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.

To divide the Meridian Line of a Sea-Chart.

A Sea-Chart, according to this Projection, may be made either General, or Particular. I call that a General Sea-Chart, whose Line E H, in the following Figure, represents the Aequinoctial, as the Line E H there doth the Parallel of 49 degr. and so I will make the Chart following to contain all Latitudes between 49 degr. and 57 degr. whose Difference of Longitude exceedeth not 8 degr.

Now to project such a Chart, having drawn the Line E F for the Meridian, and crossed it at right Angles with another Line representing the Parallel of 49 d. parallel thereto draw another Line F G, representing the Parallel of 57 degr. and another Meridian G H, parallel to F E. So shall you have made the Parallelogram E F G H.

This done, consider how far distant you would have your Degrees of Longitude upon the Aequinoctial each from other, as suppose (and as in this Chart I have made them to be) half an Inch. Take half an Inch out of a Line of Inches, and run that Distance along the Line E H from E to 1, from 1 to 2, from 2 to 3, &c. And also doe the like upon the Line F G, at the top of the Chart, drawing the Lines 1, 1; 2, 2; 3, 3; &c.

Now for the Dividing of the Meridians E F and H G, re∣pair to the foregoing Figure, taking in your Compasses the Distance that is between Degree and Degree of the Aequino∣ctial,

which in our Example is half an Inch. With this Di∣stance, set one foot of the Compasses in the Point D, and with the other describe the Arch m m; by the very Edge whereof draw the Line A G: so is your Figure prepared to divide the Meridian-line of a Sea-Chart whose Degrees of Longitude are half an Inch distant.

Now in respect that your first Parallel of Latitude E H in your Chart is drawn for 49 degr. your next Parallel must be 50 degr. Wherefore set one foot of your Compasses upon 50 degr. in the Line A B, and with the other take the nearest Distance to the Line A G: that is done by turning the Com∣passes about till the moveable foot do onely touch the Line A G; which when it so doth, that Distance at which your Com∣passes then are, being set upon the Meridian of your Chart, will reach from 49 degr. to 50, which being set upon your Chart, on both sides thereof, from 49 draw the Line 50. 50 will give you the Parallel of 50 d. of Latitude. In like man∣ner for the Parallel of 51 degr. Set one foot of the Compasses in 51 degr. upon the Line A B of the former Figure, and with the other take the least Distance to the Line A G: this Distance set upon the Meridian of your Sea-Chart, on both sides thereof, will reach from 50 to 51; and there draw the Parallel 51, 51.—Likewise for the Parallel of 52 degr. Set one foot of the Compasses in 52 degr. in the Line A B, taking the nearest Distance to the Line A G: that Distance set upon the Meridian of your Sea-Chart, on both sides thereof, will reach from 51 to 52; and there draw the Parallel of 52, 52. Doe thus with all the Degrees, as 53, 54, 55, 56, and 57. So shall the Meridians of your Chart E F and H G be divided into whole Degrees.

For the Sub-divisions of these Degrees, they may be divi∣ded each of them into equal parts, as the Divisions at the top and bottome of the Chart ought to be; but the Degrees of the Meridian, as they grow higher, they ought still to grow greater. But the Difference is so small, that it cannot produce

any considerable Errour, though the Sub-divisions be all made equal between Degree and Degree. You may therefore di∣vide them either into 60 Minutes or English Miles, or into 20 Leagues, or into 100 parts of Degrees, as you shall best like of.

But if you would make a Chart that the Distance between De∣gree and Degree upon the Aequinoctial should be an Inch, or any other Distance less then A D in the foregoing Figure; take that Di∣stance (as suppose an Inch) in your Compasses, and setting one foot in D, with the other describe the Arch o o, and draw the Line A H onely to touch the Arch o o. The least Distance taken from each Degree to this Line A H shall give you the Distance of the Degrees upon the Meridian of a Sea-Chart, whose Distance of Degrees up∣on the Aequinoctial are an Inch from each other.

Your Chart being thus prepared, I will now come to shew you how to resolve severall Problems upon it.