Encyclopædia Britannica, Ninth Edition/Measurement of Time

by John Louis Emil Dreyer

TIME, Measurement of.Time is measured by successive phenomena recurring at regular intervals. The only astronomical phenomenon which rigorously fulfils this condition, and the most striking one, the apparent daily revolution of the celestial sphere caused by the rotation of the earth, has from the remotest antiquity been employed as a measure of time. The interval between two successive returns of a fixed point on the sphere to the meridian is called the sidereal day; and sidereal time is reckoned from the moment when the "first point of Aries" (the vernal equinox) passes the meridian, the hours being counted from 0 to 24. Clocks and chrono meters regulated to sidereal time are only used by astronomers, to whom they are indispensable, as the sidereal time at any moment is equal to the right ascension of any star just then passing the meridian. For ordinary purposes solar time is used. In the article ASTRONOMY (vol. ii. p. 771) it is shown that the solar day, as defined by the successive returns of the sun to the meridian, does not furnish a uniform measure of time, owing to the slightly variable velocity of the sun's motion and the inclination of its orbit to the equator, so that it becomes necessary to introduce an imaginary mean sun moving in the equator with uniform velocity. The equation of time (loc. cit., pp. 772-773) is the difference between apparent (or true) solar time and mean solar time. The latter is that shown by clocks and watches used for ordinary pur poses. Mean time is converted into apparent time by applying the equation of time with its proper sign, as given in the Nautical Almanac and other ephemerides for every day at noon. As the equation varies from day to day, it is necessary to take this into account, if the apparent time is required for any moment different from noon. The ephemerides also give the sidereal time at mean noon, from which it is easy to find the sidereal time at any moment, as 24 hours of mean solar time are equal to 24 h 3 m 56 -5554 of sidereal time. About 21st March of each year a sidereal clock agrees with a mean-time clock, but it gains on the latter 3 m 56 a- 5 every day, so that in the course of a year it has gained a whole day. For a place not on the meridian of Greenwich the sidereal time at noon must be corrected by the addition or subtraction of 9" - 8565 for each hour of longitude, according as the place is west or east of Greenwich.

While it has for obvious reasons become customary in all civilized countries to commence the ordinary or civil day at midnight, astronomers count the day from noon, being the transit of the mean sun across the meridian, in strict conformity with the rule as to the beginning of the sidereal day. The hours of the astronomical day are also counted from to 24. An international conference which met in the autumn of 1884 at Washington, to consider the question of introducing a universal day (see below), has recommended that the astronomical day should commence at midnight, to make it coincide with the civil day. The great majority of American and Continental astronomers have, however, expressed themselves very strongly against this change; and, even if it should be made in the British Nautical Almanac, it appears very doubtful whether the other great ephemerides will adopt it, the more so as astronomers have hitherto felt no in convenience from the difference between the astronomical and the civil day.

Determination of Time.—The problem of determining the exact time at any moment is practically identical with that of determining the apparent position of any known point on the celestial sphere with regard to one of the fixed (imaginary) great circles appertaining to the observer's station, the meridian or the horizon. The point selected is either the sun or one of the standard stars, the places of which are accurately determined and given for every tenth day in the modern ephemerides. The time thus determined furnishes the error of the clock, chronometer, or watch employed, and a second determination of time after an interval gives a new value of the error and thereby the rate of the timekeeper.

