1911 Encyclopædia Britannica/Mensuration

by William Fleetwood Sheppard

MENSURATION (Lat. mensura, a measure), the science of measurement; or, in a more limited sense, the science of numerical representation of geometrical magnitudes.

1. Scope of the Subject.—Even in the second sense, the term is a very wide one, since it comprises the measurement of angles (plane and solid), lengths, areas and volumes. The measurement of angles belongs to trigonometry, and it is convenient to regard the measurement of the lengths of straight lines (i.e. of distances between points) as belonging to geometry or trigonometry; while the measurement of curved lengths, except in certain special cases, involves the use of the integral calculus. The term “mensuration” is therefore ordinarily restricted to the measurement of areas and volumes, and of certain simple curved lengths, such as the circumference of a circle.

2. This restriction is to a certain extent arbitrary. The statement that, if the adjacent sides of a rectangle are represented numerically by 3 and 4, the diagonal is represented by 5, is as much a matter of mensuration as the statement that the area is represented by 12. The restriction is really determined by a difference in the methods of measurement. The distance between two points can, at any rate in theory, be measured directly, by successive applications of the unit of measurement. But an area or a volume cannot generally be measured by successive applications of the unit of area or volume; intermediate processes are necessary. the result of which is expressed by a formula; The chief exception is in the use of liquid measure; this is of importance from the educational point of view (§ 12).

3. The measurement is numerical, i.e. it is representation in terms of a unit. The process of determining the area or volume of a given figure therefore involves two separate processes; viz. the direct measurement of certain magnitudes (usually lengths) in terms of a unit, and the application of a formula for determining the area or volume from these data. Mensuration is not concerned with the first of these two processes, which forms part of the art of measurement, but only with the second. It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

4. It is also convenient to regard as coming under mensuration the consideration of certain derived magnitudes, such as the moment of a plane figure with regard to a straight line in its plane, the calculation of which involves formulae which are closely related to formulae for determining areas and volumes.

5. On the other hand, the scope of the subject, as described in § 3, is limited by the nature of the methods employed to obtain formulae which can be applied to actual cases. Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or geometrical methods. Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus. These investigations lead, in turn, to further formulae, which, though not obtainable by elementary methods, are nevertheless simple in themselves and of practical utility. If these are included in the description “mensuration,” the subject thus consists of two heterogeneous portions—elementary mensuration, comprising methods and results, and advanced mensuration, comprising certain results intended for practical application.

6. 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.

7. 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. On the other hand, it is worth noticing that the words “quadrature” and “cubature” are originally due to geometrical rather than numerical considerations; the former implying the construction of a square whose area shall be equal to that of a given surface, and the latter the construction of a cube whose volume shall be equal to that of a given solid.

8. There are two main groups of subjects in which practical needs have tended to develop a separate science of mensuration. The first group comprises such subjects as land-surveying; here the measurements in the elementary stages take place in a plane, and the consideration of volumes necessarily constitutes a later stage; and the figures to be measured are mostly not movable, so that triangulation plays an important part. 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.

9. Relation to other Subjects.—As a result of the importance both of the formulae obtained by elementary methods and of those which have involved the previous use of analysis, there is a tendency to dissociate the former, like the latter, from the methods by which they have been obtained, and to regard mensuration as consisting of those mathematical formulae which are concerned with the measurement of geometrical magnitudes (including lengths), or, in a slightly wider sense, as being the art of applying these formulae to specific cases. 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. This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.

11. It is in reference to the measurement of areas and volumes that it is of special importance to illustrate geometrical truths by means of concrete cases. That the area of a parallelogram is equal to the area of a rectangle on the same base and between the same parallels, or that the volume of a cone is one-third that of a cylinder on the same base and of the same height, may be established by a proof which is admitted to be rigorous, or be accepted in good faith without proof, and yet fail to be a matter of conviction, even though there may be a clear conception of the relative lengths of the diagonal and the side of a square or of the relative contents of two vessels of different shapes. The failure seems (§ 2) to be due to difficulty in realizing the numerical expression of an area or a solid in terms of a specified unit, while the same difficulty does not arise in the case of linear measure or liquid measure, where the number of units can be ascertained by direct counting. The difficulty is perhaps less for volumes than for areas, on account of the close relationship between solid and fluid measure.

12. The main object to be aimed at, therefore, in the study of elementary mensuration, is that the student should realize the possibility of the numerical expression of areas and volumes. The following are some important points.

(i) The double aspect of an area should be borne in mind; i.e. area should be treated not only as length multiplied by length, but also as volume divided by thickness. There are, indeed, certain advantages in preferring the latter to the former, and in proceeding from volumes to areas rather than from areas to» volumes. 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. This method is unconsciously adopted by the teacher who illustrates the equality of area of two geometrical figures by cutting them out of cardboard of uniform thickness and weighing them.

(ii) The very earliest stages of mensuration should be directly associated with simple arithmetical processes.

(iii) Association of solid measure with liquid measure, presenting numerical measurement in a different aspect, should be retained by testing volumes as found from linear dimensions with the volumes of the same bodies as found by the use of measures of capacity. Here, as usual, the British systems of measures produce a difficulty which would not arise under the metric system.

(iv) Solids of the same substance should be compared by measuring and also by weighing; the comparison being then extended to areas of uniform thickness (see (i) above).

(v) The idea of an average may be introduced at an early stage, methods of calculating an average being left to a later stage.

13. Classification.—The methods of mensuration fall for the most part under one or other of three main heads, viz. arithmetical mensuration, geometrical mensuration, and analytical mensuration.

14. The most elementary stage is arithmetical mensuration, which comprises the measurement of the areas of rectangles and parallelepipeds. This may be introduced very early; square tablets being used for the mensuration of areas, and cubical blocks for the mensuration of volumes. The measure of the area of a rectangle is thus presented as the product of the measures of the sides, and arithmetic and mensuration are developed concurrently. The Commutative law for multiplication is directly illustrated; and subdivisions or groupings of the units lead to such formulae as (a+α) (b+β) =ab+aβ+αb+αβ. Association with other branches of science is maintained by such methods as those mentioned in § 12.

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.

15. The next stage is geometrical mensuration, where geometrical methods are applied to determine the areas of plane rectilinear figures and the volumes of solids with plane faces. The ordinary process involves three, separate steps. The first step is the establishment of the exact equality of congruence of two geometrical figures. In the case of plane figures, the congruence is tested by an imaginary superposition of one figure on the other; but this may more simply be regarded as the superposition, on either figure, of the image of the other figure on a contiguous plane. In the case of solid figures a more difficult geometrical abstraction is involved. The second step is the conversion of one figure into another by a process of dissection, followed by rearrangement of parts; the figure as rearranged being one whose area or volume can be calculated by methods already established. This is the process adopted, for instance, for comparison of the area of a parallelogram with that of a rectangle on the same base and of the same height. The third step is the arithmetical calculation of the area or volume of the rearranged figure. These last two steps may introduce magnitudes which have to be subtracted, and which therefore have to be treated as negative quantities in the arithmetical calculation.

