In all this there is no steady adherence either to one notion, or to one class of facts. The distinction of power and act is introduced to modify the idea of transparency, according to the formula of the school; then colour is made to be something unknown in addition to visibility; and the distinction of proper and improper colours is assumed, as sufficient to account for a phenomenon. Such classifications have in them nothing of which the mind can take steady hold; nor is it difficult to see that they do not come under those conditions of successful physical speculation, which we have laid down.

CHAPTER III.

EARLIEST STAGES OF HARMONICS.

AMONG the ancients, the science of Music was an application of arithmetic, as optics and mechanics were of geometry. The story which is told concerning the origin of their arithmetical music, is the following, as it stands in the Arithmetical Treatise of Nicomachus.

Pythagoras, walking one day, meditating on the means of measuring musical notes, happened to pass near a blacksmith's shop, and had his attention arrested by hearing the hammers, as they struck the anvil, produce sounds which had a musical relation to each other. On listening further, he found that the intervals were a fourth, a fifth, and an octave; and on weighing the hammers, it appeared that the one which gave the octave was one-half the heaviest, the one which gave the fifth was two-thirds, and the one which gave the fourth was three-quarters. He returned home, reflected upon this phenomenon, and finally discovered, that if he stretched musical strings of equal length, by weights which have the same proportion as those above described, they also produced the intervals above mentioned. This observation gave an arithmetical measure of the principal

musical intervals, and made music an arithmetical subject of speculation.

This story, if not entirely a philosophical fable, is undoubtedly inaccurate; for the musical intervals thus spoken of, would not be produced by striking with hammers of the weights there stated. But the experiment of the strings is perfectly correct, and is to this day the groundwork of the theory of musical concords and discords.

It may at first appear that the truth, or even the possibility of this history, by referring the discovery to accident, disproves our doctrine, that this, like all other fundamental discoveries, required a distinct and well-pondered idea as its condition. In this, however, as in all cases of supposed accidental discoveries in science, it will be found, that it was exactly the possession of such an idea which made the accident possible.

Pythagoras, assuming the truth of the tradition, must have had an exact and ready apprehension of those relations of musical sounds, which are called respectively an octave, a fifth, and a fourth. If he had not been able to conceive distinctly this relation, the sounds of the anvil would have struck his ears to no more purpose than they did those of the smiths themselves. He must have had, too, a ready familiarity with numerical ratios; and, moreover, (that in which, probably, his superiority most consisted,) a disposition to connect one notion with the other the musical relation with the arithmetical,

if it were found possible. When the connexion was once suggested, it was easy to devise experiments by which it might be confirmed.

"The philosophers of the Pythagorean school', and in particular, Lasus of Hermione, and Hippasus of Metapontum, made many such experiments upon strings; varying both their lengths and the weights which stretched them; and also upon vessels filled with water, in a greater or less degree." And thus was established that connexion of the idea with the fact, which this science, like all others, requires.

I shall quit the Physical Sciences of Ancient Greece, with the above brief statement of the discovery of the fundamental principles which they involved; not only because such initial steps must always be the most important in the progress of science, but because, in reality, the Greeks made no advances beyond these. There took place among them no additional inductive processes, by which new facts were brought under the dominion of principles, or by which principles were presented in a more comprehensive shape than before. Their advance terminated in a single stride. Archimedes had stirred the intellectual world, but had not put it in progressive motion: the science of mechanics stopped

where he left it. And though, in some subjects, as in Harmonics, much was written, the works thus produced consisted of deductions from the fundamental principles, by means of arithmetical calculations; occasionally modified, indeed, by reference to the pleasures which music, as an art, affords, but not enriched by any new scientific truths.