The ancient astronomers, although they have left us very ample information about their dials, water or sand clocks (clepsydræ), and similar timekeepers, are very reticent as to how these were controlled. Ptolemy, in his Almagest, states nothing whatever as to how the time was found when the numerous astronomical phenomena which he records took place; but Hipparchus in the only book we possess from his hand gives a list of forty-four stars scattered over the sky at intervals of right ascension equal to exactly one hour, so that one or more of them would be on the meridian at the commencement of every sidereal hour. In a very valuable paper^[1] Schjellerup has shown that the right ascensions assumed by Hipparchus agree within about 15 or one minute of time with those calculated back to the year 140 B.C. from modern star-places and pro per motions. The accuracy which, it thus appears, could be attained by the ancients in their determinations of time was far beyond what they seem to have considered necessary, as they only record astronomical phenomena (e.g., eclipses, occupations) as having occurred "towards the middle of the third hour," or " about 81/3 hours of the night," without ever giving minutes.^[2] The Arabians had a clearer perception of the importance of knowing the accurate time of phenomena, and in the year 829 we find it stated that at the commencement of the solar eclipse on 30th November the altitude of the sun was 7° and at the end 24°, as observed at Baghdad by Ahmed ibn Abdallah, called Habash.^[3] This seems to be the earliest determination of time by an altitude; and this method then came into general use among the Arabians, who on observing lunar eclipses never failed to measure the altitude of some bright star at the beginning and end of the eclipse. In Europe this method was adopted by Purbach and Regiomontanus, apparently for the first time in 1457. Bernhard Walther, a pupil of the latter, seems to have been the first to use for scientific purposes clocks driven by weights: he states that on 16th January 1484 he observed the rising of the planet Mercury and immediately attached the weight to a clock having an hour-wheel with fifty-six teeth; at sunrise one hour and thirty-five teeth had passed, so that the interval was an hour and thirty-seven minutes. For nearly two hundred years, until the application of the pendulum to clocks became general, astronomers could place little or no reliance on their clocks, and consequently it was always necessary to fix the moment of an observation by a simultaneous time determination. For this purpose Tycho Brahe employed altitudes observed with quadrants; but he remarks that they are not always of value, for if the star is taken too near the meridian the altitude varies too slowly, and if too near the horizon the refraction (which at that time was very imperfectly known) introduces an element of uncertainty. He therefore preferred azimuths, or with the large "armillary spheres" which played so important a part among his instruments he measured hour-angles or distances from the meridian along the equator.^[4] Transits of stars across the meridian were also observed with the meridian quadrant, an instrument which is alluded to by Ptolemy and was certainly in use at the Maragha (Persia) observatory in the 13th century, but of which Tycho was the first to make extensive use. It appears, however, that he chiefly employed it for determining star-places, having obtained the clock error by the methods already described.

In addition to these methods, that of "equal altitudes" was much in use during the 17th century. That equal distances east and west of the meridian correspond to equal altitudes had of course been known as long as> sun-dials had been used; but, now that quadrants, cross-staves, and parallactic rules^[5] were commonly employed for measuring altitudes more accurately, the idea naturally suggested itself to determine the time of a star's or the sun's meridian passage by noting the moments when it reached any particular altitude on both sides of the meridian. But Tycho's plan of an instrument fixed in the meridian was not for gotten, and from the end of the 17th century, when Roemer invented the transit instrument, the observation of transits across the meridian became the principal means of deter mining time at fixed observatories, while the observation of altitudes, first by portable quadrants, afterwards by reflecting sextants, and during the 19th century by portable alt-azimuths or theodolites, has been used on journeys. During the last fifty years the small transit instrument, with what is known as a "broken telescope," has also been much employed on scientific expeditions; but great caution is necessary in using it, as the difficulties of getting a perfectly rigid mounting for the prism or mirror which reflects the rays from the object-glass through the axis to the eye piece appear to be very great, for strange discrepancies in the results have often been noticed. The gradual development of astronomical instruments has been accompanied by a corresponding development in timekeepers. From being very untrustworthy, astronomical clocks are now made to great perfection by the application of the pendulum and by its compensation, while the invention of chronometers has placed a portable and equally trust worthy timekeeper in the hands of travellers.

We shall now give a sketch of the principal methods of determining time.

