HOROLOGIOGRAPHIA. The Art of Dialling: teaching an easy and perfect way to make all kinds of Dial's upon any plain plat howsoever placed: With the drawing of the twelve Signs, and hours unequal in them all. Whereunto is annexed the making and use of other Dial's and Instruments, whereby the hour of the day and night is known: Of special use and delight not only for Students of the Arts Mathematical, but also for divers Artificers, Architects, Surveyors of buildings, free-Masons and others.  
By T. Fale.  
AT LONDON Printed by Thomas Orwin dwelling in Pater noster-Row ou●… against the sign of the chequer. 1593.   

SINGULIS ARTIUM MATHEMATICARUM STUDIOSIS IN CELEBERRIMA CANTABRIGIENSI ACADEMIA, THOMAS FALUS EIUSDEM ALUMNUS, ET VERAE MATHESEOS STUDIOSUS, EXIGWM HOC GRATI ANIMI MONUMENTUM DD. ANNO. 1593.   

Simonis Muri ad lectorem carmen.  

Scire cupis certa cur machina tangitur umbra?  
Et brevis aethereum line a fignat iter?  
Haec, licet evoluas, facili praecepta libello  
Tradita, sed voto non leviora tuo.    

To the friendly Readers, and namely to such as be well-willers to the Mathematiks.  
THe Arts Mathematical (gentle Reader) in regard of their antiquity and excellency may be compared with any other of the liberal Sciences whatsoever. For Seth who lived in the first age of the world is commended of josephus, 
Lib. 1. c. 3.  and Abraham of Berosus, to have been skilful masters in these mysteries. And the very name importeth, that in old time these of all other were esteemed worthy to be taught, being called for their excellency Mathemata, that is, Sciences 
Aul. Gell. de ●oct▪ Att. l. 1 Ram. Scho. Math. lib. 4.  meet to be learned. These be Arithmetic, Geemetrie, and Astronomy, from which this Art of Dialling taketh his beginning: a knowledge also ancient and necessary, and therefore practised by Princes and famous men of former ages. The first Dial that histories remember is Lib. 2. cap. 20. of the Kings in the holy 
2. King. 20. vitrvuius de Archit. lib. 9 Munst. Hor. in praef. Lib. 2. c. 76.  scripture, where the Lord turned the Sun back 10. degrees for Hezechias sake, whereby it had gone down in the Dial of Ahaz. This Ahaz was King of jerusalem, and reigned in the 3200. year after the creation of the world, and in the first olympiad of the Grecians. Afterward as Pl●●e writeth, Anaximenes Milesius the scholar of Anaximander, first found out the reason and proportion of shadows amongst the Lacedæmonians, and there taught the Art of Dialling, who lived 200. year after the reign of Ahaz, and was a famous Philosopher in Greece before Plato's time, as Diogenes Laertius reporteth. But some affirm that it was Anaximander himself that found 
Cal. R●●digin.  out this Art, and set forth the first Map of the earth.  
Herodotus saith, that the Grecians learned this art, and the division 
Lib. 2.  of the day into twelve hours, of the Chaldeans. Diodorus writeth that one Hyperion first observed the hours. But if we may believe Macrobius, it seemeth that this Science came from the Egyptians: for they called the Sun Horus, which by his motion limitteth to each hour his appointed time.  
vitrvuius rehearseth sundry inventors of this Art of Dialling, 
Lib. 9  as Berosus the Chaldean, Aristarchus Samius, Eudoxus, Theodosius with others, who were renowned, and lived many years before the birth of Christ. I need not here remember Architas Tarentinus, who by art made a Dove of wood to fly in the air: neither Archimedes, who defended Syracuse against Marcellus, 
Plutar●. in Marcel.  and affirmed that if he had a place to stand on, he would move the earth with his engines: both of them no doubt skilful in this Science.  
It was long after the invention that this Art was known in Rome: for in the twelve tables was only mentioned the rising and setting of the Sun, and after certain years the midday was added. Then in the first Punic war the Romans obtaining victory, there was a Dial brought amongst other spoils out of Sicily. But in process of time they began to be more common in Rome: for 100 years before Cicero his time, the parasite in a Comedy, being hungry, spoke against the multitude of Clocks and Dial's which were then in the city, Optans ut suus cuique venter sit horologium.  
Pliny also telleth of a Dial placed in the field of Flora at 
Lib. 36. c. 10  Rome, which by the space of thirty years had not agreed with the Sun: and the reason was as he thought, because that either the Sun had taken a new course, or else the earth was slipped from his Centre, wherein at the first it stood: or the stile was put awry by the shaking of the city. Since which times learning spreading itself into divers parts of the world, this Art hath been amongst the rest in great account.  
Concerning the profit of this Art, daily experience teacheth how needful it is in a well ordered Commonwealth, seeing nothing can be done in due and convenient season, where this Science is neglected: for the division of the day into certain parts or hours, (which this Art teacheth) doth limit and allot to each action his due time. This Art being then so ancient, and the use so necessary, I trust none will think this labour superfluous, unless they be rude without civility, or such as have always at hand a Dial of natures framing, of whom this verse seemeth to be made:  * Si tuus ad solem statuatur nasus, hianti 
Morus.  Ore, bene ostendas dentibus hora quota est. 
Many have promised (but none as yet performed) to write of this Science in our English tongue, which hath been published in other languages, as D. Record long since, M. Digges, M. Blagrave with other, who if they would take the pains, I know could do it with great commendation.  
divers have written hereof in the Latin tongue, as Munster, Schonerus, Orontius, Witekindus, Clavius and others: yet every one differing from other in precept. Some teach the making of Dial's by the help of the Globe, as Gemma Frisius: some by the Astrolabe, as the same Gemma Frisius and M. Blagr. which Instruments every man have not. Some use the table of Sines and Arithmetic, as Witekindus, which way as most plain and easy, is observed in this book; though in some kinds for want of trial Witekindus deceived himself. Munster useth a Rectificatorie with a circle, which is unfit for small plaits, and faileth in greater, without great heed. Schoner wandereth in a wilderness of lines, that a man know not where to begin, or when to end. Vlmer hath not the Delineation of all kinds.  
Othersome observe the rules of Geometrical proportion: which order also we thought to have observed in all kinds, as we have done in the South and North erect declining: but that sundry precepts of the same thing would have bred tediousness and trouble to the learner, and the cutting of the Figures would have been very chargeable. By means whereof we contented ourselves with this one way here set down, not doubting, but that every one with small pains may attain to the making of all kinds of Dial's in this book expressed. As for a great part of them, every Artificer may easily understand.  
Only thus much I advertise the unlearned, that they must acquaint themselves with some few Mathematical principles, as to know what the Elevation of the Pole meaneth, how a squire line is to be drawn, and such like, which (if they want a teacher) they may sufficiently learn by themselves out of Records Castle, his pathway and ground of Arts, published in the English tongue: for these terms could not be avoided, neither plainly described without much tediousness.  
We have here added also examples and figures to every kind, that so the precepts might appear more plain and easy: so that there is no plain plat or wall howsoever it standeth, or be placed either Declining, Reclining, or Inclining, but by the help of this book you may draw a Dial upon it.  
If any man complain of obscurity, he must know, that Difficilia quae pulchra, and yet small pains overcometh all.  
The making of the horological Cylindre, and the Ring, with some other Instruments, we have presently omitted, partly for their curiosity in cutting and delineation, and partly because (if occasion serve) we will entreat of these kinds of horological Instruments by themselves, together with the making of all kinds of plain Dial's in this book, prescribed by the way of Geometrical proportion. In the mean while (gentle Reader) committing this book to thy favourable acceptation, and thyself to the protection of the Almighty, I end.  
Tho. Fale.   

