THE DESCRIPTION AND USE OF THE SPHERE. Divided into three principal Parts. WHEREOF The first intreateth especially of the Circles of the uppermost movable SPHERE, and of the manifold uses of every one of them severally. The second showeth the plentiful use of the uppermost Sphere, and of the Circles thereof jointly. The third containeth the Description of the Orbs whereof the Spheres of the Sun and Moon have been supposed to be made, with their Motions and Uses.  
By EDWARD WRIGHT.  
The Contents of each Part are more particularly set down in the Table.  
LONDON, Printed by B. A. and T. Faucet, for john Tap, and are to be sold at his Shop at S. Magnus' corner. 1627.   

A TABLE OF THE CONTENTS OF this Book.  

The first Part. Of the Circles of the uppermost SPHERE, and their several uses.  

THe Definition and division of this Sphere, Chap. 1.  
The description of the Horizon, Chap. 2.  
The uses of the Horizon, Chap. 3.  
The description of the Meridian, Chap. 4.  
The uses of the Meridian, Chap. 5.  
The description of the Houre-circle, and Poles of this Sphere, Chap. 6.  
Of the Equinoctial circle, and why it is so called and how divided, together with his manifold uses, Chap. 7.  
The description of the Zodiac of this Sphere, Chap. 8.  
The uses of the Zodiac, Chap. 9  
The description of the two Colours, together with the uses common to them both, Chap. 10.  
The uses of the Equinoctial colour, Chap. 11.  
The uses of the Solstitial colour, Chap. 12.  
The description of the two Tropickes, Chap. 13.  
The uses of the Tropickes, Chap. 14.  
The Polar circles, Chap. 15.  
Uses of the Polar circles, Chap. 16.  
Of the Zones, Chap. 17.  
The difference of shadows that the Sun maketh in these Zones, Chap. 18.    

The second Part. Of the uses of the uppermost Sphere, and of the Circles thereof jointly.  

TO rectify the Sphere, that is to set the Sphere to the Latitude of that place for which you would use it, Prop. 1.  
To know▪ the place of the Sun by this Sphere, Prop. 2.  
To know the declination of the Sun, or of any point of the Ecliptic, Prop. 3.  
To know the right Ascension of the Sun, or any point of the Zodiac, Prop. 4.  
To know the oblique ascension of the Sun, or of any Star or point in the Zodiac, Prop. 5.  
To find the difference of Ascension, Prop. 6.  
To find at what time the Sun riseth or setteth, Prop. 7.  
To find the length of the Artificial Day or Night, Prop. 8.  
To know the time of the Sun rising, or Sun setting, Prop. 9  
To find the length of the artificial Day or Night, otherwise by the Sphere, Prop. 10.  
To know the Meridian altitude of the Sun at any place, whose latitude is known, Prop. 11.  
To know how high the Sun is about the Horizon at any time of the day, Prop. 12.  
To find the hour of the day by the height of the Sun, the place of the Sun, and height of the Pole being given, Prop. 13.  
To find the breadth of the Sun's rising or setting, that is, how far he riseth or setteth from the point of true East or West at any time, Prop. 14.  
To find the place of the Sun, his declination, and the quarter of the year being known, Prop. 15.  
To find what day of the month it is by knowledge of the Sun's declination, Prop. 16.  
The day of the month being known to find at what time the day breaketh, Prop. 17.  
To find how long the twilight continueth, Prop. 18.  
To find how much the declination of the Sun must alter at any time of the year, to make the day one hour longer or shorter, Prop. 19  
To find how many days it is ere the day lengthen or shorten an hour, Prop. 20.  
To make an horizontal Dial by the Sphere, Prop. 21.  
How to make a direct mural Dial by the Sphere, Prop. 22.  
To make any direct inclining, or direct reclining Dial by the Sphere, Prop. 23.  
To know at what time the Moon, or any other of the Planets, or fixed Stars that are within the breadth of the Zodiac rise or set, or come to the Meridian: as also with what degree of the ecliptic they rise set or mid Heaven, together with their declinations, and their right and oblique ascensions, and descensions, and their amplitudes or bredths of rising or setting, Prop. 24.  
To know how long the Moon, or any of the Planets or fixed Stars do shine, or continue above the Horizon, Prop. 25.  
To find which of the Planets of fixed Stars (that are within the compass of the Zodiac) are above or under the Horizon at any time of the day or night, Prop. 26.  
To find in what time any Sign, or part of the Ecliptic, riseth or setteth, Prop. 27.  
To find the hour of the Night by any of the Planets, or fixed Stars in the Zodiac, that appear above the Horizon, Prop. 28.  
To know at any time of the year, what Stars in the Zodiac arise or set Cosmically, Achronycally, or Heliacally, Prop. 29.     


The Meridian Line.    

Of the use of the SPHERE and GLOBE. Pars, 1.  
The Description of the Sphere and Globe, divided into three principal parts..  
Whereof this first intreateth specially of the Circles of the uppermost movable Sphere and of their peculiar uses.  

CHAP. I. The definition and division of the Sphere.  
THis Sphere, is nothing else but a representation of the Celestial orbs and circles, that have been imagined for the easier understanding, expressing, and counting of the motions and appa●ences, either common to the whole Heavens, or proper to the Sun and Moon.  
The circles of this Sphere are either inmoveable, as the two greatest and utmost circles, the Horizon and Meridian, (whereto is adjoined the little hour circle that is fixed to the Meridian) or else movable▪ as all the rest contained within these.   

CHAP. II. The Description of the Horizon.  
THe greatest and utmost circle of the Sphere that lieth level on all sides from the ground, is called the Horizon, which is divided into 7. limbs, or borders. The first and utmost of them containeth the 32. points of the Compass, or the winds (as they are at this day divided and used by Seamen) with their Latin names adjoined unto them. The second limb containeth the names and divisions of the 12. Winds as they were wont to be divided in old time. The third is divided into the months and days of the new Calendar, first established by Pope GREGORY the XIII. and now used in many places beyond the Seas. In the fourth limb are set down the months and days of the ordinary Calendar used in England. Next within this, are placed the 12. Signs and degrees of the Zodiac, that so the place of the Sun might be presently known for any day of the year given, or chose, that the day of the Month might be readily found by the place of the Sun. After this, followeth the sixth limb containing the 32. Winds or points of the Compass, with letters representing the names now in use amongst English Mariners. The seventh and last limb next the innermost edge of the Horizon, is divided into 360. degrees with figures set to every tenth degree, beginning from the points of East and West, and ending at North and South; that so the number of any degree of the Horizon might be the easilier known: Which Circle appeareth most plainly to them that are in a plain Champion Country, or upon the Sea close by the water in a clear calm day.   

CHAP. III. The uses of the Horizon.  
1. IT deuideth the upper and visible part of the Heavens from the neither half that is hidden out of our sight.  
2. It showeth partly the difference of a right and oblique Sphere, for when this circle and the Equinoctial, cross each other at the right Angles, it is said to be a right Sphere, otherwise when they make oblique Angles one with another, it is called an oblique Sphere.  
3. In an oblique Sphere this circle severeth those Stars which never rise nor set, but are always either above or beneath the Horizon, from such Stars as rise and set in every 24. hours. For all the Northerly stars that are no further distant from the North pole, than the North pole is from the Horizon, do never set, but are always above the Horizon: And chose, those Stars that be about the South pole, no further distant from it than it is from the Horizon, do never rise, but are always hidden out of sight under the Horizon.  
4. In respect of this circle, the Sun, Moon and Stars, or any other part or point of the Heavens, are said to rise or set: For when they come up from under the Horizon, they are said to rise; otherwise when they go from above the Horizon down underneath the same, they are said to set.  
5. And hereof it cometh that the ascendent, and descendent are found by this Circle: for that part of the Ecliptic that is at the East part of the Horizon arising, is the Ascendent; and the point opposite to this at the West part of the Horizon, may be called the Descendent.  
6. This Circle partly showeth the difference of ascension of any part or point of the Heavens.  
7. In this Circle we reckon how far the Sun, the Moon, or any Star, or point of Heaven, ariseth from the point of due East.  
8. The Horizon determineth the time of the artificial day and night: for we call the time wherein the Sun abideth above the Horizon, an artificial day: And the time that he continueth under the Horizon, is the artificial night.  
9 This Circle showeth the reason of the equality of artificial days and nights, in a right Sphere: and of the inequality of them in an oblique Sphere. For in a right Sphere, the Horizon deuideth all the parallels of the Sun or Circles of the natural days, into equal parts: But in an oblique Sphere, it deuideth them into unequal parts.  
10. By means of this Circle, we know what Stars, and what Eclipses, Conjunctions, or other aspects of the Planets may be seen in our Hemisphere at any time.  
11. From the Horizon is measured the twilight: For in the morning the Sun being under the Horizon about 18. degrees of the vertical, Circle, the twilight beginneth: And when the Sun is so much under the Horizon at Evening, the twilight endeth.  
12. This Circle is of especial use in Geography, for from it we begin to account the elevation of the Pole, and of the Equinoctial circle, whereby the Latitude of any place is known.  
13. In Astrology for erecting a figure, this Circle showeth the beginning of the first and seaventh Houses.   

CHAP. FOUR The description of the Meridian.  
NExt the Horizon, succeeds the Meridian standing upright on edge, and crossing the Horizon at right angles in the points of North and South. This circle is divided on both sides at the inner edge into 360▪ Degrees, with figures set to every tenth degree, beginning at the Equinoctial, and ending at the Poles with 90. and beginning also at the Poles, and ending at the Equinoctial with 90. The numbers beginning at the Pole, serve to set the Sphere readily to any elevation desired. The other numbers beginning at the Equinoctial, show presently the declination of any degree of the Zodiac, or of any point assigned in the Sphere, one quarter of the Meridian on either side thereof from the Equinoctial to both Poles, showeth the Climates, and the quantities of the longest days.   

CHAP. V. The uses of the Meridian.  
1. IT deuideth the World into two halves or Hemisphaeres: that is, the East and the West hemisphaeres. The Easterly hemisphere is all that part of the world which is on the East-side of the Meridian, and the other half may be called the West hemisphere.  
2. It showeth the North and South parts of the world, for the two intersections of the Meridian with the Horizon, show the very points of North and South. The South point is that which is directly under the Sun at noon: And the point right over against this, is called the North-point.  
3. It deuideth the arches of the Equinoctial, and of all his Parallels, into two equal parts both above and beneath the Horizon.  
4. And therefore it deuideth the artificial Day and Night into two equal parts.  
5. And consequently, it showeth midday and midnight.  
6. In an oblique Sphere it serveth in stead of a right Horizon (that is) an Horizon that maketh right angles with the Equinoctial.  
7. Therefore the Astronomers begin their account of times (which are measured by the equal motion of the Equinoctial) from the Meridian: the principal of which times, is the natural day which is usually begun from midday, or midnight.  
8. This Circle showeth the highest and lowest heights of the Sun and Stars, which is most manifest in those Stars that are always above the Horizon. These heights are called the Meridian altitudes of the Sun or Stars, which heights are chiefly observed by Astronomers and Navigators with great diligence.  
9 In this Circle, we observe the distance of the Tropickes, and the greatest obliquity of the Zodiac.  
10. In this Circle, we observe and count the Latitudes of places, the height of the Pole, and of the Equinoctial. For the height of the Pole or Equinoctial, is nothing else but the arch of the Meridian contained between the Pole or Equinoctial and the Horizon. The height of the Pole is always equal to the Latitude of the place. The height of the Equinoctial is equal to the Compliment of the Latitude and therefore it being substracted out of 90. 〈◊〉 shall remain the height of the Pole.  
11. The Meridian showeth the longitude of places in Geographie.  
12. In the Meridian, are measured the breadth of the Zones and Climates.  
13. This circle in Astrology, showeth the highest and lowest parts of Heaven, which are the beginnings of two principal Houses: that is, the fourth and the tenth houses.   

CHAP. VI The description of the Houre-circle, and Poles.  
THe little Circle fastened to the Meridian is called the Houre-circle, which is divided into 24. equal parts, signifying and representing unto us so many equal hours, whereof both the 〈◊〉 hours are fixed just upon the Meridian, becaus●●●hen the Sun cometh to the Meridian, it is just twelve a clock: the upper XII. serveth for the Day; and the other XII. beneath serveth for the Night.  
The Index, or the Painter in form of an Arrow, fastened upon the Pin that cometh through the midst and Centre of this circle, is made to show and point out the said hours as need shall requite, in the use of the Sphere.  
The use of this hour Circle shall be showed hereafter, when we shall speak of the common use of many circles of the Sphere together. And these two Circles (that is, the Meridian and Horizon) are called immoveable, because they keep themselves always, and in all places over the same parts of the Earth; where as all the rest (contained within these two) move round about altogether with one motion in the space of four and twenty hours.  
This motion (being common to the whole Heavens) is made about two Points or Poles, represented in this Sphere, by the two Wyre pins about which the Sphere is turned; whereof the one that cometh through the midst of the little Circle fastened to the Meridian (which we call the hour Circle) representeth unto us the Pole Arctic or the North Pole: the other because it is opposite to this, is called the Antarctic pole, that is the right opposite, to, or right over against the North pole, which is also the South pole.   

