Chapter IX: Section II: showed that the shorter filled distances are (5)

Any form of color-wheel may be used, but preferably one which is driven by electricity or clock-work, so that a fairly constant speed is assured. Several pairs of paper discs are needed, of the ordinary interpenetrating kind which permit a ready readjustment of the ratios between the two sectors, as follows: one pair consisting of a white and a black disc, one of a light-and a dark-colored disc (light green and dark red have been found admirably suited to the purpose), and a pair of discs distinctly different in color, but equal in luminosity.

The rod should be black and not more than a quarter of an inch broad. It may be passed before the rotating disc by hand. For the sake of more careful study, however, the rod should be moved at a constant rate by some mechanical device, such as the pendulum and works of a Maelzel metronome removed from their case. The pendulum is fixed just in front of the color-disc. A further commendable simplification of the conditions consists in arranging the pendulum and disc to move concentrically, and attaching to the pendulum an isosceles-triangular shield, so cut that it forms a true radial sector of the disc behind it. All the colored bands of the illusion then appear as radial sectors. The radial shields should be made in several sizes (from 3 to 50 degrees of arc) in black, but the smallest size should also be prepared in colors matching the several discs. Such a disposition, then, presents a disc of fused color, rotating at a uniform rate, and in front of this a radial sector oscillating from side to side concentrically with the disc, and likewise at a uniform rate. Several variations of this apparatus will be described as the need and purpose of them become clear.

II. PREVIOUS DISCUSSION OF THE ILLUSION.

Although Jastrow and Moorehouse (_op. cit._) have published a somewhat detailed study of these illusion-bands, and cleared up certain points, they have not explained them. Indeed, no explanation of the bands has as yet been given. The authors mentioned (_ibid._, p. 204) write of producing the illusion by another method. "This consists in sliding two half discs of the same color over one another leaving an open sector of any desired size up to 180 degrees and rotating this against a background of a markedly different color, in other words we substitute for the disc composed of a large amount of one color, which for brevity we may call the 'majority color,' and a small amount of another, the 'minority color,' one in which the second color is in the background and is viewed through an opening in the first. With such an arrangement we find that we get the series of bands both when the wire is passed in front of the disc and when passed in back between disc and background; and further experimentation shows that the time relations of the two are the same. (There is, of course, no essential difference between the two methods when the wire is passed in front of the disc.)" That is true, but it is to be borne in mind that there is a difference when the wire is passed behind the disc, as these authors themselves state (_loc. cit._, note):--"The time-relations in the two cases are the same, but the _color-phenomena_ considerably _different_." However, "these facts enable us to formulate our first generalization, viz., that for all purposes here relevant [_i.e._, to a study of the _time-relations_] the seeing of a wire now against one background and then immediately against another is the same as its now appearing and then disappearing; a rapid succession of changed appearances is equivalent to a rapid alternation of appearance and disappearance. Why this is so we are unable to say," etc. These authors now take the first step toward explaining the illusion. In their words (_op. cit._, p. 205), "the suggestion is natural that we are dealing with the phenomena of after-images.... If this is the true explanation of the fact that several rods are seen, then we should, with different rotation rates of disc and rod, see as many rods as multiplied by the time of one rotation of the disc would yield a constant, _i.e._, the time of an after image of the kind under consideration." For two subjects, J.J. and G.M., the following tabulation was made.

J.J. G.M.
Av. time of rot. of disc when 2 images of rod were seen .0812 sec. .0750 sec.
" " " " 3 " " " " .0571 " .0505 "
" " " " 4 " " " " .0450 " .0357 "
" " " " 5 " " " " .0350 " .0293 "
" " " " 6 " " " " .0302 " .0262 "

