stance, or the monochord) cannot sustain the vibratory impulse at first imparted to it, it breaks off in some of its parts into a doubly quicker but feebler rate of vibration (2 : 1); then to a treble rate (3:1); then to a quadruple (4 1); then to a quintuple (5 : 1); and so on, till the vibrations, or pulsations, growing feebler as they augment in speed, the sonorous body at length relapses into rest and silence.-Such is the series of Harmonics; and if we reject the intermingled octaves from it, which are just the preceding notes on a higher pitch, we shall see that the essential notes progress unalterably in the ratio of 2, 3, 5—the notes produced by which are called respectively the tonic, dominant, and mediant, and which, in union with the key-note, form what is termed the Harmonic notes in music.

Having thus seen how it is that these Harmonic sounds are produced in nature, the next question to be solved is, Why is it that the combinations of these notes give rise to the finest concord? There is no real mystery here, any more than in the former problem; but first let us venture a word or two on a subject which lies at the root of this, as well as of many other inquiries. Why is it that so many leading minds have attached a mighty significance to Numbers? And how is it that this old faith, of which Plato and Pythagoras are the best known exponents, instead of waning with the lapse of two thousand years, is now springing into new life, and assuming a definiteness and wideness of application at which modern inquirers stand amazed, but which are only a realisation of the sublime conceptions of those sages of the Past? The deeper modern Science goes, the more does it come in contact with laws which express themselves by Numbers. The more the human mind, passing by weary stages through an infinity of superficial phenomena, approaches the heart of things-the inner shrine of Nature,Simplicity begins to dawn like a light through the maze, and by-and-by is itself seen to resolve into an embodiment of numbers. Numbers, in truth, lurk like a shaping spirit beneath

almost all the forms and phenomena of Matter. Cast your eyes up to the purple skies of night, and as you watch the stars and planets in their courses, Kepler will show you that their bright wanderings own the sway of numbers, and Newton expound how a numerical proportion marks alike the gravitation of worlds and the fall of an apple. Light and Heat, those life-sustaining and all-pervading forces of the universe, act by similar laws. And in Chemistry, the science of dead matter, which searches into the composition of all material things-air, earth, and water, rocks, plants, metals, and all that is therein-Dalton and his followers will show you that the power of Numbers is omnipotent,-regulating the organism of each, and being, in truth, the grand influence which makes one thing to differ from another. Turn to M'Cosh's elaborate analysis of the Typical Forms of Creation, and in that work, and still more in the profound speculations of Dr Macvicar, we find such astonishing examples of numerical order pervading all departments and shaping the forms of Nature, that we are irresistibly reminded of the saying of Plato, that the Deity proceeds by geometry. The Pythagoreans represented numbers, the significance of which is so clearly seen in music, as in some mysterious sense the principia of the Universe. They described numbers as being in the Divine Mind prior to the exist ence of the worlds, -as being used as a model or pattern (agáderya) in the formation of objects, and as that by which all things were brought together and linked in order.

Let us ask, then, Whence comes this significancy of Numbers-this grand part which they play alike in Nature and in Art? Their agency is marvellous, and has been called mysterious; and in more than one age and country, they have even been supposed to have a magical power. But the mystery connected with these hieroglyphics is not insoluble; and, when once explained, the truth will appear abundantly simple. Consider what Numbers are. Those hieroglyphics, the sole ones pervading modern languages, what do they express? Are they

not simply symbols of Proportion? And is not Proportion that quality by which all things fitly unite and work together, -by which Matter springs into shapes at once useful and comely-which characterises the motions of mind as well as the forms of matter-which in single objects is recognised as Beauty, and in the mass produces that Divine Order which pervades the universe? Numbers, then, are the symbols of Proportion, and proportion is the basis of Order and Beauty. Surely it is no wonder, no mystery, then, that we should find Numbers operating in all parts of a universe of which order and beauty are the prime characteristics,—since numbers are but another word for, an analytical expression of, that order and beauty themselves.

