TRANSACTIONS OF THE SECTIONS. 3 



the Reports, I do not see that the Association need be put to any expense ; and if 

 it were thought well to sell the Reports independently of the yearly volumes of 

 the Association, probably a good part even of this expense might be recovered. 



The mutual relations subsisting between the two great groups of sciences which 

 we discuss in this Section under the names Mathematics and Physics offer so many 

 deeply interesting points for consideration, that, at the risk of reminding you how 

 admirably aud with what fulness of knowledge the same subject has been treated 

 by more than one of my predecessors in this Chair, I venture to ask your attention 

 once more to a few remarks on this topic. 



The intimate connexion between Mathematics and Physics arises out of the fact 

 that all scientific knowledge of physical phenomena is based upon measurements— 

 that is to say, upon the discovery of relations of number, quantity, and position — of 

 the same kind as those which form the subject-matter of mathematics. It is true 

 that in studying physics we have to learn much about the quality of phenomena 

 and of the conditions under which they occur, as well as about their purely quan- 

 titative relations ; but even in the qualitative study of physical phenomena we 

 find it impossible to determine what is really characteristic and to distinguish the 

 essential from the accidental, except by the aid of measurements. In fact if we 

 take the most elementary treatise upon any branch of physics that we can meet 

 with (a book, it may be, which aims at giving a purely descriptive account of 

 phenomena), we find, when we examine it, that numberless careful measurements 

 have been required to establish the truth of the merely qualitative statements 

 which it contains. To take a simple and well-known example, the old question, 

 whether the ascent of water in a pump was due to the pressure of the atmosphere 

 or to Nature's horror of a vacuum, was not conclusively settled by Torricelli's di- 

 C3very that mercury would not rise beyond a certain height in a glass tube, even to 

 prevent a vacuum being formed at the top of it, for the same thing was already 

 known about the water in a pump. But when he measured the height of the 

 mercury column in his tube and found that, if he multiplied it by the speciiic 

 gravity of mercury, the product was equal to 32 feet, the height to which, as 

 Galileo said (probably between jest and earnest) nature's abhorrence of a vacuum 

 in a pump extended, it was clear that the ascent both of water and of mercury 

 depended upon the particular depth of each liquid that was needed to produce some 

 definite pressure ; and when Pascal had persuaded his brother-in-law to carry a 

 Torricelli's tube to the top of the Puy de Dome, and he had measured the height 

 of the mercury column at the top of the mountain as well as at the foot, the proof 

 was completed that the pressure which determined the height of both the water 

 and the mercury was the pressure of the atmosphere. 



Again, let us examine a still more familiar phenomenon, the falling of heavy 

 bodies to the ground. So long as we consider this merely under its general or, 

 as we may call them, its qualitative aspects, we might reasonably infer that it 

 is the result of some inherent tendency of bodies ; and, so far from its seeming 

 to be true, as stated in Newton's First Law of Motion, that bodies have no power 

 to alter their own condition of rest or of motion, we might infer that, however 

 indifferent they may be as regards horizontal motion, they have a distinct tendency 

 to move downwards whenever they can, and a distinct disinclination to move 

 upwards. But when we measure the direction in which bodies tend to fall [and 

 the amount of the tendency in different places, and find that these vary in the 

 way that they are known to do with geographical position and distance from the 

 sea-level, we are obliged to conclude that there is no inherent tendency to motion 

 at all, but that falling is the result of some mutual action exerted between the 

 earth and the falling body ; for if we suppose falling to be due to any internal cause, 

 we must imagine something much more complicated than a mere tendency to 

 motion in one direction, else how could a stone that has always fallen in one 

 direction in England fall in almost exactly the opposite direction as soon as it is 

 taken to New Zealand ? 



These two simple examples illustrate a principle that we meet with throughout 

 Physics, namely that, in the investigation of the causes of physical phenomena, 

 or, in other words, of the connexion between these phenomena and the conditions 

 under which they occur, the really decisive guidance is afforded by the study of 

 their measurable aspects. 