"Multiplying the number of rods by the rotation rate we get for J.J. an average time of after image of .1740 sec. (a little over 1/6 sec.) with an average deviation of .0057 (3.2%); for G.M. .1492 (a little over 1/7 sec.) with an average deviation of .0036 (2.6%). An independent test of the time of after-image of J.J. and G.M. by observing when a black dot on a rotating white disc just failed to form a ring resulted in showing in every instance a longer time for the former than for the latter." That this constant product of the number of 'rods' seen by the time of one rotation of the disc equals the duration of after-image of the rod is established, then, only by inference. More indubitable, since directly measured on two subjects, is the statement that that person will see more 'rods' whose after-image persists longer. This result the present writer fully confirms. What relation the 'constant product' bears to the duration of after-image will be spoken of later. But aside from all measurement, a little consideration of the conditions obtaining when the rod is passed _behind_ the disc will convince any observer that the bands are indeed after-images somehow dependent on the rod. We may account it established that _the bands are after-images_.

From this beginning one might have expected to find in the paper of Jastrow and Moorehouse a complete explanation of the illusion. On other points, however, these authors are less explicit. The changes in width of the bands corresponding to different sizes of the sectors and different rates of movement for the rod and disc, are not explained, nor yet, what is more important, the color-phenomena. In particular the fact needs to be explained, that the moving rod analyzes the apparently homogeneous color of the disc; or, as Jastrow and Moorehouse state it (_op. cit._, p. 202): "If two rotating discs were presented to us, the one pure white in color, and the other of ideally perfect spectral colors in proper proportion, so as to give a precisely similar white, we could not distinguish between the two; but by simply passing a rod in front of them and observing in the one case but not in the other the parallel rows of colored bands, we could at once pronounce the former to be composite, and the latter simple. In the indefinitely brief moment during which the rod interrupts the vision of the disc, the eye obtains an impression sufficient to analyze to some extent into its elements this rapid mixture of stimuli." The very question is as to _how_ the eye obtains the 'impression sufficient to analyze' the mixture.

It may be shown at this point that the mistake of these authors lies in their recognition of but one set of bands, namely (_ibid._, p. 201), 'bands of a color similar to that present in greater proportion' on the disc. But, on the other hand, it is to be emphasized that those bands are separated from one another, not by the fused color of the disc, as one should infer from the article, but by _other bands_, which are, for their part, of a color similar to that present in _lesser_ proportion. Thus, bands of the two colors alternate; and either color of band is with equal ease to be distinguished from the fused color of the main portion of the disc.

Why our authors make this mistake is also clear. They first studied the illusion with the smaller sector of the disc open, and the rod moving behind it; and since in this case the bands are separated by strips not of the minority but of the fused color, and are of about the width of the rod itself, these authors came to recognize bands of but one sort, and to call these 'images of the rod.' But now, with the rod moving in front of the disc, there appear bands of two colors alternately disposed, and neither of these colors is the fused color of the disc. Rather are these two colors approximately the majority and minority colors of the disc as seen at rest. Thus, the recognition of but one set of bands and the conclusion (_ibid._, p. 208) that 'the bands originate during the vision of the minority color,' are wholly erroneous. The bands originate as well during the vision of the majority color, and, as will later be shown, the process is continuous.

Again, it is incorrect, even in the case of those bands seen behind the open sector, to call the bands 'images of the rod,' for images of the rod would be of the color of the rod, whereas, as our authors themselves say (_ibid._, p. 201), the bands 'are of a color similar to that present in greater proportion' on the disc. Moreover the 'images of the rod' are of the most diverse widths. In fact, we shall find that the width of the rod is but one of several factors which determine the width of its 'images,' the bands.

Prejudiced by the same error is the following statement (_ibid._, p. 208): "With the majority color darker than the minority color the bands are darker than the resulting mixture, and lighter when the majority color is the lighter." If this is to be true, one must read for 'the bands,' 'the narrower bands.'

Another observation found in this article must be criticised. It is asserted that difference of shade between the two sectors of the disc, as well as difference of color, is essential to the illusion. To support this, four cases are given: two in which the sectors were so similar in luminosity as to bring out the illusion but faintly; two in which like luminosities yielded no illusion at all. The present writer agrees that if the two sectors are closely similar in luminosity, the illusion is fainter. He also selected a red and a green so near each other in brightness that when a rod 4 mm. broad (which is the largest rod that Jastrow and Moorehouse mention having used) was passed by hand before the disc, no trace of a band could be seen. The pendulum, however, bearing a shield considerably wider than 4 mm. (say of 15 degrees) and moving before the very same red and green shades, mixed in the same proportions, yielded the illusion with the utmost clearness. Colors of like luminosities yield the illusion less strikingly, nevertheless they yield it.

