STUDIES IN PERCEPTION. — часть 7

Fig 2.

Now in Fig. 2 let the

Now

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

then

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

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,

Since

W = Z - 2w,

we have

or

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

and

therefore

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

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

and, again,

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.

Fig. 4.

We have

and

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

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

Now when pendulum and disc move in the same direction,

so that to include both cases we may say that

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

r'(w-p) = rw,

w (r'-r) = pr',

Fig. 5.

Fig. 6.

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

w = BB',

r'(p - w) = rw ,

w(r' + r) = pr' ,

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

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

and

therefore, by dividing,

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_τW must be positive or greater than zero for all values of r which lie in question.

Now

and

or reduced,

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_τW > 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

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

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)

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

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

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

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'),

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 Marbe3 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 1σ, 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."

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 = 1σ. 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

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."

Fig. 7.

Fig. 8.

Fig. 9.

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. Helmholtz4 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 1σ, 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.

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).

We have now reviewed all the phenomena so far enumerated of the illusion-bands, and for every case we have identified these bands with the bands which must be generated on the retina by the mere concealment of the rotating sectors by the moving rod. It has been more feasible thus far to treat these deduction-bands as if possibly they were other than the bands of the illusion; for although the former must certainly appear on the retina, yet it was not clear that the illusion-bands did not involve additional and complicated retinal or central processes. The showing that the two sets of bands have in every case identical properties, shows that they are themselves identical. The illusion-bands are thus explained to be due merely to the successive concealment of the sectors of the disc as each passes in turn behind the moving pendulum. The only physiological phenomena involved in this explanation have been the persistence as after-images of retinal stimulations, and the summation of these persisting images into characteristic effects—both familiar phenomena.

From this point on it is permissible to simplify the point of view by accounting the deduction-bands and the bands of the illusion fully identified, and by referring to them under either name indifferently. Figs. 1 to 9, then, are diagrams of the bands which we actually observe on the rotating disc. We have next briefly to consider a few special complications produced by a greater breadth or slower movement of the rod, or by both together. These conditions are called 'complicating' not arbitrarily, but because in fact they yield the bands in confusing form. If the rod is broad, the bands appear to overlap; and if the rod moves back and forth, at first rapidly but with decreasing speed, periods of mere confusion occur which defy description; but the bands of the minor color may be broader or may be narrower than those of the other color.

VII. FURTHER COMPLICATIONS OF THE ILLUSION.

9. If the rod is broad and moves slowly, the narrower bands are like colored, not with the broader, as before, but with the narrower sector.

The conditions are shown in Fig. 9. From 1 to 2 the deduction is increasingly green, and yet the remainder of the characteristic effect is also mostly green at 1, decreasingly so to the right, and at 2 is preponderantly red; and so on to 8; while a like consideration necessitates bands from x to 16. All the bands are in a sense transition-bands, but 1-2 will be mostly green, 2-3 mostly red, and so forth. Clearly the widths of the bands will be here proportional to the widths of the like-colored sectors, and not as before to the oppositely colored.

It may reasonably be objected that there should be here no bands at all, since the same considerations would give an increasingly red band from B' to A', whereas by hypothesis the disc rotates so fast as to give an entirely uniform color. It is true that when the characteristic effect is A' A entire, the fusion-color is so well established as to assimilate a fresh stimulus of either of the component colors, without itself being modified. But on the area from 1 to 16 the case is different, for here the fusion-color is less well established, a part of the essential colored units having been replaced by black, the color of the rod; and black is no stimulation. So that the same increment of component color, before ineffective, is now able to modify the enfeebled fusion-color.

Observation confirms this interpretation, in that band y-1 is not red, but merely the fusion-color slightly darkened by an increment of black. Furthermore, if the rod is broad and slow in motion, but white instead of black, no bands can be seen overlying the rod. For here the small successive increments which would otherwise produce the bands 1-2, 2-3, etc., have no effect on the remainder of the fusion-color plus the relatively intense increment of white.

It may be said here that the bands 1-2, 2-3, etc., are less intense than the bands x-9, 9-10, etc., because there the recent or weighted unit-effects are black, while here they are the respective colors. Also the bands grow dimmer from x-9 to 15-16, that is, as they become older, for the small increment of one color which would give band 15-16 is almost wholly overridden by the larger and fresher mass of stimulation which makes for mere fusion. This last is true of the bands always, whatever the rate or width of the rod.

