Example 1.

If the difference of latitude made by a ship in 24 hours be 177.4 miles north, and the departure 102. 6 miles east, required the course and distance made good?

Solution. The tabular difference of latitude and departure, nearest corresponding to those given, are 177.5 and 102.5 respectively: these are found in the compartment under or over 205, and opposite to 30 degrees; hence the course made good is N. 30 E., and the distance 205 miles.

Example 2.

The difference of latitude made by a ship in 24 hours, is 98.5 miles south, and the departure 140.6 miles west; required the course and distance made good?

Solution. The tabular difference of latitude and departure, nearest to those given, are 98.7 and 140.9 respectively: these are found in the compartment under or over 172, and opposite to 55 degrees; hence the course made good is S. 55: W., and the distance 172 miles.

Example 3.

The difference of latitude is 700 miles south, and the departure 928 miles west; required the course and distance?

Solution. Since the difference of latitude and the departure exceed the limits of the Table, take therefore any aliquot part of them, as one fourth, and they will be 175 and 232 respectively: now, the tabular numbers, answering nearest to those, are 175. 1 and 232. 4; these are found in the compartment under or over 291, and opposite to 53 degrees: hence the course is S. 53: W., and the distance 291 x 4 = 1164 miles, as required.

Remark. Whenever it becomes necessary to take aliquot parts of the difference of latitude, the same must be taken of the departure, whether it falls without the limits of the Table or not; and, vice versa, whenever it becomes necessary to take aliquot parts of the departure, the same must be taken of the difference of latitude.

And, in all cases where the tabular numbers differ considerably from those given, proportion must be made for that difference.

PROBLEM III.

Given the proper Difference of Latitude between two Places, the Meridional Difference of Latitude, and the Departure, to find the Course, Distance, and Difference of Longitude.

RULE.

With the proper difference of latitude and the departure, find the course and distance by Problem II.; then, with the course thus found and the meridional difference of latitude, (in a latitude column,) take out the corresponding departure, and it will be the difference of longitude required; as thus: run the eye along the horizontal line answering to the course, from where the proper difference of latitude was found, (always to the right hand,) and find, in a latitude column, the tabular difference of latitude answering nearest to the given meridional difference of latitude ; abreast of which, in the departure column, will be found the difference of longitude.

Example,

The proper difference of latitude between two places, is 142 miles north, the departure 107 miles west, and the meridional difference of latitude 169 miles; required the course, distance, and difference of longitude ?

Solution. The tabular difference of latitude and departure answering nearest to those given, are 142. 2 and 107.3 respectively; these are found in the compartment under or over 178, and opposite to 37 degrees: hence the course is N. 37: W., and the distance 178 miles. Now, with the course 37 degrees, and the meridional difference of latitude 169 miles, the difference of longitude is found, as thus: from where the proper difference of latitude was found, run the eye along the horizontal line answering to 37 degrees, (always towards the right hand,) and the tabular difference of latitude answering nearest to the given meridional difference of latitude will be found in the compartment under or over 212, viz. 169. 3; corresponding to which, in the departure column, is 127.6; and which, therefore, is the difference of longitude, as required.

PROBLEM IV.

Given the proper Difference of Latitude, the Meridional Difference of Latitude, and the Difference of Longitude, to find the Course and Distance.

RULE..

With the meridional difference of latitude and the difference of longitude, esteemed as difference of latitude and departure, find the course by Problem II.; then with the course, thus found, and the proper difference of latitude, the distance is to be obtained, as thus: run the eye (always to the left hand) along the horizontal line answering to the course, from where the meridional difference of latitude was found, and seek, in the proper column, the difference of latitude answering nearest to that given; over or under which, at the top or bottom of the column, will be found the required distance.

Note.-When the meridional difference of latitude exceeds the difference of longitude, the course is to be taken from the left-hand column; but otherwise from the right.

Example.

The proper difference of latitude between two places is 78 miles south, the meridional difference of latitude 107 miles south, and the difference of longitude 119 miles east; required the course and distance?

Solution. The tabular difference of latitude and departure, answering nearest to the meridional difference of latitude and the difference of longitude, are 107. 1 and 118, 9 respectively; these are found in the compartment under or over 160, and opposite to 48 degrees: hence the course is S. 48: E. Now, the eye being run along the horizontal line answering to 48, (towards the left hand,) the nearest tabular difference of latitude, answering to the proper difference of latitude, will be found in the compartment under or over 117: hence the distance is 117 miles.

PROBLEM V.

Given the middle Latitude, and the Meridian Distance or Departure, to find the Difference of Longitude,

RULE.

Enter the Table with the middle latitude, taken as a course, and the departure in a latitude column; run the eye along the horizontal line

answering to that course (towards the right hand or the left, according as the first tabular difference of latitude which meets the eye therein is greater or less than the given departure), and find a difference of latitude that either agrees with, or comes nearest to, the given departure; then the distance corresponding to this, at the top or bottom of the column, will be the difference of longitude.

Example.

The middle latitude between two places is 20: north, and the meridian distance or departure 140 miles; required the difference of longitude?

Solution. The middle latitude, 20 degrees, taken as a course, and the departure 140, as difference of latitude, will be found to correspond in the compartment under or over 149: hence the difference of longitude is 149 miles, as required.

PROBLEM VI.

Given the middle Latitude, the Difference of Latitude, and the Difference of Longitude between two Places, to find the Course and Distance.

RULE.

Enter the Table with the difference of longitude, esteemed as distance, at the top or bottom of the page, and the middle latitude, taken as a course, in the left or right-hand column; answering to which, in the difference of latitude column, will be found the departure. Now, with this departure and the given difference of latitude, the course and distance are to be found by Problem II.

Example.

The middle latitude is 26 degrees north, the difference of latitude 200 miles north, and the difference of longitude 208 miles east; required the course and distance?

Solution. In the compartment under or over 208 miles (the given longitude), and opposite to 26 degrees (the middle latitude taken as a course), stands 186.9 in the difference of latitude column, which, therefore, is the departure. Now, the tabular numbers answering nearest to the given difference of latitude and the departure, thus found, are 200. 4 and 186.9 respectively; these are found in the compartment under or over 274, and opposite to 43 degrees: hence the course is N. 43: E., and the distance 274 miles.

Remark. The numbers in the general Traverse Table were computed agreeably to the following rule; viz.,

As radius is to the distance, so is the co-sine of the course to the difference of latitude; and so is the sine of the course to the departure.

Example.

Given the course 35 degrees, and the distance 147 miles; to compute the difference of latitude and the departure.

This Table contains the meridional parts answering to each degrée and minute of latitude from the equator to the poles; the arguments of which are, the degrees at the top, and the minutes in the left or right hand marginal columns; under the former, and opposite to the latter, in any given latitude, will be found the meridional parts corresponding thereto, and conversely. Thus, if the latitude be 50:48', the corresponding meridional parts will be 3549.8 miles.

Remark.-The Table of meridional parts may be computed by the following rule; viz.,

Find the logarithmic co-tangent less radius of half the complement of any latitude, and let it be esteemed as an integral number; now, from the