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 Proposition. Questions of Sailing by the Plain, Ordinary Sea-Chart.

I. Sailing 57 Leagues upon the first Rhomb, How much shall I alter my Parallel of Latitude?

THe Angle that any Point makes with the Meridian, we call the Rhomb; but the Angle that it makes with any Parallel, is called the Complement of the Rhomb. Unto every Point of the Compass there answers 11 deg. 15 min. therefore the fifth Rhomb from the Meridian makes Angles therewith of 56 deg. 15 min. namely, S. W. b. W. S. E. b. E. N. W. b. W. N. E. b. E. whose Complement 33 deg. 45 min. is the Angle of the same Rhomb with every Parallel.

Now admit I sail from A to D, S. W. b. W. 57 Leagues, I demand the difference of Latitude EA.

[illustration] geometrical diagram

First, by the following Traverse-Table, at the Head of the Table, over every Co∣lumn, is put the Figure of Halfs, Quarters, and whole Rhombs; and in one of the Columns over head is N. S. and at the foot E. W. and so is numbred at the Head, from the left hand to the right. N. S. stands for Northing. Then the Rhombs are reckon∣ed at the bottom, from the right hand back again; The Margent of the Tables shews the Leagues sailed; and over E. W. or under E. W. shews how much you have sailed East or West from the Meridi∣an. N. S. shews North or South from the Latitude. As in this Example, The di∣stance sailed is 57 Leagues on the fifth Rhomb; therefore under

3 Rhomb.
N S	W E
47 39	31 67
E W	N S
5 Rhomb.

Distance Sailed, in the Side, I enter with 57 Leagues, and in the Common Angle or Line of Meeting, I find 31. 67/100 over N. S. in the Foot; and in the next Column, over E. W. is 47. 39, as you see in the Table in the Side: So that the Difference of Lati∣tude is 31 Leagues and 67/100 Parts of a League. And if it were required to find the Departure, you see it to be 47 Leagues and 39/100 Parts. This is very plain and easie, you need no farther Precept.

By the Traverse-Scale.

EXtend the Compasses in the Line of Numbers from 100 to 57, the same Distance will reach from 5 Points to 31, and about 7/10 in the Line of Numbers.

By the Artificial Sines and Numbers on the Ruler.

EXtend the Compasses from 100 in the Line of Numbers, to 57, as before; the same Distance will reach from the Sine-Complement of the Rhomb, to the Diffe∣rence of Latitude, which is the same way as by the Traverse-Scale.

By the Tables of Artificial Sines and Numbers, by the Fourth Case of Plain Triangles.

As the Radius, which is the Sine of 90 deg. or Angle at E	1000000
Is to the Distance run 57 Leagues AD	175587
So is the Sine Complement of the Rhomb at D 33 deg. 45 min.	974473
To the Difference of Latitude required AE 31 Leag. 7/••••	150060

In like manner you may find the Difference of Latitude for any Distance run upon any Point of the Compass: But remember to add the second and third Numbers together, and from it to substract the first or uppermost.

II. Sailing 57 Leagues upon the first Rhomb, How far am I de∣parted from the Meridian of the Place from whence I came?

By the Traverse-Table.

THis Question was answered in the last Example, and found over E. W. to be 47 Leagues and 39/100, as you may see in the small Table in the foregoing Side. In the like manner you may find the Difference of Latitude and departure from the Me∣ridian, for any Distance run upon any Point of the Compass, which is the Use of that Table.

By the Traverse-Scale.

EXtend the Compasses from 100 in the Line of Numbers, to the Distance run 57 Leagues; so is the Sine of the Rhomb; that is, put one Point of the Compass on 5 Points, in the Line of East and West of the Scale, and the other will reach to the Departure from the Meridian 47 Leagues 39/100 Parts.

By the Tables of Sines and Numbers, by the Fourth Case of Plain Triangles.

As the Radius or Sine of 90. deg. at E	1000000
Is to the Distance run 57 Leagues AD	175587
So is the Sine of the Rhomb 56 deg. 15 min. A	991084
To the Departure from the Meridian to 47 39/100 ED	166671

By the Artificial Lines on the Ruler.

EXtend the Compasses from 90 deg. to 57; the same Distance will reach from 56 deg. 15 min. to 47 39/100 Leagues.

Or, Extend the Compasses from 90, to 56 deg. 15 min. the same Distance will reach from 57 Leagues, to 47 39/100, as before.

III. Sailing upon the fifth Rhomb, until I alter my Latitude 1 deg. 35 min. I demand how far I have Sailed?

AS sailing from A to C, S. W. b. W. till the Difference of Latitude be 31 Leagues 67/100, I demand the Distance run AC.

[illustration] geometrical diagram

First, By the Traverse-Table, Look in the Foot of the Table for the fifth Rhomb, and over N. S. in that Column, look for 31 Leagues 67/100, and in the Common Angle of Meeting, to the left hand, under Distance Sailed, you will find Distance Sailed 57 Leagues AC required.

By the Line of Sines and Numbers.

EXtend the Compasses from the Complement-Sine 33 deg. 45, to 31 67/100 the Diffe∣rence of Latitude; the same Extent will reach from 90 deg. to 57 Leagues.