To my loving kinsman Thomas Osborne.  
THis book, which seven years since, was in a manner perfected (as you know) doth now upon just occasion present itself to the view of the world: wherein you have taken such pains for the trial of each example, that I think none can find any great fault, but such as can see far into other men's faults, and forget their own. For after we found some precepts in Witekindus to be false, we were enforced to try and examine with great care each figure and example in the Sun. And therefore if any receive benefit by this our travel, I would you should have your due praise you deserve.  
I have altered some few things, and added the making of the South and North Erect Declining Dial's, by the way of Geometrical proportion: because those kinds be most in use, and I would the learner should have his choice of the easiest way.  
The graver of the Figures was one M. jod. Hondius, who hath showed himself an excellent workman in the great Globes set forth by M. Mullineux, and the Maps of England for M. Camden's book: and whether he hath performed like diligence in these, I refer it to yourself to judge. If any be desirous to have the Instrument mentioned in the beginning of this book for the trial of plaits, I hope you will help them to it: for being of your own invention, I know none so fit as yourself to make it: on which Instrument also, it were convenient to draw the Quadrant horological, that so it might serve for divers uses.  
I trust you will not be offended in that I leave under our names this small monument unto the world, as a speaking witness of our thankful hearts to this our Country, and a testimony of our affection towards the Arts Mathematical. Thus beseeching the Lord (who hath endued you with extraordinary knowledge in all Manuel Sciences) to finish that good work of his heavenly grace already begun in you, to his glory and your own comfort, I take my leave. Commend me many times to yourself & all our good friends.  
From London, january. 3. 1593. Yours, assuredly, T. F.   

Advertisements to the Reader.  
NOte always, that in every Dial the one end of the Style must be placed directly towards the North Pole, and the other end towards the South Pole: for about these two stars the whole heavens are moved, they remaining immovable. The North Pole is a star in the North part of the heaven, being raised above the earth or our Horizon 52. degrees. And this changeth his height, if you go Northward or Southward one degree in 60 miles: but eastward or westward it altereth not. The Elevation of the Pole is the height of this star above the earth.  
Observe also, that the Substile is the line or place over which the Style or Gnomon in your Dial directly hangeth. The space between the Style and Substile is the just height thereof.  
One line cutteth another squirewise, when they make right and equal angles.  
The Contingent or touch line, is that which is drawn by any point of another line or circle, so that it toucheth the same: and this line commonly in all Dial's is drawn squirewise to the Substile.  
A quadrant is the fourth part of a circle.  
The Meridian and twelve a clock line are all one.  
I call these lines parallel which are every where of like distance one from another: example whereof you may see in the Est and West Dial, where all the hour lines be parallel.  
You may make all kinds of plain Dial's upon one stone, if you prepare it first to be square like a die, and then take off the eight corners, and all the sharp sides, so shall you have 25. plain plaits besides the base, or foot whereon your Dial must stand.  
If any be desirous to have this Instrument ready made, let him inquire at the Printers, and he shall hear of them.   


The making of an Instrument to find out the situation of any plat or Dial, and to place them already prepared. CAP. I.  
FOr as much as it is necessary before you can draw my Dial, to know how your plat is already or aught afterward to be placed, it shallbe expedient to show the making and use of an Instrument, whereby you may examine and try all plain plaits and place all dials being ready made and prepared.  
Provide therefore an half circle of Peare-trée, Walnut-trée, Borachia, or any other close grained and solid wood, being well seasoned, so that the alteration of the weather at any time may not make it change from the first proportion ther of. Let it be perfectly tried on both sides of an even thickness, half or three quarters of an inch thick or more if you will, and sir or eight inches broad as you think good. The edge or side A. B. must be very right. Then draw the Line C. D. three quarters of an inch equally distant from the side A. B. place one foot of your compasses in the centre E. which is the midst of the line C. D. and with the other draw half a circle from C. to A. divide it into two equal parts or quadrants, and laying your ruler upon the centre E. and upon this division draw the line E. F. this done, divide each of these quadrants into 90. equal degrees or parts, which you may thus do: First, divide each of them into three equal parts, and every of these into three other parts, so shall you have 9 divisions in either quadrant, whereof every one shall represent ten degrees. Again, part every one of these into two, and each of those into five (if you can) and so is each quadrant divided into 90. degrees.  
Moreover, it were necessary if your Instrument were large enough, to part each degree into 60. minutes, or at the least into 3. whereof each part may contain 20. minutes. This done, fasten a thread well warred in the centre E. with a plummet of lead on the end, so that it may move at free liberty.  
You may also (if you will) upon this side of the quadrant between E. D. draw the dial whose delineation is taught in the 28. Chapter.   dialing diagram 
It were necessary to have here a little hole with a cover to put in your pl●●●…et & thread.  
The ●omer ●●● of ●●●● Instrument thus finished, let the other be altogether li●e unto it, except that in stead of the the thread with the pl●●●●…t, you must have a plombrule made in this manner.  
Prepare a piece of very good wood, try it perfectly on both sides to an equal thickness, let it be about half an inch thick, and twice so broad as the space between the line C. D. of your instrument and the edge A. B. in the midst thereof draw the right line G. H. and placing one foot of your compasses in the point G. with the other make a half circle according to the breadth of the ruler, whereby you may cut the end round, make the other end with a sharp point right in the end of the line G. H. as you see in this figure.  
Let the distance between the point G. and the end H. be so much as is the semidiameter of the quadrant, make by G. an hole so great that a plommet of lead hanging by a thread from the sharp end, may move therein at liberty: Make also an other round or square hole, wherein you may place a needle touched with the Loadstone, as in the compasses or dials you see. Finally, make a little hole in the point or centre G. and with a broad headed na●e of brass fasten it upon the centre E. of your Instrument, so as it may move & turn which way you will, and then your Instrument is finished.  
Note that you may make your Instrument in form of a quadrangle rightangled, as Witekindus teacheth, cutting a round furrow without the circle and degrees made, for the plommet to hang at liberty in: so shall you need but one side of your instrument to use, with a compass or needle touched with the Loadstone for declinations. But the form of the other, whose making we have taught, seemeth more commnodious, behold the Figure.   

The use of this instrument. CAP. 2.  
 dialing diagram  
The use of this instrument is in examining and placing the plat or dial. In immoveble plaits, to know how they stand, as if you would make a Dial upon a wall, first you must understand whether the wall be erect or else recline or incline: then whether it be direct East, West, North, or South, or else decline, which you cannot know without the help of this Instrument. Again, in movable plaits for the placing of them, as if you have made a South dial or any other kind, by this Instrument you shall easily and perfectly place them.  

