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.

SECT. VII. To finde the Number of Miles answering to one Degree of Longitude, in each several Degree of Latitude.

In Sailing by the Compass, the Course sometimes holds upon a great Circle, some∣time upon a Parallel to the Aequator, but most commonly upon a crooked Line, winding towards one of the Poles, which Lines are well known by the Name of Rhombs.

If the Course hold upon a great Circle, it is either North or South under some Meridian; or East or West under the Aequator.

Deg.	Min.	Miles.
00	00	60
18	12	57
25	15	54
31	48	51
36	52	48
41	25	45
45	34	42
49	28	39
53	08	36
56	38	33
60	00	30
63	01	27
66	25	24
69	30	21
72	32	18
75	31	15
78	28	12
81	23	09
84	15	06
87	08	03

In these Cases every Degree requires an allowance of 60 Miles, or 20 Leagues; every 60 Miles or 20 Leagues will make a Degree difference in the Sailing; therefore as was shewn in the first Di∣agram, and use of the Line of Sines, may be sufficient here, which is the Rule of Proportion.

But if the Course hold East or West, on any of the Parallels to the Aequator,

As the Radius is to 60 Miles, or 20 Leagues, the Measure of one Degree of the Aequator:

So is the Sine-Compl. of the Latitude, to the Measure of Miles or Leagues to one Degree in that Latitude.

But if you would know by the foregoing Quadrant the Miles answering to a Degree in each Parallel of Latitude, it is thus.

Set the Ear E on the Center-Pin, and reckon the Degrees of Latitude from D: to which set the Edge of the Index, and note the Parallel-Line that is at the Degree; carry your Eye on it to the Side CD, and from the Center to that Line you have the Number of Miles answering a Degree in that Latitude.

EXAMPLE.

In the Latitude of 18 degrees 12 min. set the Index 18 gr. 12 min. from D, and the Parallel-Line rising with that Degree, with your Eye or a Pin follow to the Edge, and you will find it to be 57 Miles, the Miles answering one Degree of Longitude and 51 Miles, in the Latitude of 31 gr. 48 min. as in the foregoing Table; and so work for any other Latitude in like manner.

But if the Course hold upon any of the Rhombs between the Parallel of the Aequator and the Meridian, we are to consider besides the Aequator of the World to which we Land, which must be always known.

• First, The difference of Longitude, at least in general.
• 2. The difference of Latitude, and that in particular.
• 3. The Rhomb whereon the Course holds.
• 4. The distance upon the Rhomb, which is the distance we are here to consider, and is always somewhat greater than the like distance upon a great Circle. The first follows in the next Proposition.

I. To find how many Leagues do answer to one Degree of Latitude, in every several Rhomb.

In this Table is the Degrees of every quarter Point, ½, and whole Point in the Quadrant; as the first quarter is 2 gr. 49 m. so the half Rhomb is 5 gr. 37 m. the third is 8 gr. 26 m. and the first Point from the Meridian is 11 gr. 15 m. and so you may plainly see the rest.

Rhombs.	Inclinati∣on to the Meridian.	Number of Leagues.
	Gr. Min.	Leag par.
	2 49	20 2
	5 37	20 10
	8 26	22 22
1	11 15	20 39
	14 4	20 62
	16 52	20 90
	19 41	21 24
2	22 30	21 65
	25 19	22 12
	28 7	22 68
	30 56	23 32
3	33 45	24 05
	36 34	24 90
	39 22	25 87
	42 11	26 99
4	45 00	28 08
	47 49	29 78
	50 37	31 52
	53 26	33 57
5	56 15	36 00
	52 4	38 90
	61 5	42 43
	64 41	36 76
6	67 30	52 26
	70 19	59 37
	73 7	68 90
	75 56	82 31
7	78 45	102 52
	81 34	136 30
	84 22	205 14
	87 11	407 60
8	90 00	ad infinit.

As the Sine-Complement of the Rhomb from the Meri∣dian, is to 20 Leagues or 60 Miles, the Measure of 1 Degree at the Meridian:

So is the Radius or Sine of 90, to the Leagues or Miles answering to one Degree upon the Rhomb.

Suppose by the Quadrant it were required to answer this Question,

Sailing N. N. E. from 40 Degrees of North-Latitude, How many Leagues shall the Ship run before it can come to 41? By reason this is the second Rhomb from the Meridian, and the Inclination thereof is 22 deg. 30 m.

Therefore set the side of the Index EL to the second Point from the Meridian N. N. E. 22 d. 30 m. and reckon from C 20 Leagues towards D, and with your Eye or a Pin fol∣low the Parallel-Line to the Index, and you will find it cut 21 Leagues 65 parts (or better than ½ more) the number of Leagues you must Sail before you can reach 1 Degree.

You may do the same by the Travis-Scale thus. Extend the Compasses from 2 Points nearest the end of the Scale, and greatest Number of the Line of Numbers that is N. N. E. 2, and E. N. E. 6 Points, unto 20 Leagues on the Line of Numbers; remove the Compasses to 100 in the Line of Num∣bers, and the other Point of the Compass will reach to 21 Leagues 6/10 ½ or 65 parts, as before in the Line of Numbers.

