A New Method of Finding Time by Equal Altitudes. By Alexander Aubert, Esq. F. R. S.

IV. A new Method of finding Time by equal Altitudes. By Alexander Aubert, Esq. F. R. S. R. Nov. 9, 1775. Among the various methods practised for finding time, that by equal altitudes of the Sun or of a star, hath hitherto been esteemed the most eligible for observers who are not furnished with a good and well-adjusted transit instrument. But this method, although one of the best, is generally attended with inconveniencies, which render the practice of it more difficult, and the result less perfect than one could wish. If the Sun or stars near the equator are made use of, as usual, and the altitudes are taken near the prime vertical, where the change of altitude is the quickest, the interval of time between the observations must, in most latitudes, be of so many hours, that the observer cannot always attend to the corresponding altitudes; the weather may prove variable, so as to disappoint him at last; the clock or watch may go irregularly during so long an interval; and if the altitudes cannot, on account of their great distance from the meridian, be taken very high; an alteration in the state of the atmosphere may produce a variation of the refraction, and occasion the horary arcs to be different, although the apparent altitudes will be the same. To which may be added, the difficulty of making making the instrument follow the object in its motion in azimuth, without danger of disturbing its adjustment in regard to altitude. To remedy all these inconveniencies, the following method was thought of; and having been practised with constant success, it is presumed, the communication of it may be acceptable to astronomers. If a star is selected, of which the polar distance is very little less than the complement of the latitude of the place of observation, it will, at equal distances from the meridian, come to vertical circles, which touch its parallel of declination. The star, when in these vertical circles, will be near the meridian, near the prime vertical, and near the zenith; and consequently, if it be observed there, the interval between the Eastern and the Western altitudes will be short; the alteration in altitude will be quick; the star cannot be affected by a different refraction; besides, it will have no motion in azimuth. To observe the star in these vertical circles, two things are necessary; the first is, to be provided with an astronomical quadrant, having three or more horizontal wires in the telescope, and if it have also a speculum at the eye-end of the telescope, to bring the vertical ray horizontal, it will be found very convenient. The next thing is, to make a computation of the apparent zenith distance of the star in the vertical circles which touch its parallel of declination; for if the telescope be fixed to this zenith distance, as soon as the star is found found to come to it, it will be in the proper vertical circle. The true zenith distance is found easily in the following manner; and if diminished by the refraction belonging to it, it will give the apparent zenith distance wanted. Let \( \text{HMOM} \) represent the horizon; \( z \) the zenith; \( MM \) the meridian; \( P \) the pole of the equator; \( \gamma \gamma \) the parallel of declination of a star, intersecting the meridian very near the zenith between \( z \) and \( P \). Then, the vertical circles \( ZE \) on the Eastern side of the meridian, and \( zw \) on the Western side, being drawn to touch the path of the star, the hour circles passing through the points of contact \( E, W \), will be at right angles to the circles of azimuth \( ZE, zw \). Now in the right-angled spheric triangle \( PEZ \); \( pz \) the complement of the latitude, and \( PE \) the star's polar distance, being known, the other parts are easily found; viz. \( ZE \) the zenith distance, \( PZE \) the azimuth, \( ZPE \) the horary angle. So that the telescope may be fixed properly, the position of the instrument is ascertained, and the time known when the star may be expected in the field field of the telescope; and if the vertical wire is once brought near to the star, the star will appear to move parallel to it, and will pass successively over the horizontal wires, while the instrument remains undisturbed. The advantage of this method will appear in the following example of equal altitudes, taken the 15th July, 1773, at Loam-pit Hill, near Deptford, in latitude $51^\circ 28' 7''$ N. and longitude $5''$ in time W. of the Royal Observatory at Greenwich. The star selected was $\gamma$ Draconis, having $38^\circ 28' 21''$ apparent North polar distance, being very little less than the complement of the latitude $38^\circ 31' 53''$. Then $\cos. PE : \text{rad.} :: \cos. PZ : \cos. ZE = 2^\circ 19'$ the zenith distance, and $\sin. PZ : \text{rad.} :: \sin. PE : \sin. PZE = 87^\circ 5' 20''$ the azimuth, also rad. : tan. PE :: cotan. PZ : $\cos. ZPE = 3^\circ 43' 13''$ the horary arc = $14^\circ 52',9$ in sidereal time, or $14^\circ 50',5$ in mean time. The true zenith distance being $2^\circ 19'$, the same was diminished by $2''$ for refraction, and the telescope fixed to $2^\circ 18' 58''$, the apparent zenith distance; and when the star came to the wires, the times by the clock were as follow: | Eastern altitudes | Western altitudes | Meridian passage | |-------------------|-------------------|------------------| | 1st wire at | | | | 9 55 43 | 2' 14'' | 10 29 46 | | 2d | | | | 9 57 57 | 2 12 | 10 27 32 | | 3d | | | | 10 0 9 | | 10 25 20 | so that in about $34'$ the compleat set of altitudes was obtained near the prime vertical, free from the effects of a different refraction, and any motion in azimuth. The horary Mr. Aubert's Method of horary arc observed by the middle wire not turning out exactly according to the computation, is of no consequence to the observations. Some little difference may arise in it from small inaccuracies in the estimation of the star's apparent polar distance, the latitude of the place, or the error of the line of collimation; or from not setting the telescope exactly to the proper zenith distance; but as the chief intention of the computation is to find the vertical circles in which the star hath no motion in azimuth, the other parts of it need not be strictly attended to. The following manner of inferring mean time from the star's meridian passage being more convenient and concise than the usual one, may also be acceptable. From the star's apparent right ascension, increased by 24 hours if necessary, subtract the Sun's apparent right ascension for apparent noon; diminish the remainder by the proportional part of the star's acceleration, at the rate of $3'55''$,91 for 24 hours, of which a table is easily computed; to this last remainder apply the equation of time for apparent noon, according as it is additive or subtractive; the result will be the mean time of the star's passing the meridian. Example. finding Time by equal Altitudes. EXAMPLE. The apparent $A$ of $\gamma$ Draconis the 15th July, 1773, was — the apparent $A$ of the Sun at apparent noon, First remainder, — the star’s acceleration for $10^h 11^m 25^s$, at $3^h 55^m 9^s$ for 24 hours Second Remainder, + the equation of time at apparent noon, additive, Star’s meridian passage in mean time, But the clock shewed $10^h 12^m 44^s$, when the star passed, consequently it was $2' 28''$, too slow for mean time. Observers, who are not furnished with tables of the Sun’s right ascension and of the equation of time for the apparent noon of their meridian, may apply both as they are given in the Nautical Ephemeris for the meridian of the Royal Observatory at Greenwich; the result will be the mean time of the star’s passing the Greenwich meridian. And by applying the proportional part of the foregoing acceleration of $3' 55''$, belonging to the difference of longitude in time of the place of observation from Greenwich, the mean time of the star’s passing the meridian of the place of observation will be found. If the place be to the East of Greenwich, the acceleration will be additive; if to the West, subtractive. Vol. LXVI. In a similar manner, the mean time of any observation made with a clock, regulated to sidereal time, may be inferred, provided the preceding transit of the Sun hath been observed; for if from the time of the observation by the clock, increased, if necessary, by 24 hours, the time of the observed transit of the Sun be subtracted, the remainder, diminished by the proportional part of $3'55''$, and duly corrected by the equation of time for the preceding noon, will give the mean time required. It is understood, that the clock keeps the rate of sidereal time exactly; for if not, a further correction for the loss or gain since noon must be applied.