To the logarithmic sine of the sum of this arch and the sun's reduced declination, add the logarithmic sine of their difference; half the sum will be the logarithmic sine of the latitude of the place of observation.

Example 1.

July 13th, 1824, in north latitude, and longitude 120 west, the sun's lower limb, at the time of its setting, was observed to touch the horizon at 7:59:58 apparent time, and the upper limb at 8:44; required the latitude of the place of observation?

Apparent time of sun's setting = 7:59:58:8:44:÷2=

Longitude 120: west, in time =

Greenwich time of sun's setting =

8 2 1:

8. 0. 0

16. 2. 1

=

Sun's semi-diameter 15:45". 8, in seconds 945". 8 Log. 2.975799 Interval of time between the setting of the sun's

=

lower and upper limbs 476, or 246: Log. ar. comp. 7.609065 Constant log. (the ar. comp. of the prop.

log. of 24 hours esteemed as minutes) =

9.124939

Latitude of the place of obs. 50:46:34"N. Log. sine =

Example 2.

October 1st, 1824, in north latitude, and longitude 105: east, the sun's upper limb, at the time of its rising, was observed to emerge from the horizon at 6:343', and the lower limb at 6:632'; required the latitude of the place of observation?.

Apparent time of sun's rising =63436:6:32 +2= 6 5 7: Longitude 105: east, in time =

Greenwich time past noon, September 30th =

7. 0. 0

11: 5: 7:

Sun's declination at noon, September 30th, 1824, = 2:52:46% S.

Correction of ditto for 11:574:=

Sun's reduced declination =

Sun's semi-diameter

+ 10.48

16:1". 2, in seconds 961".2 Log.=2.982814

Interval of time between the rising of the

sun's upper and lower limbs=249, or 169: Log. ar. comp. 7.772113 Constant log. =

9.124939

Latitude of the place of observation=40°31' N. Log. sine. 9.812691

Remark. In this method of finding the latitude, it is indispensably necessary that the interval of time (per watch) between the instants of the sun's lower and upper limbs touching the horizon be determined to the nearest second; otherwise the latitude resulting therefrom may be subject to a considerable error, particularly in places where the limbs of that object rise or set in a vertical position; which is frequently the case in parts within the tropics.

SOLUTION OF PROBLEMS RELATIVE TO APPARENT TIME.

Time, as inferred directly from observations of the sun, is denominated either apparent or mean solar time. Apparent time is that which is deduced from altitudes of the sun, moon, stars, or planets. Mean time arises from a supposed uniform motion of the sun: hence, a mean solar day is always of the same determinate length; but the measure of an apparent day is ever variable,-being longer at one time of the year, and shorter at another, than a mean day; the instant of apparent noon will, therefore, sometimes precede, and at other times follow, that of mean The difference of those instants is called the equation of time; which equation is expressed by the difference between the sun's true right ascension and his mean longitude, corrected by the equation of the Equinoxes in right ascension, and converted into time at the rate of 1 minute. to every 15 minutes of motion, &c. &c. The equation of time is always equal to the difference between the times shown by an uniform or equable going clock, and a true sun-dial.

noon.

The sun's motion in the Ecliptic is constantly varying, and so is his motion in right ascension; but since the latter is rendered further unequal, on account of the obliquity of the Ecliptic to the Equator, it hence follows that the intervals of the sun's return to the same meridian become unequal, and that he will gradually come to the meridian of the same place too late, or too early, every day, for an uniform motion, such as that shown by an equable going watch or clock.

It is this retardation, or acceleration of the sun's coming to the meridian of the same place, that is called the equation of time; which implies a correction additive to, or subtractive from, the apparent time, in order to reduce it to equable or mean time.

