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.

The First Scituation.

FIrst, If any two Places being proposed, the one under the Aequinoctial, the other may be in any other Latitude given, either North or South, and the Difference of Longitude of the Places being known; you may find the three things before spoken of in any Question, by the following Directions. We call the

Angle that the Rhomb leading from one Place to another, makes with the Meridians, the Position of these Places: But in regard the Arch of a Great Circle, drawn between two Places, is the most neer distance from the one Place to the other; therefore the Angles which that Arch makes with the Meridian of those Places, we call the An∣gles of Direct Position: or direct way of two Places one from the other.

* 1.1Now in the following Diagram, let A be the Entrance of the great River of Ama∣zones, under the Aequator; AQ the Arch of the Aequator, or Difference of Longi∣tude; and let C represent the Island of Lundy, lying in Latitude 51 deg. 22 min. Northerly, and CQ the Meridian thereof: and suppose the Difference of Longitude AQ to be 41 deg. 22 min.

[illustration] geometrical diagram

How to do these Questions Geographically.

FIrst, With an Arch of 60 Degrees describe the outward Meridian AEECQIF.

Secondly, Draw AEQ the Aequinoctial Line. Thirdly, Take 51 deg. 22 min. of the Line of Chords, and lay it from Q to C; and draw the Line CD through the Center, and the Line EF at Right Angles thereunto. Fourthly, Take off your Scale of ½ Tangents, counting from 90,-41 deg. 22 min. and lay from Q to A, for A re∣presents the River of Amazones. Now draw the Circle CAD through A, the Hori∣zon thereof is EF; then measure FK, and apply it to the Line of ½ Tangents, as before directed: and you will find the Angle of Direct Position to be 48 deg. 25 min. Take that Number out of your Line of ½ Tangents, from 1 Degree forwards towards 90, and lay it from H to L for the Pole, and draw a Line from L through A, it will cut the Line in I; so measuring CI on the Line of Chords, it will be 61 deg. 57 min. for the Distance, which is 1237 ⅓ Leagues and 3712 Miles.

By the Tables.

THen in this Triangle CAQ, Right-Angled at Q, there is required CA the nearest Distance of the two Places in the Arch of a Great Circle; and the Angle ACQ of Direct Position from the Island Lundy to the Amazones: and the Angle

CQO being the Complement of the Angle of the Direct Position of the Island of Lundy.

For the nearest Distance CA,

As Radius, is to Co-sine of Difference of Longitude 41 d. 22 m.	987534
So is Co-sine of the Latitude or Difference 51 deg. 22 min.	979699
To the Co-sine of the Distance 61 deg. 57 min.	967233

which 61 deg. 57 min. converted into Leagues, is 1237 ⅓ as before, the nearest Di∣stance between those two Places.

For the Angle of Direct Position from the Amazones toward Lundy, NAER,

As the Radius, to the Sine of the Differ. of Longitude 41 d. 22 m.	982011
So is the Co-tangent of Difference of Latitude 51 deg. 22 min.	990267
To the Co-tangent of the Angle of Position 27 d. 50 m. NAER	972278

For the Angle of Position ACQ,* 1.2

As Radius 90, is to Co-sine of Differ. of Lat. 51 d. 22 m. QC	989273
So is Co-tangent of Differ. of Longitude 41 deg. 22 m. AQ	1005522
To the Co-tang. of the Angle of Direct Position 48 d. 25 m. ACQ	994795

The same Proportions will hold by the Artificial Lines on the Scale.

And thus you see, he which will sail the nearest way from the Amazones to the Lizard, shall at first shape his Course 27 deg. 50 min. from the Meridian to the Eastward; that is, N. N. E. almost ½ a Point Easterly. Now if the Wind should serve that you might sail this Course, it is to be understood, that in this kind of sail∣ing he is not to continue this Course long; but to shift it, and incline more and more to the Eastward, as often as occasion requires: which how it may be done, shall be shewed in the following Discourse.

PROBL. II. How to find the Great Circle's greatest Latitude N. or S. or Obliquity.

NOte, Without the knowledge of the true Quantity of the Obliquity or Latitude of that Great Circle which will pass directly over the Places propounded, there can be no compleat Demonstration, much less Arithmetical Calculation of things per∣taining thereunto; therefore it is needfull that the true Quantity of each Great Cir∣cle's Obliquity be diligently found to exact certainty: which to do, in some Cases is very easie, and in some again more difficult. Therefore I will propound Rules for the several Scituations following, except those that are scituate under the Aequator, or under the same Meridian.

If one Place be under the Aequator and hath no Latitude, and the other hath any Quantity of Latitude, and the Difference of Longitude being less than 90 deg. as before 41 deg. 22 min. it is easily found, thus:

The greatest Obliquity in the foregoing Diagram is HRV,

As the Sine of the found Distance 61 deg. 57 min.	994573
Is to the Sine of the Latitude 51 deg. 22 min.	1989273
So is the Radius (added to the last Number) To the Sine of the greatest Obliquity 62 d. 16 m.	994700

So 62 deg. 16 min. is the greatest Obliquity or Latitude from the Aequator, of that Great Circle extended over those two Places.

But if the Difference of Longitude be 90 deg. as AEH, and one of the Places have no Latitude, and the other have any Quantity of Latitude; then it is evident to reason, as in the foregoing Diagram may appear, that the second Place is scituate in the very Point of the greatest Obliquity, which is never above 90 Degrees, as HN; and the other Place is in the very Point of Intersection of the said Great Circle with the Aequator: For note, That every Great Circle that passeth over any two Places propounded, cuts the Aequator in two opposite Points 180 deg. from each other, as the Ecliptick Line doth in the two Points of Aries and Libra; and the greatest Obli∣quity of that Circle is 23 deg. 30 min. the Sun's greatest Declination, and never any more.

Now if one Place have no Latitude 00 deg. 00 min. and the other have any Quantity of Latitude, the Difference of Longitude being more than 90 deg. to find the Obliquity of the Great Circle passing over those Places.

As admit one Place Latitude 00 deg. 00 min. and the other 51 deg. 22 min. Diffe∣rence of Longitude 138 deg. 38 min. Distance betwixt them is near 107 deg. There∣fore take the Distance 107 deg. out of 180, and there remains 73 deg. Then,

As the Sine of the Remainer 73 deg.	998251
Is to the Sine of the Latitude 51 deg. 22 min.	989273
So is Radius	10
To the Sine of the greatest Obliquity 54 deg. 25 min.	991022

So that 54 deg. 25 min. is the greatest Obliquity of the Great Circle extended over these two Places. And so you may work for any Questions of this nature.

Notes

• * 1.1

  To draw a Great Circle from the Ama∣zones over Lun∣dy, put the Dif∣ference of Lon∣gitude on the West Side from AE to G; and draw the prickt Line NS, N for the North, and S for the South Pole: and through G draw the Azimuth Circle NGS, and draw the Parrallel of Latitude CZC; and through Z draw the Arch AEZRVQ, for the Circle which passeth from AE Amazones to Z Lundy.
• * 1.2

  Measure NR on the half Tan∣gents, and the Angle of Posi∣tion, and the Distance is all one as before.