Again (_op. cit._, p. 205), these authors say: "It has been already observed that the distance between the bands diminishes as the rotation rate and the rate of movement of the rod increases." But what had been said before is (_ibid._, p. 203) that 'the bands are separated by smaller and smaller spaces as the rate of movement of the rod becomes slower and slower'; and this is equivalent to saying that the distance between the bands diminishes as the rate of movement of the rod decreases. The statements are contradictory. But there is no doubt as to which is the wrong one--it is the first. What these authors have called 'distance between the bands' has here been shown to be itself a band. Now, no point about this illusion can be more readily observed than that the widths of both kinds of band vary directly with the speed of the rod, inversely, however (as Jastrow and Moorehouse have noted), with the speed of the disc.

Perhaps least satisfactory of all is their statement (_ibid._, p. 206) that "A brief acquaintance with the illusion sufficed to convince us that its appearance was due to contrast of some form, though the precise nature of this contrast is the most difficult point of all." The present discussion undertakes to explain with considerable minuteness every factor of the illusion, yet the writer does not see how in any essential sense contrast could be said to be involved.

With the other observations of these authors, as that the general effect of an increase in the width of the interrupting rod was to render the illusion less distinct and the bands wider, etc., the observations of the present writer fully coincide. These will systematically be given later, and we may now drop the discussion of this paper.

The only other mention to be found of these resolution-bands is one by Sanford,[2] who says, apparently merely reiterating the results of Jastrow and Moorehouse, that the illusion is probably produced by the sudden appearance, by contrast, of the rod as the lighter sector passes behind it, and by its relative disappearance as the dark sector comes behind. He thus compares the appearance of several rods to the appearance of several dots in intermittent illumination of the strobic wheel. If this were the correct explanation, the bands could not be seen when both sectors were equal in luminosity; for if both were dark, the rod could never appear, and if both were light, it could never disappear. The bands can, however, be seen, as was stated above, when both the sectors are light or both are dark. Furthermore, this explanation would make the bands to be of the same color as the rod. But they are of other colors. Therefore Sanford's explanation cannot be admitted.

[2] Sanford, E.C.: 'A Course in Experimental Psychology,'
Boston, 1898, Part I., p. 167.

And finally, the suggestions toward explanation, whether of Sanford, or of Jastrow and Moorehouse, are once for all disproved by the observation that if the moving rod is fairly broad (say three quarters of an inch) and moves _slowly_, the bands are seen nowhere so well as _on the rod itself_. One sees the rod vaguely through the bands, as could scarcely happen if the bands were images of the rod, or contrast-effects of the rod against the sectors.

The case when the rod is broad and moves slowly is to be accounted a special case. The following observations, up to No. 8, were made with a narrow rod about five degrees in width (narrower will do), moved by a metronome at less than sixty beats per minute.

III. OUTLINE OF THE FACTS OBSERVED.

A careful study of the illusion yields the following points:

1. If the two sectors of the disc are unequal in arc, the bands are unequal in width, and the narrower bands correspond in color to the larger sector. Equal sectors give equally broad bands.

2. The faster the rod moves, the broader become the bands, but not in like proportions; broad bands widen relatively more than narrow ones; equal bands widen equally. As the bands widen out it necessarily follows that the alternate bands come to be farther apart.

3. The width of the bands increases if the speed of the revolving disc decreases, but varies directly, as was before noted, with the speed of the pendulating rod.

