# Popular Science Monthly/Volume 6/February 1875/The Personal Equation

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THE
POPULAR SCIENCE
MONTHLY.
FEBRUARY 1875
* THE PERSONAL EQUATION.
By Prof. EDWARD S. HOLDEN,
OF THE UNITED STATES NAVAL OBSERVATORY, WASHINGTON, D. C.
IN an attempt to explain clearly some of the phenomena which have led to the consideration of what astronomers call the "personal equation," it will, perhaps, he most advantageous to consider the subject somewhat in an historical manner. In this way we shall, it is true, lose something in directness, but it will assist in gaining a definite conception of the whole subject if we consider it in the order in which astronomers have been forced to do.
To make the meaning of the term plain, it will be necessary to premise a brief account of the methods of observation with astronomical instruments, and of some of the refinements which have gradually been found necessary in these methods.
Nearly every astronomical observation has for an object to fix the relative position of two bodies at a given time. If, then, a second observation of a similar kind is made, these two, taken together, will suffice to give some idea of the apparent relative motion of one body, referred to the other. If, for example, the design is to determine the orbit of a new comet, the mode of proceeding is, or might be, something as follows: Some star, whose place is known (or whose place is subsequently determined), is chosen in the vicinity of the comet, and the distance of the comet from this star is measured. This may be done in several ways, by a sextant, with which we can measure this distance directly, or, more usually, by one of the fixed instruments of an observatory, with which we can determine two things: 1. The distance of the comet east or west of the star; and, 2. Its distance north or south of it. The distance north or south is usually determined by a direct measure of the celestial arc included between the respective parallels on which the star and comet are at a given time; while the distance east or west is usually measured by the interval of time required for the earth's rotation to carry a body from the ​meridian of the star to that of the comet. To make this measure, it is customary to fix in the focus of the telescope some uneven number of fine filaments of spider's-web at (say) equal distances apart, and to allow the telescope to remain fixed while the diurnal rotation of the earth carries the body first to be observed into the field of the telescope and slowly across this. As it crosses each of the threads, the time at which it is exactly on the thread is noted. Now, when the second body enters the field of the telescope (which is supposed to remain fixed in its former position) the times of its passage over the various threads are noted.
The mean of the times for the first body gives the time at which this body was on the middle thread (these being at equal intervals), while the mean of the times for the second body gives the corresponding time for the second body, and the difference of these two times gives evidently the distance which one of them is, east or west, of the other, expressed in time. This may be easily reduced to degrees, etc., by the rule that twenty-four hours is equal to 360 degrees.
If it were possible for an astronomer to note the exact instant of the transit of a star over a thread, it is plain that one thread would be sufficient; but, as all estimations of this time are, from the very nature of the case, but approximations, several threads are inserted in order that the accidental errors of estimations may be eliminated, as far as possible. The method of making these estimations will be better understood from the two following figures, 1 and 2. Fig. 1 represents
Fig. 1.—Transit-threads in Telescope.
the reticle of a transit-instrument as it would be viewed by an observer, where twenty-five threads are placed arranged in groups or tallies of five. The star may enter on the left hand in the figure, and may be supposed to cross each of these wires, the time of its transit over each ​of them, or over a sufficient number, being noted. The method of noting this time may be best understood by referring to Fig. 2.
Suppose that the line in the middle of the figure is one of the transit-threads, and that the star is passing from the right hand of the figure toward the left: if it is on this wire at an exact second by the clock (which is always near the observer, beating seconds audibly), this second must be written down as the time of the transit over this thread. As a rule, however, the transit cannot occur on the exact beat of the clock, but at the seventeenth second (for example) the star will be on the right of the wire, say at a; while, at the eighteenth second, it will have passed this wire and may be at b. If the distance of a from the wire is six-tenths of the distance a b, then the time of transit is to be recorded as—hours—minutes (to be taken from the clock-face), and seventeen and six-tenths seconds; and in this way the transit over each wire is observed. This is the method of "eye-and-ear" observation, the basis of such work as we have described, and it is so called from the part which both the eye and the ear play in the appreciation of intervals of time. The ear catches the beat of the clock, the eye fixes the place of the star at a; at the next beat of the clock the eye fixes the star at b, and subdivides the space a b into tenths, at the same time appreciating the ratio which the distance from the
Fig. 2.—Passage of Star across the Thread.
