A perfect and easy Treatise of the use of the celestial Globe: written as well for an Introduction of such as be yet unskilful in the study of Astronomy: as the practice of our Countrymen, which be exercised in the Art of Navigation. Compiled by Charles Turnbull: And set out with as much plainness as the Author could: to the end it might of every man be understood.  Psalm. nineteen. The heavens declare the glory of God, and the firmament showeth the work of his hands. Day unto day uttereth the same, and night unto night teacheth knowledge. 
¶ Imprinted at London for Simon Waterson.   

INSIGNISSIMO VIRO, AC OMNI VIRtutum genere clarissimo Magistro Henrico Nocllo, Carolus Turnbullus salutem ac foelicitatem perpetuam comprecatur.  
cum antiquissimorum scriptorum singularem industriam in studijs Mathematicis iam ante seriùs, accuratiúsque cognouerim: & recentium quorundam depravatam lectionem summa cura, summáque diligentia viderim & obseruauerim: ne infestus error damnum inferret gravius: curavi (vir insignissime) ea faciliùs explicare, quae si fideliter discantur, omnis erroris tollent difficultatem. Eorum verò te patronum esse volui, ut meo exemplo ad eadem vota multos invitem, tanquam indicem te, in florentissimo hoc regno non solùm tanti negotij virum celeberrimum: sed omnium regiarum virtutum laudúmque patronum. Itaque minusculam opus hoc A●…, qualecunque sit, (quod scio quam sit indignum) tibi dedico: ut qui imposterum non solum obseruationes astronomicas elaboratas magis, minúsque usitatas: (sive sint theorica sive horalogica) sed opticas quoque & qua sunt geometriae communis praecepta, (ut tempus postulat) sis receptarus. Quibus ut es spiritus nobilissimi: ita mirificè quoque delectaris. unum hoc interim precibus etiam atque etiam obsecrans: ut qua humanitate erga me semper usus es. (cum maxime possis:) eadem iam indies confirmata magis (libertissime quoque velis patrocinari.  
Vale.   

To the Reader.  
MEn which are more desirous of public Fame and Renow ne, then studious of common profit: with great curiosity set a glass upon such things, as being discovered, would show dismembered & misshapen. (As for myself I seek no higher dignity, then to be reported to set forth a truth: and therefore without any wrested eloquence, I make bold to offer the use of the Globe, to the exquisite view of your curious eyes: (though far inferior as I suspect to some men's expectation) to the end such gross enormities might be amended, as often times in divers have been discovered. Who for want of right conceit of things by them attempted: have in the air built such fortresses, as have without assaults wounded their loving enemies. But happily such men will now retire, and arm themselves better against the next assault, lest they be like to the dogs in the capital of Rome: which were placed to the end, that if by night spoilers should arrive, they might sound a warning. For, true it is, that by night these bark out false Allaromes at their enemies: but if by day, they bark likewise at friends: I hope ye will judge them worthy to have their legs broken. which things I leave to your gentle interpretation. Nothing misdoubting, but if in this tract either anything be escaped contrary to my will: or omitted not satisfying your expectation: ye will equally suffer the same. For if ye receive the fruits of my labour and care of your commodity, I require no more. Wherefore my travail I bequeath to your discreet consideration, and yourselves to the protection of almighty God.  
Farewell.   

DEFINITIONS to be praemised necessary for the understanding of the Globe.  
THe Sphere or Globe, is a perfect round & solid body, contained under one superficies or face; in whose middle is a point, from which point all lines that are drawn to the superficies and face of the same: are equal the one to the other.  
The Centre of the Sphere, is the middle point of the same.  
The Axe of the Sphere, is a right line passing from one side of the same, (by his Centre) to the contrary side, about which line the Sphere is carried: but the line itself standeth still.  
The Poles of the Sphere, be the ends of his Axe.  
The Pole of any Circle, is a point without the compass of the same: (and yet is equally distant from all points, of the circuit or borders of the Circle whose Pole it is:) & from which the same Circle is drawn.   


OF THE NAME of the Sphere, and his divers and sundry kinds of divisions: together with the motion of each one in his kind.  
THE NAME OF the Sphere, is taken either generally or particularly. Generally, and so it is said to contain all perfect round bodies, whether they be solid or not: whether contained under one only Superficies or more. And so may every Orb be called a Sphere. But if we take the word Sphaera, in his particular and proper signification: then nothing is a Sphere: but a perfect round body being solid, contained under one, etc. as the former definition declared. This Sphere is divided either according to his substance: or according to certain properties and affections which he is capable of.  
According to his substance he is divided into two parts: the one Elemental, the other ethereal. The Elemental, containeth the four Elements, Fire, Air, Water and Earth: and is subject to alterations, by reason of their effectual working. The ethereal, compasseth in round, the Elemental part, in his hollowness, and is lightsome by nature, and unchangeable: and containeth ten Spheres. The first and highest from the earth, being called the first movable, containeth in his hollowness all the other: and by his natural motion is moved directly from the East to the West, and so to East again, in the space of 24. hours continually, and carrieth about with him by violence, all the other Spheres. The next under this is the ninth Sphere, called the crystal heaven, and by his natural motion is carried from West toward East, but very slowly, in many years passing but one degree: and this motion hath caused the Stars to alter their Longitudes. The third Sphere is the Firmament or Sphere of the fixed Stars: whose motion by nature is upon two little Circles: the one being described about the head of Aries, and the other of Libra: which motion is called the motion of Trepidation. The other seven Spheres be of the seven Planets: the highest of Saturn, which moveth by nature from West toward East, and that in 30. years one perfect revolution. The next of jupiter, moving from West to East by nature, and that in twelve years. The other of Mars, making his revolution from West toward East in two years. Under Mars is the Sun, moving by nature from West toward East, making one perfect revolution in 365. days and 6. hours almost. Under the Sun is Venus, and then Mercury, moving from West to East about the same time as the Sun. The last is the Moon, making one perfect revolution from West toward East in 27. days. 7. hours. 43′. 7″. yet all these are carried by violence of the first movable from East to West, as is before said.   