10. In general, equal sectors give equal bands, but if one sector is considerably more intense than the other, the bands of the brighter color will, for a broad and swiftly-moving rod, be the broader. The brighter sector, though equal in width to the other, contributes more toward determining the fusion-color; and this fact is represented by an intrusion of the stronger color into the transition-bands, at the expense of the weaker. For in these, even the decreased amount of the stronger color, on the side next a strong-color band, is yet more potent than the increased amount of the feebler color. In order to observe this fact one must have the rod broad, so as to give a broad transition-band on which the encroachment of the stronger color may be evident. The process is the same with a narrow rod and narrow transition-bands, but, being more limited in extent, it is less easily observed. The rod must also move rapidly, for otherwise the bands overlap and become obscure, as will be seen in the next paragraph.

11. If the disc consists of a broad and narrow sector, and if the rod is broad and moves at first rapidly but more slowly with each new stroke, there are seen at first broad, faint bands of the minority-color, and narrow bands of the majority-color. The former grow continuously more intense as the rod moves more slowly, and grow narrower in width down to zero; whereupon the other bands seem to overlap, the overlapped part being doubly deep in color, while the non-overlapped part has come to be more nearly the color of the minor sector. The overlapped portion grows in width. As the rate of the rod now further decreases, a confused state ensues which cannot be described. When, finally, the rod is moving very slowly, the phenomena described above in paragraph 9 occur.

The successive changes in appearance as the rod moves more and more slowly, are due to the factors previously mentioned, and to one other which follows necessarily from the given conditions but has not yet been considered. This is the last new principle in the illusion which we shall have to take up. Just as the transition-bands are regions where two pure-color bands overlap, so, when the rod is broad and moves slowly, other overlappings occur to produce more complicated arrangements.

These can be more compactly shown by diagram than by words. Fig. 10, a, b and c (Plate VI.), show successively slower speeds of the rod, while all the other factors are the same. In practice the tendency is to perceive the transition-bands as parts of the broad faint band of the minor color, which lies between them. It can be seen, then, how the narrow major-color bands grow only slightly wider (Fig. 10, a, b) until they overlap (c); how the broad minor-color bands grow very narrow and more intense in color, there being always more of the major color deducted (in b they are reduced exactly to zero, z, z, z). In c the major-color bands overlap (o, o, o) to give a narrow but doubly intense major-color band since, although with one major, two minor locus-bands are deducted. The other bands also overlap to give complicated combinations between the o-bands. These mixed bands will be, in part at least, minor-color bands (q, q, q), since, although a minor locus-band is here deducted, yet nearly two major locus-bands are also taken, leaving the minor color to predominate. This corresponds with the observation above, that, ' ... the non-overlapped part has come to be more nearly the color of the minor sector.'

A slightly slower speed of the rod would give an irreducible confusion of bands, since the order in which they overlap becomes very complicated. Finally, when the rod comes to move very slowly, as in Fig. 9, the appearance suffers no further change, except for a gradual narrowing of all the bands, up to the moment when the rod comes to rest.

It is clear that this last principle adduced, of the multiple overlapping of bands when the rod is broad and moves slowly, can give for varying speeds of the rod the greatest variety of combinations of the bands. Among these is to be included that of no bands at all, as will be understood from Fig. 11 (Plate VII). And in fact, a little practice will enable the observer so to adjust the rate of the (broad) rod to that of the disc that no bands are observable. But care must be taken here that the eye is rigidly fixated and not attracted into movement by the rod, since of course if the eye moves with the rod, no bands can be seen, whatever the rate of movement may be.

Thus, all the phenomena of these illusion-bands have been explained as the result solely of the hiding by the rod of successive sectors of the disc. The only physiological principles involved are those (1) of the duration of after-images, and (2) of their summation into a characteristic effect. It may have seemed to the reader tedious and unnecessary so minutely to study the bands, especially the details last mentioned; yet it was necessary to show how all the possible observable phenomena arise from the purely geometrical fact that sectors are successively hidden. Otherwise the assertions of previous students of the illusion, that more intricate physiological processes are involved, could not have been refuted. The present writer does not assert that no processes like contrast, induction, etc., come into play to modify somewhat the saturation, etc., of the colors in the bands. It must be here as in every other case of vision. But it is now demonstrated that these remoter physiological processes contribute nothing essential to the illusion. For these could be dispensed with and the illusion would still remain.

Fig. 10.

If any reader still suspects that more is involved than the persistence after-images, and their summation into a characteristic effect, he will find it interesting to study the illusion with a camera. The 'physiological' functions referred to belong as well to the dry-plate as to the retina, while the former exhibits, presumably, neither contrast nor induction. The illusion-bands can be easily photographed in a strong light, if white and black sectors are used in place of colored ones. It is best to arrange the other variable factors so as to make the transition-bands as narrow as possible (p. 174, No. 4). The writer has two negatives which show the bands very well, although so delicately that it is not feasible to try to reproduce them.

