Innumerable are the illustrations which may be given of errors of sensory judgment, but none are more striking than the various figures which may be drawn with converging or diverging lines. The mind under-estimates acute and over-estimates obtuse angles. It is impossible to convince oneself that in [Fig. 36] the line A bisects a symmetrical arch. Equally difficult is it to believe that in [Fig. 37] the line with diverging terminal segments and the line with converging terminal segments are of exactly equal length. In the Ruskin Museum at Sheffield there is a sketch by the master of the façade of a church which shows a vertical tower to one side of a triangular pediment, or, rather, this is what the sketch was meant to show, and does show, when measured on an architect’s table. In effect the tower appears to be leaning towards the pediment. Errors of judgment of this type have been attributed to the curvature of the lines of a rectilinear image on the retina, the mind judging the distance between two points by the length of the chord, and not the length of the arc which joins them. This is very simply illustrated by the example of the apparently greater length of a filled space than of a vacant one.

A B looks longer than B C. If A B C be represented as a curved line, the arc A B will, of course, be longer than the chord B C. But it is not safe to suppose that the mind compares the length of an arc with the length of a chord. Judgment is based upon experience, and probably the illusion is due to more subtle causes than the curvature of the retina. The mind does not look at the retina. If it did, it would find the reversal of the picture the least of the inaccuracies which it had to correct. It would find it very difficult, for example, to superpose in its stereoscope the photographs of a vertical tower taken simultaneously by the right eye and the left. The curved images on the retina of the vertical lines which define the angles of the tower, as seen with one eye, could not be made to correspond with the images focussed by the other eye. The Greeks felt this when they settled the form of a column. The canon of the swelling entasis and increasing taper above it did not destroy the appearance of uniform thickness which the shaft presented. It gave to the eye just the slight help which it needs to enable it to picture the shaft as of the same thickness from base to capital.

CHAPTER XIV
HEARING

The ear, like the eye, records amplitude of vibration; loudness. It also records rapidity of vibration, musical pitch, which corresponds with colour. But it seems to have a more difficult task than the eye, since it has to analyse, or at any rate has to transmit information regarding the form of compound vibrations. The meanings of these distinctions may be illustrated by reference to a tracing on the cylinder of a phonograph. A needle attached to the posterior surface of the thin metal plate against which one speaks scratches the surface of a rotating cylinder of hardened wax. Examined with a lens, the record is seen to be an irregularly changing line. The depth of the marks is a measure of loudness. Their varying number in a given time indicates the changing pitch of the voice which produced them. Their form is a record of the quality of its tone. The work of the ear, so far as it consists in the estimation of the amplitude and rapidity of pulsations of sound, is easy to describe, but the acoustics of form are complicated.

Light is transmitted as vibrations of æther. They are transverse to the direction in which the light is travelling. Sound cannot travel through a vacuum, since it is dependent upon displacements of material particles. The particles move forwards and backwards in the direction in which sound is progressing. Sound is a sequence of pulsations, alternate condensations and rarefactions of the media which conduct it. Their particles are first pressed together, and then rebound to positions farther apart. A sequence of to-and-fro movements, each smoothly continuous throughout the whole duration of a pulsation, would produce a pure musical tone. Tuning-forks carefully bowed settle down after a few seconds into unbroken oscillations, which convey to the air the to-and-fro movements of pure tones. Such tones vary in nothing but loudness and pitch. If their pulsations are slow, we speak of the pitch as “low”; if they are rapid, we say that their pitch is high. But if the sound produced by tuning-forks (and low-toned stopped organ-pipes) be omitted from the list, no pure tones reach our ears. The notes of flutes, fiddles, trumpets, pianos, have each a certain “quality” characteristic of the instrument. Even in a violin the G string has not the same timbre as the D string. Owing to the elasticity of the substances which originate and of the substances which transmit sound, its pulsations are not simple to-and-fro movements, uninterrupted from beginning to end. Each pulsation is partially broken at intervals; and the quality of the sound depends upon the number and relative accentuation of these partial interruptions. Sound travels through air at the rate of 1,100 feet per second. This figure, divided by the number of vibrations per second of a tone, gives the wave-length in air of a tone of that particular pitch. For example, the middle C has a vibratory rate of 256. Its wave-length is, therefore, somewhat over 4 feet. The lowest tone of an organ has a wave-length of 37 feet; its highest of 3½ inches. These figures give no information, however, regarding the movement of the particles which pass on the sound. When air is transmitting a note—say the middle C—its separate molecules do not move through a distance of 4 feet. Each molecule moves but a short distance, varying with the loudness of the tone; but the “wave” of crowding runs straight forward from the piano-string to the ear, the molecules at the end of each stage of 4 feet taking on a backward movement, so that the crowding, so far as the molecules of that particular section are concerned, returns to its starting-point. Between the piano-string and the ear there is a crowding and forward movement at 0, 4, 8, 12 ... feet; a spreading and backward movement at 2, 6, 10, 14 ... feet. Most illustrations which are intended to aid the mind in forming a definite picture of the transmission of sound are liable to be misinterpreted, because they translate rectilinear movements into waves. They represent the movements of the string, and not the movements of the molecules of air between the string and the ear; but with the aid of the imagination one may picture the positions of the particles in this path. The pulse, we will suppose, has just reached the limit of 12 feet. Half-way from its 8-foot halting place the molecules are again crowded, although not so densely. One-third of the distance from the same point there again appears a tendency to crowd. This latter point marks an interval of one-third of this wave plus the wave which led up to it. At the end of the ninth foot there is a crowding, though less marked—this wave plus the two preceding waves, divided into fourths. Within these intervals are other points at which the molecules have closed together, the distances from a nodal point depending upon the number of waves involved, and, speaking generally, growing less marked as the number increases. Such are the very complex pulsatile movements which reach the ear.

Every musical sound produced by a piano, a violin, or other instrument, is compounded of a fundamental or prime tone, and overtones, partial tones, or harmonics. The following table shows the more important partial tones which accompany the prime tone when the middle C on a pianoforte is struck:

Note. Number of
Vibrations. 		Interval. Ratio. Number of
Overtone.
C‴2,048 7th
Super-Second8/7
B″♭1,792 6th
Sub-minor third7/6
G″1,536 5th
Minor third6/5
E″1,280 4th
Major third5/4
C″1,024 3rd
Fourth4/3
G′768 2nd
Fifth3/2
C′512 1st
Octave2/1
C256 Fundamental