thread to a bears to the distance a b. This is recorded as above. Now, if the action of the eye and the ear and the coördinating action of the brain (which must associate some spot in the field of view with some second) were all instantaneous in their action, the phenomenon of personal equation would not exist. As a matter of fact, when the clock beats and the star is really at a, the mind refers it to some point farther on in the field as a'; and when the clock again beats, the star, which truly is at b, is by the mind referred to a point b'. The distance a b is the same as a' b'; but the distance from the thread to a is greater than the distance from the thread to a'. Hence, instead of recording the time of transit as 17$^s$.6, an observer, whose habit is correctly represented by the figure, might record this time as 17$^s$.4, and the correction $+$ 0$^s$.2 would be required to be applied to his times of transit to reduce them to the exact truth: $+$ 0$^s$.2 is then his absolute personal correction. But, in general, we have no means of determining where a and b, in our field of view, are, and hence the knowledge of the absolute personal equation has to be gained by some special ​devices, to be hereafter spoken of. A little consideration will show, however, that, although every transit observed by our astronomer is too early by 0$^s$.2, yet, in ordinary cases, this correction is of no account, provided only that it is constant. If he observes the star too early by 0$^s$.2, and the comet also too early by that amount, the difference in the times will be absolutely correct. But suppose one observer to note the transit of the star, and another that of the comet: each may have a peculiar habit, so that where one would note 0$^s$.2 top early, another might note 0$^s$.3 too early, and the difference of their absolute personal equations, 0$^s$.1, it would be necessary to apply to the observations of A to reduce them to homogeneousness with those of B. This difference of absolute personal equations is relative personal equation, which, when once truly known, enables us to reduce the observations of one skillful astronomer to what they would have been had another made them.
We say "skillful," because it is only among skillful observers that the phenomenon in question is truly found. In astronomical observations the senses are trained to a fine delicacy, and old observers acquire a constancy of habit which gives to their work a homogeneousness that is wanting in that of younger men.
We have given a brief account of the early method of estimating the time of a star's transit across a spider-line in the field of the telescope by the method of eye and ear; there is yet another method now in common use, which it is necessary to understand before we pass to the consideration of the means of determining personal equation.
This second method is the American or chronographic method; this consists, in the present practice, in the use of a sheet of paper wound about and fastened to an horizontal cylindrical barrel, which is caused to revolve by machinery once in one minute of time. A pen of glass which will make a continuous line is allowed to rest on the paper, and to this pen a continuous motion of translation in the direction of the length of the cylinder is given. Now, if the pen is allowed to mark, it is evident that it will trace on the paper an endless spiral line. An electric current is caused to run through the observing clock, through the pen, and through a key which is held in the observer's hand.
Fig. 3.—Portion of a Chronographic Record
The upper line of figures numbers the seconds; and as twenty of them are shown, and the cylinder revolves once every sixty seconds, about one-third of the length of the strip of papaer is here represented. The middle column of figures numbers the lines, and, therefore, the minutes. The jogs in the lines are represented so exactly at intervals of one second as to give the effect of columns across the paper. As explained in the text, it is all done by one pen, which makes one continuous line round and round, the machine making regular jogs as a time-scale, and the observer making the regular jogs at the instant the star reaches the thread; its time being thus exactly recorded.—[Ed.
​minute, and second, taken from the clock on the sheet at its appropriate place, and the translation of the spaces on the sheet into times may be done at leisure. This will be plainer if we examine Fig. 3, which is a fac-simile of a portion of a chronographic record.
