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.

CHAP. IX. The Projection of the Meridian-Line by Geometry, and how to make a Scale of Leagues for to measure Distances in any Latitude.

THE Projection is the ground-work of Mr. Wright's Table of Latitudes, in his Book called, The Correction of Errors in Navigation, where he sheweth how to make it, and hath also made a Table by the continual addition of the Secants of every Minute, which shews how much you are to lengthen the De∣grees of Latitude in your Chart, that so there may be a true proportion between the

Degrees of Longitude and Latitude in all Places. Which Table I have abridged, and made it more plain and easie, by reducing it into Leagues and Tenth parts, as hath been shewed before. We will here shew you how to do the same by Geometry, and also how to make a Meridian-line answerable to any Line of Longitudes, and a Scale of 100 Leagues to measure any Distance in any Latitude.

First, Make the Quadrant ABC, of what largeness you please, and divide the Limb thereof into 90 Degrees, and number them from B towards C; Then divide the Side of the Quadrant into 5 Equal parts, which are five Degrees of the Aequi∣noctial. Then divide the first Degree from the Center, as AD, into 6 Equal parts, and through them draw Parallel-lines to AC: You may divide each of the other four Degrees from D to B into 20 Equal parts, which are 20 Leagues, which makes a De∣gree of Longitude at the Aequator; and so you may number them as you see, from 10 to 100: So the whole Line AB will be your Radius, and the length of 110 Leagues, or five Degrees and a half of Longitude of your Chart. And because the Degrees of Longitude are to be of one length in all Latitudes, therefore the Degrees of Lati∣tudes must encrease, as the Secants of the Latitudes increase. Therefore if you would know how long one Degree of Latitude must be in the Latitude of 50 Degrees, lay a Ruler on 50 Degrees, and on the Center A, and draw the Line AH. Now the Ra∣dius being AD, the length of one Degree of the Aequator, this Line A h, or h K,

[illustration] geometrical diagram

being both of one length, is the Secant of 50 Degrees to that Radius, and must be the length of one Degree of Latitude in a Chart from 50 Degrees to 51 Degrees, as you may presently try by the former Chart; and so the Line AC which is the Secant of 20 Degrees, is the length of one Degree of the Meridian-line in the Latitude of 20 Degrees; and so for any other Latitude. The six Lines divided in the first Degree AD, are 10 Minutes apiece; and so you have the Secant of every 10 Minutes of Latitude, and their length in every Latitude, for a particular Chart, and for a gene∣ral Chart, which hath in it North and South Latitude.

You may divide the Quadrant's Side AB into 10 Equal parts, and subdivide them into 10 more; so will D ♓ be 10 Degrees of the Aequator, and e ♑ will be

the length and Secant of 10 Degrees in the Latitude of 20 Degrees, and L ♈ the length and Secant of 10 Degrees in the Latitude of 60 Degrees, which is twice the length of one Degree of the Aequator: So that you may presently try the truth of this Proje∣ction, how it agrees with the Globe: Whereas one Degree of Latitude in the Globe, is equal to two Degrees of Longitude in that Latitude of 60 Degrees; so here AL the Secant of 60 Degrees, is twice the length of AD the measure of one Degree of Longitude in the Blank Chart; and L ♈ is twice the length of 10 Degrees D ♓: So every Degree is two of the Aequator in the Latitude of 60 Degrees of a general Chart; and by the Globe, in the Latitude of 75 deg. 30 min. one Degree of Latitude is equal to four Degrees of Longitude. So in the Quadrant, AR is four times the length of A D: and so the Proportion will hold in any other Latitude.

[illustration] geometrical diagram

A Scale of Leagues from the Latitude of 25 Degrees, to the Latitude of 57 Degrees 00 Minutes.

How to make the Scale of Leagues.

THE Quadrant being drawn, as before-directed, take 110 Leagues and lay from A to B, and draw the Line MNB at Right Angles thereunto: and if it be for a particular Chart, as that before-going, draw Lines from the Center through every particular Latitude; as you see in the Quadrant I have done, to make a Scale for the blank Chart before-going, from the Latitude of 49 deg. 30 min. to 55 deg. 30 min. So that if you would know the length of 110 Leagues in the Latitude of 50 Degrees, lay a Ruler upon 50 deg. in the Arch of the Quadrant and the Center, and draw the Line A ♌, and that is the length of 110 Leagues in that Latitude. So that if you draw Parallel-lines to MN, through every 10 Leagues in the Side AB, you will have the length of every 10 Leagues in every Latitude, as you may plainly see in the Quadrant: and so you may do for every League, as you see the little Checkers be∣twixt the Latitude of 20 deg. and 30 deg. for 10 Leagues between Latitude of 50 deg. and 56.

Suppose you would know the length of 40 Leagues in the Latitude of 50 deg. Extend the Compasses from A to K, and that Distance is 40 Leagues in that Latitude: And in like manner work by the rest in any other Latitude.

If you would make this into a Scale, as the Figure YM in N; First in the Qua∣drant extend your Compasses from the Center A, to the Intersections of the Lines drawn through every Degree MNB, and lay them down upon the Side of the extreme Latitude of your Chart, as A, O, P, Q, Y, M, with the small Arches, as you see I have done from M to NY, and that is the length of the Meridian-line of

your Scale or Degree of Latitude MY; therefore draw the Parallel-lines YN and M m for the extreme Parallels of your Scale: Then extend your Compasses from Y in the Quadrant, to each of the Intersections of these small Arches that are drawn from the Intersections on the Tangent-line MN; and from 25 deg. uppermost, lay that Extent downward for the Parallel of 55 deg. of Latitude, as the Line above the lowermost; and so lay down all your Latitudes by these small Arches, in like man∣ner; and so neatly divide the Side of your Scale MY of Deg. of Latitude: Then draw Parallel-lines to all these Degrees, as you see: Then extend the Compasses from the Center of the Quadrant to M, for the length of the lowermost Line of your Scale M m for 110 Leagues. Then extend the Compasses from the Center of the Quadrant to N, which is the length of 110 Leagues in the Latitude of 25 deg. and it is the Di∣stance of the uppermost Line YN of your Scale; and draw N m the outside of your Scale 120 Leagues: So take every 10 Leagues from the Center A, in the Line AM, in Latitude 56 Degrees, and divide the lowermost Line of the Scale; and the like do in the Latitude of 25, for to divide the uppermost Line of the Scale; and draw Lines through each of them, which will divide all the rest of the Parallel-lines in each Latitude into 10 Leagues apiece, and number them as you see I have done; and divide the first 10 Leagues by the Meridian-line of the Scale, into 10 Equal parts below and above, and draw Lines through each of the Divisions: So have you neatly divided your Scale, and every Degree of Latitude thereof, into Leagues, to 100 and 10 odd Leagues; which will measure any Distance in a Chart, made according to the Degrees of Latitude and Longitude in the foregoing Chart.

For to know the Rhomb between any two Places, shall be shewed in the Use of the general Sea-Chart following, by a Protracting Quadrant, and also how to find the Place of any Ship in Mercator's Chart, and to lay down any Traverses.