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The American journal of psychology online

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Journal of Pstchologi






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Journal of Psychology

Vol. II NOVEMBER, 1888 No. 1


E. C. 8ANF0RD.

The terms ** personal equation " and ** personal dif-
ference *' are somewhat loosely used by astronomers
to indicate such systematic errors in observation as
originate in the observer, in distinction from those that
arise from instrumental and atmospheric conditions.
But the errors thus grouped together in their place of
origin have by no ineans the same causes. Some are
purely anatomical, such as the constant and clear dif-
ference which has been found between observers in
setting the cross-wires of a microscope on the division
mark of a scale,^ or in bringing a star midway
between two parallel wires, the cause of which seems
to be asymmetry of the halves of the eye. To the
same general class would belong astigmatism and
other structural defects of the eye as far as they inter-

' For example, on the limb of a transit circle, or in microscopi-
cally comparing standards of length.

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4 sanfobd:

fere with observation, and color blindness (if that be
an anatomical defect), which has been suggested as
explaining the different magnitudes assigned by dif-
ferent observers to the same celestial object. Another
set are in part from psychic causes. Such are those
that beset observations where judgments of time or
space must be made. And others are purely psychic,
without physical admixture, like the bias for or against
special tenths of a second shown in the recorded obser-
vations of some astronomers and recognized more or
less consciously by others in themselves. It is, how-
ever, a portion of those of the second class that were
first noticed, first received the name of personal equa-
tion, have since received the most careful investiga-
tion, and yet remain the most important. Of the dis-
covery and investigation of these it is the purpose of
this paper to give an accoimt.

Every observatory has for one of its chief businesses
the fixing of the instant in which heavenly bodies
cross its meridian. On this depends the keeping of
the true time, and, in connection with the measure-
ment of the distance of these bodies north or south of
the equator, the fixing of their positions and motions
in the heavens. And in this very process the personal
equation is involved. The instrument used for these
observations is, in its lowest terms, a telescope
mounted on an east and west axis and turning in the
plane of the meridian. In the focus of its eye-piece
is a set of fine parallel wires or spider-lines from five
to twenty-five in number, called a reticle. The middle
one of these lies in the meridian. As the image of the
star moves across the field, the instant of its bisection
by each of these wires is taken, and the average of the
times, provided the intervals between the wires are

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equal, gives the time of the bisection by the central
wire with much less liability to accidental error than
if that had been used alone.

At the time of the first notice of personal equation,
the method of fixing the inst€uit when the star crossed
a wire of the reticle was that of Bradley, or, as it is
called, the " eye and ear " method. When the star is
about to make its transit, the observer reads off the
time from his clock, and then while he watches the
star in the telescope, continues to count the second
beats. He fixes firmly in mind (as the moving image
approaches the wire) its place at the last beat before it
crosses the wire and its place at the first beat after, and
from the distances of these two points from the wire,
estimates by eye the time of the crossing in tenths of
a second. A glance at the figure will make the modtts
operandi clear.



The star in most telescopes^appears to move from
right to left. If we suppose it to be at a when the
eighth second is counted, and at h when the ninth is
counted, the time of crossing the third wire will be so
many hours, so many minutes, 8.7 seconds. The role
of the mind in observations by this method is the fixing
of the exact place of the star at the first beat, the hold-
ing of the same in memory, the fixing of the place at
the second beat, the comparison of the two, and the
expression of their relation in tenths. When inst€ui-

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taneous occurrences like heliotrope or powder signals
or the occnltation or emergences of stars are to be
observed, several ways are open, but the most common
ones require the estimation of the fractional part of
the second directly by ear. A few astronomers also
were accustomed to observe transits in the same way,
treating the passage of the wire like an occultation.
But this was generally regarded as a vicious aberration
from the true method. The psychic action here is
a comparison of the two very short intervals of time
between the event and the preceding and following
clock-beats ; or, regarding the whole series of beats,
the interpolation of the sudden sensation into their
recurring series.

