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.

Two Sides and their contained Angle being given to find the third Side and the other Angles.

QUESTION III.

Two Ships set from one Port, and make an Angle of 58 deg. one sailed N b E, and the other sailed N W 1 deg. 45 min. Westerly, the Eastermost Ship sailed 70 miles and came to her Port: The Westermost Ship sailed 89 miles, and came to her Port: I de∣mand the Distance of these Ports asunder, and how they bear one from the other?

FIrst draw the North and South line A S. Let A B repre∣sent the Port you set from: Then from A as a Center, describe the Arch of 60 deg. F S D: From S to the East∣wards set off one point: for N b E the Eastermost Ships Course draw the line A D at length, and set the first Ships distance 70 miles upon it, which is A B, then from A draw a N W line 4 deg. 45 min. Westerly, and set off the Wester∣most Ships distance run upon that, which is 89 miles, and it reacheth from A to C, draw the line C B from the ends of the other two sides, which is the Distance between the Ports (for the ends of each Ships run must be the Ports) then is A C B the Angle of the bearing of the Eastermost Port from the Westermost, or A B C the bearing of the Westermost Port from the Eastermost, C A is a S E line, 1 deg. 45 min. Easterly from C, see how many points are contained between the C B and C A, by the way that is shewed in the Question before, or because B A is N b E

line (from B) you may find the Point of the Compass that B C runs upon from B, you need find it but one of these ways, and the opposite point must be the other, as I have shewed before, then take the length of the side C B, which is the distance of the Ports asunder, I find their distance and bearing to be as followeth.

[illustration] depiction of geometrical figure

The Distance between the two Ports is C B 79 miles.

The Bearing of them is East 5 deg. 35 min. Northerly, or West 5 deg. 35 min. Southerly.

By the Tables.

Here is given the Angle at A, 58 deg. 00 min. the side A B 70 miles, the side A C 89 miles, which is two sides, and their contained Angle, and here is required A C B or A B C, and the side B C.

First For the Angles.

For all Questions of this nature, this Rule is beneficial, and will work your Question. Note,

That in all plain Triangles, the Sum of two Sides are in such pro∣portion to their Difference, as the Tangent of the half Sum of their opposite Angles, is to the Tangent of an Angle, which Angle shall be the Difference between the half Sum, and the Angles: so that if it be added to the half Sum, it shall make the greater Angle; if it be subtracted, it shall make the lesser.

I have said something of this Rule before, which makes me demonstrate it no farther here.

Proportion.

As the Sum of A B and A C 159 miles	comp. arith.	6,79860
Is to their Difference 19 miles		2,27875
So is the Tan. of the half Sum of B and C 61 d. om.	10,25624
To Tangent of an Angle	12 10	9,33359

This added to the half Sum makes the Angle at B 73 d. 10 m. subtracted, it makes the Angle at C 48 deg. 50 min. which is 4 points, and 3 deg. 50 min. then say I, if C A be a S E Course, 1 deg. 45 min. Easterly from C, 'tis certain that C B must be 4 points and 38 deg. 50 min. from it, which is East 5 deg. 35 min. Northerly, the bearing of the Eastermost Port from the West∣ermost.

To find the half Sum of the Angles, Note that the three Angles are 180 d. therefore if one of them be known, subtract that from 180 d. and the Remainder must be the other two.

Now for their Distance asunder, C B.

As Sine A C B 48 deg. 50 min.	comp. arith.	0,123321
Is to B A 70 mil.		2,845098
So is Sine B A C 58 deg. co min.		9,928420
To the Distance of the Ports asunder C B 78 9/10	2,896839