478 THE POPULAR SCIENCE MONTHLY. 



observations within the limits of temperature from which we have drawn 

 our observations could equally be so satisfied. Having once ascertained 

 that such a formula is possible, it is a mere affair of arithmetic to find 

 the values of a, b, and c, which will make the formula satisfy the ob- 

 servations best. This is what physicists call an empirical formula, 

 because it rests upon mere induction, and is not explained by any 

 hypothesis. 



Such formulae, though very useful as means of describing in general 

 terms the results of observations, do not take any high rank among 

 scientific discoveries. The induction which they embody, that expan- 

 sion by heat (or whatever other phenomenon is referred to) takes 

 place in a perfectly gradual manner without sudden leaps or innumera- 

 ble fluctuations, although really important, attracts no attention, because 

 it is what we naturally anticipate. But the defects of such expressions 

 are very serious. In the first place, as long as the observations are 

 subject to error, as all observations are, the formula cannot be expected 

 to satisfy the observations exactly. But the discrepancies cannot be 

 due solely to the errors of the observations, but must be partly owing 

 to the error of the formula which has been deduced from erroneous 

 observations. Moreover, we have no right to suppose that the real 

 facts, if they could be had free from error, could be expressed by such 

 a formula at all. They might, perhaps, be expressed by a similar for- 

 mula with an infinite number of terms ; but of what use would that be 

 to us, since it would require an infinite number of coefficients to be 

 written down? When one quantity varies with another, if the corre- 

 sponding values are exactly known, it is a mere matter of mathematical 

 ingenuity to find some way of expressing their relation in a simple 

 manner. If one quantity is of one kind say, a specific gravity and 

 the other of another kind say, a temperature we do not desire to 

 find an expression for their relation which is wholly free from numerical 

 constants, since if it were free from them when, say, specific gravity as 

 compared with water, and temperature as expressed by the centigrade 

 thermometer, were in question, numbers would have to be introduced 

 when the scales of measurement were changed. We may, however, 

 and do desire to find formulas expressing the relations of physical 

 phenomena which shall contain no more arbitrary numbers than changes 

 in the scales of measurement might require. 



When a formula of this kind is discovered, it is no longer called an 

 empirical formula, but a law of Nature ; and is sooner or later made the 

 basis of an hypothesis which is to explain it. These simple formulae are 

 not usually, if ever, exactly true, but they are none the less important 

 for that ; and the great triumph of the hypothesis comes when it ex- 

 plains not only the formula, but also the deviations from the formula. 

 In the current language of the physicists, an hypothesis of this impor- 

 tance is called a theory, while the term hypothesis is restricted to sug- 

 gestions which have little evidence in their favor. There is some justice 



