The formation of the mathematical sciences which rest upon the measures of the intensity of sensible qualities took place mainly in the course of the last century. Perhaps we may consider Lambert, a mathematician who resided in Switzerland, and published about 1750, as the person who first clearly felt the importance of establishing such sciences. His Photometry, Pyrometry, and Hygrometry, are examples of the systematic reduction of sensible qualities (light, heat, moisture) to modes of numerical measurement. {336} We now proceed to speak of such modes of measurement with regard to the most obvious properties of bodies. SECT. II.--_The Musical Scale._ 3. THE establishment of the _Harmonic Canon_, that is, of a Scale and Measure of the musical place of notes, in the relation of _high_ and _low_, was the first step in the science of Harmonics. The perception of the differences and relations of musical sounds is the office of the sense of hearing; but these relations are fixed, and rendered accurately recognizable by artificial means. 'Indeed, in all the senses,' as Ptolemy truly says in the opening of his Harmonics, 'the sense discovers what is approximately true, and receives accuracy from another quarter: the reason receives the approximately-true from another quarter, and discovers the accurate truth.' We can have no measures of sensible qualities which do not ultimately refer to the sense;--whether they do this immediately, as when we refer Colours to an assumed Standard; or mediately, as when we measure Heat by Expansion, having previously found by an appeal to sense that the expansion increases with the heat. Such relations of sensible qualities cannot be described in words, and can only be apprehended by their appropriate faculty. The faculty by which the relations of sounds are apprehended is a _musical ear_ in the largest acceptation of the term. In this signification the faculty is nearly universal among men; for all persons have musical ears sufficiently delicate to understand and to imitate the modulations corresponding to various emotions in speaking; which modulations depend upon the succession of acuter and graver tones. These are the relations now spoken of, and these are plainly perceived by persons who have very imperfect musical ears, according to the common use of the phrase. But the relations of tones which occur in speaking are somewhat indefinite; and in forming that musical scale which is the basis of our science upon the subject, we {337} take the most definite and marked of such relations of notes; such as occur, not in speaking but in singing. Those musical relations of two sounds which we call the _octave_, the _fifth_, the _fourth_, the _third_, are recognized after a short familiarity with them. These _chords_ or _intervals_ are perceived to have each a peculiar character, which separates them from the relations of two sounds taken at random, and makes it easy to know them when sung or played