The '^eye and ear" method remained the accepted
one till about 1850, and is even now more or less used,
especially for slow-moving stars like the pole-star.
About 1860 the chronographic method of observation
was introduced. The chronograph consists essentially
in an evenly revolving drum, with which a writing
apparatus, under control of an electro-magnet, is
connected in such a way that as the drum revolves
the apparatus moves slowly from one end of it to the
other. If it were undisturbed the pen would trace a
spiral line upon the paper with which the drum is cov-
ered. But a clock is brought into the circuit with the
electro-magnet, and at each second-beat sends a current
through it ; the magnet draws back the pen and puts a
jag in the line for every second except the sixtieth,
which is omitted to indicate the minute. A key in the
hands of the observer enables him to record his observa-
tion by a jag in the same line or a parallel one. All that
remains to do then is to indicate the time on the clock
to which a certain one of the second-jags corresponds.

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and there is a permanent record from which the time
of the observation can be read off with ease to a small
fraction of a second. By this method of recording the
process of observation is much simplified. The astron-
omer now watches till he sees the star bisected by the
wire, then taps his key. He has simply to perceive an
event and to will a movement of his finger. The part
which the mind plays is thus nearly the same in the
observation of transits and sudden phenomena. There
is, however, here also a variant application of the
method little to be commended. Some observers aim
to tap the key so that they shall hear the click of it at
the instant of the bisection. They thus add an element
of judgment to simple perception and the willing of
movement ; for to accomplish what they intend; the
impulse of will must be given before the star is really
behind the wire, and the length of time by which the
impulse must precede must vary with the apparent
rate of the star. For sudden occurrences they are
obliged, of course, to observe like other people.

Now, in all the methods of observation which have
been mentioned, observers habitually vary both from
the true time and from each other. Their variations
from the true time are called their absolute personal
equations; their mutual differences are their relative
personal equations. It is natural that the latter should
have been first discovered.

So much of a preface has seemed necessary to show
what personal equation is. In what follows I propose
first to give a brief historical account of the discovery
and chief general studies on personal equation, then a
more detailed presentation of the circumstances which
produce variation in its amount, and, lastly, something
of the theories which have been put forward in expla-
nation of it.

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8 8ANF0BD :

The Discovery of Personal Equation.

The first record of a persistent personal diflference
between the observations of experienced astronomers
goes back a little less than a hundred years. About
1795, Maskelyne, the British Astronomer Royal, noticed
such a diflference between those of himself and his
assistant. At the end of the third volume of the
Greenwich Observations he writes as follows :

'^ I think it necessary to mention that my assistant,
Mr. David Kinnebrook, who had observed the transits
of stars and planets very well in agreement with me
all the year 1794, and for a great part of the present
year, began from the beginning of August last to set
them down half a second of time later than he should
do according to my observations ; and, in January of
the succeeding year, 1796, he increased his error to
eight tenths of a second. As he had imfortunately
continued a considerable time in this error before I
noticed it, and did not seem to me likely ever to get
over it and return to a right method of observing,
therefore, though with reluctance, as he was a diligent
and useful assistant to me in other respects, I parted
with him.

"The error was discovered from the daily rate of
the clock deduced from a star observed on one of two
days by him and on the other by myself, coming out
diflFerent to what it did from another star observed both
days by the same person, either him or myself . . .

"I cannot persuade myself that my late assistant
continued in the use of this excellent method (^rad-
ley's) of observing, but rather suppose he fell into
some irregular and confused method of his own, as I
do not see how he could have otherwise committed
such gross errors."

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To the unastronomical mind a difference of eight
tenths of a second seems small, but its real significa-
tion is more apparent when it is multiplied by fifteen,
tp give seconds of arc.

For the next twenty years this germ of a discovery
lay dormant. But in 1816, von Lindenau mentioned
the incident in a history of the Observatory of Green-
wich in the Zeitschrift fur Astronomies and there it
fell under the eye of Bessel, the celebrated Konigsberg
astronomer. Later, the English Board of Longitude
sent the latter a copy of Maskelyne's observations,
from which he got a more complete knowledge of the
facts. The case impressed him. Considering the easy
conditions of such observations with good instruments,
and that such were regarded as sure to one tenth or at
most two tenths of a second, a difference of eight
tenths seemed wellnigh incredible. Its continuance,
too, in spite of the desire that Kinnebrook must have
felt to bring his observations into harmony with those
of his superior, went to prove it involuntary, and
therefore important alike to astronomy and anthropo-
logy. Bessel desired to know whether such a differ-
ence could be found between other pairs of astron-
omers, and in 1819, while on a visit to Encke and von
Lindenau at the Observatory of Seeberg near Gotha,
he proposed to test the point with them. Each ob-
served the culmination of several stars, but no second
clear night during his stay allowed them to complete
the comparison, and the question remained unan-

