tor's Sailing, between those several successive latitudes and longitudes : then, if the sum of the several distances coincide, or nearly so, with the true spherical distance found as above, the senses must become reconciled to the propriety of adopting that high southern route at which they originally seemed to recoil.

In order to determine the several successive latitudes at which the ship must arrive, we must previously compute the vertical or polar angles ASF and BS: then, if the sum of these angles makes up the whole difference of longitude, or polar angle between the two given places, it will be a convincing and satisfactory proof that, for so far, the operations will have been properly conducted. Now, in the right angled spherical triangle ASF, given the hypothenuse AS, 55:36 the co-latitude of the Cape of Good Hope, and the perpendicular FS, 31:137134" the complement of the highest latitude at which the ship should arrive, to find the vertical or polar angle FSA. And, in the right angled spherical triangle BS F, given the hypothenuse BS, 50:23 = the co-latitude of King's Island, and the perpendicular FS, 31:13:13", to find the vertical or polar angle BS F. Hence, by right angled spherical trigonometry, Problem I., page 184,

=

90: 0 0 Log. co-secant

55.36. 0 Log. co-tangent=
31.13. 13 Log. tangent =

10.000000

9.835509

9.782550

65:28:48 Log. co-sine

= 9.618059

And, since the sum of the polar angles, thus obtained, viz., ASF 65:28:48 + BSF 59:53:12" = 125°22′0′′, makes up the whole difference of longitude between the two given places expressed by the whole angle AS B, it shows that thus far the work is right.

Now, on the equator, from Q to m, lay off the proposed changes of longitude, viz., 5°, 10, 15, 20, 25°, &c. These are to be taken respectively, in the compasses, from the scale of semi-tangents, reckoning backwards from 90 towards 0:, till the proposed changes of longitude reach the centre C; and then forwards on that scale, or from 0: towards 90, till those changes of longitude meet the point m; thus, the extent from 90:

to 85 will reach from Q to 5; the extent from 90 to 80%, will reach from Q to 10, and so on to the centre C; then, the extent from 0 to 5%, will reach from C to 95; the extent from 0 to 10%, will reach from C to 100%, and so on to the point m. Through the points S and N, and the several points made by the proposed changes of longitude on the equator, draw arcs of great circles, viz., S 1, 5; S 2, 10; S3, 15; S 4, 20%, &c. &c.; and then the arcs S1, S2, S3, &c. &c., will represent the respective complements of the several latitudes at which the ship should arrive at the given changes of longitude; the true values of which may be found in the following manner, viz.,

=

=

From the polar angle ASF, subtract the proposed changes of longitude continually; and the several polar angles made by those changes, and contained between the perpendicular FS and the co-latitude of the Cape of Good Hope = SA, will be obtained. Thus, from the polar angle ASF = 65°28'48", let 5 be continually subtracted, and the results will be FS1 = 60:28:48"; FS 2 55:28:48"; FS3 50:28:48", &c. &c. And, since the last subtraction in this triangle leaves the remainder, or polar angle, FS 125:28:48", which is 28:48 greater than the proposed alteration of longitude, therefore, in the triangle BSF, where the polar angle S is 59:53:12" (and where the several polar angles contained between the perpendicular FS and the co-latitude of King's Island are to be determined by a contrary process to that which was observed in the preceding triangle), the first polar angle is expressed by 59 28:48 = 4:31:12 the angle FS a; to which let the proposed alterations of longitude be continually added, and the sums will be FSb 9:31:12"; FSC 14:31:12", &c. &c. Those various results are to be arranged agreeably to the form exhibited in the first column of the following Table; and, since they respectively express the true measures of the several polar angles contained between the meridians of the given places and those of the several co-latitudes to which they correspond, it is, therefore, manifest that those results reduce the two right angled spherical triangles (ASF and BS F) into a series of right angled spherical triangles; to each of which the perpendicular FS is common. Then, in each of these triangles, we have the perpendicular and the angle adjacent, to find the hypothenuse or co-latitude. Thus, in the right angled spherical triangle FS1, right angled at F, given the perpendicular FS 31:13:13", and the polar angle FS 160:28:48", to find the hypothenuse or co-latitude S 1; in the right angled spherical triangle FS 2, given the perpendicular FS = 31:13:131%, and the polar angle FS 2 = 55:28:48", to find the hypothenuse or co-latitude S 2, &c. &c. Hence, by right angled spherical trigonometry, Problem IV., page 188,

=

31:13:13

90. 0. 0

To find the Hypothenuse, or Co-Latitude = S 1 :

As the perpendicular FS

Is to the radius =

Log. co-tangent = 10. 217450*

9.692607

To the co-latitude S 1 =

50.53.28

Log. co-tangent = 9.910057

First latitude =

39: 6:32 S., at which the ship should arrive.

