Sixteen Experimental Investigations from the Harvard Psychological Laboratory by Hugo Münsterberg (100 books to read .txt) 📕

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.”

[Illustration: PSYCHOLOGICAL REVIEW. MONOGRAPH SUPPLEMENT, 17. PLATE V.

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