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Moon's transit over the meridian of Greenwich, per Nautical
Almanac, April 13th, 1824, is

Correction from Table XXXVIII., answering to retardation of
transit 50%, and longitude 174:56' west =

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Moon's transit reduced to the meridian of Queen Charlotte's
Sound
Correction answering to reduced time of transit (10:3731)
in Table XXXIX., is

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Corrected time of transit
Time of high water at given place on full and change days.

Time of high water at Queen Charlotte's Sound, past noon of

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19:26:31:

Subtract 12.24. 0

the given day

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Reduction of the Moon's Horizontal Parallax on account of the Spheroidal Figure of the Earth.

Since the moon's equatorial horizontal parallax, given in the Nautical Almanac, is determined on spherical principles, a correction becomes necessary to be applied thereto, in places distant from the equator, in order to reduce it to the spheroidal principles, on the assumption that the polar axis of the earth is to its equatorial in the ratio of 299 to 300; and, when very great accuracy is required, this correction ought to be attended to, since it may produce an error of seven or eight seconds in the computed lunar distance. The correction, thus depending on the spheroidal figure of the earth, is contained in this Table; the arguments of which are, the moon's horizontal parallax at the top, and the latitude in the left-hand column; in the angle of meeting will be found a correction, expressed in seconds, which being subtracted from the horizontal parallax given in the Nautical Almanac, will leave the horizontal parallax agreeably to the spheroidal hypothesis.

Thus, if the moon's horizontal parallax, in the Nautical Almanac, be 57:58%, and the latitude 51:48; the corresponding correction will be 7 seconds subtractive. Hence the moon's horizontal parallax on the spheroidal hypothesis, in the given latitude, is 57:51.

Remark. The corrections contained in this Table may be computed by the following

Rule.

To the logarithm of the moon's equatorial horizontal parallax, reduced to seconds, add twice the log. sine of the latitude, and the constant log. 7.522879; the sum, rejecting the tens from the index, will be the logarithin of the corresponding reduction of parallax.

Example.

Let the moon's horizontal parallax be 57:58%, and the latitude 51:48; required the reduction of parallax agreeably to the spheroidal hypothesis?

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Moon's equatorial horiz. par. 57:58" = 3478" Log.= 3.541330
Latitude
Twice the log. sine 19.790688
Constant log.

51:48

7.522879

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Reduction of Latitude on account of the Spheroidal Figure of the Earth.

Since the figure of the earth is that of an oblate spheroid, the latitude of a place, as deduced directly from celestial observation, agreeably to the spherical hypothesis, must be greater than the true latitude expressed by the angle, at the earth's centre, contained between the equatorial radius and a line joining the centre of the earth and the place of observation. This excess, which is extended to every second degree of latitude from the equator to the poles, is contained in the present Table; and which, being subtracted from the latitude of any given place, will reduce that latitude to what it would be on the spheroidal hypothesis: thus, if the latitude be 50 degrees, the corresponding reduction will be 11:42", subtractive; which, therefore, gives 49:48:18" for the reduced or spheroidal latitude.

Remark. The corrections contained in this Table may be computed by the following rule; viz.,

To the constant log. .003003,† add the log. co-tangent of the latitude,

* The arithmetical complement of the log. of the earth's ellipticity assumed at + The excess of the spherical above the elliptic arch in the parallel of 45 degrees from the equator, is 11'.887, or 11′53′′ (Robertson's Navigation, Book VIII., Article 134): hence 45o — 11′ 53′′ = 44° 48′ 7′′, the log. co-tangent of which, rejecting the index, is .003003.

and the sum will be the log. co-tangent of an arch; the difference between which and the given latitude will be the required reduction.

Example.

Let it be required to reduce the spherical latitude 50:48 to what it would be if determined on the spheroidal principles; and, hence, to find the reduction of that latitude.

