Hence P=153° 49′ 49"= :(in time) 10h 15m 194s, so that twilight commenced in the morning at 1" 44m 40s, and ended in the evening at 10h 15m 1918.

2. At what time does the twilight begin at latitude 48° 38′ 56′′ N., when the sun's declination is 8° 28′ 54′′ N.?

Twilight begins at 3, 20m.

3. At what time does twilight end at latitude 52° 12′ 35′′ N., when the sun's declination is 15° 55′ 25′′ N.?

Twilight ends at 10h 121m.

PROBLEM IX.

Given the latitude of the place to determine on what day of the year

the twilight is the shortest, and its duration on that day.

Let HO represent the horizon, and ho the parallel to it, 18° below it; also let PS be the declination circle, passing through the sun at sunset, and PZ, that passing through the zenith. Conceiving these two circles to revolve with S, PS will come to PS' when S comes to S', and PZ will take some determi

2

n

10

nate position PZ'. Now, since the angles ZPS, Z' P S', are equal, we have, by taking from each the common part Z'PS, ZPZ'SPS'; but SPS', converted into time, expresses the duration of twilight, ZPZ' is therefore the least possible when the twilight is the shortest possible. Now since the sides PZ, PZ', are both given, the side ZZ' will be the shortest when the opposite angle, P, is the least; (see equa. (A) p. 74,) hence when ZZ' is the shortest, the twilight is the shortest; but as the two sides Z'S', ZS, of the triangle ZZ'S', are given, the third side will be shortest when the angle S' is the least possible, and this is the case when Z' falls on ZS', for then the angle is 0. Hence the twilight is shortest when the angle PSZ is equal to the angle PS'Z.

Let then be the proper position of Z'; we shall have Zz=Zh′· zh' zS' zh'h'S': = 18°, and because PZ Pz, the arc Pn, bisecting the angle ZPz, will also bisect the base Zz, and be perpendicular to it (54); consequently,

The declination being known by this equation, the day of shortest twilight is also known, (Naut. Alm.) The declination will be of a contrary name with the latitude as its sine is negative. Equation (1) expresses the duration of the twilight.

Since the angles ZPz, SPS', are equal, the hour angles for the beginning and ending of the morning twilight, or for the ending and beginning of the evening twilight, are ZPS', PS'. Now, in the right

The sum of (1) and (3) gives the angle ZPS', and their 'difference the angle zPS' = ZPS, and thus we have the hour angles for the beginning and end of the twilight.

EXAMPLES.

1. Required the time and duration of shortest twilight at Greenwich, lat. 51° 28′ 40′′, in the year 1832.

The declination is therefore 7° 7′ 5′′ south, which (Naut. Alm.) corresponds to March 1st and to October 12.

Also the hour angle SPS' is 29° 5′ 38′′, which, in time, is 1 56m 221, the duration sought.

To find the times of beginning and ending of the twilight, we have, from the equation (3),

The angle nPS', thus determined, is obtuse, because its opposite side is greater than PS, and this is opposite to a right-angle. This angle, converted into time, is 6h 22m 51s. Adding therefore, to this the angle ZPn, in time, that is, half the duration, or 58m 111s, we have th 20m 16, the time when the evening twilight ends. Also, by subtracting the same quantity, we have 51 23m 54s for the time when the evening twilight commences. These results respectively taken from 12 leave the times when the morning twilight begins and ends.

150

CHAPTER III.

ON THE PRINCIPLES OF NAUTICAL ASTRONOMY.

(73.) IN our chapter on Navigation we have laid down several methods of determining the place of a ship at sea, by help of the account kept on board of its progress through the water, that is, of the course and distance sailed; and, if confidence could be placed in this account, even when kept with the utmost care, the art of Navigation would be perfect. Such perfection, however, it is hopeless to expect; for it does not seem possible to measure, with strict accuracy, either a ship's rate or the direction in which she moves, both of which may indeed be continually varying. In order, therefore, to determine the place of a ship at sea, with that accuracy which the safety of Navigation requires, it is absolutely necessary that we be furnished with methods entirely independent of the dead reckoning, and these methods it is the business of Nautical Astronomy to teach.

"It must not, however, be understood that the dead reckoning is without its value; on the contrary, when combined with astronomical observations, it is of considerable utility in detecting the existence and velocity of currents, and is indispensably necessary to fill up the short intervals which may occur in unfavorable weather between celestial observations. But the too general practice of relying exclusively upon it cannot be sufficiently deprecated, and numerous instances might be adduced of the fatal consequences of this reliance, in the loss of vessels, from errors of such magnitude that they might have been detected by the most superficial knowledge of nautical astronomy, and the aid of even a good common watch." (Capt. Kater's Nautical Astronomy in the Ency. Met.)

On the Corrections to be applied to the observed Altitudes of

Celestial Objects.

(74.) The true altitude of a celestial object is always understood to mean its angular distance from the rational horizon of the observer. This is not obtained directly by observation; but is the result of certain corrections applied to the observed altitude. These we shall now enumerate and explain.

Of the Dip or the Depression of the Horizon.

(75.) Let E represent the place of the observer's eye, and S the situation of any celestial body; the first object is to obtain its apparent altitude above the horizontal line EH; that is, the angular distance SEH. Now, as to the observer, the visible horizon is EBH', the altitude given by the instrument is the angle SEH'; hence we must subtract from this ob

Η

H

served altitude the angle HEH', called the Dip or Depression of the horizon, in order to obtain the apparent altitude SEH.

The angle HEH', or its equal C, is calculated for various elevations, AE of the eye above the surface of the sea from the proportion,

CE: EB EC2 CB2 rad. sin. C;

and the results are registered in a table.

Of the Semidiameter.

(76.) When the foregoing correction for dip has been applied, the result will be the apparent altitude of the point observed above the horizontal plane through the observer's eye. If this point be the uppermost or lowermost point of the disc of the sun or moon, a further correction will be necessary to obtain the apparent altitude of the centre; that is, we must apply the angular distance due to the semidiameter. This