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 12b; 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 12ab, 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 14π 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.