The trial of a plat for a horizontal Dial.  
IF your plat seem to lie plain, making no angle with the horizon, set the side or diameter A. B. of your Instrument upon it, and erect the plombrule right up between the two quadrants precisely over the line E. F. then if the thread with the plummet hanging at liberty fall directly upon his line G. H. which way soever you turn your Instrument, such a plat is said to make no angle with the horizon: and therefore the dial which is made upon it, is called a horizontal dial. In like manner may you place this kind of dial being ready made parallel to the horizon, and by the help of the needle direct the stile towards the North pole.   

The trying of plaits for Erect, Direct, South, North, East and West Dial's.  
WHen the plat standeth upright, it maketh a right angle with the Horizon, and is called Erect, whereof some behold one priucipall part of the world directly as the South, North, East, or West: other decline as the South and North, toward the East or West, of which kinds hereafter.  
To examine an Erect plat, apply the side or diameter A. B. of your Instrument unto it, the thread with the plummet on the foreside hanging at liberty. If the thread fall upon the line E. C. or E. D. the plat is Erect.  
This done, apply the said line or diameter A. B. to the plat, your Instrument being placed equally distant to the Horizon, and the plombrule upon the line E. I. then if the needle stand directly over his character, the end touched with the Loadstone being next to the plat, that plat is called a South Erect Direct.  
In the North all the foresaid things are to be considered, except only that the end of the needle touched with the loadstone is farthest from the plat.  
These things known, you may easily find out the East and West Erect Direct, if either you have a line drawn squire wise to the character of the needle, or else if you place the plombrule over the line E. C. or E. D.  
And here note, that the East and West, are not said to decline, because the declination is accounted from the south and North, to the direct East, and West points.   

The examining of plaits for Declining Dial's.  
ALl such plaits as behold not some principal part of the world directly, are called Declining. The quantity of their declination is found out thus.  
Apply the diameter A. B. of your Instrument to the plat, remembering to hold it equally distant from the Horizon. Then move the plombrule until the needle standeth right over the character, and the point of the ruler, which toucheth the degrees in the limb, shall show how many degrees and minutes it doth decline: either toward the East if the plumb-rule lie in the quadrant C. E. F. or toward the west, if in the quadrant D. E. F.   

The trying of such plaits as recline.  
IF the plat standeth not upright, but maketh an obtuse or blunt angle with the Horizon, it is said to recline. The degrees of reclination are found out thus.  
Apply the diameter A. B. of your Instrument to the plat, the one end placed upward, the other downward: than move the plomrule (the thread with the plommet having free course) until the thread hang precisely over the line G. H. then the point of the ruler shall show the degrees of reclination.   

How Inclining plaits are tried.  
BUt if the angle which the plat maketh with the Horizon be acute or sharp, than it doth incline. The quantity of inclination is thus known.  
Apply the diameter A. B. of your instrument to the plat, the thread with the plommet of the former side hanging at liberty, and mark what degree and minute the thrad shall cut, for so much is the inclination.   

The manner of trying those plaits which recline and decline, or incline and decline.  
IF your plat shall both recline and decline, or incline and decline: First seek out the reclination as hath been showed, and then the declination as in Erect declining.    

The making of a horizontal, or plain lying Sun Dial. CAP. 3.  
YOur plat being prepared smooth and plain, draw upon it two lines, as in the figure following, the one A. B. the other C. D. cutting themselves squirewise, that is, making right angles in the point E. upon which make the quadrant of any circle from the line E. C. to the line E. A. or E B and writ at C. the North, at D. the South, at A. the East, at B. the West. And the line C. A. which here is the quadrant being divided into 90. degrees or parts, the elevation of the Pole shallbe accounted in it (which in our example is 52. d. from C. to A. and at the end of this number draw a line from the centre E. which shallbe E F representing the stile and artrée of the world. Then draw an other line K. L. by C. or by some other point of the line D. C. squire wise, so long as you can which shallbe called the touch line, or line of Contingence. Then measuring with your compasses the least distance of the point O. and the line E. F. or the Style, the one foot placed in O. which is the point of intersection, and the other extended toward E. where it shall chance to divide or be placed in the line E. C. mark that point or centre with the letter G. and draw with your compasses a half circle upon this centre for the equinoctial circle, from H. by C. to I. whose diameter must be equally distant to the line L. K. Then divide this half circle into 12. equal parts: this done, lay a ruler upon the centre G. and upon every mark or division made in the half Equator, and where the ruler shall touch the line of contingence, there make marks or pricks, by which pricks draw lines from E. for the hours. E. C. is the 12. hour, E. B. the 6. in the morning, E. A. the 6. at evening, the rest you may see in the figure.  
And whereas in Summer the 4. and 5. in the morning, and also the 7. and 8. at evening, shallbe necessary in this kind of Dial: prolong or draw the lines of 4. and 5. at evening, beyond the centre E. which shall show the hours of 4. and 5. in the morning. And likewise the 7. and 8. in the morning, for the 7. and 8. at evening.  
You may observe an order both in these and in all other erect direct dials, by dividing the one half of the Equator, drawing hour lines for the forenoon, and observing the same distance from the meridian line, on the other side for the afternoon: for the line of the 11. hour in the forenoon is of like distance from the Meridian, that the 1. is in the afternoon, and the 10. as 2. and so of the rest.  
When you would draw or make the half hours, you must divide every part of the Equator into 2. equal parts, 〈◊〉 the ruler and the line of contingence as you find in the drawing of the hour lines.  
And this remember for the drawing of the half hour-lines, not only in this kind, but also in all other kinds of dials, which afterward shall follow.  
The Style must be fixed in the centre E. hanging directly over the Meridian line E. C. with so great an angle, as the lines C. E. F. make, declining from that on neither sides.  
The Equinoctial circle, the Quadrant, the line of the Style and of Contingence, must be lightly drawn, because they ought to be put out again, in that they serve to no use but for the drawing of the Dial. And this likewise remember in all other kinds of Dial's, that the preparative or pricked lines must after the making of the Dial be omitted and extinguished, as altogether unprofitable.  
This and all other kinds of Dial's may most fitly be drawn upon a clean paper, and then with the help of your compasses placed on the plat.   dialing diagram  

The making of a South Erect, direct Dial. CAP. 4.  
THe making of this, is almost altogether like unto other before. Draw here also a line A. B. parallel to the Horizon, out of the midst whereof, let the line C. D. perpendicular to that be extended: and let the Quadrant be drawn from A. to D. or B. to D. parted into 90. degrees, as before. Above C. let Zenith be written, beneath D. Nadir; nigh unto B. the East, and by A. the West. In the Quadrant number the Elevation of the Pole (which in our example is 52. degrees) from A. toward D. And at the end hereof draw from the centre C. the line C. E. representing the Style: and by D. draw the line of Contingence, I. K. squirewise, as before. Then the least distance of the point or intersection D. and the Style being taken with your Compasses, extend th●n in the line D. C. the one foot placed in D. set the other in F. toward C. and draw by F. the Diameter of the half circle of the Equator equally distant to the line K. L. which must be made upon the centre F. from G. by D. to H. and divide it into 12. equal parts, upon every one of which and the centre F. the ruler being placed, wheresoever it shall happen to touch the line of contingence, there make marks. Then from the centre C. by these marks the hour lines must be drawn. The line C. A. shall show the 6. in the morning, C. B. the 6. at evening, C. D. the 12. etc. The Style must be placed or fixed in the centre C. hanging precisely over the line of the 12. hour, with so great a distance, as the angle D. C. E, is. This kind of Dial doth receive and show only but 12. hours at the most.  
 dialing diagram   