4. Adjacent bands are not sharply separated from each other, the transition from one color to the other being gradual. The sharpest definition is obtained when the rod is very narrow. It is appropriate to name the regions where one band shades over into the next 'transition-bands.' These transition-bands, then, partake of the colors of both the sectors on the disc. It is extremely difficult to distinguish in observation between vagueness of the illusion due to feebleness in the after-image depending on faint illumination, dark-colored discs or lack of the desirable difference in luminosity between the sectors (cf. p. 171) and the indefiniteness which is due to broad transition-bands existing between the (relatively) pure-color bands. Thus much, however, seems certain (Jastrow and Moorehouse have reported the same, _op. cit._, p. 203): the wider the rod, the wider the transition-bands. It is to be noticed, moreover, that, for rather swift movements of the rod, the bands are more sharply defined if this movement is contrary to that of the disc than if it is in like direction with that of the disc. That is, the transition-bands are broader when rod and disc move in the same, than when in opposite directions.

5. The total number of bands seen (the two colors being alternately arranged and with transition-bands between) at any one time is approximately constant, howsoever the widths of the sectors and the width and rate of the rod may vary. But the number of bands is inversely proportional, as Jastrow and Moorehouse have shown (see above, p. 169), to the time of rotation of the disc; that is, the faster the disc, the more bands. Wherefore, if the bands are broad (No. 2), they extend over a large part of the disc; but if narrow, they cover only a small strip lying immediately behind the rod.

6. The colors of the bands approximate those of the two sectors; the transition-bands present the adjacent 'pure colors' merging into each other. But _all_ the bands are modified in favor of the color of the moving rod. If, now, the rod is itself the same in color as one of the sectors, the bands which should have been of the _other_ color are not to be distinguished from the fused color of the disc when no rod moves before it.

7. The bands are more strikingly visible when the two sectors differ considerably in luminosity. But Jastrow's observation, that a difference in luminosity is _necessary_, could not be confirmed. Rather, on the contrary, sectors of the closest obtainable luminosity still yielded the illusion, although faintly.

8. A _broad_ but slowly moving rod shows the bands overlying itself. Other bands can be seen left behind it on the disc.

9. But a case of a rod which is broad, or slowly-moving, or both, is a special complication which involves several other and _seemingly_ quite contradictory phenomena to those already noted. Since these suffice to show the principles by which the illusion is to be explained, enumeration of the special variations is deferred.

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 pass 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 pass 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.

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 passes, painted them on the disc.

Clearly the bands are not formed simultaneously, but one after another as the pendulum passes 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-passu_ 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 passed 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 pass 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 passing out from behind it. Some part at least of the sector has been hidden during the entire interval in which the pendulum was passing 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.

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.

Substituting 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 passage behind P of a sector whose width is zero. Thus if, in the expression for Z already found, we substitute 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

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.

We have

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

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

W(r' + r) = rs - pr' ,

rs - pr'
W = -------- . (2)
r' + r

Now when pendulum and disc move in the same direction,

rs - pr'
W = --------- , (1)
r' - r

so that to include both cases we may say that

rs - pr'
W = -------- . (3)
r' ± r

The width (W) of the transition-bands can be found, similarly, from the geometrical relations between pendulum and disc, as shown in Figs. 5 and 6. In Fig. 5 rod and disc are moving in the same direction, and

w = BB'.

Now
W - p
t = ------- ,
r'

w
t = --- ,
r'

r'(w-p) = rw ,

w(r'-r) = pr' ,

pr'
w = ------- . (4)
r'-r

In Fig. 6 rod and disc are moving in opposite directions, and

w = BB',

p - w
t = ------- ,
r

w
t = --- ,
r'

r'(p - w) = rw ,

w(r' + r) = pr' ,

pr'
w = -------- .
r' + r (5)

So that to include both cases (of movement in the same or in opposite directions), we have that

pr'
w = -------- .
r' ± r (6)

VI. APPLICATION OF THE FORMULAS TO THE BANDS OF THE ILLUSION.

Will these formulas, now, explain the phenomena which the bands of the illusion actually present in respect to their width?

1. The first phenomenon noticed (p. 173, No. 1) is that "If the two sectors of the disc are unequal in arc, the bands are unequal in width; and the narrower bands correspond in color to the larger sector. Equal sectors give equally broad bands."