But the mystery is only half resolved. Besides the general significancy of numbers, there is the fact that certain numbers are so peculiarly significant, and exercise so important an influence, as to have acquired the title of "magical." 2, 3, 5, are peculiarly the mystical numbers, but all the numbers up to 10 have more or less claim to a similar title. Thus it is the lowest numbers that are regarded as mystical,-it is the lowest numbers that are most prevalent and powerful in nature. Does not this fact suggest the solution of the mystery? Since numbers are the symbols of proportion and order, the lower the numbers -is it not evident?-the more simple the proportion. And the more simple the proportion, the more readily is it perceived and acknowledged perfect by the mind, producing a feeling of order or beauty,-while, regarded as a cosmical influence, the more easily do the various elementary parts unite and work together in the fabric of creation. These lower or mystical numbers, then, acquire their pre-eminence from symbolising the most simple and perfect proportions. Take the numbers from 1 up to 6, for example, and arrange them as proportions. The simplest of all proportions, of course, is, or, as it may otherwise be expressed, 1: 2; and from this as a middle term, it will be seen, the proportions expressed by the adjoined

E

5:6

4:5

3:4

2:3

-1 : 2

1:3 1:4

figures increase as they ascend to, and diminish as they descend to. The further the proportions recede on either side from 1 2, the less simple becomes the ratio, and the less readily is it apprehended by the mind; while the balance of parts, so grateful to the mind, becomes more and more destroyed, on the one side, by the parts too much approximating, and, on the other, by their too widely diverging. Such approximation and divergence, when extreme, render the proportions not easily appreciable by the mind, and, when appreciated, not relished. Still more, in bad Architecture, for example, we sometimes meet with shapes and sizes which bear no definite ratios to one another at all!

1:5 1:6

Such, in brief, is the reason of the value of those "mystical" numbers in Art. Proceeding with our task, let us see how these numbers present themselves in the elements of Music, and note the manner in which beauty of sound is evolved from their combinations. It is well known that when a monochord, a bell, or other very sonorous body, is set a-vibrating, besides. the strong note first heard, a brief series of others, rapidly rising in pitch and declining in strength, may be heard as the sound dies away. In the sawing asunder of stereotype plates, composed of type-metal, amid all the harshness of the sound, I have often been struck by the peculiar acoustic phenomenon of this kind. There is given forth a fleeting pyramid of sound, which, loudest at its base (producing the fundamental note), in disappearing becomes momentarily vivid at the tapering top; and so rapid and brilliant is the phenomenon, and so analogous are the impressions of the different organs of sense upon the mind, that the effect produced at times is as if the sounds, in flying upwards, vanished like a flash of light. The sound of the word "Ring"-which, beginning with a deep guttural, flies up into the highest nasal sound-is imitative of this phenomenon. Experiment has shown that the vibrations of the several notes of the monochord are in number respectively as 1, 2, 3, 4, 5,

-

&c.-in other words, they stand to each other in the simplest proportions; and thus combine more perfectly than any other possible series of notes, their blending vibrations having the greatest number of consonances in a given time. The leading notes of this Harmonic series, and those alone perceptible to ordinarily good ears, are (after the fundamental note) the octave, with its fifth, and the third of the double octave above,-i. e., the notes which vibrate to the key-note in the ratio of 2, 3, 5,-the very numbers which have been regarded as peculiarly magical, which figure in the "golden rule," which will meet us again in Form, and which certainly produce the finest harmony in Music.

The notes in musical instruments are to some extent an artificial arrangement, adopted for convenience; but they are constructed with a reference to those laws of Numerical Proportion which we have seen to be the basis of harmony. Musical instruments are of all kinds-stringed, wind, or percussion,harp and violin, trumpet and flute, piano and organ,-the only characteristic which they must have in common being the property of producing fine sounds. All sound is the result of vibration; but the vibrations must be somewhat rapid before audible sound is produced. Comparatively few bodies, as they exist in nature, are sonorous, the component atoms of the majority vibrating too sluggishly to produce other than the mere dull rudiments of sound. And it is well that it is so; for otherwise "the air," as Caliban found it, would be ever "full of noises"-though not so sweet as those which haunted the Enchanted Island, and we should be deafened or killed outright by the universal din. At the same time, most bodies may be made resonant,-Art can make even stones "discourse sweet music;" and it is not without an acoustic purpose that the soil of earth, like a vast damper or thick carpet, spreads beneath us to deaden the world's noises, and muffle the echovoices of the underlying rocks. Sonorous bodies, properly so called, are those whose parts easily vibrate synchronously, so as