The making of a North Erect, Direct Dial. CAP. 5.  
HEre as in the South make a line Parallel to the Horizon, A. B. cut it squire wise with the Perpendicular C. D. let C. be the centre. At C. writ Zenith, at D. Nadir, ●c. From C. draw the Quadrant of a circle to D. or B. divide this into 90. Degrees, account the Elevation of the pole (which in our example is 52. d) from A. toward C. Draw at the end of of this number the line E. F for the Style. Afterward draw the line of Contingence by C. squirewise, and take the shortest distance, with your compasses between the point C. and the Style, placing the one foot in the point C. extend the other toward E. in the line C. D: making a point or prick G: whereupon (as a centre) the same wideness of the compasses remaining, describe the half Equator by C. ended with the Diameter H. I. equidistant to the line of Contingence. When you have divided thus half circle of the Equator into twelve equal parts, lay the ruler upon the centre G. and upon each division of the Equator, and where it shall touch the Contingent line, make marks. This done, draw the lines for the hours, by those marks from the centre E. but those very few, that is, two nigh unto A. and two by B. but prolong and extend them beyond the centre E. so that their contraries may be made: for in this kind of Dial's there be but only ten hours profitable, that is, 4, 5, 6, 7, 8, before noon, and 4, 5, 6, 7, 8, after noon; which show but only in Summer, from the entering of the Sun into ♈, until such t●ne as it entereth ♎.  
The Style must be fixed in the centre E. placed upward, directly over the lint E. C. with so great an angle as I. E. C. is. Let the line C. D. be placed upward perpendicularly, but so that it may not be almost seen, as afterward serving to no use, because it is only a line preparative.  
Behold the Figure.  
A North Dial erect, direct.   dialing diagram  

The making of the East and West Erect Dial's. CHAP. 6.  
IF you know the making of one of these, you may easily make them both, for they be very like, differing only in the naming of the hours, for the one containeth hours for the forenoon, and the other for the afternoon.  
You must therefore on your plat make the quadrant of a circle A. B. C. which may afterward easily be put out, as all the other lines must be, except the hour lines: let the side A. B. be Perpendiculare: B. C. Parallel to the horizon, and let the ark behold the South, which being divided into 90. degrees, number therein the elenation of the Pole downward from A. toward C. by the end of this number, and by the Centre B, draw a line so long as your plat will give you leave, whose South end shall behold precisely the Equinoctial circle. At which end draw a circle whose diameter shall be almost the third part of the line. Then draw another diameter or line in the centre, squire wise to the other, which shall show the axletree of the world, and be the line for the 6. hour. After ward at the cutward sides of the circle, draw two contingent lines, one beneath, the other above, so that they may be Parallel to the middle line. Divide each quarter of the circle into six equal parts. Then place the ruler upon the centre, and each of those marks or parts, and where it toucheth the lines of contingence there make marks in them. Afterward draw a line by those two marks which be next to the 6. hour, in the lines of contingence, which may be equally distant from the line of the 6. hour.  
In like manner do with the rest; so that you may have in the East Dial, two above the 6. hour, the 4. and 5. in the morning, and under it 7, 8, 9, 10, 11. In the West Dial like wise 7. and 8. in the evening above the 6. hour, and under it 6, 5, 4, 3, 2, 1. Neither of them do show the 12. hour, because at that time the Sun beams be Parallel to the plat.  
Fix the Style in the centre of the circle right up from the Plate, so long as the Semidiameter of the circle is: only showing the hour with the very top or end thereof. Yet it were more convenient to have it placed along over the line of the 6. hour, being a plate of iron or some other metal, being so broad as the Semidiamer of the circle is.  
Behold the figures following.  
The East dial Erect.   dialing diagram 
The West Dial erect.   dialing diagram 
Note that these five kinds of Dial's before taught may be made upon a stone cut square in form of a die.   

The making of a South Erect declining Dial, which may be placed on any upright wall whatsoever. CAP. 7.  
IN all Declining Dial's because the Style doth not hang directly over the meridianline, therefore you must first find out and place the Substile (which is the line over which the Style directly hangeth) and like wise the line of the Style: which may be both eastly and speedily performed in this manner.  
First, by your instrument seek out the Declination of the wall or plat, whereupon you would make a dial, which for example suppose I had found to be 50. degrees.  
● Therefore draw the horizontal line A. B. let the Meridian C. D. cut it squire wise in the point E. which may be in any place of the line A. B. and let the line F. G. cut the meridian squirewise in the point C. which you may like wise take in what part of the Meridian you think most convenient. Upon the centre E. describe the quadrant of a circle from C. to A. if the declination of your plat be Westward, or from C. toward B. if Eastward (as in this example it is) divide it into 90. degrees and number in it from C. toward B. the Elevation of the Equator 38. degrees, and laying your ruler 
* This is always so much as the Elevation of the Pole, wanteth of 90. degrees.  upon the centre E. and upon the end of this number draw the line E. H.  
Again account in the quadrant from C. toward B. the declination of the plat being 50. degrees, and draw the line of Declination E. I. from E. to the end of this number.  
Then take your compasses, set one foot in the point C. extend the other unto H. (which is the intersection point of the line E. H. with the line C. G.) and that same wideness remaining, place one foot in the centre E. and with the other mark the point 1. in the line E I. and from the point I. draw the line I K. squirewise to the Meridian E. C. placing K. at the intersection it maketh with E. C.  
Moreover take with your compasses the just length of the line I K. and placing one foot in C. turn the other toward F. and make the point L. in the line C. F. draw the Substile from E. by L. Let the line of Contingence be drawn squirewise to the Substile in the point L.  
This done, take the distance between the centre E. and the point K. and place it in this line of Contingence from the point L. unto M. and make the point M. that the line E. M. may be made from E. unto M. for the stile.  
Measure with your compasses the least distance between the point L. and the Style, and with the same wideness, one foot remaining in L. turn the other toward E. and make the centre O. in the Substile: upon which describe the Equinoctial circle. Then placing your ruler upon O. being the centre of the Equinoctial circle, and N. (which is the point of intersection of the Meridian and Contingent) marking where it cutteth the circumference, for there you must begin to divide it into 24. equal parts, not withstanding those 12. are only in use which are next the contingent.  
Finally, place your ruler upon the centre O. and upon the several division points of the Equator, and where it toucheth the line of Contingence make marks, by which from the centre E. draw the hour lines.  
Place the Meridian perpendicularly upon the wall, the centre E. upward, the Style pointing downward. Let the Style hang directly over the Substile making an angle equal to E. M. L.   

The making of a North Erect Declining Dial. CAP. 8.  
THe North Erect declining Dial differeth from the South only herein, that the centre is to be placed downward, the Style pointing upward to the North Pole, and that the Meridian representeth the 12. hour at midnight and not at noon. Therefore if the declination be toward the East, you must account the hour lines from that which is as it were the Meridian forthward 1, 2, 3, 4, 5, 6 etc.  
But if it decline toward the West, number them backward, 11, 10, 9, 8, etc. omitting the first 3. which are before Sun rising and after Sun setting in our elevation.  
Let the line E. D. be placed perpendicularly upon the plat, and the Style point upward to the North Pole. Compare this figure with the former.   

