Nine geometricall exercises, for young sea-men and others that are studious in mathematicall practices: containing IX particular treatises, whose contents follow in the next pages. All which exercises are geometrically performed, by a line of chords and equal parts, by waies not usually known or practised. Unto which the analogies or proportions are added, whereby they may be applied to the chiliads of logarithms, and canons of artificiall sines and tangents. By William Leybourn, philomath.

CASE XI. The three Sides A B 30 degr. B E 18 degr. 47 min. and A E 42 d. 51 min. being given, to finde the Angle at E.

First, find the Sum and the Difference of the Sides B E and A E.

	degr.	m.
Their Sum is	61	38
Their Difference is	24	04

THE Sum and Difference of the two Sides being taken, out of your Line of Chords take 24 degr. 4 min. the Difference of them, and set them from A to P. Also from your Line of Chords take 61 degr. 38 min. the Sum of the two Sides, and set them from A to X. Again, take 30 degr.

the quantity of the third Side A B, and set them from A to Q; and draw the Lines P S, Q T, and X Y, all three per∣pendicular to A C.

[illustration] geometrical diagram

This done, take the distance between Y and S, and setting one foot of the Compasses in C, with the other describe the Arch Z, and draw the Line A AE so that it may onely touch the Arch Z. Then take in your Compasses the distance be∣tween S and T, and setting one foot thereof upon the Line A C, move it gently along the same, till the other foot, being turned about, do onely touch the Line A AE; and where the Compass-point resteth, which you will find it to doe at the Point Y, upon this Point Y erect the Perpendicular Y R. So shall the distance A R, being measured upon your Line of Chords, give 38 degr. 15 min. the quantity of the Angle at E, which was required to be found.

To work this by the Canon.

The Analogie or Proportion is,

As the Rectangle contained under the Sines of the Sides is to the Square of the Radius,

So is the Rectangle contained under the Sines of the half Sum of the three Sides, and the Difference between this half Sum and the Base, to the Square of the Co-sine of half the An∣gle required.