This may be found also by a Line of Chords and Equal Parts, if you draw a Right Line, and take with your Com∣passes 20 parts, and lay it from one end on the Line; then take 60 deg. and sweep an Arch, and take 2 Points with your Compasses, and lay from the Meridian on that Arch from N. N. E. and draw the Secant or Rhomb-Line, at 20 Leagues draw a Perpendicular or Line at Right Angles there to the former, and measure the distance from the Center, to the Intersection of the Line drawn from 20, with the Rhomb-Line on the Scale of Leagues or Equal Parts, and you will find it the same as before. And so the Qua∣drant shews you all at one sight, if you understand without more words. By the Artificial Sine and Number, Extend your Compasses from the Sine of the Rhomb 67 deg. 30 to 20 in the Line of Numbers, the same Extent will reach from 8 Points, or 90 deg. or 100 in the Line of Numbers, to 21 Leagues 65 parts, as before.— This consider in general; I shall shew you more particularly in 12 Proofs (how of these four, any two being given, the other two may be found, both by Mercator's Chart, and all other ways, as is usual) when I come to treat more particularly of Na∣vigation.

II. By one Latitude, Rhomb, and Distance, To find the difference of Latitudes.

Let the place given be C in the Latitude of 40 Degrees, that is in the Center of the Quadrant, the second Latitude unknown; The distance upon the Rhomb 21 Leagues 65 parts of a League; the Rhomb N. N. E. the second from the Meridian: Therefore set the Index to the Point, and count 21 Leagues 10 parts, and run your Eye up the Parallel-Line you there meet with, and reckon the Leagues from the Center C to that Line, and you will find it 20 Leagues; and such is the difference of Latitude required.

It is easie to be understood how to lay it down by the Plain Scale; therefore I shall forbear to write any more of that Way.

As the Radius, to the Co-sine of the Rhomb from the Meridian:

So the distance upon the Rhomb, to the difference of the Latitudes.

Extend the Compasses from the Sine of 90, to the Co-sine of the Rhomb 67 deg. 30 m. the same distance will reach from 21-65 Leagues in the Line of Numbers, to the difference of Latitude 20 Leagues. In like manner you must work for all such Propositions, let the Number be greater or less, by either Instrument.

The Travis-Scale is the same manner of Work, as the Artificial Sines, Tangents, and Numbers; For extend the Compasses from 8 Points, to 2 Points, the same distance will reach from 21: 65 in the Line of Numbers, to 20 the difference required.

III. By the Rhomb and both Latitudes, To find the Distance upon the Rhomb.

As suppose the one place given were C the Center of the Quadrant, in the Latitude 40 deg. the second place in the Latitude 41 deg. and the Course the second from the Meridian.

Set the Index to the Rhomb, and account 20 Leagues, which is 41 deg. the second Latitude, and carry your Eye on that Parallel that leads to the Index; and there it will cut the distance upon the Rhomb, which in this Question is 21 Leagues 65 parts.

Extend the Compasses from the Co-sine of the Rhomb from the Meridian, to the Radius or Sine of 90—

The same Extent will reach from 20 Leagues, the difference of Latitude, to 21: 65 in the Line of Numbers, the distance upon the Course required.

IV. By the distance and both Latitudes, To find the Rhomb.

Suppose the Place given was at C, in Latitude 40 deg. and the second Place a Degree or 20 Leagues further Northward, and the distance was 21-65 Leagues upon the Course.

From the Center C reckon 20 Leagues towards D, follow that Parallel, and set the Index, and count the distance until it touch the Parallel, and look in Arch of the Quadrant, and you will find the Rhomb 22 gr. 30 m. or N. N. E. 2 Points from the Meridian.— Or, Extend the Compasses from the distance upon the Rhomb 21: 65, to the distance of Latitudes 20 Leagues; The same Extent will reach from the Radius or Sine of 90, to the Sine-Complement of the Rhomb 67 deg. 30, which was required.

V. By the difference of Meridians, and Latitude of both Places, To find the Rhomb.

As if the Place given was C the Center of the Quadrant, 40 deg. and 20 Leagues was the difference of Latitude Northward, that is 41 deg. and the difference of Longi∣tude 8 Leagues 45 parts of a League.

First, count from C the difference of Latitude 20 Leagues, on that Parallel count 8 Leagues 45 parts; to that put the Index; and in the Arch you will find the Course 22 gr. 30. from the Meridian.

Extend the Compasses from 20 Leagues to 8: 45, the same Extent will reach from 90 to the Tangent of the Rhomb 22 gr. 30 min. as before.

VI. By the Rhomb and both Latitudes, To find the difference of Longi∣tude, or departure from the Meridian.

Let the Rhomb be 2 Points from the Meridian, the one Latitude given 40 deg. the other Latitude 41 deg. the difference 20 Leagues.

Set the Index to the Point and Rhomb 20 gr. from the Meridian, and count 20 Leagues the difference; on that Parallel reckon the Leagues between the Side and the Index, and you will find it in this Question 8 Leagues 45 parts, the Meridians-distance required.— Or, Extend the Compasses from the Tangent of the Rhomb 22 gr. 30, to Radius 90, the same Extent will reach from the difference of Latitude 20 Leagues, to the departure from the Meridian 8 Leagues 45 parts.

These six last Propositions depend one upon the other, as you may plainly see; which may be sufficient for the Explanation of the Quadrant, by which may be un∣derstood much more.