by simply laying off the quantities of heat at the mean temperatures, namely 24°, and 74°, and so on. Lord Rayleigh has shown that if we have drawn such an incorrect curve, we can with little trouble correct it by a simple geometrical process, and obtain to a close approximation the true ordinates instead of those denoting areas.1 Interpolation and Extrapolation. When we have by experiment obtained two or more numerical results, and endeavour, without further experiment, to calculate intermediate results, we are said to. interpolate. If we wish to assign by reasoning results lying beyond the limits of experiment, we may be said, using an expression of Sir George Airy, to extrapolate. These two operations are the same in principle, but differ in practicability. It is a matter of great scientific importance to apprehend precisely how far we can practise interpolation or extrapolation, and on what grounds we proceed. In the first place, if the interpolation is to be more than empirical, we must have not only the experimental results, but the laws which they obey-we must in fact go through the complete process of scientific investigation. Having discovered the laws of nature applying to the case, and verified them by showing that they agree with the experiments in question, we are then in a position to anticipate the results of similar experiments. Our knowledge even now is not certain, because we cannot completely prove the truth of any assumed law, and we cannot possibly exhaust all the circumstances which may affect the result. At the best then our interpolations will partake of the want of certainty and precision attaching to all our knowledge of nature. Yet, having the supposed laws, our results will be as sure and accurate as any we can attain to. But such a complete procedure is more than we commonly mean by interpolation, which usually denotes some method of estimating in a merely approximate manner the results 1 J. W. Strutt, On a correction sometimes required in curves professing to represent the connexion between two physical magnitudes. Philosophical Magazine, 4th Series, vol. xlii. p. 441. which might have been expected independently of a theoretical investigation. Regarded in this light, interpolation is in reality an indeterminate problem. From given values of a function it is impossible to determine that function; for we can invent an infinite number of functions which will give those values if we are not restricted by any conditions, just as through a given series of points we can draw an infinite number of curves, if we may diverge between or beyond the points into bends and cusps as we think fit.1 In interpolation we must in fact be guided more or less by à priori considerations; we must know, for instance, whether or not periodical fluctuations are to be expected. Supposing that the phenomenon is non-periodic, we proceed to assume that the function can be expressed in a limited series of the powers of the variable. The number of powers which can be included depends upon the number of experimental results available, and must be at least one less than this number. By processes of calculation, which have been. already alluded to in the section on empirical formulæ, we then calculate the coefficients of the powers, and obtain an empirical formula which will give the required intermediate results. In reality, then, we return to the methods treated under the head of approximation and empirical formulæ; and interpolation, as commonly understood, consists in assuming that a curve of simple character is to pass through certain determined points. If we have, for instance, two experimental results, and only two, we assume that the curve is a straight line; for the parabolas which can be passed through two points are infinitely various in magnitude, and quite indeterminate. One straight line alone. can pass through two points, and it will have an equation of the form, y = mx +n, the constant quantities of which can be determined from two results. Thus, if the two values for x, 7 and 11, give the values for y, 35 and 53, the solution of two equations gives y = 45 × x + 3'5 as the equation, and for any other value of x, for instance 10, we get a value of y, that is 485. When we take a mean value of x, namely 9, this process yields a simple mean result, namely 44. Three experimental results Herschel Lacroix' Differential Calculus, p. 551. : being given, we assume that they fall upon a portion of a parabola and algebraic calculation gives the position of any intermediate point upon the parabola. Concerning the process of interpolation as practised in the science of meteorology the reader will find some directions in the French edition of Kaëmtz's Meteorology. When we have, either by direct experiment or by the use of a curve, a series of values of the variant for equidistant values of the variable, it is instructive to take the differences between each value of the variant and the next, and then the differences between those differences, and so on. If any series of differences approaches closely to zero it is an indication that the numbers may be correctly represented by a finite empirical formula; if the nth differences are zero, then the formula will contain only the first n I powers of the variable. Indeed