The seaman's companion being a plain guide to the understanding of arithmetick, geometry, trigonometry, navigation, and astronomy. Applied chiefly to navigation: and furnished with a table of meridional parts, to every third minute: with excellent and easie ways of keeping a reckoning at sea, never in print before. Also, a catalogue of the longitude and latitude of the principal places in the world with other useful things. The third edition corrected and amended. By Matthew Norwood, mariner.

The three Angles of a Triangle given, with one of the Sides, to find the other two sides.

QUESTION II.

Admit I set from a Head-land lying in the Latitude of 50 deg. 00 min. North Latitude, and sail W S W 38 Leagues, and then meet with a Ship that came from a Place which lies S S W from the Head-land. Now this Ship hath Sailed N W. I demand the Distance of that Place from the other, and also the Distance the Ship hath sailed that came from the Southermost Place.

FIrst draw your North and South line B s. Let B be the Head-land. From B then draw a W S W line (by describing the Arch of 60 deg. s D from the Center B, and setting it off as hath been shewed) upon this W S W line set off the Ships run 58 miles B O: From O as a Center describe the Arch of 60 deg. R n, and extend the side O B to it: Now because the Question saith that B O is a W S W line from B, then O B must be an E N E line from O. Also because the Question saith, that the Ship that met her sailed N W, the side A O is a N W line from A, which is a S E line from O to A. Now there are 6 points between an E N E and a S E Course: there∣fore from the line D B extended upon the Arch of 60 deg. from B, set off 6 points or 67 deg. 30 min. which is n R, and

draw O R a white line: Then because the Question saith that the place the other Ship set from, bears from the Head-land B S S W. Draw a S S W line from B, which is B A, and where

[illustration] depiction of geometrical figure

it cuts the line O R, which is in A, there will be the place the other Ship set from, and A B will be the distance of the Head-land and that asunder, A O will be the distance that the Ship hath sailed that came from it, which I find as followeth.

The distance between the two places is A B

The distance that the Ship sailed is A O

If you have occasion to find the Latitude the place at A lies in, let fall the Perpendicular A N upon the South line that comes from B, and it cuts it in N, and leaves the Difference of Latitude B N, see how many miles it is, and subtract it from 50 deg. and you have your desire, measure A N, and you have the Westing that A lies from the Head-land B.

By the Tables.

Here the Angle at B is an Angle of 4 points or	45 d.	00 m.
The Angle at A is an Angle of 6 points or	67 d.	30 m.
The Angle at O is an Angle of 6 points or	67 d.	30 m.
The side B O is 58 miles.

First for the Side A B.

The general Rule saith, That the Sine of every Angle is pro∣portional to its opposite side. Then from this I conclude, that B A should be equal to B O, because the Angles opposite to them are equal, and so you will find them

For as Sine A 67 deg. 30 min.	comp. arith.	0,034384
Is to O B 58 miles		1,763428
So is Sine O 67 deg. 30 min.		9,965615
To A B 58 miles, the two places distance	1,763428

You see it is so exactly, for these three numbers added toge∣ther, and Radius cast away, produceth the same Logarithm that 58 (taken out of the Book) did, and this Question I do on purpose to shew the truth of the general Rule. The same way other Questions of this nature are wrought.

For the Distance that the Southermost Ship sailed, A O.

As A O B Sine 67 deg. 30 min.	comp. arith.	0,034384
Is to A B 58 miles		2,763428
So is Sine A B O 45 deg. 00 min.		9,849485
To A O 44 4/10 miles		2,647297

If you desire to find the Difference of Latitude between the two places B and A, you have the Distance A B, and the Course A B n given to find it or the Longitude.

If you desire to find the Latitude the Ships are in when they meet, let fall a Perpendicular from O upon the South line (which is B N) and you have the distance O B given, 58 miles, and the Angle O B R 67 deg. 30 min. to find it, or the Departure between the Ships at their meeting, and the Head-land O r.

The like is to be understood of any other Question of this nature.

Notes

• 58 miles.
• 44 4/10 miles.