The difficulties to which reference has been made in § 11 are largely due to the abstract nature of the process involved in the second of the above steps. The difficulty should, wherever possible, be removed by making the process of dissection and rearrangement complete. This is not always done. To say, for instance, that the area of a right-angled triangle is half the area of the rectangle contained by the two sides, is not to say what the area is, but what it is the half of. The proper statement is that, if a and b are the sides, the area is equal to the area of a rectangle whose sides are a and 1/2b; this being, in fact, a particular case of the proposition that the area of a trapezium is equal to the area of a rectangle whose sides are its breadth and the arithmetic mean of the lengths of the two parallel sides. This mode of statement helps to establish the idea of an average. The deduction of the formula 1/2ab, where a and b are numbers, should be regarded as a later step.

Elementary trigonometrical formulae, not involving the conception of an angle as generated by rotation, belong to this stage; the additional geometrical idea involved being that of the proportionality of the sides of similar triangles.

16. The third stage is analytical mensuration, the essential feature of which is that account is taken of the manner in which a figure is generated. To prevent discontinuity of results at this stage, recapitulation from an analytical point of view is desirable. The rectangle, for instance, has so far been regarded as a plane figure bounded by one pair of parallel straight lines and another pair at right angles to them, so that the conception of “rectangularity” has had reference to boundary rather than to content; analytically, the rectangle must be regarded as the figure generated by an ordinate of constant length moving parallel to itself with one extremity on a straight line perpendicular to it. 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.

A plane figure bounded by a continuous curve, or a solid figure bounded by a continuous surface, may generally be most conveniently regarded as generated by a straight line, or a plane area, moving in a fixed direction at right angles to itself, and changing as it moves. This involves the use of Cartesian co-ordinates, and leads to important general formulae, such as Simpson’s formula.

The treatment of an angle as generated by rotation, the investigation of the relations between trigonometrical ratios and circular measure, the application of interpolation to trigonometrical tables, and the general use of graphical methods to represent continuous variation, all imply an analytical onlook, and must therefore be deferred to this stage.

17. There are certain special cases where the treatment is really analytical, but where, on account of the simplicity or importance of the figures involved, the analysis does not take a prominent part.

(i) The circle, and the solid figures allied to it, are of special importance. The ordinary definition of a circle is equivalent to definition as the figure generated by the rotation of a radius of constant length in a plane, and is thus essentially analytical. The ideas of the centre and of the constancy of the radius do not, however, enter into the elementary conception of the circle as a round figure. This elementary conception is of the figure as already existing, rather than of its method of description; the test of circularity being the possibility of rotation within a surrounding figure so as to keep the two boundaries always completely in contact. In the same way, the elementary conception of the sphere involves the idea of sphericity, which would be tested in a similar way, and is in fact so tested, at an early stage by tactual perception, and at a more advanced stage by mechanical methods; the next step being the circularity of the central section, as roughly tested (where the sphere is small) by visual perception, i.e. in effect, by the circularity of the cross-section of a circumscribing cylinder; and the ideas of the centre and of non-central sections follow later.

It seems to follow that the consideration of the area of a circle should precede the consideration of its perimeter, and that the consideration of the volume of a sphere should precede the consideration of its surface-area. The proof that the area of a circle is proportional to the square of its diameter would therefore precede the proof that the perimeter is proportional to the diameter; the former property is the easier to grasp, since the conception of the length of a curved line as the limit of the sum of a number of straight lengths presents special difficulties. The ratio 1/4π would thus first appear as the ratio of the average breadth of a circle to the greatest breadth; the interpretation of π as the ratio of the circumference to the diameter being a secondary one. This order follows, in fact, the historical order of development of the subject.

(ii) Developable surfaces, such as the cylinder and the cone, form a special class, so far as the calculation of their area is concerned. The process of unrolling is analytical, but the unrolled area can be measured by methods not applicable to other surfaces.

(iii) Solids of revolution also form a special class, which can be conveniently treated by the two theorems of Pappus (§ 33).

18. The above classification relates to methods. The classification of results, i.e. of formulae, will depend on the purpose for which the collection of formulae is required, and may involve the' grouping of results obtained by very different methods. A collection of formulae relating to the circle, for instance, would comprise not only geometrical and trigonometrical formulae, but also approximate formulae, such as Huygens’s rule (§ 91), which are the result of advanced analysis.

The present article is not intended to give either a complete course of study or a complete collection of formulae, and therefore such only of the ordinary formulae are given as are required for illustrating certain general principles. For fuller discussion reference should be made to Geometry and Trigonometry, as well as to the articles dealing with particular figures, such as Triangle, Circle, &c.

19. The most important formulae are those which correspond to the use of rectangular Cartesian co-ordinates. This implies the treatment of a plane or solid figure as being wholly comprised between two parallel lines or planes, regarded by convention as being vertical; the figure being generated by an ordinate or section moving at right angles to itself through a distance which is called the breadth of the figure. The length or area obtained by dividing the area or the volume of the figure by its breadth is the mean ordinate (mean height) or mean section (mean sectional area) of the figure.

Quadrature-formulae or cubature-formulae may sometimes be conveniently replaced by formulae giving the mean ordinate or mean section. In the early stages it is best to use both methods, so as to develop the idea of an average (§ 12). In the present article the formulae for area or volume will be used throughout.

20. Approximation.—The numerical result obtained by applying a formula to particular data will generally not be exact. There are two kinds of causes producing want of exactness.

(i) The formula itself may not be numerically exact. This may happen in either of two ways.

(a) The formula may involve numbers or ratios which cannot be expressed exactly in the ordinary notation. This is the case, for instance, with formulae which involve π or trigonometrical ratios. This inexactness may, however, be ignored, since the numbers or ratios in question can generally be obtained to a greater degree of accuracy than the other numbers involved in the calculation (see (ii) (b) below).

(b) The formula may only be approximative. The length of the arc of a circle, for instance, is known if the length of the chord and its distance from the middle point of the arc are known; but it may be more convenient in such a case to use a formula such as Huygens’s rule than to obtain a more accurate result by means of trigonometrical tables.

(ii) The data may be such that an exact result is impossible.

(a) The nature of the bounding curve or surface may not be exactly known, so that certain assumptions have to be made, a formula being then used which is adapted to these assumptions. The application of Simpson’s rule, for instance, to a plane figure implies certain assumptions as to the nature of the bounding curve. Such a formula is approximative, in that it is known that the result of its application will only be approximately correct; it differs from an approximative formula of the kind mentioned in (i) (b) above, in that it is adopted of necessity. not by choice.

(b) 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. Thus the inaccuracy in taking the measured diameter as the datum is practically of the same order as the inaccuracy in taking the grass-plot to be circular.

(iii) In dealing with cases where actual measurements are involved, the error (i) due to inaccuracy of the formula will often be negligible in comparison with the error (ii) due to inaccuracy of the data. For this reason, formulae which will only give approximate results are usually classed together as rules, whether the inaccuracy lies (as in the case of Huygens’s rule) in the formula itself, or (as in the case of Simpson’s rule) in its application to the data.

