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 0 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 inconvenience 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 (clepsydrae), 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 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 proper 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, occultations) as having occurred “towards the middle of the third hour,” or “about 8⅜ hours of the night,” without ever giving minutes. 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. 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. 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 Marágha (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 17 th 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 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 forgotten, 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 determining 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 eyepiece 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 trustworthy 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=90°-≠ (≠ being the latitude), PS=9Q o -δ (δ being the declination), and ZS or Z=90° minus the observed altitude. The angle ZPS=t is the star’s hour-angle or, in time, the interval between the moment of observation and the meridian passage of the star. We have then j , cos Z- sin φ sin δ cos t= -— , cos φ cos δ

which formula can be made more convenient for the use of logarithms by putting Z+ φ + δ= 2S, which gives 2 sin(S-≠)sin(N-δ) tan^ κt — r . — 7 rr~. λ cos £ cos(0 - Z)

According as the star was observed west or east of the meridian, t will be positive or negative. If α be the right ascension of the star, the sidereal time =Z + α, a as well as δ 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 corrected for refraction, and in the case of the sun also for parallax, while the sun’s semi-diameter must be added or subtracted, according 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 observation, but a correction may be applied to the mean of the zenith distances. Supposing n observations to be taken at the moments T v , T 2 , T a ,..., the mean of all being 7 ’ 0 ,∙and calling the z corresponding to this Z, we have ⅛=z+f( 3 ∙ 1 -r<,) + 15(¾-¾)> i and so on, t being the hour-angle answering to To- As Σ(T - 2 τ 0 ) =0, these equations give ¾+⅞+⅞+... 1 d 2 Z (T 1 -T 0 ) 2 + (T 2 -T 0 ) 2 +... . n 2 dt 2 n ⅞+⅝*Fi⅛4* « » d~Z 22sin- ⅜(T-- Tj) ~ n dt 2 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 δ sin t; sin z cos A = — cos φ sin δ + sin φ cos δ cos t; and by differentiation d 2 Z _ cos φ cos δ cos A co·&p dC 2 ~ sin Z ’ in which A and p are determined by sin t . . sin t sin A = -— cos δ and sin p = ---~ cos Φ∙ sin Z sin Z

With this corrected mean of the observed zenith distances the hour-angle and time are determined, and by comparison with 7 ’ 0 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 ∆δ, and the observed altitude h, we have sin Λ=sin φ sin (δ - ∆δ) + cos φ cos (δ - ∆δ) cos (t+«) and sin h= sin φ sin (δ + ∆δ) + cos φ cos (δ + ∆δ) cos (í -.x), whence, as cos a; may be put=l, sinae=ae, and tan∆δ=∆δ,

/tan φ tan δ∖ . s x= - ( -—- )∆δ,

X sin t tan t f 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.

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 δ Ursae 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 Durchgangs- instrument im Verticale des Polarsterns (St Petersburg, 1863 and 1874, 4to).

Longitude. — Hitherto we have only spoken of the determination of local time. But in order to compare observations 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 “Insulae Fortunatae” as being the farthest known place towards the west. 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 Feuillée, who had been sent there by the Paris Academy, to be 20° Γ 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.

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 (g.ν.) 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, somewhere 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 5 J 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...