The equation of time vanishes at four periods in the year,-which happen, at present, about the 15th of April, the 15th of June, the 31st of August, and the 24th of December; because, at these periods, there is no difference between the sun's true right ascension and his mean longitude: hence the apparent noon, at those times, is equal to the mean noon. When the sun's true right ascension differs most from his mean longitude, the equation of time is greatest: this happens, at present, about the 11th of February, the 15th of May, the 27th of July, and the 3d of November. But, since at those times the diurnal motion of the sun in right ascension is equal to his mean motion in longitude, or 59:8", the length of the apparent day, at these four periods, is, therefore, equal to that of a mean day: at all other times of the year, the lengths of the apparent and mean days differ; and it is the accumulation of those differences that produces the absolute equation of time.

The equation of time is additive from about the 25th of December to the 15th of April, and, again, from the 16th of June to the 31st of August; because, during the interval between those periods, the sun comes to the meridian later than the times indicated by a well-regulated clock: but it is subtractive from about the 16th of April to the 15th of June, and, again, from the 1st of September to the 24th of December; because, during the interval between these periods, the sun comes to the meridian earlier than the times indicated by an equable going clock.

The equation of time is contained in page II. of the month in the Nautical Almanac ; but, since it is calculated for the meridian of the Royal Observatory at Greenwich, and for noon, a correction, therefore, becomes necessary, in order to reduce it to any other meridian, and to any given time under that meridian. This correction is to be found by Problem V., page 298; or by means of Table XV., as explained in page 25.

PROBLEM I,

To find the Error of a Watch or Chronometer, by equal Altitudes of the Sun.

RULE.

In the morning, when the sun is nearly in the prime vertical, or at least when he is not less than two hours distant from the meridian, let several altitudes of his upper or lower limb be taken, and the corresponding times (per watch) increased by 12 hours, noted down in regular succession. In the afternoon, observe the instants when the same limb of the sun, taken in the morning, comes to the same altitudes, and write down each, augmented by 24 hours, opposite to its respective altitude. Take the means of the morning and of the afternoon times of observation; add them together, and half their sum will be the time of noon, per watch, incorrect. The difference between the means of the morning and afternoon times will be the interval between the observations: with this interval, and the latitude, enter Table XIII., and with the interval and the declination, corrected for longitude, enter Table XIV.; take out the corresponding equations, noting whether they be affirmative or negative, agreeably to the rule in page 23: then, with the sum or difference of those two equations, according as they are of the same or of contrary signs, and the variation of the sun's declination for the given day, compute the equation of equal altitudes, by the said rule in page 23.. Now, to the time of noon, per watch, incorrect, apply the equation of equal altitudes, by addition or subtraction, according as its sign is affirmative or negative, and the sum or difference will be the time, per watch, of apparent noon, or the instant when the sun's centre was on the meridian of the place of observation; the difference between which and noon, or 24 hours, will be the error of the watch for apparent time.

If the watch be regulated to mean solar time, such as a chronometer, let the equation of time (as given in the Nautical Almanac, and reduced to the meridian of the place of observation by Problem V., page 298,) be applied to noon, or 24 hours, by addition or subtraction, according to its title, and the mean time of noon will be obtained; the difference between which and the time, per watch, of apparent noon, will be the error of the watch for mean solar time.

Example 1.

March 1st, 1825, (civil time) in latitude 50:48:N., and longitude 30: W., the following equal altitudes of the sun were observed; required the error of the watch?

Afternoon mean= 27.59.46 Forenoon mean=20. 0.59

50:48 and interval 7:58*47:= -16"59"; negative, because the sun is

Equation, Table XIV., ans. to dec.

advancing towards the elevated pole.

7:32.25"S. and int. 75847:= 0.55; negative, because the sun's

Watch true for apparent time=0 0 0 Watch slow for

Example 2.

mean time =

12:38:

August 2d, 1825, (civil or nautical time) in latitude 50:48: N., and longitude 30: W., the following equal altitudes of the sun were observed; required the error of the watch?

* Since the morning observations belong, astronomically, to February 28th, therefore, half the sum of the variation of the sun's declination, for the days preceding and following the given one, is to be taken for the true variation of declination.