A NEW HANDLING OF THE PLANISPHERE, DIVIDED into three Sections.  
IN THE FIRST IS A PLAIN AND sensible explication of the circles of the Sphere, and such terms as appertain unto the doctrine de primo mobili; after a Method not heretofore used in any language.  
THE SECOND SHOWETH HOW UPON any plain, being Paper, Pastboord, or a Latin plate, having one circle divided into degrees, and crossed with two Diameters at right angles, most Conclusions of the Astrolabe may for all Latitudes or Countries be readily and exactly performed only with Ruler and Compasses.  
IN THE THIRD, BEING A SUPPLEMENT Organical, is contained the making of certain easy instruments for the perfecter working the former Conclusions, as to know what degrees and minutes be in any circumference given, with other necessary matters about the Meridian line and the height of the Pole.  
Pleasant and profitable generally for all men, but especially such as would get handines in using the Ruler and Compass, and desire to reap the fruit of Astronomical and Geographical documents, without being at the charge of costly Instruments.  
Invented for the most part, and first published in English by THOMAS OLIVER.  
AT LONDON Imprinted by FELIX KYNGSTON, for Simon Waterson and Ralph jacson. 1601.   

TO THE RIGHT WORSHIPFUL SIR JOHN PETER KNIGHT, at Thornedon in Essex.  
KNowing well (right Worshipful) after the dispatch of much weighty business, which for the good of others you willingly undertake, and, by report of the shire, with very great discretion, and all uprightness perform, that as it were for your recreation, you perused often books of Geography: some while since I sent the manner how three or four problems Geographical might be wrought upon a plain with Ruler and Compass only. At my leisure afterward taking that subject in hand again, I followed it so far, that I devised after the like way to handle most conclusions of the Astrolabe, or problems Astronomical concerning the Primum mobile. When I had almost brought my purpose to effect, it was my chance to light upon Clavius his Astrolabe, where I found as much promised, and performed even by Ptolemy's Analomma, the very model of johannes de Roias his Planisphere, as learned Guidus Vbaldus hath fully proved, and also the ground upon which I have framed much of my work. Notwithstanding, perceiving (as by comparison every one may see) that except the 5. and 28. taken out of Manrolycus, the 14. and 23. out of Commandine and Vbaldus, my constructions were very far differing from Clavius, amending those defects which Gemma noteth in Roias his work, besides taking the matter to be commodious, and not unpleasant for travelers, and all such in general as delight in Astronomical and Geographical devices, I set light the reproachful name of an English Scribbler, and took courage by publishing it to the view of the world, to hazard myself to every readers censuring. Upon this resolution I am bold to use your Worship's name in the front of this book, testifying thereby the earnest and unfeigned affection which I bear, and long have borne both to you and yours: in regard whereof making no doubt, but that this little mite shall be gently accepted, with remembrance of my duty to your Worship I take my leave. S. Edmunds Bury in Suffolk, 6. jan. 1600. after our English account.  
Your Worships in all duty most affectionate, THOMAS OLIVER.   

To the Reader.  
SEeing the matters here entreated of admit no eloquence, but must in plain attire be simply offered to the view, though I could (gentle Reader) I would not flourish with thee, in pleading for myself or this treatise: therefore referring all to thy friendly censure, I will spend this void paper in advertisements for the better understanding, and easier performing of that which is delivered.  
The fittest matter, whereupon you may work these conclusions following, will be a plate of brass, and the Compass, Pritchet, Pointer or Steele, wherewith you must draw your lines, would be either of the same substance, or else Iron rather than Steel: for if you make not your strokes very deep, by rubbing the plate with a wet Pumice, you may easily take all out when you will, and thereby also tarrish or darken it in such sort, that you may presently at your ease, and very exactly draw, and work upon it again.  
When you begin to practise, it will help very much to work one conclusion by itself, and to note the lines, as you draw them, with the characters, that are named in the peculiar construction: but when you are something perfect, this shall not need.  
You may also, having a quadrant divided, make any paper serve the turn for these purposes. Opening your Compass to the distance of 60. degrees, at that distance describe a circle, and subtending four times in it a line equal to that line, which subtendeth the quadrants circumference, if you join the opposite points, it shall be crossed with perpendicular Diameters, then transferring the circumferences given or found from the quadrant to the circle, or from the circle to the quadrant, you may find the quantity of any circumference as well, as though your circle were divided into degrees.  
Where you desire exactness, you must rest neither on this way, nor on the instrument in the beginning of the second section, but use that quadrant which you are taught to make in the beginning of the third section, by it you may know the just content of any given circumference in degrees and minutes exactly, and let this be a general warning for all conclusions.  
I have barely set down the constructions without demonstrations, as the fittest for their capacities, whose good I specially aim at: yet is there nothing here propounded that hath not demonstratively been examined: and for experimental trial the working by these conclusions may be compared with tables calculated for the same purpose, which, if handines and diligence be used, will agree.  
Since my writing of the second section, I have found how the conclusions therein omitted may be perfectly wrought without the help of any Elleipsis, and all contained within the compass of the limb: but at this time I thought not good to add them.  
Thus much (gentle Reader) I thought meet thou shouldest be advertised of: and so desiring thee to pardon the escapes of my pen, if thou meetest with any, I commend thee and thy honest studies to our Lords holy direction.  
Thine THOMAS OLIVER.   

A new handling of the Planisphere.  