we may sometimes obtain by the calculus of differences a correct empirical formula; for if p be the first term of the series of values, and Ap, A2p, A3p, be the first number in each column of differences, then the mth term of the series of values will be A closely equivalent but more practicable formula for interpolation by differences, as devised by Lagrange, will be found in Thoinson and Tait's Elements of Natural Philosophy, p. 115. If no column of differences shows any tendency to become zero throughout, it is an indication that the law is of a more complicated, for instance of an exponential character, so that it requires different treatment. Dr. J. Hopkinson has suggested a method of arithmetical interpolation, which is intended to avoid much that is arbitrary in the graphical method. His process will yield the same results in all hands. So far as we can infer the results likely to be obtained by variations beyond the limits of experiment, we must 1 Cours complet de Météorologie, Note A, p. 449. 2 On the Calculation of Empirical Formula. The Messenger of Mathematics, New Series, No. 17, 1872. K K proceed upon the same principles. If possible we must detect the exact laws in action, and then trust to them as a guide when we have no experience. If not, an empirical formula of the same character as those employed in interpolation is our only resource. But to extend our inference far beyond the limits of experience is exceedingly unsafe. Our knowledge is at the best only approximate, and takes no account of small tendencies. Now it usually happens that tendencies small within our limits of observation become perceptible or great under extreme circumstances. When the variable in our empirical formula is small, we are justified in overlooking the higher powers, and taking only two or three lower powers. But as the variable increases, the higher powers gain in importance, and in time yield the principal part of the value of the function. This is no mere theoretical inference. Excepting the few primary laws of nature, such as the law of gravity, of the conservation of energy, &c., there is hardly any natural law which we can trust in circumstances widely different from those with which we are practically acquainted. From the expansion or contraction, fusion or vaporisation of substances by heat at the surface of the earth, we can form a most imperfect notion of what would happen near the centre of the earth, where the pressure almost infinitely exceeds anything possible in our experiments. The physics of the earth give us a feeble, and probably a misleading, notion of a body like the sun, in which an inconceivably high temperature is united with an inconceivably high pressure. If there are in the realms of space nebulæ consisting of incandescent and unoxidised vapours of metals and other elements, so highly heated perhaps that chemical composition is out of the question, we are hardly able to treat them as subjects of scientific inference. Hence arises the great importance of experiments in which we investigate the properties of substances under extreme circumstances of cold or heat, density or rarity, intense electric excitation, &c. This insecurity in extending our inferences arises from the approximate character of our measurements. Had we the power of appreciating infinitely small quantities, we should by the principle of continuity discover some trace of every change which a substance could undergo under unattainable circumstances. By observing, for instance, the tension of aqueous vapour between o° and 100° C., we ought theoretically to be able to infer its tension at every other temperature; but this is out of the question practically because we cannot really ascertain the law precisely between those temperatures. Many instances might be given to show that laws. which appear to represent correctly the results of experiments within certain limits altogether fail beyond those limits. The experiments of Roscoe and Dittmar, on the absorption of gases in water1 afford interesting illustrations, especially in the case of hydrochloric acid, the quantity of which dissolved in water under different pressures follows very closely a linear law of variation, from which however it diverges widely at low pressures. Herschel, having deduced from observations of the double star y Virginis an elliptic orbit for the motion of one component round the centre of gravity of both, found that for a time the motion of the star agreed very well with this orbit. Nevertheless divergence began to appear and after a time became so great that an entirely new orbit, of more than double the dimensions of the old one, had ultimately to be adopted.3 Illustrations of Empirical Quantitative Laws. Although our object in quantitative inquiry is to discover the exact or rational formulæ, expressing the laws which apply to the subject, it is instructive to observe in how many important branches of science, no precise laws have yet been detected. The tension of aqueous vapour at different temperatures has been determined by a succession of eminent experimentalists-Dalton, Kaëmtz, Dulong, Arago, Magnus, and Regnault-and by the last mentioned the measurements were conducted with extraordinary care. 1 Watts' Dictionary of Chemistry, vol. ii. p. 790. |