IV. THE GEOMETRICAL RELATIONS BETWEEN THE ROD AND THE SECTORS OF THE DISC.

It should seem that any attempt to explain the illusion-bands ought to begin with a consideration of the purely geometrical relations holding between the slowly-moving rod and the swiftly-revolving disc. First of all, then, it is evident that the rod lies in front of each sector successively.

Let Fig. 1 represent the upper portion of a color-wheel, with center at _O_, and with equal sectors _A_ and _B_, in front of which a rod _P_ oscillates to right and left on the same axis as that of the wheel. Let the disc rotate clockwise, and let _P_ be observed in its rightward oscillation. Since the disc moves faster than the rod, the front of the sector _A_ will at some point come up to and pa.s.s behind the rod _P_, say at _p^{A}. P_ now hides a part of _A_ and both are moving in the same direction. Since the disc still moves the faster, the front of _A_ will presently emerge from behind _P_, then more and more of _A_ will emerge, until finally no part of it is hidden by _P_.

If, now, _P_ were merely a line (having no width) and were not moving, the last of _A_ would emerge just where its front edge had gone behind _P_, namely at _p^{A}_. But _P_ has a certain width and a certain rate of motion, so that _A_ will wholly emerge from behind _P_ at some point to the right, say _p^{B}_. How far to the right this will be depends on the speed and width of _A_, and on the speed and width of _P_.

Now, similarly, at _p^{B}_ the sector _B_ has come around and begins to pa.s.s behind _P_. It in turn will emerge at some point to the right, say _p^{C}_. And so the process will continue. From _p^{A}_ to _p^{B}_ the pendulum covers some part of the sector _A_; from _p^{B}_ to _p^{C}_ some part of sector _B_; from _p^{C}_ to _P^{D}_ some part of _A_ again, and so on.

[Ill.u.s.tration: Fig. 1.]

If, now, the eye which watches this process is kept from moving, these relations will be reproduced on the retina. For the retinal area corresponding to the triangle _p^{A}Op^{B}_, there will be less stimulation from the sector _A_ than there would have been if the pendulum had not partly hidden it. That is, the triangle in question will not be seen of the fused color of _A_ and _B_, but will lose a part of its _A_-component. In the same way the triangle _p^{B}OpC_ will lose a part of its _B_-component; and so on alternately. And by as much as either component is lost, by so much will the color of the intercepting pendulum (in this case, black) be present to make up the deficiency.

We see, then, that the purely geometrical relations of disc and pendulum necessarily involve for vision a certain banded appearance of the area which is swept by the pendulum, if the eye is held at rest.

We have now to ask, Are these the bands which we set out to study?

Clearly enough these geometrically inevitable bands can be exactly calculated, and their necessary changes formulated for any given change in the speed or width of _A_, _B_, or _P_. If it can be shown that they must always vary just as the bands we set out to study are _observed_ to vary, it will be certain that the bands of the illusion have no other cause than the interception of retinal stimulation by the sectors of the disc, due to the purely geometrical relations between the sectors and the pendulum which hides them.

And exactly this will be found to be the case. The widths of the bands of the illusion depend on the speed and widths of the sectors and of the pendulum used; the colors and intensities of the bands depend on the colors and intensities of the sectors (and of the pendulum); while the total number of bands seen at one time depends on all these factors.

V. GEOMETRICAL DEDUCTION OF THE BANDS.

In the first place, it is to be noted that if the pendulum proceeds from left to right, for instance, before the disc, that portion of the latter which lies in front of the advancing rod will as yet not have been hidden by it, and will therefore be seen of the unmodified, fused color. Only behind the pendulum, where rotating sectors have been hidden, can the bands appear. And this accords with the first observation (p. 167), that "The rod appears to leave behind it on the disc a number of parallel bands." It is as if the rod, as it pa.s.ses, painted them on the disc.

Clearly the bands are not formed simultaneously, but one after another as the pendulum pa.s.ses through successive positions. And of course the newest bands are those which lie immediately behind the pendulum. It must now be asked, Why, if these bands are produced successively, are they seen simultaneously? To this, Jastrow and Moorehouse have given the answer, "We are dealing with the phenomena of after-images." The bands persist as after-images while new ones are being generated. The very oldest, however, disappear _pari-pa.s.su_ with the generation of the new. We have already seen (p. 169) how well these authors have shown this, in proving that the number of bands seen, multiplied by the rate of rotation of the disc, is a constant bearing some relation to the duration of a retinal image of similar brightness to the bands.

It is to be noted now, however, that as soon as the rod has produced a band and pa.s.sed on, the after-image of that band on the retina is exposed to the same stimulation from the rotating disc as before, that is, is exposed to the fused color; and this would tend to obliterate the after-images. Thus the oldest bands would have to disappear more quickly than an unmolested after-image of the same original brightness. We ought, then, to see somewhat fewer bands than the formula of Jastrow and Moorehouse would indicate. In other words, we should find on applying the formula that the 'duration of the after-image' must be decreased by a small amount before the numerical relations would hold. Since Jastrow and Moorehouse did not determine the relation of the after-image by an independent measurement, their work neither confirms nor refutes this conjecture.

What they failed to emphasize is that the real origin of the bands is not the intermittent appearances of the rod opposite the _lighter_ sector, as they seem to believe, but the successive eclipse by the rod of _each_ sector in turn.