21. It is necessary, in applying formulae to specific cases, not only, on the one hand, to remember that the measurements are only approximate, but also, on the other hand, to give to any ratio such as π a value which is at least more accurate than the measurements. Suppose, for instance, that in the example given in § 20 the diameter as measured is 15 ft. 3 in. If we take π＝3·14 and find the area to be 26288·865 sq. in.＝182 sq. ft. 80·865 sq. in., we make two separate mistakes. The main mistake is in giving the result as true to a small fraction of a square inch; but, if this degree of accuracy had been possible, it would have been wrong to give π a value which is in error by more than 1 in 2000.

Calculations involving feet and inches are sometimes performed by means of duodecimal arithmetic; i.e., in effect, the tables of square measure and of cubic measure are amplified by the insertion of intermediate units. For square, measure—

12 square inches＝1 superficial prime,
12 superficial primes＝1 square foot;

while for cubic measure—

12 cubic inches＝1 solid second,
12 solid seconds＝1 solid prime,
12 solid primes＝1 cubic foot.

When an area has been calculated in terms of square feet, primes and square inches, the primes and square inches have to be reduced to square inches; and similarly with the calculation of volumes. The value of π for duodecimal arithmetic is 3+1/12+8/12²+ 4/12³+8/12⁴+ . . . ; so that, marking off duodecimal fractions by commas, the area in the above case is 1/4 of 3, 1, 8, 4, 8×15, 3×15, 3 sq. ft.＝182, 7, 10 sq. ft.＝182 sq. ft. 94 sq. in. (or 1821/2 sq. ft. approximately).

mensuration of specific figures (geometrical)

22. Areas of Plane Rectilinear Figures.—The following are expressions for the areas of some simple figures; the expressions in (i) and (ii) are obtained arithmetically, while those in (iii)–(v) involve dissection and rearrangement.

(i) Square: side a. Area＝a².

(ii) Rectangle: sides a and b. Area＝ab.

(iii) Right-angled triangle: sides a and b, enclosing the right angle. Area＝1/2ab.

(iv) Parallelogram: two opposite sides a and a, distance between them h. Area＝ha.

(v) Triangle: one side a, distant h from the opposite angle.  Area＝1/2ha.

If the data for any of these figures are other than those given above, trigonometrical ratios will usually be involved. If, for instance, the data for the triangle are sides a and b, enclosing an angle C, the area is 1/2ab sin C.

23. The figures considered in § 22 are particular cases of the trapezium, which is a quadrilateral with two parallel sides. If these sides are a and b, at distance h from one another, the area is h.1/2(a+b). In the case of the triangle, for instance, b is zero, so that the area is 1/2ha.

The trapezium is also sometimes called a “trapezoid,” but it will be convenient to reserve this term for a different figure (§ 24).

The most important form of trapezium is that in which one of the two remaining sides of the figure is at right angles to the two parallel sides. The trapezium is then a right trapezium; the two parallel sides are called the sides, the side at right angles to them the base, and the fourth side the top.

By producing the two parallel sides of any trapezium (e.g. a parallelogram), and drawing a line at right angles to them, outside the figure, we see that it may be treated as the difference of two right trapezia.

Fig. 1.

It is, however, more simple to convert it into a single right trapezium. Let CABD (fig. 1) be a trapezium, the sides CA and DB being parallel. Draw any straight line at right angles to CA and DB (produced if necessary), meeting them in M and Along CA and DB, on the same side of MN, take MA′＝CA, NB′＝DB; and join A′B′. Then MA′B′N is a right trapezium, whose area is equal to that of CABD; and it is related to the latter in such a way that, if any two lines parallel to AC and BD meet AB, CD, MN, A′B′, in E, G, P, E′, and F, H, Q, F′, respectively, the area of the piece PE′F′Q of the right trapezium is equal to the, area of the piece GEFH of the original trapezium. The right trapezium so constructed may be called the equivalent right trapezium. In the case of a parallelogram, the equivalent right trapezium is a rectangle; in the case of a triangle, it is a right-angled triangle.

24. If we take a series of right trapezia, such that one side (§ 23) of the first is equal to one side of the second, the other side of the second is equal to one side of the third, and so on, and place them with their bases in a straight line and their equal sides adjoining each other, we get a figure such as MABCDEFS (fig. 2), which has two parallel sides MA and SF, a base MS at right angles to these, and the remainder of its boundary from A to F rectilinear, no part of the figure being outside the space between MA (produced) and SF (produced). A figure of this kind will be called a trapezoid.

(i) If from the other angular points B, C, D, E, perpendiculars BN, CP, DQ, ER, are drawn to the base MS (fig. 2), the area is MN.1/2(MA+NB)+NP.1/2(NB+ PC)+. . . .+RS.1/2(RE+SF)＝ 1/2(MN. MA + MP. NB + NQ. PC+. . . .+ RS. SF). The lines MA, NB, PC, . . . . are called the ordinates of the points A, B, C, . . . . from the base MS, and the portions MN, NP, PQ, . . . . of the base are the projections of the sides AB, BC, CD, . . . . on the base.

	⁠
Fig. 2		Fig. 3

(ii) A special case is that in which A coincides with M, and F with S. The figure then stands on a base MS, the remainder of its boundary being a broken line from M to S. The formula then becomes

area＝1/2(MP . NB+NQ . PC+ . . . +QS . RE),

i.e. the area is half the sum of the products obtained by multiplying each ordinate by the distance between the two adjacent ordinates. It would be possible to regard this form of the figure as the general one; the figure considered in (i) would then represent the special case in which the two end-pieces of the broken line are at right angles to the base.

(iii) Another special case is that in which the distances MN, NP, PQ, . . . RS are all equal. If this distance is h, then

area＝h(1/2MA+NB+PC+. . .+1/2SF).

25. To find the area of any rectilinear figure, various methods are available.

(i) The figure may be divided into triangles. The quadrilateral, for instance, consists of two triangles, and its area is the product of half the length of one diagonal by the sum of the perpendiculars drawn to this diagonal from the other two angular points.

For figures of more than four sides this method is not usually convenient, except for such special cases as that of a regular polygon, which can be divided into triangles by radii drawn from its centre.

(ii) Suppose that two angular points, A and E, are joined (fig. 3) so as to form a diagonal AE, and that the whole of the figure lies between lines through A and E at right angles to AE. Then the figure is (usually) the sum of two trapezoids on base AE, and its area can be G calculated as in § 24. If BN, CP, DQ, . . . . FS, GT are the perpendiculars to AE from the angular points, the ordinates NB, PC, . . . are called the offsets from the diagonal to the angular points.

The area of the polygon, in fig. 3 is given by the expression

1/2(AP . NB+NQ . PC+PE . QD+ET . SF+SA . TG).

It should be noticed (a) that AP, NQ, . . . . SA are taken in the cyclical order of the points ABC . . . GA, and, (b) that in fig. 3, if AN and NB are regarded as positive, then SF, TG, ET and SA are negative, but the products ET . SF and SA . TG are positive. Negative products will arise if in moving from A to E along the perimeter of either side of the figure the projection of the moving point does not always move in the direction AE.

