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As Radius

10.0000000 To Cc-fine of Dif. of Long. 05° co' 9.9983442 So is the Tangent of GP = 60° oo! 10.2385606 To the Tangent of P O = 59° 54' 10.2369048 Then

GP - PO = 00° 06', that is 6 Italian Miles, is the greatest Difference of Latitude the Ship made in that Course.

PROBLEM IV. Admit a Ship fets sail from B in Latitude 18° 46', and steers her Course away between E. and N. on the Arch of a Great Circle 40° 22', and then finds her Latitude at A to be 51° 32', I demand her Course, Difference of Longitude, and Departure.

Practice. In the Oblique Triangle A B P, there is known the Side B P = 719 14', the Complement of the first Latitude B Å= 40° 22', the Distance failed ; and the Side A P = 38° 28', the Complement of the Latitude arrived at ; to find the other Parts.

This is done by Case 5 Of Oblique Triangles, and otherwise by the fifth Proposition of Chap. 9, which has been so often exemplified, that I need not here repeat it. For thus the Angle at P will be found 30% ool for the Difference of Longitude ; and the Angle at B = 289 43', the Course at first fetting Sail ; but the Angle at A = 46° 58', the Course at her Arrival ; which is 18° 15' more to the East than the first.

The Departure will be found (by the Analogies in the two foregoing Problems ) to be 18° 32' = 1112 · Miles = 370Leagues.

But if the Ship had failed from A to B, her Departure would have been 28° 22' = 1702 Miles = 567 Leagues.

Now

Now notwithstanding there is such a Difference of the Departure, yet the Proportion by Plain and Mer.. cator's Sailing, or by Middle Latitude, gives none.

PROBLEM V.

By having the Latitudes of two Places given, with their Difference of Longitude ; to calculate the Diftance in the Arch ; the Course of the Ship at fetting Sail, and at her Arrival ; the intermediate Latitudes and Longitudes by which the Arch of Distance pas. seth ; the Course and Distances between those several intermediate Points of Latitude and Longitude.

Praćtice.

1. Suppose the Summer Inands be designed by E in Latitude N. 32° 25', from whence a Ship sets fail for the Lizard's Point at F, in Latitude 50° 00'; and admit the Difference of Longitude be 70° oo'; then,

2. In the Oblique Triangle FP È there is given the Side FP = 40000; the Side EP = 57° 35'; and the Angle at P = 70° OC'; to find E F, the nearest Distance between the two Places. Now this will be found 53° 24' = 3204 = Miles = 1068 Leagues, by Case 4 Of Oblique Triangles.

3. Then by Case 3 Of Oblique Triangles, you find the Angle FEP = 48° 48', i. e. N. E. and 30 48' Easterly, for the Ship’s Course at first setting Sail ; and the Angle E FP 819 10', i. e. E. by S. and 20 25' Easterly, for the Course of the Ship at her Arrival to the Lizard Point.

4. In the next Place, let fall the Perpendicular Pa, then is the Oblique Triangle FP E reduced to two Right-angled Triangles a PF, and a P E ; now the Perpendicular a P will be found 39° 26', whose Com

plement

Sa

a Pc

e

Pe. Pf.

plement 50° 34' is the Greatest Latitude through which the Ship pafseth.

5. Having done this, next find the Angles. E P as and a P F, by the common Analogy; the first will be 58° 31', and the latter 11° 29'; these Angles being obtained, you may proceed to find the Latitudes to as many Degrees of Longitude from a as you please. Thus,

6. Suppose the Latitude at every 10 Degrees Diftance from a be required ; a Pb

Pb. Then in the

there is given the Side PC. Right-angled <a Pd a P, and Angle at P, P d. Triangle

P

to find the Side a Pf

( which Sides are the Complements of the several Latitudes the Ship pafseth by at each tenth Degree of Longitude from a.

7. You may then also find the Angles af P, a e P, ad P, ac P, a b P, which will shew the Course to be steer'd in each of the Latitudes before found. Also the Parts Ef, fe, ed, &c. may be found if desired, which are the Distances between the aforesaid Points of Latitude.

Thus I have shewn how the whole Problem may be folved in each Particular; but have left the Practice thereof to the studious young Practitioner.

These Problems are general ones, I could not branch them out into particular Cafes, for that would be thought not only a mercenary Design to stuff a Book, but would be holding a Candle to a Reader at Noon Day, which is very absurd and affronting. I rather fear he who hath regularly proceeded thus far from the Beginning, will find I have been often obliged Aftum agere, i. e. to do the same thing over and over again, than been deficient in any one Part of this Undertaking. VOL, II.

А A Ta.

Ss

A Table of Rumbs, and the Angles they make with

the Meridians.

Rumbs.
Merid.

Rumbs.
North. South Angles. South. North.

2 49
5 38

8 26
N by E S by Eu 15S by W

15 S by W N by W.

I

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125 19
28 8

30 56 NE by N SE by S33 45 S W by SNW by N. 3

136 34
139 23

IL

NEI

472 SE

145 00 SW

[blocks in formation]

47 49
50 37

153 26

NĘ by ESE by E36 15 S W by WNW by W. 5

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CH A P. V.

The Method of Making and Sailing by the

Globular Chart.

A

S I have given an Epitome of the several
Kinds of Sailing, in several Parts of this work,

which depend more immediately on Trigonometry Plain or Spherical; fo in order that the young, Student might have a compleat Idea of all kinds of

Navigation in use, I have here thought good to subjoin a little Tract of the Nature, Corftruction, and Use of the Chart which is generally called the Globular Chart ; although it hath not that direct Relation to Trigonometry, as all the other parts of Navigation have.

I am the more willing to do this, because 'cis reckoned as a very celebrated modern Improvement of this Art, and is not to be found in any Mathematical Tracts that I have yet seen ; and therefore that every one may have the Benefit or it, I shall explain it in the following Particulars.

1. First therefore you must understand, that it is peculiar to this Chart to have all the Properties of the Surface of the Globe it self; and from thence it acquires the Name of the Globular Chart; for as on the Globe all the Meridians incline to each other from the Equator to the Poles, where they all terminate, so do they on this Chart ; as on the Globe the Parallels are all equidistant lesfer Circles, so are they in this Chart. The Rumbs here, as on the Globe, are inclining Spirals, or Helical Curves intersecting the VOL. II.

Ss 2

Meria

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