APPENDIX C.

POSSIBILITY OF ERRORS IN SCIENTIFIC RESEARCHES, DUE TO
THOUGHT-TRANSFERENCE.

If the theory of Richet is true, an important error may enter many scientific researches in which an assistant is aware of facts a knowledge of which the observer intentionally avoids. An excellent example occurs in the revision of the northern stars, contained in the Durchmusterung of Argelander, which has been undertaken by the Astronomische Gesellschaft. It was provided that the observers, after familiarizing themselves with the scale of magnitudes of the Durchmusterung, should estimate the brightness of each star observed. The Durchmusterung magnitude was then read aloud by the recorder, to enable the observer to continually correct his estimates of the scale of magnitudes. Let M represent the number of cases in which the difference between the estimated and catalogue magnitudes was D. If the number of observations is large, we should in general expect that the relation between N and D would be represented by a smooth curve. If no errors entered but those due to accident, this would become the probability curve. On the other hand, if any thoughttransference occurs between the recorder and observer, we should expect an increase in the value of N when D is zero; that is, of cases in which the magnitude was estimated correctly. It is accordingly only necessary to count the values of N corresponding to various values of D. These results may then be compared with that given by the law of frequency of error; or a curve may be constructed with the various values of N and D, not including those in which D equals zero. The value of N, when D is zero, is now derived from the curve passing through the other points. If the actual value of N, when D is zero, in general exceeds that given from the curve, we may infer that thought-transference occurs, unless some other explanation can be found. The amount of material available for this discussion is very large. The number of stars to be observed exceeds a hundred thousand, each of which is measured on at least two nights. More than a dozen observatories participated in the work; so that the test may be applied to many different persons. The stars between +50° and +55° were observed at the Harvard College

Observatory. A count has been made of the residuals in 0, 6, 12, and 18 hours of right ascension. This furnishes sufficient material for the present investigation, although only about one-sixth of the entire work. Similar estimates of magnitude were also made in connection with observations with the meridian photometer, and thus the results of a number of observers and recorders could be tested. The various series employed are compared in the successive lines of Table I., where a comparison is also made with the result derived from the theory of probabilities, assuming that no error enters but that due to accident. The successive columns give a number for reference, the initial of the observer who becomes the percipient if any thoughttransference occurs, and the recorder or agent. The letters C., E., M., P., R., and W. indicate Messrs. Cutler, Eaton, McCormack, Pickering, Rogers, and Wendell respectively. When the results of various persons are combined, they are indicated by V. The fourth column gives the number of observations contained in the series; the fifth, the average value of the residual, or arithmetical sum of all the residuals divided by their number. The sixth column gives the number of cases in which the residual is zero; and the seventh, the ratio of these numbers to the numbers in the fourth column. This quantity is, therefore, the observed proportion of zeros. From the average deviation we may compute what proportion of residuals should be zero according to the theory of probabilities. The average deviation of each series was next multiplied by .845, which gives the probable error according to the formula of Peters, and .05 was divided by this quantity. The quotient gives the fraction of the probable error which an error must not exceed to give a residual zero. A table of the frequency of error then gives the proportion of the observations whose error should fall within this limit, or which should give residuals zero. These computed proportions are given in the last column but one of Table I. The last column is found by subtracting the computed from the observed proportion of cases in which the residual is zero. About four-fifths of the stars are estimated in the Durchmusterung as fainter than the magnitude 7.9; and these only are employed, since the brighter stars are much more difficult to estimate. In line 7 all the stars are included, and all are brighter than this limit. This is probably the cause of the larger average deviation.

The first four lines of the table give the results of the observations of Professor Rogers, in 0, 6, 12, and 18 hours of right ascension, respectively, with the meridian circle. It is impossible to determine whether the conditions in this case were favorable to thought-transference, as Mr. McCormack is not now living. He was instructed to

record the estimated magnitude before calling out the catalogue magnitude; and, if he did not look at the catalogue magnitude until then, no thought-transference would be indicated. Line 5 gives the observations made in series 1 to 100 with the meridian photometer, or between Feb. 28, 1882, and Jan. 23, 1883. The observer had probably not as yet acquired a fixed habit of estimating the magnitudes. Line 6 relates to series 101 to 400 between the dates Jan. 23, 1883, and Feb. 7, 1885. Line 7 relates to similar estimates of the magnitudes of the standard stars of the Uranometria Argentina, and are the only estimates not relating to the Durchmusterung magnitudes. Line 9 contains the observations contained in series 301 to 400, between July 25, 1884, and Feb. 7, 1885; and line 10, those from series 401 to 450, between Feb. 10, 1885, and April 25, 1885. The same distinction applies to lines 12 and 13. A portion of the last five series were recorded by Professor Searle, but not enough to render a subdivision desirable. Lines 15, 16, and 17 give the results of all of the observations by the three percipients respectively; and line 18 gives the results of all combined.

Table II. gives the details of the count of the number of residuals

of various magnitudes. These magnitudes are given in the first column, and the successive columns give the number of residuals in the first fourteen lines of Table I. When the residual is larger than one magnitude, it is indicated by an L in the first column. The numbers at the top of the columns of Table II. have the same meaning as those in the first column of Table I.

Every residual in the last column of Table I., with one exception, is positive. The actual number of residuals equal to zero is, therefore, in excess of that given by theory; and this effect is most marked in the cases of Professor Rogers and Mr. Wendell. It would not be safe, however, to infer from this the existence of any thoughttransference until all other explanations of this deviation have been carefully considered. If the probable error is diminished in any series of observations, the theoretical number of zero-residuals would be increased. But, in almost every series of observations, the num