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.

How to make a Line of Chords.

ACcording to the largeness of your Line of Chords you intend to make, draw a right Line, as A B, and upon the Point A (by the II. Prop.) erect the Perpendicu∣lar A C, and upon A (as a Centre) describe the Quadrant B D E C, which you must divide into 90 equal parts or De∣grees. Which that you may readily doe, your Compasses being opened to the distance A B, set one foot in B, and the other will reach to E; also set one foot in C, and the o∣ther will reach to D: so is your Quadrant divided in∣to three equal parts, each part containing 30 degr. This done, divide each of these three parts into three more; so shall you have divided your Quadrant in∣to 9 equal parts, each con∣taining 10 degr. and each of these 9 parts, being di∣vided into halves, will con∣tain 5 degr. and (if you make your Line large enough) you must divide those into 5 equal parts, which you may very well doe, if the Line A B be but two inches long, as all the Schemes and Figures in these Exercises are drawn by a Line of Chords of that length.

[illustration] geometrical diagram

Your Quadrant being thus divided into 90 degr. draw the Line B C, and parallel thereto two other Lines, one pretty close to B C, to contain the small Divisions, and the other at a larger distance, to set the Figures in. Now it is the Line B C which is called the Line of Chords, (possibly for this Reason,

the Arch or Ark B D E C representing Arcus, a Bow, and B C the String or Chord thereof) the divisions whereof are to be transferred from the degrees of the Quadrant B D E C, in this manner.—First, setting one foot of your Compasses in B, extend the other to 80 degr. in the Quadrant, and from the division of 80 degr. in the Quadrant draw the Arch 80, 80, which will cut the Chord-Line in 80; doe so with 70, 60, 50, &c. and the like with every fifth degree, as you see in the Fi∣gure. And if your Line be very large, you may doe so to every single degree, and part of a degree. And by this means have you reduced the degrees of the Quadrant B D E C to the straight Line B C, more commodious to be set upon a Ruler, then the crooked Arch B D E C.

The Uses of the Line of Chords.

THE Ʋses of this Line are principally two. The one is, To protract or lay down upon Paper an Angle of any quantity (that is, of any number of degrees) required.— The other Ʋse is, If an Angle be already protracted or laid down, to finde how many degrees and parts of a degree it containeth. —In both which I would have the Reader very perfect, because very much contained in this Book hath dependence thereupon.

And here it will be necessary that I give you the Definiti∣on of an Angle. Know therefore that an Angle is the Inclination or bowing of two right Lines the one to the o∣ther.—As the two right Lines C A and B A incline the one to the other, and touch or meet each other in the Point A, in which Point, by rea∣son of the inclination of the said Lines, is made the Angle C A B.

[illustration] geometrical diagram

And here note that an Angle is commonly signed by three Letters, the middlemost whereof signifies the angular Point. As in this Figure, when we say the Angle C A B, you are to understand the very Point at A.

I. How to protract (or lay down upon Paper) an Angle containing any number of Degrees and Minutes by the Line of Chords.

DRaw a right Line at pleasure, as A B, and from the Point A let it be required to protract or lay down an Angle containing 40 degrees.—First, open your Compasses alwaies to 60 degr. of your Line of Chords, (which is equal to the Line A C of the Quadrant,) and with this distance, set∣ting one foot of the Compasses upon the Point A, with the other foot de∣scribe the Arch B C.—Secondly, take in your Compasses 40 degr. (which is the quantity of the Angle to be laid down) out of the Line of Chords, from the beginning thereof, and setting one foot in B, the other will reach to C upon the Arch: where∣fore through the Point C draw the Line C A. So shall the Angle at A contain 40 degr. as was required.

[illustration] geometrical diagram

II. An Angle that is already protracted, how to finde the quantity of Degrees it containeth.

SUppose C A B were an Angle already protracted, and it were required to finde the quantity thereof.—First, open your Compasses to 60 degr. of your Line of Chords, and setting one foot in A, (the angular Point) with the other describe the Arch B C.—Secondly, take in your Com∣passes

the distance between B and C, which distance apply to your Line of Chords, (by setting one foot in the beginning thereof) and you shall finde the other to fall upon 40 degr. which is the quantity of the Angle at A.

Thus have you the Ʋses of your Line of Chords in protra∣cting and finding the quantities of Angles. And now it will not be impertinent, if in this place I shew you how Angles may be protracted and laid down, and also their quantities found, by an Instrument which I shall make use of towards the end of this Book, which I call a Protracting Quadrant.