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. V. How to allow for known Currents, in Estimating the Ship's Course and Distance.

THis Subject hath been largely handled by Mr. Norwood, at the end of his Sea-mans Practice; and by Mr. Philips, in his Advancement of Navigation, page 54, to 64. As also how to find them out by comparing the Reckoning homeward with the Reckoning outward, which was kept betwixt two Places: There∣fore

I shall be brief, and demonstrate by Scale and Compass, what they have done by Tables.

First, This is easie to be understood, If you sail against a Current, if it be swifter than the Ship's way, you fall a Stern; but if it be slower, you get on head so much as is the Difference between the Way of the Ship, and the Race of the Current.

EXAMPLE.

* 1.1If a Ship sail 8 Miles South in an Hour, by Log or Estimation, against a Current that sets North 3 Miles in an Hour, that substracted from 8, leaves 5 Mile an Hour the Ship goes a head South: But if the Ship's way were 3 Mile an Hour South, against a Current that sets 8 Mile an Hour North, the Ship would fall 5 Miles an Hour a Stern.

* 1.2Admit a Ship runs East 4 Miles an Hour, and the Current runs also 3 Miles an Hour, What is the true Motion of the Ship? Answer, 7 Miles an Hour a Head.

Admit a Ship cross a Current that sets North East-by-North 4 Miles an Hour; the Ship sails in a Watch, or 4 Hours, 9 Leagues East-by North, and in two Watches more she sails 13 Leagues E. N. E. by the Compass.

Now it is required what Course and Distance the Ship hath made good from the first place of setting out from A.

[illustration] geometrical diagram

First draw the Right Line AL, then with the Chord of 60 Deg. describe the Quadrant on it; to be sure take 90 deg. off the Line of Chords, and lay it from N to O; then draw the North Line AP; then set off the Ship's first Course one Point from the East from N to G, and draw the Line AG, and from A to B lay off the first Distance 9 Leagues: Then prick off the Course of the Cur∣rent, being 5 Points from N to F, and draw the Line AF, being the Course N. E. b. N. of the Current. And because the Cur∣rent in 4 Hours sets 5 ⅓ Leagues forward in its own Race, there∣fore draw the Line BC, parallel to AF, that is, take the nearest Distance from B to AF, and sweep a small Arch, and from B to the upper Edge of the Arch, draw the Line BC thereon, put from B to C 5 ⅓ the Currents Motion, and draw the Line AC, which shews the Course the Ship hath made good the first Watch.

Now for the second Course, draw CH parallel to the Line AL, and with the Ra∣dius or Chord of 60 deg. upon C as a Center, draw the Arch HZ, whereon prick 22 deg. 30 min. or 2 Points for E. N. E. for the Ship's second Course from the East; and draw CZ, whereon prick down the Distance sailed 13 Leagues from C to D; then draw DW parallel to AF, as you did BC; then because the Current sets 10 ⅔ in two Watches, therefore prick down 10 ⅔ Leagues from D to W, and draw the Line AW; which being measured upon the same Scale of an Inch divided into 10 parts, shews the Ship's direct Distance is 35 6/10 Leagues; whereas if there had been no Cur∣rent, the direct Distance had been AR 22 2/10 Leagues: Then measure the Arch NE, and you will find it 35 deg. which is a little above 3 Points from the East. So the Point the Ship hath made good is North-East-by-East a little Northerly; whereas if there had been no Current, the Course had been NS, that is, East and by North ¾ of a Point Northerly, and had been at R, but now the Ship is at W, therefore di∣stant from it equal to RW 15 6/10 Leagues. The prick'd Lines are the Courses and Arches without a Current.

This is a good way to work these Questions: If you have no Compasses, draw on a Slat or Quadrant to work Traverses by; if you have, that way is the soonest done by them after the same manner. Some will expect, that knows me, some other sort of Questions, (besides these most useful beforegoing:) For them, and their leisure-time, I have inserted these six Questions following.

QUESTION I. A Ship Sails 40 Leagues more than her Difference of Latitude, and is departed from the Meridian 80 Leagues, I demand her Diffe∣rence of Latitude.

[illustration] geometrical diagram

MAke a Right-Angled Trian∣gle, so that the Base FG be equal to her Difference 40 Leagues, and the Perpendicular GH equal to her Departure 80 Leagues: Then continue the Base FG, and find the Center point E unto H and F, so it will be E, and G 60 Leagues for the Diffe∣rence of Latitude sought.

