solar or astronomical day when the sun is on the meridian. The sidereal time of day is measured by the arc of the equator, or the angle at the pole between the first point of Aries and the meridian of the observer, estimated in the order of the signs; and the solar or apparent time of day is measured by the arc of the equator, or the angle at the pole intercepted between the meridian on which the sun is, and the meridian of the observer, estimated also in the order of the signs.

Hence the solar time added to the sun's right ascension is the sidereal time, or the sun's right ascension taken from the sidereal time leaves the solar or apparent time.

A mean solar day is longer than a sidereal one, for the sun daily advances in the ecliptic so far towards the east that the mean interval between his transits is about 3m 55s greater than the interval between the transits of a fixed star. The sidereal days are all perfectly equal, but from the variable angular motion of the sun in the ecliptic (a circle inclined to the equator) his diurnal change of right ascension is a variable quantity, and hence he comes to the meridian at unequal intervals of time. Mean time is that which would be shown by the sun if he revolved in the plane of the equator with the mean angular velocity with which he revolves in the ecliptic; the difference between mean and apparent time is called the equation of time.

Chronometers for nautical purposes, as well as clocks and watches for use in ordinary life, are regulated by mean time, but observatory clocks by sidereal time; as when an object is on the meridian, the sidereal time is its right ascension, which in observatories it is one of the chief objects to determine.

The meridian distance of a celestial object is the angle at the pole, included between the meridian on which the objectis, and the meridian of the observer.

The semidiurnal arc is the arc of the parallel of declination which a celestial object describes, between the time of its rising or setting, and that of its passing the meridian, and it is measured by the corresponding arc of the equator.

The ascensional difference is the difference between the semidiurnal arc and six hours.

From what has been said on the subject of time, it is evident that the difference between the time at Greenwich, and that at any other place, is measured by the same arc of the equator, which measures the longitude of that place, whether the time is apparent, mean, or sidereal, the circumference of the equator, or 360°, representing 24 hours of time. Hence the problem of finding the longitude by observation, requires us to be able to determine the time at the place at which we are, and the time at the same instant at the meridian of Greenwich.

The apparent altitude of a celestial object is the arc of a vertical circle

intercepted between the centre of the object and the sensible horizon, the eye being on the surface of the earth; and the true altitude is the arc of a vertical circle intercepted between the object and the rational horizon, the eye being conceived to be at the centre of the earth, The true and apparent zenith distances are the complements of the true and apparent altitudes, or the true and apparent distances of the object from the zenith. The altitude and zenith distance of an object, when on the meridian, are called the meridian altitude, and meridian zenith distance of that object.

But altitudes, if observed on or above the surface of the earth, require several corrections before the true altitudes can be deduced from them. These corrections are for semidiameter, dip, refraction, and parallax. By the semidiameter of a celestial object, is meant the angle which the radius of its apparent circular disc subtends at the eye of the observer; and by the parallax, the angle which the radius of the earth on which an observer is situated subtends at the centre of the object. When the object is in the horizon, the semidiameter and parallax are called the horizontal semidiameter and horizontal parallax; and the parallax, at any altitude, is called the parallax in altitude. The sun's semidiameter is given in the Nautical Almanac for every sixth day of the month. His mean horizontal parallax is about 83", and it never differs more from that quantity than about a quarter of a second. His parallax, at any altitude, may be taken by inspection from Table 10, and his semidiameter, with sufficient exactness, from Table 16; but, for common purposes, the semidiameter may be always taken at 16'.

The moon's horizontal parallax and semidiameter, as seen from the centre of the earth, are given in the Nautical Almanac for every noon and midnight of Greenwich time. The parallax there given is the horizontal parallax of the earth's greatest or equatorial radius, and the reduction, to adapt it to any particular latitude, may be taken from Table 14. The parallaxes of the sun and the planets are so small that this correction, with respect them, is too minute to be regarded. The moon's parallax is to its semidiameter always very nearly as 11 is to 3. Her apparent semidiameter is greater as her altitude increases, as she is then nearer to the observer. This augmentation of her semidiameter may be taken from Table 12.

When the eye is above the surface of the earth, the visible horizon appears depressed, as the line from the eye, touching the horizon, is a tangent to the earth at the point of contact. The depression of this line below the horizontal plane passing through the eye of the observer, is called the dip; and the altitude as observed, as well as its supplement, are too great by this depression. The correction for dip is therefore subtractive; it may be taken, by inspection, from Table 7.

When a ray of light passes obliquely from one medium into

another of greater density, it is bent towards the perpendicular at the point at which it enters. This bending is called refraction. Now the strata of the atmosphere, through which the rays of light pass to the earth, increase in density as they approach the surface, being compressed by the weight of the superincumbent atmosphere; and therefore the rays of light from celestial objects are bent downwards as they approach the earth; and, consequently, to us, all celestial objects appear more elevated than they would do, or their altitudes are greater than they would be, if light were not refracted.