In the winter of 1820-1, at Konigsberg, he returned
to the subject and made comparisons with Dr. Wal-
beck, by transits observed on the meridian circle of the
observatory. They observed ten stars near the equator

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10 8AKF0BD :

on several nights, each observmg five a night, and
alternating in such a way that those observed by Wal-
beck on one night were observed by Bessel the next,
and vice versa. In this way they arrived at determi-
nations of the rate of the clock which should differ by
double the amount of the personal difference, and were
thus well calculated to show it if any existed.* They
found that Bessel was always in advance :


Dec. 16 and 17 1.145

*' 17 ** 19 0.985

" 19 ** 20 1.010

** 20 " 22 1.025

In the mean, 1.041

This result Bessel considers exact within a few hun-
dredths of a second. The difference was striking on
the second day and led naturally to redoubled efforts
for accuracy. " We ended the observations," says the
astronomer, " with the conviction that it would be im-
possible for either to observe differently, even by only
a single tenth of a second."

Later he repeated the experiment with Argelander,
using a little different method. In 1821 he observed
8ev«n stars in Gemini^ each six times, under favorable
circumstances, and their mean position for 1820 was
calculated. On two evenings in March and April,
1823, Argelander observed the same stars, while Bessel
^^'mself determined the clock corrections. The result-

' Suppose the clock to be saining and that Bessel observes earlier
an Walbeck. Then for the stars which Bessel observes first and
albeck second, the clock rate found will be the real guin of the clock
OS the difference of the observers. When Walbeck observes first
d Bessel second, the rate found will be the gain of the clock
inus the difference of the observers. The difference to the two
bes of eain found will be double the personal difference. If the
>ck is losing, the case is similar.

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ing right ascensions were in excess of those previously
found by Bessel, and in excess of what they would
have been if Argelander had observed the clock stars
himself ; that is, the stars appeared to Argelander to
cross the meridian later than they did to Bessel. The
mean difference for the day in March was 1.222 s., for
the day in April 1.224, whence B — A =—1.223 s.^

Bessel, however, was not content here ; Walbeck
and Argelander were less practiced in transit observa-
tions than he, and he thought that possibly the cause
of the difference lay in this. He accordingly asked
Struve, of Dorpat, to compare observations with him
by means of comparisons with Walbeck and Argelan-
der as they passed through his city. In 1821 Walbeck
and Struve observed together on four days, with the
resulting equation :

B — W= — 0.242
whence B — S =—0.799

In July, 1823, Argelander obtained the following :


S — A = — 0.202
whence B — S =—1.021

The personal difference, therefore, did not originate
in difference in practice.

There is, however, a difference of 0.222 s. between
the two values for B — S, and, since there is very little
uncertainty in the individual determinations, is evi-
dence of change in one or another of the four observers;
most probably in B or S, for the comparison of the
intermediary with S was made each time soon after
that with B. A single direct comparison points the

^The statement of the personal difference in this form has led to
its being caUed the "personal equation,**

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12 sanfobd:

same way. In October, 1814, Struve visited Bessel,
and the two observed together ; Struve observing the
transit of one star, Bessel of two. From these by
calculation the equation B — S := — 0.044 s. is found,
and though it rests on a single transit, is not without
weight, for Struve considered the observation success-
ful, and the agreement of the single wires testifies the
same. At any rate the error was not one of eight
tenths of a second.