Arithmetically.

Square GH 80, you have 6400,* 1.3 which divide by GF 40, the Quotient is 160; from whence substract GF 40, there remains 120; the half is 60, for the Difference of Latitude sought.

QUEST. II. A Ship Sails 20 Leagues more than her Difference of Latitude, and but 10 Leagues more than her Departure from the Meridian, I demand her Distance Sailed.

[illustration] geometrical diagram

[illustration] geometrical diagram

IN the Triangle ABC, you have EB 20 Leagues more than the Difference of Latitude AC; and AD, 10 Leagues more than the De∣parture from the Meridian BC.

First, with the double of ei∣ther Number, which here I take, the double of EB 20, w^ch is 40 Leagu. and lay from F unto G; then I take the other Number AD 10, and add it thereunto, as GH. Now on the midst of FH, as at K, making it the Cen∣ter, I describe the Semicircle HIF: Then on G erect the Perpendicular which cuts the Yrch in I; then measuring

GI, it will be equal to DE 20 Leagues, which added to the two former Numbers 20 and 10, you have in all 50 Leagues for the Distance sailed, required.* 1.4

Anabically: 2 AD:X:EB = DEq 400, whose V q is 20, 〈 math 〉〈 math 〉 (20 the Root.

QUEST. III. Two Ships Sail from one Port; The first Ship Sails directly South, the second Ship Sails W. S. W. more than the first by 35 Leagues, and then were asunder 76 Leagues; The Question is, How ma∣ny Leagues each Ship Sailed.

FIrst draw the Meridian-line AB, and from A draw a W. S. W. Course as AC continued, and from C lay down the 35 Leagues unto D. Now draw the Chord-line of 6 Points, as BC; then take 76 Leagues, and lay it from D to cut the Chord-line in E. Lastly, from E you must draw a Parallel Meridian, which will cut the Rhomb-line in F; so measuring EF, you shall have 45 68/100 Leagues, that the first Ship sailed directly South: So the second Ship sailed 35 Leagues more, therefore must Sail in all 80 68/100 Leagues, which is the Distance required.

[illustration] geometrical diagram

By the Artificial Tables of Sines and Numbers.

As the Side ED	76 Leagues co: ar.	811919
To the Sine of the Angle ECD 56 deg. 15 min	991985
So is the Side CD	35 Leagues	154407
To the Sine of the Angle CED 22 deg. 31 min.	958311

which substract from 56 deg: 15 min. you have the Angle at D 33 deg. 44 min.

Then,

As the Sine of the Angle at F 67 deg. 30 min. co. ar.	003438
Is to his opposite Side	ED 76 Leagues add	188081
So the Sine of the Angle at D 33 deg. 45 min. add	974455
To his opposite Side	FE 45 ••8/100 Leagues	165974

So the South Ship Sailed 45 68/100 Leagues; and the other W. S. W. 80 68/100 Leagues.

QUEST. IV. Two Ships Sailed from one Port: The first Sails S. S. W. a certain Distance; then altering her Course, she Sails due West 92 Leagues: The second Ship Sailing 120 Leagues, meets with the first Ship. I demand the second Ship's Course and Rhomb, and how many Leagues the first Ship Sailed S. S. W.

DRaw the first Ship's Rhomb from A unto E, being S. S. W. then lay her Distance sailed West 92 Leagues from A unto C, and from C draw a S. S. W. Course, as CD continued: Next take 120 Leagues, and lay it from A, so that it shall cut the continued Line in D: so drawing AD, you shall have the second Ship's Rhomb, near W. S. W. Lastly, measuring CD equal to AB, you shall find it to be 49 ½ Leagues that the first Ship sailed S. S. W.

[illustration] geometrical diagram

For the Course,

As the Side AD	120 Leagues,	co: ar.	792082
Is to the Sine of the Angle at B 67 deg. 30 min.	996562
So is the Sine of the Side BD 92 Leagues	196379
To the Sine of the Angle BAD 45 deg. 6 min.	985023

Unto which add the Angle FAB 22 deg. 30 min. you have the second Ship's Rhomb 67 deg. 36 min. being near W. S. W. whose Complement is the Angle ADB 22 deg. 24 min.