THE FIRST SECTION, WHEREin is contained a plain and sensible declaration of the circles of the Sphere, and the terms used in the doctrine de primo mobili, after another Method than is yet in any other language.  
THE circles of the Sphere have more than once already been written of in English: notwithstanding I think it will not be accounted labour lost, though it were only for the better understanding of the conclusions following, here to handle them again as principles in the beginning of this treatise. Aristotle and Ptolomey for proof of some matters here propounded, have used many exquisite arguments, which I mean not to prosecute, but hoping from the testimony of sense in such manner to deduce the positions, that any mean capacity may have so reasonable a conceit of them, that with probability they may easily be admitted, I will refer their exact and more subtle demonstration, with the reproof of such as impugn them, to some others handling, or to some other place. Therefore requiring first the reader not to cast untimely doubts, nor hastily to judge of any part by itself, before he hath considered, and in some manner understood the whole discourse, I will according to Hypocrates his counsel begin with the notablest and easiest things. Whosoever doth but cast up his eyes unto heaven, presently perceiveth that it compasseth him round about in manner of half a globe or an Hemisphere: and if he stand upon some high place, or be at Sea far from land, where nothing can be seen but water and the Sky, his eye, if he turn himself about, will represent unto him at the lowest bounds or limits, which he seethe, the fashion of a circle, upon whose plain, as upon a base, this visible Hemisphere seemeth to be placed: therefore that Circle was first in Greek, and is now commonly in English called the Horizon, that is to say, bounding or limiting, uz. the compass of your sight. The largeness of this Circle, even as it falleth under view, is of so great a compass, as you need not restrain the centre thereof, according to the precise Mathematical definition, to one determinate, exquisite, indivisible, and very point, or prick, but without sensible error, as your eye will plainly testify, any one at pleasure may be taken in the place where you stand. From a centre so taken, if there be, or be imagined, a plumb line or line perpendicular to the plain of the Horizon extended unto the heaven, the place, which there it toucheth, is commonly called by an Arabian name the Zenith, being in truth the pole of the Horizon, and the line itself his Axis. By this Axis, and any lines crossing it, if plainness be every way extended reaching unto heaven, they mark out their circumferences, or semicircles, cutting one another in the Zenith: but in the Horizon their sections are right lines, cutting one another in his centre. Those circles are named Azimutes, and are imagined by Astronomers to be cut by other circle's parallel to the Horizon, which they term Alinicantars' and circle's of Altitude, because the portion of the Azimute, which is between the Horizon and the Parallel, showeth how his above the Horizon that point of the Parallel is, by which the Azimute passeth. Furthermore, in a clear night beholding the bright shining Stars, you shall evidently perceive how they change their places continually, some to rise and show themselves, other to go down under the Horizon, and to be hidden from your sight, only if you place yourself so, that your right hand be towards their rising, and your left hand toward their going down, looking right forth, and somewhat upward, you shall behold certain Stars, which are all times of the night to be seen, and never  depiction of constallations 
Charles Wayne.  
pole star.  
Little Bear  go down; amongst some of which placed after this fashion, there is one commonly called the Pole star, which being the last in the tail of the constellation called the little Bear, lieth in manner directly as it were in a right line, with those two in the hindermost wheels of Charles his wain. This Star seemeth little or nothing to remove out of his place, and indeed not far from it there is a point, or prick, which remaineth in one and the same place always immovable. The Azimute passing by the Orisons Axis, and a right line drawn from this immovable point, or the pole star, is properly called the Meridian circle; the common section of it and the Horizon, being a right line, is the Meridian line, or the line of North and South; the right line, which crosseth this Meridian line at right angles in the centre of the Horizon, is called the line of true East, and true West, or simply the line of East and West, which points the ends thereof extended directly fall upon. The Arimute standing upon this line, joannes de Roias, & others, specially such as write of dialling, as it were for dignity and pre-eminence, do call the Vertical circle: which name being common to all Azimutes, because they pass by the Zenith, in Latin called Vertex. or punctum eregione verticis: for a distinction Gemma Frosius hath named it the Circle of the East. The other Azimutes have no proper names, but are measured or determined in the Orisons circumference. That, and so likewise all other Circles, Astronomers do imagine to be divided into 360. equal parts, which they name degrees, that is to say, every quarter or quadrant into 90. degrees: every degree they further divide into 60. minutes, every minute into 60. seconds, every second into 60. thirds, and so continue sometimes unto tenths, and may go further if they will. By these degrees, minutes, etc. which are between that point, where any Azimute cutteth the Horizon, and the Meridian line, or the line of East and West, the Azimute is determined, assigned, or said to be given. Upon the points where the Vertical circle cutteth the Horizon, which points are the true poles of the Meridian, and the precise East and West, if by often and diligent viewing you shall see two Stars, the one in the East, and the other at the same time right against it in the West, in winter time even in one night, you may behold that which is in the East first ascending, and then again descending at length to come into the West, and that which was in the West being till that time hidden from your sight, then to appear again rising in the East: all this while continually the like shape of an Hemisphere being still represented to your eye, without any change or alteration, whereby you may gather the heaven to be a perfect Globe, or Sphere, having that part under the Horizon equal and like to that which you see above it, and that the place where you stand is the very centre thereof. A line drawn from the pole Star, or rather from the point immovable before mentioned, called the Arctic, or North pole, a line (I say) drawn, or imagined to be drawn from that point to the place of your standing, is called the Aris of the world, and extended to the other side of the celestial Sphere, which is under us, falleth there upon the South or Antarcticke pole, which in these our countries never appeareth. These things, being but a little héedily considered, be so manifest and apparent, that neither example nor figure is greatly requisite for the perfect understanding of them: yet in this beginning to remove all difficulty, and to make every thing as plain as may be, do thus much: In some plain ground, or rather upon a post, or stone, or some such like thing made exactly smooth, and lying very flat, without any manner of leaning or declining, which you must warily and diligently try by your level, and your plumb line, take the centre A, upon which describe a circle B C D E, representing the Horizon, whose circumference you may divide, or imagine only to be divided into degrees, and from A, the centre thereof, erect a Steel or wire, standing plumb upright, or perpendicular to the flat of the circle A B C D, by trial of your plumb rule: then take a board made precise square, having the side somewhat longer than the semidiameter of the circle B C D E, and set one corner thereof in the centre A, so that one of the sides containing it, may agree just with the Steel or wire, and the other with the circles plain: then looking along the upper edge, or through two sights set upon the edge, and continually lifting it up without swerving, till the side that lay upon the flat of the circle agree with the Steel: the line of your sight, or the line drawn from the centre (for in respect of the heavens mighty distance, the difference between them is not sensible) shall, howsoever the square be placed, describe  sky chart an Azimute. And if staying the edge of the square at one height, the corner be continually kept in the centre without slipping, by turning it round about the Steel, you shall describe an Almicantar, parallel to the Horizon. In this manner, carrying the square about, & lifting it up & down, till you see the Pole star, or if you can aim at it, the often spoken of immovable point, staying it fast in that situation, either of the edges without sensible error may be taken for the Axis of the world, which extended falleth on both the Poles, that which is seen being the Arctic Pole, and the Antarctick that which is under the Horizon. In lifting it up and down, you shall in the heaven point out the Meridian circle, and bringing it just to the flat of the circle B C D E, the edge will there show the right line B D, which is the Meridian line. The line C E, in A crossing the line B D, is the line of right East, and right West: the Azimute standing upon C E, is the Vertical circle, or the circle of the East: the Azimute standing upon G F, is sometime said to be of so many degrees, as are between B the North point of the Horizon, and G, where the Azimute standing upon G F cutteth the Horizon, uz. the circumference B G, denominateth the Azimute standing upon G F, and sometime that circumference G E, which containeth the number of the degrees between the same point, and the line of East and West E C, which be the very poles of the Meridian. Working by the Pole star, these things will somewhat vary from exactness, which some help by awaiting the time, when the star next the wheels of Charles wain, commonly called the filler or hill horse, is directly over or under the Pole star, so that the square, being lifted up and down close by the Steel, passeth by them both. But howsoever, this may suffice for some conceit of these terms, hereafter shall be taught other most precise ways, to describe and take them most exactly. To return again therefore to our principal purpose, in beholding of the Stars, how they continually rising in the East side of the Horizon, ascend unto the Meridian, and so from thence descend unto the West side, you shall observe, that always they keep their distance from the Pole star, or the true pole unchanged: as if you mark the two hinder wheels of Charles wain, you shall find, howsoever they be situated in the heaven, that their distance from the Pole star is always one, and the same not nearer or further of at one time then an other, and therefore every star or point, by such motion in a Globe or Sphere, his superficies describeth the circumference of a circle: so that the whole heavens must seem to turn upon the poles, and make a circular motion or revolution. This motion in divers respects hath divers names, called sometime the Diurnal motion, sometime the motion of the first movable heaven, sometime Notus captus. Again, every Star or paint by this conversion describing a circle, if you take some star, or point, which lieth between the poles just 90. degrees from either of them, that will describe a circle, passing by the points of East and West, which for some reasons hereafter to be expressed, is called the Equinoctial. The other points or stars describe circles parallel to the Equinoctial circle, which be called circles of declination, because, if from the poles of the world, by any point taken in any of the parallels, a greatest circle be drawn to the Equinoctial, that portion of it, which lieth between the Equinoctial and the Parallel, is called the declination of the point so taken. Some of these Parallels towards the North, described by revolution of the points or stars, that never set, are whole above the Horizon, and one of them, which some name the Arctic circle, toucheth it only: about the Antarcticke pole there be some wholly under the Horizon, and one, which some call the Antarcticke circle, only toucheth it in the South point: other between these two touching Parallels, which arise and set or go down, cut the Horizon, whose portion lying between that point and the true East or West, is called the Amplitude or Latitude of the rising, or the setting of the point describing the Parallel. These things hitherto rehearsed, travelers may every where and in all countries observe: and beside, in their journeys Southward or Northward, they shall see sometimes more, and sometimes fewer stars never to go down: more in going Northward, fewer in going Southward, yet always without alteration keeping one and the same distance one from another. Mary the Pole star or the true pole, proportionably to the length of the way traveled, altereth his distance from the Horizon, and the Zenith appearing, the further you go Northward, the higher, & the further you go Southward, the lower, and so accordingly that portion of the Meridian, between the Equinoctial and the Zenith, which is  sky chart called the Latitude of the country, being always equal to that which is between the Horizon and the Pole, called his height, or elevation, is of necessity greater or lesser: so that your situation is apparently changed in respect of the heaven, but yet still you keep the same distance from it, whereby you may easily infer the figure or shape of the Earth to be like and answerable to the celestial Sphere: and that how vast and huge soever it be in itself, yet in comparison of the large compass of the heaven, it is justly counted for no more but a small point, or prick: for it never hindereth you from seeing the precise half of the celestial Sphere, which, were the bigness thereof any thing of account, it would of necessity do. For better understanding of these matters, imagine the circle B C D E, described on the centre A, to represent the Meridian, and upon the line E C, a semicircle erected perpendicularly to represent the Horizon, whose Pole, or the Zenith, the point B is, and let H be the North pole. From H to A the place of your standing (being the centre of the Horizon and the whole Sphere, and so consequently of all the greatest circles in the same Sphere described) from the pole H to the centre A draw a right line, and extend it unto I: the line H I is the Axis of the world: the poles be H the North, and I the South, or Antarcticke pole. Let the point G, taken in the Meridian, be just 90. degrees from either pole, so that the circumferences G H, G I, be just quadrants: by the revolution of the Sphere upon the poles H I, which is the Diurnal motion, the line G A shall describe a circle passing by the centre of the Sphere, and perpendicular to the Meridian: therefore it must pass by the poles thereof which be in the Horizon, the points of East and West, as the circle set plumb upon the Diameter F G will do. Other points, as S, N, V, describes circles parallel to that standing upon G F, of which that which standeth upon C S, doth only touch the Horizon, and is wholly above it: the other beyond it to the pole-ward, as V, W, be wholly above it, and come not near. So likewise E, T, is always under the Horizon, and only toucheth it: the other beyond it towards I, are under it, and never come near it. Now if in going Northward the pole H grow higher, the Equinoctial must go further from the Zenith: for seeing B C and G H be equal, that is the just quarters of the same circle, if you take away the circumference B H common to them both, the residue G B, the distance of the Equinoctial from the Meridian, which is the Latitude of the country, whose Zenith is B, is equal to H C, the height of the pole above the Horizon, or the elevation of the pole, and as the pole riseth higher, so is the circle S Cremoved from the Horizon further and further: whereby you see, that by that means more stars never set. The circumferences of the Meridian G N, G S, G V, are called the declinations of the points N, S, V, taken in the parallels N Z, S C, V W, and the portion of the Horizon between the point of true East, and the point, where the parallel standing upon N Z, cu●teth the Horizon, is the amplitude or latitude of rising of the point L, or any other taken in the parallel, standing upon the Diameter N Z. These things being well conceived, let us consider again, that as apparently as any star, the Sun making his revolution, ariseth on the East side of the Horizon, and mounting up unto the Meridian, at which time it is high noon, descendeth again unto the West side, which time of his continuance above the Horizon is called the artificial day, then being hid from our sight, the night beginneth, which continueth till the Sun, passing by that part of the Meridian, which is under the Earth, when it is midnight with us, returneth again unto the East side, making up the natural day, which is the space of time from the Sun rising to the Sun rising again, and may be indifferently reckoned from Sun set to Sun set, as the jews do, or from midnight to midnight, which is our common account. The obscure light before the suns rising, is called the dawning and break of the day: after his setting is called the twilight, Crepusculum in Latin is a common name to both times. Although in the suns daily revolution we observe all this, yet is it not without much variety and change: his rising and setting is not always in one place of the Horizon, nor his height in the Meridian always the same at noon. In the midst of winter, about the 12. of our December, when the days be shortest, and the nights longest, it riseth & setteth most Southerly, and is lowest in the Meridian at noon: towards the Spring, as the days wax longer and longer, in rising and setting it continually groweth from the South, and mounteth higher and higher in the Meridian, till about the 11. of our March, making the day and night equal, it riseth and setteth just East and West, coming at noon to the common section of the Equinoctial and the Meridian: then, as Summer cometh on, the days increasing, it riseth and setteth more Northerly, mounting still higher and higher in the Meridian at noon, till in the midst of Summer, the days being at the longest, about the 12. of our june, it be at the highest in the Meridian at noon, and riseth and setteth furthest North: then, as the days shorten, it returneth back again, growing continually lower at noon, and rising and setting less Northerly, till the days and nights become equal again, about the 13. of our September, when it riseth and setteth precisely East and West again, and at noon will be in the common section of the Equinoctial and the Meridian. From that point it groweth lower and lower at noon, rising & setting more and more Southerly, as the night's increase, till about the 12. of our December the days being at the shortest, it becometh lowest again at noon, and in rising and setting most Southerly, as it was before. Of this returning from the highest in Summer, and the lowest in winter, the two Parallels, which it then describeth by the Diurnal revolution, be called tropics, and the Equinoctial was so named, because in March and September the Sun coming to it, the day is made equal to the night. By these varieties it was easily gathered, that the Sun made a peculiar course beside his Diurnal revolution, in a greatest circle touching the two Tropics, and declining from the Equinoctial, which it divideth (as all greatest circles do one another) into two equal parts, or semicircles. This declining circle, because when the Sun and Moon either meet or be opposite in it, there happeneth an Eclipse, is called the Ecliptic, and sometime the Zodiac: but the Zodiac properly is a superficies like a fillet or a girdle, laid upon this circle's circumference, comprehending on either side of it six, or (as some say) eight degrees. The poles of this circle, which decline from the poles of the world, as the circle itself declineth from the Equinoctial, describeth by the Diurnal revolution two Parallels, which some, in stead of them afore described, take for the Arctic and Antarcticke circles. That section of the Equinoctial and the Ecliptic, which the Sun cometh to in the Spring time, is called the Vernal section: that which is opposite to it, by which the Sun passeth in the end of Harvest, is the Autumnal section: the points which touch the tropics, be commonly called Solstitial points. The greatest circle, passing by the Equinoctial sections and the poles of the world, is the Equinoctial Colour: that which passeth by the poles of the world and the Solstitial points, is called the Solstitial, or Tropical Colour, the portion thereof lying between the two tropics, is the distance of the tropics, whose half is the suns greatest declination, otherwise called the Obliquity of the Zodiac, which by observation made with large instruments, is in this our age found by great Artesmen to be 23. degrees and 30. minutes, whose report it shall be sufficient to accept of, and commit to memory, without taking or trying it by any small instrument: though some writers of the use of the Astrolabe have made that one of their special conclusions. The Equinoctial and the Ecliptic, are circles of chief account: the first being the rule and measure of the first motion, or the motion of Primum mobile, otherwise called the Diurnal revolution: and the other as it were the Standard, whereby all secondary motions are examined: therefore the degrees of these circles have peculiar names, called in the Equinoctial Tempora, times, because they be the first measures of time, every 15. degrees ascending making an hour, and so the whole revolution of 360. which is performed every natural day, make 24. hours. Astronomers begin to account the parts or degrees, as well in the Equinoctial, as in the Ecliptic, from that point common to them both, which is called the Vernal section: and though in the Equinoctial there be no change of names, yet in the Ecliptic or Zodiac every thirty degrees is called a sign, and hath a peculiar name. The first 30. immedtatly following the Vernal section, is called Aries: the next 30. Taurus, the third 30. Gemini, the fourth 30. (whose first point begin a new quadrant, and toucheth the Summer Tropic) is termed Cancer: the fift 30. Leo, the sixth Virgo. These six Signs make up the semicircle of the Zodiac, which leaneth or declineth from the Equinoctial toward the North pole. The first 30. after this semicircle, beginning at the Autumnal section, make the seventh sign, named Libra: the next or eight 30. is the sign Scorpio: the ninth 30.  Sagittarius: the tenth 30. (beginning a new quadrant with the first point, which toucheth the winter Tropic) is Capricornus: the eleventh 30. is Aquarius, and the twelfth 30.  Pisces. There be characters used to express every Sign with, which are found in every Almanac. The Sun passeth in the Ecliptic from point to point, making his continual revolution in it, without swerving to the one side or the other: other Stars for the most part decline from it, but yet have their places determined by Longitude and Latitude in respect thereof thus: By any point or Star, a greatest circle being drawn from the pole of the Ecliptic to his circumference, the section there made, or the portion of the Ecliptic between that section and the Vernal section, is the Longitude of the point, or Star, by which the greatest circle is drawn: his Latitude is that portion of the same greatest circle lying between the Star and the Ecliptic. The Sun being in any point of the Ecliptic, and the point itself, with all other Stars, wheresoever placed beside, are in like manner compared to the Equinoctial, and in respect thereof, not only their declination, which hath been touched before, but also their right Ascension is considered. A greatest circle being drawn from the Poles of the world to the Equinoctial, by any point or Star, his right Ascension is that portion of the Equinoctial, taken between that point where the greatest circle cutteth, and the Vernal section. The declination is the portion of the greatest circle so drawn, which lieth between the Equinoctial and the Star, which (as before was noted) is also determined by a Parallel to the Equinoctial, passing by the Star. The right Ascension is so called, because under the Equinoctial, that is, in a Situation where the Equinoctial passeth by the Zenith, and the Horizon by the Poles of the world, the Horizon (by which once in 24. hours every point ascendeth) doth the office of any greatest circle so drawn, as is appointed, and agreeth with it most exactly. This Situation is called Sphaera recta, the right Sphere; and the Horizon in that Situation, Horizon rectus, the right Horizon: but when the Equinoctial declineth from the Zenith, and the Poles be one above, the other beneath the Horizon, that Situation is called the Obliqne, or Declining Sphere; and the Horizon, the Obliqne Horizon, which never agreeth with the foresaid greatest circle, drawn by the Poles of the world: and therefore with the Star there is another point of the Equinoctial in the Horizon, between which and the Vernal section is contained that portion of the Equinoctial, that is called the Stars Obliqne Ascension, and the portion of the Equinoctial between the ends of the right and Obliqne Ascension, is called the Difference Ascensionall. But for helping the conceit of these things, I will use a linearie example, taking the former figure, where E C is the Meridian line, and the Diameter on which the Horizon standeth, now in the midst of winter the Sun being lowest in the Meridian, in the point K, by the Diurnal revolution, describeth a Parallel standing upon the Diameter K M, which cutteth the Horizon from the East towards the South: it continually groweth higher and  sky chart higher in the Meridian pan, & riseth in the Horizon nearer and nearer the East, till about the 11. of March at noon it cometh to G, the section of the Equinoctial and the Meridian, and then by the Diurnal revolution describeth a circle just answering the Equinoctial, which passing by the points of East and West in the Horizon, standeth upon the Diameter G F: from thence, a● Summer cometh on, it ascendeth in the Meridian, and riseth more Northerly upon the Horizon, till about the 12. of our june, being at the highest in N, it describeth a Parallel standing upon the Diameter N Z, in which it riseth upon the Horizon furthest from the East to the Northward: from thence it descendeth again by G, to K, and passeth in the Horizon by the true East to the furthest Southward, than it ascendeth again till it come to N, continually returning from the highest to the lowest, & from the lowest to the highest: whereof the two Parallels standing upon the Diameters K M, L N, be called tropics. By these changes it was easily gathered, that the Sun (besides the Diurnal revolution) made another proper to itself in a circle standing upon the Diameter K L, drawn from Tropic to Tropic by A the centre of the Sphere: by reason whereof, this circle, called the Ecliptic, and the Equinoctial, divide the one the other into two semicircles. The Ecliptic is sometimes called the Zodiac, though properly the Zodiac be a superficies lying on either side of it, as you see the superficies O P Q R, parted by the line Z K. The common Diameter of the Equinoctial and the Ecliptic standing in A, perpendicular to the plain of the circle B C D E, the points K and L being the Solstitial points, the circle B C D E is the Solstitial Colour. The circle standing upon I H, and passing by the perpendicular Diameter, or the Equinoctial section, is the Equinoctial Colour: the circumference K N is the distance of the Tropic, the one half thereof G K, or G N, is the obliquity of the Zodiac, or the Sun's greatest declination, for it groweth less & less as it cometh nearer the Equinoctial section: where it hath no declination at all, it never declineth from the Ecliptic or circle standing upon K Z, but make his motion in it, his place is named of the point wherein it is. Other Stars or points without the Ecliptic (as for example, S) have their place determined by a greatest circle B C D E, drawn from X, the Pole of the Zodiac, by S ●o K, a point in the Ecliptic circumference, the portion of the Ecliptic between the Vernal section and K, or the point K is the Longitude of S, and the circumference S K, is his Latitude. In like manner, if by any star or point Y, and the Poles of the world H I, there be drawn a greatest circle B C D E, cutting the Equinoctial in G, the portion of the Equinoctial between the Vernal section and G, is the right Ascension of Y, and the circumference G Y is his declination. Now if G, the section of the Meridian and the Equinoctial, be the Zenith of any place, the Horizon must stand on I H, and pass by the Poles of the world, which situation is called the right Sphere, and the Horizon the right Horizon, wherein, by the Diurnal revolution twice in 24. hours, the circle B C D E must join close with it. But if G, the section of the Equinoctial and the Meridian, decline from B the Zenith, which situation is called the Obliqne Sphere, and the Horizon then standing on E C, being an Obliqne Horizon, by the Diurnal motion made upon the Poles H I, the circle B C D E can never agréene or match evenly with the Horizon: therefore of force some other point of the Equinoctial, and not G, must be in the Horizon with Y: the circumference of the Equinoctial between that point in the Horizon, and the Vernal section, is the Obliqne Ascension of Y: but the portion that lieth between that point and G, is the difference Ascensionall. That point of the Zodiac, where it is cut off the Meridian, is called Medium coeli, mid heaven; Culmen and cor coeli above the Horizon, & Imum coeli under the Horizon, the lesser angle, made by that section, is called the Meridian angle. The point of the Ecliptic, that is in the East side of the Horizon, is called the Ascendent, and the Horoscope, the section there make the angles, of which the lesser is called the horizontal angle: the quantity thereof is equal to the Nonagesimus Gradus of the Ecliptic, which is peculiarly given to that point of it, which is just 90. degrees from the Ascendent. There be moreover other greatest circles to be considered, which are called circles of position, or circles of Station, six of them dividing the heaven in twelve portions, which are called houses or Stations, though many times those names be restrained only to such parts of the Zodiac, as be intercepted by those circles. Though the Horizon be always taken for one, and the beginning of the first house: yet are the rest by divers diversly drawn. Some will have them all to have the Meridian line for their common Diameter, and yet agreeing in this make difference in placing them. Campanus and Gazulus divide the Vertical circle into twelve equal parts, and by every two opposite sections draw a circle of position. Regiomontanus divideth the Equinoctial into twelve equal parts, and will have every circle of position pass by two of those opposite sections: both these ways keep the Horizon and the Meridian for circles of position. Other divide the Zodiac into twelve equal parts, beginning at the Ascendent, and two of those opposite sections draw from the poles of the Zodiac their circles of position: these keep the same Ascendent that the former do, but must differ many times in the tenth house, or the Cardo medij coeli, though Gemma writeth that all agree about the foruer Cardines, that is the Horoscope, seventh house Culmen, and Imum coeli: for they cannot have always the Meridian & the Horizon for circles of position. Other draw these circles from the poles of the world, and that diversly: but the three former ways are in greatest estimation. A circle of position differeth from an angle of position, which (after Prolomey) is an angle at the Zenith, subtended of that portion of the Horizon, which lieth between the Meridian and the Azimute, passing by the Zenith of the other place, unto which the position belongeth. Gemma Frisius maketh it a right lined angle in the centre of the Horizon, subtended of the same circumference: by it topographical descriptions are made, wherein sometimes the distance of two places is measured by a right line lying between them, but in a Spherical superficies by the portion of a greatest circle, because that is the shortest that can be drawn in it. These things may be easily conceived without linearie explication, if the former matters be well understood: therefore as it were beside the principal purpose, setting down how by supputation you may readily guess somewhat near the Sun's place, being a thing for many purposes to be had in a readiness, I will end this first section: the thing is taken out of Clavius, but fitted to our common, or the julian year.  