(iii) Take any straight line intersecting or not intersecting the figure, and draw perpendiculars Aa, Bb, Cc, Dd, . . . Ff, Gg to this line. Then, with proper attention to signs,

area＝1/2(gb . aA+ac . bB+bd . cC+. . . +fa . gG).

(iv) The figure may be replaced by an equivalent trapezoid, on the system explained in § 23. Take any base X′X, and draw lines at right angles to this base through all the angular points of the figure. 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;

where x = the distance between the two lines=N₁/N₀. These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes. A line through the centroid of a plane figure (drawn in the plane of the figure) is a central line, and a plane through the centroid) of a solid figure is a central plane, of the figure.

The centroid of a rectangle is its centre, i.e. the point of intersection of its diagonals. The first moment of a plane figure with regard to a line in its plane may be regarded as obtained by dividing the area into elementary strips by a series of parallel lines indefinitely close together, and concentrating the area of each strip at its centre. Similarly the first moment of a solid figure may be regarded as obtained by dividing the figure into elementary prisms by two sets of parallel planes, and concentrating the volume of each prism at its centre. This also holds for higher moments, provided that the edges of the elementary strips or prisms are parallel to the line or plane with regard to which the moments are taken.

33. Solids and Surfaces of Revolution.—The solid or surface generated by the revolution of a plane closed figure or a plane continuous line about a straight line in its plane, not intersecting it, is a solid of revolution or surface of revolution, the straight line being its axis. The revolution need not be complete, but may be through any angle.

The section of a solid of revolution by a plane at right angles to the axis is an annulus or a sector of an annulus (fig. 5), or is composed of two or more such figures. If the solid is divided into elements by a series of such planes, and if h is the distance between two consecutive planes making sections such as ABDC in fig. 5, the volume of the element between these planes, when h is very small, is approximately h×AB × arc PQ＝h.AB.OP.θ. The corresponding element of the revolving figure is approximately a rectangle of area h.AB, and OP is the distance of the middle point of either side of the rectangle from the axis. Hence the total volume of the solid is M.θ, where M is the sum of the quantities h.AB.OP, i.e. is the moment of the figure with regard to the axis. The volume is therefore equal to S.ȳ.θ, where S is the area of the revolving figure, and ȳ is the distance of its centroid from the axis.

Similarly a surface of revolution can be divided by planes at right angles to the axis into elements, each of which is approximately a section of the surface of a right circular cone. By unrolling each such element (§ 30) into a sector of a circular annulus, it will be found that the total area of the surface is M′.θ＝L.z̄.θ, where M′ is the moment of the original curve with regard to the axis, L is the total length of the original curve, and z̄ is the distance of the centroid of the curve from the axis. These two theorems may be stated as follows:—

(i) If any plane figure revolves about an external axis in its plane, the volume of the solid generated by the revolution is equal to the product of the area of the figure and the distance travelled by the centroid of the figure.

(ii) 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.

These theorems were discovered by Pappus of Alexandria (c. A.D. 300), and were made generally known by Guldinus (c. A.D. 1640). They are sometimes known as Guldinus’s Theorems, but are more properly described as the Theorems of Pappus. The theorems are of use, not only for finding the volumes or areas of solids or surfaces of revolution, but also, conversely, for finding centroids or centres of gravity. They may be applied, for instance, to finding the centroid of a semicircle or of the arc of a semicircle.

34. Segment of Parabola.—The parabola affords a simple example of the use of infinitesimals. Let AB (fig. 6) be any arc of a parabola; and suppose we require the area of the figure bounded by this arc and the chord AB.

Fig. 6.

Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and draw the tangent at C, meeting AT and BT in a and b. Then (see Parabola) TC＝CV, AV＝VB, and ab is parallel to AB, so that aC＝Cb. Hence area of triangle ACB＝twice area of triangle aTb. Repeating the process with the arcs AC and CB, and continuing the repetition indefinitely, we divide up the required area and the remainder of the triangle ATB into corresponding elements, each element of the former being double the corresponding elements of the latter. Hence the required area is double the area of the remainder of the triangle, and therefore it is two-thirds of the area of the triangle.

The line TCV is parallel to the axis of the parabola. If we draw a line at right angles to TCV, meeting TCV produced in M and parallels through and B in K and L, the area of the triangle ATB is 1/2KL.TV＝KL.CV; and therefore the area of the figure bounded by AK, BL, KL and the arc AB, is

KL.1/2(AK+BL)+2/3KL{CM−1/2(AK+BL)}

＝1/6KL(AK+4CM+BL).

Similarly, for a corresponding figure K′L′BA outside the parabola, the area is

1/6K′L′(K′A+4M′C+L′B).

35. The Ellipse and the Ellipsoid.—For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere. Hence the area of an ellipse whose axes are 2a and 2b is πab; and the volume of an ellipsoid whose axes are 2a, 2b and 2c is 4/3πabc. The area of a strip of an ellipse between two lines parallel to an axis, or the volume of the portion (frustum) of an ellipsoid between two planes parallel to a principal section, may be found in the same way.

36. Examples of Applications.—The formulae of § 24 for the area of a trapezoid are of special importance in land-surveying. The measurements of a polygonal field or other area are usually taken as in § 25 (ii); a diagonal AE is taken as the base-line, and for the points B, C, D, . . . there are entered the distances AN, AP, AQ, . . . along the base-line, and the lengths and directions of the offsets NB, PC, QD, . . . The area is then given by the formula of §25 (ii).

Fig. 7.

37. The mensuration of earthwork involves consideration of quadrilaterals whose dimensions are given by special data, and of prismoids whose sections are such quadrilaterals. In the ordinary case three of the four lateral surfaces of the prismoid are at right angles to the two ends. In special cases two of these three lateral surfaces are equally inclined to the third.

(i) In fig. 7 let base BC＝2a, and let h be the distance, measured at right angles to BC, from the middle point of BC to AD. Also, let angle ABC＝π−θ angle BCD＝π−φ, angle between BC and AD＝ψ. Then (as the difference of two triangles)

area ABCD＝(h cot ψ+a)²/2(cot ψ−cot φ) − (h cot ψ−a)²/2(cot ψ+cot θ) ⋅

(ii) If φ＝θ, this becomes

area＝tan θ/tan² θ−tan² ψ(h + a tan θ)² − a² tan θ.

(iii) If ψ =0, so that AD is parallel to BC, it becomes

area＝2ah+1/2(cot θ + cot φ)h².

(iv) To find the volume of a prismoidal cutting with vertical ends, and with sides equally inclined to the vertical, so that φ＝θ, let the values of h, ψ for the two ends be h₁, ψ₁, and h₂, ψ₂, and write

m₁≡cot ψ₁/cot ψ₁−cot θ (a + h₁ cot θ), n₁≡ cot ψ₁/cot ψ₁+cot θ (a + h₁ cot θ),

m₂≡cot ψ₂/cot ψ₂−cot θ (a + h₂ cot θ), n₂≡ cot ψ₂/cot ψ₂+cot θ (a + h₂ cot θ),

Then volume of prismoid＝length × 1/3 {m₁n₁ + m₂n₂+ 1/2(m₁n₂ + m₂n₁)−3a²} tan θ.

mensuration of graphs

38. (A) Preliminary.—In § 23 the area of a right trapezium has been expressed in terms of the base and the two sides; and in § 34 the area of a somewhat similar figure, the top having been replaced by an arc of a parabola, has been expressed in terms of its base and of three lengths which may be regarded as the sides of two separate figures of which it is composed. We have now to consider the extension of formulae of this kind to other figures, and their application to the calculation of moments and volumes.