Another way to make the South Erect Declining Dial. CAP. 9  
ALbeit we have plainly and perfectly showed the making of the South and North erect Declining Dial's, in the two former Chapters; yet to satisfy them that delight in variety; here is also declared another way, whereby you may make them, namely, by the help of Arithmetic, and the Table of Sines which is 
Read the ●se of the Table of Sines.  placed in the end of this book for this intent.  
Therefore the Elevation of the Pole being known, and the Declination of your plat, by your Instrument found out:  
Multiply the Sine of the complement of the Elevation, by the Sine of the complement of the Declination, divide the product by the whole Sine (which is 100000) and you shall have a quotient Sine, whose ark is the distance of the Style from the Substile, which distance keep.  
Then take the complement of this distance, and the Elenation of the Pole, and multiply the Sine of the lesser by the whole Sine, parting the product by the Sine of the greater: the quotient Sine which shall come of this division, shall give you an Ark, whose complement is the distance of the line of the Substile from the Meridian, which distance you shall likewise keep. For better instruction herein, consider the example.  

Example of a South Dial declining 45. d. Elevation of the Pole. 52. d.  
FIrst I enter the Table of Sines, for the Sine of the complement of the Elevation, which is 38. d. and I find it to be 61566. Then I look for the Sine of the complement of the Declination, which is 45. d. and I find that to be 70710. This done, I multiply the one by the other, and the product is 435333 1860. which I divide by the whole Sine being 100000 whereof cometh a quotient 43533. With this quotient Sine I enter the Table, and because I find not the just number, I take that which is next either greater or less unto it: (which you must always remember to do) for so small a difference maketh no alteration, and therefore I take in stead thereof 43523. whose Ark is 25 d. 48. m. which is the distance of the Style from the substile.  
Then I take the complement of this distance, which is 64. d. 12. m. whose Sine is 90031. and the Elevation of the Pole 52. d. whose Sine is 78801. and multiplying the Sine of the lesser, (which is the Elevation of the Pole) by the whole, 100000. the product is 7880100000. which I divide by the Sine of the greater, to wit, the Sine of the complement of the distance of the Style from the Substile, whose Sine is 90031, whereof cometh this quotient 87526 whose Ark is 61. d. 5. m. The complement of which ark is 28. d. 55. m. being the distance of the substile from the Meridian. These distances being thus found out, the drawing of the dial followeth.  
First draw a line Parallel to the Horizon A. B. out of whose middle point C. draw the Meridian line squire wise C. D. upon the centre C. make the quadrant of a circle between A. and D. Here understand that generally in all kinds of declining Dial's, if the declination be towards the East, you must draw the quadrant towards the West, (except in the South reclining declining, and the North inclining Declining, where the contrary is prescribed) But if the declination be Westward, the quadrant must be drawn Eastward. Hereby you shall easily know on which side you ought always to draw the quadrant, and which way the figures following do decline. Your quadrant being made, divide it into 90. degrees: number therein from D. the distance of the Substile from the Meridian, (which is 28. d. 55. m. Draw at the end of this number the line C. E. for the Substile. Then from E. towards A. account the distance of the Style from the Substile, which is 25. d 48. m. and at the end hereof, draw the line C. F. for the Style. Afterward by the point E. or in any place of the Substile draw the Contingent line G. H. so long as you can squirewise to the Substile. Then take with your compasses the least distance between E. and the Style, the one foot remaining in E. and the other extended in the Substile toward C. place at the prick there made with the compasses, I. upon which prick as a centre, the same wideness of your compasses remaining, draw a circle by E. which shall represent the equinoctial or Equator. Then lay the ruler upon the point I. and the intersection (whose mark is K.) of the line of Contingence and the line C. D. which is always drawn so that it may cut the other) and where the ruler so placed shall touch the circle, there make a mark, and there begin to divide it into 24. equal parts: notwithstanding those 12. only are to be used, which do behold the contingent line. Then lay the ruler upon the centre I. and the Contingent line by every division of the Equator, and where it shall touch the line of Contingence, there make marks, by the which from the centre C. draw lines for the hours, so many as shallbe necessary.  
The line C. D. shall always show the 12. hour, which must hang perpendicularly. Number the residue of the lines in their place, as they follow in order.  
The line A. B. in such as do decline is unprofitable, except it so chance that some hour line falleth in it.  
Let the Style be fixed in C. hanging directly over the Substile, with so great an angle, as E. C. F. is, declining on neither side.  
Note diligently the making of this declining Dial, because in those which follow, we mean not to repeat those things which here have been taught. And this one kind well known, all the other will seem most easy. For better understanding hereof behold the figure.  
A South Erect declining Dial.   dialing diagram   

The making of a North Erect declining Dial another way. CAP. 10.  
THe distance of the Style from the Substile, and of the Substile from the Meridian, is found out altogether like to the South Erect declining. Therefore you may resort thither for the working hereof, I will only draw the Figure.  
Let the line A. B. being parallel to the Horizon, cut C. D. squire wise, placing E. at the intersection Draw the quadrant from C. to B. divide it into 90.ᵈ. accounting therein from C. toward B. the distance of the Substile from the line, which is (as it were) the Meridian, which distance is 28.ᵈ. At the end of this number draw the line E. F. for the Substile. Then number from that line the distance of the Substile and the Style, which is 25. degrees, 48 minutes toward B. Draw likewise at the end of this number the line for the Style E. G. This done, let the line of Contingence be drawn squire wise by the point F. and then taking the least distance between the point F. and the Style, extend the compasses in the line F. E. the one foot being placed in F. where the other shall divide the line, place the letter H. With the same wideness of the compasses draw upon the centre H. the Equator, and where the ruler shall touch the same, being laid upon the centre H. and the intersection of the Contingent line, and that which is (as it were) the Meridian, begin to divide it into 24. equal parts. Finish all things remaining, as in the South erect declining, only this excepted, the Style being fixed in the centre E. must be placed upward, beholding the substile with so great a distance or angle, as the letters F. E. G. do show. The line C. D. being applied to the plat perpendicularly, showeth as it were the 12. hour at midnight: therefore account from that the hour lines, as they follow in order, 1, 2, 3. which hours have no use in this kind of Dial for our Elevation, but from 4. etc.  
Let the line A. B. be parallel to the Horizon, being lightly drawn, as the other preparative lines be, serving afterward to no use.  
Note, if the plat decline towards the East, as this figure doth, it is for the forenoon: and you must account the hour lines from that which is (as it were) the Meridian, forward 4, 5, 6. etc. If it decline toward the West, account them backward 11, 10, 9 as we showed before.  
A North erect declining Dial.   dialing diagram  

The making of a Meridional or South Reclining direct Dial. CAP. 11.  
IF the Reclination of the plat be less than the complement of the Elevation of the Pole, add this complement, and the Reclination of the plat together, and with this number as if it were the Elevation of the Pole make a South erect direct dial, and it will be perfect and fit for your plat.  