▪ OF THE Circle's of the Sphere of Heaven, and of their names, and how they be made.  
AStronomers to the end they might show the motions of Heaven, and the strange and wonderful conclusions of the Celestial bodies: have imagined certain Circles in the body of the first Sphere or first movable, and principally ten: whereof some be greater Circles of the Sphere, so called because the Centre of these Circles is also the Centre of Heaven: & every such Circle divideth the whole Sphere into two equal parts. Of this sort be six: the Equinoctial, Zodiac, Horizon, Meridian, and two Colours. Some be lesser Circles of the Sphere so called, because they have not the Centre of the world for their Centre, neither divide the whole Sphere equally. Of this kind be four, the Tropic of Cancer, the Tropic of Capricorn, the Arctic and Antarctic.  
The Equinoctial, called the aequator or girdle of Heaven, is a great Circle of the Sphere, dividing the Sphere into two equal parts, and is equally distant from each Pole of the world. And took his name of the aequator, either because it is equally in the middle of Heaven, as Euclid saith in his Optics: or for that the Sun, coming to this Circle, maketh the day and night equal. & it is divided in 360. equal parts, which parts are called degrees. His Axe is the Axe of the world, and Poles, the Poles of the world.  
The Zodiac is a great Circle of the Sphere, which crosseth the Equinoctial in two points, the one being the head of Aries, the other of Libra, and swarneth from him in all other points, leaning toward each Pole of the world in the point of his greatest swarning, 23. degrees, 30. minutes. This Zodiac is of breadth 12. degrees, and of length, that is to say, in compass 360. degrees, and according to his length, is divided into 12. equal parts, which are called the 12. signs. Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius and Pisces. And each sign containeth of length 30. degrees. in the middle breadth of the Zodiac, we imagine a Circle to pass, which we call the Ecliptic Circle or line: For that that when the Sun and Moon be both under this line in a Diameter: then the Moon is Eclipsed. Under this Circle the Sun moveth daily (without declining any ways) the quantity of one degree very near in each day. the rest of the Planets are found some times on one side the Ecliptic, & some time on the other. This Zodiac taketh his name of a greek word signifying a living creature: or as the Latens will, is called Signifer, for that that it beareth the 12. Signs. the Axe of the Zodiac and the Ecliptic, is all one, being a line divers from the axe of the world, and the Poles be two points always so much distant from the Poles of the world: as the greatest declination of the Ecliptic cometh unto.  
A Colour doth generally signify any Circle passing by the Poles of the world, and hath his name of his unperfect showing himself in the motion of heaven. But now by the name of Colours, we understand two great Circles, the one going from the Poles of the world by the points where the Equinoctial and Zodiac cut themselves (which be called the Equinoctial points) and is called the Equinoctial Colour. The other passeth from the Poles of the world, by those points of the Ecliptic which serve most of all others from the Equinoctial line (which points are called the Solsticiall points) and this is called the Solsticial Colour. And here be you to know that these four greater Circles which we have defined: be still the same through the whole world, and are said to be movable Circles: for so much as in the motion of heaven, they be also moved. of which, some are movable perfectly: as the Equinoctial and Zodiac, (for they in the going about of heaven, do ascend by little and little, till the whole Circle have gone our the Horizon) some unperfectly movable, as the two Colours, which never show the whole Circle in any crooked Sphere. the other two greater Circles which follow, be called fixed, for that they never move by the motion of heaven. But they be changeable in every Region.  
The Horizon is a greater Circle dividing the half of the Heaven which we see, from the half which we see not, and is called in Latin Finitor, because it endeth our sight. The Horizon maketh four principal points, East, West, North, and South. His Axe is a line imagined to fall from the point of heaven, which is directly over our head where we be, down to the ground like a plum line, and his Poles be the ends of that line, called the Vertical point, and point opposite to the Vertical.  
The Meridian is also a great Circle, passing from the Poles of the world by our Vertical point, cutting the Horizon in the North and South points his Axe is a line going from the East point of the Horizon to the West, and his Poles be the same points, and these two Circles do always change, & are divers in every Region: for so much as the Vertical point of every Region is divers, by the which the Meridian of necessity must pass, and is the Pole also of the Horizon.   

OF THE LESSER Circles of the Sphere, and their names, and of their making.  
THe less Circles of the Sphere in number be four. The Tropic of Cancer, the Tropic of Capricorn, and the two Artickes. The Tropic of Cancer, is a less Circle of the Sphere, which is equally distant from the equinoctial, lying betwixt the Equinoctial and the North Pole, and touching the Ecliptic in the beginning of Cancer. This Circle is described by the body of the Sun in the longest day of Summer, at which time the Sun is entered the solstitial point, or beginning of Cancer. & is called the Tropic, of a Greek word, which signifieth a returning: because the Sun being brought to this point, falleth in his noon height, and returneth again.  
The Tropic of Capricorn, is a like Circle betwixt the Aequator & the South pole, and is described by the Sun in the shortest day of Winter, at which time the Sun is in the beginning of Capricorn, whereof it is called the Tropic of Capricorn.  
The Arctic Circle is a less Circle of the Sphere, described by the Northern Pole of the Ecliptic. Proclus saith it is described by the foremost foot of the great bear, and thereof taketh his name.  
The Antarctic is a like Circle described by the South Pole of the Ecliptic, & is called Antarctic of the Greek word, which signifieth Opposition, because it is opposite to the other.  

