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as the upper or lower limb has been taken :* this quantity is found for each day of the month in the "Nautical Almanac."

When observations are made at sea, an allowance must be made for the height of the eye above the horizon: this correction, termed the dip, is evidently always subtractive. And in observing with a sextant, it is always necessary to ascertain and apply its index error, which term is meant to express the deviation of the reading of the instrument from zero, when the direct and reflected images of an object are made exactly to coincide, in which case the horizon and index glasses are parallel.

The usual method of ascertaining the amount of this error of the instrument in astronomical observations, is by measuring the diameter of the sun on different sides of the true zero, and is done as follows:Set the vernier at about half a degree from zero on the graduated limb, and perfect the contact of the two limbs with the tangent screw, noting the reading: unclamp the index, and set the vernier again to about the same distance on the other side of zero, termed the arc of excess, which is divided for a few degrees for this purpose; and observe also this reading, when the contact is again perfected; half the difference will evidently be the index error,—when the reading of the arc of excess is the greatest, and when that of the limb: thus, Reading on the arc, 32′ 10′′


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These definitions are rendered more evident by reference to the figure below, taken from Sir J. Herschel's work, already quoted:

"Let C be the centre of the earth, NCS its axis; then are N and S its poles; EQ its equator; AB the parallel of latitude of the station A on its surface; AP, parallel to SCn, the direction in which an observer at A will see the elevated pole of the heavens; and AZ the prolongation of the terrestrial radius, CA that of his zenith; NAES

* When several observations are taken, the necessity for this correction is obviated, by taking alternately the upper and lower limb.

will be his meridian; NGS that of some fixed station, as Greenwich; and GE, or the spherical angle GNE, his longitude, and EA his latitude. Moreover, if ns be a plane touching the surface in A, this

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will be his sensible horizon; nAs marked on that plane by its intersection with his meridian, will be his meridian line, and n and s the north and south points of his horizon."

"Again, neglecting the size of the earth, or conceiving him stationed at its centre, and referring every thing to his rational horizon, let the annexed figure* represent the sphere of the heavens; C the spectator; Z his zenith; and N his nadir; then will HAO, a great eircle of the sphere whose poles are Z and N, be his celestial horizon; Pp the elevated and depressed poles of the heavens; HP the altitude of the pole; HPZEO his meridian; ETQ, a great circle perpendicular to Pp, will be the equinoctial; and if r represent the equinox, rT will be the right ascension, TS the declination, and PS the polar distance of any star or object S referred to the equinoctial by the hour circle PSTP; and BSD will be the diurnal circle it will appear to describe about the pole. Again, if we refer it to the horizon by the vertical circle ZSM, HM will be its azimuth, MS its altitude, and ZS its zenith distance. H and O are the north and south, and e and w the east and west points of the horizon, or of the heavens. Moreover, if Hh, Oo, be small circles, or parallels of declination touching the horizon in its north and south points, Hh will be the circle of perpetual apparition, between which and the elevated pole the stars never set; Oo that of perpetual occultation, between which and the depressed pole they never rise. In all the zone of the heavens between Hh and Oo they rise and set; any one of them, as S, remain* Page 134.


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ing above the horizon in that part of its diurnal circle represented by ABa, and below it throughout all that represented by ADa." From these figures it is evident that the altitude of the elevated pole is equal to the latitude of the spectator's geographical station, for the angle PAZ in the first, which is the H co-altitude of the pole, is equal to NCA; CN and AP being parallels whose vanishing point is the pole. But NCA is the co-latitude of the place A, whence




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the altitude of the pole must be equal to the latitude. The equinoctial intersects the horizon in the east and west points, and the meridian in a point whose altitude is equal to the co-latitude of the place.

