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.

QUEST. II. A Ship at A discovers an Island at C, lying from her directly East, but she sails from A towards B 32 Cent. or 6 ⅖ Leagues directly South; but her Compass coming to some mischance, that use cannot be made of it, she again at B discovers the same Island, and sails upon an unknown Point of the Compass directly upon the Island, and touches upon it, having sailed 8 Leagues.—I demand upon what Point of the Compass the Ship sailed from B to C, and also how far off the Island was from A, where it was first discovered.

DRAW a Line C A, representing a Line of East and West, and upon A erect a Perpendicular A B, and from A to B set off 32 Cent. or 6 ⅖ Leagues, the distance that the Ship sailed from A to B. Then take out of your Scale of equal

[illustration] geometrical diagram

parts 40 Cent. or 8 Leagues, the distance that the Ship sailed from B to the Island; and setting one foot of the Compasses in B, with the other describe an obscure Arch of a Circle m m, crossing the East and West Line in C: so is C the place of the Island.

Now first, to find upon what Point of the Compass the Ship sailed from B to the Island, you must find the quantity of the Angle at B, (either by your Line of Chords, or Pro∣tracting Quadrant,) and you shall find it to contain 33 degr. 45 min. which is three Points from the North Easterly, that is N. E. by N. and upon that Point did the Ship sail from B to the Island at C.—Then, to know how far the Island C was from A, where it was first discovered, Take in your Com∣passes the length of the Line A C, and measure it upon your Scale; so shall you find that to contain 24 Cent. or ⅘ Leagues: and so far distant was the Island from A.

The Point of the Compass that the Ship sailed upon from B to C may be found by the second Case of Right-angled plain Triangles, by this Analogie.

As the Distance which the Ship sailed from B to C is to the Radius;

So is the Distance sailed between A and B to the Co-sine of the Point that the Ship sailed upon from B to C.

The Distance that the Ship was from the Island, when first discovered, may be found by the fifth Case of Right-angled plain Triangles, by the following Analogie.

(1.) As the Distance that the Ship sailed from B to C is to the Radius;

So is the Distance that the Ship sailed from A to B to the bear∣ing of the Island from B.

(2.) As the Radius is to the Distance that the Ship sailed from C to B;

So is the Sine of the Rhumb that the Ship sailed upon from B to C to the Distance of the Island from A.