39. The plane figures with which we are concerned come mainly under the description of graphs of continuous variation. Let E and F be two magnitudes so related that whenever F has any value (within certain limits) E has a definite corresponding value. Let u and x be the numerical expressions of the magnitudes of E and F. On any line OX take a length ON equal to xG, and from N draw NP at right angles to OX and equal to uH; G and H being convenient units of length. Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively. Let KA and LB be the positions of NP corresponding to the extreme values of x. Then the different positions of NP will (if x may have any value from OK to OL) trace out a figure on base KL, and extending from KA to LB; this is called the graph of E in respect of F. The term is also sometimes applied to the line AB along which the point P moves as N moves from K to L.

To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x. It may be shown that this is the same thing as the mean distance 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.

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 1/2h(u₀, + u₁) or hu_1/2.

51. The next case is that in which u is a quadratic function of x, i.e. is of the form px² + 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 mid-ordinate. If we take these to be u₀ and u₀, and u₁, so that m＝2, we have

area＝1/6h(u₀ + 4u₁ + u₂)＝1/3h(u₀ + 4u₁ + u₂).

This is Simpson’s formula.

If instead of u₀, u₁, and u₂, we have four ordinates u₀, u₁, u₂ and u₃, so that m＝3, it can be shown that

area＝3/8h(u₀ + 3u₁ + 3u₂ + u₃).

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 u_3/2, we can express A + B; B + C; A + B + C; and B in terms of u₀, u₁, u₂; u₁, u₂, u₃; u₀, u_3/2, u₃; u₁, u_3/2, u₂ respectively. Thus we get two expressions for A + B + C, from which we can eliminate u_3/2.

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

52. Simpson’s two formulae also apply if u is of the form px³ + qx² + 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＝0, 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 u₀, u₁, u₂, u₃, u₄, we have

area＝2/45h(7u₀ + 32u₁ + 12u₂, + 32u₃, 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

mh ${\Big \{}$u_1/2m + 1/24m²h²u″_1/2m + 1/1920m⁴h⁴u^iv_1/2m + 1/1920m⁶h⁶u^vi_1/2m + 1/92897280m⁸h⁸u^viii_1/2m + . . . ${\Big \}}$,

where u_1/2m, u″_1/2m, u ‴_1/2m , . . . denote the values for x＝x_1/2m 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, u_1/2m, will be one of the given ordinates, and we can express h²u″_1/2m, h⁴u^iv_1/2m in terms of u_1/2m 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 $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$u_p + p²/6δ²u_p + 3p⁴ −5p²/360δ⁴u_p + 3p⁶ −21p⁴ + 28p²/15120δ⁴u_p + . . . $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ .

If u is of degree 2f or 2f + 1 in x, we require to go up to δ^2fu_p, so that m must be not less than 2f Simpson’s (first) formula, for instance, holds for f＝1, and is obtained by taking p＝1 and ignoring differences after δ²u_p.

(ii) If m is odd, the given ordinates are u₀, . . . u_1/2m−1/2, u_1/2m+1/2 . . . u_m. We then have

area＝mh $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ μu_1/2m + m² − 3/24μδ²u_1/2m + 3m⁴ − 50m²+135/5760μδ⁴u_1/2m + 3m⁶ − 147m⁴+1813m² − 4725/967680μδ⁶u_1/2m + . . . $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$,

where μu_1/2m, μδ²u_1/2m, . . . denote 1/2(u_1/2m−1/2 + u_1/2m+1/2), 1/2(δ²u_1/2m−1/2 + δ²u_1/2m +1/2), . . . Simpson’s second formula is obtained by taking m＝3 and ignoring differences after μδ²u_1/2m.

55. The general formulae of § 54 (p being replaced in (i) by 1/2m) 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 δ^2fu_1/2m or μδ^2fu_1/2m at least, where u is of degree 2f or 2f + 1 in x. Thus we find from (i) that Simpson’s second formula, for the case where the to 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＝3/8h(3u_1/2 + 2u_3/2 + 3u_5/2).

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.

Fig. 8.

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 the trapezette would be the greater. There 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 . 1/2(QD + RE).

To find this position, let us write QM＝MR＝θ . KM. Then

WC＝θ².VC, VW＝(1 − θ²) VC;

curved area ACB＝2/3 of parallelogram AFGB＝2/3KL . VC;

parallelogram AHJB＝KL . VW＝(1 − θ²) KL . VC.

Hence the areas of the trapezette and of the trapezium will be equal if

1 − θ²＝2/3, θ＝1/√3.

This value of θ 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 + 1, the area can be expressed in terms of p + 1 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 S₀ 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 S₁, the volume is 1/6H(S₀ + 4S₁ + S₂), 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 π(R²−x²), which is a quadratic function of x; the values of S₀, S₁, and S₂ are respectively 0, πR², and 0, and the volume is therefore 1/6 . 2R . 4πR²＝4/3πR³.

Fig. 9.

To show that the area of a cross-section of a prismoid is of the form ax² + 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 x²/h² × 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, 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−x²/h²)× 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 in x, and therefore the first moment is 1/3h(x₀u₀+4x₁u₁+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 cross section through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to S₀, S₂, and 4S₁, 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, πR², and 4.3/4πR² , placed at the extremities and the middle of its axis; i.e. the centroid is at distance 3/8R 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 λ₀v₀+y₁v₁+ . . . λ_mv_m, the qth moment of the trapezette is λ₀x₀^qu₀+λ₁x₁^qu₁+ . . . λ_mx_m^qu_m, and the mean value of x^q is

(λ₀x₀^qu₀+λ₁x₁^qu₁+ . . . λ_mx_m^qu_m)/(λ₀u₀+λ₁x₁+ . . . λ_mx_m^qu_m)

The calculation of this last expression is simplified by noticing that we are only concerned with the mutual ratios of λ₀, λ₁, . . . and of u₀, 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＝0 is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y＝0, 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, i.e. that it contains terms in x³y³, x³y², 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＝x₀+h and the planes y＝y₀+k, y＝y₀+2k, and that the 12 ordinates forming the edges of these six elements are given. The areas of the sides for which x＝x₀ and x＝x₀ +2h, and of the section by the plane x＝x₀+h, may be found by Simpson’s second formula; call these A₀ and A₂, and A₁. The area of the section by a plane at distance x from the edge x＝x₀ is a function of x whose degree is the same as that of u. Hince Simpson’s formula applies, and the volume is 1/3h(A₀+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

1/3h{3/8k(u_0,0 + 3u_0,2 + u_0,3) + 4×{3/8k(u_1,0 + . . .) + 3/8k(u₂, + . .)},

where u^θ,φ denotes the ordinate for which x＝x₀+θh, y＝y₀+φk. The result is the same as if we multiplied 3/8k(v₀ + 3v₁+3v₂+v₃) by 1/3h(u₀ + 4u₁ +u₂), and then replaced u₀v₀, u₀v₁, . . . by u_0,0, u_0,1 . . . The multiplication is shown in the adjoining diagram; the factors 1/3 and 3/8; are kept outside, so that the sum u_0,0+3u_0,1+ . . . +4u_1,0+. . . . can be calculated before it is multiplied by 1/3h, 3/8k.