Example and delineation of a South Dial, reclining 25. d.  
MAke the first line A. B. draw also an other C. D. cutting the other with right angles. Then add the Complement of the elevation of the Pole (which is 25.,) to the reclination of the plat (which is 38.ᵈ.) and the total sum shallbe 63.ᵈ. which number being in place of the Elevation of the Pole, account it in the Quadrant from A. towards D. and at the end hereof, draw the line F. for the Style. Then make the line of Contingence K L. squire wise to the line C. D. Afterward take the least distance with your compasses between the point G. and the Style, with the wideness, the one foot extended toward C. make a point or prick E. by which point draw the line H. I. equidistant from A. B. or K. L. upon E. make a half circle from H. by G. to I. which being divided into 12. parts, finish the rest as the South Erect direct.  

A South Direct reclining 25. d.   dialing diagram   

BUt if the Reclination be equal to the complement of the Pole, make your Dial on this manner, like to the East and West.  
Draw a line Parallel to the Horizon A. B. so long as the plat will give you leave, divide it into 7. equal parts, and with the same wideness of the compasses, in the midst of the line, make a circle representing the Equator. Then draw two lines of Contingence by the circumference of the circle, equidistant from the first A. B. Divide the Equator into 24. equal parts. Finish the rest like the East or West dial: except in naming the hours, for that which is in them the 6. hour line, is here the 12. etc.  
A South Direct reclining 38. d.   dialing diagram  

BUt if the reclination of the plat be greater than the complement of the Elevation of the Pole, subtract the complement of this Elevation out of the reclination, and that which remaineth shall serve in stead of the Elevation of the Pole: with which Elevation make your dial altogether like a horizontal, which you shall find fit for your plat.   

Example of a South Dial, reclining 55. d.  
FIrst I take the complement of the Elevation, which is 38.ᵈ. out of the Reclination of the plat, which is 55.ᵈ. and there remaineth 17.ᵈ. which number keep for the Elevation of the Pole.  
In Delineation of this Dial, draw a line Parallel to the horizon A. B. draw another C. D. making right angles with the line A. B. Then make the quadrant from A. to D. which being divided into 90. degrees, number in it that, which is, as it were, the Elevation of the Pole, namely 17. degrees from D. towards A. Then draw the line of Contingence K. L. in any point of the line C. D. squire wise. Take with your compasses the least distance of the point G. and of the Style: extend them forth in the line C. D. I mean, from G. toward C. make there a point marked with E. upon which point or centre with the same wideness of the compasses, draw the Equator from H. by G. to I. divide it into 12. equal parts, etc.  
Finish that which remaineth in all respects, like to the horizontal Dial, in naming the hours, drawing of the lines, fixing of the Style, etc. for in this kind of Dial, the Style must be fixed in the centre C. standing upward, with so great an angle, as C. D. F. is.  
Behold the Figure following.  
A South Reclining Erect.   dialing diagram   

The making of a North Dial Reclining Direct. CAP. 12.  
IN these kinds, if the Reclination be less than the Elevation of the Pole, subtract that from this, & keep the remainder for the Elevation of the Pole: according to which Elevation, make a North Dial Erect Direct, and it will be fit for your plat. In which the hours about Noon are placed, being profitable only in summer: and not then unless the Reclination be greater than the distance of the Trop●k of S, from the Zenith. And the more your Dial Reclineth, the longer time it will show in Summer.  

Example of a North Dial Reclining 45. degrees.  
FIrst I take the Elevation of the Pole (which we account 52.ᵈ.) 〈◊〉 the Reclination of the plat, which is 45.ᵈ. and there remaineth 7.ᵈ. which shall be in place of the Elevation of the Pole.  
Then for the drawing of it, resort to the North Erect Direct, because their Delineation is like: remembering always to call that which remaineth the Elevation of the Pole: which in this example is but 7. degrees. In the figure following you see that most part of the hour lines be drawn opposite from the Contingent, beyond the centre E. as you must do in any of this kind, if you will have the hours about noon.   dialing diagram  

Example of a North Dial Reclining 52. degrees.  
BUt if the Reclination be equal with the Elevation of the Pole: then describe a circle upon the centre E. and divide it into 24. equal parts, beginning the division at the 12. hour. Draw by those points, lines for the hours from the centre E. so many as shall be necessary: erecting the Style (being some small wire) in the centre E. right up. This kind of Dial serveth only when the Sun is in the north signs, which be ♈, ♉, ♊, ♋, ♌, ♍.  
A North Reclining Direct.   dialing diagram 
You may with the help of a small brass quadrant and a needle placed conveniently make this a portable and fit Dial for any Elevation of the Pole.  
Again, if the Reclination shall exceed the Elevation of the Pole: add the complement of the reclination to the Elevation, and with that number as if it were the Elevation of the Pole, make your Dial altogether like a horizontal, and it shall be fit for your plat. In which notwithstanding in winter, very few hours will show, and those about noon: and yet not these, unless the Reclination shall be greater, than the ●●…hest bending of the Tropic of ♑, from the Zenith.   

Example of a North Dial reclining 80. d.  
FIrst add the complement of ●he reclination of the plat, which is 10.ᵈ. to the Elevation of the Pole 52.ᵈ. and with the total number 62.ᵈ. ●●●f it were the Elevation of the Pole, make a horizontal Dial, as is showed cap. 3. Always remember to call the two numbers added together, the Elevation of the Pole.  

A North reclining Direct.   dialing diagram    

The making of an East and West Dial reclining. CAP. 13.  
FIrst multiply the Sine of the Elevation of the Pole, by the Sine of the Reclination of the plat, and divide the product by the whole Sine: whereof shall come a quotient, whose ark is the distance of the Style, from the Substile.  
Compare the Complement of this distance with the Complement of the Elevation of the Pole, and which you shall find least, multiply the Sine thereof by the whole Sine, parting the Product by the Sine of the greater. The quotient shall yield an ark, whose Complement shall be the distance of the Substile from the Meridian.  

Example of an East or West Dial, reclining 45. degrees.  
FIrst, I multiply the Sine of the Elevation of the Pole 52. degrees being 78801. by the Sine of the Reclination of the plat, which is 70710. & the product 5572018710 I divide by 100000 the whole Sine, and the quotient is 55720. the ark whereof is 33.ᵈ. 52.ᵐ. which is the distance of the Substile from the Style.  
Then I compare the Complement of this distance, which is 56.ᵈ. 8.ᵐ. with the Complement of the Elevation of the Pole, which is 38.ᵈ. and finding the Complement of the Elevation to be least, I multiply 61566 the Sine thereof, by the whole Sine: and the Product arising thereof is 6156600000: which I divide by the Sine of the greater, which is the Complement of the distance, whose Sine is 83033. And the quotient 74141 yieldeth an ark 47.ᵈ. 51.ᵐ. whose Complement is 42.ᵈ. 9 which is the distance of the Substile from the Meridian.  
First draw a line Parallel to the Horizon A. B. draw 
The delineation of the figure.  another squire wise, C. D. Then make the quadrant from A. to D. Number therein the distance of the Substile from the Meridian (which we found to be in our example 42.ᵈ. 9) from A. toward D. At the end of this number from the centre C. draw the line for the Substile, C. E. Account from this line the distance between the Style and the Substile, which is 33.ᵈ. 52.ᵐ. draw likewise at the end of this number from C. the line C. G. for the Style. Let the line of Contingence be squire wise to the Substile in any point where you will. Then take with your compasses the least distance between the point E. and the line of the Style: with that wideness the one foot remaining in E. extend the other in the line of the Substile toward C. make there the point F. upon which draw the Equator from H. by E. to I. Divide it into 12. equal parts; beginning your division where the ruler being placed upon the centre F. and the intersection of the Meridian with the Contingent shall touch the equator. Lay the ruler from F. by each of those marks, and where it shall touch the line of Contingence, there make marks: by which from the centre C. draw the hour lines so many as shall be necessary. The line A. B. is the Meridian, and showeth the 12. hour. Finish all other things, as in the rest before you were taught.  
Note that if it be an East dial, you must make the quadrant from A. to D. If a West, from B. to D. Finish the residue as before, etc.  
An East or West reclining.   dialing diagram   