The marks of the clock giving regular signals of seconds are easily distinguished; the rattles were made by the observer to attract attention to an observation to follow: and the signals of the observer are seen at the times of the transits of various stars. By applying a graduated ruler to the sheet when it is unrolled, the exact times of transit can be determined to within a hundredth part of a second, provided we have some hour, minute, and second marked on the sheet as an origin of time.
It is quite plain that the senses of the observer are not strained to so great a degree in this method as in the method of eye and ear; the eye has but one thing to do, the ear is not occupied, and the hand has only to press the key at the proper time.
In this method, we see that the origin of relative personal equation is again in the different times required for different observers to coordinate the position of the star in the field and the position of the wire.
True personal equation, considered physiologically, must arise from the personal differences between observers when they note the same phenomenon. With the chronograph it is the habit of most observers to tap the observing key at the moment at which the star is actually on the wire. There are cases, however, where astronomers of some experience are accustomed to tap the key so that the sound of the tap shall come to the ear at the time when the star is on the wire. This seems an utterly wrong habit of observing, as it is really the record of an event which has not yet taken place which such an observer makes. Astronomically, the difference between such an observer and another observer may be treated as a case of personal equation, provided the habit described above remains constant, which it is probably less likely to do than the ordinary one.
The first case of personal equation on record appears in the "Observations" of the Rev. Nevil Maskelyne, Astronomer Royal for England ("Observations" for 1796, vol. iii., p. 339). We there find the following note: "I think it necessary to mention that my assistant, Mr. David Kinnebrook, who had observed the transits of the 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 unfortunately 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."
But time has its revenges, and Kinnebrook's observations are now used as well as Maskelyne's (see "Annales de l'Observatoire de Paris; Memoires," iii., p. 307), and they are probably about as free from accidental errors as his.
In 1822 Bessel examined this subject, and we find in the Königsberg observations of that year an account of quite extended experiments on personal equation.
Bessel, after quoting from Maskelyne's own report (see extract above), considers the subject at some length. He calls attention to the fact that the accidental errors in an eye-and-ear observation certainly do not exceed two-tenths of a second, and that a careful consideration of the observations of Maskelyne and his assistant shows that there may be an "involuntary constant difference" between the estimations of various observers which far surpasses the limits of possible accidental error.
In 1819 Bessel made a visit to the Seeberg Observatory, where he observed, on two nights, transits with Von Lindenau and Encke. These observations showed no personal equation between these three celebrated astronomers. In 1820 Dr. Walbeck and Bessel made several sets of observations at Königsberg, for the purpose of determining their relative personal equation, and the results of their work are given below:
| 1820, December 16th and 17th, Walbeck later than Bessel | 1.045 |
| "17th and 19th,"""" | 0.985 |
| "19th and 20th,"""" | 1.010 |
| " 20th and 22d,"""" | 1.025 |
| Mean | 1.041 |
Bessel says that this great difference was evident from the second day, and that no pains was spared by either of them to observe carefully; and that at the end of the series each was confident that it would have been impossible for him to observe differently, by so much even as a tenth of a second. Here, then, was an enormous difference—one almost incredible. To test the reality of the phenomenon, Bessel compared with Argelander, and found that Argelander was later than he by 1$^s$.223.
Bessel remarks that neither Walbeck nor Argelander had observed as much as he had with the transit-instrument, and he therefore used all opportunities for comparing his work with that of Struve, of Dorpat. He found that in 1814 Struve was later than himself by 0$^s$.044; in 1821, by 0$^s$.799; in 1823, by 1$^s$.021. Bessel now determined to arrive at some conclusion by studying this phenomenon under different aspects.
