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for a meridian observation cannot be depended upon, it becomes an important problem to determine the latitude at sea, by observations made out of the meridian; and considerable attention has accordingly been paid, by scientific persons, to the method of finding the latitude by "double altitudes," and various tables have been computed to facilitate the operation. But the direct method, by spherical trigonometry, though rather long, involving three spherical triangles, will be more readily remembered, and more easily applied by persons familiar with the rules and formulas of trigonometry than any indirect or approximate process; we shall therefore explain the direct method.

Let P be the elevated pole, z the zenith of the ship, and s, s' the two places of the sun, when the altitudes are taken. Then, drawing the great circle arcs as in the figure, we shall have these given quantities, viz., the codeclinations PS, PS'; the coaltitudes zs, zs', and the hour angle SPS', which measures the interval between the observations; and the quantity sought is the colatitude ZP. Now, in the triangle PSS', we have given two sides

S'

S

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and the included angle to find the third side ss', and one of the remaining angles, say the angle Pss. In the triangle zss' we have given the three sides to find the angle s'sz; having then the angles Pss', s'sz, the angle zSP equal to their difference, becomes known, so that we have, lastly, two sides and the included angle in the triangle zsp, to find the third side ZP.

Before the application of the trigonometrical process, the observed altitudes must, of course, be reduced to the true altitudes, as in the preceding examples. Moreover, as the ship most probably sails during the interval of the observation, an additional reduction becomes necessary, as follows: Let z be the zenith of the ship, and s the place of the sun, at the first observation z' and s' the same at the second. Then the angle z'zs will represent the bearing of the ship's path from the sun, which may be observed with the compass; considering

this angle as a course, and the distance sailed, zz', as the corresponding distance, find by the table (or by the formula zz' cos z'zt) zt which subtracted from zs will give z's nearly, which, instead of zs, should be used with z's' in the solution before given. This must be subtracted from the first zen. dist. if the angle z'zt is less than 90°; but it must be added when the angle exceeds 90°. If the angle is 90°, no correction for the ship's change of place will be necessary.

Where great accuracy is aimed at, account should be taken of the ship's change of longitude during the interval of the observations; when converted into time it must be added to the interval of time between the observations when the ship has sailed eastward, and subtracted when she has sailed westward. This correction is very easily applied.

Having thus mentioned the necessary preparative corrections, we shall now give an example of the trigonometrical operation.

EXAMPLE.

Let the two zenith distances corrected be (see last figure but one), zs = 73° 54′ 13′′, zs′ = 47° 45′ 51′′, the corresponding declinations 8° 18' and 8° 15 north, and the interval of time three hours; to determine the latitude.

Considering ss' to be the base of an isosceles spherical triangle, of which one of the equal sides is (PS + PS')* = 81° 43' 30", and the vertical angle equal to 3* or 45°, let the perpendicular PM be drawn, then we have in the triangle PMS right angled at M, PS 81° 43' 30", and

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Which we may, without sensible error, where the base is so small.

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This angle is acute like its opposite side (see p. 196).

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IV. TO FIND THE TWO UNKNOWN ANGLES OF THE TRIANGLE ZSP.

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is employed to express the difference between two quantities, whichever may be the greater.

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Upon the same principles may the latitude be determined from the altitudes of two fixed stars, taken at the same time; in this case s, s', in the preceding figure, will represent the two stars: PS, Ps', their known polar distances, and the angle sPs', the difference of their right ascensions; the same quantities are therefore given as in the case of the sun, but, as in the case of two stars, PS, Ps', may differ very considerably, ss' cannot be considered as the base of an isosceles triangle, but must be computed from the other two sides and their included angle.

For other modes of determining the latitude, see the next Appendix.

ON FINDING THE LONGITUDE.

The determination of the longitude of a place always requires the solution of these two problems, viz.: 1st, to determine the time at the place at any instant; and, 2d, to determine the time at the first meridian, or that from which the longitude is estimated, at the same instant; for the difference of the times converted into degrees, at the rate of 15° to an hour, will obviously give the longitude.

When the latitude of the place is known (and it may be found by the methods already explained), the time may be computed from the altitude. of any celestial object whose declination is known; for the coaltitude, codeclination, and colatitude, will be three sides of a spherical triangle given to find the hour angle, comprised between the codeclination and the colatitude.

(See Art. 84.) The following example will illustrate the mode of proceeding.

At Columbia College, January 13th, 1850, the double altitude of the sun's lower limb was observed by reflexion from Mercury to be 42° 29'. Thermometer 40°, and Barometer 30 in.

Time by the watch, 10' 6" 10′ A.M.

Index error of the sextant, 52" additive.

Latitude of station 40° 42′ 40′′.

Longitude from Greenwich in time, 4" 56" 4".

Required correct time of Observation and error of the watch.

Equation of time at ap. noon, January 13th, 1850, 9′ 00′′ •27.

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Sun's declination at ap. noon, January 13th, 1850, 21° 29′ 19 5 S.

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Index error of sextant. Additive (see 2d note, p. 290).

Double altitude corrected for index error,

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10

6

10 A.M.

4 56 4

3

2 14

9

3.08

2 53 10 92

21 28

4.18 S.

P

Z

S

This must be multiplied by the time after apparent noon at Greenwich, found below, reduced to hours and decimals of an hour, and the product added to the equation of time at noon above, to obtain the equation of time at the instant of observation.

+ This is computed from the data in the third and fourth lines from the top of the page, in the same manner as the equation of time.

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