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A Table of the Angles which every Point of the Compass makes with

the Meridian.

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PLANE SAILING.

PLA

LANE SAILING is the Art of Navigating a Ship upon Principles deduced from the Notion of the Earth's being an extended Plane; and is no more than the Application of Plane Trigonometry to the Solution of the feveral Variations, or Cafes; where the Hypothenufe, or longeft Side, is always the Rhumb that the Ship fails upon.

The Perpendicular is the Difference of Latitude counted on the Meridian, and the Bafe the Departure: which is either Eafting or Wefting, counted from the Meridian.

The Angle oppofite the Bafe is the Courfe, or Angle, that the Ship makes with the Meridian; and the Angle oppofite the Perpendicular is the Complement of the Course, which being taken together, make always eight Points or Rhumbs, which is 90 Degrees.

In conftructing Figures relating to a Ship's Courfe, let the upper Part of the Paper, or what the Figure is drawn upon, always reprefent the North; the lower Part will be the South; the Right Hand Eaft; and the Left Weft.

Draw the North and South Line to represent the Meridian of the Place the Ship fails from; then if the Ships Courfe is to the Southward, mark the upper End of the Line for the Place failed from; but if the Courfe is Northward, mark the Lower End for that Place.

When the Course is Eafterly, defcribe the Arch, and lay off the Course and Departure on the Right Hand Side of the Meridian; but when Wefterly on the Left Hand Side.

When the Courfe is given in Degrees, the Degrees expreffing it must be taken from the Line of Chords; but when in Points from the Line of Rhumbs; and is always to be laid off upon the Arch, beginning at the Meridian.

When the Courfe is given in Points, it may be fet down with its correfponding Logarithm in Points in the Calculation, as found in the first Page of the Logarithms, without reducing it into Degrees, as it feems altogether unneceffary.

In all Cafes, wherever the Complement Course, or Co-fine, &c. is ufed, the Degrees or Points put down is the Course itself; yet the Logarithm belonging to the Complement, or Co-fine, &c. of that Courfe is taken.

CASE

CASE I.

Courfe and Distance failed given, to find the Difference of Latitude and the Departure from the Meridian.

Suppofe a Ship from the Lizard in the Latitude of 49° 57′ North, fails S. W. by W. 496 Miles; required the Latitude come to, and Departure from the Meridian?

By

PROJECTION.

Draw the Meridian or Difference of Latitude, with the Chord of 60° in your Compaffes, and one Foot in C, defcribe an Arch, take 56° 15' or 5 Points in your Compaffes, and lay off that Distance upon the Arch from B C towards CA; through the Point where it cuts, draw the Distance C A; upon which fet off 496: from A let fall the Perpendicular A B, the Departure, and it is done; for A B being measured on the fame Scale that A C was, will give the Departure 412.4, and B C 275.6 the Diff. of Lat.

The Calculation is the fame as Problem I. in TRIGONOMETRY.

I fhall work this Cafe by making each Side Radius.

56.15

C

Diff Lat

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By making the Distance Radius it will be by AXIOM I. The Courfe 5 Points 56° 15.

To find the Departure.

The Co-Courfe 3 Points 33° 45"
To find the Difference of Lat.

As Radius

90' 10,00000

2,69548

10,00000
2,69548 Is to the Dift. 496
9,91985 So is Co-fine Course 5 Pts. 9,74474

As Radius

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90'

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Is to the Distance 496

So is Sine Course

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To the Departure 412.4

12,44022

10,00000

2,61533 To the Diff. of Lat. 275,6 2,44022

Now as the Ship is in North Latitude failing Southerly from the

Latitude left

Take the Difference of Latitude 275,6

Gives the Latitude in

And the Departure from the Meridian is 412,4 Miles.

By making the Departure Radius it will be,

To find the Departure.
As Co-Sec. Courfe 5 Pts.
Is to the Dift. 496
So is Radius 8 Pts 90°

To the Departure 412.4

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To find the Diff. of Lat. 10,08015 As Co-Sec. Course 5 Pts. 10.08015 2,69548 Is to the Dift. 496 2,69548 10,00000 So is Co-Tan. Courfe 5 Pts. 9,82489 12,69548 10,08015

12,52037

10,08015

2,61533 To the Diff. of Lat. 275,6 2,44022

By making the Difference of Latitude Radius, it will be,

To find the Difference of Latitude.
As Sec. Courfe, 5 Pts.

To find the Departure

As Sec. Courfe 5 Pts.

Is to the Dift. 496

So is Tang. Courfe 5 Pts.

10,25526
2,69548 Is to the Dist.
10,17511 So is Radius 8 Pts. 90°

10,25526

496

2,69548

10,00000

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Here all the three Sides are made Radius, to find the Difference of Latitude and Departure; therefore, the Learner may make which Side Radius he pleases; but as for my Part, I fhall make the firft, where the Distance is made Radius, whenever the Courfe is given.

Though this Method of working by Logarithms is certain, yet the fame may be wrought by Gunter's Scale and Compafles, much more expeditiously; and exact enough in the Practice of Navigation.

NOTE. When the Courfe is given in Points, make Use of the Lines marked Sine Rhumbs, and Tangent Rhumbs, on the upper Side of the Scale; when in Degrees make Ufe of the Lines marked Sine and Tangent.

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Now to perform the last Cafe, extend from Radius, or 8 Points to Points on the Line marked S R; that Extent will reach from the Distance 496 to the Departure 412.4 on the Line of Numbers.

2dly. Extend from Radius or 8 Points to 3 Points, (the Complement of the Courfe) on the Line S R; that Extent will reach from the Distance 496 to the Difference of Latitude 275.6 on the Line of Numbers.'

Thus may all the Operations be performed in the feveral Cafes of Navigation.

By this Cafe is calculated the Tables of Latitude and Departure for every Degree, Point, and Quarter Point of the Mariner's Compafs, to the Distance of 300 Miles, which is of excellent Use in working Days Works at Sea, and may be applied both to Middle Latitude and Mercator's Sailing, as fhall be fhewn hereafter; we fhall only proceed now to the Working of the last Cafe by the Table of Diff, of Latitude and Departure.

By INSPECTION.

Find the given Course at the Top or Bottom of the Tables, either among the Points or Degrees, and in that Page, and right against the Distance taken in its Column, ftand the Difference of Latitude and Departure in their Columns.

Thus the Courfe is S. W. by W. or 5 Points, which is found at the Bottom of the Table of Difference of Latitude and Departure for Points; and as the Distance 496 is too great to be found in the Tables, divide it by 2, (or any other convenient Number) and that gives 248, which look for in the Distance Column, and right against it ftands 137.8 for the Difference of Latitude, and 206.2 for the Departure, which being doubled, (becaufe divided by 2) gives 275.6 for the Difference of Latitude, and 412.4 for the Departure, the fame as before. Any of thefe Methods will do, but the laft is chiefly practifed at Sea.

CASE II.

Courfe and Difference of Latitude given, to find the Distance run, and Departure from the Meridian.

If a Ship runs S. E. by E. from 1° 45′ North Latitude, and then by Obfervation is in 2° 50' South Latitude, what is her Distance and Departure?

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