In Groups 1 and 2 the intensity of stroke used was just sufficient to give a sharp and distinct stimulation. The intensity of the stimulation was not of a high degree of constancy from day to day, on account of variations in the electric contacts, but within each test of three stimulations the intensity was constant enough.

In experiments under Group 3 two intensities of strokes were employed, one somewhat stronger than the stroke employed in the other experiments, and one somewhat weaker--just strong enough to be perceived easily. The introduction of the two into the same test was effected by the use of an auxiliary loop in the circuit, containing a rheostat, so that the depression of the first key completed the circuit as usual, or the second key completed it through the rheostat.

At each test the subject was warned to prepare for the first stimulation by a signal preceding it at an exact interval. In experiments with the pendulum apparatus the signal was the spoken word 'now,' and the preparatory interval one second. Later, experiments were undertaken with preparatory intervals of one second and 1-4/5 seconds, to find if the estimation differed perceptibly in one case from that in the other. No difference was found, and in work thereafter each subject was allowed the preparatory interval which made the conditions subjectively most satisfactory to him.

Ample time for rest was allowed the subject after each test in a series, two (sometimes three) series of twenty to twenty-four tests being all that were usually taken in the course of the hour. Attention to the interval was not especially fatiguing and was sustained without difficulty after a few trials.

Further details will be treated as they come up in the consideration of the work by groups, into which the experiment naturally falls.

II. EXPERIMENTAL RESULTS.

1. The first group of experiments was undertaken to find the direction of the constant error for the 5.0 sec. standard, the extent to which different subjects agree and the effects of practice. The tests were therefore made with three taps of equal intensity on a single dermal area. The subject sat in a comfortable position before a table upon which his arm rested. His hand lay palm down on a felt cushion and the tapping instrument was adjusted immediately over it, in position to stimulate a spot on the back of the finger, just above the nail. A few tests were given on the first finger and a few on the second alternately throughout the experiments, in order to avoid the numbing effect of continual tapping on one spot. The records for each of the two fingers were however kept separately and showed no disagreement.

The detailed results for one subject (_Mr_,) are given in Table I. The first column, under _CT_, gives the values of the different compared intervals employed. The next three columns, under _S_, _E_ and _L_, give the number of judgments of _shorter_, _equal_ and _longer_, respectively. The fifth column, under _W_, gives the number of errors for each compared interval, the judgments of _equal_ being divided equally between the categories of _longer_ and _shorter_.

In all the succeeding discussion the standard interval will be represented by _ST_, the compared interval by _CT_. _ET_ is that _CT_ which the subject judges equal to _ST_.

TABLE I.

_ST_=5.0 SEC. SUBJECT _Mr._ 60 SERIES.

_CT_ _S_ _E_ _L_ _W_ 4. 58 1 1 1.5 4.5 45 11 4 9.5 5. 32 13 15 21.5 5.5 19 16 25 27 6. 5 4 51 7 6.5 1 2 57 2

We can calculate the value of the average _ET_ if we a.s.sume that the distribution of wrong judgments is in general in accordance with the law of error curve. We see by inspection of the first three columns that this value lies between 5.0 and 5.5, and hence the 32 cases of _S_ for _CT_ 5.0 must be considered correct, or the principle of the error curve will not apply.

The method of computation may be derived in the following way: If we take the origin so that the maximum of the error curve falls on the _Y_ axis, the equation of the curve becomes

y = ke^{-[gamma]x}

and, a.s.suming two points (x_{1} y_{1}) and (x_{2} y_{2}) on the curve, we deduce the formula

____________ D / log k/y_{1} x_{1} = --------------------------------- ____________ ____________ / log k/y_{1} / log k/y_{2}

where D = x_{1} x_{2}, and k = value of y when x = 0.

x_{1} and x_{2} must, however, not be great, since the condition that the curve with which we are dealing shall approximate the form denoted by the equation is more nearly fulfilled by those portions of the curve lying nearest to the _Y_ axis.

Now since for any ordinates, y_{1} and y_{2} which we may select from the table, we know the value of x_{1} x_{2}, we can compute the value of x_{1}, which conversely gives us the amount to be added to or subtracted from a given term in the series of _CT_'s to produce the value of the average _ET_. This latter value, we find, by computing by the formula given above, using the four terms whose values lie nearest to the _Y_ axis, is 5.25 secs.