Of the use of the Circles of the Sphere or Globle.  
THE most principal cause why Artificers invented the Equinoctial, was first, because it is the measure of the first Heaven, by a convenient, perpetual and equal swiftness. Secondly, it measureth and limitteth the time of rising of the Signs, as also the length of the Artificial days, and times of the Aequinoctials, with declinations and right ascensions of Stars, together with Longitudes of Regions. Lastly, for the exection of the twelve howsen of Heaven. In like manner, the Zodiac serveth for Latitudes & Longitudes of Stars: for distinction of the times of the year: for the motions of all the Planers and effects of the same. Not unlike be the uses of the Colours and Meridian, each showing the greatest declination of the Ecliptic: but especially the Meridian, which giveth as well all declinations of Stars, their noon height, and distinguisheth the days and nights into two; equal parts, and serveth for the Horizon of the right Sphere. It beginneth likewise and endeth all Longitudes of Regions, and showeth Latitudes and Elevations of the Pole: It helpeth to divide the 12. howsen. In like manner sundry and divers be the uses of the Horizon: As in separating the hidden part of Heaven, from that which is seen, and showeth the place of rising and setting of any Star: how far from East or West, with his height. All which points are respected of Astronomers, as the Sphere is secondarily divided: that is to say, as he is a right, or a crooked Sphere, which be his properties and affections meant in the division afore specified. By a right Sphere is meant such a kind of position of Heaven: as that neither Pole be raised above ground, but that each lie in the face of the earth. And such a kind of position have they which dwell in Bersera, and the Islands of Molucca, or such like. Contrariwise, it is said to be a crooked Sphere, when any one of the Poles is raised above ground. Such a Sphere have we at Oxford, and London: and generally all, which dwell not under the line. All which things for our better conceit, are showed to the eye in the material Globe, whose names and divisions appear at the first view: two things only being weighed. First, that the mechanical or material Globe which representeth the first movable: beareth in him the fixed Stars, (not because the Stars be in the first movable) but because their motion is so little in their own Spheres in many years, that they may seem not to have moved at all in a mans his age from their places under which they be of the first movable: therefore they may be supposed to stand in it. Secondly, the Globe representeth the Stars to us in his connexitie, which appear in Heaven in the concavity. For that our eye is not in the Globe but without. Furthermore in the Globe, besides the aforenamed Circles, be found three others of brass: the one being a perfect Circle of a little quantity, placed about the Pole which is elevated: & is called the hour Circle, whose style is called the Index. An other is a thin rule of brass, representing one quarter of a whole Circle, called the quadrant of Altitude, and is alway to be fixed (when ye use the Globe) on the middle of the half of your Meridian, which is above the Horizon: that is to say, 90. degrees above the Horizon. The third and last is a great half Circle lying at the Horizon, serving aswell for the erection of the scheme of Heaven, as any Circle of position. All which things being advisedly considered of, ye may proceed in the use of the Globe: As followeth.    

▪ HOW THE Globe is to be placed, ready for his use and practise.  
THe placing of the Globe ought to be such, that the Horizon of the Globe may stand parallel or levied to the true Horizon: and the Meridian of the Globe stand in the Superficies of the true Meridian of Heaven, and the Poles of the Globe and his Axe, answer exactly to the Poles and Axe of Heaven. Now to the levying of the Horizon: there ought to be at your Globe a hanging plummet. and for the Meridian, a Needle touched of the load stone, and touching the rectifying of the Poles and Axe of the Globe, the elevation of the Pole of heaven is first to be known. the means to perform and accomplish the same, being such as follow.  

Propositio. 1. To find a Meridian line in any place appointed.  
SEt up on your Horizon or some plain levied board, a Gnomon of any reasonable length then, (at such time as the same shineth, describe from the top of your Gnomon a Circle by the tip of his shadow. and make a mark in the Circle where the shadow ended, at your observation (which must be before noon.) Then mark in the afternoon at what time the end of the shadow returneth into the same Circle again, and make a mark at his point of falling, so shall ye have a portion of the said Circle enclosed betwixt the two points: If then ye divide this portion into two equal parts, and draw a line from this middle point, by the point in which the Gnomon standeth, it shallbe a Meridian line.   

Propositio. 2. To take the height of any Star.  
From the point of Heaven, which is directly over our heads (being called the Vertical point, or Zenith:) are imagined divers Circles to fall by every degree and minute of the Horizon: all which Circles are called Verticals, & serve for the height of Stars, for so much as the altitude of Sun or star is the portion of the V●…ticall Circle, enclosed betwixt the Centre of the Sun or Star (in the time of his observation) and the Horizon, which height is thus found.  
Take your Astrolabe, and let him hang freely by his ring, then turn up his Dioptral so long, that ye see the Star (whose height ye seek) thorough his sights: for then, how many degrees and minutes are enclosed betwixt the Dioptrall and the line of your Astrolabe, which is parallel to the Horizon: so many hath that Stars of height. as the seventh day of januarie. Anno. 1585. under the Meridian of Oxford, at 9 of the clock I sought the height of the Sun. taking then my Astrolabe, and hanging him toward the Sun, and raising his Dioptrall till I espied the Sun. I found betwixt the Dioptral and the line representing the Horizon seven degrees, and 16. minutes, so much was the height of the Sun at that time.   

Propositio. 3. To take the altitude of the Pole in in any place or country.  
THe altitude of the pole is the portion of the Meridian Circled enclosed betwixt the Pole and the Horizon. and is thus found. Find a Meridian line, and draw him in the Horizon by the first proposition: then take the height of any fixed Star which setteth not (and that at the fore part of the night) at such time as he is pointed upon your Meridian line, by the second proposition. Again the next morning, or any other morning take the height of the same star at such time as he is pointed with your Meridian line: then subduct the less altitude from the greater, and divide the difference into two equal parts. Lastly, add half the difference to the less altitude, so the whole number made giveth the altitude of the Pole. As at Oxford I took the height of a star in the fore part of a night in winter, being the tenth of December. 1584. at what time he was pointed with my Meridian, and found his height. 55. degrees. 59 minutes. Again the next morning I took his height, and found it 47. degrees 41. minutes. This less altitude I subourted from the greater 55. degrees. 59 minutes, and the difference was 8. degrees. 18. minutes, which being parted, had 4. for his half, and 9 minutes. This half added to 47. degrees. 41. min. (the less altitude) giveth 51. degrees. 50. minute. for the true elevation of the Pole at Oxford.   

Propositio. 4. To rectify the Globe perfectly for to be used.  
KNow first the elevation of the Pole of Heaven, for the place where ye use the Globe, by the third proposition, then erect the Pole of the Globe so many degrees above his Horizon, as the Pole of heaven is elevated. Again level the Horizon of your Globe by his hanging plummet, lastly turn his Meridian to the South by help of his Needle, and put his Quadrant of altitude upon the 90. degree from his Horizon: for then the Meridian answereth to the Meridian of heaven. Axe to axe, and Pole to Pole as is required. But this way of setting him South, albeit it be of antiquity: yet hath it imperfection by reason of the variation of the needle. but of that ye shall hear more hereafter.   