In the spherical triangle ZPS between the zenith, the pole, and a star the side ZP=QO-<f> (< being the latitude), PS =90 -5 (5 being the declination), and ZS or .=90 minus the observed alti tude. The angle ZPS=t is the star's hour-angle or, in time, the interval between the moment of observation and the meridian pass age of the star. We have then, cos Z- sin sin 5 cost = -^, cos <t> cos o which formula can be made more convenient for the use of logarithms by putting Z+<f> + 6=2S, which gives tan 2 U = s ( s ~ & S (S - V cos S cos(& - Z) According as the star was observed west or east of the meridian, t will be positive or negative. If a be the right ascension of the star, the sidereal time =t + a, a as well as d being taken from an ephemeris. If the sun had been observed, the hour-angle t would be the apparent solar time. The altitude observed must be cor rected for refraction, and in the case of the sun also for parallax, while the sun's semi-diameter must be added or subtracted, accord ing as the lower or upper limb was observed. The declination of the sun being variable, and being given in the ephemerides for noon of each day, allowance must be made for this by interpolating with an approximate value of the time. As the altitude changes very slowly near the meridian, this method is most advantageous if the star be taken near the prime vertical, while it is also easy to see that the greater the latitude the more uncertain the result. If a number of altitudes of the same object are observed, it is not necessary to deduce the clock error separately from each observa tion, but a correction may be applied to the mean of the zenith distances. Supposing n observations to be taken at the moments T lt Tp T 3, . . ., the mean of all being TO, -and calling the 2 corre sponding to this Z, we have As Z(T-T ) 2 3P*-** and so on, t being the hour-angle answering to = 0, these equations give ^ =?L 2 dt 2 1i di* n

But, if in the above-mentioned triangle we designate the angles at Z and S by 180 - A and p, we have

sin z sin A cos 5 sin't; sin z cos A = - cos $ sin S -f sin <p cos 5 cos t; and by differentiation d?Z_ cos cos S cos A ~~ in which A and p are determined by sin't sin't sin A = -. ^ cos o and sin p = -^^ cos a>. sin Z * sin Z

With this corrected mean of the observed zenith distances the hour-angle and time are determined, and by comparison with T the error of the timekeeper.

The method of equal altitudes gives very simply the clock error equal to the right ascension minus half the sum of the clock times corresponding to the observed equal altitudes on both sides of the meridian. When the sun is observed, a correction has to be applied for the change of declination in the interval between the observations. Calling this interval 2t, the correction to the apparent noon given by the observations x, the change of declination in half the interval AS, and the observed altitude h, we have

$\sin h=\sin \phi \sin(\delta -\Delta \delta )+\cos \phi \cos \delta -\Delta \delta )\cos(t+x)$

and $\sin h=\sin \phi \sin(\delta +\Delta \delta )+\cos \phi \cos(\delta +\Delta \delta )\cos(t-x)$,

whence, as cos x; may be put=1, sin x= x, and tan Δδ = Δδ,

$x=-\left({\frac {\tan \phi }{\sin t}}-{\frac {\tan \delta }{\tan t}}\right)\Delta \delta$

which, divided by 15, gives the required correction in seconds of time. Similarly an afternoon observation may be combined with an observation made the following morning to find the time of apparent midnight.

The observation of the time when a star has a certain azimuth may also be used for determining the clock error, as the hour-angle can be found from the declination, the latitude, and the azimuth. As the azimuth changes most rapidly at the meridian, the observation is most advantageous there, besides which it is neither necessary to know the latitude nor the declination accurately. In the article Geodesy (vol. x. p. 166) it has been shown how the observed time of transit over the meridian is corrected for the deviations of the instrument in azimuth, level, and collimation. This corrected time of transit, expressed in sidereal time, should then be equal to the right ascension of the object observed, and the difference is the clock error. In observatories the determination of a clock's error (a necessary operation during a night's work with a transit circle) is generally founded on observations of four or five "clock stars," these being standard stars not near the pole, of which the absolute right ascensions have been determined with great care, besides observation of a close circumpolar star for finding the error of azimuth and determination of level and collimation error.^[6]

