PCP' the hour circle at right-angles to the meridian, is called the Six o'clock Hour Circle.

PO the Altitude of the Pole.

Since PE ZO = 90°, if ZP be omitted ZE = PO, but ZE measures the latitude of the place whose Zenith is Z, therefore, PO, or the altitude of the elevated pole is equal to the latitude of the observer.

PZ is the Co-latitude, or Complement of Latitude.

S is a Celestial Body.

SB is its Declination, PS its Polar Distance.

SA its True Altitude, ZS its Zenith Distance.

The angle ZPS is the Hour-angle or Meridian Distance.

SZP its Azimuth from the North in North Latitude, or from the South in South Latitude.

The arc EB also measures the Hour-angle.

HA or AO also measures the Azimuth.

If an object rise or set at R, CR or the angle CZR is called its Amplitude.

PROJECTION ON THE PLANE OF THE MERIDIAN.

A body X having S. declination, seen in

the Northern Hemisphere.

Fig. 46.

A body X having N. declination, seen in the Southern Hemisphere.

a

a'

a

A PROJECTION UPON THE PLANE OF THE HORIZON.

The following figure represents the projection of the celestial sphere on the plane of the horizon.

The principal advantage of this projection is that it exhibits at one view the whole hemisphere which is above the horizon.

The horizon, NES'W, in this case appears as a circle, and the spectator must suppose himself looking down upon it from the Zenith Z; and all circles passing through the Zenith will appear as straight lines: thus we have

N, the North Point; E, the East Point; S', the South Point; W, the West Point.

P is the Elevated Pole, and NP the Elevation of the pole equal to the latitude of the place of observation.

NZS' is the Meridian of the observer's position, cutting the horizon in the North and South points; E and W are the poles of the meridian, for they are 90° distant from every point in that circle.

WZE, the vertical circle at right-angles to the meridian, is called the Prime Vertical.

WQE is the Equator or Equinoctial.

WPE is the Six o'clock Hour Circle.

ZoT, ZMR are Circles of Altitude or Vertical Circles.

oT is the Altitude of a body o.

Zo is Zenith Distance of complement of altitude of o.
PoD is a Circle of Declination.

o D is the Declination of a body o.

Po is its Polar Distance.

PZo is the Azimuth.

oPZ is the Hour-angle, or Meridian Distance.

e Ac is the Ecliptic, and A the Equinoctial Point.

o L is the Latitude of o.

AL is the Longitude of o.

AD or the angle APD is the Right Ascension.

If an object rose at R and set at S, then ER would be its rising amplitude, and WS its setting amplitude.

PRELIMINARY RULES IN NAUTICAL ASTRONOMY.

CIVIL AND ASTRONOMICAL DAY.

138. The Civil Day, or common method of reckoning time, begins at midnight, and ends the following midnight, the interval being divided into two periods of 12 hours each; the first twelve hours, from midnight to noon, are denoted by A.M. (ante meridian); the latter, from noon to midnight, are styled P.M. (post meridian); thus we say 10 A.M. when an event occurred at 10 o'clock in the morning, and 10 P.M. when it occurred at 10 o'clock in the evening.

139. The Astronomical Day begins at noon and ends at the following noon, and is later than the civil day by twelve hours. The hours are reckoned throughout, or continuously from oh to 24h. The distinction of A.M. and P.M. is not recognised in astronomical time. Thus 11 o'clock in the forenoon of the second of January in the civil reckoning of time corresponds to January 1 day 23 hours in the astronomical reckoning; and 1 o'clock in the afternoon of the former to January 2 days 1 hour of the latter reckoning.

I

140. Since the civil day commences at the midnight preceding the noon which commences the astronomical day, it is evident that the civil mode of reckoning is always twelve hours in advance of the astronomical reckoning, and hence we have the following rule for converting civil into astronomical time.

Given civil time at ship, to reduce it to astronomical time.

RULE LIV.

1o. If the civil time at ship be P.M., it will also be astronomical time, P.M. being omitted.

2o. If the civil time be A.M., add twelve to the hours and subtract one from the days of the month; also omit A.M. The result in each case is the Astronomical Date.

EXAMPLES.

Ex. 1. May 10th, at sh 30m P.M., civil time is 5h 30m astronomical time of the same date; because the 10th astronomical day begins at noon of the roth civil day, and 5h 30m have elapsed since that noon. But 5h 30m A.M. civil time on May 10th is 17h 30m astronomical time on the 9th of May, for the 9th day of the month, according to the astronomical reckoning, commences at noon of the 9th civil time, and ends at noon of the 10th civil day (the hours being reckoned up to 24), and 5h 30m A.M. of the 10th is 17h 30m from noon of the 9th.

Ex. 2. October 7th, at 3h 20m P.M., civil time, is October 7th, at 3h 20m astronomical time. (See 1° of Rule LIV.)

Ex. 3. October 7th, at 3h 20m A.M., civil date, is October 6a 15h 20m astronomical date; since 7d less rd is 6a, and 12h added to 3h 20m is 15h 20m. (See 2° of Rule.)