Or, Extend the Compasses from 33 deg. 45 min. to 90; the same Distance will reach from 31 67/100 Leagues, to 57 Leagues, the Distance AC, as before.

Say by the second Case in Plain Triangles,

As the Sine-Complement of the Rhomb, 33 deg. 45	9,744739
Is to the Difference of Latitude 31 67/100 Leagues	3500648
So is the Sine of 90 deg. Radius	10000000
To the Distance run AC 57 67/100 Leagues	3755909

IV. Sailing upon the fifth Rhomb, until I have altered my Latitude 31 67/100, or 1 deg. 35 min. How much am I departed from my first Meridian?

AS sailing from A to C, S. W. b. W. till the Difference of Latitude AB be 31 67/100 Leagues, I require BC my departure from my Meridian.

By the Traverse-Table.

AS in the last Case, find 31 67/100 Leagues over the fifth Rhomb, in the Foot, and in the next Column to the left hand, over E. W. is 47 39/100 Leagues, the Departure required.

By the Line of Sines and Numbers.

EXtend the Compasses from the Complement-Sine of the Rhomb, to 33 deg. 45, to 31 67/100 Leagues; the same Distance will reach from 56 deg. 15 min. the Sine of the Rhomb, to 47 39/100 Leagues, the Departure from the Meridian.

By the fourth Case of Plain Triangles.

As the Sine of 90 deg.	10000000
To the Difference of Latitude AB 31 67/100	2501059
So is the Tangent of the Rhomb 56 deg. 15	10175107
To the Departure from the Meridian 47 39/100 Leagues	2676166

In the like manner, by the Departure from the Meridian, you may find the Diffe∣rence of Latitude.

V. Sailing upon some Rhomb between the South and the West 57 Leagues, and finding I have altered my Latitude 1 deg. 35 m. I demand upon what Point I have sailed.

SUppose I had sailed from A to C (being a Rhomb between the West and South) 57 Leagues, and then find the Difference of Latitude 31 67/100 Leagues, I demand the Angle BAC.

By the Traverse-Table.

NUmber 57 Leagues in the Column of Distance Sailed, and in that Line or Com∣mon Angle of Meeting, you must find the Difference of Latitude 31 67/100 Leagues, at the Foot of the Table in the fifth Rhomb, which was required.

By the Line of Sines and Numbers on the Scale.

EXtend the Compasses from the Distance run 57 Leagues, to the Sine of 90; the same Distance will reach from the Difference of Latitude, to the Sine-Comple∣ment of the Rhomb 33 deg. 45 min.

By the fifth Case of Plain Triangles.

OR, Open the Compasses from 57 Leagues the Distance, to 31 67/100 the Difference of Latitude; the same Distance will reach from the Sine of 90, to the Sine of 33 deg. 45 min. the Sine-Compl. Rhomb.

As the Distance on the Rhomb AC 57 Leagues	2755874
Is to the Difference of Latitude 31 7/10 Leagues AB	2501059
So is the Sine of 90 deg. B	10000000
To the Compl. Sine of the Rhomb at C 33 d. 45 m. the Sum	12501059
The first Number substract	2755874
The Sine of the Angle	9745185

The Sine-Complement of the Rhomb is C 33 deg. 45, substracted from 90 degrees, there remains the Angle of the Rhomb at A 56 deg. 15 min. which is five Points, namely, S. W. b. W. We neglect some part of a Minute, which is not to be regarded.

VI. Sailing upon some Rhomb between the South and the West 57 Leagues, and finding I have altered my Latitude 1 deg. 35 min. I demand my Departure from my first Meridian.

By the Traverse-Table.

NUmber 57 Leagues in the Column of Distance Sailed, and in that Line or Angle of Meeting find 31 67/100 Leagues, and in the Column to the left hand you will have 47 39/100 the Departure from the Meridian.

By the Sixth Case of Plain Triangles.

Distance run AC 57 Leagues	Sum 88 Leagues	2947923
Diff. of Lat. AB 31 67/100 Leagues	Remain 26 7/10 Leagues	2424881
		5372804
Departure from the Meridian BC 47 ⅔ Leagues	2681402

This is thus done. To the Distance run, add the Difference of Latitude, and also substract it from the same, noting the Sum and Remainder; then add together the Lo∣garithm of this Sum and Remain, and half that is the Logarithm of the Distance from the first Meridian.

By the Line of Numbers.

EXtend the Compasses from the Distance 57 Leagues, to 31 ⅔ the Difference of La∣titude; the same Distance will reach from 88 the Sum, to the Departure, as be∣fore, 47 ⅔ Leagues.

Or, Extend the Compasses from 57, to the Sum 88 ⅔ Leagues; the same Distance will reach from 31 ⅔, to 47 ⅔, as before, which is the Departure required.

All things that have been done by the Artificial Sines and Numbers, are done by the Traverse-Scale, or Artificial Points, Halfs, and Quarters, and Tangent-Rhombs, with the Line of Numbers in the Traverse-Table; and this agreeing very well in Leagues and 100 Part of a League.