CHAP. VII. Of the Equinoctial Circle.  
THat Circle which compasseth about the midst of the Sphere, 
Why this Circle is called the Equinoctial or Equator.  and is every where of equal distance from both Poles, is called the Equinoctial circle, or the Equator; either because it is equally distant from both Poles of the world; or ●●se because the Sun coming under this Circle maketh equality of days and nights throughout the world.  
It is divided at the utmost edge, or both sides thereof into 360. Degrees, with figures 〈◊〉 to every tenth degree, beginning at the beginning of Aries, and proceeding Eastwards, till you become about to the same point again.  
This Circle hath many uses.  
1. It is the measure of the first 〈◊〉. For this only amongst all the Circle of the Sphere is moved equally both in a right and 〈…〉 Sphere, because ● alone being perpendicular to the 〈…〉 world, about which the Sphere is equally turned, is divided into two halts by every Horizon in the same points.  
2. It is the measure of time; because it measureth the quantity of the artificial and natural days, of which Months and Years are made: It measureth also the quantity of Hours and of other times which the Sun maketh going under the Zodiac. And therefore the degrees of the Equinoctial are called tempora (that is) times.  
3. It showeth the two Equinoctial points in the Ecliptic, cutting the Ecliptic in two places, which are the beginnings of Aries and Libra: and the Sun when he cometh to those two points, is equally distant from both Poles of the World, and maketh equality of days and nights in all places; which happeneth in our time about the 10. or 11. day of March, and the 13. or 14. of September.  
4. The irregularity of the Zodiac, and of all the Signs and degrees thereof, is measured by this Circle. For seeing the most part of the apparences of the first motion are referred to the Zodiac, which is not turned about his own Poles, but about the Poles of the Sphere, and therefore must needs be unequally turned about; it was needful that this inequality should be ruled and measured by some other equal motion.  
5. It deuideth the Sphere into two halves (which they call Hemisphaeres) that is, into the North half or hemisphere, wherein is the North pole, and into the South hemisphere, wherein is the South pole.  
6. So it deuideth the Zodiac into the North half, and the South half; or into the North signs, and the South signs.  
7. From this Circle are numbered the declinations of the Stars, and of the degrees and parts of the Ecliptic, and of any other point of Heaven.  
8. And in this Circle are counted the right ascensions of the same Degrees and Stars, etc. For the right ascension of any star or point of the Heavens, is nothing else but the Arch of the Equinoctial circle contained between the beginning of Aries and the Meridian, the same Star or point being first brought under the Meridian.  
9 In the Equinoctial is counted the ascentionall difference and the oblique Ascension and Descension of any point of Heaven. And from the same Circle is reckoned the distance of the Sun rising from the true East point. For the oblique ascension or descension is nothing else but the arch of the Equinoctial, contained between the beginning of Aries, and that point of the Equinoctial Eastwards, which ariseth or setteth together with the Star or point that is given, in an oblique Sphere. And the difference ascentionall or descentionall is nought else but the arch of the Equator, whereby the right and oblique ascension or descension of a Star, or any other point in Heaven do differ each from other. The distance of the sun's rising from the true East point (which in Latin is called Amplitudo ortiva,) is the arch of the Horizon contained between the Equinoctial and the parallel of the Sun, or his Centre when he riseth.  
10. In Geography we count the Longitudes of places in this Circle; and from it we reckon the Latitudes, in the Globe of the earth, and in Maps, and sea Charts. For the longitude of a place is nothing else but the arch of the Equinoctial circle contained between two Meridian's, whereof one goeth by the canary Lands, and the other by the place that is given, And the latitude of a place is the arch of a Meridian contained between the Equinoctial, and the Zenith of the place that was given.  
11. In Dialling this Circle is of especial use. For by means of it the spaces of the hours are divided in all kinds of Dial's, horizontal, erect, direct, declining, inclining, reclining, etc.  
12. In Astrology the twelve Houses are set out by the equal divisions of this Circle into twelve parts, according to the way devised by Regiomontanus, which way is commonly called rational or reasonable. And this Circle governeth the directions, whereby things to come are artificially foretell.   

CHAP. VIII. The description of the Zodiac.  
THe great broad Circle that compasseth about the Sphere obliquely, coming nearer the Pole of the Sphere in one place then in another, is called the Zodiac.  
Round about through the midst of this Circle, is drawn the Circumference commonly called the Ecliptic line, dividing the whole Sphere, and the whole breadth of the Zodiac throughout, into two equal parts.  
In this Sphere there are represented unto us two Ecliptic lines. The one may be called the middle, or fixed Ecliptic, which keepeth always the same distance or obliquity from the Equinoctial. The other may be called the true or movable Ecliptic, because it maketh not always the same angles of intersection with the Equator, but sometimes greater, sometimes less. For the greatest obliquity of the Zodiac, which not long before Ptolomees time was observed to be 23. Degrees and 52. Minutes; in Copernicus his time, was hardly found to exceed 23. degrees 28. minutes, according to his observation, and therefore he thought that the difference between the greatest and least obliquity of the Zodiac, was 24. Minutes: and the middle or mean obliquity between both these, to be 23. Degrees 40. Minutes.  
The manner of the variation of this obliquity may in some sort be showed by this Sphere, if we suppose the fixed Ecliptic drawn round about through the midst of the Zodiac to be 23. degrees 40. min. distant from the Equinoctial at the beginning of Cancer and Capricorn: and the movable Ecliptic (fastened as it were upon two Poles at the beginning of Aries and Libra, and so having always the same points of intersection with the middle Ecliptic and Equinoctial) to be moved up and down above and beneath the middle Ecliptic, by the space of 12. Minutes at the beginning of Cancer and Capricorn: and this motion to finish his revolution once in 3432. julian years.  
The breadth of the Zodiac is bounded by the greatest latitudes of the Planets, especially of Venus and Mars, which sometimes hath almost 7. degrees of latitude.  
The Zodiac is divided by the Equinoctial into two semicircles.  
The one above the Equinoctial is called the Northerly semicircle: the other half beneath the Equinoctial, is the Southern semicircle of the Zodiac.  
So long as the Sun moveth under the first of these semicircles, the days are longer than the nights, otherwise they are shorter.  
Each of these semicircles is again divided into two parts, and so the whole Zodiac into four quarters: the first from Aries to Cancer, may be called the vernal or Spring-quarter, which in this Sphere is also showed by the word Ver (signifying the Spring:) The next from Cancer to Libra, the Summer quarter, wherein is written the word Ae●tas signifying the Summer. The 3. from Libra to Capricorn, is the Harvest quarter, wherein you shall find in this Sphere the word Autumnus which signifieth Autumn or Harvest. The fourth and the last, from the beginning of Capricorn to Aries, is called the winter Quarter, which in this Sphere is showed by this word Hiems, which signifieth the Winter. And these four quarters of the Zodiac are thus called by the names of the Quarters of the year, because the Sun moving under those quarters of the Zodiac, maketh those four Quarters of the year. Every one of these quarters of the Zodiac is again divided into three parts, and so the whole compass of the Zodiac into 12. which are called the 12. Signs, whereof every one containeth 30. Degrees in length from West to East, and is in breadth equal to the breadth of the Zodiac. These Signs, and the Zodiac itself have their beginning from that common meeting, or crossing of the Ecliptic, and the Equinoctial, where the Ecliptic beginneth to arise above the Equinoctial towards the North pole: and they are called by these names; Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, sagittary, Capricorn, aquary, Pisces. That is to say, The Ram, the Bull, the Twins, the Crab, the Lion, the Virgin, the Balance, the Scorpion, the Shooter, the Goat, the Waterpourer, the Fishes. And they are signified by these Characters, ♈ ♉, ♊, ♋, ♌, ♍, ♎, ♏, ♐, ♑, ♒, ♓. This division of the Zodiac into 12. Signs and of every sign into 30. Degrees, nature itself seemeth to have showed by the motions of the Sun and Moon. For in what time the Sun moveth once about the whole compass of the Zodiac, the Moon maketh twelve revolutions through the same. Therefore as the time of a year is divided into 12. Moons, so the Zodiac is divided into 12. Signs: And as every Month containeth 30. days, so every sign is divided into 30. parts, which they call Degrees, which signifieth as much as steps, because the Sun steppeth, or goeth forwards almost so much as a degree in every day, from the West Eastwards under the Zodiac.  
The Zodiac is otherwise also divided into two semicircles, the one (from Capricorn to Cancer) ascending, because that so long as the Sun or any of the Planets are in that semicircle, they still ascend and rise higher and higher above the Horizon. The other semicircle of the Zodiac, from Cancer to Capricorn, is called descending, because the Sun or Planets being in that semicircle, come down every day lower than other.  
The 12. Signs are by the Astrologians diversely divided, first into chief, mean, and common signs. The chief signs (which are also called Cardinal, that is the principal signs) are Aries, Cancer, Libra, and Capricorn, because they come next after the principal points of the Zodiac, that is, the two Equinoctial points at the beginnings of Aries, and Libra; and the two solstitial points of Cancer and Capricorn. The mean signs (which are also called fixed) are Taurus, Leo, Scorpio, and Aquarius. They are called mean, because they are placed between the chief or principal, and the common signs. They are called fixed signs, because that when the Sun is in those signs, we find a more perfect temperature of the Air, then when he is in the other signs.  
The common signs (which are also called double bodied) are Gemini, Virgo, sagittary, and Pisces. They are called common, because they take part of the nature of the fixed signs going before them, and of the Cardinal signs following after them. They are called double bodied, by reason of their Images, as they are imagined in the eight Sphere, which are compounded of two bodies: For there be two Twins; and the Virgin holdeth an ear of corn in her hand; sagittary is made of a Man and an Horse; and there are two Fishes. The placing, and nature of these signs brought in this division.  
The Astrologians also divide the 12. Signs into four trigons of triplicities, so called because they are distant the third part of a Circle, one from another. The first triplicity containeth Aries, Leo, and Sagitarius; and is called the fiery trigon, or triplicity: The second triplicity containeth Taurus, Virgo, and Capricorn; and is called the earthly trigon. The third triplicity containeth Gemini, Libra, and Aquarius; and is called the airy trigon. The fourth triplicity containing Cancer, Scorpio, and Pisces; is called the watery trigon. Nature itself is the cause of this division of the Signs also. For into these Trigons of the signs, she hath distributed the Conjunctions of the three superior Planets: especially the conjunctions of Saturn and jupiter, which the Astrologians call great conjunctions. For example's sake, if there be a great conjunction in Aries, the same shall be twenty years after in sagittary, and other twenty years after in Leo; and after as many more years, it returneth again into Aries. The revolution of one Trigon containeth almost 200. years, after which time the same great conjunctions remove into the next trigon.   

CHAP. IX. The use of the Zodiac.  
1. THe Zodiac is the measure of the second motions, as the Equinoctial is the measure of the first motion.  
2. For in this Circle we reckon the longitudes, and from it we count the latitudes of all the Stars. For the longitude of a Star is nothing else but the arch of the Ecliptck contained between the beginning of Aries, and the circle of the Stars latitude. And the latitude is the Arch of a great circle, drawn by the poles of the Ecliptic, contained between the Star and the Ecliptic.  
3. According to this circle, the whole Heaven, yea, the whole world is divided into twelve Signs. Whereof it cometh that because of this Circle, aswell the fixed, as the wand'ring Stars which we call Planets, yea, and those Stars also that appear of a sudden, as blazing Stars or Comets, and other Meteors, are said to be in this or that sign; and that three manner of ways.  
First, to be in a sign, is to be under some one of the 12. parts of the Ecliptic. Thus the Stars which are under the Ecliptic, but especially the Sun which runneth always under it, are said to be in the Signs.  
Secondly, because the Zodiac hath latitude, those Stars are said to be in a sign, which although they be beside the Ecliptic, yet are under the Zodiac, and so any of the other Planets, which for the most part wander beside the Ecliptic, may be said to be in some sign.  
Thirdly, if we understand six great Circles to be drawn by the beginnings of the twelve signs, and by the poles of the Ecliptic; by these circle's the whole heaven (or rather the whole world) is divided into twelve parts, which with a general name are called signs: Thus all the Stars aswell fixed as Planets and Comets, which are without the Zodiac in any of these parts, may be said to be in some sign.  
4. In this circle are noted the degrees of the signs, with which the Stars do rise and set, as well in a right as in an oblique Sphere. For because this circle is the chiefest, all Celestial appearances (or at least the most part of them) are referred unto it, and not unto the Equinoctial. But the Equinoctial measureth the times of their risings and settings.  
5. The obliquity of the Ecliptic is the cause of the inequality, aswell of natural days in both Spheres, as of artificial days in an oblique Sphere. For seeing it is moved unequally, because it is moved upon other Poles than his own, the Sun which is the author, and maker of times moving under it, must needs make unequal days.  
6. The chief times are defined by this circle, as the time of a year, by the motion of the Sun; the time of a month by the motion of the Moon, through the whole compass of this circle. Also the four quarters of the year, Spring, Summer, Autumn, and Winter, whereto may be added Plato his great year, which is the time wherein the fixed Stars make one revolution about the axtree and poles of the Zodiac, if God would have the world to last so long.  
7. The Ecliptic line showeth the places, and times of the Eclipses: For the Sun and Moon, are Eclipsed only under it, or near unto it.  
8. As the description of the Tropics dependeth on the obliquity of the Ecliptic, so the polar Circles are described by the Poles thereof.  
9 Hereof it cometh, that by reason of the same obliquity, the Zones and Climates are set forth and bounded.  
10. This Circle is of especial use in Astrology, for it distinguisheth the points of the 12. Houses, and in it the Aspects and configurations of the Planets are observed. The chiefest judgement aswell in casting Figures as in revolutions and directions is taken from this circle.   

CHAP. X. The description of the two Colours.  
THe two circles crossing each other at right Angles in the poles of the Sphere, are called the Colours: whereof the one that passeth by the common meeting of the Ecliptic and Equinoctial, is called Colurus aequinoctiorum, that is the Equinoctial Colour, or the colour of equal days and nights. The other passing by the poles of the Ecliptic, and the Solstitial points, is called Colurus solstitiorum, the Solstitial colour, or colour of the Sunne-standing.  

Uses common to both Colours.  
1. By means of these two Colours, all the movable circles of the material Sphere are framed together, that so they might be turned about, like as the whole Heavens are moved.  
2. The Poles are fastened in the common meeting of these two Circles: and the Poles are also showed by the same common meetings.  
3. They show the four principal points of the Ecliptic; that is, the two Equinoctial, and the two Solstitial points.  
4. These circles show those points of the Ecliptic, wherein the Sun is either equally distant from both poles of the Sphere, or cometh nearest to either of them: In which points the Sun maketh the day's longest or shortest, or of a mean length between both these in an oblique Sphere.  
5. They divide the Ecliptic into four quarters, in which the Sun maketh four quarters of the year, the Spring, the Summer, Autumn, and Winter.  
6. They divide the Ecliptic and Equinoctial into such four quarters, as in a right Sphere do rise together in equal time.    

CHAP. XI. Uses of the Equinoctial Colour.  
1. THe section of this Circle with the Ecliptic; showeth the Equinoctial points, wherein the Equinoctial and the Ecliptic do divide and cross each other. In which points the Sun maketh equality of days and nights throughout the whole world: whereof this circle is also called Colurus Aequinoctiorum; that is, the colour of equal days and nights, or the Equinoctial colour.  
2. It deuideth the Ecliptic into the North and South halves.  
3. It deuideth the Signs wherein the Sun maketh the days longer than the nights, from those signs wherein the days are made shorter than the nights.  
4. It showeth which halves of the Ecliptic and Equator, do arise together in equal time in an oblique Sphere.  
5. It showeth the two high Sunne-standing in a right Sphere, in the time of which Sunne-standing, the Sun passeth by the Zenith.   