1/3×3/8	1	4	1
1	1	4	1
3	3	12	3
3	3	12	3
1	1	4	1

62. The above is a particular case of a general principle that the obtaining of an expression such as 1/3h(u₀+4u₁, +u₂) or 3/8k(v₀ +3v₁ +3v₂+v₃) is an operation performed on u₀ or v₀, and that this operation is the sum of a number of operations such as that which obtains 1/3hu₀ or 3/8kv₀. 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 u_θ,φ, 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 1/3h(1 + 4E + E²), where E 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 3/8k(1+3E′+3E′²+E′³), where E′ denotes the change of y into y + k. The result of performing 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 distance 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 quadrature-formula 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.u] or ∫udx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[V_x,y.u]] or ∬udxdy 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 u₀, u₁, . . . u_m, the figure formed by joining the tops of these ordinates is a trapezoid whose area, is h(1/2u₀ +u₁ + u₂ + . . . +u_m−1; + 1/2u_m). This is called the trapezoidal or chordal area, and will be denoted by C₁. If the data are u_1/2, u_3/2, . . . u_m−1/2, 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_1/2+u_3/2;+ . . . +u_m−1/2). This is called the tangential area, and will be denoted by T₁. The

tangential area may be expressed in terms of chordal areas. If we write C_1/2 for the chordal area obtained by taking ordinates at intervals 1/2ℎ, then T₁＝2C_1/2−C₁. If the trapezette, as seen from above, is everywhere convex or everywhere concave, the true area lies between C₁ 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 chosen 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−1 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 ≃ 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≃1/3ℎ(𝑢₀ +4𝑢₁ +𝑢₂) + 1/3ℎ(𝑢₂ + 4𝑢₃ +𝑢₄) + . . . ≃ 1/3ℎ(𝑢₀ +4𝑢₁ + 2𝑢₂ + . . . + 2𝑢_𝑚−2 + 4𝑢_𝑚−1 + 𝑢_𝑚)

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≃3/8ℎ(𝑢₀ + 3𝑢₁ + 3𝑢₂ + 2𝑢₃ + 3𝑢₄ + . . . + 3𝑢_𝑚−4 + 2𝑢_𝑚−3 +3𝑢_𝑚−2 + 3𝑢_𝑚−1 + 𝑢_𝑚).

(iii) If mid-ordinates are given, and 𝑚 is a multiple of 3, the repeated application of the formula of § 55 will give

A ≃3/8ℎ(3𝑢_1/2 + 2𝑢_3/2; + 3𝑢_5/2 + 3𝑢_7/2; + . . . + 2𝑢_𝑚−3/2 + 3𝑢_𝑚−1/2).

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(𝑣₃ + 3/2δ²𝑣₃ + 11/20δ⁴𝑣₃ + 41/840δ⁶𝑣₃)＝6ℎ(𝑢₃ + 3/2δ²𝑢₃ + 11/20δ⁴𝑢₃ + 41/840δ⁶𝑢₃).

If we replace 41/840δ⁶𝑢₃ in this expression by 42/840δ⁶𝑢₃, the method of § 68 gives

A ≃3/10ℎ(𝑢₀+ 5𝑢₁ + 𝑢₂ + 6𝑢₃ + 𝑢₄ +5𝑢₅ +𝑢₆);

the expression on the right-hand side being an approximate expression for B, and differing from it only by 1/840δ⁶𝑢₃. 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

4/3ℎ(1/2𝑢₀ + 𝑢₁ + 𝑢₂ + 𝑢₃ + . . . 𝑢_𝑚−2 + 𝑢_𝑚−1 +1/2𝑢_𝑚)−2/3ℎ(1/2𝑢₀+𝑢₂ + 𝑢₄ + . . . + 𝑢_𝑚−2 + 1/2𝑢_𝑚)

Now, if 𝑝 is any factor of 𝑚, there is a series of equidistant ordinates 𝑢₀, 𝑢_𝑝, 𝑢_2𝑝, . . . 𝑢_𝑚−𝑝, 𝑢_𝑚; and the chordal area as determined by these ordinates is

𝑝ℎ(1/2𝑢₀ + 𝑢_𝑝 + 𝑢_2𝑝 + . . . . + 𝑢_𝑚−𝑝 1/2𝑢_𝑚),

which may be denoted by C_𝑝. With this notation, the area as given by Simpson’s rule may be written in the form 4/3C₁−1/3C₂ or C₁+1/3(C₁−C₂). 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 ≃ 1/3(4C₁ − C₂≃C₁ + 1/3(C₁ − C₂)

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

A ≃ 1/8(9C₁ − C₃) ≃ C₁ +1/8(C₁ − C₃)

(iii) 𝑚 a multiple of 4.

A ≃ 1/45(64C₁ − 2OC₂+C₄) ≃ C₁ + 4/9(C₁ − C₂) − 1/45(C₁ − C₄).

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

A ≃1/10(15C₁−6C₂+C₃) ≃ C₁ +1/2(C₁ − C₂) −1/10(C₂ − C₃).

(v) 𝑚 a multiple of 12.

A ≃ 1/35(56C₁ − 28C₂ + 8C₃ − C₄)
⁠≃C₁+3/5(C₁ − C₂) − 1/5(C₂ − C₃) + 1/35(C₃ − C₄).

There are similar formulae in terms of the tangential areas T₁, T₂, T₃. Thus (iii) of § 68 may be written A ≃ 1/8(9T₁ − T₃).

71. The general method of constructing the formulae of § 70 for chordal areas is that, if 𝑝, 𝑞, 𝑟, . . . are 𝑘 of the factors (including 1) of 𝑚, we take

A≃PC_𝑝 + QC_𝑞, +RC_𝑟, + . . . ,

where P, Q, R, . . . satisfy the 𝑘 equations

P + Q + R + . . .	＝1,
P𝑝+Q𝑞2+R𝑟2+ . . .	＝0,
P𝑝4 + Q𝑞4 + R𝑟4 + . . . .	＝0,
	: :
P𝑝2𝑘−2 + Q𝑞2𝑘−2 + R𝑟2𝑘−2 + . . .	＝0.