Observers in the field with portable instruments often find it inconvenient to wait for the meridian transits of one of the few close circumpolar stars given in the ephemerides. In that case they have recourse to what is known as the method of time determination in the vertical of a pole star. The alt-azimuth is first directed to one of the standard stars near the pole, such as a or 5 Ursse Minoris, using whichever is nearest to the meridian at the time. The instrument is set so that the star in a few minutes will cross the middle vertical wire in the field. The spirit-level is in the meantime put on the axis and the inclination of the latter measured. The time of the transit of the star is then observed, after which the instrument, remaining clamped in azimuth, is turned to a clock star and the transit of this over all the wires is observed. The level is applied again, and the mean of the two results is used in the reductions. In case the collimation error of the instrument is not accurately known, the instrument should be reversed and another observation of the same kind taken. The observations made in each position of the instrument are separately reduced with an assumed approximate value of the error of collimation, and two equations are thus derived from which the clock error and correction to the assumed collimation error are found. This use of the transit or alt-azimuth out of the meridian throws considerably more work on the computer than the meridian observations do, and it is therefore never resorted to except when an observer during field operations is pressed for time. The formulae of reduction as developed by Hansen in the Astronomische Nachrichten (vol. xlviii. p. 113 sq.) are given by Chauvenet in his Spherical and Practical Astronomy (vol. ii. pp. 216 sq., 4th ed., Philadelphia, 1873). The subject has also been treated at great length by Döllen in two memoirs, Die Zeitbestimmung vermittelst des tragbaren Durchgangsinstrument im Verticale des Polarsterns (St Petersburg, 1863 and 1874, 4to).

Longitude.—Hitherto we have only spoken of the de termination of local time. But in order to compare ob servations made at different places on the surface of the earth a knowledge of their difference of longitude becomes necessary, as the local time varies proportionally with the longitude, one hour corresponding to 15. Longitude can be determined either geodetically or astronomically. The first method supposes the earth to be a spheroid of known dimensions. Starting from a point of departure of which the latitude has been determined, the azimuth from the meridian (as determined astronomically) and the distance of some other station are measured. This second station then serves as a point of departure to a third, and by repeating this process the longitude and latitude of places at a considerable distance from the original starting-point may be found. Referring for this method to the articles Earth (Figure of the), Geodesy, and Surveying, we shall here only deal with astronomical methods of determining longitude.

The earliest astronomer who determined longitude by astronomical observations seems to have been Hipparchus, who chose for a first meridian that of Rhodes, where he observed; but Ptolemy adopted a meridian laid through the "Insulse Fortunatae" as being the farthest known place towards the west.^[7] When the voyages of discovery began the peak of Teneriffe was frequently used as a first meridian, until a scientific congress, assembled by Richelieu at Paris in 1630, selected the island of Ferro for this purpose. Although various other meridians (e.g., that of Uranienburg and that of San Miguel, one of the Azores, 29 25 west of Paris) continued to be used for a long time, that of Ferro, which received the authorization of Louis XIII. on 25th April 1634, gradually superseded the others. In 1724 the longitude of Paris from the west coast of Ferro was found by Louis Feuillee, who had been sent there by the Paris Academy, to be 20 1 45"; but on the proposal of Guillaume de Lisle (1675-1726) the meridian of Ferro was assumed to be exactly 20 west of the Paris observatory. Modern maps and charts generally give the longitude from the observatory of either Paris or Greenwich according to the nationality of the constructor; the Washington meridian conference of 1884 has recommended the exclusive use of the meridian of Greenwich. On the same occasion it was also recommended to introduce the use of a "universal day," beginning for the whole earth at Greenwich midnight, without, however, interfering -with the use of local time.^[8]

The simplest method for determining difference of longitude consists in observing at the two stations some celestial phenomenon which occurs at the same absolute moment for the whole earth. Hipparchus pointed out how observations of lunar eclipses could be used in this way, and for about fifteen hundred years this was the only method available. When Regiomontanus (q.v.) began to publish his ephemerides towards the end of the 15th century, they furnished other means of determining the longitude. Thus Amerigo Vespucci observed on 23d August 1499, some where on the coast of Venezuela, that the moon at 7 h 30 m P.M. was 1, at midnight 5| east of Mars; from this he concluded that they must have been in conjunction at 6 h 30 m, whereas the Nuremberg ephemeris announced this to take place at midnight. This gave the longitude of his station as roughly equal to 5J hours west of Nuremberg. The instruments and the lunar tables at that time being very imperfect, the longitudes determined were very erroneous; see Navigation (vol. xvii. p. 251), to which article we may also refer for a history of the long-discussed problem of finding the longitude at sea. The invention of the telescope early in the 17th century made it possible to observe eclipses of Jupiter's satellites; but there is to a great extent the same drawback attached to these as to lunar eclipses, that it is impossible to observe with sufficient accuracy the moments at which they occur.