In formula 3, _W_ represents the width of a band, and _s_ the width of the _oppositely colored_ sector. Therefore, if a disc is composed, for example, of a red and a green sector, then

rs(green) - pr'
W(red) = ------------------ ,
r' ± r
and
rs(red) - pr'
W(green) = ------------------ ,
r' ± r

therefore, by dividing,

W(red) rs(green) - pr'
--------- = ------------------- .
W(green) rs(red) - pr'

From this last equation it is clear that unless _s_(green) = _s_(red), _W_(red) cannot equal _W_(green). That is, if the two sectors are unequal in width, the bands are also unequal. This was the first feature of the illusion above noted.

Again, if one sector is larger, the oppositely colored bands will be larger, that is, the light-colored bands will be narrower; or, in other words, 'the narrower bands correspond in color to the larger sector.'

Finally, if the sectors are equal, the bands must also be equal.

So far, then, the bands geometrically deduced present the same variations as the bands observed in the illusion.

2. Secondly (p. 174, No. 2), "The faster the rod moves the broader become the bands, but not in like proportions; broad bands widen relatively more than narrow ones." The speed of the rod or pendulum, in degrees per second, equals _r_. Now if _W_ increases when _r_ increases, _D_{[tau]}W_ must be positive or greater than zero for all values of _r_ which lie in question.

Now
rs - pr'
W = --------- ,
r' ± r
and
(r' ± r)s [±] (rs - pr')
D_{[tau]}W = -------------------------- ,
(r ± r')

or reduced,
r'(s ± p)
= -----------
(r' ± r)²

Since _r'_ (the speed of the disc) is always positive, and _s_ is always greater than _p_ (cf. p. 173), and since the denominator is a square and therefore positive, it follows that

D_{[tau]}W > 0

or that _W_ increases if _r_ increases.

Furthermore, if _W_ is a wide band, _s_ is the wider sector. The rate of increase of _W_ as _r_ increases is

r'(s ± p)
D_{[tau]}W = -----------
(r' ± r)²

which is larger if _s_ is larger (_s_ and _r_ being always positive). That is, as _r_ increases, 'broad bands widen relatively more than narrow ones.'

3. Thirdly (p. 174, No. 3), "The width of The bands increases if the speed of the revolving disc decreases." This speed is _r'_. That the observed fact is equally true of the geometrical bands is clear from inspection, since in

rs - pr'
W = --------- ,
r' ± r

as _r'_ decreases, the denominator of the right-hand member decreases while the numerator increases.

4. We now come to the transition-bands, where one color shades over into the other. It was observed (p. 174, No. 4) that, "These partake of the colors of both the sectors on the disc. The wider the rod the wider the transition-bands."

We have already seen (p. 180) that at intervals the pendulum conceals a portion of both the sectors, so that at those points the color of the band will be found not by deducting either color alone from the fused color, but by deducting a small amount of both colors in definite proportions. The locus of the positions where both colors are to be thus deducted we have provisionally called (in the geometrical section) 'transition-bands.' Just as for pure-color bands, this locus is a radial sector, and we have found its width to be (formula 6, p. 184) pr' W = --------- , r' ± r

Now, are these bands of bi-color deduction identical with the transition-bands observed in the illusion? Since the total concealing capacity of the pendulum for any given speed is fixed, less of _either_ color can be deducted for a transition-band than is deducted of one color for a pure-color band. Therefore, a transition-band will never be so different from the original fusion-color as will either 'pure-color' band; that is, compared with the pure color-bands, the transition-bands will 'partake of the colors of both the sectors on the disc.' Since pr' W = --------- , r' ± r

it is clear that an increase of _p_ will give an increase of _w_; _i.e._, 'the wider the rod, the wider the transition-bands.'

Since _r_ is the rate of the rod and is always less than _r'_, the more rapidly the rod moves, the wider will be the transition-bands when rod and disc move in the same direction, that is, when

pr'
W = --------- ,
r' - r

But the contrary will be true when they move in opposite directions, for then

pr'
W = --------- ,
r' + r

that is, the larger _r_ is, the narrower is _w_.

