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Half sum
58:40" - 37.50" (33! + 4:50") = 20:50? Prop. log.
Constant log.
Correction

1:51" Prop. log. Approximate time of setting = . 21:41:28:

3. 87494 0.9365 1. 1761

1.98755

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Apparent time of moon's setting = 21:39737!

Note.— The direct method of solving this and the three preceding Problems, by spherical trigonometry, is given in some of the subsequent pages of this work.

TABLES LI. AND LII.

For computing the Meridional Altitude of a Celestial Object. Since it frequently happens, at sea, that the meridional altitude of the sun, or other celestial object, cannot be taken, in consequence of the interposition of clouds at the time of its coming to the meridian ; and since it is of the utmost importance to the mariner to be provided at all times with the means of determining the meridional altitude of the heavenly bodies, for the purpose of ascertaining the exact position of his ship with respect to latitude, these Tables have therefore been carefully computed; by means of which the meridional altitude of the sun, or any other celestial object whose declination does not exceed 28 degrees, may be very readily obtained to a sufficient degree of accuracy for all nautical purposes, provided the altitude be observed within certain intervals of noon, or time of transit, to be governed by the meridional zenith distance of the object: thus, for the sun, the number of minutes and parts of a minute contained in the interval between the time of observation and noon, must not exceed the number of degrees and parts of a degree contained in the object's meridional zenith distance at the place of observation. And since the meridional zenith distance of a celestial object is expressed by the difference between the latitude and the declination when they are of the same name, or by their sum

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when of contrary names; therefore the extent of the interval from noon (within which the altitude should be observed) may be determined by means of the difference between the latitude and the declination when they are both north or both south, or by their sum when one is north and the other south. Thus, if the latitude be 40 degrees, and the declination 8 degrees, both of the same name, the interval between the time of taking the altitude and noon must not exceed 32 minutes ; but if they be of different names, the altitude may be taken at any time within 48 minutes before or after noon: if the latitude be 60 degrees, and the declination 10 degrees, both of the same name, the interval between the time of observation and noon ought not to exceed 50 minutes; but if one be north and the other south, the interval may be extended, if necessary, to 70 minutes before or after noon, and so on.

The limits within which the altitudes of the other celestial objects should be observed, may be determined in the same manner ; taking care, however, to estimate the interval from the time of transit or passage over the meridian, instead of from noon.

Now, if the altitude of the sun or other celestial object be observed at any time within the limits thus prescribed, and the time of observation be carefully noted by a well-regulated watch, the meridional altitude of such object may then be readily determined, to every desirable degree of accuracy, by the following rule; viz.,

Enter Table Ll. or LII., according as the latitude and the declination are of the same or of contrary names, and with the latitude in the side column, and the declination (reduced to the meridian of the place of observation) at the top or bottom; take out the corresponding correction in seconds and thirds, which are to be esteemed as minutes and seconds ;then,

To the proportional log. of this correction,* add twice the proportional log. of the interval between the time of observation and noon, or time of transit, and the constant log. 7. 2730; and the sum will be the proportional log. of a correction, which, being added to the true altitude deduced from observation, will give the correct meridional altitude of the object.

Note I.-In taking out the numbers from Tables LI. and LII., proportion must be made for the excess of the given latitude and declination above the next less tabular arguments.

* When the object either comes to, or within, one degree of the zenith, the angle of meeting made by the latitude and declination will fall within the zigzag double lines which run through the body of Table Ll., and through the upper left-hand corner of Table LII: in this case, since the interval between the time of observation and noon, or meridional passage, must not exceed one minute, the corresponding number will be the correction of altitude direct, independently of any calculation whatever.

2.-The interval between the time of observation and noon may be always known by means of a chronometer, or any well-regulated watch; making proper allowance, however, for the time comprehended under the change of longitude since the last observation for determining the error of such watch or chronometer.

Example 1.

In latitude 45? north, at 34"40! before noon, the sun's true altitude was found to be 54:12:49", when his declination was 10: north ; required the meridional altitude ?