Parergon primum. What month, and what day of it, the Sun entereth every sign of the Zodiac.  
FIrst tell the months upon your fingers, seeking Innuarie on your thumb, February on your forefinger, and so the other in order, till you come to your month, which mark on which finger it falleth: in like manner tell the Signs, beginning with Aquarius on your thumb, Pisces on your forefinger, and so the rest in order, till you come to the finger, on which your month fell: the Sign which falleth on that finger, the Sun entereth that month. Then learning these two verses by heart:  

Inclyta Laus justis Impenditur, Haeresis Horret  
Garrula, Grex Gratus Faustos Gratatur Honours.   
In like manner set Inclyta on your thumb, and word by word in order, till you come to the finger, on which your month fell: then telling from A in the cross row, till you come to the first letter of your word, take that number and subtract it from 20. the residue is the day of the month, when the Sun entereth the Sign. For example: I would know what Sign the Sun entereth in October, and what day it entereth, telling from januarie upon my fingers the months in order: at the second going over I find October on my little finger: then taking the Signs, and telling from Aquarius, at the second going over my fingers, Scorpio falleth on my little finger, so I say that the Sun entereth Scorpio in October: then taking the verses, and setting the words one by one in like manner on my fingers, I find in the second going over, Faustos on my little finger: now telling in the cross row the letters from A to F, I find six: this six I subtract from 20. so rest 14. which the day of October the Sun entereth Scorpio.   

Parergon secundum. To know the suns place in the Zodiac upon any day given.  
Find the Sign answering your month, and the word likewise, then to the day of the month put ten, and the number the first letter showeth in the cross row: if this be less than 30. the number is the degree of the Sign next before that, which the Sun entereth in your month; if more, cast away 30. and the residue is the degree of the Sign entered in your month. Example, I would know where the Sun is the 9 of October, I find in October he entereth Scorpio, and the word for it to be Faustos, whose first letter F is the sixth in the cross row, I take ten and 6. uz. 16. which I put to, 9 there is made 25. therefore I say the Sun is in the 25. degree of the Sign before Scorpio, which is Libra. If I would know for the 18. day, putting 16. to 18. there would amount 34. from which abating 30. the 4. remaining showeth me, that the Sun is in the 4. degree of the Sign belonging to October, which is Scorpio.   

Parergon tertium. The place of the Sun being given, to find the day of the month.  
TAke the day and month the Sun entereth the Sign given, the number of the degrees given put to the day, wherein the Sun entereth the Sign, that product being less than the days in your month, show the day: but being more, cast away the days of your month, the residue is the day of the month following. Example, I find the Sun in the 4. degree of Scorpio. I would know what day of the month it is; I find that the Sun entereth Scorpio the 14. of October, I put 4. to 14. there amounteth 18. so I say it is the 18. day of October. Again, let it be in the 25. of Libra, I find that the Sun entereth Libra 13. Septembris: I put 13. to 25. there amounteth 38. from which I take 30. the days of September, and there remaineth 8. the day of October. These are but guesses, as you may see much after the way commonly taught by the back of the Astrolabe. Let him that desireth exactness, seek the suns place in some Ephemerideses, or in the Regiment of the Sun, published by E. W. painfully calculated upon his own diligent and exact observations.  
The end of the first Section.    

THE SECOND SECTION OF THE NEW HANDLING THE PLANISPHERE; SHOWING HOW THE CONclusions of the Astrolabe may readily and exactly for any Country be performed only with Ruler and Compasses.  
ALthough for working the conclusions following any circle divided, or to transfer out of any divided quadrant, the given circumferences would be sufficient: yet for the easier dispatch, I would a plate of Latin to be provided, according to the figure which you see on the other side. There, upon one centre A, be six circles described one within another, of which the two outermost containeth the narrowest space, which must be divided into 360. equal parts or degrees, as the common manner is. At every 30. degrees, in the space contained under the second and third circles, make divisious, and beginning at E writ Ariosto, or Aries: at the next, Tau, or Taurus, and so for the rest of the 12. Signs: this circle shall be called the Zodiac. The third space, contained under the third and fourth circle, must be divided  sky chart at every 15. degrees: beginning at B you must set down 12. at the next division toward C set down 1. at the next 2. and at every division the numbers in order, till coming to D, you are to set down 12. again: and so in the other semicircle DEB, set 1. at the division next D, at the second 2. till coming to the next before B, you are to set down 11: this fourth circle shall be called the hour circle. The fourth space, contained under the fourth and fift circle, must be divided at every 10. degrees: at the first division from E towards B set 10. at the second 20. and so increasing continually at every division by 10. till you come again to E, where you are to set down 360. this circle shall be named the Equinoctial. The space contained under this and the sixth, or innermost circle, must from E by D unto C, have divisions at every 20. degrees, and at every division a number set increasing by 10. till you come to 90: in the quarters B C, B E, you must make divisions at every 10. degrees: at the first, from B towards E, set 10, and likewise at the next unto C towards B, at the next 20. and so in order, till coming to E and B, you are there to set 90: this inner circle shall be called the limb. If you will set two sights on the outer edge, so that the line, drawn from lope to lope, be parallel to the line subtending the arch B C, and hang a plummet from B, you may by your plate begin.  

The first Conclusion. To take the height of the Sun, or any Star, above the Horizon.  
TVrning your plate towards the Sun, lift it up and down, till his beams shine through both the lopes of your sights, or (if it be a Star) till you see it through them: the thread showeth in the semicircle E D C, how many degrees the Sun or Star is above the Horizon.   

The second Conclusion. To find the distance of the Suns or any Stars Azimute, from any determinate point in the Horizon.  
LEt your plate be fixed level with the Horizon, and direct E to any point: then a ruler being turned and lift up by a steel or wire, standing plumb upright from the centre A, till you see the star, by the edge of it, or rather through two sights set upon the edge, or if it be the Sun, till his beams pass through the lopes, then without stirring the end next your eye bring it close to the centre, the edge showeth in the Equinoctial the distance of the Azimute from the point respected by E. The Azimute is said simply without addition to be given, when his distance from the Meridian's North end is given.   

The third Conclusion. From a point given in the circumference of a circle, at any side assigned, to apply or subtend a distance or a right line not greater than the Diameter.  
LEt the point given be C, from which upon the side towards D, I am to apply a line equal to A Q, draw the Diameter C A E: now if the line A Q be equal to the Diameter, I have done that which is required: but being less, I take the distance A Q with my Compass, and setting one foot in C, with the other I mark Q in the limb, and so under the circumference C Q, I have subtended a distance equal to A Q, which was required.   

The fourth Conclusion. From a point given in any circle's circumference, to take at any side assigned, an arch equal to any arch given in the same, or any equal circle.  
FRom the point given C, draw the Diameter C A E: now if the arch given be a semicircle, the circumference C D E, taken at the side assigned D, is the arch desired: but if it be lesser, as F G, I take the distance F G, and setting one foot of the Compass in C, with the other towards D, I mark 3 in the limb, the circumference 3 C is that, which is desired: but if it be greater than a semicircle, as 6 G, I take in like manner with my Compass the distance 6 G, and setting one foot in C, on the other side towards B, not on that which was assigned towards D, I mark in the limb with the other foot L, the circumference C D L is that, which is required.   

The fift Conclusion. The heights or Almicanters, together with the distance between the Azimutes of any two Stars given, or taken by observation, to find the distance one from another.  
Upon the centre A describe a circle B C D E, and let the Diameters B D, C E, be perpendiculars one to another. This construction shall be always assumed hereafter without further expressing: from E to H reckon the distance of the Azimutes, and draw the line A H: then taking the circumference E F, equal to the greatest altitude, and making E G equal to it, upon F and G laying a ruler, it will cut A E in N: again, taking H I equal to the other altitude, make H K equal to it, a ruler laid on I and K, will cut A H in P, take the distance N P, and setting one foot of the Compass in A, with the other in A B mark Q: again, taking the distance F N, and setting one foot in A, take the line A S: again make the line S T equal to the distance P I, opening your Compass to the distance T Q, setting one feote in C, with the other in the limb mark 4, the circumference C 4 is the distance sought. This construction is general, but in some cases may be abridged: for if the Altitudes were equal, it should suffice to apply the distance N P in the limb without any more ado: if both were in one Azimute between the Horizon and the Zenith, as E F being the greater, and F L the lesser, the distance should be L E: but if the Zenith were between the height, whereof E M were the one, laying a ruler on M and A, it would cut the limbs beyond A in 6. then toward B, taking the other height 6.1. the distance shall be E B. 1.   