The making of a South reclining declining Dial. CAP. 14.  
FIrst, in this kind, as in other which follow, you must find out and place the Meridian, the Substile, and the Style, which being done, that which remaineth is finished as in those before.  
Therefore the Declination and the Reclination of the plat being known by your instrument, multiply the Sine of the Declination by the Sine of the complement of the Reclination, dividing the product by the whole Sine. The quotient Sine shall yield an Ark, whose complement shall be named the Complement to be repeated.  
Afterward augment the Sine of the Complement of the Declination, by the whole Sine, divide the Product by the Sine of the Complement to be repeated: where of shall come a Sine, whose ark shallbe the distance of the Meridian from the Horizon.  
Again, multiply the Sine of this distance, by the Sine of the Complement of the Reclination, part the Product by the whole Sine, the ark of whose quotient shallbe called the Elevation of the Meridian.  
Then compare this Elevation of the Meridian, with the Elevation of the Pole, and which you shall find least, subtract that from the greater, and that which remaineth keep, (for it shall be called the difference kept) being mindful which of them was the greatest.  
This done, multiply the Sine of the Complement to be repeated, by the Sine of the difference kept: dividing the Product by the whole Sine, whereof shall come a Sine, whose ark shallbe the distance of the Style from the substile.  
To conclude, compare the Complement of this distance, with the complement of the difference kept, and which you shall find least, multiply the Sine thereof by the whole Sine: part the Product by the Sine of the greater, and thereof shall come a Sine, the Complement of whose ark shallbe the distance of the Substile from the Meridian.  

Example of a South Dial  
Declining 18. d.  
Reclining 25. d.   
First look out in the Table the Sine of the Declination, which is 30901. Then the Complement of the Reclination being 65.ᵈ. the Sine hereof is 90630. Afterward multiply 90630: by 30901. and the product 2800557630 divide by the whole Sine, which is 100900. the quotient shallbe 28005. whose ark is 16.ᵈ. 16ᵐ. The complement whereof being 73.ᵈ. 44.ᵐ. is the complement to be repeated.  
This done, multiply 95105 the Sine of the Complement of the Declination 72.ᵈ. by the whole Sine 100000. the product shallbe 9510500000. which being divided by 95996 the Sine of the complement to be repeated, the quotient shallbe 99071, whose ark is 82.ᵈ. 11.ᵐ. which is the distance of the Meridian from the Horizon.  
Afterward augment the Sine of this distance 99071. by the Sine of the complement of the Reclination, which is 90630. and the product 8978804730 part by the whole Sine: and the quotient 89788 shall yield an ark 63.ᵈ. 53.ᵐ. which is the Elevation of the Meridian.  
Then compare the Elevation of the Meridian, with the Elevation of the Pole, which in this example is 52.ᵈ. and finding the Elevation of the Pole to be least, subtract this namely 52. degrees) from 63.ᵈ. 53.ᵐ. the Elevation of the Meridian, and there remaineth 11.ᵈ. 53.ᵐ. which is the difference kept.  
Now multiply 95996 being the Sine of the Complement to be repeated, which is 20591. the product is 1976653636. which divided by the whole Sine, the quotient shallbe 19766, whose ark 11.ᵈ. 24.ᵐ. is the distance of the Style from the Substile.  
Then to conclude. the Complement of this distance, which is 78.ᵈ. 36.ᵐ. being compared with the Complement of the difference kept, which is 78.ᵈ. 7.ᵐ. you shall find the Complement of this difference kept to be least, wherefore multiply 97856 the Sine thereof, by the whole Sine, and the product 9785600000. part by the Sine of the greater, to wit, of the Complement of the distance of the Style from the Substile, which is 98027. the quotient shallbe 99825. whose ark is 86.ᵈ. 37.ᵐ. The Complement whereof is 3.ᵈ. 23.ᵐ. which is the distance of the Substile from the Meridian.  
There be three sundry kinds of these Dial's, the one differing from the other.  

The first kind.  
Mark therefore, if the Elevation of the Meridian be greater than the Elevation of the Pole, draw a line parallel to the Horizon A. B. out of the midst whereof extend an other C. D. squire wise to the line A. B. make the quadrant from A. to D. number therein from A. towards D. the distance of the Meridian from the Horizon, which is 82.ᵈ. 11.ᵐ. at the end hereof from the centre C. draw the line C. E. for the Meridian. From which account backward towards A. the distance of the Substile from the Meridian, which is 3.ᵈ. 23.ᵐ. draw at the end of this number the line C. F. for the Substile. Then from the Substile toward A. number the distance of the Substile from the Style 11.ᵈ. 24ᵐ. and at the end thereof extend the line C. G. which shall represent the Style. This done, in the line of the Substile in the point F. (which you may take where you will in the Substile) draw the line of Contingence squirewise to the Substile. Then take the least distance of the point F. from the Style. Afterward extend your compasses, the one foot being placed in F. in the line of the Substile toward C. unto H. Describe upon the centre H. the Equinoctial circle by F. Divide it into 24. equal parts, beginning where the ruler shall touch the Equator, being placed upon the centre H and upon every mark of the Equator, and where the ruler shall touch the line of Contingence, there make marks, by which, from the centre C. draw the hour lines somany as shall be necessary, C▪ E. always she wing the 12. hour. Let the Style hang directly over the Substile with so great an angle as F C. G. is: fixing it in the centre C. pointing▪ downward to the Pole Antarctic.  
Place the line A. B. parallel to the Horizon, having afterward as other preparative lines no use.  
A South Declining Reclining.   dialing diagram  

The second kind.  
But if the Elevation of the Pole, and the Elevation of the Meridian be found equal, the making of your Dial differeth from the former: yet the finding out of the Meridian and the Substile, etc. is wrought as in the other before, whither you may resort. It shall be sufficient here to show an example of this kind.  

