pS REPORT—1874. 7 
sciences which are already recognized as belonging to its legitimate province ? 
Are we trying to perfect the mathematical treatment of such sciences as optics or 
electricity, which have been already brought under the sway of mathematics ? 
Are we trying to extend its sway by bringing under it sciences (chemistry, for 
example, or biology) in which as yet its power has been but little felt? Or 
have we come to the conclusion, to which some writers would lead us, that we 
have already pushed the use of mathematics too far? Is it true, for example, and 
do we feel it to be true, that in our anxiety to bring physical optics completely 
under the power of mathematical science, we have abandoned the principles of the 
inductive philosophy, and substituted mere hypotheses for true knowledge? And 
are we convinced, at least, that every chemist is bound, as he values the truth and 
reality of his science, to resist the introduction into chemistry of the methods of 
mathematical analysis, if any such attempt should be made ? 
This latter is the opinion of Comte, whose severe strictures on the application 
of mathematical analysis to physical optics I shall have to consider further on; 
for the present I would confine your attention to the inquiry, What indications on 
this subject are presented by the actual progress of physical science? Does its 
history exhibit a tendency to widen or to contract the field of mathematical 
analysis ? 
ie reviewing, with this purpose, the history of physical science, we may leave 
out of sight those sciences, or parts of a science, to which the methods and language 
of mathematics are applicable without the aid of hypotheses. No scientific man 
doubts the advantage of applying, as far as our analytic powers enable us so to 
do, the methods of mathematical analysis to such sciences as plain optics or plain 
astronomy. Even physical astronomy, although in strict logical precision not 
wholly independent of hypothesis, has been long recognized as, in the most proper 
sense of the word, a mathematical science. Wherever, in fact, the fundamental 
equations rest either on direct observation (as in plain optics) or (as in physical 
astronomy) upon an hypothesis, if we may venture to call it an hypothesis, so 
entirely accepted as universal gravitation, the extension of the methods of mathe- 
matics is only limited by the weakness of mathematical analysis itself. But there 
are other sciences, as, for example, physical optics, to which mathematical analysis 
cannot be applied without the intervention of hypotheses more or less uncertain. 
And if we would appreciate the true character of scientific progress, the question 
which we must put to scientific history is this, Is science becoming more or less 
tolerant of such hypotheses? A principle is assumed, possessing in itself a certain 
amount of plausibility, and capable of mathematical expression, from which we are 
able to deduce, as consequences and by mathematical reasoning, phenomena whose 
reality may afterwards be proved by direct experiment. And from this experi- 
mental verification we infer, with more or less probability, the truth of the original 
assumption. The question, then, which we have to put to scientific history is 
this, Do the records of science indicate a greater or a less tolerance of this kind 
of logic? Is the mode of physical investigation which I have shortly sketched 
gaining or losing the favour of scientific men ? 
Passing over sciences like astronomy, which, though not wholly free from 
hypothesis, do not give us very extended information on this point, I come to a 
part of scientific history to which we may put the question with every probability 
of obtaining (so far, at least, as one science is concerned) a decisive answer—I mean, 
the history ef physical optics. 
We have here a science whose basis is purely hypothetical. The definition of 
light is an hypothesis, the nature of the vethereal motion is an hypothesis, even 
the very existence of the ether is an hypothesis—hypotheses, indeed, which have 
led to conclusions amply verified by experiment, but hypotheses still. Does the 
history of optical science indicate a desire to discard this hypothetical base ? 
Does the history of this science betray a tendency on the part of scientific men to 
abandon or neglect mechanical theories of light? Have physicists given up as 
hopeless, or perhaps unphilosophical, the attempt to reduce, by the intervention of 
a supposed ether, the phenomena of light under the mathematical laws which 
govern motion? Are they even abandoning the reasoning or the phraseology of 
the undulatory system? The answer to these questions is not doubtful. Com- 