For the Distance,

As the Angle	BAD 45 deg. 6 min. co. ar.	014976
Is to the Side	BD 92 Leagues	196379
So is the Angle	ABD 22 deg. 24 min.	958101
To the Side AB 49 5/10 Leagues required	169456

QUEST. V. Two Ships Sail from one Port 7 Points asunder: The one Sails in the S. W. Quadrant, and departs from the Meridian 57 Leagues; and the other Sailed in the S. E. Quadrant, and was departed from the Meridian but 25 Leagues, and then are both fallen into one Latitude; I demand the Rhomb or Courses of each Ship.

FIrst draw an East and West Line continued; and making choice of a Point at D, upon D erect a Perpendicular, which will be a Meridian-line, as DA continued. Now from D lay down the West Ship's Departure DB 57 Leagues; also the East Ships Departure 25 Leagues DC: so their whole Distance will be CB 82 Leagues. Now upon the Point at B, or else as here at C, draw an Angle of the Complement of 7 Points, or one Point, which is W. b. N. as CF the prickt Line; but if their Courses had been more than 8 Points, then you must lay it to the Southward of the West Line.

[illustration] geometrical diagram

Now from the midst between B and C, at E, draw another Meridian-line, until it cut the former Rhomb-line CF in the Point G: So taking the Distance from the Point G unto C, lay the same from G until it cut the Meridian-line in the Point A, which is the Place and Port you Sailed from.* 1.5 Lastly, From A you shall draw their Rhombs or Courses, as AB, which is 4 ½ from H to N from the South, Westwards; and the Eastward Ships Course is AC 2 ½ Point from P to N, from the South, Eastward.

As the Sum of their Departures CB 82 Leagues	191381
To the Difference of their Departure SB 52 Leagues	150515
So is the Sine of the Sum of their Courses CAB 78 deg. 45 min.	999080
To the Sine of the Difference of their Courses, Sum	1149595
SAB 22 deg. 30 min. the Sum	958214

Now 22 deg. 30 min. added to 78 deg. 45 min. the half is 50 deg. 37 min. ½; that is 4 Points ½ or S. W. ½ W. for the one Ships Course Sailed from A to B: and 22 deg. 30 min. substracted from 78 deg. 45 min. the half is 28 deg. 07 min. ½; that is, 2 Points and ½ S. S. E. ½ a Point Easterly, for the other Ship's Course.

QUEST. VI. From the Port at A I Sail S. S. W. unto B, and from B I Sail N. W. b. W. unto C, and from C I Sailed unto my first Port at A, E. b. N. Now having Sailed in all 120 Leagues, I would know how many Leagues I have Sailed upon each Point.

FIrst draw AB a S. S. VV. Course, at any convenient distance; then from B draw a N. VV b. VV. Course, and from A draw the opposite Course of E. b. N. which is VVest by South, which will cut BC in C; so continue the Sides of the Tri∣angle AB unto E, and AC unto F. Then lay BC from B unto D, and AC from D unto E. Then take 120 Leagues, and lay the same from A unto F: Next draw the Line EF, and from D and B draw Parallels thereunto, which will cut AF in G and H. Lastly, measuring AH, you shall have 33 ⅔ Leagues that you have Sailed S. S. VV. And measuring HG, you shall have 39 Leagues 6/10 parts that you have Sailed N. VV. b. VV. Also measuring GF, you shall have 46 ¾ Leagues near, that you have Sailed E. b. N. which makes in all near 120 Leagues.

[illustration] geometrical diagram

Arithmetically, By the Table of Natural Sines in the Sea-mans Kalendar.

First, Add up all the Sines of the Angles together, Which is

deg.	min.
45	00	7071
56	15	8314
78	45	9790
		25175

Then by the Rule of Three,

As 25194, to 120 Leagues: So	45	00	To the Distan∣ces sailed	S. S. W. 33 68/106 AB.
56	15	N. W. b. W. 39 60/100 Leag.
78	45	E. b. N. 46 72/100 CA.

I might have added several other Questions of this nature, but I hold these sufficient; for those that understand how these are done, may do any of the like nature: But the way of demonstrating and laying of them down, as you see in the Figures, I ne∣ver saw before of any other Mans Work. Therefore now we will come to the true way of Sailing, and Use of the true Sea-Chart.

Notes

• * 1.1

  	mil.
  South	8
  Current	3
  Goes a head	58
• * 1.2

  	mil.
  Current	8
  Ships Way	3
  Falls a stern	5
• * 1.3

  〈 math 〉〈 math 〉
• * 1.4

  20

  10

  200

  200

  The Square 400
• * 1.5

  1 Course S. W. ½ W. 2 Course S. S. E. ½ E.