SECTION 16. DEFINITE, EXACT, AND MATHEMATICAL PROCEDURE. 125 



panies the attraction ; it might be replied that these phenomena are them- 

 selves subject to exact laws, and that when all the laws have been taken 

 into account, the actual motion will exactly correspond with the calculated 

 motion. . . . 



"The word 'exact '-has a practical and a theoretical meaning. When a 

 grocer weighs you out a certain quantity of sugar very carefully and says 

 it is exactly a pound, he means that the difference between the mass of 

 the sugar and that of the pound weight he employs is too small to be 

 detected by his scales. If a chemist had made a special investigation, 

 wishing to be as accurate as he could, and told you this was exactly a 

 pound of sugar, he would mean that the mass of the sugar differed from 

 that of a certain standard piece of platinum by a quantity too small to 

 be detected by his means of weighing, which are a thousandfold more 

 accurate than the grocer's. But what would a mathematician mean, if he 

 made the same statement? He would mean this. Suppose the mass of 

 the standard pound to be represented by a length, say a foot, measured 

 on a certain line ; so that half a pound would be represented by six inches, 

 and so on. And let the difference between the mass of the sugar and 

 that of the standard pound be drawn upon the same line to the same 

 scale. Then, if that difference were magnified an infinite number of times, 

 it would still be invisible. This is the theoretical meaning of exactness; 

 the practical meaning is only very close approximation; how close, de- 

 pends upon the circumstances. The knowledge, then, of an exact law in 

 the theoretical sense would be equivalent to an infinite observation. I do 

 not say that such knowledge is impossible to man, but I do say that it 

 would be absolutely different in kind from any knowledge that we possess 

 at present." (Op. cit., pp. 27-29 ) 



The argument in favour of the preferability of mathematical 

 procedure in science is therefore complete. As Lord Kelvin 

 says: "In physical science a first essential step in the direction 

 of learning any subject is to find principles of numerical reckoning 

 and practicable methods for measuring some quality connected 

 with it. I often say that when you can measure what you are 

 speaking about, and express it in numbers, you know something 

 about it; but when you cannot measure it, when you cannot 

 express it in numbers, your knowledge is of a meager and 

 unsatisfactory kind: it may be the beginning of knowledge, 

 but you have scarcely, in your thought, advanced to the stage of 

 science, whatever the matter may be." (Constitution of Matter, 

 1891, pp. 80-81.) Yet much maybe accomplished without recon- 

 dite mathematical formulae. We have it on Jevons' authority 

 that Faraday "has made the most extensive additions to human 

 knowledge without passing beyond common arithmetic". (Prin- 

 ciples of Science, p. 579.) And Tyndall, who concurs in Jevons' 

 estimate concerning Faraday's lack of mathematical equipment, 

 says of him: "Taking him for all in all, I think it will be 

 concede^ that Michael Faraday was the greatest experimental 

 philosopher the world has ever seen." (Faraday as a Discoverer, 

 p. 147.) It should not be supposed, therefore, that every distin- 

 guished discoverer is ipso facto a great mathematician, or, to 

 consider the reverse side of the shield, that there is no extensive 

 scope for measurement and computation in the ordinary practical 

 affairs of life. 



