sible to trace a line among the points which will approximate to the true law more nearly than the points themselves. The accompanying figure sufficiently explains Perkins employed this graphical method with much care in exhibiting the results of his experiments on the compression of waterk. The numerical results were marked upon a sheet of paper very exactly ruled at intervals of one-tenth of an inch, and the original marks were left in order that the reader might judge of the correctness of the curve drawn, or choose another for himself. Regnault carried the method to perfection by laying off the points with a small screw dividing engine1; and he then formed a table of results by drawing a continuous curve, and measuring its height for equidistant values of the variable. Not only does a curve drawn in this manner enable us to assign by measurement numerical results more free from accidental errors than any of the numbers obtained directly from experiment, but the form of the curve sometimes indicates the class of functions to which our results belong k 'Philosophical Transactions,' 1826, p. 544. Engraved sheets of paper ready prepared for the drawing of curves may be obtained from Mr. Stanford, at 6 and 7 Charing Cross, or from Messrs. W. and A. K. Johnston, of London and Edinburgh. When we do not require great accuracy, paper ruled by the common machine-ruler into equal squares of about one-fifth or onesixth of an inch square will serve well enough. I have found Vere Foster's Exercise Book, No. 12 m, which is ruled in this way, very useful for statistical or other numerical purposes. I have also met with engineers' and surveyors'. memorandum books ruled with one-twelfth inch squares. When a number of complicated curves have to be drawn, I have found it best to rule a good sheet of drawing paper with lines carefully adjusted at the most convenient distances, and then to prick the points of the curve through it upon another sheet fixed underneath. In this way we obtain an accurate curve upon a blank sheet, and need only introduce such division lines as are requisite to the understanding of the curve. In some cases our numerical results will correspond, not to the height of single ordinates, but to the area of the curve between two ordinates, or the average height of ordinates between certain limits. If we measure, for instance, the quantities of heat absorbed by water when raised in temperature from o° to 5°, from 5° to 10°, and so on, these quantities will really be represented by areas of the curve denoting the specific heat of water; and, since the specific heat varies continuously between every two points of temperature, we shall not get the correct curve by simply laying off the quantities of heat at the mean temperatures, namely 24, 7, and so on. Mr. J. W. Strutt has shown that if we have drawn such an incorrect curve, we can with little trouble correct it by a simple m Published by Whittaker & Co., London. geometrical process, and obtain to a very close approximation the true ordinates instead of those denoting areas". Interpolation and Extrapolation. When we have by experiment obtained two or more numerical results, and endeavour, without further resort to experiment, to infer and 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 to a certain extent the same in principle, but differ in practicability. It is a matter of great scientific importance to apprehend precisely how far we can interpolate or extend experimental results by extrapolation, and on what grounds we proceed. In the first place, if the interpolation is to be more than empirical and speculative, 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 fair position to anticipate the results of any 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 more or less affect the result. Even 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 n 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. results will be as sure and accurate as any we can attain to. But such a complete procedure is more than we generally mean by interpolation, which generally denotes the employment of some general method of estimating in a merely approximate and probable manner the results which might have been expected independently of any complete 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 always invent an infinite number of functions which would give those values if we are not restricted by any other conditions, just as through a given series of points we can always draw an infinite number of curves, if we may diverge between or beyond the points into bends and cusps as we think fit. In any process of 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, and we must be guided accordingly in the choice of mathematical formulæ. Supposing, for the present, that the phenomenon is nonperiodic, we next 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 formula, we can 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 formula; and interpolation, as commonly understood, consists in assum Herschel, Appendix to Translation of Lacroix' Differential Calculus,' P. 551. ing 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 must 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 readily 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 simple equations gives y=45x+3'5 as the equation, and for any other value of x, for instance 10, we get a value of y, 48'5. When we take an exactly intermediate value of x, namely 9, this process yields a simple mean result, namely 44. Three experimental results being given, we may assume that they fall upon a portion of a parabola, and simple algebraic calculation readily 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 Kæmtz' Meteorology P. When we have, either directly by experiment or by the use of a curve, a series of values of the variant for exactly equidistant values of the variable, it is often very 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-1 powers of the variable. Indeed we may sometimes obtain by the Cal P Cours complet de Météorologie,' traduit par Martins, Note A, du Traducteur, p. 449. |