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in

A General Traverse Table; or Difference of Latitude and Departure. This Table, so exceedingly useful in the art of navigation, is drawn up a manner quite different from those that are given, under the same denomination, in the generality of nautical books; and, although it occupies but 38 pages, yet it is more extensive than the two combined Tables of 61 pages, which are contained in those books. In this Traverse Table, every page exhibits all the angles that a ship's course can make with the meridian, expressed both in points and degrees; which does away with the necessity of consulting two Tables in finding the difference of latitude and the departure corresponding to any given course and distance. If the course be under 4 points, or 45 degrees, it will be found in the left-hand compartment of each page; but that above 4 points, or 45 degrees, in the right-hand compartment of the page. The distance is given, in numerical order, at the top and bottom of the page, from unity, or 1, to 304 miles; which comprehends all the probable limits of a ship's run in 24 hours; and, by this arrangement, the mariner is spared the trouble of turning over and consulting twenty-three additional pages. Although the manner of using this Table must appear obvious at first sight, yet since its mode of arrangement differs so very considerably from the Tables with which the reader may have been hitherto acquainted, the following Problems are given for its illustration.

PROBLEM I.

Given the Course and Distance sailed, or between two Places, to find the Difference of Latitude and the Departure.

RULE.

Enter the Table with the course in the left or right-hand column, and the distance at the top or bottom; opposite to the former, and under or over the latter, will be found the corresponding difference of latitude and departure: these are to be taken out as marked at the top of the respective columns if the course be under 4 points or 45 degrees, but as marked at the bottom if the course be more than either of those quantities.

Note.-If the distance exceed the limits of the Table, an aliquot part thereof may be taken, as a half, third, fourth, &c.; then the difference of latitude and departure corresponding to this and the given course, being multiplied by 2, 3, 4, &c., (that is, the figure by which such aliquot part was found,) the product will be the difference of latitude and departure answering to the given course and distance.

Example 1.

A ship sails S.S.W. W. 176 miles; required the difference of latitude and the departure?

Opposite 24 points and under 176 miles, stand 155. 2 and 83.0: hence the difference of latitude is 155. 2, and the departure 83.0 miles.

Example 2.

A ship sails N. 57: E. 236 miles; required the difference of latitude and the departure?

Opposite to 57%, and under 236 miles, stand 128.5 and 197.9: hence the difference of latitude is 128. 5, and the departure 197.9 miles.

Example 3.

The course between two places is E. b. S. & S., and the distance 540 miles; required the difference of latitude and the departure?

Distance divided by 2, gives 270 miles ; under or over which, and opposite to 64 points, stand .. 91.0 and 254.2

Multiply by

Products

2

2

182.0 and 508.4: hence the difference

of latitude is 182. 0, and the departure 508. 4 miles.

Example 4.

The course between two places is N. 61 W. and the distance1176 miles; required the difference of latitude and the departure ?

Distance 1176 divided by 4, gives 294 miles; under or over which, and opposite to 61, stand 142.5 and 257.1

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Multiply by

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570.0 and 1028.4: hence the differ

ence of latitude is 570. 0, and the departure 1028.4 miles.

PROBLEM II.

Given the Difference of Latitude and the Departure, to find the Course and Distance.

RULE.

With the given difference of latitude and departure, enter the Table and find, in the proper columns abreast of each other, the tabular difference of latitude and departure either corresponding or nearest to those given; then the course will be found on the same horizontal line therewith in the left or right-hand column, and the distance at the top or bottom of the compartment where the tabular numbers were so found.

Note. If the difference of latitude be greater than the departure, the course will be less than 4 points, or 45 degrees; and, therefore, it is to be taken from the left-hand column: but when the difference of latitude is less than the departure, the course will be more than 4 points or 45 degrees, and, consequently, it must be taken from the right-hand column.

Note, also, that when the difference of latitude and the departure, or either of them, exceed the limits of the Table, aliquot parts are to be taken, as a half, third, fourth, &c., with which find the course and distance as before; then the distance, thus found, being multiplied by 2, 3, 4, &c., the product will be the whole distance corresponding to the given difference of latitude and departure. The course is never to be multiplied, because the angle will be the same whether determined agreeably to the whole difference of latitude and the departure, or according to their corresponding aliquot parts.

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