To find the Hypothenuse, or Co-Latitude = S 2 :

Hence, the first latitude at which the ship should arrive, is 39:6:32′′S.; and the second latitude 43:4:31 S.: and, since it is the latitude itself, and not its complement, that is required, if the log. tangent of the sum of the three logarithms be taken, it will give the latitude direct; and, by rejecting the radius, the work will be considerably facilitated. Proceeding in this manner, the several successive latitudes corresponding to the proposed alterations of longitude will be found, as in the third column of the following Table.

Now, let the several successive longitudes be arranged (agreeably to the proposed change, and to the measure of the corresponding polar angles,) as given in the second column of the following Table; and find the difference between every two adjacent longitudes, as shown in the fourth column of that Table. Find the difference between every two successive latitudes, and place them in the fifth column of the Table. Take out from Table XLIII. the meridional parts corresponding to the several successive latitudes, as given in column 6, and find the difference between every two adjacent numbers, as given in the seventh column. Then find, by Mercator's Sailing, Problem I., page 238, the respective courses and distances between the several successive latitudes and longitudes; and let those courses and distances, so found, be arranged as in the two last columns of the following Table: viz.,

The log. co-tangent is used, so as to avoid the trouble of finding the arithmetical complement of the log. tangent.

A TABLE,

Now, the sum of the several successive differences of longitude = 7522 miles, makes up the whole difference of longitude between the two given places; the sum of the successive differences of latitude = 2612.53 miles, is equal to the whole difference of latitude comprehended under the highest latitude at which the ship should arrive, and the latitudes of the two given places, viz. 34:24'0" S., 58:46:461" S., and 39:37:0? S.-And, the sum of the several meridional differences of latitude = 3973. 85 miles, coincides exactly with the whole meridional difference of latitude corresponding to the highest latitude, and the latitudes of the two given places; which several agreements, form an incontestable proof that the work has been carefully conducted.

The sum of the several distances measured on the consecutive rhumb lines intercepted between the successive latitudes and longitudes, as exhibited in the last column of the Table, is 5426. 46 miles ;-but the true spherical distance on the arc of a great circle is 5426. 30 miles; the difference, therefore, is only O'. 16; or, about of a mile; which is very trifling, considering the extent of the arc.-The distance by Mercator's sailing is 6011.2 miles; which is 585 miles more than by great circle sailing.

Hence, it is evident that the shortest and most direct route from the Cape of Good Hope to King's Island is by the latitude of 58:46:464" S.; and that the ship must make, successively, the several longitudes and latitudes contained in the 2nd and 3rd columns of the Table, in the same manner, precisely, as if they were so many headlands, or places of rendezvous, at which she was required to touch. The first course, therefore, from the Cape of Good Hope is S. 40:22. E. distance 371 miles, which will bring the ship to longitude 23:32. E. and latitude 39:6:32" S.;-the second course is S. 43:31 E. distance 328 miles, which brings the ship to longitude 28:32: E. and latitude 43:4:31 S.; the third course is S. 46:56. E. distance 292 miles, which brings the ship to longitude 33:32 E. and latitude 46:23:39 S;-and so on of the rest.-Whence, it is evident that if the ship sails upon the several courses, and runs the corresponding distances respectively set forth in the two last columns of the Table, she will, most assuredly, arrive at the several successive longitudes and latitudes pointed out in the 2nd and 3rd columns of that Table; and thus will she reach King's Island, the place which it is intended she shall make, by a track 585 miles shorter than if such track had been determined agreeably to the principles of Mercator's sailing.

And, in a long voyage, like the present, in which ships generally experience a great scarcity of fresh water, particularly those bound to His Majesty's colony at New South Wales with troops, or convicts, the saving of 585 miles run at sea becomes a consideration of no inconsiderable import

ance.

Nor is there any more difficulty in sailing on the arc of a great circle,