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.

PROBL. VII. The Latitude of two Places being given, together with the Difference of Longitude, To find,

• First, The nearest Distance in the Arch of a Great Circle.
• Secondly, The Direct Position from the first Place to the se∣cond. And,
• Thirdly, From the second Place to the first. And,
• Fourthly, The Circles greatest Obliquity that passeth over those two Places.

ADmit L be the Latitude of Lundy 51 deg. 22 min. and Longitude 25 deg. 52 min. and B is the Latitude of Barbadoes 13 deg. 10 min. and Longitude 332 deg. 57 m. and Difference of Longitude 52 deg. 55 min.

[illustration] geometrical diagram

Lay down first the Latitude 51 deg. 22 min. from Q to L; secondly, the Diffe∣rence of Longitude QF 52 deg. 55 min. and draw the Meridian-circle P through F to S; then lay down the Latitude of Barbadoes from Q to 13, and take of the half

Tangent Line QF 57 deg. 55 min. and lay from C to K, and draw the prick'd Line, and he will cut the Meridian PFS in B, the Latitude of Barbadoes, 13 deg. 10 min. through B draw the Circle LBN, so the Angle of Position is BLQ, whose measure is RO 67 deg. 51 min. that is, W. S. W. 21 min. Westerly; which taken off the Line of half Tangents of your Scale, and laid from C to Pole, and draw the Line from Pole through B, and it will cut the Limb in G: Therefore measure GL on the Line of Chords, you have 57 deg. for the Distance, or 1140 Leagues, or 3420 Miles.

To find the Distance in Questions of this Nature,

1 As the Radius, Is to the Co-sine of the Diff. of Longit. 52 d. 55 m.	978030
So is the Co-tangent of the greater Latitude 51 deg. 22 min.	990267
To the Tangent of the first Arch 25 deg. 44 min.	968297

Take 25 deg. 44 min. from 76 deg. 50 min. the Complement of 13 deg. 10 min. the less Latitude, and the Remain is 51 deg. 6 min. the second Arch.

2 As the Sine of the first Arch 25 d. 44 m.	995464	To find the Distance.
Is to the Co-sine of the second Arch 51 d. 06 m.	979793
So is the Sine of the greater Latitude 51 d. 22 m.	989273
	1969066
To the Co-sine of the Distance 57 d. 00 m.	973602

* 1.1

which is the nearest Distance in the Arch of a Great Circle, by 17 Leagues less than Mercator's Chart by the Rhomb, and less by 166 Leagues than by the Rhomb on the Plain Chart; which confirms this to be the nearest of all ways of Sailing betwixt any two Places.

To find the Angle of Position,

3 As the Sine of the Distance 57 deg. 00 min.	992359
Is to the Sine of the Difference of Longitude 52 deg. 55 min.	990187
So is the Co-sine of the Latitude 13 deg. 10 min.	998843
Add the two last: Substract the first Numbers	1989030
There remains the Sine of the Angle of Direct Position	996681

Which is 67 deg. 51 min. from the South part of the Meridian Westward, as name∣ly, W. S. W, 20 m. Westerly.

Now to know the Distance and Angle of Position, you must put Barbadoes at B, on the West side of the Circle 13 deg. from AE, and draw the Parallel of Latitude 51 deg. 22 min. and lay off the Difference of Longitude from AE to F, and draw the Meridian-circle PFS, and it will cut the Parallel of Latitude in L; therefore from B draw the Circle from L to K: And if you follow your former Directions, BD will be the Measure of BL the Distance found, as before, 57 deg. 00 min. and LBP the Angle of Position, whose Measure is RG 36 deg. 26 min. and the Great Circles greatest Obliquity is CO 54 deg. 40 min. For,

4 As the Sine of the second Arch 51 deg. 6 min.	989111
Is to the Sine of the first Arch 25 deg. 44 min.	963767
So is the Tangent of Difference of Longitude 52 deg. 55 min.	1012157
	1975924
To the Tangent of the Direct Position 36 deg. 26 min.	986813

From B towards L, which is 3 Points, 2 deg. 41 min. from the Meridian, namely, N. E. b. N. 2 deg. 41 min. Easterly, you must sail first from B towards L; but al∣ter your Course, still increasing toward the Eastward, as shall be shewed.

[illustration] geometrical diagram

For the Obliquity, to find that,

5 As Radius, To Co-sine of less Latitude 13 deg. 10 min.	998843
So is the Sine of the Angle of Position 36 deg. 26 min.	977370
To the Co-sine of the greatest Obliquity 54 deg. 40 min.	976213

These are the Scituations of all Places upon the Terrestrial Globe; so that there can∣not be any two Questions, but, in respect of each other, they will be found in one of these four kinds; except they fall in one Meridian, or on the Aequator: and these Directions you have in the Tenth Chapter: Therefore if you will seriously observe these short Directions already given, and as follows, you shall never have your Expe∣ctation deceived.

Notes

• * 1.1

  1306 P.

  1157 M.

  1140 G.

  Leag. 17

  166