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. XX. Having the Meridian-Altitude of an unknown Star, and the distance there∣of from a known Star; to find the Longitude and Latitude of the unknown Star.

IN the 16th Chapter of the second Book of Harmonicon Coeleste, Mr. Vincent Wing hath this Example, and Observation, made by the Phoenix of Astronomy Ticho-Braghe in the year 1577, which we will borrow for an Example; it being a useful Rule for all Ingenious Navigators, for by it they may find the Longitude and Latitude, and consequently, by the foregoing Rules, the Right-Ascension, and Declination of those good Stars for their use, that are in the South Hemisphere, viz. as they have been named by the Portugals; the South-Triangle, which Constellation hath 5 Stars, one of the Eastermost corner, which comes last to the Meridian of the second Magnitude. The Crane, in which there is 13 Stars on the left Wing, and another on the right side the back of the second Magnitude. The Phoenix, 15 Stars, the Water-Serpent hath 15 Stars. The Dorado, or Gilt-head-Fish, situate in the very Pole of the Ecliptick; and in that Constellation is 4 Stars. The Chamelion, with the Flie, in which is 13 Stars; The Bird of Paradise, in which is 12 Stars; the Peacock, in which is 15 Stars; one in the head of the second Magnitude; the Naked Indian, in which is 12 Stars; and also the Bird Taican, or Brasilian Pye, in which Constellation is 7 Stars, two of them of the third Magnitude: Also two useful Stars for Navigators; in one Constellation, which are Noah's Dove, which containeth 11 Stars, of which there are 2 in the back of it, of the second Magnitude, which they call the Good Messengers, or Bringers of good News, and those in the right-Wing are consecrated to the appeased Deity; and those in the left to the retiring of the Waters, in the time of the Deluge; and they come to the Meri∣dian about half an hour before the great Dog; and by the Globes are about 21 degr. 30′ distant, from the nearest in the back; but I would have the Sea-men take him exact, as likewise a good Constellation called the Crane Grus, or the Flamengo, as the Spa∣niards call it; this Astensine consisteth of 13 Stars, and hath 3 Stars of the second Magnitude, that in the head is called the Phaenicopter Eye, and the other are on his Back, and the other in his left Wing; These Stars I would desire those Mariners that Sail to the East or West-Indies; to take the Meridian-Altitude thereof, and their distance from any known Stars, and by it you shall have all the rest; for many times I have been Sailing between the Tropicks, and for 12 days together have had no Meri∣dian-Altitude of the Sun, by reason of close and cloudy weather, which is bad for those that are bound to small Islands, and Cape-Lands; therefore to the Southward, as well as to the Northward; these Stars will stand them in great stead, and serve their turn, as well as the Sun, to find the Latitude thereof.

The Rule is thus; About the end of the Year 1577, Ticho observed the distance of rhe little Star in the breast of Pegasus from the bright Star of the Vulture, to be exactly 45 degrees 31′, and by the Meridian-Altitude thereof, he found the Declination thus.

Ticho observed at Ʋraniburge the Star in the breast of Pegasus, and found his Meridian-Altitude	56 d. 32 m.
The Latitude of Uraniburge is 55 d. 54′ the Compl.	34 06	Substract the Alti∣tude of the Equa∣tor.
The Declination is found to be	22:26
To it add the Complement of 56 deg. 32′, which is	33:28
And the sum is the Latitude of Uraniburge	55:54 m.

So the Declination is found to be 22 deg. 26 min. North, which being given, the Lon∣gitude of the said Star is to be inquired.

Therefore in the Oblique-Angled Triangle (of this Diagram) FOL is known.

[illustration] geometrical diagram

First,

The Complement of the Declination of the bright Star of the Vulture 82 degrees 8 min. FL.

Secondly,

FO, the Complement of the Declination of the Star in the Breast of Pegasus 67 degr. 34′.

Thirdly,

OL the distance of them 45 degr. 31′.

By this Rule the Angle at F, which is the difference of their Right-Ascension, will be found to be 44 degr. 54′, as will be here demonstrated.

The Co-Sign of Declination FL,	is 82 deg. 08 min.	999589
The Co-Sign of Declinat. FO, is	67 − 34	996582
The difference is	14 − 34
(1.) The sum is		1996171
(2.) The Quadrat or (2) the Rad.	2000000

The Base or distance of the Stars LO	45 d. 31
The difference of their Declin. FL and FO	14:34
The Sum	60:05
The difference	30:57
The half of the Sum	30 d. 02 30″ Sign	969951
The half of the difference	15:28 30 Sign	942621
The (3) Sum is	1912572

Then say,

As the first Sum	1996171
is to the double or Quadrat of the Radius	2000000
So is the third Sum	1912572
to the double or Quad. of the Sign of half ☞	1916401

The Angle sought, which Bi-sected, gives the Sign of 22 d. 27′ 17″ 958200

Which doubled, is 44 degr. 54′ 34″, is the Angle LFO, which is equal to the Arch DE, the difference of their Right-Ascension, which I add to the Right-Ascen∣sion of the bright Star of the Vulture, 292 degr. 35′, and the Sum is 337 degr. 29′ 34″, is the Right-Ascention of the little Star in the breast of Pegasus.

Then having the Declination of this Star 22 degr. 20′, and the Right-Ascension 337 degr. 29′ 34″, the Longitude of the said Star by the last Problem (19) will be found to be 18 degr. 36 ♓, and the Latitude thereof 29 degr. 24 min. North.

Now to draw the Diagram by Chords, and half-Tangents Geometrically, with the Chord of 60 degr. draw the Circle; then draw ♑, ♋, the Ecliptick 23 degr. 31′ ♑, S, and by C draw the Equator; then by the Parallel of Declination, and Right-Ascen∣sion of the Vulture, will find LO; therefore if you put the difference of Ascension from ♈ to E 22 degr. 30′ 20″, from the nearest Equinoctial-Point, and draw the Meridian-Circle FES, and it will cut the Parallel of Declination at O; draw through O, as PON the Circle of Longitude, and measure ♈ X on the Line of half-Tan∣gents, and it is 11 degr. 24 min. from the nearest Vernal Equinox, Substracted from 30 degr. leaves 18 degr. 36′ of ♓ for the Stars Longitude; and as before directed, you may find the Latitude ♑ ♄ to be 29 degr. 24′.