To find the polar distance of a celestial object, proceed according to the following rule :—

RULE LVIII.

When the latitude of the place, and declination of the object, are of the same name subtract the declination from 90°; but when the latitude and declination are of contrary names, add the declination to 90°; the result in either case is the polar distance.

When the latitude is o, the declination, either added to or taken from 90°, is the polar distance.

TO FIND THE EQUATION OF TIME.

APPARENT SOLAR DAY, is the interval between two successive transits of the actual sun's centre over the same meridian; it begins when that point is on the meridian. The apparent solar day is variable in length from two causes; first, the sun does not move uniformly in the ecliptic -its apparent path sometimes describing an arc of 57', and at other times an arc of 61' in a day; second, the ecliptic twice crosses the equinoctial-the great circle whose plane is perpendicular to the axis of rotation—and hence is inclined to it in its different parts; at the points of intersection the inclination is about 23° 27', at two other limiting points they are parallel. A uniform measure of time is obtained by the invention of the Mean Solar Day.

MEAN SOLAR DAY is the interval between two successive transits of the mean sun over the same meridian; it begins when the mean sun is on the meridian. This fictitious body is conceived to move in the equinoctial with the mean motion of the actual sun in the ecliptic. The length of the mean solar day is the average length of the apparent solar days for the space of a solar year.

EQUATION OF TIME is the difference between apparent and mean time, It is measured by the angle at the pole of the heavens between two circles passing, the one through the apparent sun's centre, the other through the mean sun. The Equation of Time is so called because it

enables us to reduce apparent to mean, or mean to apparent time. In consequence of the motion of the sun in the ecliptic being variable, and the ecliptic not being perpendicular to the axis of the earth's rotation, apparent time is variable, and this fluctuation is considerable, amounting to upwards of half an hour-apparent noon sometimes taking place as much as 161m before mean noon, and at others as much as 144" after. These are the greatest values of the equation of time; it vanishes altogether four times a year-this occurring about April 15th, June 15th, September 1st, and December 24th. It is calculated and inserted in the Nautical Almanac for every day in the year. On page I of each month the equation of time given is that to be used in deducing mean from apparent time; that on page II is to be used in deducing apparent from mean time. The difference in the value of the two arises from the one being that at apparent noon, and the other that at mean noon. As these may be separated by an interval of more than a quarter of an hour, the equation of time given in pages I and II may differ by a quarter of the "Diff. for 1 hour" given in the adjoining column. The equation of time is itself a portion of mean time.

I

RULE LIX.

By hourly difference.

1o. Get a Greenwich date, as before.

Take out of Nautical Almanac, page II of the month, the equation of time for the noon of Greenwich date, and mark it additive or subtractive, according to the heading of equation of time in page I of the month; also take from the column in page I, the "Diff. for 1 hour."*

3°. Multiply the "Diff. for 1 hour" by the hours, and when great precision is necessary, by the fractional parts of an hour also. The result is the correction to be applied to the equation of time taken from the Nautical Almanac, and is to be added when equation of time is increasing but subtracted when equation of time is decreasing.

(a) When the correction, being subtractive, exceeds the equation of time itself, subtract the equation of time from the correction; the remainder is the reduced equation of time sought-and it is to be subtracted from apparent time when equation of time at noon is directed to be added, but added to apparent time when equation of time at noon is directed to be subtracted.

The Greenwich middle time being found, the hourly difference may be reduced to it in the same way as directed for the declination (see Note, pages 151 and 152). This is a refinement wholly unnecessary.

EXAMPLES.

1872, January 29th, 6h 53m 49s mean time at Greenwich: find equation of time to be applied to apparent time (in working the chronometer).

Ex. I.

In working this example we take the "Diff. for 1 hour" from the Nautical Almanac from the column on page I of the month, and against the given day. The Greenwich date being mean time we take the equation of time from page II of the month, and mark it add as directed in page I; also note that the equation of time is increasing. The Greenwich date being nearly 7, the Hourly Difference is multiplied by 7, and we mark off three decimal places to the right: the product is the correction to be applied to the equation of time at noon.

Ex. 2. 1872, September 30th, 10h 15m, mean time at Greenwich: find equation of time to be applied to app, time in working the chronometer.

Eq. of time, page II, N.A.

