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 Use of the Quadrant in Astronomy.

SECT. I. Having the Latitude of the Place of the Sun's Declination, It is required to find the Time of the Sun-Rising and Setting.

The Latitude 51 Deg. 30 Min. Northward, and the Declination 20 Degrees, the difference of Ascension will be found thus.

First, Lay the Center Ear at E of the Index, over the Brass-Pin in the Center at A of the Quadrant, and lay the Edge of the Index to EL, to the Latitude of the Place on the Arch DM; and take of the Tangent-Line on the Edge of the Quadrant 20 Degrees the Sun's Declination; and lay that distance from the Center at A towards D, at that distance run with your Eye along the Parallel-Lines, and mark where it toucheth the Edge of the Index; there follow that Parallel-Line to the Arch, and reckon the Degrees from B to that Parallel-Line will be 27 Deg. 14 Min. the diffe∣rence of Ascension between the Sun's Rising and Setting, and hour of 6, according to the time of the Year.

The Degrees resolved into Hours and Minutes, is 1 Hour 49 Min. which is 4 of the Clock and about 11 Min, for the Sun Rising in the Morning, and 7 of the Clock 49 Min. his Setting in the Evening. In the same manner you must work for all La∣titudes.

SECT. II. Having the Latitude of the Place, and the Distance of the Sun from the next Aequinoctial Point, To find the Amplitude.

So the Latitude being 51 Deg. 30 Min. and the place of the Sun in one Degree of Aquarius, that is 59 Degrees from the next Aequinoctial Point; therefore set the Ear at S of the Line of Sines of the Index on the Pin at A, and the Edge thereof to the Latitude, and reckon 59 Degrees the Sun's distance from the first Aequinocti∣al Point, from the Center to C along the Line of Sines of the Index; there note the Line that cuts the 59 Degrees following with your Eye, to the Degrees in the Arch, and reckon the Minutes of Degrees from M to the Edge of the Index, and you will find it about 33 Deg. 20 Min. the Amplitude required.

SECT. III. Having the distance of the Sun from the next Aequinoctial Point, To find his Declination.

The Sun being either in 29 Degrees of Taurus, or 1 Deg. of Aquarius, or 1 Deg. of Leo, or 29 Deg. of Scorpio, that is 59 Degrees from the next Aequinoctial Point, To find his Declination do thus: Put the Ear of the Line of Sines on the Pin and Edge of the Index; put to 23 Deg. 30 Min. in the Sun's greatest Declination, reckoned from M on the Arch; then count the Sun's distance 59 Deg. on the Deg. of Sines of the Index: From the Center put one Foot of your Compasses by the Degree, with the other take the nearest distance to the Line or Side CM; apply that distance in the Line of Sines of the Index, from S along, and the other Foot will reach to 20 Degrees, the Declination required when the Sun is in the aforesaid Degrees and Sines. In like manner you must do for any other Degrees of the Sun's Place.

SECT. IV. Having the Latitude of the Place, and the Declination of the Sun, To find the Sun's true Amplitude from the true East and West.

This is a most excellent ready way by this Quadrant, and as near the Truth as any Man can make any rational use of this Problem at Sea: It is thus. Suppose the Latitude to be 13 Degrees, and the Sun's Declination 20 Degrees Northward, the Sun's true Amplitude of Rising and Setting is required, from the true East and West.

Set the Ear of the Side of the Index on which is the Line of Sines on the Center, and Edge to the Latitude 13; then count from M 20 Degrees of Declination, and carry your Eye upon the Parallel-Line from that Degree of the Arch, and mark what Degree it cuts of the Index and Line of Sines, as in this Question it doth 20 De∣grees 23 Minutes, and that is the true Amplitude required.

Secondly, Suppose you was about the Cape of Virginia, in Latitude 37 Degrees and 30 Min. and Declination 10 Deg. If you work as before-directed, you may find the true Amplitude to be 12 Deg. and about 36 Min. You may estimate the Min. but you cannot Steer by a whole Deg. when you have rectified your Compass by this; therefore this is sufficient for that Use, to shew you the difference between the true Compass and the Steering Compass, if you observe his Rising and Setting by it.

Note, The Amplitude is the distance of Rising or Setting of the Sun or Stars from the true East and West Points upon the Horizon.

As for the foregoing Work,—In the Latitude of 13 Deg. the Sun or Star having North-Declination 20 Deg. therefore they will rise 20 Deg. 33 Min. to the Northward of the East, and set 20 Deg. 33 Min. to the Northward of the West. But if the De∣clination had been 20 Deg. South, then they would have risen 20 Deg. 33 Min. Southward of the East, and set 20 Deg. 33 Min. to the Southward of the West.

And so if you bring these Degrees and Minutes into Points and Quarters, and use the Variation-Compass upon the Instrument of the Moon in the First Book, you may readily rectifie the Compass you Steer by.

SECT. V. The Ʋse of the Quadrant and Variation-Compass in the First Book, on the Instrument of the Moon for shifting of Tydes.

