To the use of the experimental method, physics can often join that of mathematical analysis. But in the employment of the latter it must be extremely cautious, and we must only have recourse to this application of mathematics after having “carefully considered the reality of the starting point,” which alone can guarantee the solidity of the deductions. In a word, the spirit proper to physical investigation, must constantly direct the use of this powerful instrument. Now, this condition has not always been fulfilled. Too often the preponderance of mathematical analysis has been the cause of the neglect of experimental studies. Not only has mathematical analysis in this way retarded the progress of physics but it has even tended to vitiate the conception of that science, and to bring it back to a state of obscurity and uncertainty which, says Comte, notwithstanding the apparent severity of the forms differs little, at bottom, from its old metaphysical state.[136] For this reason, the application of analysis to physics must not be left to geometers who are chiefly concerned with the instrument. It must belong to the physicists who before all things consider the use to be made of it. Mathematicians have often encumbered physics with a quantity of analytical labour founded upon very doubtful hypothesis; they must give way to physicists trained in experimental studies, and, nevertheless, with sufficient knowledge of mathematics to make use of the analysis whenever it is possible. Within these limits mathematical analysis will render the greatest service to the science of physics. Would optics, acoustics, the theories of heat and of electricity have reached the point where we see them to-day without the powerful help of analysis? Yet even here, physical researches are almost always so complex that, in order to assume a mathematical form, they demand the setting aside of a more or less essential portion of the conditions of the problem. Indeed we are here in presence of the general problem of the translation of the concrete into the abstract. This problem, which is admirably solved in mathematics, and sufficiently in astronomy, is only imperfectly solved in physics. The art of closely combining experience and analysis, says Comte, is still almost unknown. It constitutes the final progress of the method proper to the deeper study of physics.[137] We may add, and this is in Comte’s mind, that conversely the progress made by this art would be useful to analysis itself. II. Astronomy has reached a perfect state of “positivity.” All trace of the metaphysical spirit has disappeared from it. Can we say as much of physics? It would not seem so, when we see the hypotheses which play so great a part in this science, and of which a few are keenly contested by Comte.