The sixth Conclusion. In any Country unknown to find the Diameter of the Parallel of the Sun, or any unknown Star.  
Three times in one day, or for any Star in the night, take the height and distance of the Azimute from any certain point, whereby you may have the distance of every Azimute from other, by the former conclusion: take the distance of the points observed one from another, which let be C 6. 6 B and B K of three lines equal to the chords, of those circumferences, make a triangle C B A, and about it describe a circle A B C, whose Diameter A C is the Parallels Diameter.  
 sky chart   

The seventh Conclusion. The Diameter of the Suns, or any Stars parallel being given, to find the declination.  
FRom C on either side apply a distance equal to A C, as C 1. C 2. upon 1 & 2 a ruler laid, will cut A C in R: upon R, as a centre at the distance A R, describe a semicircle A B C (a semicircle thus described, having one angle in the centre A, and his Diameter the Semidiameter of the limb, as A C is, shall hereafter be called the sinical arch of the limb). Now from C towards D apply at 12 a distance equal to A C, the Parallels Diameter given, upon C and 2. laying a ruler it will cut the Semicircle A B C in D, then laying a ruler on A and D, it will cut the limb in 5.5 C is the declination sought for. In the Sun you may know, by the time of the year, whether it be Northerly or Southerly, and to what quadrant it belongeth, saving when he is near the tropics, or Equinoctial sections: for distinguishing thereof, make your observations two, or three, or four sundry days: then the Sun being about the tropics, comparing two observations, if the declination increase, the first observation was before it came to the Tropic: but if it decrease, the second was after it came to the Tropic, the Tropic will show North and South, from Aries by Cancer to Libra North, from thence by Capricornus to Aries South. About the Equinoctial sections, if the declination increase, the Sun is passed the section: if it decrease, it is not come to it, and so you may know certainly unto which quadrant the declination belongeth. You may find whether the Star you observe, declineth North or South, thus: Observe two Stars, of which by marking the situation of Charies wain, and the Pole star, you may determine the more Southerly, or the more Northerly: then if you find their declinations equal, the more Southerly hath South declination, and the more Northerly, North declination: but if they be unequal, the more Southerly, being less, it declineth Southward: but if it have the greater declination, the more Northerly declineth North, which is to be understood of observations taken on this Northside of the Equinoctial: but you may make this doctrine general, thus: If that which is nearest the bidden pole be less, it declineth toward that pole: if greater, then that, which is nearer the apparent pole, declineth toward the apparent pole. This distiction shall not always be precisely observed hereafter.   

The eight Conclusion. The declination of the Sun, or any Star being given, together with the Azimute and any height, to find the Meridian height.  
LEt the circumferences E F, C G, be made either of them equal to the height given, and laying a ruler on F and G, it will cut A B in P: make the circumference E X equal to E F, and laying a ruler on P and X, it will cut A E in Y: upon Y, as a centre, at the distance Y A, describe a semicircle R V A (a semicircle thus described, having one angle in the centre A, and the Diameter equal to the semidiameter of another circle, as R A equal to F P, which is the semidiameter of the circle, whose Diameter is the distance F G) this semicircle shall be called the sinical arch of that line, or distance, as R V A is the Sinical arch of the distance F G. Let the distance of the Azimute from the East, and that in this figure Southward, be the circumference E W: upon A and W, laying a ruler, it will cut the semicircle R V A in V, take the distance R V, and setting one foot of your Compass in P, with the other in the line P F, mark a point O. Now let K L be the Diameter of the suns parallel, and divide K L so in O, as the greater segment O L may be to O G, the greater segment of F G, as F O, the lesser segment of F G, is to the lesser segment K O (how this may be done shall be taught hereafter:) then opening your Compass to the length of the segment lying toward the quadrant E F B, as namely to O K, set one foot in O, and with the other mark in the quadrant E F B the point K, the circumference E K is the Meridian height.  

A Lemma.  
Between F O and O G, by 13. P. 6. take a mean proportional H N, and to that mean proportional given, in K L find the two extremes, K O, O L, by the addition of Peletarius to 13. P. 6. but in the English Euclyde the figure is drawn wrong, & the demonstration distorted: you may help yourself in the same author, where 17. p. 10. you have out of Campane the same matter truly propounded.    

The ninth Conclusion. In any Country unknown, two different heights with the Azimutes of the Sun, or any known Star being given, to find their Meridian height, and declination.  
LEt the two heights given be E F, and E H, make C G equal to E F, and C I equal to E H, draw the lines F G, and A I, cutting A B in the points P & N: let the semicircle R V A be the sinical arch to F G, and S T A, the Sinical arch to H I. Moreover, let E W be the distance of the Azimute, belonging to the height E F, from the East South, and E X, the distance of the Azimute, belonging to the height E H: likewise from the East South, a ruler laid on A and W, will cut the semicircle R V A in V: take the distance R V, and setting one foot in P, with the other in P F mark O: again, laying a ruler on X and A, it will cut the semicircle S T A in T: take the distance S T, and setting one foot in N, with the other in the line N H, mark a point M: upon M and O laying a ruler, draw the line K M O L, which is the Diameter for the parallel of declination, whereby you may know the declination, and E K is the Meridian height.  
 sky chart   

The tenth Conclusion. The Meridian height & declination of the Sun, or any Star being given, to find the latitude.  
FIrst you must know whether the height be Northerly or Southerly, which Nonnius thinketh the magnetical Index, or Compass will show readily, and exactly enough: Clavius determineth of it, by marking on what hand the rising or East side of the Horizon is: but in the night it may be certainly known, by looking upon the stars about the North pole, then must you mark, whether after the height taken, the Star grow higher, or lower. Now let the Meridian height being Southward and highest, be E K, from K towards the pole, which the declination respecteth not, as if the declination be North, than South ward: if South, then Northward: and here supposing the declination Northward, from K towards E, if must be taken, as K Z, the circumference Z K B is the latitude, if the height bade been lowest, the declination should have been taken at the other end of the parallels Diameter, between B and C, for so it should have been above the Horizon wholly.   

The eleventh Conclusion. The declination of the Sun or any known Star being given, in a Country of known latitude, to find the Meridian height.  
IF the complement of the Stars declination, or the distance from the Pole be less than the latitude, there be two Meridian heights, one highest, and the other lowest. Let the latitude given, be K Z, and let the declination be Z K, opening your Compass to the distance B C, keep one foot in B, and with the other towards K, mark in the limb the point E, the circumference E Z K, is the Meridian height.   

The twelfth Conclusion. The Meridian height of the Sun, or any Star being given, in a Country of known latitude, to find the Declination.  
LEt E K be the Meridian height, opening your Compass to the latitude given, set one foot in B, and with the other from the pole, mark in the limb the point Z: Z K is the declination sought for, taking denomination of that Pole, between which, and Z, it is placed.   

The thirteenth Conclusion. In a Country of known latitude, the declination, and height of the Sun or any known Star being given, to find the Azimute.  
LEt the height given be E H, make the circumference C I equal to E H, and draw the line H I, and let S T A be sinical arch of the line H I, let the latitude given be B Z, upon Z and A laying a ruler, it will cut the limb on the other side in Q, make either of the circumferences Z K, Q L equal to the declination given, and laying a ruler on K and L, it will cut N H in M: with your compass take the distance N M, and setting one foot in S, in the semicircle S T A take with the other foot a point T, upon A and T laying a ruler, it will cut the limb in X: the circumference E X is the distance of the Azimute from East, or West: Southward, because M fell in the quadrant B Z E, whereas if it had fallen in the quadrant B I C, it should have been Northward, whether it be to be accounted from East or West, the rising in height, or decreasing will show.   

The fourteenth Conclusion. In a Country of known latitude, the Azimute and declination of the Sun, or any Star being given, to find the height.  
LEt K L be the Diameter of the parallel of declination, cutting A B axis of the Horizon in Q, and let B D, be equal to the Azimutes distance from the East, or West towards the South taken in the quadrant B E, if it were Northward, it were to be taken in the quadrant B C: between D and B take any point 1, and make the circumference D 3 equal to 1 B, laying a ruler upon 3 and 1, draw the line 2 O, then making A P equal  sky chart to 2 O, upon O and P, laying a ruler, draw the line P O F, a ruler laid on A and D, will cut O 2 4: in P F take a line P 6, equal to the distance A 4, then laying a ruler upon Q and 6, it will cut the limb in H, the circumference E F H is the height desired. In some cases this construction will not be contained within the limb, but it is generally true.   

The fifteenth Conclusion. In a Country of known latitude the Azimute, and height of the Sun, or any Star being given, to find the declination.  
LEt B Z be the latitude given, and let the altitude be E H, make C I equal to E H, and draw the line H I, cutting A B, in N: let the distance of the Azimute from the East be E X; and let S T A be the sinical arch of H I: now laying a ruler on A and X, it will cut S T A in T, take the distance S T, and setting one foot in N, with the other in N H, because the Azimute is Southward (otherwise in N I) take a point M, upon M and A laying a ruler it will cut the limb in 5, make Q 7 equal to Z 5, & draw the line A 7, make A 8 equal to A M, upon M and 8, laying a ruler, draw the line K M 8 L, the circumference Z K, or Q L, is the declination sought.   

The sixteenth Conclusion. The suns declination being given, to find his place in the Zodiac.  
LEt E H be the suns greatest declination, videl. as I said in the first section, to be received upon their credit, which by observation made with large instruments, affirm it in this our age to be 23. degrees, 30. minutes: draw the line A H, and let either of the circumferences, E F, C G, be equal to the declination given, upon F and G laying a ruler, it will cut A H in K: in E M A the sinical arch of the limb, apply from E at M a distance equal to A K, upon M and A lay a ruler, which will cut the limb in O, a circumference taken in the quadrante, to which the declination belongeth, from the next Equinoctial section, equal to E O, as E O in the quadrante D E, showeth the place of the Sun, accounting from E, the beginning of Aries by B and C, to O.   

The seventeenth Conclusion. Another way to find the suns place by his declination given.  
Upon I, and H, the suns greatest declination reckoned from E, and C, lay a ruler, which will cut the line A B in P, upon A as a centre, at the distance A P, describe a circle Q P R: now let either of the circumferences E F, C G be equal to the declination given, upon F and G lay a ruler, which will cut the circle Q A R in S, and likewise on the other side in with, but imagining the declination to appertain to the Quadrant E H B, take S, the point in it, and upon S and A laying a ruler, it will cut the limb in T, reckoning from E the beginning of Aries to T, you have the suns place you sought for.   

The eighteenth Conclusion. The suns place in the Zodiac being given, to find his declination.  
FRom E towards D take a circumference E O equal to the suns distance from the Equinoctial section, that is nearest to his place, and let E M A be the sinical arch of the limb, E H, C I equal to the suns greatest declination, and draw the lines H A, A I, then laying a ruler on A, and O, it will cut E M A in M: make A K equal to the distance E M, and A V equal to A K, then laying a ruler on K and V, it will cut the limb in F, and G, either of the Circumferences E F, C G, are the declination sought for.   

The nineteenth Conclusion. Another way to find the suns declination by his place given.  
Upon T the suns place, and A lay a ruler, which will cut the circle Q P R in S: make the circumference R W, equal to QUEEN'S S, and laying a ruler on S and with, it will cut the limb in F and G, either of the circumferences E F, or C G is the declination sought for.   

The twentieth Conclusion. The suns declination being given, to find his right Ascension.  
LEt either of the circumferences E F, and CG, be equal to the suns declination given, upon G, and F laying a ruler, it will cut A H in K, and B A in X, let Y L A be the sinical arch of the distance F G, and from Y to L applite in the Semicircle Y L A, a distance equal to X K; upon L and A laying a ruler, it will cut the limb in N. In the quadrant, to which the declination belongeth, from the next Equinoctial section, a circumference taken equal to E N, as E N in the quadrant E N D, showeth the right ascension from E by B to N.   

The 21. Conclusion. The right ascension of any point in the Zodiac being given, to find the Meridian angle.  
LEt E O be equal to the distance of the right ascension from the nearest Equinoctial section, upon A & O a ruler laid will cut E M A t●e Sinical arch of the limb in M, in the lines H A, A I take A K, and A V equal to the distance E M, upon K V, laying a ruler, it will cut A B in X: in Y L A the sinical arch of the distance F G apply at Z, from A  sky chart a distance equal to X V, a ruler laid on A and Z, cutteth the limb in 4, the circumference E 4 is the Meridian angle.   

The 22. Conclusion. The right ascension of the Sun being given, to find his place in the Zodiac.  
LEt E O be equal to the right ascensions distance from the nearest Equinoctial section, and make D 3 equal to D O▪ and draw the line O 3, let the Meridian angle taken by the former conclusion be E 4, and make D 5 equal to D 4 a ruler laid upon the points 4 and 5 will cut A D in 7, open your compass to the distance A 7, and keeping one foot in A, with the other towards 3, in the line O 3, mark a point 6, a ruler laid on A and 6, cutteth the limb in 2, the circumference C 2 is equal to the distance of the suns place from the Equinoctial section, in the quadrant, wherein the right ascension was.   

The 23. Conclusion. The suns right ascension being given, to find his declination.  
FRom D and B towards E, take circumferences D O, B T equal to the distance of the right ascension from the nearest Equinoctial section, & laying a ruler on O and T, it will cut the lines A, A H in the points B & Q: in A B take A D, equal to the distance B Q, and draw the line D C, in it take a line C 8, equal to C A, and then taking the distance A 8, set one foot of your compass in E, and with the other in the quadrant E B, mark a point F, the circumference E F, is the declination sought for.   

The 24. Conclusion. The declination of any star being given, with his latitude, which never altereth, to find his longitude or place in the Zodiac.  
LEt L be the pole of the Zodiac, B the pole of the world, and make the circumferences L H, L I equal to the complement of the latitude given, that is, let them be equal to that portion of a quadrant, whereby it exceedeth the latitude, draw the line H I. Again let either of the circumferences E F, C G be equal to the declination given, a ruler laid upon F and G, will cut H I in K: again a ruler laid on A and L, will cut it in M: with your compass take the distance M K, & in the semicircle 6 N A, which is the sinical arch for the line H I, from 6 apply at N a distance equal to M K, a ruler laid upon the points A N, will cut the limb in O: a distance equal to E O, applied in the quadrant C D, to which I suppose the declination belongeth, from C the Equinoctial section, unto P, showeth in the Zodiac from E by B to P the place of the star.   