CHAP. XII. Uses of the Solstitial Colour.  
1. THe common meetings of this circle with the Ecliptic, show the Solstitial or Tropical points; in which points the Sun seemeth to stand, and then returneth back again: for which cause this circle is called the Colour of the Sun-standing. These points are called tropical (which is as much to say as turne-points, or points of return) because that when the Sun going always under the Ecliptic cometh to these points, which are furthest distant from the Equinoctial circle, it returneth again towards the same circle. But they were called Solstitial or Sun-ding points, because that whilst the Sun is about those points, the difference of the Sun's returning is (for certain days) insensible. Hereof the Sun is said to make his station, or to stand, when he cometh to either of those points. They that dwell without the Tropickes, have two sunne-standing, that is the Summer sun-standing, or high sun-standing (when the Sun in Summer time is at the highest, and next unto our Zenith being in the beginning of Cancer) and the winterly, or low sun-standing, when the Sun in Winter time is lowest in the Meridian, and furthest from our Zenith. But they that dwell within the Tropics (by a certain similitude taken from our sun-standing, wherein the Sun is either highest or lowest) are said to have four sun-standing; that is two high sun-standing, when the Sun passeth by their Zenith (the highest point in the Heavens) which happeneth twice every year in two places, equally distant from the beginnings of Cancer and Capricorn: and two low sun-standing, when the Sun is in the beginning of Cancer and Capricorn.  
2. In this Circle by the arch contained between the Equator and Ecliptic, we measure the greatest declination of the Sun, or obliquity of the Ecliptic, which in Ptolomees time was 23. degrees 51. minutes, and one third part of a minute. But ever since that time it hath been found by observation to decrease; so as in this our age, it is no more than 23. degrees and one half, or little more: Notwithstanding Copernicus thought that the greatest obliquity was 23. degrees 28. minutes.  
3. It showeth the places of the Ecliptic, in which the Sun (coming nearest to our Zenith) maketh the artificial day longest; or going furthest from the same point, maketh the same shortest.  
4. It deuideth the Zodiac into two halves, the one ascending, and the other descending.  
5. Hereby also the signs are distinguished, which do rise rightly, and which rise obliquely in an oblique Sphere. For the descending half riseth rightly, and the ascending half riseth obliquely.  
6. So the points of the Ecliptic are showed by this Circle, wherein the greatest difference of right and oblique ascensions happeneth. It distinguisheth those signs in which when the Sun moveth, the artificial days are increased, and the nights decrease; from those signs wherein the days are diminished, and the night's increase.  
7. In this circle are the bredths of the Zones bounded; for the obliquity of the Ecliptic doubled, showeth the breadth of the torrid or burnt Zone: the distance of the poles of the Ecliptic, and of the Poles of the Equator, show the breadth of the cold or frozen Zones; and the other two Arches remaining, show the bredths of the temperate Zones.   

CHAP. XIII The Description of the two Tropickes.  
THe two smaller Circles Equidistant in all places from the Equinoctial, and coming under these Solstitial points of the Ecliptic on both sides, are called the Tropics, that is circles of return.  
And they are so called, because that when the Sun cometh to them, it beginneth to return back again towards the Equinoctial circle. Or else they may be so called, because they are described by the turning about of the Tropical points of Cancer and Capricorn. They are also called solstitial Circles; that is Circles of the sun-standing; because that by reason of the insensible alteration of the declination of the Ecliptic, for some space both before, and after the Tropical points, the Sun (in respect of his Meridian altitudes, or in respect of the motion he hath towards the North or South, by reason of the obliquity of the Ecliptic) seemeth to stand (as it were) for certain days in those places.  
There be two Tropics, the Tropic of Cancer, and the tropic of Capricorn.  
The tropic of Cancer, toucheth the Ecliptic in the beginning of Cancer, which is the most northerly point of the Ecliptic: or it is the Tropic described in the first movable Sphere, by the Summer solstitial point.  
This circle is called the Tropic of Cancer, because it toucheth the Ecliptic in the beginning of Cancer.  
It is also called the Summer Tropic, and the Tropic of the Summer sun-standing, because that when the Sun cometh to it, the Summer beginneth. It is called the North tropic, because it is in the North part of the world: and the Circle of the high sunne-standing, because the Sun coming to it, is highest in the Meridian, and next unto our Zenith which dwell in the North part of the world, without the Tropics. The Tropic of Capricorn is the Tropic which toucheth the Ecliptic in the first point of Capricorn. It is called the Tropic of Capricorn, because it toucheth the Ecliptic in the beginning of Capricorn. It is called the winter Tropic and Tropic of the Winter sun-standing, because the Sun cometh to it in Winter.  
It is also called the circle of the lowest Sunne-standing, because that when the Sun cometh to this Tropic, it is furthest distant from our Zenith, and hath his lowest height in the Meridian.   

CHAP. XIIII. Uses of the Tropickes.  
1. THe Tropics show the Tropical, or Solstitial points of the Ecliptic: that is, the points wherein the Sun seemeth to stand, and beginneth to return back again.  
2. They bound out the greatest declinations of the Sun, which in our times is about 23. degrees and an half.  
3. Therefore they do also bound out the obliquity of the Ecliptic, for they are the bounds of the Sun's way, beyond which the Sun goeth not at any time.  
4. The Sun coming to either of these circles, is either nearest, or furthest distant from our vertical point.  
5. In an oblique Sphere, they measure out the shortest, and longest artificial day and night.  
6. The Tropics (aswell in Heaven as in Earth, contain betwixt them the Torrid Zone, and separate it from the temperate.   

CHAP. XV. The Polar Circles.  
THe two smallest circles that are next about the poles of the Sphere, are called the polar circles. They are drawn by the poles of the Ecliptic, and are every where Equidistant from the Equinoctial, and from the poles of the Sphere.  
They are called polar Circles, either because they are near the poles of the Sphere, or else because they are described by the motion of the poles of the Ecliptic.  
And therefore there be two polar Circles, that is, so many as there are poles of the Ecliptic: the Polar circle Arctic, and the Polar Antarctic.  
The Arctic polar circle, is that which passeth by the North pole of the Ecliptic, or which is described by the North pole of the Ecliptic being carried about with the motion of the first movable Sphere.  
The Antarctic polar circle, is that which goeth by the South pole of the Ecliptic, being described with the first motion by the Antarctic pole of the Ecliptic.  
The distance of these polar Circles from the poles of the Sphere, is equal to the distance of the tropicks from the Equinoctial, which in our time is about 23. degr. and an half: for so much as is the obliquity of the Zodiac (whereto the distance of the Tropics from the Equinoctial is always equal) so much are the poles of the Ecliptic distant from the Poles of the world.   

CHAP. XVI. Uses of the Polar Circles.  
1. THe Polar Circles show the poles of the Zodiac, and show their distance from the poles of the Equinoctial.  
2. The temperate Zones are bounded by these polar circles; for the Arctic circle boundeth the North side of the North temperate Zone; and the Antarctic circle boundeth out the South side of the South temperate Zone.  
3. The Polar circles separate the temperate Zones, from the cold Zones which they compass round about, and enclose within them.  
Therefore the four lesser circles, that is the two Polar circles, and the Tropics, divide Heaven and Earth, into five Zones.   

CHAP. XVII. Of the Zones.  
A Zone is a space of Heaven, or Earth, contained between two of the smaller Circles; or enclosed within the compass of either Polar circle.  
They are called Zones (that is as much to say as girdles) because they compass about Heaven or Earth like a girdle.  
The Zones are divided by ancient Writers into two kinds; that is into temperate, and untemperate Zones.  
A temperate Zone is the space of Heaven or earth, contained between either of the Tropics, and the next Polar circle.  
There be two temperate Zones; the one North, the other South.  
The North temperate Zone is contained between the Tropic of Cancer, and the Arctic polar circle.  
The South temperate Zone is that which is contained between the Tropic of Capricorn, and the Antarctic polar circle.  
They are called temperate Zones, because they have a better temperature of the air for the most part, and more mere for habitation, than the untemperate Zones. The breadth of either temperate Zone is always equal to the compliment of the distance of the Tropics, and therefore in this age is about 43. degrees, that is 2580. English miles.  
There be two kind of untemperate Zones, the one exceeding in heat, the other in cold, for the most part.  
The hot untemperate Zone, (called also the Torrid; that is, the burnt or broiled zone) is that space of Heaven or Earth, which is contained between the tropicks.  
It is called the burnt Zone, because that by reason of the Sun's continual going over that zone; and casting his beams directly down thereupon, it is scorched with overmuch heat, and is not so meet to be inhabited as the temperate zones.  
The breadth of this Zone is always equal to the obliquity of the Zodiac, or greatest declination of the Sun, doubled; which in our time is about 47. degrees, that is 2820. English miles.  
The cold or frozen zones, are the spaces of Heaven or earth, contained within the Polar circles.  
There be two cold zones, the one North, contained within the compass of the Arctic circle: the other South, contained within the compass of the Antarctic Polar circle.  
These zones exceed in cold, because they want the sight of the sun for a great part of the year, and when the Sun appeareth unto them, his beams fall so obliquely upon them, that they can (in all likelihood) receive but small heat thereby for the most part.  
The breadth of these Zones is measured from the Poles of the world to the Polar circles, and therefore must always be so much as the Polar circles are distant from the Poles: that is, in our age about 23. Degrees and a half, which make 1410. English miles.   

CHAP. XVIII. The difference of Shadows that the Sun maketh in these Zones.  
THey that dwell in the torride Zone, do cast their shadows which the Sun maketh at noon (which we may therefore call their noon shadows) both towards the North, and towards the South: towards the North, when the sun is betwixt their zenith and the south point of the Horizon; and towards the South, where the sun is between their Zenith and the North.  
For seeing the zenith of them that dwell in that Zone is between the Tropics, the sun must needs be sometimes Northwards from their zenith, and so make a south shadow: and sometime Southwards, and then make a north shadow. For which cause they that inhabit this Zone are called Amphiscij; that is, such as cast their noon shadows on both sides.  
But they that dwell in the temperate Zones, are called Heteroscij; that is, such as cast their shadows at noon, one way only. For they that dwell in the North temperate Zone, have the Sun always at noon from their Zenith Southwards, and therefore must needs always cast their noon shadows Northwards. Whereas chose they that inhabit the South temperate Zone, having the Sun at noon always Northwards from their Zenith, must needs have their shadows at noon, always towards the South.  
And they that are in the cold Zones, are called Periscij; that is, such as cast their shadows round about them. For seeing the Sun continueth every year for certain days together, always above their Horizon, and therefore moveth round about them without setting: it must needs be, that their shadows also are carried round about them, falling towards all parts of the world in the space of 24. hours. * ⁎ *    

THE SECOND PART. Of the uses of the uppermost SPHERE, and of the Circles thereof jointly.  

PROP. I. To rectify the Sphere to the Latitude. etc.  
FIrst find by observation, or otherwise the height of the Pole, or Latitude of that place for which you would rectify the Sphere. Then (by turning about the Meridian of the Sphere; lift up or put down the North Pole of the Sphere (about which the hour circle is fastened) till the arch of the Meridian from the North part of the Horizon upwards unto the Pole, be just so many degrees as the elevation of the Pole or latitude of the place was found to be: for so have you the Sphere duly rectified.  
As for example, the Latitude of the City of London is 51. degrees and 32. minutes, therefore if you lift up the North Pole of the Sphere, above the North part of the Horizon, so many degrees and minutes you shall have your Sphere rectified for that place.   

PROP. II. To know the place of the Sun; etc.  
Look the day of the month (for which you desire to know the place of the Sun) in the Horizon, and see what sign and degree of the Zodiac upon the Horizon answereth thereto; for there have you the place of the Sun.  
Take for example the 25. of December: look this day therefore in the Horizon, and you shall find answerable thereto 13. degrees, and about 40. minutes of Capricorn, which is the place of the Sun at that time.   

PROP. III. To know the declination of the Sun, etc.  
BRing the point whose declination you desire to know, unto the Meridian of the Sphere, and look what number of degrees and minutes of the Meridian is contained between that point, and the Equinoctial, for so much is the declination.  
As if you would know the declination of the 10. degree of Taurus, bring that degree to the Meridian, and you shall find the arch of the Meridian between that degree and the Equinoctial, to be 14. degrees and about 51. minutes.   

PROP. FOUR To know the right ascension of the Sun, etc.  
BRing that point (as before) to the Meridian, and see then how many degrees and minutes of the Equinoctial are contained between the beginning of Aries and the Meridian: for that is the right ascension of that point. So you shall find the right ascension of the 10. degr. of Taurus to be 37. degr. 35. min. for if you bring that degree of Taurus to the Meridian, you shall find so many degrees and min. between the beginning of Aries, and the Meridian.   

PROP. V. To know the oblique ascension of the Sun, etc.  
SEt the Sphere to the elevation of the place for which you desire to know the oblique ascension; then bring the Sun, Star, or point whose oblique ascension you would know, unto the East semicircle of the Horizon, and look how many degrees and minutes of the Equinoctial circle, are contained between the East point of the Horizon, and the beginning of Aries; for so much is the oblique ascension desired: As for example, if you see the Sphere to the Latitude of London 51. degr. 32. min. and then bring the 10. degree of Taurus to the East part of the Horizon, you shall find about 19 degrees and an half of the Equinoctial, at the same East part of the Horizon; which is the oblique ascension of that degree of Taurus, for the Latitude of the City of London.   

PROP. VI To find the difference of Ascension.  
COmpare the right and oblique ascensions of the Sun, (or of any point of the Zodiac) together, and subtract the less from the greater, for the remainder shall be the difference of ascension. As for example, the right ascension of the 10. degree of Taurus, being found by the 4. Propo. to be 37. degrees, 35. min. and the oblique ascension of the same degree at London, by the 5. Prop. 19 degree 30. min. by subtraction of the less out of the greater, the difference shall be found to be 18. degr. and 5. minutes, which is the difference of ascension sought for.   

PROP. VII. To find at what time the Sun riseth or setteth.  
REduce the difference of Ascension into hours and minutes (taking for every 15 degrees 1. hour, and for every one degree that remaineth 4. minutes, and for every minute of a degree 4 seconds) for these hours, minutes and seconds, being added to 6. hours, if the Sun be in any of the South signs; or subtracted, if he be in the North signs, showeth the time of the Sunrising. And chose, the same hours and minutes subtracted from six hours when the Sun is in the South signs, or added when he is in the North signs, showeth the the time of the Sunne-setting.  
As for example, the Sun being in the 10. degree of Taurus which happeneth about the 20. or 21. day of April) I would know at what hour and minute the Sun riseth, and setteth at London: Having therefore found by the former Proposition the difference of ascension to be 18. degr. and 5. minutes I take for 15. degrees thereof one hour, and for the three degr. remaining, 12. minutes of an hour, and for the 5. minutes, 20. seconds of an hour. Which hour, minutes and seconds being subtracted out of 6. hours, because the Sun is in a North sign, there remaineth the time of the Sun's rising at 4. a clock 47. minutes, 40. seconds. And adding the same hour, min. and seconds to 6. hours, you have the time of the Sunsetting that day at 7. a clock, 12. min. and 20. seconds.   

PROP. VIII. To find the length of the artificial day or night.  
THe artificial day, is the time contained between the Sunrising and the Sunsetting: and the artificial night is the time between Sunne-setting and Sunrising. The length of both these is found after this manner: having found the difference of ascension, and reduced it into hours and minutes (as in the former Proposition) double th●se hours and minutes, and add them to 12. hours if the Sun be in the North signs, or subtract them from 12. hours if the Sun be in the South signs, for so shall you have the length of the day: But (chose) subtract the same hours and minutes (being doubled) from 12. hours, the Sun being in the North signs; and add them to 12. hours when he is in the South-signes; so have you the length of the night.  
Or else, double the time of the Sunsetting, so have you the length of the day. And double the time of the Sunrising, so have you the length of the right.  
As the time of the Sunrising being found by the former Proposition to be 4 hours 48. minutes after mignight at London, the Sun being in the 10. degr. of Taurus, by doubling the time of the Sunrising, the length of the night shall be found to be 9 hours and 36. minutes. And doubling the time of the sunset that is 7. hours, and 12. minutes, you have the length of the day, 14. hours; and 24. minutes.   