This is sufficient to establish the variability of the
personal equation ; but later comparisons (leaving
BessePs first study for the moment) give further
evidence of the same thing. In 1825 the visit at
Konigsberg of another astronomer, Knorre, who had
just compared with Struve, gave opportimity for
repeating the determination of B — S. The result was,
B — S=— 0.891 8. A direct comparison in 1834 gave
B — S = — 0.77 s. Taking all together we have :

1814, B—S=— 0.044, direct comparison.
1821 = — 0.799, indirect "

1823 = — 1.021 " '*

1825 = — 0.891 " "

1834 = — 0.770 direct "

Bessel's next thought after having established the
fact of a personal diflference, was to find its cause. To
that end he began to vary the conditions. He first
substituted the sudden disappearance or reappearance
of a star, as in occultations and emergences, for its
steady motion across the reticle. Seventy-eight com-
parisons of this kind gave for Bessel and Argelander,
B' — A':= — 0.222 s.; another set of twenty-one gave
B' — A'= — 0.289 s. A comparison of Struve and
Argelander on these sudden phenomena developed no

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significant personal diflference. Observations of this
kind are less certain than transit observations, but
they seemed to Bessel to indicate that the trouble lay
in combining the steady advance of the star with the
sudden beat of the clock, and his next experiment was
therefore with a variation in the clock. On two
nights he observed a chosen series of stars with a
clock beating half seconds, with the following result
(indicating by B" his observations with the half -second
clock) :


On the first night B — B" =— 0.520
'* " second " B — B"=— 0.467

That is, he observed transits later, on the average, by
about half a second in this way than with the whole-
second clock. Argelander's observations on the half-
second clock compared with those of Bessel made in
the ordinary way showed no particular change:
B — A" = — 1.246 s., or, in another series, B— A" =
— 1.208 s.

Observations with a half -second clock at Dorpat
gave 8" — A" :=— 0.227 ; from all of which it appears
that Bessel alone had his personal equation changed
by the alteration of the rapidity of the beat. One
other point the Konigsberg astronomer investigated,
namely, the eflfect of the apparent rate of the star,
which varies with its declination, on the personal
equation. This is of great importance, for if it be
found that the rate has no effect, then, provided the
personal equation is constant for the time being, it
will affect equally the times of transit of all stars
observed by the same observer, and will not change at
all their relative times of transit, on which their right
ascensions depend. Bessel varied the apparent rate of

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iDKX>c« bj ihe use of izSerenx powsers in Lis eye-fi-roej*
aikd C3«fcc::idrf thatt liie r^vt h4»i no inf :Je2:^e, ^ kast
fK^ diSemMces eq^^al to itjee frotn the €q:2£S<:^ to
wzthiii hj' of tbe poLe.

In brief, Bessel established these pc-ints : the fact cf
personal equation, its spontaaeoos Tanat>:a in cc-n*
siderabSe periods of toDe, and its ardncial charge, f or
himself at ka«t, with chazige of the c>jck beat and
frc«n transitB to sodden phenocnena, and he trkd. wiih
negative results, the influence of the rate of mz-ticn-
How important these discoveries are in relation to
present knowksdge will appear as the narrative p>rv>-

B cmcV b tbeoTj of the psrchical caose of the persc*nal
eqaaxkm which he had disoorered will be considered
elsewhere. In brief it is that the work of the mind is
the comparison or s up erposing of the unlike impres-
sions on the eye and ear, and that obserrers difiFer in
the readiness with which they accomplish this : an
additi^^ial difference coming in when one of them
goes over from seeing to hearing and the other from
hearing to seeing.

It has more than once been noticed as a fortunate
coincidence for the knowledge of this matter, that the
discorerer of the personal equation should himself
hare had so large a <me ; and such it probably was.
But its rery size has provoked incredulity. It seems
simply impossible that two practiced astronomers
should observe the transit of a star differently with a
clock beating seconds by almost a beat and a quarter.
Encke has contended, and after him Wolf, that Bessel
must have differed from other observ^v in the count*
ing of his seconds ; counting, for example, a transit
which occurred seven tenths of a second after the

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pointer of the clock had passed the fourteenth division
of the face, as 13.7 seconds instead of 14.7 with other
astronomers; in other words, adding the fractional
part to the second completed instead of the second
beg^un.^ The strongest argument in support of this
view is that furnished by Bessel's own experiment
with the half -second clock, where the average of the

Online LibraryG. Stanley (Granville Stanley) HallThe American journal of psychology → online text (page 1 of 64)