Propositio. 5. To find the place of the Sun at any tyme.  
BY this place, is understood the degree of the Ecliptic line, in which he is, and this place is thus found. In the Horizon of your Globe be set, the winds, the signs, and months with their days. find therefore the day of your month in which ye would have the place of the Sun in the Horizon of the globe. For look what sign and degree of sign is right against the day, the same is the place of the Sun. As on the twelfth day of December: Anno. 1584. I sought the place of the Sun. this day being had in the Horizon, I found the first degree of Capricorn, 32. minutes to answare against it, and therefore that was then the place of the Sun.   

Propositio. 6. To find the declination of any point of the Ecliptic or of the Sun, at any tyme.  
THe declination of any point of the Ecliptic, Sun, or any Star, is the portion of the Meridian Circle, enclosed betwixt the Equinoctial and the say● point, Sun, or star, and is found thus. Turn the point whose Declination yet seek, to the Meridian of the Globe, and there see how many degrees and minutes there be of your Meridian enclosed betwixt the said point and the Equinoctial. For so much is the Declination. so hath the Sun in the 5. of Aries 2. degrees declination. And the 7. of Taurus' 13. degrees. 52. minutes. and this declination is called Northern, when the point is of the North side the Equinoctial, and Southern if of the South side. Here must ye also know, that two points of the Ecliptic want declination, and are the two Aequinoctials. Two have greater declination than anyother, and be the two Solstitials. of the rest four have like declination.   

Propositio. 7. To find the right ascension of the Sun, or any point of the Ecliptic line.  
THE right ascension of any Star, Sun, or any point of the Ecliptic, is the portion of the Equinoctial Circle from the head of Aries, (where the Equinoctial taketh his beginning) and that point or degree of the same, which meeteth with the said Star, Sun, or Ecliptic point, under the Meridian Circle in a crooked Sphere: being numbered orderly in the Equinoctial, and is thus found. turn the Star, Sun, or any point whose ascension ye look, under the Meridian of the Globe, and see then what portion of the aequator is from the head of Aries to that point of the aequator which standeth then under the Meridian: for the same portion, is the right ascension of the Star, Sun, or point looked for. So do I find the right ascension of Boötes a Star, to be 209. degrees. 1. minute. And the right ascension of the Sun when he is in the first of Taurus, to be 27. degrees. 54. minutes. And the right ascension of the first of sagittary, to be 237. degrees. 48. minutes.   

Propositio. 8. To find the crooked ascension of the Sun, Star, or any point of the Ecliptic.  
THE crooked ascension of the Sun, is that Ark of the aequator which is enclosed betwixt the beginning of the aequator, and the point of the same which cometh up with the Sun in a crooked Sphere, & is found thus. Take the Sun Star, or point, whose crooked ascension ye desire: and put him to the East side of the Horizon till it touch: Then mark what part of the aequator is enclosed betwixt the beginning of it, and the point now in the Horizon, for so much is the crooked ascension of the Sun, Star, or point. Th●… do I find the crooked ascension of the Sun in the first of Taurus, to be the 12. degrees. 48. minutes. All this being in the Elevation 52. degree. 0. minute.   

Propositio. 9 To find the difference of ascension, or increase of the day.  
THE Sun being in one and the self same point of the Ecliptic (except in the Equinoctial intersections) hath one degree of the aequator that cometh up with him above the Horizon in any crooked Sphere, and an other, (not the same) that cometh up with him in a right Sphere. And therefore the portion of the aequator, betwixt the point of the said, that cometh up with the Sun in the right Sphere: and the point rising with the same in the crooked Sphere, is called the difference of ascension. As in a right Sphere the Sun being in the first of Taurus, there riseth with him, the 27. degree. 54. minute of the Equinoctial. (Which point also meeteth him under the Meridian in a crooked Sphere: for that the Meridian of any crooked Sphere, showeth the same that the Horizon doth in the right Sphere.) but in the crooked Sphere, where the Pole is elevated 52. degrees, there riseth with the Sun the same day, the 12. degree. 48. minute of the aequator. Subducting now the less from the greater, the difference is 15. degrees. 6. minutes, and is called the difference of ascension. And because the Artificial day of the crooked Sphere, is longer or shorter than the Equinoctial day by twice this difference: therefore the difference of ascension is called also the increase of the day. And this difference is thus found. Find the right ascension of the Sun by the 7. proposition: and again find his crooked ascension by the 9 proposition: then subduce the less from the greater, for the remains is the difference of ascension.   

Propositio. 10. To find the length of the Artificial day in any Region or Country.  
Find out the difference of ascension of the place of the Sun by the 9 proposition, and double the same then convert is all into hours and parts of hours, allowing for one hour 15. degrees, and for a half 7. degrees. 30. minutes, etc. This time which cometh of the difference of ascension add to 12. hours (if the place of the Sun be any degree betwixt Aries and Libra: or subduct it from 12. if he be betwixt Libra and Aries,) for the number made or left, is the length of the day. As the Sun being in the first of Tanrus, his difference of ascension is 15. degrees. 6. minutes: this double and converted into time, maketh 2. hours and 12. equinoctial minutes. And because Tanrus is a Northern Sign, ye must add this difference to 12. hours, so do ye make 14. hours and 12. Equinoctial minutes, for the length of that whole day.   

Propositio. 11. To find the hour of the Sun rising, or of his setting.  
Know the length of the Artificial day by the 10. proposition: and take half of the same day, for that showeth the hour of Sun setting. But if ye reckon so much from noon forward, it giveth Sun rise. As the Sun being in the first of Taurus, the day is 14. hours and 12. minutes. The half is 7. hours and 6. minutes. I say then the Sun setteth after 7. of the clock, 6. minutes. Again, thus much taken from noon forward, showeth the Sun to rise before 5. of the clock, 6. minutes.   

