid of terms in 0 2. If we use c 1 to represent the chord of the whole arc, c 2 the chord of half the arc, and c 4 the chord of one quarter of the arc, then corresponding to (i) and (iii) of § 70 or § 79 we have a (8c 2 - c i) and4 5 (256c 4 - 40c2+ci) as approximations to the length of the arc. The first of these is Huygens's rule. REFERENCES. - FOr applications of the prismoidal formula, see Alfred Lodge, Mensuration for Senior Students (1895). Other works on elementary mensuration are G. T. Chivers, Elementary Mensuration (1904); R. W. K. Edwards, Elementary Plane and Solid Mensuration (1902); William H. Jackson, Elementary Solid Geometry (1907); P. A. Lambert, Computation and Mensuration (1907). A. E. Pierpoint's Mensuration Formulae (1902) is a handy collection. Rules for calculation of areas are also given in such works as F. Castle, Manual of Practical Mathematics (1903); F. C. Clarke, Practical Mathematics (1907); C. T. Millis, Technical Arithmetic and Geometry (1903). For examples of measurement of areas by geometrical construction, see G. C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907). Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g. E. Goursat, A Course in Mathematical Analysis (1905; trans. by E. R. Hedrick). For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION. On calculation of moments of graphs, see W. P. Elderton, Frequency-Curves and Correlation (1906); as to the formulae of § 82, see also Biometrika, v. 450. For mechanical methods of calculating areas and moments see CALCULATING MACHINES. (W. F. SH.)