To this end Argelander and himself noted the times of 78 ​disappearances or reappearances of a material object, and. he found that Argelander was later than himself by 0$^s$.222. Again, in the observation of the occultations of stars (an instantaneous phenomenon), Argelander was slower than Bessel by 0$^s$.281. Here was some light: for it was now evident that not only had each astronomer a different habit of estimating time, but that this habit was only constant so long as the same phenomenon was observed; that a personal equation for transit observations would not serve for observations of occultations.
Bessel next investigated the question whether there was any difference in his own absolute personal equation in observations with a clock beating whole seconds, or with a chronometer beating half-seconds; he found that he observed 0$^s$.494 later when the clock beat half-seconds than when it beat whole seconds, while Argelander and Struve did not change their habits in this regard.
Bessel's whole investigation is very complete, especially when we consider that it was the first published research on a subject which had escaped attention until his time. The principal points established were:
1. A personal equation subsists in general between two observers.
2. For limited periods of time this equation is probably constant between two observers for the same class of work.
3. The absolute personal equation of any one observer varies with the class of observation; i. e., from transit observations to sudden phenomena like occultations.
4. The rapidity with which the star (in transit observations) traversed the field of the telescope had no influence on Bessel's personal equation.
Bessel does not seem to have supposed that there would be any different personal equation for stars and for the moon. This we now know to have been erroneous, and we shall see that the apparent velocity with which a star moves through the field of the telescope is also held by some observers to have an influence on the magnitude of their personal equation.
All of the preceding results referred simply to the personal equation between observers who were. using the eye-and-ear method. As soon as the chronographic method of registering transits was introduced, it was seen that the personal equation became smaller. This is undoubtedly due to the smaller amount of work which the brain has to perform; the phenomena to be appreciated are, in this latter case, far more simple than in the former, and the effect of this is shown in the amount of personal difference.
We must now give a brief account of the ordinary methods for determining the amount of the relative personal equations of various observers, in order that we may proceed to the determination of the absolute equation, which is of great interest physiologically and psychologically, although not of capital importance to astronomy. As ​we have seen, to reduce the observations of A to what they would have been if B had made them, it is simply necessary to know how much later B is in the habit of observing than A, and to apply this as a constant correction to A's work.
This may be done in practice by A and B observing the same star in the same telescope; A over the first ten wires (see Fig. 1), and B over the second ten.
A knowledge of the distances of the various wires from the middle wire enables us to compare A's work with B's, and A—B is the relative personal equation.
There is, however, a strong objection to this process: if personal equation is any thing, it is the difference between established habits; and, if A observes over ten wires, and then hastily rises to allow B to take his place at the instrument, both A's habits and B's are broken in upon, and the resulting personal equation is likely to be affected by this fact. In general, the way adopted is to allow A to observe several stars leisurely, and from them to determine the error of the clock; B does the same, and from his observations also a clock-error is found; the difference of these clock-errors, reduced to the same epoch, gives the relative equation of A and B.
Now if, instead of A registering his own observations on the chronograph (for example), we could have the star register its own transit, then B's observation, compared with this, would give at once an absolute equation. We cannot use the real star for this purpose; but several attempts have been made to construct an apparatus which should register the transit of an artificial star, which star could, at the same time, be observed. The principle of all of these machines is, in general, the same, and we will merely give a brief account of one which is now under trial by the Coast Survey.
The artificial star is produced by lamp-light falling upon a small hole in a blackened plate; this plate is given a motion laterally, and the small point of light passes from one side to the other of a plate of ground glass, upon which lines are ruled to represent the spider-lines of the reticle. As the artificial star passes each wire, an electric signal is recorded on the chronograph, and the observer can also record his signal; and thus on the same chronograph-sheet many observations of absolute personal equation can be permanently recorded. Any velocity can be given to the star, so that it may pass through the field of view as slowly as the pole-star, or as rapidly as a star at the equator.
​An apparatus similar to this was invented and used by Wolf, of the Paris Observatory, and we owe to him much the fullest account of personal equation which we have. We cannot do better than to give a brief abstract of his memoir ("Mémoires de l'Observatoire de Paris," tome viii., p. 153), as the results obtained by the American device have not been made public.