Eclipses of the sun and occultations of stars by the moon were also much used for determining longitude before the invention of chronometers and the electric telegraph offered better means for fixing the longitude of observatories. These methods are now hardly ever employed except by travellers, as they are very inferior as regards accuracy. For the necessary formulæ see venet's Spherical and Practical Astronomy, vol. i. pp. 518–542 and 550-557.

We now proceed to consider the four methods for find ing the longitudes of fixed observatories, viz., by (1) moon culminations, (2) rockets or other signals, (3) transport of chronometers, and (4) transmission of time by the electric telegraph.

1. Moon Culminations.—Owing to the rapid orbital motion of the moon the sidereal time of its culmination is different for different meridians. If, therefore, the rate of the moon's change of right ascension is known, it is easy from the observed time of culmina tion at two stations to deduce their difference of longitude. Let the right ascension of the moon a and its differential coefficients be computed for the Greenwich time T, and let the culmination be observed at two places whose longitudes from Greenwich are X and X, the time of observation being T+t and T+t Greenwich time, or in local time T+t + = and T+t + =d; we have then

and, as the difference of longitude is -=(9 / -0)-(^-), we have only to determine t -t from the first equation. This is simply done by a suitable selection of T. Calling T+^(t + t )=T, we have to put T -^(l -t) and r + (t -t) for T+t and T+t . It is then easy to see that and, solving this equation by first neglecting the second term on the right side and then substituting the value of t t, thus found . o -e ire -ff- 3 cPa m that term, t - t=

In order to be as much as possible independent of instrumental errors, some standard stars nearly on the parallel of the moon are observed at the two stations; these " moon-culminating stars " are given in the ephemerides in order to secure that both observers take the same stars. As either the preceding or the following limb, not the centre, of the moon is observed, allowance must be made for the time the semi-diameter takes to pass the meridian and for the change of right ascension during this time. This method was proposed by Pigott towards the end of the 18th century, and has been much used; but, though it may be very serviceable on journeys and expeditions to distant places where the chronometric and tele graphic methods cannot be employed, it is not accurate enough for fixed observatories. This is due, not only to the difficulties attend ing the observation (the difference of personal error in observing the moon and stars, the different apparent enlargement of the moon by irradiation in different telescopes and under different atmo spheric circumstances, &c.), but chiefly to the large coefficient with which ff - d has to be multiplied in the final equation for X - X. Errors of four to six seconds of time have therefore frequently been noticed in longitudes obtained by this method from a limited number of observations: the longitude of the Madras observatory was for many years assumed to be 5 n 21 m 3 8 77, but subsequently by a telegraphic determination this was found to be 4 S 37 too great.

2. Signals.—In 1671 Picard determined the difference of longi tude between Copenhagen and the site of Tycho Brahe's observa tory by watching from the latter the covering and uncovering of a fire lighted on the top of the observatory tower at Copenhagen. Powder or rocket signals have been in use since the middle of the 18th century; they are nowadays never used for this purpose, although several of the principal observatories of Europe were con nected in this manner early in the 19th century.^[9]

3. Transport of Chronometers.—This means of determining longi tude was first tried in cases where the chronometers could be brought the whole way by sea, but the improved means of communication on land led to its adoption in 1828 between the observatories at Greenwich and Cambridge, and in the following years between many other observatories. A few of the more extensive expedi tions undertaken for this object deserve to be mentioned. In 1843 more than sixty chronometers were sent sixteen times backwards and forwards between Altona and Pulkowa, and in 1844 forty chronometers were sent the same number of times between Altona and Greenwich.^[10] In 1844 the longitude of Valentia on the south west coast of Ireland was determined by transporting thirty pocket chronometers via Liverpool and Kingstown and having an intermediate station at the latter place. The longitude of the United States naval observatory has been frequently determined from Greenwich. The following results will give an idea of the accuracy of the method.^[11]

