Approximation

“ It is evident that if we repeat our calculations, using in our fundamental equation intervals shortened in the same ratio as the calculated intervals have come out too long, the intervals calculated from the second solution of the fundamental equation must agree almost exactly with the desired values. If necessary, this process may be repeated, and thus any required degree of accuracy may be obtained, whenever the solution of the uncorrected equation gives an approximation to the true positions. ”

“ When we have thus determined, by the numerical solution of our fundamental equation, approximate values of the three positions of the body, it will always be possible to apply a small numerical correction to the equation, so as to make it agree exactly with the laws of elliptic motion in a fictitious case differing but little from the actual. After such a correction the equation will evidently apply to the actual case with a much higher degree of approximation. ”

“ A very much closer approximation may be obtained, if we regard at any given point of the medium and for light of a given period, as linear functions of and the nine differential coefficients We shall write and diff. ”

“ A last important class of applications of the calculus of finite differences, which deserves to be distinguished from the preceding, consists in the eminently useful employment made of it in geometry for determining by approximation the length and the area of any curve, and in the same way the cubature of a body of any form whatever. ”

“ The distance, b, c, being known, that of the base, b, j, and the height of the tree, j, a, will be best found by measurement of equal parts, but bear in mind that the result can only be an approximation to truth, for every additional operation involves an increase of possible error. ”

“ Conclusions.—We see that experience plays an indispensable rôle in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science.
If it were experimental, it would be only approximative and provisional. And what rough approximation!
Geometry would be only the study of the movements of solids; but in reality it is not occupied with natural solids, it has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids.
The notion of these ideal solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them. ”

“ It is true that these approximations are susceptible of almost unlimited accuracy; but still they are, and ever will continue to be, only approximations. Optics and all other branches of natural science are liable to the same observations. ”

“ Similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form u = A + Bx + Cx2 + Dx3 + . . . The various forms that interpolation-formulae take are due to the various principles on which ordinates are selected for determining the values of A, B, C . . .
B. Inverse Interpolation.
5. To find the value of x when u is given, i.e. to find the value of θ when uθ is given, we use the same formula as for direct interpolation, but proceed (if differences beyond the first are involved) by successive approximation. ”

“ It should be noted that the mathematical calculations provide data which are really only approximations, and consequently it is often found that a system constructed on such data requires modification before it fulfils the practical requirements. ”

“ We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles. ”

“ Ordinary measurement by a foot rule, a surveyor’s chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars, are sufficient instances of this procedure.
In the second case, where p y/x = q, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. ”

“ Three such generalizations as the foregoing afford such facilities for the indirect measurement of magnitudes, (by supplying us with known lines or angles which are marks of the magnitude of unknown ones, and thereby of the spaces which they inclose,) that it is easily intelligible how from a few data we can go on to ascertain the magnitude of an indefinite multitude of lines, angles, and spaces, which we could not easily, or could not at all, measure by any more direct process. ”

“ The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ This equation, or equations, will be referred to as the principal equation, or principal equations, of the problem.
The problems and method of the calculus admit of more exact formulation as follows: We confine our attention to the case where the sought curve is plane, and the function F contains no differential coefficients of order higher than the first. Then the problem is to determine a curve joining two fixed points (x0, y0) and (x1, y1) so that the line integral
taken along the curve may be a maximum or a minimum. ”

“ We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the comparison.
If we adopt arbitrary units of length, time, and mass (or quantity of matter) , we can express the measure of all other quantities in terms of these so-called fundamental units. A unit of any other quantity, thus expressed, is called a derived unit. ”

“ UNITS AND STANDARDS OF MEASUREMENT. As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios between any series of objects will never tell us their absolute magnitudes; we must have at least one ratio for each, and we must have one absolute magnitude. ”

“ The same exactitude is found in the applications of differential calculus to geometry and mechanics. If we imagine a curve cut by a secant at two adjacent points, naming E the interval of the ordinates of these two points, E will be the increment of the abscissa from the first to the second ordinate. It is easy to see that the corresponding increment of the ordinate will be the product of E by the first ordinate divided by its subsecant; augmenting then in this equation of the curve the first ordinate by this increment, we shall have the equation relative to the second ordinate. ”

“ All numbers, for instance, which represent physical measurements, are limited in their accuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations. ”

“ To establish a system of measurement on a line we must assign a distinct number to each point of it, and for this purpose we shall plainly have to suppose the numbers carried out into an infinite number of places of decimals. ”

“ 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. ”

“ In practice, the mathematician does not trust to so rough a method as construction, but applies processes of calculation.
Now, it is clear that in this plan everything depends on the base-line. It must not be too short in comparison with the distance of the inaccessible object; for then, if we make the least error in observing the bearings of the object, we get an important error in the resulting determination of the distances. ”

“ 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. ”

“ Every mathematician knows that scores of methods, differing altogether from each other in process, all end in this mysterious 3.14159..., which insists on calling itself the circumference to a unit of diameter. A reader who is competent to follow processes of arithmetic may be easily satisfied that such methods do actually exist. I will give a sketch, carried out to a few figures, of three: the first two I never met with in my reading; the third is the old method of Vieta. ”

“ Moreover, we are usually dealing with more than one variable at once, and thinking of the way in which one variable depends on the other: for instance, we think of the way in which the height reached by a projectile depends on the time of attaining that height. Or we are asked to consider a rectangle of given area, and to enquire how any increase in the length of it will compel a corresponding decrease in the breadth of it. Or we think of the way in which any variation in the slope of a ladder will cause the height that it reaches, to vary. ”

“ When numbers, however, are taken in a concrete sense, and are applied to express peculiar modes of quantity,—such as angular, linear, superficial, and solid magnitudes,—a new set of numerical relations arise, and a large number of computations are required.
To express angular magnitude, and the various relations of linear magnitude with which it is connected, involves the consideration of a vast variety of Geometrical and Trigonometrical tables; such as tables of the natural sines, co-sines, tangents, secants, co-tangents, &c. &c.; tables of arcs and angles in terms of the radius ”

“ Not all physical quantities, in the present state of our knowledge, can be assigned a definite dimensional formula, and this indicates that not all of physics has as yet been reduced to a clearly established mechanical basis. The dimensional formula thus affords a valuable criterion of the extent and boundaries of our strictly definite knowledge of physics. Within these boundaries we are on safe and easy ground and are dealing, independent of all speculation, with the relations between precisely defined quantities. ”

“ In one of those splendidly printed books, bound in blue, published by the University Press, and sold at about a guinea as a guide to some advanced branch of pure mathematics, one may read, even in the second edition published in 1900, the words: “Representation [of a complex variable] on a plane is obviously more effective for points at a finite distance from the origin than for points at a very great distance.” ”

“ 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 most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, this numerical resolution is none the less a very imperfect method, even when it is strictly sufficient. ”

“ A great many practical problems occur where we have to deal with functions of such a nature that they have a greatest (maximum) value or a least (minimum) value, [1] and it is very important to know what particular value of the variable gives such a value of the function. For instance, suppose that it is required to find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius 5 inches. ”

“ A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. ”