Propositio. 12. another way to find the same more mechanically.  
Find the place of the Sun by the 5. proposition, and turn the said place directly under the Meridian: them put the Index of the hour Circle precisely on 12. of the clock. Lastly, turn the said place of the Sun to the East side of the Horizon: for when he is there, then shall the Index show the time of the Sun rising. And contrariwise, putting the place of the Sun to the West, it showeth his setting.   

Propositio. 13. To find how far the Sun riseth or setteth from the true East or West point any day.  
FIrst find the place of the Sun by the 5. proposition: then turn the same place to the East side of the Horizon till he touch the same, for then the number of degrees in the Horizon, (enclosed betwixt the true East point and the place of the Sun,) show how far he riseth and setteth from the true East: And this portion of the Horizon is called his breadth of rising: and is called Northern breadth if the Sun rise beyond the East point toward the North, & Southern if contrary. Likewise are ye to know, that of the Ecliptic two points Aries and Libra have no breadth of rising. Two points also as Cancer and Capricorn have greater than any other: and of the rest four points have the like.   

Propositio. 14. To rectify the Index of the hour Circled every day as he ought.  
Find the place of the Sun every day in which ye use the Index, by the 5. proposition, and put the said place under the Meridian: this being done, them put the Index on 12. of the clock, for afterward in the motion of the Globe he will go true as he ought.   

Propositio. 15. To find the noon height of the Sun for any day to come, or gone in any place whose elevation is known.  
THe height of the Sun, is the portion of the vertical Circle enclosed betwixt the Centre of the Sun and the Horizon. But for as much as at noon the Meridian, and the Vertical of the Sun be all one Circle: therefore his noon height is the portion of the Meridian betwixt the Centre of the Sun and the Horizon. this height is thus to be known. Find the place of the Sun for the day proposed, and turn the same place under the Meridian, for then the portion of the Meridian betwixt the said place and the Horizon is his noon height. Thus found I the height of the Sun at noon in Oxford, whose Pole is raised 51. degrees. 50. min. on the 2. day of May to be 59 degrees. 47. mi. and on the twelfth of june, to be 61. degrees. 41. minutes.   

Propositio. 16. To find the depression of the Sun at midnight.  
AS the Meridian altitude is the portion of the Meridian from the Centre of him, to the Horizon when he is above the earth: so is his depression the part of the Meridian betwixt the Centre and the Horizon when he is under ground, and may thus be known. Find the place of the Sun, and put it to the Meridian under the Horizon: for then the portion of the Meridian betwixt it & the Horizon, showeth his depression. So find I the depression of the sun, at Oxford (his place being the first of Taurus) to be 27. degrees. 40. min. but his place being the first of Scorpius, to be 50. degrees. 0. min.   

Propositio. 17. To find what height the Sun shall have at any certain hour of any artificial day.  
TAke the place of the sun by the 5. proposition: & rectify the Index by the 14. pro. then turn the Globe, till the Index of the hour circled be on the hour, for whom ye desire the height of the Sun, and scopping the Globe there, put the quadrant of altitude to the place of the Sun, for his portion betwixt the place of th● Sun & the Horizon, giveth his height. So find I the height of the Sun at Oxford, at 9 of the clock the 7. day of March. to be 24. dedrees. 25. min. and at one of the clock the same day, to be 34. degrees. 51. min.   

Propositio. 18. By any height of the Sun given and his place: to find the hour of the day.  
LEt it be, that either ye take the height of the Sun at some time of the day by the second proposition: or that ye have some height of him given by supposition, and ye would know by it what it is of the clock that day at that time. Find therefore the place of the Sun for that day▪ by the 5. proposition & rectify the Index by the 14. proposition. Lastly put the place of the Sun to the Quadrant of Altitude, and move them both up and down, till ye allow him the same height in your Quadrant, as ye found or supposed him in truth to have. For then the Index of the hour Circle showeth what was or is of the clock, as finding the height of the Sun before Noon on the seventh of March, at Oxford, to be twenty four degrees. 25. min. I found it to have been then nine of the clock.   

Propositio. 19 By the hour known, and the height of the Sun at that hour: together with the Index, rectified as he ought: to find the place of the Sun at that tyme.  
Move your Globe till his Index stand on the hour which was known before. Then fix the Globe for removing: Lastly turn your Quadrant of altitude to the Ecliptic line, and look what degree of the Ecliptic agreeth in your Quadrant with the height that was before known, and that is the place of the Sun on that day.   

Propositio. 20. The hour and place of the Sun being given: to find how far the Sun is gone from the true East point.  
THe place of the Sun being given by supposition: rectify the Index by the 14. proposition: then turn the Globe till the Index show the hour given. This being done, fix the Globe that he move not away, and set the edge of the Quadrant of altitude to the place of the Sun: and withal mark how many degrees of the Horizon are enclosed betwixt the true East point, and the edge of the Quadrant, at such time as he standeth on the place of the sun: for so much is he distant in the Horizon from true East.   

Propositio. 21. The distance of the Sun being given, from true East, together with his height at the same time, and the height of the Pole for the same region: to find the true place of the Sun, at any time.  
TO the end we make not unnecessary repetitions of the first principles: know this, that in all the propositions following, we always suppose before the working, the Globe rightly rectified as is specified in the beginning. For the performance therefore of this practice: first consider diligently in what quarter of the year ye be in▪ that is, whether it be betwixt the equinoctial of March, and height of Summer: or betwixt height of Summer, and equinoctial of September. Likewise whether betwixt equinoctial of September and dept of Winter: or betwixt dept of winter and equinoctial of March. For then set the edge of the quadrant of altitude at the true distance of the Sun from the East: and turn the Globe till that quarter of the Ecliptic come under him, which serveth for the quarter of the year in which ye be: and see what degree of that part of the Ecliptic agreeth with the height proposed: For that is the place of the Sun at that time.  
Note therefore here, that to the Spring (which is from the equinoctial of March till the height of Summer) answereth the part of the Zodiac from Aries to Cancer. To summer which is from the height till the equinoctial of September: answereth the part from Cancer to Libra. The Autumn is guided by the quarter from Libra to Capricorn: and Winter by the signs from Capricorn to Aries.   