PROP. IX. To know the time of the Sun rising and Sun setting.  
THe place of the Sun being found by the 2. Proposition, bring the same to the Meridian, and withal set the point of the Index of the hour circle, to the 12. hour in the same circle: Then bring the place of the Sun to the Horizon Eastwards; and the point of the hour Index shall show you in the hour circle, the time of the Sunrising. But if you bring the place of the Sun to the Horizon Westwards, the point of the Index will show in the hour circled the time of the Sunsetting.  
As for example, the Sun being in the 10. degree of Taurus, bring the same degree to the Meridian, and bring the point of the hour Index also to the Meridian: then (the Sphere being set to the Latitude of London) bring the same 10. degree of Taurus to the East part of the Horizon, for then the hour Index will show you in the hour circle, that the Sun riseth at 4. of the clock and 48. minutes.  
And bringing the same degree to the West semicircle of the Horizon, the same Index will show the time of the Sunsetting to be 7. hours and 12. min. after noon.   

PROP. X. To find the length of the artificial day or night.  
BRing the place of the Sun (being found as before to the East semicircle of the Horizon: set the hour Index 12. a clock in the Hour circle: turn about the Sphere from the East Westwards, till the place of the sun come to the Horizon, and mark how many hours the Index hath run over upon the Hour circle in the mean time, for so much is the length of the day.  
And to find the length of the night: Bring the place of the sun to the West semicircle of the Horizon, and set the Index to 12. a clock as before; Then turning forwards the Sphere from East Westward till the place of the sun come to the East semicircle of the Horizon; see how many hours the Index passeth over in the Houre-circle, for so many hours long is the night.  
As for example; supposing the Sun to be as before in the 10. degree of Taurus, bring the same degree to the East part of the Horizon, and the point of the Index to the Meridian: then turning about the Sphere, till the same degree come to the West part of the Horizon; you shall find that in the mean time, the point of the Index shall pass over 14. hours and 24. minutes, which is the length of the day. Likewise, if you bring the same 10. degr. of Taurus to the West part of the Horizon, and the Index to the Meridian, and turn about the Sphere, till that degree come to the East semicircle of the Horizon, the number of hours that the Index runneth over in the mean time upon the Hour circle, shall be found to be 9 Degrees, and 36. Minutes.   

PROP. XI. To know the Meridian altitude, or the height of the Sun at noon, for any time and place.  
SEt the Sphere to the latitude of the place where you desire to know the Sun's height at noon: bring the place of the Sun (being found as before by the 2. Prop.) to the Meridian, then see how many degrees of the Meridian, are contained between the Horizon, and the place of the sun, for so much is the height of the Sun at noon.  
In like sort it may be known how much the Sun is under the Horizon at midnight, after this manner: Bring the place of the sun in the Zodiac to the Meridian under the Horizon, and see how many degrees of the Meridian, are contained between the upperside of the Horizon, and the place of the Sun downwards: and so shall you have that you sought for.  
Or else if you cannot well come to the Meridian under the Horizon: bring that point of the Ecliptic which is opposite to the place of the sun, unto the Meridian above the Horizon; for the arch of the Meridian, or the number of degrees and minutes of the Meridian, between that point and the Horizon showeth how much the sun is under the Horizon at midnight.  
After this manner▪ the Sun being in the 10. degr. of Taurus, you shall find that his Meridian altitude at London is 53. degrees, and about one half.  
As also that he is under the Horizon at midnight about 23. degrees and a half at London.   

PROP. XII. To know how high the Sun is above the Horizon at any time of the day.  
BRing the place of the Sun (found by the 2. Prop.) to the Meridian: set the hour Index to 12. a clock upon the hour circle: turn the Sphere about till the Index come to the hour at which you desire to know the height of the Sun above the Horizon; take the distance of the place of the Sun from the Horizon with a large pair of Compasses: then set both feet of the Compasses in the Ecliptic, and look how many degrees are contained between them, for so much is the height of the Sun.  
Thus may you find by the Sphere, that when the Sun is in the tenth degree of Taurus, his height at 10. of the clock in the forenoon (the Sphere being duly rectified by the first Proposition) shall be about 45. degrees and an half at London.   

PROP. XIII. To find the hour of the day by the height of the Sun; etc.  
SEt the pole Arctic of the Sphere to his elevation for that place where you desire to know the hour of the day: bring the place of the Sun in the Zodiac to the Meridian, and the hour Index to 12. a clock of the hour circle: take so many degrees of the Ecliptic between the feet of your Compasses, as the height of the Sun amounteth unto.  
Then set one foot of your Compasses in the place of the Sun, and turn the Sphere about, Eastwards, if it be in the forenoon, or Westwards, if in the afternoon, till you can but only touch the Horizon with the other foot of your Compasses; for then the Index pointeth out the hour of the day in the Hour circle.  
As suppose you observe the height of the Sun being in the 10. degr. of Taurus, and find him to be 30. degrees high in the forenoon: you shall find (following the directions prescribed in this Proposition) that it shall then be about 8. of the clock in the morning.   

PROP. XIIII. To find the Amplitude or breadth of the Sun's rising, or setting, etc.  
THe pole of the Sphere being set to his elevation, and the place of the Sun to the East semicircle of the Horizon: see how many degrees of the Horizon, are contained between the place of the Sun, and the true East point, for so you shall have the breadth of the sun's rising.  
Thus the sun being in the 10. degree of Taurus, you shall find by the Sphere, that (for the latitude of London) he riseth about 23. degr. and a half Northwards, from the true East point, and that he setteth as many degrees towards the North, from the true West point.   

PROP. XV. To find the place of the Sun, etc.  
THe quarter of the year being known, bring the quarter of the Ecliptic that is answerable thereto, under the Meridian; and turn the Sphere to or fro, till there be so many degrees and minutes of the Meridian, contained between the Ecliptic and the Equator, as the declination cometh to: then look what degree of the Ecliptic is under the Meridian, for that is the place of the Sun.  
As suppose the declination of the Sun in some day of the Springtime of the year be found to be 14. degr. 51. min. (turning therefore the Sphere to and fro, till some part of the spring quarter of the Ecliptic, come right under that degree and minute of declination in the Meridian) you may find that the Sun is then in the tenth degree of Taurus.   

PROP. XVI. To find what day of the month it is, etc.  
THe place of the Sun being found by his declination (as is already showed) seek the place of the Sun in the Horizon of the Sphere, and look what day is answerable thereto, for that is the day of the month which was sought for.  
As the place of the Sun being found by his declination (as is showed in the former Proposition) to be in the 10. degree of Taurus, the day of the month shall thus be found to be the 21. of April.   

PROP. XVII. The day of the month being known, to find at what time the day breaketh.  
Find the place of the Sun (by the 2. Prop.) and bring it to the Meridian, then bring the hour Index, to 12. a clock upon the hour circle.  
Find out also the point of the Ecliptic that is right over against the place of the Sun: then take between the feet of your Compasses 17. degrees of the Ecliptic, and setting one foot of the Compasses in the point opposite to the place of the Sun, turn the Sphere Westwards, till you can but only touch the Horizon with the other foot, for then the Index showeth in the hour circle at what time the day breaketh.  
So the 21. of April, the Sun being in the 10 degr. of Taurus, you shall find that the day breaketh about half an hour past 2. of the clock in the morning.   

PROP. XVIII. To find how long the twilight continueth.  
Find out by the former Prop. at what time the day breaketh, and learn also at what time the Sun riseth by the 7. or 9 Prop.  
Then subtract the lesser from the greater, and there shall remain the length of the twilight.  
Or else thus: having brought the point that is opposite to the place of the Sun to be 17. degrees above the Horizon Westwards, in such sort as is showed in the former Proposition; and keeping the Sphere in that position, bring about the point of the hour Index unto 12. a clock upon the hour circle; then tune the Sphere Westwards until the degree or point of the Ecliptic that is opposite to the place of the sun come to the Horizon: and see how many hours the point of the Index hath run over in the mean time upon the hour circle: for so many hours continueth the twilight.  
By either of these ways, the Sun being in the 10. degr. of Taurus, you shall find that the twilight (that is the time from the break of the day till Sun-rise) is about 2. hours and 20. minutes.   

PROP. XIX. To find how much the declination of the Sun must alter at any time of the year, to make the day an hour longer or shorter.  
BRing the place of the Sun (found by the second Prop.) to the East semicircle of the Horizon, and mark what degree or point of the Horizon it falleth upon; bring one of the Colours to the same degree or point, and there make a prick in that colour; and holding the Sphere immoveable,) mark withal what degree of the Equinoctial, or of either of the Tropickes is then at the Horizon: Then turn the Sphere 7. degrees and an half forwards, towards the West, if the days shorten: but chose if the days lengthen; and holding the sphere there immoveable, make another prick in the colour at the Horizon: for the distance of these two pricks in the colour taken with the Compasses and brought to the Ecliptic, or Equinoctial, showeth how much the Sun's declination must alter to make the day an hour longer, if the day's increase; or shorter, if they decrease.  
After this manner you shall find that the sun being in the 10. degree of Taurus, his declination must increase about 5. degrees, (or little more) to make the day an hour longer; but when the sun is in the 20. degree of Pisces, his declination, or rather his Meridian altitude, must increase about 6. degrees to make the day an hour longer: and when he is in the beginning of Capricorn, his declination decreaseth scarce 5. degrees to make the day an hour longer.   

PROP. XX. To find how many days it is ere the day lengthen or shorten an hour.  
BRing the foresaid pricks (made in the Colour by the former Proposition) unto the Meridian, and there make two marks justly answerable unto those pricks in the Colour: turn about the Sphere till the Ecliptic line come just under one of those marks, and there make a prick in the Ecliptic: then again turn the Sphere till the Ecliptic come just under the other mark made in the Meridian, and there make another prick in the Ecliptic: (But here it is to be noted, that whereas the Ecliptic may be brought under that mark whether way soever you turn the Sphere, it must (I say be noted that the Sphere must be turned that way which may soon bring the Ecliptic under that mark. (Last, find out amongst the signs and degrees described upon the Horizon, the like arch to this, that is contained between these pricks in the Ecliptic: For the number of days answerable to this arch in the Horizon, is the time wherein the day groweth an hour longer or shorter.  
Thus shall you find, that when the Sun is in the beginning of Aries, it will be about 18. days after, ere the day be one hour longer. But when the Sun is in the beginning of Capricorn, you shall find that it will be almost twice so much, that is near 34. days before the day will be an hour longer.  
Hereby therefore the error of them manifestly appeareth, which think that in every 15. days, the day is lengthened or shortened an hour, whereas indeed the lengthening or shortening of the days, keepeth no such rule. For when the Sun is about the Equinoctial points, the days lenghthen or shorten very fast: but when he is near the Tropical points, they grow longer or shorter very slowly.   

PROP. XXI. To make an horizontal Dial.  
SEt the Sphere to the elevation of the place for which you would make the Dial turn about the Sphere, till the solstitial Colour be 15. degrees (measured in the Equinoctial) from the Meridian; and where the Colour crosseth the Horizon, there make a prick; then turn the Colour yet 15. degr. further, that is 30. degrees from the Meridian; and where the Colour crosseth the Horizon, there make an other prick: again turn the Colour forwards yet 15. deg. more, (that is 45. degrees from the Meridian) and at the common meeting of the Colour and Horizon, make the third prick in the Horizon; and so proceed with the rest, till you have made so many pricks on that side of the Horizon as there are hours in halfe the longest day. Then look how many degrees the first, second, third, fourth pricks, etc. are from the Meridian, for so many degrees must the hour lines of 11. a clock, and one a clock; of 10. and 2, of 9 and 3. of 8. and 4. etc. be from the 12. a clock line in the horizontal Dial.  
After this manner in an horizontal Dial made for the Latitude of London, (which is 51. degr. and 32. minutes (you shall find the distances of all the rest of the Houre-lines from the 12. a clock line as followeth: Betwixt twelve and 11. and twelve and 1. are conceyned 12. degrees almost: Between 12. and 10. and 12. and 2. there are contained 14. degr. and an half: Between 12. and 9 and 12. and 3. 38. degr. Between 12. and 8. and 12. and 4. 53. degr. Between 12. and 7. and 12. and 5. 70. degrees and an half.  
Between 12. and 6. both before and after noon, 90. degr. The other hour spaces before 6. in the morning; and after 6. in the evening, are equal to the Hour spaces after six in the morning, and before 6. in the afternoon.   

PROP. XXII. How to make a direct mural Dial.  
SEt the Pole arctic of the Sphere so much under the Horizon as is the compliment of the Poles elevation: the Horizon therefore being thus set as it were to the Zenith of the Sphere, and so representing the vertical Circle of East and West (that is the plain supper fices of a direct mural Dial) you shall find the distances of all the houre-lines, (both before and after noon) from the 12. a clock line, in such sort as you did before for the horizontal Dial.  
So you shall find the distances of the houre-lines in an erect direct mural Dial made for the Latitude of London to be as followeth: Between the twelve a clock line and the lines of 11. and 1. 9, degr. and about one third part of a degree: Between 12. and 10. and 12. and 2. 19 degrees and one quarter; Between 12. and 9 and 12. and 3. 32. degr. or little more: between 12. and 8. and 12. and 4. 48. degrees: between 12. and 7. and 12. and 5. 67. degr. or little more: between 12. and 6. both before and afternoon 90. degrees.   

PROP. XXIII. How to make any direct inclining, or direct reclining Dial.  
REckon from the Equinoctial upwards in the Meridian, so many degrees as the height of the Pole cometh to at that place where you would make your Dial; for there is the vertical point or Zenith of that place: from this Zenith reckon Southwards in the Meridian, the inclination of south Dial's, and the reclination of North dials; but contrariwise, reckon from the Zenith Northwards the inclination of North dials, and the reclination of South dials. Then bring that degree of the Meridian, where this reckoning endeth to the Horizon, for so the Horizon representeth unto you the plain or the flat superficies of the Dial which you would make. Therefore you shall find how 〈…〉 every one of the hour lines should be dist●●● from the 12. a clock line, in such sort as you did before in making the horizontal Dial.  
Thus in a South direct Dial inclining 30. degr. or in a North direct reclining 30. degrees made for the Latitude or elevation of the Pole at London, you may find the distances of the eleven a clock line, and of the one a clock line, from the 12. a clock line, to be about 14. degrees.  
But the hour lines of 10. in the forenoon, and of 2. in the afternoon, are distant from the 12. a clock line 28. degr. & one half; From 12. to 9 and to 3. you shall find 43. deg. From 12. to 8. in the forenoon, and 4. in the afternoon, you shall have 58. deg. & an half: also from 12. to 7. & to 5. shall be about 74. deg. And from 12. to 6. in the morning, & 6. afternoon 90. degrees.  
Likewise in a South direct reclining, or North direct inclining 20. deg. for the elevation of London; the spaces between 12. & 11. & 12. & 1. shall be about 5. deg. or little less: Between 12, & 10. & 12, & ●. about 10. deg. & ●. third parts. From 12, to 9 in the forenoon, & 3. in the afternoon, 18. almost▪ From 12. to 8. & 4. ●g. deg. or little more. From the 1●. a clock line; to the line of 7. a clock in the forenoon, & 5. in the afternoon, 50. deg. or thereabouts. From 12. to 6. both before and afternooe▪ 90. deg. as in the former kinds of D●. In all which it is to be noted, that there is always 〈◊〉 distance between the hour lines of 5, & ●. & 4, & 6. that there is between 7, & 6. and 8, & 6. in the forenoon, and between 5, & ●. & 4, & ●. and 7, & 6. & 8, & 6. in the afternoon. So as the distances of all the hour lines from the 12. a clock line being found from 6. in the morning, till 6. at night, the distances of the other Houre-lines before 6. in the morning, and after 6. at night shall easily be had.   

