one second would be sufficient to bring out sound, though its pitch would be so low as to be hardly perceptible. For practical purposes, the lowest tone we hear is produced by 30 double vibrations in one second, the highest by 4000. Between these two lie the usual seven octaves of our musical instruments. It is said to be possible, however, to produce perceptible musical sounds through 11 octaves, beginning with 16 and ending with 38,000 double vibrations in one second, though here the lower notes are mere hums, the upper notes mere clinks. The A' of our tuning-forks, as fixed by the Paris Academy, requires 437-5 double, or 875 single1 vibrations in one second. In Germany the a' tuningfork makes 440 double vibrations in one second. It is clear that beyond the lowest and the highest tones perceptible to our ears, there is a progress ad infinitum, musical notes as real as those which we hear, yet beyond the reach of sensuous perception. It is the same with the other senses. We can perceive the movement of the pendulum, but we cannot perceive the slower movement of the hand on the watch. We can perceive the flight of a bird, but we cannot perceive the quicker movement of a cannon-ball. This, better than anything else, shows how dependent we are on our senses; and how, if our senses are our weapons for the discovery of truth, they are likewise our chains that keep us from soaring too high. Up to this point everything, though wonderful enough, is clear and intelligible.

1 It is customary to reckon by single vibrations in France and Germany, although some German writers adopt the English fashion of reckoning by double vibrations or complete excursions backwards and forwards. Helmholtz uses double vibrations, but Scheibler uses single vibrations. De Morgan calls a double oscillation a "swing swang."

As we hear a note, we know, with mathematical accuracy, to how many vibrations in one second it is due; and if we want to produce the same note, an instrument, such as the siren, which gives a definite number of impulses to the air within a given time, will enable us to do it in the most mechanical manner.

When two waves of one note enter the ear in the same time as one wave of another, the interval between the two is an octave.

When three waves of one note enter the ear in the same time as two waves of another, the interval between the two notes is a fifth.

When four waves of one note enter the ear in the same time as three waves of another, the interval between the two notes is a fourth.

When five waves of one note enter the ear in the same time as four waves of another, the interval between the two notes is a major third.

When six waves of one note enter the ear in the same time as five waves of another, the interval between the two notes is a minor third.

When five waves of one note enter the ear in the same time as three waves of another, the interval between the two notes is a major sixth.

All this is but the confirmation of what was known to Pythagoras. He took a vibrating cord, and, by placing a bridge so as to leave

of the

cord on the right, on the left side, the left portion vibrating by itself, gave him the octave of the lower note of the right portion. So, again, by leaving on the right, on the left side, the left portion vibrating gave him the fifth of the right.

But it is clear that we may hear the same tone, i. e. the result of exactly the same number of vibrations in one second, produced by the human voice, by a flute, a violoncello, a fife, or a double bass. They are tones of the same pitch, and yet they differ in character, and their difference is called their quality. But what is the cause of these various qualities? By a kind of negative reasoning, it had long been supposed that, as quality could neither arise from the amplitude nor from the duration, it must be due to the form of the vibrations. Professor Helmholtz, however, was the first to prove positively that this is the case, by applying the microscope to the vibrations of different musical instruments, and thus catching the exact outline of their respective vibrations, a result which before had been but imperfectly attained by an instrument called the Phonautograph. What is meant by the form of waves may be seen from the following outlines:

In pursuing these inquiries, Professor Helmholtz made another most important discovery, viz., that the different forms of the vibrations which are the cause of what he calls quality or color are likewise the cause of the presence or absence of certain harmonics, or by-notes; in fact, that varying quality and varying harmonics are but two expressions for the same thing.

Harmonics are the secondary tones which can be perceived even by the unassisted ear, if, after lifting the pedal, we strike a key on a pianoforte. These harmonics arise from a string vibrating as if its motion were compounded of several distinct vibrations of strings of its full length, and one half, one third, one fourth, &c., part of its length. Each of these shorter lengths would vibrate twice, three times, four times as fast as the original length, producing corresponding tones. Thus, if we strike c, we hear, if listening attentively, ɗ, Gʻ, c", E", G", B" flat, c", &c.

That the secondary notes are not merely imaginative or subjective can be proved by a very simple and amusing experiment. If we place little soldiers

very light cavalry—on the strings of a pianoforte, and then strike a note, all the riders that sit on strings representing the secondary tones will shake, and possibly be thrown off, while the others remain firm in their saddles, because these strings vibrate in

sympathy with the secondary tones of the string struck. Another test can be applied by means of resounding tubes, tuned to different notes. If we

apply these to our ear, and then strike a note the secondary tones of which are the same as the notes to which the resounding tubes are tuned, those notes will sound loudly and almost yell in our ears; while if the tubes do not correspond to the harmonics of the note played, the resounding tubes will not answer in the same manner.

We thus see, again, that what seems to us a simple impression, the one note struck on the pianoforte, consists of many impressions which together make up what we hear and perceive. We are not conscious of the harmonics which follow each note and determine its quality, but we know, nevertheless, that these by-notes strike our ear, and that our senses receive them and suffer from them. The same remark applies to the whole realm of our sensuous knowledge. There is a broad distinction between sensation and perception. There are many things which we perceive at first and which we perceive again as soon as our attention is called to them, but which, in the ordinary run of life, are to us as if they did not exist at all. When I first came to Oxford, I was constantly distracted by the ringing of bells; after a time I ceased even to notice the dinner-bell. There are ear-rings much in fashion just now — little gold bells with coral clappers. Of course they produce a constant jingling which everybody hears except the lady who wears them. In these cases, however, the difference between sensation and perception is simply due to want of attention. In other cases