Mensuration
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," th& 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 quadratureformulae 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 vSlumes. 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 + a) (b + /3) = ab + a/3 + a& +o/3. Association with other branches of science is maintained by such methods as those mentioned in 1 2. ' 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 1 1 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 36; 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 %ab, 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 Jir would thus first appear as the ratio of the average breadth of a circle to the greatest breadth; the interpretation of ir 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, etc.
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 it 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).
(6) 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) (6) 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 (ft) above are closely related. Suppose, tor 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 (it) due to inaccuracy of the data. For this reason, formulae which will on 'y g' ve 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 ir 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 ir= 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 ir a value which is in error by more than i 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 = i superficial prime, 12 superficial primes = i square foot; while for cubic measure 12 cubic inches = I solid second, 12 solid seconds = i solid prime, 12 solid primes = i 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 for duodecimal arithmetic is 3 + i/i2+8/i2 2 + 4/i2 3 +8/i2 4 -(- . . . ; so that, marking off duodecimal fractions by commas, the area in the above case is i of 3, I, 8, 4, 8Xi5> 3X15, 3 sq. ft. = 182, 7, 10 sq. ft. = 182 sq. ft. 94 sq. in. (or 1825 sq. ft. approximately).
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