Previous to 1849, 373 chronometers 5 8 12- 52 Expedition of 1849, Bond's discussion 11 - 20 Walker's 12* 06,, Bond's second result 12 -260 20 1855, 52 chronometers, 6 trips, Bond . . 13 490s-19

The value now accepted from the telegraphic determination is 5^h 8^m 12^s·09. The probable errors of the results for PulkowaAltona and Altona-Greenwich were supposed to be S- 039 and S- 042. It is of course only natural that the uncertainty of the results for the trans-Atlantic longitude should be much greater, considering the length of time which elapsed between the rating of the chronometers at the observatories of Boston (Cambridge, Massachusetts) and Liverpool. The difficulty of the method con sists in determining the "travelling rate." Each time a chrono meter leaves the station A and returns to it the error is determined, and consequently the rate for the time occupied ly the journeys from A to B and from to A and by the sojourn at B. Similarly a rate is found by each departure from and return to B, and the time of rest at A and B is also utilized for determining the station ary rate. In this way a series of rates for overlapping intervals of time are found, from which the travelling rates may be interpolated. It is owing to the uncertainty which necessarily attaches to the rate of a chronometer during long journeys, especially by land, where they are exposed to shaking and more or less violent motion, that it is desirable to employ a great number. It is scarcely neces sary to mention that the temperature correction for each chrono meter must be carefully investigated, and the local time rigorously determined at each station during the entire period of the operations.

4. Telegraphic Determination of Longitude.—This was first suggested by the American astronomer S. C. Walker, and owed its development to the United States Coast Survey, where it was employed from about 1849. Nearly all the more important public observa tories on the continent of Europe have now been connected in this way, chiefly at the instigation of the "Europaische Gradmessung," while the determinations in connexion with the transits of Venus and those carried out in recent years by the American and French Governments have completed the circuit of the greater part of the globe. The telegraphic method compares the local time at one station with that at the other by means of electric signals. If a signal is sent from the eastern station A at the local time T, and received at the western station B at the local time 2, then, if the time taken by the current to pass through the wire is called x, the difference of longitude is X^T-Ti+x, and similarly, if a signal is sent from B at the time T z and received at A at TS, we have = T 3 - T 3 - x, from which the unknown quantity x can be eliminated.

The operations of a telegraphic longitude determination can be arranged in two ways. Either the local time is determined at both stations and the clocks are compared by telegraph, or the time determinations are marked simultaneously on the two chronographs at the two stations, so that further signals for clock comparison are unnecessary. The first method has to be used when the tele graph is only for a limited time each night at the disposal of the observers, or when the climatic conditions at the two stations are so different that clear weather cannot often be expected to occur at both simultaneously, also when the difference of longitude is so considerable that too much time would be lost at the eastern station waiting for the arrival of the transit record of one star from the western station before observing another star. The independent time determination also offers the advantage that the observations may be taken either by eye and ear or by the chronograph, and that the signals may be either audible beats of a relay or chronographic signals, the rule being to have observations and signals made by similar operations. The best way of using audible beats of a relay is to let the circuit pass through an auxiliary clock, which from second to second alternately makes and breaks the current, the making of the current being rendered audible by the tapping of the relays at both stations. If, now, the auxiliary and the observing clocks are regulated to a different rate, the coinci dences of the beats of the relay with those of the observing clock can be noted with great accuracy, from which the difference between the two observing clocks is found. It has been proved by experience that the degree of accuracy with which the clock comparison can l>e made by one coincidence is exactly equal to that of one chrono graph signal, the probable error being in both cases about S 015. It should, however, be mentioned that the interval between two consecutive coincidences cannot be made less than two minutes, whereas the chronograph signals may be given every second, and, as the observations made with the chronograph are also somewhat more accurate than those made by eye and ear, the chronograph should be used wherever possible. The other method, that of simultaneous registration at both stations of transits of the same stars, has also its advantages. Each transit observed at both stations furnishes a value of the difference of longitude, so that the final result is less dependent on the clock rate than in the first method, which necessitates the combination of a series of clock errors determined during the night to form a value of the clock error for the time when the exchange of signals took place. When using this method it is advisable to select the stars in such a manner that only one station at a time is at work, so that the intensity of the current can be readjusted (by means of a rheostat) between every despatch and receipt of signals. This attention to the intensity of the current is necessary whatever method is employed, as the constancy of the transmission time (x in the above equations) chiefly depends on the constancy of the current. The probable error of a difference of longitude deduced from one star appears to be^[12]

