The mariners magazine, or, Sturmy's mathematical and practical arts containing the description and use of the scale of scales, it being a mathematical ruler, that resolves most mathematical conclusions, and likewise the making and use of the crostaff, quadrant, and the quadrat, nocturnals, and other most useful instruments for all artists and navigators : the art of navigation, resolved geometrically, instrumentally, and by calculation, and by that late excellent invention of logarithms, in the three principal kinds of sailing : with new tables of the longitude and latitude of the most eminent places ... : together with a discourse of the practick part of navigation ..., a new way of surveying land ..., the art of gauging all sorts of vessels ..., the art of dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties and forfeitures, by acts of parliaments appointed, relating to the customs and navigation : also a compendium of fortification, both geometrically and instrumentally / by Capt. Samuel Sturmy.

CASE X. Having two Sides, and the Angle contained between them given, To find either of the other Angles.

FOr the performance of this Problem, Suppose there were given the Side RS 406, and the Side RQ 185, and the Angle comprehended by them, namely the An∣gle at R, 45 deg. 20 min. and it were required to find either of the other Angles.

First, Take the Sum and Difference of the two Sides given; their Sum is 591, and their Difference is 221. Then knowing, that the three Angles of all Right-lined Triangles, are equal to two Right Angles, or 180 Degrees (by the 17th Theor. of Chap. 3.) Therefore the Angle SQR being 45 deg. 20 min. if you substract this Angle from 180 deg. the Remainder will be 134 deg. 40 min. which is the Sum of the two unknown Angles at Q and S; the half thereof is 67 deg. 20 min.

			deg.	mi.
The Side RS	406 Paces.	Two Right Angles	180	00
The Side RQ	185 Paces.	The Angle at R	45	20
The Sum	591 of the Sides given, RS and RQ.	The Sum of the two opposite Angles	134	40
The Difference	221 of the Sides.	The Half-Sum	67	20

The Sum and Difference of the Sides being thus found, and also the Half-Sum of the two unknown Angles, The Proportion by which you must find the Angles se∣verally is,

As the Logarithm of the Sum of the Sides 591	2771587
Is to the Logarithm of the Difference of the Sides 221	344392
So is the Tangent of the Half-Sum of the two unknown Angles 67 deg. 20 min.	10379213
The Sum of the second and third Number	12723605
The first Number substracted from the Sum	2771587
The Tangent of 41 deg. 50 min. (is this)	9952018
Which added to the Half-Sum, makes	109 deg. 10 min. Greater Angle.
The greater of the Angles required, Substract 41 deg. 50 min. from the Half-Sum, leaves the lesser Angle at S	25 deg. 30 min. Lesser Angle.

* 1.1

[illustration] geometrical diagram

By the Line of Sines and Numbers.

EXtend the Compasses from the Sum of the Sides 591, to the Difference of the Sides 221; the same Extent upon the Line of Tangents, will reach from the Half-Sum, to the Tangent of the found Angle 41 deg. 50 min.

Or else extend the Compasses from the Difference 221, to the Tangent of the Half-Sum of the unknown Angles; the same Distance will reach from the Half-Sum 67 deg. 20 m. in the same Line, to the Tangent of 41 deg. 50 min. which added to, or substracted from the Half-Sum, as before is shewn, will give the Quantity of either of the two unknown Angles.

Notes

• * 1.1

  ½ Sum	67	20
  Tang.	41	50
  	109	10
  Lesser Angle	25	30