The 25. Conclusion. The declination of any star being given, with his latitude, which never altereth, to find the right Ascension thereof.  
AS before let L be the pole of the Zodiac, B, the pole of the world, L I, L H, the compliments of the stars latitude, E F, CG, equal to the declination, draw the line H I, and lay a ruler on F and G, which will cut A B in Q, and H I in K; let the semicircle A R S be the Cynical arch for the distance F G, and from S in the semicircle A R S, apply at R a distance equal to the distance Q K, a ruler laid on R & A, will cut the limb in T, the distance, C T, in the quadrant, to which the declination belongeth, applied from the Equinoctial section, showeth in the Equinoctial the right ascension sought for.   

The 26. Conclusion. The longitude and latitude of any star being given, to find the declination and right Ascension thereof.  
LEt the poles & latitude be as before, & let E O be equal to the distance of the stars place from the nearest Equinoctial section: a ruler laid on A and O, will cut the Circumference 6 N A, which is the Sinical arch to the line H I, in the point N; again a ruler laid on A & I, will cut H I in M, take the distance 6 N, & setting one foot in M, with the other towards I, if the place be nearer the pole B, or towards H, if nearer D, but here supposing it nearer B, in M I mark K, upon A and K laying a ruler, it will cut the limb in V, make the circumference B W equal to B V, and draw the line A W, and in it make A X equal to the distance A K, a ruler laid on X and K, will cut the limb in F and G, and the line A B in Q, either of the circumferences E F, C G is equal to the declination. Now in the semicircle S R A, which is the Cynical arch for the distance F G, apply a distance equal to QUEEN'S K from S to R: upon A, and R laying a ruler, it will cut the limb in T, the circumference C T, is the distance of the right ascension, from the Equinoctial section, taken in the quadrant, to which the longitude belongeth.   

The 27. Conclusion. The declination and right ascension of any star being given, to find the longitude and latitude thereof.  
THis differeth not in working from the former, only change the names, let L be the pole of the world, B the pole of the Zodiac, H I joineth the compliments of the declination given, E O is equal to the distance of the right ascension, from the nearest Equinoctial section, 6 N A the sinical arch for H I, is cut in N by a ruler laid on O and A, make M K equal to 6 N, upon A and K a ruler laid cutteth the limb in V, the circumferences B V, B W being equal, and the live A with drawn, A X must be made equal to A K, a ruler lying on X and K cutteth the limb in F and G, so either of the circumferences E F, C G be equal  sky chart to the latitude sought for. In the sinical arch of F G, which is A R S, a distance equal to M K applied from S to R, and then a ruler laid on A and R, cutteth the limb in T, so is C T the distance of the longitude, from the nearest Equinoctial section taken in the quadrant, to which the right ascension belongeth.   

The 28. Conclusion. The longitudes and latitudes of any two stars, or any two places in the earth being given, to find the distance of the one from the other.  
IF they be both on one side of the Equinoctial, or of the Ecliptic what case soever fall out, you shall take their latitudes, as heights above the Horizon, and the difference of their longitudes, as the distance of their Azimutes; by which working, after the 5. Conclusion, you shall find their distance: but if one be on the one side of the Equinoctial, and the other on the other, or, of the Ecliptic, then if they differ not in longitude, from E towards B, take one latitude E L, and make L K equal to the other latitude, the circumference E K in the Equinoctial showeth their distance. If they differ in longitude a whole semicircle, from E towards D take E M one latitude, and laying a ruler on M and A, it will cut the limb on the other side in 6, take 61, the other latitude, and from E to 1, in the Equinoctial, you shall find the distance: but let the difference of the longitudes be E H; and draw the line A H, let the circumference F E be equal to the greater latitude, H I to the less, then making E G equal to E F, and H K equal to H I, upon F and G a ruler laid, will cut A E in N, likewise laid on I K, it will cut H A in P. In the line A B take A Q equal to the distance N P, and in A C make A S equal to F N, and S W equal to I P, in the quadrant C D from C unto 3 apply a distance equal to Q W, the Circumference C 3, is the distance sought for.   

The 29. Conclusion. The Latitudes and distance of any two Stars, or any two places in the Earth being given, to find the difference of their Longitudes.  
LEt the places first be on the one side of the ecliptic, or the Equinoctial, and let the distance given be C 4, let the greater Latitude be E F, the lesser E L; make the Circumferences E G equal to OF, and E M equal to E L, then laying a ruler on F and G, it will cut A E in N, and laid upon L and M, it will cut it in O, make A S equal to N F, and S V equal to L O, then opening your compass to the distance C 4, set one foot in V, and with the other mark Q in the line A B, taking the distance A Q, set one foot in N, and with the other describe an arch at P, then opening your compass to the distance A O, and keeping one foot in A, with the other cross the arch at P, a ruler laid on A and P, cutteth the limb in H, the Circumference E H is the difference of their Longitudes; when the Circumferences meet in the line A, they differ not in Longitude, otherwise the construction is general in this case. But if one be on the one side, and the other on the other, then must you make S W equal to L O, and opening your compass to the distance given, which suppose were C 3, set one foot in with, and with the other, in the line A B, mark the point Q, and then on N, as a Centre at the distance A Q, describe an arch at P, which being crossed with another arch, described upon the Centre A, at the distance A O, a line drawn from A by P to H, showeth E H, the difference of their Longitudes.   

The 30. Conclusion. The Longitudes and Latitudes of any two Stars, or any two places in the Earth being given, to find the angle of Position, or the bearing of the one from the other.  
BY the Longitudes and Latitudes you may find the distance, and first let both the Latitudes be on the one side of the Ecliptic, or the Equinoctial, & their distance less than a Quadrant, as B L, and E F the Latitude of the place from which the other beareth; the complement of the other place whose position is sought C 4, make E G equal to E F, and E M equal to E L, which is the complement of the distance B L, that is given: laying a ruler on G and F, it will cut A E in N, in A C take a line A S equal to F N, then laying a ruler on L and M, it will cut A E in O, make S V equal to L O. Now take the distance C 4, which is the complement of the Latitude belonging to the other place, and setting one foot in V, with the other mark in A B, a point Q. taking the distance A Q. set one foot of your compass in N, and with the other make an arch at P, then upon A as a centre, at the distance A O, describe an arch at P, crossing the former, a ruler laid on A and P, will cut the limb in H, and so E H is the angle of position sought for. But if the distance be greater than a Quadrant, as for example D L, then making E M equal to L E, the excess of the distance above a Quadrant, upon L and M lay a ruler, which will cut A E in O, then having taken in A C, a line A S equal to the distance F N, take also in S C a line S W equal to L O, and the other Latitude being D 3, take the distance C 3, subtended under the complement thereof, and setting one foot in with, with the other mark in the line A B a point Q, opening your compass to the distance A Q, upon N as a centre, describe an arch at P, which crossed with another arch described on the centre A, at the distance A O, a ruler laid on A, and the crossing at P, cutteth from the limb the circumference E H, which is the angle of position sought for.  
But now let one Latitude be on the one side, and the other on the other of the Ecliptic, or Equinoctial, and let the distance be less than a quadrant, than must you take the lines A S, and S V, as in the first case, but in steed of C 4, the complement of the other Latitude D 4, you must take E D 4, compounded of the Latitude D 4, and the Quadrant E D, and opening your compass to the distance E 4, set one foot in V, then take a point in A B, and so prosecute the construction as in the first case.   sky chart  

The 31. Conclusion. The Latitude of one Star, or one place in the Earth being given, with the distance, and angle of position, or bearing towards another, to find the Latitude of this other, and the difference of their Longitudes.  
LEt the angle of Position be equal to the Circumference H E, and let the Latitude given be E F, and the distance being less than a Quadrant, let his complement be H I, make the Circumference E G equal to E F, and H K equal to H I, a ruler laid on F, and G, will cut A E in N: again laying a ruler on I and K, it will cut A H in in P, in the line A B take A Q, equal to the distance N P, and in A C take a distance equal to Q V, which being less than a Quadrant, is the Latitude of the other Star or place, and on the same side of the Ecliptic, or Equinoctial that the former Latitude is: but if it had been greater, than a Quadrant as E D 4, then should D 4 the excess above a Quadrant, be the Latitude sought for, and on the other side of the Ecliptic, or Equinoctial: thus having the two Latitudes by them and the distance given, you may find the difference of their Longitudes.  
But if the distance be greater than a Quadrant, as 6 B H is, then must you take H I, the excess of it, above the Quadrant I B 6, and making H K equal to H I, and E G equal to F E, the Latitude given, lay a ruler on F G, which will cut the line A E in N, and on I K, which will cut A H in P, then make A Q equal to N P, and A S equal to F N, and S W equal to I P, then in the limb from C, apply at 3 a distance equal to with Q, the Circumference C 3, being less than a Quadrant, the Complement thereof D 3, is the Latitude sought for: but if it be greater than a Quadrant, as E D 3, then is D 3 the excess of it above a Quadrant, the Latitude sought, but on the side contrary to the Latitude given, as it was on the same side being less than a Quadrant.   

Adnotandum.  
PTolomey in the first book of his Geography, Chap. 3. promiseth with his Meteoroscopium, by the Latitudes of two places given, and the angle of Position to find the difference of their Longitude, which as Nonnius hath truly noted, cannot possibly by any instrument be performed, no nor by any tables, though Regiomontanus Tabul. prim. mob. Prop. 46. hath undertaken it: this 31. Conclusion expresseth a general truth, and Regiomontanus in his treatise de Meteorosc. quite digressing from Ptolomey, hath in like manner expressed it.   

A Caution.  
IN these four last Conclusions, following the construction here set down, the circle will not be sufficient to contain the lines required, which you may help by the sinical arch of the limb, applying your lines in the limb from the concourse thereof and the sinical arch, it would be too tedious to exemplify all particulars, therefore for your better understanding I will only set down for.   

The 32. Conclusion. Another way to find the distance of two Stars, or places in the Earth, having their Latitudes, and the difference of their Longitudes given.  
LEt the Semicircle A B C be the sinical arch of the limb, and from the concourse of it and the limb C, apply at 7 a distance equal to N P, a ruler laid on C  sky chart and 7 will cut the Semicircle A B C in Z, take the distance C Z, and setting one foot in A, with the other in the line A B mark a point X, again making A S equal to F N, and S V equal to I P, apply from C to D a distance equal to V A, a ruler laid on C D will cut A B C in Y, make T V equal to C Y, and from C in the arch A B C apply at B a distance equal to T X, a ruler laid on C B will cut the limb in 4, the Circumference C 4 is the distance sought for: the other cases may be all handled after this fashion.   

The 33. Conclusion. The longitudes and latitudes of any three Stars, or any three places in the earth being given, to know whether they be in one greatest circle or no.  
TAke the distance between the first and the middlemost, which let be E L, & likewise between the middlemost and the third, which let be L I, and let the distance between the first and the third be E G, now if E G be equal to E I they be in one greatest circle, otherwise not.   

The 34. Conclusion. Any unknown star being seen, to find the right ascension and declination of it.  
Having thrice observed the Azimute, and Almicantar or height, you may by the seventh conclusion find the declination of it, then take the distance of it, from any known star, by the declinations taken as latitudes and their distance, you may find the difference of their right ascensions, as the difference of their longitudes, which added or substracted accordingly from the known stars right Ascension, yieldeth the right Ascension of the unknown Star. Gemma Frisius his chapter answering this conclusion, includeth Comets, but their Parallaxis being neglected, which he regardeth not, neither his working nor this conclusion, no less exact than his working, can duly perform that he propoundeth. In the Moon besides her Parallar, the swiftness of her motion, will be also a let, although Ptolomeye Almag. lib. 7. c. 2. Regiomontanus in Epit. and Copernicus lib. 1. c. vlt. by their Astrolabe or Armillae which Pappus Comment in 5. Alm. calleth Meteoroscopion, take the place of the Moon to find the longitude of the stars, and peradventure the method used of Copernicus lib. 2. c. 2. is fit for that purpose.   

The 35. Conclusion. The latitude or amplitude of the Suns or any known Stars rising being given with their declination, to know the height of the Pole.  
ON the land it is easy to mark in what points of the Horizon the Sun or any star riseth and setteth, and it is a common practice at Sea, to set the Compass with the Sun, at his rising, and going down, the half of the circumference intercepted between those two points, is the amplitude of rising, reckoned from the Meridian. In the Sun about the Equinoctial sections some little error may happen, which although M. Digges in Scal. Mathem. thinketh worth the correcting, may in handling these common instruments be nothing regarded. But to the conclusion: let E F, and C G, be either of them equal to the declination given, and draw the line F G, let the circumference E H, be equal to the amplitude of rising, counted from the East, a ruler laid on H, and A, will cut E I A the sinical arch of the limb in I, with your Compass take the distance E I, and setting one foot in A, with the other toward G, or F, it skilleth not which, take a point K, a ruler laid on A and K, will cut the limb in L, the circumference B L, is the height of the pole.   

The 36. Conclusion. In a Country of known Latitude, the declination of the Sun or any Star being given, to know the amplitude or Latitude of rising.  
LEt the Circumferences E F, C G be equal to the declination given, and B L the height of the Pole, a ruler laid on A and L, will cut F G in K, in E I A the sinical arch of the limb, apply from E to I a distance equal to A K, and laying a ruler on A and I, it will cut the limb in H, the circumference E H, is the distance of rising from the East Northward, if G F fall between the Equinoctial and the North Pole South ward, if between the Equinoctial and the Antarctic Pole.   