His first experiences showed him that his absolute personal equation, when he used the chronographic method of recording, was extremely small (from three to four hundredths of one second); and, although this was an interesting fact, yet the very smallness of this equation showed that it was hopeless to attempt to discover the laws of variation of so minute a quantity.
These laws would be masked by the accidental errors: so that all the observations of M. Wolf have been by eye and ear. It should be stated that M. Wolf is an observer of experience. In his own experiments he proposed to himself to determine the effect on his equation—
* (a.) Of the position of the observer (sitting or standing, etc.).
* (b.) Of the magnifying power of the telescope.
* (c.) Of the direction of motion of the star (i. e., whether from right to left, or the reverse).
* (d.) Of the brightness of the star.
His personal equation he found was, at first, about +0$^8$.3; and in a short time this fell to +0$^8$.1; this was undoubtedly due to the fact that the observer felt in what direction his observations had to be modified, in order to bring them nearer to the truth, and that he unconsciously so modified them. This, however, did not continue without limit; his personal equation remained, for all the time he observed, at this lower limit, and this fact gave him the first clew to the physiological explanation of the phenomenon.
M. Wolf finds that the brilliancy of the star has no sensible effect on personal equation, a conclusion identical with that derived by Mr. Dunkin, of the Royal Observatory at Greenwich ("Monthly Notices, Royal Astronomical Society," vol. xxiv., p. 158).
With regard to the influence of the direction of motion of the artificial star, M. Wolf finds in his own case a mean constant difference of 0$^8$.04 obtained from over 400 transits: this he subsequently explains by the fact that, if his right eye be fixed on two dots equidistant from a line drawn on a sheet of paper, one of these dots always appears nearer to the line than the other by a small quantity. This, of course, is a defect in the symmetry of the eye, and it is quite a common defect, which probably many of the readers of The Popular Science Monthly have, perhaps without knowing it.
The influence of the apparent velocity of the star Bessel states to have been nothing in his own case, provided the star was situated more than 20° from the pole. Wolf's experiments do not agree with this, and he confirms the researches of Dr. Pape and of Dunkin.
​Pape finds (Astronomische Nachrichten, vol. xliv., p. 179) that the error of a transit observation is composed of two parts: one is constant, and the other depends on the polar distance of the star. Dunkin likewise considers the probable error of a transit observation as depending upon the polar distance of the star, and Wolf's experiments corroborate these results, and show that his own personal equation became larger as the velocity of the star increased. It is evident that this rule must be held true only within limits, and probably these limits are not very far apart. Wolf further made experiments to determine whether the position of the observer affected his personal equation, and he concluded that, for his own case, there was no effect due to this cause. It is probable that most astronomers would differ with Wolf in this respect: observers of double stars, especially, have noticed a constant influence in their measures due to the position of the head.
After having recited the results of his experiments, M. Wolf comes to the consideration of the really important question, "What is the origin of the phenomenon known as personal equation?" Before he discusses this, he considers the remarkable personal differences between Bessel and other astronomers which we have noticed, showing that this is undoubtedly the largest personal equation on record, and expressing his opinion that it was really due to an erroneous counting of the whole seconds, and that the fractional part of his enormous personal equation with Argelander (1$^s$.223) was alone a case of true physiological personal difference. Let us recall the fact that Bessel and Argelander differed in observations of sudden phenomena only by 0$^s$.222, or 0$^s$.281; and again, that Bessel observed transits with a chronometer beating half-seconds so much as 0$^s$.494 (nearly a whole beat) later than with a clock beating seconds; and it seems impossible to avoid Wolf's conclusion that Bessel counted his seconds differently from other observers. The only thing which militates against this theory is, that Bessel must have examined this question of enumeration himself; and again, that, in two nights' observation with Von Lindenau and Encke, he found no signs of personal equation. Encke, however, in speaking of this large personal equation of Bessel's, says that there is no doubt that he had a different method of counting the strokes of the clock from other observers. M. Wolf, too, mentions the case of an assistant at the Paris Observatory, whose transit observations were earlier by one second than those observed by his fellow-assistants (Bessel's habit), but, in this case, a few experiments on artificial transits sufficed to show him that his habit was wrong, and led him to change it.