Propositio. 22. The distance of the Sun being given from true East, and the place of the same: to find the height of the Sun which he hath at the same time.  
PLace the quadrant of altitude at the true distance from East, so shall he cut the place of the Sun by the 21. proposition: and therefore the portion of the Quadrant betwixt the place of the Sun, and the Horizon, is his height.   

Propositio. 23. The distance of the Sun from true East being given, and his place: to find the hour of the day.  
FIrst having his place: rectify your Index by the 14. proposition: again setting the Quadrant of altitude in the distance from true East, reduce the place of the sun, till he fall in the edge of the Quadrant, for then the Index doth show the hour.   

Propositio. 24. The distance of the Sun being given from true East, and his height, to find the time of his rising.  
THe distance being given, find his place by the 21 proposition: and then rectify the Index by the 14 proposition: Lastly put the place of the Sun to the East side of the Horizon: for then the Index will show the Sun rising.   

Propositio. 25. The distance of the Sun being given from true East, and his height, to find his Declination.  
THE distance being given, his place is found by the 21. proposition: & his place being known, giveth his Declination by the 6. proposition: So may we likewise by the said distance (finding his place) find his right or crooked ascension, or difference of ascensions, and length of Artificial days.   

Propositio. 26. The declination of the Sun being known: to find the place of the Sun.  
COnsider first diligently in what quarter of the year ye be in, as was expressed before: then take that quarter of the Ecliptic which answereth to your quarter of the year: and move it still under the Meridian of your Globe, till ye find no more of the Meridian enclosed betwixt the aequator and Ecliptic, than the declination that is given cometh unto: for then look what degree of the Ecliptic is under the Meridian, that is the place of the Sun. As the declination of the Sun in the quarter of the year betwixt the Equinoctial of March, and height of Summer was given to be 11. degrees. 50. minutes. And to this quarter of the year, answereth the quarter of the Ecliptic from Aries to Cancer. Therefore moving the said quarter under the Meridian, I found the first of Taurus to answer to this declination: and therefore that was the place of the Sun.   

Propositio. 27. The declination of the Sun being known: to find the day of the month.  
BY the declination given, find the place of the Sun by the 26. proposition: them take the said place in the Horizon of your Globe: for look what day answereth against it, that is the day of the month.   

Propositio. 28. The day of the month being known, to find the length of the Planetary hour.  
THE Artificial day is from Sun rise to Sun set: and the 12. part of this day, whether it be longer or shorter than an hour by the clock, is the Planetary hour: and may thus be known. The day being gi●en, find the length of that day by the 10. proposition: and divide all by 12. The Quotient is the length of a Planetary or Artificial hour of that day. As the day being 15. hours by the clock, I divide it by 12. the Quotient is one hour and a quarter, and so much is a Planetary hour of that day.   

Propositio. 29. The day of the month being given: to find the dawning of the day.  
BY the day known, find the place of the Sun by the 5. proposition: and then rectify your Index by the 14. proposition. Again, take the degree of the Ecliptic which is opposite in a Diameter to the place of the Sun: and move him toward the West together with the Quadrant of Altitude, till ye have 18. degrees of height: for then the Index showeth the beginning of the dawning or spring of the day.   

Propositio. 30. To find the length of the whole dawning.  
Find the beginning of the dawning by the 29. proposition, and then the Sun rise by the 11. or 12. proposition: for the difference of those times is the whole dawning. And thus far have I followed such conclusions, as have a more orderly coherence: it remaineth now to show some others, whose coherence is not so natural.   

Propositio. 31. another way to find the length of the Artificial day or night.  
Find the time of the Sun rising for your day proposed by the 12. proposition: then double all those hours and parts of time which be from Sun rise till noon, for it giveth the Artificial day. Or if ye number all the hours and parts from Sun rise to his setting, it giveth the same.   

Propositio. 32. To find the hour of the day.  
PLace the Globe in the Sun shine, and rectify him to his use by the 4. proposition, then find the place of the Sun by the 5. proposition. Again, rectify his Index by the 14. proposition. Lastly, 〈◊〉 the needle or pin directly up in the place of the Sun: then turn the Globe up till the pin cast no shadow, for then the Index showeth what is then of the clock.   

Propositio. 33. To find the elevation of the Pole, in any place.  
Draw in the open air upon some table that is level, a Meridian line by the 1. proposition, and place the Globe so on it, that his Meridian Circle hang directly ●uer it: then having the place of the Sun, set a pin right up in it, and put the said place and pin close to the Meridian circle. Lastly, lift up the Pole and Meridian Circle, till the pin cast no shadow: for then the degrees betwixt the Pole and the Horizon, be the true elevation of that place. But this practice is to be performed at noon only or height of the day.   

Propositio 34. another way to do the same.  
TAke the height of any fixed Star (whom ye know) by the 2. proposition, at such time as he pointeth with the Meridian line: then take the same Star on the Globe, and by help of your Quadrant or Meridian Circle, cause him to have the same Altitude in the Globe, and withal to be under the Meridian of the Globe: for them is the Pole at his true Elevation. So did I find the Pole Star (making my observation at Oxford, the 11. of December 1584.) by the plain Sphere, to have 55. degrees▪ 59 minutes in Altitude, being then in the Meridian of Heaven: and when I set him at the same in my Globe, I found the Pole elevated there 51. degrees. 50. minutes. And here ye are to know, that when soever ye have by any way, the elevation of the Pole in any place: if ye subduct the same elevation from 90. degrees, it shall leave and she●… the elevation of the aequator in the said place. So then the elevation of the aequator at Oxford, is 38. degrees. 10. minutes.   

Propositio. 35. another way of working the same, with more praecisenes.  
FIrst learn by some good Ephemeris the precise place of the Sun at noon in the day of your observation: then again learn the exact declination of the said place. Lastly, with your 〈◊〉 take the Meridian height of the Sun that day: And if the declination be Northern, then subduct it from the Meridian Altitude: but if it be Southern, then add it to the Meridian Altitude: so shall we bring forth the Altitude of the aequator: and this Altitude being subducted from 90. degrees, leaveth the Altitude of the Pole: but if the Sun in the time of observation be in the Equinoctial point, then is the Meridian Altitude, the Altitude also of the aequator, and it subducted from 90. degrees, leaveth the Altitude of the Pole.   