The 37. Conclusion. In a Country of known Latitude, the Amplitude or Latitude of the Suns or any Stars rising being given, to find the Declination.  
LEt B L be the height of the Pole, and draw the line A L, let the distance of the rising from the East, be E H, a ruler laid on A and H, will cut E I A, the sinical arch of the limb in I, take the distance E I, and setting one foot in A, with the other in the line A L, mark K, then make B M equal to B L, and draw the line A M, in it take A N, equal to A K, a ruler laid on N and K, will cut the limb in F and G; either of the Circumferences E F, C G are the declination sought for.   

The 38. Conclusion. In a Country of known Latitude, the declination of the Sun or any Star being given, to find the semidiurnal arch, or half the time it continueth above the Horizon.  
LEt the height of the Pole be L B, either of the Circumferences E F, C G, equal to the declination given,  sky chart drawing the line G F, on A and L lay a ruler, which will cut G F in K, take the distance O K, and from R in R P A the sinical arch of G F, apply at P a distance equal to O K, a ruler laid on A and P, cutteth the limb in Q, if the line F G be between the Equinoctial and the apparent pole, the semidiurnal arch shall be B E Q, but if it be between the Equinoctial and the hidden Pole, the semidiurnal diurnal arch shall be Q D, you may easily see in the hour circle, what hours they contain.  

A Corollary.  
THe point Q. if the line G F, be between the apparent Pole and the Equinoctial, showeth the time of the rising: but if it be between the hidden Pole and the Equinoctial, it showeth the setting: if from C towards D, you take in the limb a distance equal to E Q, that point will show the setting, when F G is between the apparent Pole and the Equinoctial, and the rising, when it is between the hidden Pole and the Equinoctial.    

The 39 Conclusion. The declination and semidiurnal arch of the Sun, or any Star being given, to find the height of the Pole.  
LEt the Circumferences E F, C G, be equal to the declination given, and let B E Q, or D Q, be the semidiurnal arch, a ruler laid on A and Q will cut R P A, the sinical arch of the line G F in P, make O K equal to the distance R P, and laying a ruler on A and K, it will cut the limb in L, the Circumference B L is the height of the Pole.   

The 40. Conclusion. By two known Stars, whereof the one is in the Meridian, the other in the Horizon, to know the height of the Pole or Latitude.  
LEt the declination of that which is in the Horizon, be equal to either of the Circumferences E F, C G draw the line G F, cutting the Axis A B in O, and let R P A be the sinical arch of the line GF, and let the right ascension of that, which is in the Meridian, be distant from the nearest Equinoctial section, so much, as Q is from E; now laying a ruler on A and Q, it will cut R P A the sinical arch of the line F G in P, make O K, or ON, it skilleth not which, equal to R P, a ruler laid on K and A, will cut the limb in L, the Circumference B L is the height of the Pole.   

The 41. Conclusion. In a Country of known Latitude, the height and declination of the Sun being given, to know the hour of the day, or what it is a clock.  
LEt the Circumferences E F, C G be equal to the declination of the Sun, and B L and D T the height of the Pole, make either of the Circumferences L S, T H, equal to the height given, a ruler laid on S and H, will cut F G in V, in R P A the sinical arch of F G, apply from R to W, a distance equal to O V, a ruler laid on A and W, will cut the limb in X. Before Noon the observation being taken, and if the Axis of the world fall between V and the Horizon, then in the quadrant C D, taking from C, a Circumference equal to EX, you shall have the hour pointed forth, but if V fall between the Axis and the Horizon, then take in the Quadrant B C from C, a Circumference equal to E X, and the hour shall be pointed out. The observation being taken after Noon, you must use the Quadrants E D and E B in like manner.  

A Caution.  
BEcause when the Sun draweth near the Meridian, you cannot perfectly discern, whether it be before Noon, or after Noon, you must help that matter thus. Take two Observations with some pretty distance between, and if the last be greater than the first, thou was the first before Noon, but if the last be lesser, than was the last after Noon.    

The 42. Conclusion. In a Country of known Latitude, the Azimute and declination of the Sun being given, to find the hour of the day.  
BY the Azimute and declination given, you may by the 14. Conclusion find▪ the height thereof, that being had by it and the declination you shall by the former conclusion find the hour.   

The 43. Conclusion. In a Country of known Latitude, the height and declination of any Star being given, to find the hour of the night.  
LEt the Circumferences E F, C G, be equal to the declination of the Star, draw the line F G, cutting B A in O; let either of the Circumferences L B, D T Bee the height of the Pole, and make the Circumferences L S, T H equal to the height given: on S and H laying a ruler, it will cut F G in V: in R W A apply from R to W, a distance equal to O V; and laying a ruler on with and A, you shall cut the limb in X, if the observation be taken before the Star come to the Meridian, which you shall know by the caution of the 41. Conclusion, then take a Circumference equal to EX on the one or the other side of C, according to the falling of the point V showed in the 41. Conclusion, and so shall you have the distance of the Star from the Meridian, or the hour of the Star, which imagine you find to be pointed out by T: from T toward the point E, take a Circumference  sky chart TEG, equal to the Stars right ascension, and from G towards B take a Circumference G M equal to the Sun's right ascension, a ruler laid on A and M in the hour circle pointeth forth the hour.   

The 44. Conclusion. In a Country of known Latitude, the Azimute and declination of any Star being given, to know the hour of the night.  
THe Azimute and declination being given, you may by the fourteenth Conclusion find the height, then by the height and declination by the former Conclusion you may find the hour.   

The 45. Conclusion. Any day of the year in a Country of known Latitude, to find the beginning, continuance, and end of the Crepusculum, that is, the dawning in the morning, and twilight in the evening.  
LEt B L be the height of the Pole, and either of the Circumferences E F, C G, equal to the declination of the suns place for the day given, draw the line F G, upon A and L laying a ruler, it will cut F G in K, and the limb on the other side in T: from L and T take under the Horzon, that is towards C and D, the Circumferences L Y, T Z of eighteen degrees apiece, upon Z and Y laying a ruler, it will cut F G in 2: in R P A the sinical arch of F G, apply from R to P a distance equal to O K, again from R to 4 apply in R P A a distance equal to O 2, now a ruler laid upon A and P, will cut the limb in Q, but laid on A and 4, will cut it in 6, the Circumference Q 6 in the hour circle will show the continuance of the Crepusculum. This added to the suns rising, which you are taught to know by the 38. Conclusion, Cor. showeth, when it beginneth in the morning, and added to the time of his setting, showeth when it endeth in the evening: I need not tell that it endeth in the morning with the rising of the Sun, and beginneth in the evening with his going down.   

The 46. Conclusion. In a Country of known Latitude any day in the year at any hour assigned, to know the height of the Sun.  
LEt B L be the height of the Pole, and the Circumferences E F, C G equal to the suns declination for the day given, and let E X be the hours distance from six, on X and A laying a ruler, it will cut R P A the sinical arch of the line G F, in with: with your compass take the distance R W, and setting one foot in O, mark in the line F G a point toward G, if in the morning it be before six, or in the evening after six, but if it be after six in the morning, or before six in the evening, then towards F, as here it is supposed in the morning after six, and therefore let it be V, on A and V laying a ruler, it will cut the limb in 5, make the circumference T 7, equal to L 5, and draw the line 7 A; in it take a line A 3 equal to A V, and laying a ruler on V 3, it will cut the limb in S and H, either of the circumferences L S, or T H are the height desired.   

The 47. Conclusion. In a Country of known latitude, for any day of the year, to find in what Azimute the Sun is in.  
BY the former conclusion you learned how to find the height for the time assigned, in the 13 conclusion was taught by the height and declination given in a country of known latitude to find the Azimute, and so by these two conclusions you may perform that, which is here propounded.   

The 48. Conclusion. In a country of known Latitude any day of the year at any hour assigned, to know how far any determinate point is distant from the Meridian.  
LEt the hour assigned be M, and from M by B Eastward take a circumference M B G equal to the  sky chart suns right ascension for the time given, again from G by E take a circumference G E T equal to the right ascension of the point or star given, the circumference T D is the distance thereof from the Meridian, which you desired.   

The 49. Conclusion In a Country of known Latitude at any hour assigned any day of the year, to know the height of any known Star.  
TAke the distance of the point or star at the time given from the Meridian, and so consequently from the hour of six, which let be E X, and let E F, C G be equal to the stars declination, draw the line F G, cutting A B in O, then upon A and X laying a ruler, it will cut R W A, the sinical arch of F G, in with, with your compass take the distance R W, and in the line F G from O towards G, if the distance from the Meridian be more than six hours, but if less than six hours, as here it is supposed, then towards F take V O, equal to R W, then laying a ruler on A and V, it will cut the limb in 5, make the circumference T 7 equal to L 5, and draw the line 7 A, therein take A 3 equal to the distance A V, and laying a ruler on V and B, it will cut the limb in the points F and G, either of the circumferences E F, C G is equal to the height you desire.   

The 50. Conclusion. In a Country of known Latitude at any hour assigned any day of the year, to know the Azimute of any known Star.  
BY the former conclusion you knaw how to find the height for the time assigned, and by the height and declination, the 13. conclusion will teach you to find the Azimute as you desire.   

The 51. Conclusion. In a Country of known Latitude, to find the difference Ascensional, or the difference of the right and obliqne ascension of any point of the Ecliptic or any known Star.  
LEt E W be the height of the Pole, and let either of the Circumferences W I, and W H be equal to the complement of the declination given, draw the line HI, cutting the Horizon E C in Y, and the Axis of the world A W in 4, let 6 Z A be the sinical arch for the line H I, and therein from 6 to Z, apply a distance equal to 4 Y, upon A and Z laying a ruler, it will cut the limb in 5, the circumference E 5 is the difference Ascensionall which you require.   

The 52. Conclusion. In a Country of known Latitude to find the obliqne ascension of any point of the Ecliptic, or any known Star.  
LEt the right ascension of the point taken by the 20. Conclusion, be the Circumference E B G, accounted from E, the beginning of Aries, and the difference ascensional of the point, whose line of declination is H I, let be E 5: now if H I be between the apparent Pole and the Equinoctial, take from G towards E a Circumference G 1, equal to E 5, and E B 1 shall be the obliqne ascension: but if H I be between the Equinoctial and the hidden Pole, then from G towards C, take the Circumference G 3 equal to E 5, and then shall E G 3 be the obliqne ascension sought for.  

A Corollary.  
IF you add a semicircle to the obliqne Ascension found, that is, if you lay a ruler on 1, or 3 and A, it will in the opposite quadrant E D, point out the obliqne descension of the point, that is opposite to the point whose obliqne Ascension you look.   

Another Corollary.  
YOu may also find the obliqne descension of any point by this Conclusion, if you add, when here you are appointed to subtract, and subtract when you are appointed to add.    

The 53. Conclusion. The obliqne Ascension of any point of the Ecliptic, or any known Star being given, to find the Latitude.  
LEt W be the Pole of the world, W I and with H the compliments of the points declination, draw the line H I, cutting A with the Axis of the world in 4, and let the difference Ascensionall (which you may know by comparing of the right and obliqne Ascensions) be E 5, a ruler laid on the Centre A and 5, will cut 6 Z A the sinical arch of the line H I in Z, in the line H 4, take Y 4 equal to the distance 6 Z; then laying a ruler upon A and Y, it will cut the limb in E, the Circumference E W is the height of the Pole, or the Latitude sought for.  
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The 54. Conclusion. In a Country of known Latitude, any day assigned to know what point of the Ecliptic is in the Meridian in the morning.  
TAke the obliqne ascension for that point of the Ecliptic, wherein the Sun is that day which is assigned, as for example the point V, then opening your Compass to the distance B C, set one foot in V, and with the other from the West towards the East, or according to the succession of the degrees or times of the Equinoctial mark in the limb T, that point is the point of the Equinoctial which is in the Meridian: now if by the 22. Conclusion you take the point answerable to that right ascension in the Ecliptic, it shall be the same that is in the Meridian.   

The 55. Conclusion. In a Country of known Latitude any day of the year, at any hour assigned, to find what point of the Ecliptic is in the Meridian.  
LEt the right ascension answering the degree of the Ecliptic, which serveth the day given, be E B V, accounted from E the beginning of Aries, now take the hour distance of the Sun from the Meridian, which let be B W: if the Sun be not yet come to the Meridian, from V towards C, or from West towards East, according to the succession of the degrees of the Equinoctial, take a Circumference V 7 equal to B W, 7 showeth the point of the Equinoctial, that is in the Meridian, the place of the Zodiac answerable to the right ascension, is also in the Meridian with it. But if the Sun be past the Meridian, you must take a Circumference equal to B W, contrary to the succession of the Equinoctials degrees, as from V the Circumference B V, and then the point of the Ecliptic, which answereth the right Ascension B, shall be in the Meridian, and the same is to be understood of the opposite points.   

The 56. Conclusion. In a Country of known Latitude any day of the year, to know what time any point of the Ecliptic, or any known Star riseth or setteth.  
TAke the obliqne Ascension of the point, which let be E B G, and the right Ascension answering the point, wherein the Sun is the day given, which let be E B T, from the hour of the suns rising according to the succession of the hour circle, take a Circumference equal to G T, and it shall point out the hour, when the Star cometh to the Horizon.   