VIII. SOME CONVENIENT DEVICES FOR EXHIBITING THE ILLUSION.

The influence of the width of sector is prettily shown by a special disc like that shown in Fig. 12 (Plate VII.), where the colors are dark-red and light-green, the shaded being the darker sector. A narrow rod passed before such a disc by hand at a moderate rate will give over the outer ring equally wide green and red bands; but on the inner rings the red bands grow narrower, the green broader.

The fact that the bands are not 'images of the rod' can be shown by another disc (Fig. 13, Plate VII.). In all three rings the lighter (green) sector is 60° wide, but disposed on the disc as shown. The bands are broken into zigzags. The parts over the outer ring lag behind those over the middle, and these behind those over the inner ring—'behind,' that is, farther behind the rod.

Another effective variation is to use rods alike in color with one or the other of the sectors. Here it is clear that when the rod hides the oppositely-colored sector, the deduction of that color is replaced (not by black, as happens if the rod is black) but by the very color which is already characteristic of that band. But when the rod hides the sector of its own color, the deduction is replaced by the very same color. Thus, bands like colored with the rod gain in depth of tone, while the other pure-color bands present simply the fusion-color.

IX. A STROBOSCOPE WHICH DEPENDS ON THE SAME PRINCIPLE.

If one produce the illusion by using for rod, not the pendulum of a metronome, but a black cardboard sector on a second color-mixer placed in front of the first and rotating concentrically with it, that is, with the color-disc, one will observe with the higher speeds of the rod which are now obtainable several further phenomena, all of which follow simply from the geometrical relations of disc and rod (now a rotating sector), as discussed above. The color-mixer in front, which bears the sector (let it still be called a 'rod'), should rotate by hand and independently of the disc behind, whose two sectors are to give the bands. The sectors of the disc should now be equal, and the rod needs to be broader than before, say 50° or 60°, since it is to revolve very rapidly.

First, let the rod and disc rotate in the same direction, the disc at its former rate, while the rod begins slowly and moves faster and faster. At first there is a confused appearance of vague, radial shadows shuffling to and fro. This is because the rod is broad and moves slowly (cf. p. 196, paragraph II).

As the velocity of the rod increases, a moment will come when the confusing shadows will resolve themselves into four (sometimes five) radial bands of one color with four of the other color and the appropriate transition-bands between them. The bands of either color are symmetrically disposed over the disc, that is, they lie at right angles to one another (if there are five bands they lie at angles of 72°, etc.). But this entire system of bands, instead of lying motionless over the disc as did the systems hitherto described, itself rotates rapidly in the opposite direction from disc to rod. As the rod rotates forward yet faster, no change is seen except that the system of bands moves backward more and more slowly. Thus, if one rotate the rod with one's own hand, one has the feeling that the backward movement of the bands is an inverse function of the increase in velocity of the rod. And, indeed, as this velocity still increases, the bands gradually come to rest, although both the disc and the rod are rotating rapidly.

But the system of bands is at rest for only a particular rate of the rod. As the latter rotates yet faster, the system of bands now commences to rotate slowly forward (with the disc and rod), then more and more rapidly (the velocity of the rod still increasing), until it finally disintegrates and the bands vanish into the confused flicker of shadows with which the phenomenon commenced.

Fig. 11.

Figs. 12. and 13.

This cycle now plays itself off in the reverse order if the speed of the rod is allowed gradually to decrease. The bands appear first moving forward, then more slowly till they come to rest, then moving backward until finally they relapse into confusion.

But let the rate of the rod be not decreased but always steadily increased. The bands will reappear, this time three of each color with six transition-bands. As before, the system at first rotates backward, then lies still, and then moves forward until it is dissolved. As the rod moves still faster, another system appears, two bands of each color forming a diameter and the two diameters lying at right angles. This system goes through the same cycle of movements. When the increased velocity of the rod destroys this system, another appears having one band of each color, the two lying on opposite sides of the center. The system goes through the same phases and is likewise dissolved. Now, at this point the rod will be found to be rotating at the same speed as the disc itself.

The explanation of the phenomenon is simple. The bands are not produced by a single interruption of the vision of a sector by a rod, but each band is made up of successive superpositions on the retina of many such single-interruption bands. The overlapping of bands has been already described (cf. Fig. 10 and pp. 196-198); superposition depends of course on the same principle.