Ex. 4. January 31st, at 7h 20m P.M., civil time, is January 31st, at 7h 20m astronomical time. (Rule LIV, 1°.)

Ex. 5. February 1st, at 6h 18m A.M., civil date, is January 31d 18h 18m astronomical date; since February 14, diminished by 1a, gives January 31a, and 12h added to 6h 18m is 18h 18m. (Rule LIV, 2°).

Ex. 6. What is the astronomical date corresponding to 1889, January 1st, 8h A.M.? The corresponding astronomical date is 1888, December 31d 20h. In this case the year is diminished by 1, since in diminishing the day of the month by 1, the reckoning throws us back into the last month of the previous year, ie., the day before January 1st, 1889, also 12h added to 8h is 20h.

141. Given the astronomical date, to find the corresponding civil date.

RULE LV.

If the hours of astronomical time be less than 12" write P.M. after it, and it will be the required civil time; but if the astronomical time be greater than 12", add I to the days, diminish the hours by 12 and write A.M. after it: the result will be the required civil time.

Express the following astronomical dates in civil time:

142. The earth rotates uniformly on her axis once in twenty-four hours, and thus every spot on her surface describes a complete circle, or 360°, in that space of time; hence the longitude of any place is proportional to the time the earth takes to revolve through the angle between the first meridian and the meridian of the place, and thus the longitude of a place may be expressed either in arc or in time.* Longitude in arc and longitude in time are easily convertible, for since 360° is equivalent to 24 (360 ÷ 24 = 15°), 15° is equivalent to 1h, 15' to 1", and 15" to 1"; whence

1° is equivalent to 4m (i.e., the 15th part of 1 hour or 60m)

and the following rules are sufficiently clear.

* In reckoning by arc, each degree is divided into sixty minutes, and each minute into sixty seconds. In reckoning by time, each hour is also divided into sixty minutes, and the minutes into sixty seconds. But a distinct notation for each of these has been adopted, degrees, minutes, and seconds being represented by °'", and hours minutes, and seconds by hms; and care should be observed not to use the same marks for both, great confusion arising from so doing.

† A third is the name given to the sixtieth part of a second,

143. To convert arc (or longitude) into time.

RULE LVI.

Multiply the degrees, minutes, &c., by 4; this turns the degrees (°) into minutes (TM) of time, minutes (') into seconds (") of time, and the seconds (") into thirds (t) of time; or in other words, mark the resulting figures thus:-Those under seconds (") thirds (†), those under minutes (') seconds (*), those under degrees (°) minutes (TM), and those to the left of the latter, hours (1).

NOTE.-Instead of thirds it is customary to use tenths of seconds, in which case the thirds must be reduced to tenths by dividing by 60.

Four times 15" are 60", which contains 60 once and o over; write this remainder down under the seconds (") and mark it thirds (t) as directed in the Rule, carrying the 1. Again 4 times 18′ are 72, and the 1' carried makes 73; 60 goes in 73 once and 13 over; write this remainder (13) under the minutes (') and call them seconds (•) and carry the 1. Again, 4 times 12 are 48, and 1 carried makes 49; write this under degrees () and mark it minutes (): whence the time corresponding to arc 12° 18′ 15′′ is 49m 13o ot.

25° 15′ 16′′

4

41m 1°

4t

Four times 16" are 64", which contains 60 once and 4 over, and according to Rule this remainder placed under seconds (") becomes thirds (t), and the i is to be carried. Again, four times 1,' are 60 and I carried makes 61, which contains 65 once and 1 over; write the remainder 1 under minutes (') and call them seconds (•), and carry 1: four times 25 are 100 and 1 carried gives 101, and 60 into 101 goes once and 41 remainder, which remainder being placed under degrees (°) gives minutes () and the I carried on being placed to the left of the latter is marked hours (b); whence 1h 41m 14t is the time corresponding to the arc 25° 15′ 16′′.

or, sh 8m 8s.66

6,0)71,5m 3o 32t

or, 11h 55m 353

EXAMPLES FOR PRACTICE.

Reduce the following arcs into time:

1. 18° 54′; 12° 40′ 45′′;

2.

0° 58'6; 49° 4′ 20′′;

137° 27′; 96° 10′ 45′′; and 89° 16′.

0° 26'·8; 2° 18′ 12′′; and 130° 19′.

3. 0° 13′′5; 51° 10′ 12′′; 156° 52′; 178° 49′ 45′′; and o° 41′7.

TO CONVERT TIME INTO LONGITUDE.

144. It has been shown (No. 142, page 176) that 4" of time are equivalent to 1° of arc; hence it is evident that if we bring any given time into minutes, and divide by 4, we shall have the corresponding arc in degrees, minutes, and seconds. This is the reverse of the last process.

RULE LVII.

Reduce the hours and minutes into minutes, after which place the seconds, &c., then divide all by 4, and the quotient will be the degrees, minutes, &c., of the corresponding arc; or, in other words, after dividing by 4, mark the resulting figures thus:-Those under minutes (") degrees (°), those under seconds (*) minutes (), those under thirds () seconds (").