The opinion of most astronomers has been, that personal equation is not purely a physiological phenomenon, but likewise a psychological. The time required for the sound of the clock to reach the observer's brain, and the time required for the light to pass from the image of ​the star, so as to excite the nerves of vision, are both very small: it is the coördinating power of the brain that works slowly—and absolute personal equation is largely the measure of the time required for the brain to superpose two different sensations, to coördinate impressions derived from different sets of nerves.
This view M. Wolf combats, and maintains, on the contrary, that the phenomenon in question is purely physiological, and arises from the duration of the luminous impression of the image of the star on the retina. To prove this, he has applied his apparatus to the observing of transits in which the seconds of the clock were not marked by audible beats, but by flashes of light appearing in the field of the telescope.
In this case, and also in the case where the seconds of the clock were not heard, but were marked by light taps on his hand, his equation remained almost constant (see table):
| $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | The | second | marked | by | sound, | : ε $=+$ 0$^s$ | .10 | 80 | observations. |
| $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | “ | “ | “ | “ | sight, | : ε $=+$ 0 | .08 | 80 | “ |
| $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | The | second | marked | by | sound, | : ε $=+$ 0$^s$ | .11 | 80 | “ |
| $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | “ | “ | “ | “ | feeling, | : ε $=+$ 0 | .11 | 80 | “ |
This table seems to bear out M. Wolf’s view; but, in this connection, it will be interesting to refer to a paper by Mr. T. C. Mendenhall, of Columbus, which appeared in the American Journal of Science, vol. ii., p. 157. This gentleman says: “An attempt was made to determine the relative rapidity with which responses are made to impressions made upon the different senses. . . . Time is measured on a register similar to the astronomical chronograph, in which I have been able to move a slip of paper with great regularity at the rate of about one and one-half inch per second, the seconds being registered upon the slip by a seconds pendulum according to the electric method. The person on whom the experiment is being made is seated at a table, having his hand on a key; by pressing this, the time of the action is registered on the paper. I made an apparatus, by means of which the circuit is completed for an instant the moment that there appears at a circular opening, about three-fourths of an inch in diameter, a card, red or white, as I choose, which completely fills the opening. The subject is instructed to watch this opening, and to press the key immediately on seeing the card. The actual appearance of the card and his closing the circuit in response are marked on the slip of paper by two dots about one-fifth of an inch apart (two-fifteenths of a second). This is a measure of the time occupied by the somewhat complex operation of his perceiving the object, and acting in response to that perception. I introduce the exercise of judgment by giving him two keys, one for each hand, and by instructing him that, when a white card appears, he is to close with his right hand, and when a red card appears, with his left. . . . According to the same general plan, I made trials concerning the sense of hearing. I arranged that, by ​pressing upon the key, unseen by the subject, I could at the same time close the circuit and produce a clear and distinct sound, upon hearing which he made a response registered as before. I connected my apparatus with the key-board of a piano-forte in such a way that I was able to introduce an exercise of judgment in the comparison of two tones differing in pitch much or little, as I chose. . . . With different persons, as many as 2,000 individual trials have been made, and the errors of experiment eliminated as far as possible by averages. . . . As was anticipated, different individuals furnished, in some cases, strikingly different results, but, in general, they all followed the order given in the table:"
| CASE OF A. G. F. | Time in seconds. |
| Response to appearance of a white card | 0.292 |
| """"an electric spark (in the dark) | 203 |
| ""sound | 138 |
| ""touch on the forehead | 107 |
| """"hand | 117 |
| "when required to decide between white and red | 443 |
| """""tones C and E | 335 |
| """""C and C above (octave) | 428 |
One cannot but be struck with the additional time required when the phenomenon to be observed becomes even slightly more complex. This is evidently not entirely a physiological effect, but is truly psychological in part. Just what bearing this has on the question of the cause of personal equation it would be difficult to say: at the same time we must admit that the slightest additional exercise of judgment requires additional time. This is forcibly shown by the smallness of chronographic personal equation as compared to eye-and-ear equation.
