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 VIII. Having two Angles, and a Side opposite to one of them, To find the Side opposite to the other.

IN the Triangle QRS, is given the Angle QSR 25 deg. 30 min. and the Angle QRS 45 deg. 20 min. and the Side QS 305 Feet; And it is required to find the Side QR.

Here note, That in Oblique-Angled Plain Triangles, as well as in Right-Angled, the Sides are in proportion one to the other, as the Sines of the Angles opposite to those Sides: Therefore,

As the Sine of the Angle QRS 45 deg. 20 min.	9, 851997
Is to the Logarithm of the Side QS 305	2, 484299
So is the Sine of the Angle QSR 25 deg. 30 min.	9, 633984
The Sum of the second and third Terms	12, 118283
The first Term substracted	9, 851997
To the Logarithm of the Side QR	2, 266286

The nearest Absolute Number answering to this Logarithm is 185; and so many Feet is the Side QR.

By the Line of Sines and Numbers.

THe Line of Sines and Numbers will resolve the Triangle by the same manner of Work, as in the other before. For if you extend the Compasses from the Sine of 45 deg. 20 min. to 305 Foot, the same Distance will reach from 25 deg. 30 min. to 185 Foot, and so much is the Side QR.

Or, Extend the Compasses from the Sine af 45 deg. 20 min. to 25 deg. 30 min. the same Distance will reach from 305 to 185, the Length of the Side inquired.

[illustration] geometrical diagram

In like manner if the Angle RQS, 109 deg. 10 min. and the Angle QRS 45 deg. 20 min. and the Side QS 305 Foot, had been given, and the Side RS required, the manner of Work had been the same: For,

As the Sine of the Angle QRS 45 deg. 20 min.	9, 851997
Is to the Logarithm of the Side QS 305	2, 484299
So is the Sine of RQS 109 deg. 10 min. (or 70 deg. 50 min.)	9, 975233
The Sum of the second and third Terms	12, 459532
The first Term substracted	9, 851997
To the Logarithm of the Side RS 405	2, 607535

The Absolute Number answering to this Logarithm is 406, and so much is the Side RS.

In this Case, because the Angle RQS is more than 90 Degrees, you must there∣fore take the Complement thereof to 180 deg. so 109 deg. 10 min. being taken from 180 deg. there remains 70 deg. 50 min. whose Sine is the same with 109 deg. 10 min. And so you must work with all Angles above 90 Degrees; and so will the Comple∣ment to 180, as before-directed, effect the same thing.

By the Line of Numbers and Sines.

EXtend the Compasses from the Sine of 45 deg. 20 min. to 305 Feet, the same Di∣stance will reach from 70 deg. 50 min. to 405.

Or, The Compasses extended from the Sine of 45 deg. 20 min. to 70 deg. 50 min. the same Extent will reach from 305, to 406 in the Line of Numbers, which is the Side RS required.