The present writer could not be sure whether or not the width of transition-bands varied with _r_. He did observe, however (page 174) that 'the transition-bands are broader when rod and disc move in the same, than when in opposite directions.' This will be true likewise for the geometrical bands, for, whatever _r_ (up to and including _r_ = _r'_),

pr' pr'
---- > ----
r'-r r'+r

In the observation, of course, _r_, the rate of the rod, was never so large as _r'_, the rate of the disc.

5. We next come to an observation (p. 174, No. 5) concerning the number of bands seen at any one time. The 'geometrical deduction of the bands,' it is remembered, was concerned solely with the amount of color which was to be deducted from the fused color of the disc. _W_ and _w_ represented the widths of the areas whereon such deduction was to be made. In observation 5 we come on new considerations, _i.e._, as to the color from which the deduction is to be made, and the fate of the momentarily hidden area which suffers deduction, _after_ the pendulum has passed on.

We shall best consider these matters in terms of a concept of which Marbe[3] has made admirable use: the 'characteristic effect.' The Talbot-Plateau law states that when two or more periodically alternating stimulations are given to the retina, there is a certain minimal rate of alternation required to produce a just constant sensation. This minimal speed of succession is called the critical period. Now, Marbe calls the effect on the retina of a light-stimulation which lasts for the unit of time, the 'photo-chemical unit-effect.' And he says (_op. cit._, S. 387): "If we call the unit of time 1[sigma], the sensation for each point on the retina in each unit of time is a function of the simultaneous and the few immediately preceding unit-effects; this is the characteristic effect."

[3] 'Marbe, K.: 'Die stroboskopischen Erscheinungen,' _Phil.
Studien._, 1898, XIV., S. 376.

We may now think of the illusion-bands as being so and so many different 'characteristic effects' given simultaneously in so and so many contiguous positions on the retina. But so also may we think of the geometrical interception-bands, and for these we can deduce a number of further properties. So far the observed illusion-bands and the interception-bands have been found identical, that is, in so far as their widths under various conditions are concerned. We have now to see if they present further points of identity.

As to the characteristic effects incident to the interception-bands; in Fig. 7 (Plate V.), let _A'C'_ represent at a given moment _M_, the total circumference of a color-disc, _A'B'_ represent a green sector of 90°, and _B'C'_ a red complementary sector of 270°. If the disc is supposed to rotate from left to right, it is clear that a moment previous to _M_ the two sectors and their intersection _B_ will have occupied a position slightly to the left. If distance perpendicularly above _A'C'_ is conceived to represent time previous to _M_, the corresponding previous positions of the sectors will be represented by the oblique bands of the figure. The narrow bands (_GG_, _GG_) are the loci of the successive positions of the green sector; the broader bands (_RR_, _RR_), of the red sector.

In the figure, 0.25 mm. vertically = the unit of time = 1[sigma]. The successive stimulations given to the retina by the disc _A'C'_, say at a point _A'_, during the interval preceding the moment _M_ will be

green 10[sigma],
red 30[sigma],
green 10[sigma],
red 30[sigma], etc.

Now a certain number of these stimulations which immediately precede _M_ will determine the characteristic effect, the fusion color, for the point _A'_ at the moment _M_. We do not know the number of unit-stimulations which contribute to this characteristic effect, nor do we need to, but it will be a constant, and can be represented by a distance _x_ = _A'A_ above the line _A'C'_. Then _A'A_ will represent the total stimulus which determines the characteristic effect at _A'_. Stimuli earlier than _A_ are no longer represented in the after-image. _AC_ is parallel to _A'C'_, and the characteristic effect for any point is found by drawing the perpendicular at that point between the two lines _A'C_ and _AC_.

Just as the movement of the disc, so can that of the concealing pendulum be represented. The only difference is that the pendulum is narrower, and moves more slowly. The slower rate is represented by a steeper locus-band, _PP'_, than those of the swifter sectors.

