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 2 1/2°, and 7 1/2°, 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.12
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 which might have been expected independently of a theoretical investigation.