Let us now consider personal equation in things other than the estimation of time. We stated that the distance of one star, north or south of another, was usually measured directly; i. e., by graduated circles for large distances, and with micrometers for small ones. Prof. Coffin, now Superintendent of the American Ephemeris, has shown that in his own case, and in the case of two other observers, at the United States Naval Observatory of Washington, a marked personal difference appears in the observations of a Lyræ, and one or two other stars which pass near the zenith of Washington, depending on the direction in which the observer faced, whether north or south. It is plain that a star near the zenith may be observed as a south star or as a north star, and it appears that each position gives a different polar distance to the star: the difference of polar distance is small but constant.
In reading microscopes, and, in short, in performing any operation where the senses are strained to appreciate small differences of time, space, or position, and particularly where the judgment has to be exercised, personal differences are present. In general, these are constant with the same observer, and in astronomy they are usually eliminated in the determination of the zeros. For example, if an ​server reads the microscopes of a Transit Circle habitually too large, when he is determining the zenith-distance of a star, it is likewise his habit to read them too large when determining the position of the zenith-point from which zenith-distances are counted; and the resulting quantity is likely to be free from all but accidental errors.
Occasionally there arise cases where these differences (in the same observer) are not eliminated, but multiplied.
In the measurement of a base-line, for example, the various rods are brought into contact under a microscope: if an observer judges these rods to be in contact when they are not, it is evident that his error, originally small, will augment with the number of contacts, and it may become serious.
In the comparison of the national standards of length, undertaken by the English Ordnance Survey, an annoying case of personal difference was found.
These comparisons were made by bringing a movable cross of spider-lines to bisect one of the lines engraved, on the various bars, and it was found that Captain Clarke, R. E., and Quartermaster Steel, R. E., who made the greater number of comparisons, differed in their estimation of a bisection by a constant amount which was annoyingly large: so that "the probable error of the final results is nearly double what might be expected from errors of observations only." This error cannot be eliminated, and it still remains in the published results.
We must constantly bear in mind that the quantities of which we have all along been speaking are extremely small, and that in fact they are masked by accidental errors for inexperienced observers in most cases. Still they exist, and they are among the most curious of phenomena: their careful study would well repay physiologists.
We can never be sure we have eliminated them so long as the human mind or body is a part of the machine by means of which we are comparing or registering events; and, just so long as mind or body is employed, we can be sure that personal differences will not only exist, but that they will vary from day to day. We must use for eliminating personality those values which are the best attainable, and assume these values to be constant over extended periods of time—weeks or months. In astronomy of precision, however, we have other errors to fear much more variable than personal equation, and it is to the elimination of these that attention should be directed. In other branches of research less exact in method, personality becomes of more importance, and an attentive consideration of its effects may be well worth while undertaking.
* ↑ The chief objection to this apparatus is, that there is a constant error in its indications; i. e., it can never be adjusted so as to give its signal at the exact moment of transit, but it is always too soon or too late. This is sought to be eliminated by allowing the artificial star to travel first from right to left, and then from left to right, and using the mean of the two determinations. It is still a question whether the observer's habit is the same no matter which way the star is moving.
* ↑ The writer has recently had occasion to examine drawings of the same nebula by different observers, with telescopes which are quite similar, and the enormous differences which exist in the representations show personal differences of the most marked kind, for nothing is more certain than that all the changes shown by the drawings have not taken place.