At the moment when a system of four bands of either color is seen at rest, the rod is moving just one fifth as rapidly as the disc; so that, while the rod goes once around, either sector, say the green one, will have passed behind it exactly four times, and at points which lie 90° apart. Thus, four red bands are produced which lie at right angles to one another. But the disc is revolving at least 24 times in a second, the rod therefore at least 4.8 times, so that within the interval of time during which successive stimuli still contribute to the characteristic effect the rod will have revolved several times, and with each revolution four red bands at right angles to one another will have been formed. And if the rod is moving exactly one fifth as fast as the disc, each new band will be generated at exactly that position on the disc where was the corresponding band of the preceding four. The system of bands thus appear motionless on the disc.

The movement of the system arises when the rate of the rod is slightly less or more than one fifth that of the disc. If slightly less, the bands formed at each rotation of the rod do not lie precisely over those of the previous rotation, but a little to the rear of them. The new set still lies mostly superposed on the previous sets, and so fuses into a regular appearance of bands, but, since each new increment lags a bit behind, the entire system appears to rotate backward. The apparatus is actually a cinematograph, but one which gives so many pictures in the second that they entirely fuse and the strobic movement has no trace of discontinuity.

If the rod moves a trifle more than one fifth as fast as the disc, it is clear that the system of bands will rotate forward, since each new set of bands will lie slightly ahead of the old ones with which it fuses. The farther the ratio between the rates of rod and disc departs from exactly 1:5, whether less or greater, the more rapid will the strobic movement, backward or forward, be; until finally the divergence is too great, the newly forming bands lie too far ahead or behind those already formed to fuse with them and so be apperceived as one system, and so the bands are lost in confusion. Thus the cycle of movement as observed on the disc is explained. As the rate of the rod comes up to and passes one fifth that of the disc, the system of four bands of each color forms in rapid backward rotation. Its movement grows slower and slower, it comes to rest, then begins to whirl forward, faster and faster, till it breaks up again.

The same thing happens as the rate of the rod reaches and exceeds just one fourth that of the disc. The system contains three bands of each color. The system of two bands of each color corresponds to the ratio 1:3 between the rates, while one band of each color (the two lying opposite) corresponds to the ratio 1:2.

If the rod and the disc rotate in opposite directions, the phenomena are changed only in so far as the changed geometrical relations require. For the ratio 1:3 between the two rates, the strobic system has four bands of each color; for 1:2, three bands of each color; while when the two rates are equal, there are two bands of each color, forming a diameter. As would be expected from the geometrical conditions, a system of one band of each color cannot be generated when rod and disc have opposite motions. For of course the rod cannot now hide two or more times in succession a sector at any given point, without hiding the same sector just as often at the opposite point, 180° away. Here, too, the cycle of strobic movements is different. It is reversed. Let the disc be said to rotate forward, then if the rate of the rod is slightly less than one fourth, etc., that of the disc, the system will rotate forward; if greater, it will rotate backward. So that as the rate of the rod increases, any system on its appearance will move forward, then stand still, and lastly rotate backward. The reason for this will be seen from an instant's consideration of where the rod will hide a given sector.

It is clear that if, instead of using as 'rod' a single radial sector, one were to rotate two or more such sectors disposed at equal angular intervals about the axis, one would have the same strobic phenomena, although they would be more complicated. Indeed, a large number of rather narrow sectors can be used or, what is the same thing, a second disc with a row of holes at equal intervals about the circumference. The disc used by the writer had a radius of 11 inches, and a concentric ring of 64 holes, each 3/8 of an inch in diameter, lying 10 inches from the center. The observer looks through these holes at the color-disc behind. The two discs need not be placed concentrically.

When produced in this way, the strobic illusion is exceedingly pretty. Instead of straight, radial bands, one sees a number of brightly colored balls lying within a curving band of the other color and whirling backward or forward, or sometimes standing still. Then these break up and another set forms, perhaps with the two colors changed about, and this then oscillates one way or the other. A rainbow disc substituted for the disc of two sectors gives an indescribably complicated and brilliant effect; but the front disc must rotate more slowly. This disc should in any case be geared for high speeds and should be turned by hand for the sake of variations in rate, and consequently in the strobic movement.

It has been seen that this stroboscope is not different in principle from the illusion of the resolution-bands which this paper has aimed to explain. The resolution-bands depend wholly on the purely geometrical relations between the rod and the disc, whereby as both move the rod hides one sector after the other. The only physiological principles involved are the familiar processes by which stimulations produce after-images, and by which the after-images of rapidly succeeding stimulations are summed, a certain number at a time, into a characteristic effect.

FOOTNOTES.

1 Jastrow, J., and Moorehouse, G.W.: 'A Novel Optical Illusion,' Amer. Jour. of Psychology, 1891, IV., p. 201.

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

3 'Marbe, K.: 'Die stroboskopischen Erscheinungen,' Phil. Studien., 1898, XIV., S. 376.

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