PROP. XXIIII. To know at what time the Moon, or any other of the Planets or fixed Stars, that are within the breadth of the Zodiac; rise or set, or come to the Meridian, etc.  
Find the place of the Moon, or any other of the Planets, both in longitude and Latitude, by the Ephemerises: and find the place (that is, the Longitude and Latitude) of any of the fixed Star●es in the Zodiac by some table of the fixed Stars, or otherwise; and mark the same place of the Moon; Planet or Star, in the Zodiac of the Sphere: and having set the Sphere to the Latitude of the place, bring the place of the Sun (found by the 2. Proposition) to the Meridian, and the Houre-index to 12. a clock upon the houre-circle●, then turn the Sphere till the place of the Moon, Planet or Star marked in the Zodiac, come to the East semicircle of the Horizon; for then the Index showeth the time when the Moon, or that Planet or fixed Star riseth.  
Al●● the number of degree in the Horizon, contained between the point of the Moons, Planets, or Stars rising, and the point of true East, showeth the breadth, ●●denesse, or amplitude of rising; And you may at the same instant, 〈◊〉, what degree of the Ecliptic riseth with any of them, and what the oblique ascension of any of them is: For if you tell hour many degrees of the Equinoctial are 〈…〉 between the beginning of Aries in the Horizon, proceeding Eastwards, or according to the order of the signs: you shall have the oblique ascension of the Moon, Planet or Star, that 〈…〉 for.  
But bring the same place of the Moon, Planet, or Star to the Meridian, and the Index showeth in the hour circle at what time they come to the Meridian: where you may also see, first what degree of the Zodiath middeth Heaven (that is, 〈◊〉 to the Meridian) with any of them; secondly, you may see how much the declination of any of them is; for count how many degrees of the Meridian are contained between the Equinoctial and the place of the Moon, Planet, or fixed Star, and so much is the declination. Thirdly, you may there see what the right ascension of any of them is: for the place of any of them being brought to the Meridian and there 〈◊〉, r●ckon Eastwards how many degrees of the Equinoctial are contained between the beginning of Aries and the Meridian, so have you the right ascension. Lastly, bring the place of the Moon, Planet or Star, to the West semicircle of the Horizon; for then the Index showeth the time of their setting; and the number of the degrees of the Horizon between the point where any of them setteth, and the Equinoctial, or true West point (where the Equinoctial, and Horizon cross each other) is the amplitude or breadth of the setting of any of them, showing how much they set from the true West point.  
You may there also see what degree; either of the Ecliptic, or of the Equinoctial, setteth with any of the● and consequently you may know the oblique descension of any of them, by reckoning how many degr. of the Equinoctial there are from the beginning of Aries Eastwards, till you come about to the West part of the Horizon.  
Take for example the great Star called the Bull's eye, whose place in longitude is about the 4. degr. of Taurus, and his latitude about 5. degrees and an half Southwards.  
Following therefore the directions prescribed in this Proposition, you shall find that upon the first day of April this present year, 1600. the same Star riseth here at London about half an hour past 7. of the clock in the morning, and setteth about a quarter of an hour past 10. at night, and cometh to the Meridian about 3. a clock afternoon: Also you shall find that it riseth with the 15. degree of Gemini, and setteth with the last degr. of Taurus, and cometh to the Meridian, or middeth Heaven, with the 5. deg. of Gemini: Thirdly you shall find his declination to be about 15. deg. and 2. third parts, his right ascension 63. degr. and a quarter, his oblique ascension 43. degr. and his oblique descension about 84. deg. and an half: and lastly his amplitude of breadth of rising or letting about 25. degr. and an half from the true East and West points towards the North.   

PROP. XXV. To know how long the Moon, or any of the Planets of fixed Stars do shine or continue above the Horizon.  
THe Sphere bring set up the latitude of the place, and the place of the Moon, Planet, or fixed Star, being found and marked in the Zodiac, both in Longitude and Latitude, (as in the 〈◊〉 Prop.) bring the place of the Moon, Planet, or Star, 〈◊〉 East semicircle of the Horizon, and the Index of hours to 12. a clock: Then 〈◊〉 about the Sphere Westwards, till the same place of the Moon, or 〈◊〉 the same Planet, or Star, come to the West semicircle of the Horizon, and mark 〈◊〉 how many hours the Index runneth over in the mean time upon the hour circle, for so many hours continueth the Moon, Planet, or Star above the Horizon.  
Thus shall you find that the foresaid 〈…〉 The Bull's eye 〈…〉 the Horizon at London, about 14. 〈…〉 and 3. quarters.   

PROP. XXVI. To find which of the Planets or fixed Stars are above or under the Horizon at any time, etc.  
THe place of the plants or fixed Stars being marked in the Zodiac of the Sphere, as in the former propositions, and the place of the Sun brought to the Meridian, and then the Index to 12. 〈…〉 Sphere 〈◊〉 the Index 〈◊〉 to that hour upon the hour 〈◊〉 at which you desire to know what Planets are above or under the Horizon; and then hold still the Sphere, and mark what Planets or Stars are above or under the Horizon in the Sphere for the same Planets or Stars are above or under the Horizon in the Heavens.  
As for example: the 1. of April 1600 at 9 of the clock at night, you may by the Proposition find; that the most part of the fixed Stars, that are in the constestation of Taurus, Gemini, Cancer, Leo, Virgo, and Libra, together with the three superior Planets, 〈◊〉, ●upiter, & Mars, are at 〈…〉 to be seen above the Horizon, and that the rest of the Planets and fixed Stars, that are within the compass of the Zodiac, are under the Horizon, and cannot then be seen.   

PROP. XXVII. To find in what time any Sign or part of the Ecliptic riseth or setteth.  
BRing the beginning of the Sign, or part of the Ecliptic to the East semicircle of the Horizon, if you would know in how long time it riseth, or to the West part of the Horizon, if you would know in what time it setteth; then set the Index to 12. a clock, and turn forwards the Sphere, till the whole sign or part of the zodiac be risen, or set: For then the Index showeth upon the hour circle in how long time, that sign or part of the Zodiac riseth or setteth.  
Thus you may find (for example) that the whole sign of Aries here at London riseth in one hour or somewhat less, and setteth in two hours and three quarters, or something more: And that the whole quarter of the Zodiac, from the beginning of Aries to the beginning of Cancer, riseth in less than four hours, but setteth in more than 8. hours.   

PROP. XXVIII. To find the hour of the night by any of the Planets or fixed Stars in the Zodiac, etc.  
THe place (that is to say, the longitude & latitude) of any Planet, or fixed Star in the Zodiac, that is above the Horizon, being first found, and marked in the Zodiac of the Sphere; bring the place of the Sun (found by the 2. Proposition) to the Meridian, and the Index to 12. a clock upon the hour circle: Then having found the height of the Star, or Planet by observation, and the Sphere also being set to the Latitude of the place of observation, take between the feet of your Compasses, so many degrees of the Ecliptic, or Equinoctial, as the height of the Planet, or Star observed, cometh to; and setting one foot of your Compasses in the place of the Planet, or fixed Star that you observed in th' Zodiac, turn the Sphere forwards or backwards, till you can but only touch the Horizon with the other foot: for then the Index in the hour circle, shall show you the hour of the night.  
Suppose (for example) I should observe the height of the foresaid Bulls eye, and should find the same to be 29. degrees the first day of March at evening: finding therefore the place of that Star in the Zodiac of the Sphere, and bringing it (with help of the Compasses) to the height observed (having first set the place the Sun and houre-Index both together to the Meridian) the Index of the hours will show, that when that Star hath that height of 29. degrees, it is about 9 of the clock at night.   

PROP. XXIX. To know at any time of the year, what Stars in the Zodiac, arise or set, Cosmically, Achronically, or Heliacally.  
Such Stars as rise together with the Sun, are said to rise cosmically: and such Stars as set when the Sun riseth, are said to set cosmically; But those Stars which set together with the Sun, set achronycally; and those Stars that rise when the Sun setteth, are said to rise achronically. Lastly, those Stars that rise a little before the Sun, rise heliacally; and those that set a little after the Sun, set heliacally.  
All which may thus be found: Bring the place of the sun to the East semicircle of the Horizon: for the Stars that are then a little above the Horizon rise heliacally; but those that are in the Horizon in the East, rise cosmically; and they that are in the West semicircle of the Horizon set cosmically: But bring the place of the Sun to the West semicircle of the Horizon, for those Stars as are at the West part of the Horizon at the same time, set achronycally; but those that are then in the East semicircle of the Horizon, rise achronycally: and they which are a little above the West semicircle of the Horizon set heliacally.  
Thus you may know that upon the 26. or 27. day of May (it our latitude of London) the Bulls eye riseth cosmically, and the Stars in Serpentarius his right foot, set cosmically, you may see also that the same day the Star in the Bull's South horn setteth achronycally: and the Northermost star in Serpentarius his right foot, riseth achronycally: and lastly, you may find that about the same time the Pleiades and the Star in the Bull's North home, rise heliacally, and that the same Star also, and the former Twins feet set heliacally.   

PROP. XXX. To find the four principal or Cardinal points of Heaven (as the Astrologians call them) at any time.  
THese four Cardinal points are nothing else but four points of the Ecliptic, whereof one is at the East part of the Horizon, ascending, and is therefore called the Ascendent: another is at the upper part of the Meridian above the Horizon, and is called the midst of Heaven, and the hart of Heaven: the third is at the West part of the Horizon descending, and may be therefore called the descendent: the fourth point is that which is at the neither part of the Meridian under the Horizon. Which four points are the beginnings of the first, tenth, seventh, and fourth Houses. Therefore to find these points at any time by the Sphere, bring the place of the Sun (being found for that time by the 2. Proposition) to the Meridian, and the Index to 12. a clock: then turn the Sphere till the Index come to that hour at which you desire to know those four points, and there hold the Sphere that it move not: and look withal, what points of the Ecliptic are at the East and West semicircle of the Horizon; and at the upper and neither parts of the Meridian: for those be the four principal or Cardinal points you sought for.  
Take for example the time of the Sun's entrance into Aries this present year 1600. which was upon the tenth day of March about eight of the clock in the morning, or little after with us here at London, Having therefore brought the beginning of Aries together with the hour Index to the Meridian, and then turned back the whole Sphere till the Index come to 8. of the clock upon the hour circle: you shall find the ascendent at that time, to be the 27. degree of Taurus; the midst or hart of Heaven, the 27. of Capricorn▪ the descendent, the 27. deg. of Scorpio; and the lowest part of Heaven the 27. degree of Cancer.   

PROP. XXXI. To find out the breadth of any climate, etc.  
LIft up, or put down the pole of the Sphere, till you find that there are 7. deg. and an half of the Tropic of Cancer, more or less above the Horizon, than there were before; and mark with all how much the pole of the Sphere is raised, or let fall in the mean time, more than it was before; for so much is the breadth of that climate.  
As far example: having set the Sphere to our Latitude of London of 51. deg. and an half, with the point of your Compasses, holding and guiding some point of the Tropic of Cancer right under the Horizon; then lifting up the Pole till you find 7. degrees and an half more above the Horizon than were before, you shall find the Pole elevated about 2. degr. and an half more than it was before.  
Likewise, if you put down the Pole till there be 7. degrees and an half of the Tropic of Cancer, fewer above the Horizon than was before; you shall find the elevation of the Pole to be about 3. degrees less than before.   

PROP. XXXII. The reason of the inequality of natural days, etc.  
THe reason hereof is showed partly by the inequality of the differences of right ascensions answerable to equal arcks of the Zodiac; and partly by the unequal apparent motion of the Sun. For the first: the differences of right ascensions answerable to the parts of the Ecliptic, about the Tropical points of Cancer and Capricorn, are much greater than about the Equinoctial points of Aries and Libra.  
In so much that whereas the difference of right ascension answerable to one sign, or 30. degrees taken about those Tropical points, is more than 32. degrees and an half: about the Equinoctial points it is little more than 27. degrees and an half; as it may appear by the Sphere. So as you may hereby gather, that the difference of ascension answerable to one degree, which about the beginning of Capricorn is one degree, and about 6. minutes; about the beginning of Aries, or Libra, is only 55 minutes. Secondly, the apparent motion of the Sun is much swifter about his Parig●●●, in the sign of Capricorn, then about his Apogaeum in Cancer, or in other parts of the Zodiac: so that whereas the Sun being in Capricorn moveth 61. minutes and something more in a day: in Aries or Libra he moveth but 59 min. or very little more in the same time. Therefore seeing the natural day is nothing else, but the time wherein the Sun moveth from the Meridian about, till it return again to the same part of the Meridian; it must needs be that always in one natural day, there is made one whole revolution of the Equinoctial circle, and so much more as is the difference of right ascension answerable to the apparent motion of the Sun in the mean time: which differences of ascension because they be unequal, for the two causes before alleged; the natural days must needs also be unequal, the motion of the Equinoctial circle about his own centre being (as it hath been always supposed to be) equal, that is moving always an equal space in equal time.  
Which by this example may most plainly appear: The Sun being in Capricorn moveth 61. minutes in a natural day: difference of ascension agreeable thereto is 67. minutes, or something more. Therefore at that time, in the space of one natural day, the Equinoctial circle must make one full revolution, and 67. minutes more. But when the Sun is in Aries, moving only 59 minutes in a day, and the difference of right ascension answerable thereto, scarce 54. minutes more than one revolution of the Equinoctial circle; there shall pass only 54. minutes more in a natural day; so as here the Equinoctial circle moveth not about so much in one day as before by 13. minutes. Seeing then that 15. degr. or little more of the Equinoctial circle do pass the Meridian in every hour, and consequently one degree of the Equinoctial passeth the Meridian in 4. minutes of an hour, and one minute of a degree in 4. seconds of an hour; therefore 13. minutes of the Equinoctial shall pass the Meridian in 52. seconds: that is, almost in one minute of an hour: Whereby it manifestly appeareth that the natural day, that is to say, the space of 24. hours, which is the time wherein the Sun moveth from the Noonestead to the same noonestead again, is in our age greater almost by one minute of an hour, when the sun is in Capricorn, then when he is in Aries or Libra▪   

PROP. XXXIII. To find by the Sphere how much the natural days are longer at one time of the year then at another.  
FOr this purpose it will be best to take a good number of days together; as for example, take the whole month of December, and the whole month of March: both which months consist of the same number of 31. natural days: find the place of the Sun for the beginning and ending of both months, which you may find by the second Proposition to be for the beginning of March this present year 1600. about 20. degrees and 13. minutes of Pisces; and for the ending about 20. degr. 48. minutes of Aries: Also for the beginning of December the same year 18. degr. 46. minutes of S●gitarie; and for the ending, 20. degrees 24. minutes of Capricorn: Then seek out the right ascensions of the same places of the Sun for the beginnings and end of both those months by the fourth Proposition, and the differences of ascension answerable to the motion of the Sun in each month, by the sixth Proposition; which you may find by the Sphere to be about 33. degrees, 24▪ minutes for December, and 28. degrees, 39 minutes for March. Lastly, find out the difference of these differences of ascension by substracting the lesser out of the greater; which in this example is 4. degrees 45. minutes; which resolved into minutes of an hour, by taking for every degree 4. minutes of an hour, and for every 15. minutes of a degree, one minute of an hour shall amount to 19 minutes of an hour, that is a quarter of an hour and 4. minutes. And so much is the month of December longer than the month of March; Notwithstanding both of them consist of the same number of 31. natural days.    