The natural standards of the measurement of time are the tropical year and the solar day, and these are in a manner forced upon us by Nature, though, from their "incommensurability and want of perfect uniformity," they occasion great inconvenience, and oblige us, while still retaining them as "standards," to have recourse to other artificial divisions. In all measures of space the subdivisions are aliquot parts, but a year is no exact number of days, or even an integer with an exact fractional part; and before the introduction of the new style into England in 1752, an error of as much as 11days had thus crept into the calendar. By the present arrangement, every year whose number is not divisible by 4 without remainder, consists of 365 days; every year which is so divisible, but is not by 100, consists of 366 days; every year, again, which is divisible by 100, but not by 400, consists of only 365 days; and every year divisible by 400, of 366. The possibility of error is thus so far guarded against, that it cannot amount to one day in the course of 3000 years, which is sufficient for all civil reckoning, of which, however, astronomy is perfectly independent.

The three divisions of time for civil and astronomical purposes are the apparent solar, mean solar, and sidereal day. The apparent solar day is the interval between two successive transits of the sun over the same meridian; and from the path of the sun lying in the ecliptic inclined at an angle to the equator, upon the poles of which the earth revolves, and the carth's orbit not being circular, it follows that the length of this day

is constantly varying; so that, although it is the only solar time which can be verified by observation, it is quite unfit for application to general use.

The mean solar day, which is purely a conventional measure of time, is derived from the preceding, and is the average of the length of all the apparent solar days in the year, as nearly as it can be divided; and this is the measure of all civil reckoning. Mean time is in fact that which would be shown by the sun if he moved in the equator instead of the ecliptic with his mean angular velocity.

The difference on any day between apparent and mean time is termed the equation of time, and is given for every day of the year at mean and apparent noon in the first and second pages of each month in the "Nautical Almanac," additive or subtractive, according to the relative positions of the real and the imaginary mean sun.*

A sidereal day is the time employed by the earth in revolving on its own axis from one star to the same star again, or the interval between two successive transits of any fixed star, and is always so nearly the same length, that no difference can be perceived except in long intervals of time,† particularly in stars situated near the equator. A sidereal day is 3 minutes 55.91 seconds shorter than a mean solar, and is also less than the shortest apparent solar day, as must be evident from the figure, where the earth, moving in its orbit, and revolving on its own axis, after any point on its surface A, has by its revolution brought the star S' again on its meridian, must move also through the angle S'ES, before the arrival of the sun S on the same meridian.

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Both sidereal and apparent solar time are measured on the equinoctial, the former being at any particular instant the angle at the pole between the first point of Aries and the meridian of the observer, and the latter that contained between this meridian and the meridian where the sun is at the moment of observation, both reckoned westward;

*For a most lucid explanation of this varying equation, see Woodhouse's "Astronomy," chap. xxii., commencing at page 537; and also Vince's "Astronomy," &c.

For the causes of this almost imperceptible variation in the length of a sidereal day, see Woodhouse, page 106; there is, in fact, a mean and an apparent sidereal day.

hence the apparent solar time added to the sun's right ascension is the sidereal time; when any object is on the meridian, the sidereal time and the apparent right ascension of that object are the same.

It is evident that the difference between the times at any two places on the earth's surface is measured by the same arc of the equator, which measures the difference of their longitudes, the circumference of the circle representing either 360° or 24 hours, and making 15° of longitude one hour of time. To find the difference of longitude then between any two places, only requires us to be able to determine exactly the local time at each place at the same instant, and for this purpose chronometers, whose rates are known, and which have been set to, or compared with Greenwich mean time, are used, particularly at sea, where other means more to be depended upon cannot, from the motion of the ship, and the constant change of place, be resorted to.

From these explanations it will easily be seen that of the five following quantities, any three being given, the other two can be found by the solution of a spherical triangle, viz. :

1. The latitude of the place.

2. The declination of the celestial object observed.

3. Its hour angle east or west from the meridian.

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the spherical triangle ZPS. PZ being the co-latitude, PS the co-declination, or north polar distance, ZS the co-altitude or zenith distance, the angle ZPS the hour angle, and PZS the azimuth.

The further application of this triangle will be seen in the following astronomical problems.

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