for eye and ear transits ±0^[13]·08,

for chronograph transits ±0^s·07;

while the probable error of the final result of a carefully planned and well executed series of telegraphic longitude operations is generally between ±0^s·015 and ±0^s·025.

It is evident that the success of a determination of longitude depends to a very great extent on the accurate determination of time at the two stations, and great care must therefore be taken to determine the instrumental errors repeatedly during a night's work. But, in addition to the uncertainty which enters into the results from the ordinary errors of observation, there is another source of error which becomes of special importance in longitude work, viz., the so-called personal error. The discovery of the fact that all observers differ more or less in their estimation of the time when a star crosses one of the spider lines in the transit instrument was made by Bessel in 1820^[14]; and, as he happened to differ fully a second of time from several other observers, this remarkably large error naturally caused the phenomenon to be carefully examined. Bessel also suggested what appears to be the right explanation, viz., the co-operation of two senses in observing transits by eye and ear, the ear having to count the beats of the clock while the eye com pares the distance of the star from the spider line at the last beat before the transit with the distance at the first beat after it, thus estimating the fraction of second at which the transit took place. It can easily be conceived that one person may first hear and then see, while to another these sensations take place in the reverse order; and to this possible source of error may be added the sensible time required by the transmission of sensations through the nerves to the brain and for the latter to act upon them. As the chronographic method of observing dispenses with one sense (that of hearing) and merely requires the watching of the star's motion and the pressing of an electric key at the moment when the star is bisected by the thread, the personal errors should in this case be much smaller than when the eye and ear method is employed. And it is a fact that in the former method there have never occurred errors of between half and a whole second such as have not unfrequently appeared in the latter method.

In astronomical observations generally this personal error does not cause any inconvenience, so long as only one observer is employed at a time, and unless the amount of the error varies with the declination or the magnitude of the star; but when absolute time has to be determined, as in longitude work, the full amount of the personal equation between the two observers must be care fully ascertained and taken into account And an observer's error has often been found to vary very considerably not only from year to year but even within much shorter intervals; the use of a new instrument, though perhaps not differing in construction from the accustomed one, has also been known to affect the personal error. For a number of years this latter circumstance was coupled with another which seemed perfectly incomprehensible, the personal error appearing to vary with the reversal of the instrument, that is, with the position of the illuminating lamp east or west But in 1869-70 Hirsch noticed during the longitude operations in Switzerland that this was caused by a shifting of the reflector inside the telescope, by means of which the field is illuminated, which produced an apparent shifting of the image of the spider lines, unless the eye-piece was very accurately focused for the observer's sight. The simplest and best way to find the equation between two observers is to let one observe the transits of stars over half the wires in the telescope, and the other observe the transits over the remainder, each taking care to refocus the eye-piece for himself in order to avoid the abovementioned source of error. The single transits reduced to the middle wire give immediately the equation; and, in order to eliminate errors in the assumed wire-intervals, each observer uses alternately the first and the second half of the wires. Another method is in vogue at Greenwich, where each observer with the transit circle from a series of stars determines the clock error and reduces this to a common epoch (O h sid. time) by means of a clock rate found independently of personal error. The differences between the clock errors thus found are equal to the personal equations. This method cannot, however, be recommended, as the systematic errors in the right ascensions of the stars and any slight variation of the clock rate would affect the personal equation; the first method is there fore generally used in longitude work. It is advisable to let the observers compare themselves at the beginning, middle, and end of the operations and, if possible, at both the instruments employed. A useful check on the results is afforded by simultaneous experiments with one of the instruments contrived by C. Wolf, Kaiser, and others (sometimes called "time collimators"), by which the absolute personal error of an observer can be determined. Though differing much in detail, these instruments are all constructed on the same principle: an artificial star (a lamp shining through a minute hole in a screen mounted on a small carriage moved by clockwork) passes in succession across a number of lines drawn on oiled paper, while an electric contact is made at the precise moment when the star is bisected on each line by the carriage passing a number of adjustable contact makers. The currents thus made register the transits automatically on a chronograph, while the observer, viewing the apparatus through his telescope, can observe the transits in the usual manner either oy eye and ear or by chrono graph, thus immediately finding his personal error. On the Continent these contrivances have frequently been used to educate pupils learning to observe, and experience has repeatedly shown that a considerable personal error can be generally somewhat diminished through practice.