We are now able to consider geometrically deduced bands as 'characteristic effects,' and we have a graphic representation of the color-deduction determined by the interception of the pendulum. The deduction-value of the pendulum is the distance (_xy_) which it intercepts on a line drawn perpendicular to _A'C'_.

Lines drawn perpendicular to _A'C'_ through the points of intersection of the locus-band of the pendulum with those of the sectors will give a 'plot' on _A'C'_ of the deduction-bands. Thus from 1 to 2 the deduction is red and the band green; from 2 to 3 the deduction is decreasingly red and increasingly green, a transition-band; from 3 to 4 the deduction is green and the band red; and so forth.

We are now prepared to continue our identification of these geometrical interception-bands with the bands observed in the illusion. It is to be noted in passing that this graphic representation of the interception-bands as characteristic effects (Fig. 7) is in every way consistent with the previous equational treatment of the same bands. A little consideration of the figure will show that variations of the widths and rates of sectors and pendulum will modify the widths of the bands exactly as has been shown in the equations.

The observation next at hand (p. 174, No. 5) is that "The total number of bands seen at any one time is approximately constant, howsoever the widths of the sectors and the width and rate of the rod may vary. But the number of bands is inversely proportional (Jastrow and Moorehouse) to the time of rotation of the disc; that is, the faster the disc, the more bands."

This is true, point for point, of the interception-bands of Fig. 7. It is clear that the number of bands depends on the number of intersections of _PP'_ with the several locus-bands _RR_, _GG_, _RR_, etc. Since the two sectors are complementary, having a constant sum of 360°, their relative widths will not affect the number of such intersections. Nor yet will the width of the rod _P_ affect it. As to the speed of _P_, if the locus-bands are parallel to the line _A'C'_, that is, of the disc moved _infinitely_ rapidly, there would be the same number of intersections, no matter what the rate of _P_, that is, whatever the obliqueness of _PP'_. But although the disc does not rotate with infinite speed, it is still true that for a considerable range of values for the speed of the pendulum the number of intersections is constant. The observations of Jastrow and Moorehouse were probably made within such a range of values of _r_. For while their disc varied in speed from 12 to 33 revolutions per second, that is, 4,320 to 11,880 degrees per second, the rod was merely passed to and fro by hand through an excursion of six inches (J. and M., _op. cit._, pp. 203-5), a method which could have given no speed of the rod comparable to that of the disc. Indeed, their fastest speed for the rod, to calculate from certain of their data, was less than 19 inches per second.

The present writer used about the same rates, except that for the disc no rate below 24 revolutions per second was employed. This is about the rate which v. Helmholtz[4] gives as the slowest which will yield fusion from a bi-sectored disc in good illumination. It is hard to imagine how, amid the confusing flicker of a disc revolving but 12 times in the second, Jastrow succeeded in taking any reliable observations at all of the bands. Now if, in Fig. 8 (Plate V.), 0.25 mm. on the base-line equals one degree, and in the vertical direction equals 1[sigma], the locus-bands of the sectors (here equal to each other in width), make such an angle with _A'C'_ as represents the disc to be rotating exactly 36 times in a second. It will be seen that the speed of the rod may vary from that shown by the locus _P'P_ to that shown by _P'A_; and the speeds represented are respectively 68.96 and 1,482.64 degrees per second; and throughout this range of speeds the locus-band of _P_ intercepts the loci of the sectors always the same number of times. Thus, if the disc revolves 36 times a second, the pendulum may move anywhere from 69 to 1,483 degrees per second without changing the number of bands seen at a time.

[4] v. Helmholtz, H.: 'Handbuch d. physiolog. Optik,' Hamburg
u. Leipzig, 1896, S. 489.

And from the figure it will be seen that this is true whether the pendulum moves in the same direction as the disc, or in the opposite direction. This range of speed is far greater than the concentrically swinging metronome of the present writer would give. The rate of Jastrow's rod, of 19 inches per second, cannot of course be exactly translated into degrees, but it probably did not exceed the limit of 1,483. Therefore, although beyond certain wide limits the rate of the pendulum will change the total number of deduction-bands seen, yet the observations were, in all probability (and those of the present writer, surely), taken within the aforesaid limits. So that as the observations have it, "The total number of bands seen at any one time is approximately constant, howsoever ... the rate of the rod may vary." On this score, also, the illusion-bands and the deduction-bands present no differences.

