Alt. of the star A = 16. 45.34 Nat. co-v. sine 711648

Remainder, or difference of nat. co-versed sines 686912 L. 5.836901

Diff. of dec. and alt.. 25:31:16 Nat. v. sine 097574
Natural number

.

. 277479 Log. 5. 443231

375053=38:40:42" N.

Natural co-versed sine of the latitude
which is the latitude of the place of observation, as required.

Note.-The difference betwixt the declination and altitude of the star R, is taken in this example because the latitude and declination of that star are of the same name: in the preceding example, the sum was taken because the latitude and declination were of contrary names. And, as this is the only case to which the rule is subject; it is presumed that any further illustration, by way of examples, would be

unnecessary.

Hence, it is manifest that in the above problem the mariner is provided with a direct and most accurate method of finding the latitude at sea; and, since it prevents the uncertainty and confusion arising from an error in the assumed latitude, or that by account, and, besides, being free from all ambiguity, restriction, and variety of cases whatever,—it may, therefore, be employed with a certainty of success, at any hour

of the night, whenever two known fixed stars are visible. Indeed, if the altitudes of the objects be determined with but common attention, the latitude resulting therefrom will be always true to the nearest second of a degree, without the necessity of repeating the operation, or of applying any correction whatever to the result.

Remarks.-Although it is at all times advisable for two observers to take the altitudes of the stars at the same moment of time, yet, should one person be desirous of going through the whole operation himself, he is to proceed as follows:-viz., let the altitude of one star be taken, and the time of observation noted by a watch that shows seconds; then let the altitude of the other star be observed, and the time noted also; and let the altitude of the first observed star be again taken, and the time of observation noted.

Now, find the difference between the first and last times of observation, and between the altitudes of the first observed star; and find, also, the difference between the first time of observation of the first star, and the time of observing the second star. Then say, as the interval or difference of time between the two observations of the first star, is to the difference of altitude in that interval; so is the interval, or difference of time, between the observations of the first and second star, to a correction; which, being applied by addition or subtraction, to the first observed altitude of the first star, according as it may be increasing or decreasing, the sum or difference will be the altitude of that star reduced to the time that the altitude of the second star was taken. This part of the operation may be readily performed by proportional logarithms ;-see Example, page 75. The interval between the observations ought, however, to be as much contracted as possible, on account of guarding against any irregularities in the change of altitude. With the altitudes, thus found, and the other requisite elements, the latitude is to be computed as above.

PROBLEM IX.

Given the Latitude by Account, the Altitude of the Sun's lower or upper Limb observed near the Meridian, the Mean Time of Observation, and the Longitude; to find the true Latitude.

Since it frequently happens at sea, particularly during the winter months of the year, that the sun's meridional altitude cannot be taken, in consequence of the interposition of clouds, fogs, rain, or other causes; and since the true determination of the latitude becomes an

object of the greatest importance to the mariner when his ship is sail ing in any narrow sea trending in an easterly or a westerly direction, such as the British Channel; the present problem is, therefore, given, by means of which the latitude may be very readily and correctly inferred from the sun's altitude taken at a given interval from noon, within the following limits; viz.,-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 its declination and the latitude of the place of observation, when they are of the same name, or by their sum, when of contrary names, the extent of the interval from noon within which the altitude should be observed, may, therefore, be readily ascertained, 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 60 degrees, and the declination 23 degrees, both of the same name, the interval between the time of observation and noon ought not to exceed 37 minutes; but if one be north and the other south, the interval may be extended, if necessary, to 83 minutes before or after noon. The altitude, however, may be taken as near to noon as the mariner may think proper; the only restriction being, that the observation must be made within the above-mentioned limits.

The interval between the mean time of observation and noon must be accurately determined: this may be always done, by means of a chronometer or any well-regulated watch showing seconds; proper allowance being made for the difference of time answering to the change of longitude, if any, since the last observation for determining its

error.

Now, if the sun's altitude be observed at any time within the abovementioned limits, the latitude of the place of observation may then be determined, to every degree of accuracy desirable in nautical operations, by the following rule; which, being performed by proportional logarithms, renders the operation nearly as simple as that of finding the latitude by the meridional altitude of a celestial object.

See explanation to Tables LI. and LII., between pages 138 and 143.

RULE.

Reduce the sun's declination to the mean time and place of obser vation, by Problem XIV., page 357, and let the observed altitude of

the sun's lower limb be reduced to the true central altitude, by Problem XXIII., page 374. Then, with the sun's reduced declination, and the latitude by account, enter Table LI, or LII. (according as the latitude and the declination are of the same or of a contrary denomination), and take out the corresponding correction in seconds and thirds, which are to be esteemed as minutes and seconds, agreeably to the Rule in page 139. Now,

To the proportional logarithm of this correction, add twice the proportional logarithm of the interval between the time of observation and noon, and the constant logarithm 7.2730; the sum of these three logarithms, abating 10 in the index, will be the proportional logarithm of a correction, which being added to the true altitude of the sun's centre, the sum will be the meridional altitude of that object: hence the sun's meridional zenith distance will be known; to which let its reduced declination be applied by addition or subtraction, according as it is of the same or of a contrary name, and the sum or difference will be the latitude of the place of observation.

Remark. In taking out the equation from Table LI. or LII., proportion must be made for the excess of the given degrees of latitude and declination above the next less tabular correction; agreeably to the formula, or mode of computation, at the bottom of page 141.

Example 1.

At sea, January 1st, 1836, at 22:48" 3: mean time, in latitude 51:36 north, by account, and longitude 10:45:30" west; the mean of several observed altitudes of the sun's lower limb reduced to the true central altitude, was 13:33:58" south required the true latitude of the place of observation?

Mean time of observ. 22:48′′3: || Sun's declination at
Longitude 10:45:30%

west, in time. .. +0.43.2

noon, Jan. 1st= Cor. for 23*31*5!=

23: 4.16 S.

4.49

23:315 Sun's reduced dec.. 22:59:27:S.

. 22:48:3:

Apparent time of observation 2244′′0!; which is 1:16′′0: from appa

rent noon.

The cor. in Table LII., corresp. to lat. 50: N., and dec. 22:59:27" S., taken as 23:, is

Reduction for 1:36% of latitude

Tabular correction =

1129

Apparent time from noon =1:16:0:
Constant logarithm =

Correction of the sun's altitude =
True altitude of the sun's centre =

.

7.2730

Latitude of the place of observation 51:36 1" North.

Note. In all problems relating to the sun, the mean time of observation is to be reduced to apparent time, as above; so that the interval from noon may be expressed in apparent time.

Example 2.

At sea, March 21st, 1836, at 0:57:39: mean time, in latitude 51:5 north, by account, and longitude 35:45 west, the mean of several observed altitudes of the sun's lower limb, reduced to the true central altitude, was 38:13:37" south of the observer; required the correct latitude of the place of observation ?

Mean time of observ. = 0:57" 39: ||'s declination at noon,
Longitude 35:45! W.,

March 21st = 0:22: 4′′N. Correc. for 3 20:39: + 3.18

3:20:39:'s reduced declina. 0:25:22 N.

Apparent time of observ. 0:50:25"; which is the apparent time from