Except near the horizon, the refractions at different altitudes are nearly proportional to the cotangents of the altitudes. In a mean state of the atmosphere, the refraction at the horizon is 33'. Table 9 contains the mean refractions, and Table 11 their corrections for any variation in the temperature or density of the atmosphere.

To explain more clearly the effect of these corrections on altitudes, let MIKL be a section of the rational, and N B a section of the sensible horizon, A B the elevation of the eye of an observer above the surface of the earth, and A C a plane parallel to M L. Let N be the place of a celestial object when in the horizon, S its true place above the horizon, S a A the course of a ray of light from it to the eye at A, A S' a tangent to the curve S a A at the point A, and A H a tangent drawn from A to

E

F

G

B

H

M

K

the earth's surface at H; then S A S' will be the refraction, C A H= BKH the dip, BNK the horizontal parallax, and B S K the parallax in altitude; S'A H the observed, S B N the apparent, and SK M the true altitude, which is measured by the arc G I. Now even in the case of the moon, the parallactic angle B S K will not in any case much exceed a degree; and as it is subtended by BK, which is nearly 4000 miles, the angle A S B subtended by A B, which on ship board cannot exceed a few feet, may be considered as evanescent; and therefore the angles S A C, and S D C, or S B F, may be considered as equal. Whence S' A H (the observed altitude) CAH (the dip) - S'AS (the refraction) = SACSBF; and SBF+BSK=SEF=SKM the true altitude. And N K and SK being equal, NK: BK::SK: BK; but NK: BK:: rad sin BNK and SK: BK:: sin S B K, or cos S BN : sin B S F; consequently, rad sin horizontal parallax : cos apparent altitude: sin parallax in altitude; whence, from the horizontal parallax, the parallax in altitude may be computed. The joint effect of the moon's parallax and refraction may be taken by inspection from Table 24,

:

With respect to the semidiameter of the object, as it is the altitude of the centre that is required, the semidiameter must be added to the observed altitude when the lower edge, or lower limb is observed, and subtracted from it when the upper limb is observed.

When the altitude of an object is taken by what is called a back observation, the depression of the point diametrically opposite to the object is measured; the effect of the dip is therefore to diminish this angle, as it increases the measure of its supplement; in this case therefore the correction for dip is additive.

The altitude corrected for semidiameter and dip is the apparent altitude, and the difference between the remaining corrections, viz. the parallax in altitude and the refraction, is called the correction of altitude.

The correction for parallax being additive, and that for refraction subtractive, their difference is additive when the parallax, and subtractive when the refraction, is the greater; but it is only in the case of the moon that the parallax exceeds the refraction, and from the distance of the fixed stars their parallax is insensible, therefore the moon's correction of altitude is additive, and that of the sun or of a star subtractive; or the true place of the moon is above, and that of any other celestial object below its apparent one. Altitudes are sometimes observed on land by reflection, from what is called an artificial horizon, which is in general merely the horizontal surface of a fluid. Thus in figure 2, page 206, if the reflecting surface A C were horizontal, the angle E, or its equal S B S', would be double of S B A, the altitude of S. Altitudes can be taken with great exactness by this method, and the seaman should be familiar with the practice of it, as from altitudes so taken on shore he may, at any time, with ease and accuracy, find the error and rate of his chronometer. When the altitude of a celestial object is increasing, it is marked with+, and with when it is decreasing; hence + affixed to an altitude shows that the object is east, and that it is west of the meridian.

The same marks added to a number taken from a table show whether the number is increasing or decreasing. For example, if I take the sun's right ascension for July 4, 1825, from Table 21, thus 6h 52m 43s + 4m 7's, I mean that the right ascension is 6h 52m 43s at noon, and that its next value given in the table is 4m 7s greater.

If I take out the moon's horizontal parallax for midnight, Greenwich time, September 9, 1823, thus 55′ 55′′-21", I mean that her parallax is 55′ 55′′, and that the next following one is 21" less.

signifies the sun's lower, and his upper limb; 2 the moon's lower, and her upper limb,

PREPARATORY PROBLEMS.

PROBLEM I.

To convert longitude into time.

MULTIPLY the longitude by 4, divide the degrees of the product by 60, and the quotient will be the hours, the remainder minutes, and the other parts of the product seconds, &c. of the corresponding time.

EXAMPLE.

What time corresponds to 49° 4' 20" of longitude?

Reduce the hours into minutes and divide by 4, and the quotient will be the degrees, minutes, &c. of the corresponding longitude.

EXAMPLE.

What longitude corresponds to 7h 2m 46s?

h m S 72 46

60

4)422

105°41' 30"