The last 𝑘−1 of these equations give

1/P : 1/Q : 1/R : . . ＝ 𝑝²(𝑝² − 𝑞²)(𝑝²−𝑞²)(𝑞²−𝑟²) . . . : 𝑞²(𝑞²−𝑝²)(𝑞²−𝑟²) . . . : . . . 𝑟²(𝑟² − 𝑝²)(𝑟² − 𝑞²)

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

The same method applies for tangential areas, by taking

A ≃PT_𝑝 +QT_𝑞, +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 C₂, C₃, &c. (§ 70), is troublesome. (ii) This assignment of different coefficients means that different weights are given to different ordinates; and the relative weights ma 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 C₁ 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 C₁ 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 changes its direction. We have therefore in the first place to see whether the difference can be expressed in terms of the directions of the tangents.

Fig. 10.

Let KABL (fig. 10) 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＝1/2ℎ, and the triangles AVD and BVE are equal.

The area of the trapezette is less (in fig. 10) than the area of the trapezium KABL by two-thirds of the area of the triangle ATB (§ 34). This latter area is

∆BTE − ∆ATD ＝ ∆BTG − ∆ATD＝1/8ℎ² tan GTB − 1/8ℎ² tan DAT.

Hence, if the angle which the tangent at the extremity of the ordinate 𝑢_θ makes with the axis of 𝑥 is denoted by ψ_θ, we have

area from	⁠𝑢0 to 𝑢1＝1/2ℎ(𝑢0 + 𝑢1) − 1/12ℎ2(tan ψ1 − tan ψθ),
area from	⁠𝑢1 to 𝑢2＝1/2ℎ(𝑢1 + 𝑢2) − 1/12ℎ2(tan ψ2 − tan ψ1),
	⁠: :
area from	𝑢𝑚−1 to 𝑢𝑚 ＝1/2ℎ(𝑢𝑚−1 + 𝑢𝑚) − 1/12ℎ2(tan ψ𝑚 − tan ψ𝑚−1);

and thence, by summation,

A＝C₁ − 1/12ℎ²(tan ψ_𝑚 − ψ₀).

This, in the notation of §§ 46 and 54, may be written

A＝C₁ + [− 1/12ℎ²𝑢′] 𝑥＝𝑥_𝑚
𝑥＝𝑥₀.

Since ℎ＝H/𝑚, the inaccuracy in taking C₁ as the area varies as 1/𝑚².

It might be shown in the same way that

A＝T₁+1/24ℎ²(tan ψ_𝑚−ψ₀)＝T₁ + [1/24ℎ²𝑢′]𝑥＝𝑥_𝑚
𝑥＝𝑥₀.

75. The above formulae apply only to a parabolic trapezette Their generalization is given by the Euler-Maclaurin formula A = $\int _{x_{0}}^{x_{m}}$, udx = C₁ + [1/24h²u′ + 1/720h⁴u′′′ − 1/30240h⁶u^v + 1/1209600 − . . .]𝑥＝𝑥_𝑚
𝑥＝𝑥₀;

and an analogous formula (which may be obtained by substituting 1/2ℎ and C_1/2 for ℎ and C₁ in the above and then expressing T₁ as 2C_1/2 −C₁)

A= 3f;"udx=T +, § h'u'-rfm, h'u”'+w$-§ f6h°uvf n, g § %8 Whsuvi1+

To apply these, the differential coefficients have to be expressed in terms of differences.

76. If we know not only the ordinates u₀, u₁, . . . or ui, ug, , 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

A = C1 477' ' i12l"5u'l"712h5 l45'“.'rf5ifzs#55“ +ra'2*a°s'mrH5'“ " ~ - - x =x" x=x0

A=T»+h aaa- saeu+f¢° ° 1  »°u-aasiswiiu+ .. 3253"

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 {§ Au0-,1;A'u0+-,1, i6A3u°-, § UA4u0+ . +i1, A'u, ,, -,1;A'1'u, ,, +»,1§ , ;A'3u, ,, —f~2¢, A"u, ,, +

A ='r, +h{ -,1, Au, +,1, A=u; -, w, , a=u¢+,1, °, #, ,A4u;- —{;A'14m-{+ s11A"u -§ - =*»f"n%A'”u»..; + s't°§ %A"'u»

where ∆, ∆², have the usual meaning (∆u₀=u1-un, A'u0= Au; - Aug, . .), and A', A”, denote differences read backwards, so that A'u, ,.= u, ,, 1 -u, ,, , A"u, ,, =u, ,, 2- 2u, n,1-i-ilm, . . The calculation of the expressions in brackets may be simplified by taking the pairs in terms from the outside; 'i.e by finding the successive differences of ug -|- u, ,, , 'ui + um-1, , or of ui + u, ,, , *, 1¢§ 'i'Um-gi - —

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 Ci or T1 are sometimes found specially. Parmenliefs rule, for instance, assumes that in addition to up 'ui . . u, ..,4, we know no and um; and u*-ue and u, ,, -u, ,, § are taken to be equal to #huh and %hu', ,, respectively. These methods are not to be recommended except in s ecial cases.

79. By replacing ℎ in § 75 by 2ℎ, 3ℎ, . . . and eliminating ℎ²u', h'u"', ., we obtain exact formulae corresponding to the proximate formulae of § 70. The following are the results (for tffe 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=x0 and x=x, ,., as in §§ 75 and 76.

(i) A=§ (4C1-Cz)+[-;§ »5h'u"'+-f§ fgh°u'f-T;}1;5h°uvil+ ] =t(4C₁'C₂)+ℎ[−1/180u+rs"'r1rl15'1¢-ro°v°r'1r6#5'14+. . .].

(ii) A=%(9C1-Ca)+[-s*rh'“"'+ rhh°“Y'rr1¢“rsh“1¢"“+ - - -l =i(9Ci-Cs)+h[-n'iw5'u+t?ir#5°1¢-rn"*s°&roM5'14+- - -]

(iii) A=, 1g(64C1- 20C2+ Cr) +[-;»{, h°u"+;;-2, ~¢;h”u”“- ] =I1s(@4C1'“20C2+C4}+hl-9iirP»55U 'l'tiii%rM5"1¢- ~ - - l

(iv) A={~, ,(l5C1-6Cg+C3)+[-§ }5h°u" 1-;;H, -;;h“u"“- ] =#a(I5Ci-6Cz+ C=)-l-hi- tiUM5°1¢'i'sis"x'v6M5714- » - - ]

(v) A=|1¢(56C1-28Cz-l-SC;-C4)'l'l ~{1Hr5/z8u"“-Q' . ] =9l;(56Ci-28C¢-I-8C;—C4)+h[—2-;1n;tE7u+

The general expression, if p, q, r, . are k of the factors of m, is A=PC, +QC, + RCf+ . + ()'=b, h='=§§ f,2 + - d“'°+1u x = x

k+1 z1=+z . fn

() b'”'1h dx2"+1+°" x=xu

where P, Q, R, have the values given by the equations in § 71, and the coefficients bk, b;, +1, are found from the corresponding coefficients in the 'Euler-Maclaurin formula (§ 75) by multiplying them by Pp'-"+Qq”'+Rr”°+ ., Pp2"*'+Qq”'*'+I{r”'*“2+. ,

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 ug, ul, u¢, are given, we have only to find the area of a trapezette whose ordinates are xgpug, xipul, xgpuz,

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 AQ, Ap . Am-; of the strips -bounded by the consecutive ordinates ug, ul, . . 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 x=x0 and for x=x, ,, . The method adopted in these cases is to treat the areas Ar Ai, . . as if they were ordinates placed at the points for which x=x*, x=x§ , , 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 ρ₁, ρ₂, , we have

pi = x;A;+ xgA; -- .' + x, ,, ;A, ,, ; me x'iA=i + x'eAs + . . - + x', . iA ..§ p, ,=xPiA§ -I-xPgAg+ +xP, ,, ;A, ,, ;-

denoted by vi, va, - .