But outside of this range it can indeed be _observed_ that the number of bands does vary with the rate of the rod. If this rate (_r_) is increased beyond the limits of the previous observations, it will approach the rate of the disc (_r'_). Let us increase _r_ until _r_ = _r'_. To observe the resulting bands, we have but to attach the rod or pendulum to the front of the disc and let both rotate together. No bands are seen, _i.e._, the number of bands has become zero. And this, of course, is just what should have been expected from a consideration of the deduction-bands in Fig. 8.

One other point in regard to the total number of bands seen: it was observed (page 174, No. 5) that, "The faster the disc, the more bands." This too would hold of the deduction-bands, for the faster the disc and sectors move, the narrower and more nearly parallel to _A'C'_ (Fig. 7) will be their locus-bands, and the more of these bands will be contained within the vertical distance _A'A_ (or _C'C_), which, it is remembered, represents the age of the oldest after-image which still contributes to the characteristic effect. _PP'_ will therefore intercept more loci of sectors, and more deduction-bands will be generated.

6. "The colors of the bands (page 175, No. 6) approximate those of the two sectors; the transition-bands present the adjacent 'pure colors' merging into each other. But _all_ the bands are modified in favor of the moving rod. If, now, the rod is itself the same in color as one of the sectors, the bands which should have been of the other color are not to be distinguished from the fused color of the disc when no rod moves before it."

These items are equally true of the deduction-bands, since a deduction of a part of one of the components from a fused color must leave an approximation to the other component. And clearly, too, by as much as either color is deducted, by so much must the color of the pendulum itself be added. So that, if the pendulum is like one of the sectors in color, whenever that sector is hidden the deduction for concealment will exactly equal the added allowance for the color of the pendulum, and there will be no bands of the other color distinguishable from the fused color of the disc.

It is clear from Fig. 7 why a transition-band shades gradually from one pure-color band over into the other. Let us consider the transition-band 2-3 (Fig. 7). Next it on the right is a green band, on the left a red. Now at the right-hand edge of the transition-band it is seen that the deduction is mostly red and very little green, a ratio which changes toward the left to one of mostly green and very little red. Thus, next to the red band the transition-band will be mostly red, and it will shade continuously over into green on the side adjacent to the green band.

7. The next observation given (page 175, No. 7) was that, "The bands are more strikingly visible when the two sectors differ considerably in luminosity." This is to be expected, since the greater the contrast, whether in regard to color, saturation, or intensity, between the sectors, the greater will be such contrast between the two deductions, and hence the greater will it be between the resulting bands. And, therefore, the bands will be more strikingly distinguishable from each other, that is, 'visible.'

8. "A _broad_ but slowly-moving rod shows the bands lying over itself. Other bands can also be seen behind it on the disc."

In Fig. 9 (Plate V.) are shown the characteristic effects produced by a broad and slowly-moving rod. Suppose it to be black. It can be so broad and move so slowly that for a space the characteristic effect is largely black (Fig. 9 on both sides of _x_). Specially will this be true between _x_ and _y_, for here, while the pendulum contributes no _more_ photo-chemical unit-effects, it will contribute the newer one, and howsoever many unit-effects go to make up the characteristic effect, the newer units are undoubtedly the more potent elements in determining this effect. The old units have partly faded. One may say that the newest units are 'weighted.'

Black will predominate, then, on both sides of _x_, but specially between _x_ and _y_. For a space, then, the characteristic effect will contain enough black to yield a 'perception of the rod.' The width of this region depends on the width and speed of the rod, but in Fig. 9 it will be roughly coincident with _xy_, though somewhat behind (to the left of) it. The characteristic will be either wholly black, as just at _x_, or else largely black with the yet contributory after-images (shown in the triangle _aby_). Some bands will thus be seen overlying the rod (1-8), and others lying back of it (9-16).