The 57 Conclusion. In a Country of known Latitude to know with what point of the Ecliptic, any known Star cometh to the Meridian.  
TAke the right ascension of the Star, and then the point of the Zodiac that is answerable thereunto, and that shall be the point wherewith the Star cometh to the Meridian.   

The 58. Conclusion. The obliqne ascension being given, to know the situation of the Equinoctial sections in respect of the Meridian.  
LEt the obliqne ascension given be E B 1, with your compass take the distance B C, and setting one foot in 1, with the other mark a point L, which is the point of the Equinoctial in the Meridian: now for as much as E is beyond the Meridian, it is in the West quarter above the Horizon, if it had fallen between L and 1, it should have been in the East, the opposite section is always in the opposite Quadrant.  
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The 59 Conclusion. In a Country of known Latitude to know the horizontal angle, made by the Section of the Horizon and Ecliptic in any point given.  
TAke the obliqne Ascension of the point given, and by it find the situation of the Equinoctial section. Now if the Section be on the East side of the Meridian, take the compliments of the obliqne ascension, and the point given as Latitudes; the suns greatest declination is the difference of their Longitudes, and by the 30. Conclusion seek the horizontal angle, as the bearing of the obliqne ascension from the point of the Ecliptic, that is in the Horizon, and so have you the horizontal angle, but if the Equinoctial Section be on the West side, you must take the compliments to the opposite points of the Ecliptic, and obliqne ascension, and with them work by the Conclusion as is aforesaid.   

To the Reader.  
THere remaineth for perfecting this Section almost 20. Conclusions more, whereof many depend upon this problem: viz. In a Country of known Latitude, any obliqne Ascension being given, to find the point of the Ecliptic Coascending, which yet I have not found how to perform generally, otherwise then by taking such help of an Eleipsis, as is used in the fouretéenth Conclusion: in which manner of working the construction will very often exceed the compass of the limb, for which and some other causes I omit it and the rest at this present. Notwithstanding if I perceive hereafter, that those matters be much desired, they shall be adjoined at the next impression: the mean while I must entreat the reader to take thus much in good part.   
The end of the second Section.   

THE THIRD SECTION OF THE NEW HANDLING OF THE PLANISPHERE; WHICH IS A SUPPLEMENT Organical.  

CHAP. 1. To make a Quadrant, whereby you may readisie know what Degrees and Minutes are contained in any Circumference given.  
Upon the Centre A, describe a quadrant A B C, whose limb you must divide, as the common manner is, into 90. equal parts or degrees, & every degree into halves; in the limb take the quantity of 30. degrees, which let be C D, and draw the line A D. In either of the lines A D, or A C, take some reasonable distance from the limb, as A E, or A F; and upon the Centre A at that distance, describe the Circumference F E, then enlarging your compass a little, describe another Circumference, as you see at F and E: again, opening your compass somewhat larger, describe another Circumference G H, then extending or contracting the distance of your compass, describe a Circumference near it, as you see at G H. In like manner describe two other Circumferences somewhat near together, and distant from  sky chart G H, as you see at I K; and so make between E C, F D 30 such lists, as you see at F E, G H and I K. Then take an arch of 59 degrees, which you must divide into 60 equal parts, the sixty part of which division, will divide two degrees into two unequal parts: take the two degrees next D, and from that point towards D, take a Circumference equal to the greater segment you made by dividing 59 degrees in 60 parts, and upon that point and the Centre A laying a ruler, where it cutteth the list next the limb, as that beneath L M is, draw a line, dividing the list beneath L M, and from that division apply so many degrees in order, as the circumference will admit, till you come so near 1, as you can. Then take 58 degrees, and divide that circumference into 60 equal parts, the sixty part of this division, falling between two degrees divideth them into two unequal parts, again taking 2 degrees next D, from that point towards D, take the greatest portion, and laying a ruler on that point and A, where it cutteth the second list from the limb, draw a line cutting that list, and from that division mark so many degrees in that list, as the residue of the circumference will receive, then divide 57 degrees for the third list, 56 for the fourth list, and so the degrees in order, till you come to thirty degrees, and the thirtieth list, although if the limb be divided into half degrees, the thirtieth list, & the division of thirty degrees is needles. Having thus divided 30 lists, in the space without the quadrant to the side A C draw two parallel lines including two spaces, & extend one circumference of every fift list to the uttermost parallel, in the space next the side of the quadrant A C, at the list next the limb, set one, or 1. at the next circumference extended from the fift list set five, or 5 in the same space, under that which is extended from the tenth, set 10, and so increasing by 5 to thirty, then descending in the second space, at the fift list, set 35, at the tenth 40, and so in order by 5 to 60: then may you thus use the quadrant.  

To find what degrees and minutes are contained in any circumference given.  
Upon the centre A, describe in the quadrant A B C, an arch N P O, having A N, the semidiameter equal to the semidiameter of the circumference given, & take from it, so many degrees, that you have a remainder less than thirty, that is take from it so many times 30 degrees as you can, and if the remainder less than thirty, have a portion of a degree greater than half a degree, taking that remains with your compass, set one foot in Q, and take Q P equal to your remainder; then laying a ruler on A and P, mark the list by whose division the edge passeth most exactly, then in the utter space, on the outside of A C telling the numbers answering that list, you shall have the number of minutes above the just degrees. But if your remainder exceed just degrees by less than half a degree, then taking that portion with your compass, set one foot in N, and with the other mark P, making N P equal to the circumference given, then laying a ruler on A, & P, mark the list whose division is cut by the edge of the ruler, and then in the space next A C tell the number of that list from the limb, the number so found is the number of minutes, above the just degrees. If the semidiameter of the circumference given be greater, than the semidiameter of the quadrant, upon the centre of the circumference given, describe another, having the semidiameter equal to the quadrants semidiameter, & laying a ruler on the centre, and the point given, mark where it cut the circumference, having the semidiameter equal to the quadrants semidiameter, and instead of the circumference given, transfer that circumference into the limb of the quadrant, and then by it prosecute your construction, as is before appointed. Thus may you make one quadrant serve instead of many used by Clavius in his Gnomonia and Astrolabe, and supply with 30. divisions those 60. which in his construction of the like quadrant out of jacobus Curtius he appointeth.    

CHAP. 2. To describe a triangle for enlarging or contracting of Scales, and assigning any parts, or proportion given in any rightline.  
DRaw o●●he out side of the quadrant, two lines pacallell to A B, & divide the outward space into what number of equal parts you will, which you may do at pleasure before you describe the limb of the quadrant, and in the space next A B at every fift division write 5, 10, 15, & so forth, according to your number of divisions, whereby you may readily tell from A, the parts contained in the divided space, then drawing the line D B you shall use it thus.  

To cut from a right line given any part appointed.  
LEt the parts be these, which are contained in A 2, and make A R equal to A 2, draw the line R 2, now the line given is either greater, or less than A 2: first suppose it less, & let it be equal to R Y, make R W equal to R Y, draw the line W Y, now laying a ruler on R and 1, one of the parts contained in A 2, it will cut W Y in X, the line X Y is contained in the line W Y, so often, as the line A 1, is contained in the line A 2. But if the line given be greater than A 2, then make A S equal unto it, and A R equal to A 2 containing the parts assigned, draw the lines 2 R, and in 2 R take 2 V equal to A 1, one of the parts given, then laying a ruler on A, and V, it will cut S 3 in T, the line 3 T is so often contained in 3 S as 1 A is contained in A 2: and so in both cases you have cut off from the line given, the part which was required.  

A Corollary.  
IF any proportion given in a right line as the proportion that is between 1, 2 and 2 A, you may in like manner find in any line given, for both with Y is to X Y and S T to T 3, as 2, 1, is to 1 A.  
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Another Corollary.  
IN like manner you may enlarge, or contract any Scale at pleasure: first taking the half, third, fourth or what part else you will of it, by this problem, and then again divide that part so taken into the parts whereinto the Scale is divided, and by this Scale so contracted you may make a plat, whose sides correspondent to the sides of the first plat, shall be half or a third or a fourth part, or any number you first took, likewise by the second case you may enlarge them at your pleasure.  
You may also divide any line given into more parts than are in the side A B, which how to perform with some cautions for avoiding error in mechanical practice, I will reserve till another edition.     

CHAP. 3. Of the Meridian line, the variation of the Compass and the height of the Pole.  

A lemma About a Circle given to describe a Quadrant.  
LEt the Circle given be B C D E, whose centre let be A, and let the Diameters B D, C E be perpendicular one to another, then extending one of them, namely B D, make A G equal to the distance B C, upon G as a Centre at the distance G B, describe an arch F B H, and from G draw two touch lines G L F, and G M H: I say that G F H is a Quadrant, circumscribed about the circle B C D E, uz. the lines F G, G H, & the circumference F B H, do only touch it.  
If you make the semidiameter of the quadrant to another line, as D B the Diameter of any circle given, is to the distance B C, subtending a quadrant of the same circle, the excess of the Dameter above that line, is the semidiameter of the circle to be inscribed in that quadrant.   

The use and making of an instrument tractable as well at Sea, as at Land, wherein the shadow of the Sun shall at one instant point out his magnetical Azimute and height.  
Upon the centre A, describe a circle B C D E, and by the former Lemma, about it describe a Quadrant G F H: and draw the lines A L, A M parallel to the sides F G, and G H, then divide the limb F B H into 90. degrees, beginning at F, this being done, cut out a circle somewhat less than the circle B C D E, but concentricke unto it. Then take an Isoscheles triangle G H O; having the angle at H aright angle, and either the sides including it equal to the semidiameter of the Quadrant, this triangle must be placed upon the line G H, in such manner, as the point G agree just with the centre of the quadrant, and so that the plains of the quadrant and traingle be perpendicular one to the other: then must there in some place of the line F G be set a standard or plate (it will be best between L and G) so that one of the edges standing just upon the line G F, be perpendicular unto it, and the whole plate itself parallel to the forenamed rectangle triangle. On this standard, or some where in the triangle on that side, which respecteth F, you must hang a plumb line, and if you will set another plate upon F G, some where about F, making his section with the quadrant in the line G F, and also perpendicular unto the plain of the quadrant, you may by a plumb line hanged upon this plate, or standard, and the other plumb line hold the quadrant G F H, both at Sea and Land precisely level with the Horizon. Besides this you must make a round box, having the circle in the bottom divided into 360 degrees, or every quadrant into 90, as you like best, which box must be fastened unto the quadrant in such manner, as the two Diameters crossing one another at right angles, have the one, one end directly answering the point L, & the other another end likewise precisely answering the point M; in the centre of this boar, you must set a pin to bear a needle touched with a very good Loadstone, making the needle of such length, as it may freely play in the box; this boar must be covered with a glass to keep it from dust and other hurt, as in dials and compasses is used. The standards and the triangle may be so shouldered to the quadrant, that you may take them off, and put them on when you will; the manner how, I need not to stand upon, every workman can easily devise how, and also supply every particular circumstance in the making  sky chart of this instrument required, which I very willingly refer to their handines and skill, only I must warn them very carefully to place the triangle precisely perpendicular to the quadrant, with the point of the acute angle agreeing with the centre, and that the ends of the Diameters in the bottom of the box, do very exactly respect the points L and M in the sides of the quadrant, you shall use this instrument thus. Hold the quadrant by the plumb lines level with the Horizon, and turn the triangle towards the Sun, till the shadow of the plate or Standard that is next G, do fall precisely upon the line G F, or (if you will place it so) upon some line parallel unto it, the shadow in the limb of the quadrant will cut off, accounting from F unto it, a circumference equal to the height of the Sun. At the same time if you mark the point M, and the circumference that is between it and the South end of the needle, that is the distance of the Sun from the South of the Magnet, or the Magnetical Azimute of the Sun reckoned from the South.   

By the instrument now described, at noon tide in any Country to find the Meridian line, the variation of the Compass, and (having the suns declination given) the height of the Pole.  
ALthough it be a matter not very easy to find precisely when it is noon, or the Sun at the highest, yet may you guess when it cometh somewhat near it, as it is commonly practised by Seamen, which use at that time with their Astrolabe or cross staff, to make four or five sundry observations, and amongst them take the highest for the Meridian height: in like manner when it draweth to be about noove, make sundry observations with this instrument, marking the height of the Sun, and his Magnetical Azimute, as was even now showed in the use of the instrument, the greatest height is the Meridian height, and the Diameter A M doth precisely answer the Meridian line, the circumference intercepted between it and the south point of the needle, is the variation of the compass, whether East of West, is easily determined by the side of A M, on which the needle resteth. Now the Meridian height being given, with the declination, you have been taught in the tenth Conclusion of the second Section, how to find the Latitude or height of the Pole. The common manner of finding it by adding or subtracting of the declination is more artificially handled by Nonnius lib. 2. cap. 9 de Reg. Navig. then hath to my knowledge been hitherto published in English, and therefore I think it not amiss to translate his words, which be thus:  
We must observe the sun, when he is highest above the Horizon, which is at noon, then if the shadows of bodies perpendicular to the Horizon be cast that away, that the Sun declineth, in the day of your observation, you must add the complement of his greatest height to the declination so have you the degrees and minutes of the latitude, North, if the suns declination be North, South, if it be South.  
But if the shadows be cast to the contrary part, than you must compare the suns declination with the complement of his height, which if you find equal, the Zenith is in the Equinoctial: but if they be unequal, subtract the less from the greater, & the remainder is the latitude, named of the declination, if the declination be greater, but if less, of the contrary part or side.  
When the sun hath no declination, the complement of his greatest height is the latitude, and toward that part or pole, towards which the shadows are cast: you may know by the Mariner's Compass whether the shadows be cast North or South. When the Sun is in the Zenith, the declination, if it have any, is the latitude: thus much Nonnius.   