This Instrument contains two Parts or Rundles, which are the two uppermost in the aforesaid Instrument made of Wood or Brass, moving one upon the other, as there you may see. The biggest of the two uppermost Rundles represents the Compass you Steer the Ship by, which is subject to Variation: but the upper Compass doth repre∣sent the true Compass that never varieth, whereby you have a most necessary Instru∣ment to rectifie the Compass, as Mr. Wakely hath made Tables to be used with it; but this will serve for use as near by the Quadrant.

Admit I am in the Latitude of 27 Deg. and Declination 20 Deg. Northward, and I observe the Sun's Rising and Setting to be due East and by North, and West by South Point of my Steering or Variation-Compass; the Variation in that Latitude is required.

The Sun having North-Declination, and in that Latitude of 27 Deg. if there be no Variation the Sun will rise (as you may presently find his Amplitude by the Qua∣drant and Index, 22 Deg. 34 Min. which is but 4 Min. not to be taken notice of,* 1.1 above) E. N. E. and sets W. N. W. But according to the foregoing Propositions, the Sun did rise at E. b. N. and set at W. b. N. Therefore it plainly appeareth that there is a full Point Variation.

Therefore on the Variation-Compass on the Instrument of the Moon, you must al∣ways bring the true Point of Rising and Setting on the upper Compass, to touch the false Point or Rising and Setting, found by Observation and Steering-Compass, on the

middle Rundle, being set in this Position. You will find the E. b. N. to be the true E. N. E. and the W. and by N. to be the true W. N. W. and the N. b. E. to be the true N. and the S. b. E. to be the true S. and the S. E. ½ Point Southerly, to be the true S. E. b. S. ½ Southerly; and the South 3.1/4 East, to be the South ¼ West: And so you may do with ease in all other Observations, in like manner as you have been shewn, by Points, Halfs, and Quarters, which is on the two Trundles; and be sure nearer than ¼ of a Point I never did see any man Steer or sheape a Course.

SECT. VI. To know the Variation by the Quadrant.

You may do the same thing by the Quadrant, without the help of the Rectifier before spoken of, if you will remember, That this Quadrant hath eight Points, or ¼ of the whole Compass, by which you may orderly reckon the whole, and set the Index to the greatest difference either from the East, Southward, or Northward, or West. In like manner as in the foregoing Proposition, the true Amplitude of Rising and Setting was 2 Points, or 22 Deg. 34 Min. E. N. E. Set the Index to the Degrees and Points, reckoning the Deg. from D on the Arch of the Quadrant, toward E the Scale of Leagues of the Index; then reckon the Point and Degree taken by Observa∣tion, which is 11 Deg. and 15 Min. a just Point of the Compass: therefore it being but E. by N. short a Point of the true Amplitude, therefore the E. b. N. of the Steer∣ing-Compass, respecteth the true E. N. E. and the N. b. E. respecteth the true N. and so account all round the Compass a Point more than the Steering-Compass sheweth: And if you would know which way the Variation is, you see it is a Point more from the E. than your Compass sheweth Northerly.

But if the Steering or Azimuth-Compass, had shewn a Point more than the true Amplitude found by the Quadrant and true Point, the Variation had been Westerly.

But suppose the Amplitude found had been a Point Southerly, E. b. S. and the Sun's Rising and Setting had been a Point Northerly; by Observation of the Ship-Com∣pass, you see there is two Points difference: therefore set the Index to two Points, from M the East or West side of the Quadrant, as in this Proposition you must reckon it, and you may see plainly that the East Point by the Steering-Compass is the true East-South-East Point; and the South Point is the true South-South-West Point; and the North is the true North-North-West Point; and so of all the rest: And the Va∣riation is Southerly. So that you see how readily this Quadrant doth these things, when the Points of the Compass is imprinted in a Mans mind, which must be, and is in all Masters and Mates.

Suppose I would know by the Quadrant the true Point of the Compass, when Bootes Arcturus riseth and setteth: In the Latitude of 40 Deg. Bootes Arcturus De∣clination is 20 Deg. 58 Min.

Set the Side of the Index and Sine to the Latitude of 40, and count the Declina∣tion 21 Degrees almost, from M on the Arch; and run your Eye up the Parallel, and it will cut the Index about 27 Deg. 50 Min. which is reduced into Points and Quar∣ters by allowing Gr. 15 Min. to a Point, his Rising will be almost E. N. E. ½ a Point Northerly, his Setting W. N. W. ½ Westerly. But if the Declination of a Star of the South side the Aequinoctial, the Rising had been E. S. E. ½ Southerly, and his Setting W. S. W. ½ Southerly.

In the like manner you may know the Rising and Setting of any Star in an instant, by this Quadrant and Index, which I hold to be as necessary an Instrument as Seamen can use, in respect of its plainness, and brevity, and portability, so made as you see the Figure, the larger the better: And on it you may work all manner of Travisses to the distance of 60 Leagues or Miles which is on the side of the Index. It being so plain and easie, I need not write any thing thereof; but for the Learner's sake, take these few Rules following.

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.

Notes

• * 1.1

  11 d. 15. to each Point.