Remark 2.-The angle B may be also very readily computed by the following general Rule; viz.,

To twice the log. co-sine of half the given side, comprehended between the two given angles, add the log. sines of those angles, and the sum (rejecting 30 from the index), will be the log. of a natural number.-Now, the sum of twice this natural number and the natural versed sine of the difference of the angles, will be the natural versed sine of the required angle.

Thus, to find the angle B in the last example.

Half the given side A C= 43:45:20" twice the log. co-sine 19.717432

Diff. of the ang.= 5:10:10% nat. versed sine 004067

Angle B

the former Rule.

78:42' 2" nat. versed sine = 804063; the same as by

PROBLEM V.

Given the Three Sides of a Spherical Triangle, to find the Angles.

RULE.

Add the three sides together and take half their sum; find the difference between this half sum and the side opposite to the required angle, which call the remainder; then,

To the log. co-secants, less radius, of the other two sides, add the log. sines of the half sum and the remainder :-half the sum of these four logs. will be the log, co-sine of an arch, which being doubled will be the required angle.

One angle being thus found, the remaining angles may be computed by Rule 3, Problem II., page 200.

Example.

70.11.457

In the spherical triangle A B C, let the side A B be 70:11:45, the side A C 81:59:55", and the side BC 120:10:50%; required the angles A, B, B and C?

A

81:59. 557

To find the Angle A:

120:10:50

81.59.55 Log. co-secant, less radius=0.004248 70.11.45 Log. co-secant, less radius=0.026477

272.22.30

Remark. The required angle of a spherical triangle (when the three sides are given), may be also found by the following general Rule; viz.,

Add the three sides together and take half their sum: find the difference between this half sum and each of the sides containing the required angle, and note the remainders.-Then,

To the log. co-secants, less radius, of those sides, add the log. sines of the two remainders :-half the sum of these four logs. will be the log. sine of ' half the required angle.

Thus, to find the angle A in the last example.

Which being doubled, shows the angle A to be 126:10:16"; the same as by the former rule.

PROBLEM. VI.

Given the Three Angles of a Spherical Triangle, to find the Sides.

RULE.

Add the three angles together and take half their sum; find the difference between the half sum and the angle opposite to the required side, which call the remainder.-Then,

To the log. co-secants, less radius, of the other two angles, add the log. co-sines of the half sum, and the remainder; half the sum of these four logs. will be the log. sine of half the required side.

One side being thus found, the remaining sides may be computed by Rule 3. Problem I., page 198.

Example.

In the spherical triangle A B C, let the angle A be 125:16:25; the angle B 84:20:50", and the angle C72:40:15"; required the sides B C, AB, and A C?

P

Half the side BC.. 62:37:13" Log. sine =

The double of which gives 125:14.26%, for the whole side B C.

9.948402

Remark. The required side of a spherical triangle (when the three angles are given,) may be also found by the following general rule; viz.,

Add the three angles together and take half their sum; find the difference between the half sum and each of the angles comprehending the required side, and note the remainders.-Then to the log. co-secants less radius, of those angles, add the log. co-sines of the two remainders: half the sum of these four logs, will be the log. co-sine of half the required side.

Thus, to find the side BC in the last example.

Half Side B C =

62:37:13" Log. co-sine. 9.662648

Which being doubled gives = 125:14:26%, for the side BC; the same as by the former rule.

THE RESOLUTION OF PROBLEMS IN NAVIGATION BY LOGARITHMS; AND, ALSO, BY THE GENERAL TRAVERSE TABLE.

Lest the mariner should feel some degree of disappointment in not finding a regular course of navigation in this work the author thinks it right to remind him, that his present intention carries him no farther than merely to show the proper application of the Tables to some of the most useful parts of the sciences on which he may touch :-it being completely at variance with the plan of this work, to enter into such parts of the sciences as could reasonably be dispensed with, without entirely losing sight of their principles. Hence it is, that the cases of plane sailing, usually met with in books on navigation, will not be noticed in this. However, since it is not improbable that this volume may fall into the hands of persons not very deeply versed in nautical matters; it therefore may not be deemed unnecessary to give a few introductory definitions, &c. for their immediate guidance, previously to entering upon the essentially useful parts of the sailings.

NAVIGATION is the art of conducting a ship, through the wide and pathless ocean, from one part of the world to another.-Or, it is the method of finding the latitude and longitude of a ship's place at sea; and of thence determining her course and distance from that place, to any other given place.