Propositio. 36. To make a horizontal Dial by the Globe.  
A horizontal Dial is such a one as is made in a plain Superficies, and lieth level with the Horizon. For making whereof ye are to consider, that from one Pole of the Globe to the other go twelve great Circles, called hour Circles, and divide the aequator into 24. equal parts: And two of these be two Colours. Put therefore the Solstitial Colour precisely under the Meridian of your Globe, (the Globe being first perfectly rectified) and fix the Globe so that he cannot move. Now mark how many degrees of the Horizon are enclosed betwixt the Meridian and the next hour Circle toward the East (which for distinction sake I call the second hour Circle) so likewise betwixt the first & third, the first & fourth, the first and fifth, the first and sixth, the first and seventh: (which is he that cutteth in the true East point) and set them all down in tables: then draw on some plain thing a Circle, and divide it into four quarters, by drawing two cross lines: Now take the one end of any of the two lines, and term it the North point, so shall his other end be the South point, and the ends of the other line East and West. Again, divide that quarter of this Circle which is betwixt the North point and East into 90. equal parts, and let 90. stand at the East. So do by the quarter betwixt North and West. Lastly, reckon from the North point toward East, so many degrees as your tables show to have 〈◊〉 betwixt the first and second hour line: and from the point where they end, draw a line by the Centre of the said Circle: and so do by all the numbers of your tables: for so shall ye have your hour lines drawn for a horizontal Dial. In whose Centre must be a style erected, according to the elevation of your Pole. But this I leave obscure, as meaning to set out an ample treatise of Dialling by itself.   

Propositio. 37. How the Stars may be known by the Globe of Heaven.  
Rectify your Globe in the open air by the 4. proposition, them take the height of any known Star by your Instrument, afterward look the same Star on the Globe, and by help of your Quadrant of Altitude put the same Star at his height taken before, and in the same Coast, & then fix the Globe. Now if ye would know any other Star of Heaven, then take the same Star his height with your Instrument: lastly, turn your Quadrant of Altitude toward the same Coast of the Globe in which the Star was in: & look what Star ye find in that Coast, to have that Altitude, the same is he whom ye seek. The like is to be done by all others.   

Propositio. 38. To find the Longitude of any fixed Star.  
THE Longitude of a Star, is the portion of the Ecliptic line, taken from the head of Aries, (according to the order of the Signs) to the point of the Ecliptic, cut by a Circle which passeth from the Pole of the Ecliptic, by the Centre of the said Star: and is thus found. Take the Globe from his Horizon, and take of his Meridian Circle, and fix the same Circle by some means on the Poles of the Zodiac, then turn the Star whose Longitude ye seek, under the Circle: and reckon all the Signs and parts from the head of Aries, to that point of the Ecliptic which is under the Circle with the Star: for so much is his Longitude. And the same point of the Ecliptic which is so under the Circle, shallbe called the place of that Star. And the Star is said to be under that Sign, of whom the aforesaid point is a part. The Longitude may also be taken, if ye do but fix the Quadrant of Altitude in the Pole of the Ecliptic, and stir not at all the Meridian Circle.   

Propositio. 39 To find the Latitude of any Star.  
THE Latitude of a Star, is the portion of the Circle that passeth from the Poles of the Ecliptic line, by the Centre of any Star, which is enclosed betwixt the Ecliptic line and the Centre of the Star, and is found thus. Your Circle standing in the Poles of the Zodiac as before: turn the Star under the said Circle: for then the portion of that Circle betwixt the Star and the Ecliptic, is his Latitude. And this Latitude is Northern, when the Star is North from the Ecliptic: and Southern of contrary.   

Propositio. 40. To find what Stars be above ground at any time of the day or night.  
IF ye would know it in the day time when the Sun shineth, then take the height of the Sun by the 2. proposition: afterward find his place by the 5. proposition: lastly, by help of the Quadrant of Altitude, set the Sun at his own height & Coast, and then all the Stars above the Horizon do appear in the Globe. Now if it be in the night, and the Stars appear, then take the height of some known Star, and place the same Star at his due height in the Globe, so shall ye see the same.   

Propositio. 41. To do the same without Sun, or appearance of any Star.  
YE must know what it is of the clock at that time when ye would work this conclusion: then rectify the Index by the 14. proposition: Lastly, turn the Globe till the Index come on the same hour as is given by the clock, for then the Stars appear as they should.   

Propositio. 42. To find what Stars will pass directly over our heads in the motion of the heaven.  
AFter that the Quadrant of Altitudes is fixed in his due place, as is spoken of before, so that he now do show the Zenith or Vertical point: then move about the Globe, and mark what Stars pass under the Zenith in this motion, for those be such as go by our heads, & are called sometimes Culminant stars, sometimes Vertical stars, and have their chiefest use in Astrology.   

Propositio. 43. To know with what degree of the Ecliptic any Star rised, cometh to the Meridian: or setteth.  
Move the Globe till the Star whom ye propose, ascend above the Horizon, and then mark the degree of the Ecliptic that riseth with him. Again move him to the Meridian, and mark the degree of the Ecliptic, so do by him in the West side of the Horizon, and ye shall have your intent.   

Propositio. 44. To know the hour of any Stars rising.  
Rectify the Index by the 14. proposition, then turn the Globe till the said Star (whose time of rising ye desire,) touch the East side of the Horizon: for then the Index giveth his time of rising. And if ye turn him to the Meridian, the Index will show his time of coming the there: or if ye turn him to the West side of the Horizon, the Index showeth his setting.   

Propositio. 45. To find in how long time any whole sign ariseth.  
Rectify the Index by the fowerteenth proposition: then put the beginning of the sign (whose time of rising ye seek) to the East part of the Horizon: and mark what the Index standeth on then: again, put the last degree of the said sign to the Horizon, and see what the Index showeth: for the difference of the two times by the Index, is the time in which that sign riseth.   

