"Notwithstanding the eminent difficulties of the mathematical theory of sonorous vibrations, we owe to it such progress as has yet been made in acoustics. The formation of the differential equations proper to the phenomena is, independent of their integration, a very important acquisition, on account of the approximations which mathematical analysis allows between questions, otherwise heterogeneous, which lead to similar equations. This fundamental property, whose value we have so often to recognize, applies remarkably in the present case; and especially since the creation of mathematical thermology, whose principal equations are strongly analogous to those of vibratory motion. This means of investigation is all the more valuable on account of the difficulties in the way of direct inquiry into the phenomena of sound. We may decide the necessity of the atmospheric medium for the transmission of sonorous vibrations; and we may conceive of the possibility of determining by experiment the duration of the propagation, in the air, and then through other media; but the general laws of the vibrations of sonorous bodies escape immediate observation. We should know almost nothing of the whole case if the mathematical theory did not come in to connect the different phenomena of sound, enabling us to substitute for direct observation an equivalent examination of more favorable cases subjected to the same law. For instance, when the analysis of the problem of vibrating chords has shown us that, other things being equal, the number of oscillations is hi inverse proportion to the length of the chord, we see that the most rapid vibrations of a very short chord may be counted, since the law enables us to direct our attention to very slow vibrations. The same substitution is at our command in many cases in which it is less direct."

“The positive philosophy of Auguste Comte, freely translated and condensed by Harriet Martineau, in the Cornell University Library Historical Monographs Collection”

“Social positivism only accepts duties, for all and towards all. Its constant social viewpoint cannot include any notion of rights, for such notion always rests on individuality. We are born under a load of obligations of every kind, to our predecessors, to our successors, to our contemporaries. These obligations then increase or accumulate, for it is some time before we can return any service. … Any human right is therefore as absurd as immoral. Since there are no divine rights anymore, this concept must therefore disappear completely as related only to the preliminary regime and totally inconsistent with the final state where there are only duties based on functions.”

“In the final, positive state, the mind has given over the vain search after Abolute notions, the origin and destination of the universe, and the cause of phenomenon, and applies itself to the tudy of their laws, - that is, their invariable relations of succession and resemblance. Reasoning and observation, duly combined, are the means of this knowledge. What is now understood when we speak of an explanation of the facts is simply the establishment of a connection between single phenomena and some general facts, the number of which continually diminishes with the progress of science.”

“The first condition of unity is a subjective principle; and this principle in the Positive system is the subordination of the intellect to the heart: Without this the unity that we seek can never be placed on a permanent basis, whether individually or collectively. It is essential to have some influence sufficiently powerful to produce convergence amid the heterogeneous and often antagonistic tendencies of so complex an organism as ours.”

“The principal means of realizing it will be the formation of an alliance between philosophers and the working classes, for which both are alike prepared by the negative and positive progress of the last five centuries. The direct object of their combined action will be to set in motion the force of Public Opinion.”

“Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena. . . . but if the employment of mathematical analysis should ever become so preponderant in chemistry it would occasion vast and rapid retrogradation....”