Sept. 30th, noon, subt. 10m 10s 9 incr.
Corr. for 10

Red. eq. of time

+ 8.2

10 19:2

H. diff., Sept. 30th, noon 05799

(To be subtracted from A.T.)

Ex. 3. 1872, December 23rd, 22h 56m, mean time at Greenwich: find equation of time to be applied to apparent time.

In this example the equation of time is taken for the nearest noon to Greenwich date, viz. :December 24th. To obtain the correction and date we go back one hour, and since the equation of time is increasing at noon December 24th, it was less at one hour earlier, therefore the correction is subtractice.

Ex. 4. 1872, August 12th, 15h 28m, mean time at Greenwich: find the equation of time to be applied to apparent time in working the chronometer.

We multiply by 15, and for the half hour take half the hourly difference. This example may also be worked by taking out the equation of time for August 13th, at noon, and correcting backwards for the interval 8 hours which the Greenwich time, August 12th, 15h 28m is wanting of noon, August 13th. The correction thus obtained is additive, for since the equation of time is decreasing, it will evidently be greater, 8 hours before noon.

In each of the following examples it is required to find the equation of time corresponding to the given Greenwich date:

THE ALTITUDE of a Celestial body is the angular distance of the body from the horizon. It is measured by the arc of a circle of Azimuth (which is hence generally called a circle of altitude) passing through the plane of the body, or by the corresponding angle at the centre of the sphere.

The corrections necessary to reduce an altitude observed from the sea-horizon with a quadrant or a sextant, &c., to the true altitude, consist of the index correction, the dip, the correction of altitude, or the joint effect of refraction and parallax, and, in certain cases, of the semi-diameter.

The altitudes of heavenly bodies are observed from the deck of a ship at sea, with the sextant, for the purpose of finding latitude, longitude, &c. Such an altitude is called the "observed altitude." There are certain instrumental and circumstantial sources of error by which this is affected:-(a) The sextant (supposed otherwise to be in adjustment) may have an index error: (b) The eye of the observer being elevated above the surface of the sea, the horizon will appear to be depressed, and the consequent altitude in reality too great: and (c) One of the limbs of the body may be observed instead of its centre. When the correction for these errors and method of observing are applied-"the index correction," "correction" for dip, and "semidiameter," the observed is reduced to the apparent altitude. But, again, for the sake of comparison and computation, all observations must be transformed into what they would have been, had the bodies been viewed through a uniform medium, and from one common centre-the centre of the earth. The altitude supposed to be so taken is called the "true altitude;" it may be deduced from the apparent altitude by applying the corrections called "corrections for refraction" (Table V, Norie, or XXXI, Raper), and "correction for parallax" (Table VI, Norie, or XXXIV, Raper), which, however, are sometimes given in tables combined under the names "correction of altitude" (Tablo XVIII. Norie). (a) "Correction for refraction;" when a body is viewed through the atmosphere, refraction will cause the apparent to be greater than the true altitude; hence the correction for refraction is subtractive in finding the true from the apparent altitude. (b) "Correction for parallax;" the position of the observer on the surface, especially for near bodies, will cause the apparent to be less than the true altitude; hence the correction for parallax is additive in finding the true from the apparent altitude.

TO CORRECT THE SUN'S ALTITUDE.

RULE LX.

1o. Correct the observed altitude of the sun for index error, if any.

2o. Subtract the dip answering to height of eye (Table V, Norie, and Table XXX, Raper); the remainder is the apparent altitude of the limb observed.

3. Subtract the refraction (Table IV, Norie, and XXXI Raper), add the parallax (Table VI, Norie, XXXIV, Raper); or take out the "correction in altitude of sun (Table XVIII, Norie), and subtract it; the remainder is the true altitude of the observed limb.

4. Take from page II of the month in the Nautical Almanac the sun's semi-diameter, adding it when the sun's lower limb (L.L.) is observed, but subtracting it when the sun's upper limb (U.L.) is observed; the result thus obtained is the true altitude of the sun's centre.

Table 9, Norie, and Table 38, Raper, contain the gross correction of altitude, or the corrections for dip, refraction, sun's semi-diameter, and parallax-exclusive of index error, which are sometimes used in solving questions in nautical astronomy when great precision is not necessary.