These are called the raw moments. Then, if the true moments are their values are given by

ν₁≃ ρ₁

V2 £132 '- 32 hfpo

Va¥Pa '” U12/Ji

ν₄≃ ρ₄ −h²ρ₂ +7/240h⁴ρ₀

1's£Ps - ihzpa +;ZsVh"P1

where ρ₀ (or ν₀) is the total area A; -|- A; + . + A, ,, ;; the general expression being

!

1"p£Pp", H 2 !|(p 2 lhzpp-2 "l" M 4! P Zn h'Pp-4- . where ~

>~i=1'r» }'2=?1'6v }s=r§ izn 7=a's%»?s=a”r"a“t"r, - - » The establishment of these formulae involves the use of the integral ca cu us. 1

The position of the central ordinate is given by x=v1/po, and therefore is given approximately by x-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,1r2,1r3, . . ., the true moments ui, 112, pa, with regard to the central ordinate are given by

H1 = 0

1. flirt-1'r7l'po

/-¢a&'Ifs

u4& vu °-' ihirz +512-6 h*/Ja

1/»r%1fa"~§ h21fs

83. These results may be extended to the calculation of an expression of the form f;;"u¢(x)dx, where ¢(x) is a definite function of ~x, and the conditions with regard to u are the same as in § 82. (i) If ¢>(x) is an explicit function of x, we have f;;"'u¢(x)dx4-@Ai¢(xt)+A¥//(x&)+ - + Am-ilf(x, ..i) where ., o(x)§ ¢(x)-§ h2¢"<x>+§§ h4¢fv(x>- ,

the coefficients M, M, ... having the values given in § 82. (ii) If 4>(x) is not given explicitly, but is tabulated for the values xy xg, . of x, the formula of (i) applies, provided We take i0(x)E(1-'s'x5'+r%r5'-n'r<:5“+ - - -)¢(x)-The formulae can be adapted to the case in which ¢(x) is tabulated for x=x0, xl, f -,

84. In cases other than those described in § 82, the pth moment with regard to the axis of u is given by

VP = x"mA -Psp-1,

where A is the total area of the original trapezette, and S, , 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are . 0. x="“A;, x2'“"(Ai+~Ai), . - ~ x£ il(As+ AH- - - - + Am-i)» x5"A» The value of S, ,, has to be found by a quadrature-formula. The generalized formula is

f§§ 3'“¢»<x>dx = A¢<x > - T. T where T is the area of a trapezette whose ordinates at successive distances h are 0, A_1/2φ′ (x₁), (A;+A;)¢'(x2), (A=+A; + . . + A, , 3)¢' (x, ,, , 1), A¢»'fx, ,,); 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=x₁, and y=y₀, y=y₁, . . . The method of constructing the formulae is explained in § 62. If we write

S_p,q, ≡ $\int _{x_{0}}^{x_{m}}\int _{y_{0}}^{y_{n}}$ x^py^q u dx dy,

we first calculate the raw values σ_0,1, σ_1,0, σ_1,1, . . . of S_0,1, S_1,0, S_1,1, 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.)

S_1,1≅σ_1,1

S_2,1≅σ_2,1−1/12h²σ_0,1

1§ k2rf1, o

52,2-£012 fl? k20'2, o -°{3h2Uo,2'f'r}; ll'k2¢70, o S3»l&U3»l ihzvm

S_3,2≅σ_3,2 illzvi 12 " k2¢fa,0 'l'4J'5h2k2¢1i,

where om 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

if f3;, ';'x1=yvu dx dy = K + L + R - x:', , yg.1, ,, where KExI, ,><qth moment with regard to plane y=0, LE y2Xpth moment with regard to plane x=0, and R is the volume of a bri uette whose ordinate at 's q' (x, ., y,)1 found

by multiplying by PQ xf"“' J/3" the volume of that portion of the original briquette which lies between the planes x=x0, x=x, , y=y0, y=y, . The ordinates of this new briquette at the points of intersection of x=x0, x=x1, . . . with y=y0, y=y1, . . 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 ex ressions, we have a formula which may be written in a more general)form !b[°u¢(x, y)dxdy =fb f°udxdy . ¢(b, g)

-1* Q rr fr dx dy 1 ff-W d»

irq 3 fl” u dx dy d—¢(33, y) dy

4 ' if dx d ¢l2¢(x|)

+f! 14 y -#dy dxdy.

The second and third expressions on the right-hand side represent areas of trapezettes, which can be calculated from the data; and the fpurtéi 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 £ormula (§ 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 impossiblte to express u as an algebraical function of x and y if some value of du/dx or du/dy 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. 11; 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

Fig. 11.

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 'udx may be expressed in terms of x, it is often more convenient, or 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

fzudx = h.;to1.t-1-hi(-ilwlpou-l-»,1215;t¢$'u - .) =;i¢r (hu-fs 62hu +»f'2Jq, f54hu - . Q),

fudx = h.ou+h (, ;146u - 5§ }U6“u +. . .)

= o (hu -1- Q; 6”hu - 5-!;5U5'hu + . .).

The second of these is usually the more convenient. T hus, to. construct a table of values of fudx 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 (1+21162-gi}§ 064+ . .) hu for these values of x, and then construct the table of ffudx 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 ” udx.

The process may be repeated. Thus we-have

fzfzudxdx=(o+g;6-5-Hq;6°+...)2h“u ' 1

= (02 +112 " ei'1152 +s15”4'1ru 54 * nights# 'l' - —) hz" = o'(h'u + -112 6'h2u - 515 54h'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, hd-v/dx or h2d"u/dx” can be conveniently expressed in terms of 11 andx 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 o 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. 1 », 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 Sim son s rule or a rule based on Gauss's theorem for two ordinates (§ p 56). In the latter case the two sections are taken at distances i si-I/V 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 8. 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 θ into m equal parts, and L1 denotes the sum of the corresponding. chords, so that L1=2ma sin (6/2m), the true length of the arc is L₁ +ao 3%-§ , +. . ., where ¢=o/sm. similarly, if L, represents the sum of the chords when m (assumed even) is replaced by ém, we have an expression involving L2 and 2¢. The method of, § 71 then shows that, by taking =§ (4L1-Lg) as the Value of the arc, we get rid of terms in ¢“. I we use of to represent the chord of the whole arc, cz the chord of half the arc, and c, the chord of one quarter of the arc, then corresponding to (i) and (iii) of § 70 or § 79'we have 1/2(8c₂-c₁) and £g€256c4-4oc2+c;) 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. Pieroint'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.)