In Table II are given similar computations for each of the nine subjects employed, and from this it will be seen that in every case the standard is overestimated.

TABLE II. _ST_= 5.0 SECS.

Subject. Average ET. No. of Series.

_A_. 5.75 50 _B_. 5.13 40 _Hs_. 5.26 100 _P_. 5.77 38 _Mn_. 6.19 50 _Mr_. 5.25 60 _R_. 5.63 24 _Sh_. 5.34 100 _Sn_. 5.57 50

This overestimation of the 5.0 sec. standard agrees with the results of some of the experimenters on auditory time and apparently conflicts with the results of others. Mach[4] found no constant error. Horing[5]

found that intervals over 0.5 sec. were overestimated. Vierordt,[6]

Kollert,[7] Estel[8] and Gla.s.s,[9] found small intervals overestimated and long ones underestimated, the indifference point being placed at about 3.0 by Vierordt, 0.7 by Kollert and Estel and 0.8 by Gla.s.s.

Mehner[10] found underestimation from 0.7 to 5.0 and overestimation above 5.0. Schumann[11] found in one set of experiments overestimation from 0.64 to 2.75 and from 3.5 to 5.0, and underestimation from 2.75 to 3.5. Stevens[12] found underestimation of small intervals and overestimation of longer ones, placing the indifference point between 0.53 and 0.87.

[4] Mach, E.: 'Untersuchungen uber den Zeitsinn des Ohres,'

_Sitzungsber. d. Wiener Akad._, Math.-Nat. Kl., Bd. 51, Abth.

2.

[5] Horing: 'Versuche uber das Unterscheidungsvermogen des Horsinnes fur Zeitgrossen,' Tubingen, 1864.

[6] Vierordt: _op. cit._

[7] Kollert, J.: 'Untersuchungen uber den Zeitsinn,' _Phil.

Studien_, I., S. 79.

[8] Estel, V.: 'Neue Versuche uber den Zeitsinn,' _Phil.

Studien_, II., S. 39.

[9] Gla.s.s R.: 'Kritisches und Experimentelles uber den Zeitsinn,' _Phil. Studien_, IV., S. 423.

[10] Mehner, Max: 'Zum Lehre vom Zeitsinn,' _Phil. Studien_, II., S. 546.

[11] Schumann, F.: 'Ueber die Schatzung kleiner Zeitgrossen,'

_Zeitsch. f. Psych._, IV., S. 48.

[12] Stevens, L.T.: 'On the Time Sense,' _Mind_, XI., p. 393.

The overestimation, however, is of no great significance, for data will be introduced a little later which show definitely that the underestimation or overestimation of a given standard is determined, among other factors, by the intensity of the stimulation employed. The apparently anomalous results obtained in the early investigations are in part probably explicable on this basis.

As regards the results of _practice_, the data obtained from the two subjects on whom the greatest number of tests was made (_Hs_ and _Sh_) is sufficiently explicit. The errors for each successive group of 25 series for these two subjects are given in Table III.

TABLE III.

_ST_ = 5.0 SECONDS.

SUBJECT _Hs_. SUBJECT _Sh_.

CT (1) (2) (3) (4) (1) (2) (3) (4) 4. 2.5 2.5 1.5 2.5 0. .5 0. .5 4.5 6.0 3.0 3.5 7.0 5.0 3.5 2.0 .5 5. 14.0 11.0 11.0 11.0 8.5 11.5 4.0 7.0 5.5 11.5 11.5 6.0 12.5 11.0 16.0 14.0 15.0 6. 12.0 9.0 6.5 6.0 3.5 2.0 1.5 1.0 6.5 4.0 3.5 4.0 3.5 4.0 .5 0. 0.

No influence arising from practice is discoverable from this table, and we may safely conclude that this hypothetical factor may be disregarded, although among the experimenters on auditory time Mehner[13] thought results gotten without a maximum of practice are worthless, while Meumann[14] thinks that unpracticed and hence unsophisticated subjects are most apt to give unbiased results, as with more experience they tend to fall into ruts and exaggerate their mistakes. The only stipulation we feel it necessary to make in this connection is that the subject be given enough preliminary tests to make him thoroughly familiar with the conditions of the experiment.

[13] _op. cit._, S. 558, S. 595.