The third Part.  
Of the Orbs whereof the SPHERES of the Sun and Moon have been imagined to be made, and of their Motions and Uses.  

CHAP. I. Of the Orbs whereof the Sphere of the Sun is made.  
WIthin the Sphere or Orb containing all the Circles that we have hitherto spoken of, and representing unto us the Primum mobile; that is, the first and highest movable Heaven, that hath been imagined by the Astronomers, to show the reason of that daily motion, which appeareth to be in all the Heavens, and of all the apparences that follow thereupon, are included the Spheres and Orbs of the Sun and Moon.  
The sphere of the Sun containeth three Orbs: The uppermost of them (which in this Sphere is signified by the yellow Circle that cometh next within the compass of the Zodiac) is called Deferens apogaeum Solis; that is, the Orb which carrieth about that point, wherein the sun is furthest distant from the earth.  
Next within this Orb is placed the Eccentrick carrying about the body of the Sun; which in this Sphere is represented by the green coloured circle that cometh next under the Deferens apogaeum.  
Again, within this Eccentrick is included the third Orb of the Sphere of the Sun called Deferens Perigaeum solis; that is, the Orb carrying about that point wherein the Sun is nearest to the Earth. This is the nethermost of the three Orbs of the sun, and in this Sphere is represented unto you by the yellow coloured circle next under the sun's Eccentricke.   

CHAP. II. Of the uppermost and nethermost Orbs of the Sphere of the Sun, more particularly.  
IN the uppermost and nethermost of these three Orbs, there be 4. points especially to be considered: That is, the points where they be narrowest and where they be broadest, and where they are of a mean breadth betwixt the narrowest and broadest. For at the narrowest part of the uppermost Orb, where you may see written Aux solis, and the broadest part of the nethermost Orb, is the place of the suns Apogaeum; so that whensoever the Sun cometh there, he is furthest distant from the earth. As you may easily try, if (with a pair of Compasses, or otherwise) you take the distance betwixt the Earth and the Sun, being brought about to that place, and compare the same with the distances that the Sun hath from the Earth in other places. This point is called Aux Solis, and Longitude longior, that is, the point of the sun's furthest distance from the earth. But under the broadest part of the uppermost and uttermost Orb, where you see printed PERIGAEUM, and right above the narrowest part of the nethermost Orb, is the place where the Sun cometh nearest to the Earth, as you may easily find (with your Compasses, or otherwise) in like sort as before was showed. The point where the Sun cometh nearest to the earth, is called oppositum A●gis, and longitudo propior, that is, the point opposite to the Apogaeum, and the nearest distance. And at those parts of this Orb, which are in the midst between the former; the Sun hath a mean distance from the earth: a mean (I say) between the least, and greatest distance. The very point wherein this mean or middle distance happeneth, is showed by the points that are just in the midst between the short lines AB, and IK, which are drawn overthwart on either side of this Orb. These points are called longitudines media; that is, the mean distances of the Sun, because the sun coming to these points, hath a mean distance between the least and the greatest. About these points also, the true motion of the sun, is as it were in a mean between the slowest, which happeneth the sun being about the Apogaeum, and the swiftest, which happeneth about his Perigaeum.  
Moreover the lines A, and K, show the places wherein there is the greatest Prosthaphaerisis, or Equation of the sun: that is, the greatest difference between the true, and middle, or mean place of the sun.  
Lastly, the distance between the lines I, and K, or A, and B, show how much the eccentricity of the sun's eccentricke is, that is, how far the Centre of the eccentricke is distant from the Centre of Earth.   

CHAP. III. To find how much the Sun is nearer or further from the earth, at one time then at another.  
BY means of this Circle, you may easily find with your Compasses, how much the Sun is nearer to, or further from the earth at one time, then at another: for having set one foot of the Compasses upon the utmost edge of the Deferens Apogaeum, under the place of the Sun in the Zodiac, found by the second Prop. stretch out the other foot, to the innermost edge of the same Orb; for then, if you set one foot of your Compasses, upon the utmost edge of this Orb, at the Apogaeum, the other foot turned inwards towards the centre of the Sphere, will show you how much the Sun is nearer to the Earth at that time, then when he is in his Apogaeum: for so much as that foot reacheth within the inner edge of the Orb, so much is the sun nearer. Likewise if you set one foot of your Compasses, upon the uttermost edge of this Orb, at the Perigaeum, and turn the other foot towards the centre of the Sphere, so much as this foot of the Compass, is from the inner edge of the Deferens Paerigaeum, so much is the Sun further distant from the earth at that time, then when he is in his Paerigaeum.   

CHAP. FOUR Of the situation and motion of the uppermost, and nethermost Orbs of the Sun.  
THe uppermost, and nethermost of these three Orbs, called Deferens Apogaeum, & Perigaeum solis, do always answer each to other, in such sort that the broadest part of the one, is always against the narrowest part of the other: And therefore both of them are moved thgether, with one motion about the Axtree and poles of the Ecliptic, making one revolution under the Zodiac, in the space of 17000. years almost. For in Ptolemee his time (that is about the year of our Lord 134.) the place of the Suns Apogaeum, was about the midst of the 6. deg. of Gemini; as it may appear by the 4. Chapter of the 3. book of his Almagest. But in our time we find that it cannot exceed the 7. degr. of Cancer, although after the account of Copernicus, and of the Prutenicke tables, it should be in the 9 degr. of Cancer. So as, if the rest of the motion of the Suns Apogaeum, that is to come hereafter, be proportionable to that is past, the whole revolution thereof shallbe finished in 1699. years under the Zodiac. For in 1463. years betwixt Ptolemee his time and ours, it hath moved about 31. degn therefore it shall move 300. degrees, (that is, the compass of the whole circle) in 16990. years.  
Which number of years being divided by 360. it shall appear that the Apogaeum of the Sun moveth one degr. in little more than 47. years whereby the yearly motion thereof may be sound to be little more than one minute and a quarter.   

CHAP. V. How to find the place of the Suns Aux or Apogaeum, etc.  
THerefore the place of the Suns Apogaeum, being found for the year 1600. to be about 7. degr. in Cancer, the place thereof for any other year before or after, may easily be found in our age, only by subtracting, or adding for every 4, years 5, min. and for every single year 1. minute and a quarter, Although indeed we need not stand so precisely neither upon quarters of minutes, neither yet upon whole minutes, in the place of the suns Apogaeum, which cannot be by any Art so exactly found, but that the diligentest man that is, may err many minutes therein.  
Take for example the year of our Lord 1558. (in which our gracious Q. Elizabeth began her happy reign, which is now 42. years since) taking therefore for every 4. years 5. minutes, that is, for 40. years 50. minutes, and for the two years remaining 2. minutes and one half; that is in all 52. minutes and an half, and subtracting the same out of 7, deg of Cancer, there shall remain the place of the suns Apogaeum at the beginning of her Majesty's reign, in 6. degr. and about 8. min. of Cancer.  
The uses of these two Orbs are these.  
1. First to make the sphere of the Sun concentrical; for these Orbs be framed together, that the narrowest part of the one, answereth always to the broadest part of the other: it cometh to pass by this means, that both the outside, and the inside of the Sphere of the sun, have always the same centre, that the world itself hath.  
2. The second use is to show the reason, and manner of the motion of the Suns Apogaeum and Perigaeum.   

CHAP. VI Of the eccentrick of the Sun, etc.  
THe Orb contained between the two former, and carrying about the body of the Sun itself, is called the eccentricke of the Sun; because it hath another centre, than the centre of the world. The especial reason, that moved the skilful in this celestial science, to make this Orb (wherein the body of the sun is carried) eccentrical, was because they found the apparent motion of the sun under the ecliptic line to be unequal, that is, swifter in the Southerly Signs: and slower in the Northerly.  
For Hipparehus, and Ptolemee found in their times, that the sun continued in the Northern semicircle of the ecliptic, from Aries to Libra, 187. days: and in the other half of the Zodiac, that is Southward from Libra to Aries, 178. days and a quarter only. But in our time by diligent observation it is sound, that the time of the sun's continuance in the first of those semicircles from Aries to Libra, is 186. days 14. hours and an half: and consequently in the other semicircle, from Libra to Aries, 178. days 15. hours and an half. Taking it therefore for a ground, according to the doctrine of Aristotle, that the motion of the celestial bodies is circular and equal; it must needs follow, that a greater part of the circle described by the proper motion of the sun must be contained under the Northerly semicircle of the ecliptic, then under the Southerly: and consequently that the circle or Orb that carrieth about the body of the Sun under the ecliptic, hath another centre than the centre of the Ecliptic.  
2. Another reason to prove, that the Sun is carried in an eccentrical Circle, is the unequal apparent, bigness of the Sun's diameter, the Sun being of the same height above the Horizon and the air alike affected, and alike clear; so as if there were any refraction by reason of the thickness of the air, it must needs be the same in both places. For in Summer, when the Sun is at, or near his Apogaeum, his apparent diameter hath been found by exquisite observation to be 13. minutes 48 seconds. But in winter being about his Perigaeum 33. min. 54. seconds, as it may appear in Copernicus his revolutions 4. book 21. Chapter.  
Therefore seeing every visible object appeareth greater when it is near, and less when it is further removed from us, it is manifest that the Sun appearing greater in winter, then in summer, must needs be nearer to the earth in winter, then in summer.  
The reason of which appearance is most easily showed, by supposing the Sun to be moved, in an eccentrical Orb.  
3. A third reason may be the unequal greatness and continuance of the eclipses of the Moon, even at those times when she hath had the same latitude, or distance from the Ecliptic, and the same distance from the Centre of the earth: which argueth that the conical sharp pointed shadow of the earth, in the place where the Moon in time of the Eclipse passeth through that shadow, at the same distance from the earth, is sometimes greater, and sometimes lesser: whereof there can no cause be showed more reasonable than this, that the Sun is sometimes further distant from the earth, and the maketh the shadow greater and sometimes nearer, and so maketh it lesser. Whereby it is also manifestly proved, that the Sun is moved about another centre than the centre of the earth, and therefore that the circle or Orb, wherein the Sun is moved, is an Eccentricke.   

CHAP. VII. Of the uses of the Sun's eccentrical Orb.  
THerefore the uses of the Sun's eccentricke may be these:  
1. First to show the reason of the apparent inequality, which seemeth to be in the motion of the Sun: for although the Sun mo●e equally in his own Obbe, and about his own centre; yet to them that are at the centre of the world, or upon the earth, he shall seem to move unequally; that is, swiftly when he is in that part of his eccentricke which is nearest unto the earth; and slowly when he is farthest from the earth. And therefore in summer, when the Sun is about his Apogaeum, and in his greatest distance from the earth, he seemeth to move little above 57 min. in one day. But in winter, being about his Perigaeum and nearest unto the earth, he seemeth to move more than 16. minutes: whereas notwithstanding he moveth equally in his Eccentricke, every day about nine and fifty minutes and 8. seconds; and so finisheth his revolution in 365. days, and six hours almost.  
2. The second use of the Sun's Eccentricke, may be to show the reason why the Sun appeareth greater at one time then at another; for the Sun being in those parts of the eccentrick that are nearest unto us, seemeth greatest, and when he is in those parts of his eccentrick that are furthest from us, he appeareth to be least.  
3. And lastly the inequality of the Sun's distance from the earth, caused by his eccentrick, is one especial cause of the inequality of the Eclipses, both of the Sun and Moon.   