Propositio. 46. To find in what coast any star is, and how many degrees from the Meridian.  
Find the Stars above ground by the 40. proposition: then the Globe being fixed: put the Quadrant of altitude to any State. Then shall the foot of the Quadrant show in the Horizon, how far the same Star is from East, West, North or South. But if ye first rectify your Index by the 14. proposition: and then find the stars above by the 40. proposition. Again at the same time mark where the Index standeth: lastly put any Star under the Meridian, and again note the standing of the Index: the difference of those two times showed by the Index, is the distance of that star from the Meridian, in the time of your observation.   

Propositio. 47. To find what Stars rise or set any day, Cosmically, Achronically or Helically.  
Such stars as be near to the sun in any day, and ascend above the Horizon a little before the appearance of the same: are said to rise helically, and such Stars as set very little after the Sun, are said to set helically. Again such Stars as ascend together with the Sun, and such as set at the same time, are said to rise and set cosmically. Lastly, such stars as set together with the Sun: and such as rise at the same time: are said to set and rise achronically, and such may be thus found. Rectify the Index by the 14. proposition: and turn the place of the Sun to the East side of the Horizon: for the stars going immediately before the Sun, rise helically. and those in the Horizon rise cosmically: and they that are in the Horizon in the west, do set cosmically, and such as immediately rise after the Sun, do set helically. Lastly turn the Sun to the West point of the Horizon, and look what stars touch the Horizon with him, such set achronically: and such as are at the same time rising in the East, rise Achronically.   

Propositio. 48. To know the hour of the night, at any time by the Stars.  
Rectify the Index by the 14. proposition, than again find what stars be above ground at the same time, when ye would know the hour, by the 40. proposition: for then the Index will show the hour.   

Propositio. 49 To find the four Cardinal points of heaven at any time of day or night.  
THe four Cardinal points, be the four degrees of the Ecliptic, where of one is in the East rising: an other is in the South or under the Meridian, above at the same time: the third in the West setting, and the fourth under the Meridian beneath ground, all at one instant, in the time of any geniture, or motion of any question, and are thus known. If ye seek them in the day (the Sun shining) then find the stars above ground by the 40. proposition: and with all mark the degree of the Ecliptic in the East. so likewise in the South, West and North, for those be then the Cardinal points. Again, if it be in the night, then find the stars above by the same 40. proposition. and the points shall likewise appear. Lastly, if neither the Sun shine, or star: then know the hour by the clock, and afterward find the stars above ground at the said hour by the 41. proposition. so shall the points be given as before.   

Propositio. 50. To find the bigness of the angle, made betwixt the Meridian Circle, and any Circle of position.  
Circle's of Position be all such, as are drawn from the North point of any Horizon by the Centre of any star, and so to go to the South point of the same Horizon, to return to the North again. And every one of these Circles doth make some with the Meridian, and the said angle hath his bigness showed, by a portion of the fixed Vertical: so that to find the bigness of the angle made betwixt the Meridian and any Circle of position: is to find the portion of the fixed Vertical, enclosed betwixt the Meridian and the said Circle of position. that portion is thus found. Put your quadrant of altitude, to the true East point, then raise up your Brass half Circle as high above the Horizon as ye please: so that it may now represent some circle of position. for then the degrees of the Quadrant of altitude from the Meridian to this circle: be the bigness of the angle made betwixt the Meridian, and the Circle of position. but if your circle of position fall on the West side of the Meridian, than put the Quadrant to the West point; and work as before.   

Propositio. 51. To find the beginnings and ends of the 12 houses of Heaven.  
COncerning the erecting the scheme of heaven, or as we commonly call it the twelve howsen: though four divers ways have been received, touching the howsen, how they ought to be taken: yet it is not our intent to discourse of that question, but to show how they ought to be erected, according to the most usual way, set down by Regiomontanus, & called reasonable. Wherefore first ye are to know, that in any Horizon wheresoever we be, we do imagine six circles to be drawn from the North point of the Horizon to the South of the same, and dividing the Equinoctial into 12 equal parts, and the 12. spaces betwixt these circles, are called the twelve houses. (& two of the 6. circles are always the Meridian and Horizon.) in every one of these howsen is enclosed some portion of the Zodiac, and one portion is greater than an other. so that to erect the twelve howsen, is to find out the portion of the Ecliptic enclosed in each space, & to do it, we thus proceed. First find out the four Cardinal points by the 49. proposition, for those be the beginnings of 4. howsen of the twelve: the Cardinal point under the Meridian above ground, is the beginning of the tenth house. This done, fix the Globe, then reckon from the degree of the aequator (being then under the Meridian) 30. degrees toward the East point, & raise up your brass half circle to stand on the point of the aequator on which ye left. For look then what degree of the Ecliptic is cut then by the brass half Circle: the same is the end of the tenth house, and beginning of the eleventh. Again, yet reckon 30. degrees more in the aequator toward the East, and put the brass half circle to it, and then take the degree cut in the Ecliptic, for that is the end of the eleventh house, and beginning of the twelfth. Again the Cardinal point of the East, is the end of the twelfth house, and beginning of the first house. Now if in like sort ye go from the degree of the aequator under the Meridian, by each 30. degree of the same toward the West point, and still observe the degrees cut in the Ecliptic: ye shall have the beginnings and ends of the ninth, eighth and seventh house. Thus having erected six howsen, the degrees of the Zodiac which are opposite to these in a Diameter, (one to an other) be the beginnings and ends of the other six howsen, which were to be found. And here must ye note, that the first house beginneth at the East point, and goeth under the ground toward the Meridian Circle, the second and the third succeed. the fourth, beginning at the Meridian under ground coming toward West, the fifth and sixth succeed, the seventh beginneth in the West, and goeth above ground toward the Meridian, the eight and ninth succeed. Other conclusions less profitable I wittingly avoided: and the more excellent, deferred to a more convenient time.  
FINIS.     

Errata.  
Pa: 3. li: 13. indicem, lege judicem.  
Pa: 23. li: 1. same, lege Sun.  
Pa: 24. li: 1. lege Ver.  
Pa: 46. li: 16. till ye, league till it.