By the same instrument two observations being taken; when the Sun hath equal altitudes, to find the Meridian line, the variation of the compass, and (the suns declination being given) the height of the Pole.  
THe Sun doth not always shine at noon, & therefore at other times you may help yourself thus. Make an observation with your instrument marking the height of the Sun, and the circumference intercepted by the point of the needle, and that diameter, which is perpendicular to that side of the quadrant, on which the triangle standeth, and do this if you will four or five times before noon: in the after noon mark, when the Sun cometh again to the same height, which it had any time, when you made your observations in the forenoon, and again how many degrees are between the Diameter that is perpendicular to the side of the Triangle, and the North point of the needle, then compare that found in the fore noon, with that you found in the afternoon: if they be both on the same side of the needle, subtract the less from the greater, and the half of the residue pointeth forth the right North, to which if you add the excess of the greater magnetical Azimute above the less, you have the variation of the Compass, Eastward, if the first observation were lest, Westward, if the second. But if the one be on the one side, and the other on the other, then add both the circumferences together, & of them both take the half, that point determineth the precise north: with this half compare the last observation, and if it be greater, the variation is Eastward, but Westward if it be less. Now the Meridian line being found, you know the Azimute, wherein the Sun was at the time of your observation, and you also took his height, which two being given with his declination, you may find the latitude, as was showed in the 8 and 10 conclusions of the second Section.   

By the same instrument three observations being taken, whereof two be, when the Sun hath equal altitudes, to find the Meridian line, the variation of the Compass, the declination of the Sun, and the height of the Pole.  
YOur three observations being taken as is required, you have two of them when the suns heights be equal, by which you are taught in the former problem how to find the Meridian line, and the variation of the Compass, the Meridian line being found, you may know the distance of both those Azimutes from it, which have different Altitudes, by which two Azimutes and their altitudes, the ninth and tenth Conclusions of the second Section will show you, how to find the declination of the Sun, and the Latitude.  
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By the same instrument three observations being taken howsoever, to find the Meridian line, the variation of the Compass, the declination of the Sun, and the height of the Pole.  
Having taken by your instrument in three observations, the heights and distances of their Azimutes: from any quarter of the limb, whose Diameter you must account for the first Azimute, number the distance of the second, and from that point so taken, the distance of the third, and let those points of the Azimutes be E F G, draw the lines F A, G A; again, let E H be the height belonging to the Azimute E, and E K the height for the Azimute F, and E N the height for the Azimute G; now laying a ruler upon B and H, it will cut A E in I, again laying a ruler on B and K, it will cut A E in L, in A F make A M equal to A L, again laying a ruler on B and N, it will cut A E in O, in A G take A P equal to A O, now unto the three points I, M, P, find the centre, which let be Q, on A and Q laying a ruler, it will cut the limb in R, that point R is the right North, or South, and so have you the distance of the Azimutes from it, that is E R is the distance of the first, F R of the second, G R of the third, the difference of these Azimutes, and the Magnetical Azimute belonging to every of them, is the variation of the Compass: by any two of these Azimutes, so their height be different, in the second Section you are taught to find the declination and height of the Pole, which two things in the problem required, Nonnius, from whom thus much is borrowed, Lib. 2. Navig. c. 16. prosecuteth after a far different manner, yet suitable to the construction here set down already: but for that his construction will seldom be contained within the compass of the limb, I thought good to omit it: this finding of the Meridian line, being very ingenious and commodious, though it sometime will incur the like inconvenience, I thought worthy to be known. Clavius hath another way to perform the things required, but nothing easier than this of Nonnius. For both of them about the Equinoctial will produce the construction hugely without the compass of the limb, and in Nonnius his way, the suns parallel will differ little from a right line, so that you must use a peculiar instrument for to describe it, and yet are you nothing the nearer for finding of the centre. Therefore if at Sea the Azimutes and almicanters of the Stars might be taken, Nonnius his construction would be of far more use: for then working by such stars as be near the Poles, their Parallels would easily be described. I have thought of an instrument for that purpose, as to use a quadrant with an index, and sights, instead of the triangle, or to fit your box with the needle, at the lower end of the Mariners Astrolabe, for want of experience I dare not affirm much, but I think that Art, handiness, and diligence might do some thing with such instruments, the better consideration whereof, I willingly refer to the Seaman's experience.   

The suns declination being given with the height of the pole, to find the Meridian line by one observation.  
PLacing your quadrant with the triangle upon any plain made level with the Horizon, turn it as is appointed for the finding of the height, and by the side, on which your standards are placed, draw a right line: by the height, which you then find, in a country of known Latitude the suns declination being given, you are taught in the second section Conclusion 13, to find the Azimute, which being done, describe a circle upon a centre taken in the former line, and from the line taking accordingly a circumference answering the Azimute, the line which joineth that point, and the centre, is the Meridian line. Clavius hath two other ways to do this: one Gnomon. lib. 1. cap. 23. the other in his Astrolabe lib. 3. Can. 12. and a third I have found, but none to be preferred before that which is here propounded, notwithstanding if this book be printed again, those three also shall be adjoined.  
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CHAP. 4. Of Maps or charts and Nautical directions.  

Any line being given, to find another line, unto which the line given shall have that proportion, that the Meridian hath to any Parallel.  
Upon the centre G describe a semicircle F H Y, and let H G be perpendicular to F Y. Let F B, Y Z be equal to the distance of the Parallel from the Equinoctial, and draw the lines B G, G Z: let G A be equal to the line given, and in G Z take G S, equal to G A, laying a ruler on A, and S, it will cut G H in M, draw the line A M. I say G A, or the line given hath the same proportion to A M, that the Meridian hath to the Parallel, whose distance from the Equinoctial is F B, which was required to be given. I thought good to set down this exact and ready way, because that, which Mercator describeth in the poem of his reformed tables for Ptolomey, is more troublesome, and less exact.   

To divide any line, or the side of your quadrant, or your ruler as that line is divided, which in Gerard Mercators' General map, and his directorium doth answer the Meridian.  
Divide the line you take for the ground of this work first into 10 equal parts, and then every one into five parts, imagining every one of the fift parts to be divided into twenty: besides take every fift part for a degree in longitude: then to find the degrees of latitude answerable unto it do thus: in G Y the semidiameter of the quadrant GH Y take G I equal to one degree of longitude, and by I, draw the line I P parallel to G H; then laying a ruler on G and every degree of the limb, accounting from Y mark where it cutteth I P, the line intercepted between those sections and G do answer the correspondent degree: as for example, I take a degree at Z, uz. the 45 degree, than I lay a ruler on G and Z, which cutteth I P in 2, the line G 2 in the Meridian answereth the point Z, or is the 45 degree: Now having marked these sections for every degree in the quadrant, transfer them one after another into your ruler, or the side of your quadrant, that is at one end of your quadrant, take a line equal to that, which answereth one degree, add to it, that which answereth the second, to that put that, which answereth the third, and so till you come to the end of that line, which not being able to receive all, draw another line, and prosecute the like construction in that, and so in the third, and forth till you have transferred all the lines answering your degrees. If in this transferring, your divisions agree not just with the ends of your lines, you must take only the overplus of the last division in the beginning of the next line, and so proceed as is aforesaid. Now having your lines thus divided, and another divided to the same parts that the side of your quadrant is, being parallel to them, & all perpendicular to one right line, you may readily know what equal parts are contained between any two of your degrees taken howsoever, whereby you may perform all the uses of Mercators' Directorium. Thus for example.   

The Longitudes and Latitudes of any two places being given, to find their direction, commonly called the Rumbe.  
TAke the equal parts contained, as well in the difference of the latitudes, as in the difference of their longitudes, and see whether both be less than the side of your quadrant, or no: first suppose they be less, and let the geater be X G, the lesser G M: draw the line X M, divide G X into two equal parts in V, then opening your Compass to the distance V G, and keeping one foot in V, with the other mark T, in M X; on T and G laying a ruler, it will cut the limb in 4. Now if the difference of the Latitudes be less than the difference of the longitudes, 4 Y is the distance of the rumbe from the Meridian, but if the difference of the longitudes be less, then is 4 P the distance of the rumbe from the Meridian. But now suppose that one or both the differences, be greater than the side of your quadrant, then by the rule of proportions, you must find a line, unto which the lesser is, as the greater is to the side of the quadrant, or make the side of the quadrant to another line, as the greater is to the lesser, namely let G Y be unto G Q, as the greater difference is to the less, and draw the line Q Y: divide G Y into two equal parts in with, then opening your Compass to the distance W G, and keeping one foot in with, in Y Q mark R, a ruler laid on G and R, will cut the limb in 3: now as before, if the difference  sky chart of the longitudes be the greater, than is H 3 the distance of the rumbe from the Meridian, South, if the place respected be South, North, if it be North, and so likewise East, or West.    

CHAP. 5. Of the back of your plate.  

To inscribe in the back of your plate the fixed stars according to their Longitudes and Latitudes, or declinations and right ascensions, as you please.  
LEt the utter edge of the back side of your plate be divided into 360 degrees, which you may use for the Zodiac, or the Equinoctial; the Zodiac, if you inscribe the Stars according to their Longitudes and Latitudes, or the Equinoctial, if you will place them according to their declinations and right ascensions, which is the best, for those are oftenest in use, both are done after one manner thus. From the beginning of Aries reckon the right ascension, as from E by B unto T, and draw the line A T, then from B to V, reckon the declination B V, laying a ruler on V and A, it will cut the sinical arch of the limb A S C, in S, in the line A T take A Q equal to the distance A S, and the point Q is the place of the star according to his right ascension and declination: this being done, set down by it the name, and S or N, to signify, whether it decline North, or South. This may be done so slightly, that you may rub it out when you will with a wet pumice, and yet deep enough to continue a great while for your use.   

To find the longitude and latitude, or the declination and right ascension of such stars as be placed on the back of the plate.  
Upon the centre of the plate, and the point of the star lay a ruler, where it cutteth the limb, is the right ascension reckoned from the vernal Section, or the longitude, if they be placed for the Zodiac. As for example:  sky chart 
Place this figure in stead of the figure in folio 35.  lay a ruler on A and Q, it will cut the limb in T, the circumference E B T is the right ascension of Q. Now if in A S C the sinical arch of the limb, you apply from A to S, a distance equal to A Q, and lay a ruler on A and S, it will cut the limb in V, the circumference B V is the declination of the star placed at Q, or the latitude, if they were placed according to the longitudes, & latitudes: by the letter S or N you shall know whether it be North, or south. And that you may be furnished for these two last problems, I have set down on the other side a table out of Clavius, containing certain fixed stars, both with their longitudes and latitudes, and their declinations and right ascensions, calculated for this year 1600. And thus much at this time, for the Planisphere, which hereafter I mean to increase with more conclusions, & problems, and could have now enlarged it with handling mensurations, sinical calculations and dialling: but those things seemed somewhat far from my principal purpose, and therefore I will entreat the reader to accept thus much only as now is delivered, and so for this time I end.     
The end of the new handling of the Planisphere.   


A Table of fixed Stars out of Cla●●ius, calculated for the year 1600. complete.  Bigness Names. Place in the Zodiac or Longitude Their Latitude The part The declination. The part. The righ● Ascension 3 The Rams former horn. Aries 28 5 7 20 N 17 39 N 23 20  Medusa's head. Taur 21 5 23 0 N 40 5 N 40 55  Bulls eye. Gem. 4 5 5 10 S 15 56 N 63 6  Orion's right shoulder. Gem. 23 25 17 0 S 6 21 N 83 41  The Goat. Gem. 16 25 22 30 N 45 9 N 72 6  The great dog. Can. 9 5 39 10 S 15 ●4 S 97 19  Hydra's bright Star. Leo. 21 25 20 30 S 5 4 S 137 19  Lions heart. Leo. 23 50 0 10 N 13 44 N 146 19  Lions tail. Virg. 15 55 11 50 N 16 26 N 171 49  The Virgin's spike. Lib. 18 5 2 0 S 8 58 S 195 55  Arcturus. Lib. 18 25 31 30 N 21 49 N 209 23  Scorpions heart. Scor. 4 5 4 0 S 24 57 S 241 16  Harp. Cap. 8 45 62 0 N 38 40 N 275 15  Last in Aquarius water. Aqu: 28 25 23 0 S 33 24 S 339 56  Swans tail. Pisc. 0 35 60 0 N 44 8 N 307 22  Pegasus leg. Pisc. 23 35 31 0 N 25 44 N 341 0   

Errata.  
Page 3. line vlt. Analemma. p. 6. l. 6. Maurolycus. p. 10. l. 9 Frisius. p. 13. l. vlt. Motus raptus. p. 14. l. 33. Zenith, appearing. p. 16. l. 33. from the Zenith, which. p. 17. l. 2.5.7. for Z make L. p. 22. l. 1. blot out pan. and l. 11. N L. p. 23. l 13.25. for Z make L. ibid. l 20 tropics. p. 24. l. 26. there makes angles. p. 25. l. 12. and by two of those. p. 26. l. 9 setting januarie. p. 32 l. 26.27. for Q make 7. p. 33. l. 10. for 3 & 3: make 8 & 8. ibid. l. 13. for G make E. p. 34 l. 4.5. for T make V l. 13 for F make E. l. 27. which let be C 1, 1 l, and l K. l.: 8: 29. for B make P. p. 26 l. 2.7. for B make P. l. 5. for 12. make 2. l 8. for 2. and D make P. p. 38. l. 18. and H l, cutting. p. 40. l. 23. for K make B. in the figure of the 39 page place 3. on the other side of D.