CHAP. VIII. The definitions of certain Astronomical words of art, for the better understanding of the Theoric of the Sun.  
1. WHat the Aux or Apogaeum of the Sun is it hath been partly showed already: that ●●mely it is that part, or rather point of the Orb carrying the Sun's Apogaeum, wherein the said Orb is thinnest, or narrowest: Or it is that point of the eccentrick which is furthest distant from the earth, and is always showed by a right line understood to be drawn from the centre of the world, by the centre of the eccentrick, unto the Orb carrying the Sun's Apogaeum. Which line is therefore called the line of the Sun his Aux or the line of the Suns Apogaeum.  
2. The motion of the Aux, or of the Apogaeum of the Sun (which is also called the Suns Aux in the second signification) is nothing else but the arch of the Ecliptic, contained between the beginning of Aries, and the line of the Suns Apogaeum, drawn forth to the Zodiac; where this line also showeth the place of the Suns Apogaeum.  
3. The middle or mean place of the Sun in the Zodiac, is showed by a line drawn from the centre of the world unto the Zodiac, equidistant from the centre of the Eccentricke, and of the Sun.  
4. This line is therefore called the line of the mean or middle place of the Sun.  
5. The middle or mean motion of the Sun is the arch of the ecliptic between the beginning of Aries, and the middle place of the Sun.  
6. The true place of the Sun is showed by a straight line drawn from the centre of the earth by the centre of the Sun unto the Zodiac, which line is therefore called the line of the true place of the Sun.  
7. The true motion of the Sun is the arch of the ecliptic from the beginning of Aries, unto the true place of the Sun.  
8. The argument of the Sun (at the 〈◊〉 ●erme it) or the motion of the Sun's Anomaly (as Copernicus calleth it) is the arch of the ecliptic contained between the place of the Suns Apogaeum and the middle place of the Sun according to the order and succession of the Signs. This arch is called the argument, or motion of the sun's Anomaly or irregularity, because that by it is always found how much the suns true motion which (is unequal and irregular) differeth from his middle motion; which difference they call the Sun's equation, or prosthapheresis.  
9 The equation, or prosthapheresis of the Sun is nothing else but the arch of the ecliptic contained between the true, and middle places of the sun.  
This arch is called the sun's equation, because it maketh the suns middle motion equal to his true motion, being added to it or subtracted from it, as occasion requireth: for which cause it is more significantly and fitly called Prosthaphaeresis, that is as much to say, as that which is to be added to or subtracted from the middle motion, that so we might have the true motion. For so long as the Sun is in the semicircle of his eccentrick, descending from his Apogaeum to his ●●●gaeum, so long this Prosthapheresis is to be subtracted from the middle motion: but the Sun being in the other half of his eccentrick ascending, the Prosthapheresis or equation of the Sun must be added to the middle motion, that 〈…〉 motion and place of the Sun may be found, Because that in the first semicircle of the eccentricke descending, the middle place of the Sun goeth before the 〈◊〉, and the middle motion is 〈…〉 greater 〈…〉 the Sun, and therefore the difference of these 〈◊〉 motions, (that is to say, the 〈◊〉 or Prosthaphaeresis) must be subtracted, to findeth 〈…〉 for the true place of the Sun goeth always 〈…〉 motion and place of the same,   

CHAP. IX. Of the uppermost Orb of the Sphere of the Moon carrying the Dragon's head and tail.  
NExt within the Orbs of the Sun in this Sphere are contained the Orbs of the Sphere of the Moon: which 〈…〉 in number.  
The uppermost of them (which in this Sphere is next under the Orb that carrieth the Sun's Perigaeum and is coloured with red) is called the carrier of the Dragon's head and tail, or 〈…〉 which is as much to say as the carrier of the knots, that is of the two intersections, or points wherein the rest of the Orbs of the Moon, do cross overthwart this Orb. This Orb is divided into four nineties of degree, for the easier reckoning of the motion and place of the Dragon's head or tail in this Sphere. And it is moved about in 18. julian years 224. days 3. hours and 5. minutes almost, from the East Westwards, under the ecliptic. By reason of this motion it cometh to pass, that the Eclipses, or rather the places wherein the eclipses of the Sun or Moon do happen in the Heavens, are removed continually more backwards in the Zodiac, contrary to the order and succession of the Signs.  
As for example; the eclipse of the Moon happening this present year 1600. the 20. of januarie near unto the Dragon's tail about the 9 degree and 40. min. of Leo; the next eclipse that shall happen near the same intersection of the Dragon's tail, in the year 1601. the 29. of November, shall be in 17. degrees and an half of Gemini; And that eclipse which shall be the next year after near the same intersection the 19 of November in the morning, shall be about the 6. degree and 40. minutes of Gemini, etc.  
All this removing of the Eclipses backwards cometh to pass, by reason of the motion of this Orb carrying the Dragon's head and tail, contrary to the course and order of the Signs.  
This Orb continueth always right under, 
The situation of the Orb carrying the Dragon's head and tail.  and even with the Orbs of the Sphere of the Sun which abide always in all parts just under the ecliptic line, and hath his centre agreeing, and all one with the centre of the world, and of the ecliptic: And therefore the poles and axtree, about which this Orb is turned, agree justly with the axtree of the Ecliptic.  
The rest of the Orbs of the Moon, 
The situation of the rest of the Orbs.  that are contained within this, have all their plains agreeing in one, and lying even one with another. But the one half of all their plains, ariseth above the plain of the former Orb, and of the Ecliptic, towards the North pole of the Zodiac: and the other half descendeth beneath the plain of the ecliptic, toward the South pole: even as the one half of the Zodiac ariseth above the Equinoctial circle towards the North: and the other half descendeth towards the South. And as the angle of intersection, or obliquity of the ecliptic with the Equinoctial circle, is 23. degr. and an half or little more: so the angle of intersection, or obliquity of the plains of these Orbs of the Moon, from the plain of the Ecliptic, and of the former Orb carrying the Dragon's head and tail, is 5. degrees, or (according to Tig●● Brahe his observation) 5. degr. and a quarter almost sometimes, and sometimes less than 5. degr.  
That point or intersection of these Orbs with the former, from which they begin to arise about the plain of the ecliptic towards the North, proceeding East-wards, is called the Dragon's head; and is signified by this character ☊: and the other point or intersection diametrally opposite unto this, is called the Dragon's tail, which is also signified by the former character turned up side down after this manner, ☋.  
The two points of these Orbs that are furthest distant from the plain of the 〈◊〉, are called the bounds or limits of the Moon's latitude, and they are 90. deg. from the Dragon's head and tail, and 5. deg. and a quarter almost from the plain of the Ecliptic, according to the obliquity, or greatest declination of the plains of these Orbs, from the plain of the ecliptic: Of these two points, that which is in the north side of the ecliptic, is called the North limit, or bound of the Moon's latitude; and chose, the other point opposite to this on the south side of the Ecliptic, is called the South limit of the Moon's latitude. And when the Moon cometh to either of these two points, she hath her greatest latitude.   

CHAP. X. Of the Orbs carrying the Moons Apogaeum and Perigaeum.  
NExt within the Orb carrying the Dragon's head and tail, is contained the Orb called Deferens Apogaeum lunae which is the point wherein the Moon is furthest distant from the earth.  
And under this Orb is placed the Moon's Eccentrick, which is also called Deferens Epiculum Lunae; that is the Orb carrying the Moon's Epicycle.  
Again within this eccentrick of the Moon, is contained the least and lowest Orb, of all that are in this Sphere, Which they call Diferens Perigaeum Lunae; that is, the Orb carrying the Moons Perigaeum, which is the point wherein the Moon cometh nearest to the earth.  
The uppermost and nethermost of these three Orbs, that is to say, the Orbs carrying the Moons Apogaeum and Perigaeum (both which Orbs in this Sphere are coloured with blue) are always placed in such sort, that the nar●●west part of the one, is continually answerable to the broadest part of the other; whereby it cometh to pass, that the Sphere of the Moon is made concentrical, that is to say, to have the same centre with the world: which also is one especial use, why these Orbs were divided.  
Another use of these Orbs, is to show the reason of the motion of the Moons Apogaeum and Perigaeum: Therefore both these Orbs are moved together with one motion equally, about the centre of the world, in the same time from the East Westwards, in the space of 32. days 3. hours and 5. minutes almost: So moving in one day 11. deg. 12. min. and 1. third part almost. The axtree, about which these Orbs are moved equally, passeth through the centre of the world and of the ecliptic: but the poles of these Orbs differ from the poles of the Ecliptic and of the Orb carrying the Dragon's head and tail, by the space of 5. degr. and a quarter, or thereabouts, which poles are carried about the poles of the Orb carrying the Dragon's head and tail, with the motion of the same Orb, in the space of 19 years almost. Whereby it cometh to pass, that the poles of the Orb carrying the Apogaeum and Perigaeum of the Moon, describe certain little circles about the poles of the Orb that carrieth the Dragon's head and tail, even as the Arctic, and Antarctick circle in the ordinary Sphere, are described by the motion of the poles of the Ecliptic, carried about daily with the motion of the first and highest movable Sphere, in the space of 24. hours almost.   

CHAP. XI. Of the eccentricke of the Moon.  
THe Eccentrick of the Moon contained between the two former Orbs and coloured with a sad yellow colour in this Sphere, is moved equally about the centre of the same Orbs, from the West towards the East, finishing his motion under the Zodiac, in the space of 27. days, and 8. hours almost: and with this motion, it carrieth about the Moon's Epicycle equally, under the Zodiac.  
Therefore the motion of this Orb, about his own centre, must needs be unequal, that is to say, swifter in those parts that are about the Apogaeum, and slower in the lower parts about the Perigaeum: Because that greater arches of the Eccentrick, do answer to equal arches of the Zodiac about the Apogaeum, then about the Perigaeum of the Eccentrick.  
The axtree about which this Orb is moved, is always in all places equidistant from the axtree of the Orb carrying the Apogaeum of the Moon: and the poles of the axtree of the Moon's eccentrick, are fastened in the Orb carrying the Moons Apogaeum, equidistantly from the poles of the same Orb: therefore these poles together with the whole axtree of the eccentrick, are carried and equally moved about the poles and axtree of the Orb carrying the Apogaeum from the East, towards the West. With this motion therefore, the poles and centre of the eccentrick, describe certain little circles of equal bigness, about the poles, and centre of the Orb carrying the Apogaeum, from the East Westwards. And therefore also the Apogaeum of the eccentrick, is moved about equally, under the ecliptic, contrary to the order of Signs from the East Westwards. Whereby it cometh to pass, that both the Apogaeum and centre of the eccentricke, are sometimes under the Ecliptic, that is, when they are under the Dragon's head or tail: but for the most part they are beside the plain of the ecliptic, either towards the North, or else towards the South.  
Hereby also it appeareth, that the plain of the Ecliptic doth not always divide the plain of the eccentricke into epqall parts or halves; but then only, when the Centre and Apogaeum of the Eccentrick, is right under the Dragon's head or tail; for then only the plain of the Ecliptic deuideth the plain of the Eccentrick, by the centre thereof; and consequently deuideth it precisely into two halves. Otherwise, if the Apogaeum of the eccentrick, be not under the Dragon's head or tail, look on which side of the plain of the ecliptic the Apogaeum is, for on the same side of the ecliptic is the greater part of the eccentrick.   

CHAP. XII. In what proportion the Moon's eccentrick, and Orb, carrying her Apogaeum are moved.  
NOw the Eccentrick of the Moon, and the Orb carrying her Apogaeum, are moved in such sort, that the middle place of the Sun, is always right in the midst between the centre of the Epicycle carried in the eccentrick, and the Apogaeum of the Eccentrick; except it be when the centre of the epicycle is in conjunction, or opposition to the middle place of of the Sun. For in every middle conjunction and opposition of the Sun and Moon, the centre of the Epicycle, and the Apogaeum of the eccentrick are united together; But in the conjunction they are both conjoined with the middle place of the Sun; and in the opposition they are both together opposite to the same. Whereof it followeth, that in the first and last quarters of the Moon, the centre of her epicycle is diametrally opposite to the Apogaeum of her eccentrick.  
Hereof it cometh to pass, that although the Moon have the same position in her epicycle at the time of the new and full Moon, and of the first and last quarters; yet the equation, or prosthaphaeresis of the Moon's Argument (as they call it) that is the difference between the true, and middle places of the Moon, is always greater in the first and last quarter, then in the full and new Moon. Hereby likewise it appeareth that in the time contained between new Moon and new Moon (which they call Mensem synodicum, that is the month coniunctional, or the time from conjunction to conjunction) the centre of the epicycle maketh two complete revolutions, under the Orb carrying the Apogaeum of the Moon's eccentrick.  
And therefore in every month, the centre of the epicycle cometh twice to the Apogaeum and twice to the Perigaeum of the eccentrick; and so the monthly motion of the centre of the epicycle, describeth an oval figure: the ends whereof are always towards the place of the full and new Moon, and the ●ides towards the places of the first and last quarter.  
By this that hath been spoken, it is also manifest, that if the middle motion of the Sun, be subtracted out of the middle motion of the Moon, there remaineth the middle motion of the Moon's longitude from the sun, and that if this longitude again be doubled, you shall have the motion of the centre of the Moon's Epicycle from the Apogaeum of her eccentrick, which motion they call the centre of the Moon,   

CHAP. XIII. Of the Epicycle of the Moon, and how it is moved.  
THe little Orb placed in the Eccentrick, is called the Epicycle of the Moon; in the circumference whereof is also placed the body of the Moon, represented by the round Beade, set into the Moon's Epicycle in this Sphere.  
The plain superficies of this epicycle agreeth even with the plain of the eccentrick: and the axtree about which it is moved, is perpendicular to the plain of the eccentrick. This Epicycle is moved equally from his middle Apogaeum, about his own centre and axtree from the East Westwards, contrary to the motion of the eccentrick, carrying forwards the body of the Moon with this motion 13. degrees and almost 4. min. every day, and finishing his revolution in 27. days 13. hours and 19 minutes almost.  
The middle Apogaeum of the Epicycle is showed by a right line, imagined to be drawn, from that point of the little circle (described by the motion of the centre of the Moon's eccentrick) which is opposite to the centre of the Eccentrick, by the centre of the Epicycle unto the upper part of the Epicycle.  
But the true Apogaeum of the Epicycle, is showed by a right line, understood to be drawn from the Centre of the earth by the centre of the Epicycle, unto the upper part of the circumference thereof.  
By the motion of this Epicycle it may easily be conceived why the Moon seemeth to move sometimes swifter and somteimes slower: 
Why the Moon seemeth sometimes to move swifter, sometimes slower.  For seeing that the upper part of the Epicycle, moveth contrary to the motion of the Eccentrick from the East Westwards, when the Moon cometh in that part, she must needs seem to move more slowly, to them that are at the centre of the world.  
But when the Moon cometh in the neither part of the Epicycle, the Eccentrick carrieth the Epicycle and the Epicycle carrieth the body of the Moon both one way: that is, from the West East-wards, and therefore at that time the Moon seemeth to move more swiftly. According as you may see in Ephemerideses, the d●●●ne motion of the Moon to be sometimes little more than 11. degrees and sometimes again little less than 15. degrees. The true motion of the Moon seemeth then to be swifter, when the Moon is in the Perigaeum of her Epicycle, and the Epicycle in the Perigaeum of the Eccentrick; because than she is not only carried forwards the same way both by her Epicycle and Eccentrick, but she is also at that time nearest unto us for which cause her motion shall seem swifter, than when the Epicycle is in other parts of the Eccentrick. 〈…〉   

CHAP. XIIII. The definitions of certain Astronomical words of Art, for the better understanding of the Theoric of the Moon.  
1. THe line of the Moon's middle motion, is a line understood to be drawn from the centre of the earth, by the centre of the Moon's Epicycle, unto the Zodiac.  
2. This line showeth the middle place of the Moon in the Zodiac.  
3. And the middle motion of the Moon, is the arch of the Zodiac, from the beginning of Aries, unto the same line.  
4. So likewise the line of the true motion, or of the true place of the Moon, is drawn from the centre of the world, by the centre of the Moon, to the Zodiac.  
5. This line therefore showeth the true place of the Moon in the Zodiac.  
6. And the true motion of the Moon, is the arch of the Zodiac, from the beginning of Aries, unto the true place of the Moon.  
7. The middle longitude of the Moon from the Sun, is the arch of the Zodiac, from the middle place of the Sun Eastwards unto the middle place of the Moon.  
8. And this arch doubled, is called the doubled longitude of the Moon from the Sun, or the centre of the Moon (as the Alphonsines call it) which is nothing else but the arch of the Zodiac, between the place of the Apogaeum of the Eccentrick and the middle place of the Moon. It is called the doubled longitude of the Moon from the Sun, because it is always twice so much, as is the middle longitude of the Moon from the Sun.  
9 And it is called the centre of the Moon, because it showeth the distance of the centre of the Moon's Epicycle from the Apogaeum.  
10. The equation, or Prosthaphaeresis of the centre, is the arch of the Epicycle, between the middle and true Apogaeum of the Epicycle.  
This equation or Prosthaphaeresis is nothing at all, when the centre of the Epicycle is in the Apogaeum, or Perigaeum of the Eccentrick. But the Epicycle being in any other part of the Eccentricke there is always some equation of the centre; yea in some parts thereof, where it groweth greatest, it is 13. degr. 9 minutes: and so long as the centre of the Epicycle, is in the half of the Eccentrick descending from the Apogaeum to the Perigaeum, that equation is to be added to the motion of the Epicycle; but in the other hall of the Eccentricke ascending;, it must be subtracted; that so the true Argument 〈…〉 the Epicycle may be had.  
11. The Argument, or Anamalie of the Moon is nothing else, but the motion of the Moon's Epicycle.  
12. The true or middle argument, 〈…〉 is the arch of the Epicycle from the true or middle Apogaeum of the Epicycle, unto the centre of the body of the Moon, reckoned that way, which the epicycle moveth.  
13. The equation of the argument, or Prosthaphaeresis of the epicycle, is the arch of the Zodiac, 〈…〉 the middle, and 〈◊〉 place of the Moon. This equation is 〈◊〉, when the Moon is in the true Apogaeum, or Perigaeum of her epicycle. But it is greatest, when the centre of the Moon cometh 〈…〉 of the world, and touching the epicycle, when it is in the Perigaeum of the eccentrick.  
And the Moon being in the first, that is in the 〈…〉 of the Epicycle, 〈…〉 from the true Apogaeum thereof, the middle place of the Moon go 〈◊〉 before the true place, and the equation of the argument must therefore be subtracted: but when the Moon is 〈…〉 semicircle of the epicycle, 〈…〉 before the middle place, and to that equation must be added to the Moon's 〈◊〉 middle motion, that the true motion and place of the 〈…〉 may be found.   