Corr. in Table Ll., ans. to lat. 45. and dec. 10:, is 2"23". 1; the propor. log. of which is

1.8778 Interval between time of obs. and noon, 34:40:, twice prop. log.=1.4308 Constant log.

7. 2730

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In latitude 48: north, at 1:5"48. past noon, the sun's true altitude was found to be 20:25:5", when his declination was 20 degs, south; required the meridional altitude ?

Corr. in Table LII., answering to lat. 48.N. and dec. 20:S., is 1!19".9, the propor log. of which is

2. 1308 Interval between time of obs. and noon 1!5'48:, twice prop. log.=0.8740 Constant log.

7. 2730

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At sea, March 22d, 1824, in latitude 51:16. north, at 50132' past noon, the sun's true altitude was found to be 38:20:56" ; required the meridional altitude, the declination being 0:43:51" north?

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Corr, in Table LI., answering to lat. 51:16'and dec. 0:43:51"
is 1"35".6,* the propor. log. of which is .

2.0530 Interval between time of obs. and noon 50:32:, twice prop. log.= 1. 1034

7. 2730

Constant log:

Correction of altitude . 1. 6:58"
True alt. at time of obsery. 38. 20.56

Prop. log. =

0.4294

Sun's meridional altitude
from the truth.

39:27:54"; which differs but three seconds

Example 4. At sea, December 21st, 1924, in latitude 60.22! north, at 10:36:10: A.M., or 1.2350: before noon, the sun's true altitude was found to be 4:26'38".; required his meridional altitude, the declination being 23:27:45" south?

Corr. in Table LII., ans, to lat. 60:22! and dec. 23:27:45",
is 0?53". 8,7 the propor. log. of which is

2.3026 Interval between time of obs. and noon 1:23:50:, twice prop.log.=0.6638

7.2730

Constant log.

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After this manner may the meridional altitude of the moon, a planet, or a fixed star be obtained, when the declination does not exceed the limits of the Table.

Remarks, &c.

From the above examples it is manifest, that by means of the present Tables the meridional altitude of a celestial object may be readily inferred

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from its true altitude observed at a known interval from noon (within the limits before prescribed), with all the accuracy to be desired in nautical operations, and that it is immaterial whether the observation is made before or after noon, or time of transit, provided the time be but correctly known; and, since most sea-going ships are furnished with chronometers, there can be but very little difficulty in ascertaining the apparent time to within a few seconds of the truth.

It is to be observed, however, that the nearer to noon or time of transit the observation is made, the less susceptible will it be of being affected by any error in the time indicated by the watch : thus, in example 4, where the interval or time from noon is 1:23:50:, an error of one minute in that interval would produce an error of 24 minutes in the sun's meridional altitude; but if the observation had been made within a quarter of an hour of noon, an error of five minutes in the time would scarcely affect the meridional altitude to the value of 2 minutes : hence it is evident, that although the observation

may be safely made at any time from noon to the full extent of the interval, when dependance can be placed on the time shown by the watch, yet when there is any reason to doubt the truth of that time, it will be advisable to take the altitude as near to noon, or the time of transit, as circumstances may render convenient.

In all narrow seas trending in an easterly or westerly direction, where the meridional altitude of a celestial object is of the greatest consideration, such as in the British Channel, the mariner will do well to avail himself of this certain method for its actual deterinination ; particularly during the winter months, when the sun is so very frequently obscured by clouds at the time of its coming to the meridian.

These Tables were computed by the following rule; viz.,

To the constant log. 0.978604,* add the log. co-sines of the latitude and the declination; the sum, rejecting 20 from the index, will be the log. of a natural number, which, being subtracted from the natural co-sine of the difference between the latitude and the declination, when they are of the same name, or from that of their sum if of contrary names, will leave the natural co-sine of an arch; now, the difference between this arch, and the difference or sum of the latitude and the declination, according as they are of the same or of contrary names, will be the change of altitude in one minute from noon.

Example 1. Let the latitude be 13 degrees, and the declination of a celestial object 2 degrees, both of the same name; required the variation or change of altitude in one minute from noon?

* This is the log, versed sine, or log, rising, of one minute of time.

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