Literature. General treatises on spherical astronomy, such as Brunnow's LehrbuchderspharischenAstronomie(3d ed., Berlin, 1871; translated into English and several other languages) and Chauvenet's Manual, treat very fully of the numerous methods of determining time by combination of altitudes or azimuths of several stars. The best handbook of telegraphic longitude work is Albrecht's already mentioned; but any one engaging in practical work of this kind should consult the accounts of the numerous longitude determinations carried out during recent years, particularly the Publicationen des kon. preussischeii, geoddtischen Institute; Telegraphic Determination of Differences of Longitude by Officers of the United States Navy (Washington, 1880); Telegr. Determ. "of Longitudes in Mexico, Central America, and on the West Coast of South America (Wash ington, 1885); the Reports of the United States Coast ami Geodetic Survey; vol. ix. of the Account of the Great Trigonometrical Survey of India; and vol. iii. of Dun Echt observatory Publications. A discussion of all the investigations on personal errors up to 1875 was published by Dreyer hi Proc. R. Irish Acad.. 2d series, vol. ii., 1876, pp. 484-528. (J. L. E. D.)

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1. "Recherches sur l'Astronomie des Anciens: I. Sur le chronomètre céleste d'Hipparque," in Copernicus: An International Journal of Astronomy, i. p. 25.
2. For astronomical purposes the ancients made use of mean-time hours—ώραι ίσημεριναι, horæ equinoctiales—into which they translated all indications expressed in civil hours of varying length—ώραι καιρικαί, horæ temporales. Ptolemy counts the mean day from noon.
3. Caussin, Le livre de la grande table Hakémite, Paris, 1804, p. 100.
4. See his Epistolæ astronomicæ, p. 73.
5. Navigation, (vol. xvii. pp. 251 and 253).
6. The probable error of a clock correction found in this way from one star with the Dunsink transit circle was ±0^s·52.
7. This was probably first done in the first century by Marinus of Tyre.
8. This proposal was chiefly dictated by a wish to facilitate the international telegraph and railway traffic. In the United States, where the large extent of the country in longitude makes it impossible to use the time of one meridian, four standard meridians were adopted in 1883, viz., 75, 90, 105, 120 west of Greenwich, so that clocks showing "Eastern, Central, Mountain, or Pacific time" are exactly five, six, seven, or eight hours slower than a Greenwich mean-time clock.
9. For instance, Greenwich and Paris in 1825 (Phil. Trans., 1826). The result, 6^m2l^s-·6, is only about 0^s·6 too great.
10. As a great many of the chronometers used in 1844 were made by Dent and were of superior excellence, a smaller number was considered sufficient.
11. Gould, Transatlantic Longitude, p. 5, Washington, 1869.
12. Albrecht, Bestimmung von Längendifferenzen mit Hülfe des electrischen Telegraphen, p. 80, Leipsic, 1869, 4to.
13. s
14. Maskelyne had in 1795 noticed that one of his assistants observed transits more than half a second later than himself, but this was supposed to arise from some wrong method of observing adopted by the assistant, and the matter was not further looked into.