If, in Fig. 2, we have a disc (composed of a green and a red sector) and a pendulum, moving to the right, and if _P_ represents the pendulum at the instant when the green sector _AOB_ is beginning to pa.s.s behind it, it follows that some other position farther to the right, as _P'_, will represent the pendulum just as the last part of the sector is pa.s.sing out from behind it. Some part at least of the sector has been hidden during the entire interval in which the pendulum was pa.s.sing from _P_ to _P'_. Clearly the arc _BA'_ measures the band _BOA'_, in which the green stimulation from the sector _AOB_ is thus at least partially suppressed, that is, on which a relatively red band is being produced. If the illusion really depends on the successive eclipse of the sectors by the pendulum, as has been described, it will be possible to express BA', that is, the width of a band, in terms of the widths and rates of movement of the two sectors and of the pendulum. This expression will be an equation, and from this it will be possible to derive the phenomena which the bands of the illusion actually present as the speeds of disc and rod, and the widths of sectors and rod, are varied.

[Ill.u.s.tration: Fig 2.]

Now in Fig. 2 let the width of the band (_i.e._, the arc BA') = Z speed of pendulum = r degrees per second; speed of disc = r' degrees per second; width of sector AOB (_i.e._, the arc AB) = s degrees of arc; width of pendulum (_i.e._, the arc BC) = p degrees of arc; time in which the pendulum moves from P to P' = t seconds.

Now arc CA'

t = -------; r

but, since in the same time the green sector AOB moves from _B_ to B', we know also that arc BB'

t = -------; r'

then arc CA' arc BB'

------- = -------, r r'

or, omitting the word "arc" and clearing of fractions,

r'(CA') = r(BB').

But now CA' = BA' - BC, while BA' = Z and BC = p; therefore CA' = Z-p.

Similarly BB' = BA' + A'B' = Z + s.

Subst.i.tuting for _CA'_ and _BB'_ their values, we get

r'(Z-p) = r(Z+s), or Z(r' - r) = rs + pr', or Z = rs + pr' / r' - r.

It is to be remembered that _s_ is the width of the sector which undergoes eclipse, and that it is the color of that same sector which is subtracted from the band _Z_ in question. Therefore, whether _Z_ represents a green or a red band, _s_ of the formula must refer to the _oppositely colored_ sector, _i.e._, the one which is at that time being hidden.

We have now to take cognizance of an item thus far neglected. When the green sector has reached the position _A'B'_, that is, is just emerging wholly from behind the pendulum, the front of the red sector must already be in eclipse. The generation of a green band (red sector in eclipse) will have commenced somewhat before the generation of the red band (green sector in eclipse) has ended. For a moment the pendulum will lie over parts of both sectors, and while the red band ends at point _A'_, the green band will have already commenced at a point somewhat to the left (and, indeed, to the left by a trifle more than the width of the pendulum). In other words, the two bands _overlap_.

This area of overlapping may itself be accounted a band, since here the pendulum hides partly red and partly green, and obviously the result for sensation will not be the same as for those areas where red or green alone is hidden. We may call the overlapped area a 'transition-band,' and we must then ask if it corresponds to the 'transition-bands' spoken of in the observations.

Now the formula obtained for Z includes two such transition-bands, one generated in the vicinity of OB and one near OA'. To find the formula for a band produced while the pendulum conceals solely one, the oppositely colored sector (we may call this a 'pure-color' band and let its width = W), we must find the formula for the width (w) of a transition-band, multiply it by two, and subtract the product from the value for Z already found.

The formula for an overlapping or transition-band can be readily found by considering it to be a band formed by the pa.s.sage behind P of a sector whose width is zero. Thus if, in the expression for Z already found, we subst.i.tute zero for s, we shall get w; that is,

o + pr' pr'

w = ------- = ------ r' - r r' - r Since W = Z - 2w, we have rs + pr' pr'

W = -------- = 2 ------, r' - r r' - r or rs - pr'

W = -------- (1) r' - r

[Ill.u.s.tration: Fig 3.]

Fig. 3 shows how to derive _W_ directly (as _Z_ was derived) from the geometrical relations of pendulum and sectors. Let _r, r', s, p_, and _t_, be as before, but now let

width of the band (_i.e._, the arc _BA') = W_;

that is, the band, instead of extending as before from where _P_ begins to hide the green sector to where _P_ ceases to hide the same, is now to extend from the point at which _P_ ceases to hide _any part_ of the red sector to the point where it _just commences_ again to hide the same.

Then W + p t = ------- , r and W + s t = ------- , r'

therefore W + p W + s ------- = ------- , r r'

r'(W + p) = r(W + s) ,

W (r' - r) = rs - pr' , and, again, rs - pr'

W = -------- .

r' - r

Before asking if this pure-color band _W_ can be identified with the bands observed in the illusion, we have to remember that the value which we have found for _W_ is true only if disc and pendulum are moving in the same direction; whereas the illusion-bands are observed indifferently as disc and pendulum move in the same or in opposite directions. Nor is any difference in their width easily observable in the two cases, although it is to be borne in mind that there may be a difference too small to be noticed unless some measuring device is used.

From Fig. 4 we can find the width of a pure-color band (_W_) when pendulum and disc move in opposite directions. The letters are used as in the preceding case, and _W_ will include no transition-band.

[Ill.u.s.tration: Fig. 4]

We have

W + p t = -----, r and s - W t = -----, r'

r'(W + p) = r(s - W) ,