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.