CHAP. XV. The reason of the variety of the Moon's equation, etc.  
THis equation becometh lesser or greater, according as the epicycle 〈…〉 or nearer to the centre of the world. The least equations are, when the epicycle is in the Apogaeum of the eccentrick, and contrariwise, the greatest must happen, the epicycle being in the Perigaeum of the eccentrick.  
The difference between these greatest, and least Equations, Ptolemee and Copernicus call the excess: but 〈◊〉, and the Alphonsines call it the diversity of the Diameter; because that difference of the equations ariseth by reason of the divers apparent highness of the diameter of the Epicycle, according as it is nearer to us, or further from us.  
Therefore in the Astronomical tables, they use to set down those equations only, which happen when the Epicycle is in the Apogaeum of the eccentrick, which are the least equations, whereto they also adjoin the excess, or diversity of diameter, showing how much those equations, which happen when the Epicycle is in the Perigaeum of the Eccentrick, exceed those which happen, the epicycle being in the Apogaeum of the eccentrick. Moreover, there are annexed certain min. which they call Scrupula, or minuta proportion●alia: that is, proportional minutes: whereby is found, how much of the said excess, is to be added to the foresaid equations, when the epicycle is in any other part of the eccentrick, then in the Apogaeum: that so the true equation of the argument, for the same part of the eccentrick might at any time be found. For then only is that whole excess to be added, when the epicycle is in the Perigaeum of the eccentrick. But if the epicycle be in any other part of the eccentrick; then look what proportion 60. hath to the whole excess, the same proportion have the proportional minutes, answerable to that part of the eccentrick, wherein the epicycle is, unto the part proportional of the excess, which (part proportional) being added to the equation before found, shall give you the true equation.   

CHAP. XVI. The reason 〈◊〉 Moons proportional minutes, etc.  
THe reason of those proportional minutes, may in some sort be showed, by those concentrical arches of circles, which you see drawn upon the Moon's eccentrical Orb, in this Sphere: but indeed all those arches must be understood, to have always the same centre with the world, and not to be moved about together with the Eccentrick. The uppermost of them is to be drawn by the centre of the Epicycle being in the Apogaeum of the Eccentrick, and the nethermost is drawn by the same centre when it is in the Perigaeum of the Eccentrick: so as the distance of these two arches, or Peripheryes, is just twice so much as the eccentricity; that is the distance of the centre of the Eccentrick, from the centre of the world, showed by the distance of the short lines NO, or FF, upon the Orb carrying the Apogaeum; or of PQ, or GH, upon the carrier of the Perigaeum of the Moon.  
The whole distance, between these two peripheries, from the uttermost to the innermost, is understood to be divided into 60. equal parts, imagining every one of these to contain 10. as may appear by the figures set to every one of them, from the uppermost to the nethermost, in this order: 10. 20. 30. 40. 50. 60.  
Now the intersections of these Peripheries with the Eccentrick (that is) with the uppermost of the two divided Peripheries, which are drawn round about through the midst of the Moon's eccentrical Orb) do show what proportionable min. answer to any part of the eccentrick, 
To find the Moon's proportional minutes.  after this manner: In the uppermost of the two foresaid graduated Peripheries, look that distance of the centre of the moons Epicycle from the Apogaeum of the Eccentrick, (that is to say) that doubled longitude of the Moon as the Alphonsines call it) which you desire: Then look which of the consentricall arches before mentioned passeth by the the term, or end of that distance or doubled longitude: And thirdly, look about in the same arch, what number is set thereupon, for that showeth you the number of the proportional minute. answerable to the situation of the Epicycle, at that distance from the Apogaeum of the Eccentrick.  
These proportional min. therefore may be defined to be nothing else, 
What the proportional minutes of the Moon are.  but the sixtieth parts of the diversity of diameter, or of the excess wherewith the equations of the argument, or Prostaphaeresis, of the Epicycle are to be augmented when the Epicycle is any other part of the Eccentrick, then in the Apogaeum.  
Otherwise also, these proportional min. may be defined, to be sixtieth parts of the excess, wherewith the line drawn from the centre of the earth, to the Apogaeum of the Moon's Eccentrick, exceedeth the line drawn from the same centre to the Perigaeum of the Eccentrick: For these sixtieth parts also may not unfitly be called proportional min. because that always look how many of these parts there are left without the circumference of the Eccentrick, or beyond the centre of the Epicycle; so many of the former sixtieth parts of the diversity of diameter, or of the excess of the Prostaphaerses of the Epicycle, must be added to the Equation of the argument, that the true equation of the argument may be had, for that position, or situation of the Epicycle, in the Eccentrick.   

CHAP. XVII. The reason of the Eclipses of the Sun and Moon, etc.  
NOw by this Sphere, it may easily be conceived, why there is not an Eclipse, in every conjunction or opposition of the Sun and Moon. For seeing that the Moon hath for the most part a greater apparent latitude, than the visible or apparent conjoined semediameters of the Sun and Moon in the conjunction: and because the true latitude of the Moon, is also for the most part greater than the apparent semediameters of the Moon and shadow of the earth (at that place where the Moon should pass through that shadow) in the oppossition, to make an Eclipse: it cometh to pass, that in most conjunctions and oppositions of the Sun and Moon, there is no Eclipse. And the reason hereof is this, because that the Moon cometh under the way of the Sun (which we call the Ecliptic line) only twice in a month, and those 2. points (wherein the ways of the Sun and Moon cross each other) only twice in a synodical month, which two points we called the Dragon's head and tail; whereof we have also spoken before,) Wherefore, seeing the Sun (going but once only through the compass of the Ecliptic in a year) can come but once in a year to either of those points; the Moon for the most part, when she cometh to be in opposition, or conjunction with the Sun, must needs be foe far wide from the Ecliptic line, or way of the Sun, either towards the North or South: that she can neither come betwixt us and the Sun in the conjunction, nor yet within the compass of the shadow of the earth in the opposition.  
But when the Sun cometh near either of those points (which happeneth once in six months (there must needs for the most part be some Eclipse, either of the Sun or Moon, or both.   

CHAP. XVIII. Of the diversity of the bounds or spaces, within which an Eclipse may happen: and the reason of that diversity.  
THe bounds or distances from the Dragon's head or tail, within which there may happen an Eclipse of the Moon, are sometimes greater and sometimes less, by reason of the diverse distances of the Sun, or Moon, or both of them, from the earth. For seeing the body of the Sun is greater than the globe of the whole earth (as it is manifestly demonstrated by Ptolemee and Copernicus) it must needs be, that the greater distance the Sun hath from the earth, the greater shadow must the earth have; and the nearer the Sun is to the earth, the less shadow shall the earth have, at the place of the Moon's passage through the shadow, at equal distances from the earth.  
chose, the further that the Moon is from the earth, the less shall the shadow of the earth be, and the nearer the Moon is to the earth, the greater shall the shadow be, at the place where the Moon is to pass through the shadow.  
The greatest distance therefore from the Dragon's head or tail, wherein there can at any time happen any Eclipse of the Moon, is about 13. degrees. And the least distance at which it is possible for the Moon to avoid an Eclipse, is about 10. degr. and one third part of a degree; which happeneth when the Moon is in the Apogaeum of her Epicycle, in her greatest distance from the earth, and the Sun in his Perigaeum, in the time of his greatest eccentricity, for then the Sun cometh nearest to the earth and maketh the least shadow: as contrariwise at the same time of his greatest eccentricity, being in his Apogaeum, he hath his greatest distance from the earth, and so maketh the earth cast forth her greatest shadow. At which time, if the Moon also chance to be in the Perigaeum of her Epicycle, and so in her nearest distance from the earth, she may be something Eclipsed, although she be full 13. degrees or something more from the Dragon's head or tail.   

CHAP. XIX. How to find the place of the Dragon's head and tail for any time.  
NOw the place, and time of the full Moon, being easily known by some Almanac, or Prognostication; it shall not be hard, to give a reasonable near estimate, and to foretell both the time, and quantity of the Eclipse of the Moon, the place of the Dragon's head and tail, being first known after this manner.  
The place of the Dragans head, being first given for any time, for every year before the same time, add to the same place: and for every year after the same time subtract 19 degrees and one third part of a degree and for every month a degree and an half and a tenth part of a degree and for every day 3. minutes and the remainder shall show you the place of the Dragon's head after the same time: or the sum before that time without any great error.  
As for example, The 30. of june this present year 1600. suppose you would know the place of the Dragon's head: The place therefore of the Dragon's head being first given, for the beginning of the same year, in 0. degree 45. minutes, of Aquarius: and six months only of that year being passed, I take for those six months 6. degrees and 6. half degrees that is 9 degrees and sixteenth parts of a degree that is 36. min. the sum of all which is 9 degrees and 36. minutes.  
Which being subtracted out of 0. degree, 45. minutes of Aquarius, there remain 21. degrees 9 min. of Capricorn, for the place of the Dragon's head at that time.   

CHAP. XX. A table for finding the place of the Dragon's head and tail more exactly and the declaration thereof.  
But if you would have the place of the Dragon's head more exactly, you may find the same most easily, by means of the table following, for any time within the space of these 20. years, yet to come.  
This table containeth three principal parts or columns, the first part showeth you in what sign, degr. and min. the Dragon's head is, at the beginning of any year; from this present year 1600. till the year 1620. The second part showeth how much the Dragon's head moveth, in any number of months of the year: the third part giveth you the motion of the Dragon's head, in any number of days of the month.  
The place of the Dragon's head.  Year Sign. Deg. Mt. Month's Complete. De. Mi. Da. De. Mi. 1600 Aquarius 0 45 january 1 38 1 0 3 1601 Capricorn 11 21 February 3 8 2 0 6 1602 Sagittar. 22 2 March 4 46 3 0 10 1603 Sagittar. 2 42 April 6 22 4 0 13 1604 Scorpio 13 22 May 8 0 5 0 16 1605 Libra 23 59 june 9 36 6 0 19 1606 Libra 4 39 july 11 14 7 0 22 1607 Virgo 15 19 August 12 53 8 0 25 1608 Leo 25 59 Septemb. 14 28 9 0 29 1609 Leo 6 35 October 16 7 10 0 32 1610 Cancer 17 15 Novemb. 17 42 11 0 35 
●t the beginning of the 〈◊〉 of our ●●rd.  1611 Gemini 27 55 Decemb. 19 21 12 0 38 1612 Gemini 8 35   13 0 41 1613 Taurus 19 12   14 0 44 1614 Aries 29 52   15 0 48 1615 Aries 10 32   16 0 51 1616 Pisces 21 12   17 0 54 1617. Pisces 1 49   18 0 57 1618. Aquarius 12 29   19 1 0 1619 Capricorn 23 9   20 1 4 1620 Capricorn 3 49   21 1 7       22 1 10       23 1 13       24 1 16       25 1 19       26 1 13       27 1 26       28 1 29       29 1 32       30 1 35   

CHAP. XXI. To find the place of the Dragon's head or tail, by the former table.  
Find out in the former table, the month next going before the month given; find out also the day of the month, Add together the numbers of degrees and minutes answerable to that month and day of the month, and subtract the same out of the place of the Dragon's head at the beginning of the year, adding thereto 30. degr. (●●at is the whole sign next going before resolved in to degr.) if the Sun aforesaid be greater than the number of degr. showing the place of the Dragon's head at the beginning of the year: so shall you have the place of the Dragon's head for the time given. And the point of the Zodiac opposite to this, is the place of the Dragon's tail.  
Take for example, The 29. of November 1601. I find therefore against October (the month going next before November) 16. degrees 7. minutes and against the 29. day 1. degree 32. minutes, the sum of both these added together is 17. degrees 39 min. the place of the Dragon's head for the beginning of the year 1601. is 11. deg. 21. min. of Capricorn, which because they be less than 17. deg. 39 min. I add unto them 30. deg. that is the whole sign of sagittary, and the sum of both is 41. deg. 12. min. out of which subtract 17. deg. 39 min. and there shall remain 23. deg. 42. minutes of sagittary; the place of the Dragon's head at that time. And the point of the Zodiac which is opposite hereto (that is the 2● 〈◊〉 42. minutes of Gemini) is the place of the 〈◊〉 tail.   

CHAP. XXII. To know at what time there shall be an Eclipse of the Moon.  
THe place of the Dragon's head being thus known, find out the same place upon the horizon of the Sphere, and see what day and month answereth thereto find out also the place of the full Moon, which happeneth next before or after that day, which place if it chance to be within 11. or 12. deg. either before or after that point of the Zodiac which is opposite to the Dragon's head, there must needs be for the most part in Eclipse of the Moon.  
Likewise if you find what day and month is answerable to the place of the Dragon's tail upon the horizon of the Sphere if the place of the full Moon which happeneth next before or after that day chance to be within 11. or 12. degrees of the Dragon's head, for the most part there shall be an Eclipse of the Moon.  
As for example, The 20. of januarie last this present year 1600. the place of the Dragon's head was found (by the former Chapter) to have been in 29. deg. 41. min. of Capricorn; whereto there answereth in the horizon the 10. day of januarie the place of the full Moon happening next after, upon the 20. of the same month in the morning must needs be in the place opposite to the place of the Sun the same 20. 〈◊〉 Therefore because 〈◊〉 Sun that day is in 9 deg. 〈◊〉 one half of Aquarius, therefore the place