Example of a South Dial, whose  
Declination is, 32 d.  
Reclination is, 33.d. 30.ᵐ.  
Elevation of the Pole is, 52.d. 2.ᵐ.   
First I multiply 52991 the Sine of the declination, by 83388 the Sine of the Complement of the Reclination, and the product 4418813508 arising thereof, I divide by the whole Sine: the quotient Sine 44188 yieldeth an ark 26.ᵈ. 14.ᵐ. whose complement being 63.ᵈ. 46.ᵐ. is the complement to be repeated.  
Then I increase the Sine of the Complement of the declination, which is 84804, by the whole Sine, and the product is, 8480400000: which I part by 89700 being the Sine of the complement to be repeated. The ark of whose quotient Sine 94541 being 70.ᵈ. 59 is the distance of the Meridian from the Horizon.  
This done, I multiply this Sine 94541, by the Sine of the complement of the reclination, which is 83388, and the product arising hereof, being 7883584908 I divide by the whole Sine: the quotient thereof is 78835, whose ark 52. 2ᵐ. is the Elevation of the Meridian, which I compare with the Elevation of the Pole, and finding them equal, I end my work here. For this shallbe sufficient in this kind of Dial, as you may more plainly perceive by the delineation of the Figure.  
Wherefore if the Elevation of the Pole, and of the Meridian be found equal, (as in this example it is) make a line parallel to the Horizon A. B. Draw another C. D. making right angles with the line A. B. Draw the quadrant from A. to D. number therein the distance of the Meridian from the Horizon from A. toward D. which is 70.ᵈ. 59 being here in place of the Substile. Then draw the line of Contingence squire wise to the Meridian C. E. (which is also the Substile C E. in any point thereof, as before. Place one foot of your Compasses in the intersection E. then set the other foot in the Meridian (or Substile) being of any wideness: make there the point or centre F. upon which centre draw the half Equator from G. by E. to H. Divide the one half thereof from E. to H. into 90.ᵈ. Account therein from the Meridian E. towards H. the declination of the plat 32.ᵈ. place the ruler upon the centre F. and the end of this number, and where it shall touch the Contingent line, there make a mark, by this mark you must draw a line squire wise to the Contingent line, which shall be for the 12 hour. And where the ruler shall touch the Equator, there begin to divide it into 12. equal parts, by which lay the ruler from the centre F. making in the line of Contingence marks for the other hour lines, all which shall cut the Contingent line squirewise. Note that sometime in dividing of the Equator, the two parts at both ends next to the Semidiameter G. H shall both of them make but one whole part unless you will make a whole circle for the Equator, and divide it into 24. parts.  
Let the Style be a small wire standing right up in the point E. being so long as the semidiameter of the Equator. It may be also a plate of iron or brass fastened in the Substile so broad as the Semidiameter is, as in the East and West erect dials, etc. Let the line A. B. be parallel to the Horizon. Finish all other things as before.  
A South reclining declining.   dialing diagram  

The third kind.  
If the Elevation of the Meridian be less than the Elevation of the Pole, the making of this Dial i● unlike to both the other mentioned before, yet the finding out of the distance of the Style from the Substile is done like to the first kind, as by the example following may appear.  

Example of a South Dial  
Declining 45. d.  
Reclining 45. d.  
Elevation of the Pole 52. d.   
First I multiply the Sine of the declination being 7071▪ by the Sine of the complement of the reclination which is likewise 70710, and the product 4999904100 I part by the whole Sine. The quotient is 49999 whose ark is 30.ᵈ. The Complement of this ark is 60.ᵈ. which is the Complement to be repeated.  
Then I increase the Sine of the Complement of the declination, which is likewise 70710, by the whole Sine, and the product thereof 7071000000 I divide by the Sine of the Complement to be repeated 86602: the quotient Sine 81649 yieldeth an ark 54.ᵈ. 44.ᵐ. which is the distance of the Meridian from the Horizon.  
Afterward I multiply this Sine 81649, by the Sine of the Complement of the Reclination, which is 70710: the product arising 5773400790 I part by the whole Sine, the quotient is 57734. The ark whereof is 35.ᵈ. 16.ᵐ. the Elevation of the Meridian: which I compare with the Elevation of the Pole (being in our example) 52.ᵈ. and finding the Meridian Elevation to be least, I subtract it out of the Elevation of the Pole, & there remaineth 16.ᵈ. 44.ᵐ. which is the difference kept.  
This done, I augment 86602 the Sine of the Complement to be repeated, by 28791 the Sine of the difference kept, and the product 2493358182 coming hereof, I divide by the whole Sine, the quotient Sine 24933 yieldeth an ark 14.ᵈ. 26.ᵐ. the distance of the Style from the Substile.  
Now comparing the Complement of this distance being 75.ᵈ. 34.ᵐ. with the Complement of the difference kept, which is 73.ᵈ. 16.ᵐ. and seeing the Complement of the difference kept to be least: I multiply 95765 the Sine thereof, by the whole Sine, and the product 9576500000 I divide by 96843 the Sine of the complement of the distance, and 98886 shallbe the quotient thereof, whose ark is 81.ᵈ. 27.ᵐ. The Complement of this ark is 8.ᵈ. 33.ᵐ. which is the distance of the Substile from the Meridian.  
In the Delineation of this dial, draw first as before a line Parallel to the Horizon A B. extend another C. D. making right angles with the line A. B. Draw the Quadrant from A. to D. divide it into 90. degrees. In which account the distance of the Meridian from the Horizon, from A. towards D. which is 54.ᵈ. 44.ᵐ. draw at the end of this number the Meridian line from the centre C. which shall show the 12. hour. Account from this toward D. the distance of the Substile from the Meridian, which is 8.ᵈ. 33.ᵐ. at the end hereof extend from C. the line E. for the Substile. From this, number the distance of the Style being 14.ᵈ. 26.ᵐ. toward D. if there be so much space, if not, account it from the Substile toward A. make at the end of this number the line C. F. for the Style. Let the line of Contingence cut the Substile squirewise, in what point you will. Finish all other things, as in the first of these three kinds of South reclining declining Dial's.  
In this kind of reclining Dial's, whose Style must be placed from the centre C. upward toward the Pole artic: if at any time you cannot draw to both ends of the Contingent line so many hour lines as shall be necessary: then prolong beyond the centre C. the lines opposite on the other side. As if you would make the 8. hour line for the morning, draw the 8. for the evening beyond the centre C. and you shall have your desire.  
Note that moreover, in this kind contrary to the other before, if the plat decline toward the West, (as in this example it doth) then draw the quadrant towards the West, etc. But if the Declination be towards the East, make the quadrant towards the East.  
A South reclining, declining.   dialing diagram   

The making of a North reclining declining Dial. CAP. 15.  

The first kind.  
IN this kind as in the South before: first multiply the Sine of the Declination, by the Sine of the Complement of the Reclination: parting the product by the whole Sine. The quotient Sine thereof shall yield an ark, whose Complement shall be called the Complement repeated.  
Then increase the Sine of the Complement of the Declination by the whole Sine, divide the product by the Sine of the Complement repeated: the quotient Sine thereof will give an ark, which shall be the distance of the Meridian from the Horizon.  
The same quotient Sine multiply by the Sine of the Complement of the reclination: and the number arising part by the whole Sine: the ark of this quotient is the Elevation of the Meridian.  
Now add the Elevation of the Pole to the Elevation of the Meridian, and the total number or ark hereof, we will call the compounded ark. And if the compounded ark be less than 90. degrees: then multiply the Sine thereof, by the Sine of the Complement repeated, dividing the product by the whole Sine: and hereof shall arise a quotient Sine, whose ark shallbe the distance of the Style from the Substile.  
Now compare the Complement of this distance, with the Complement of the compounded ark: and increase the Sine of the lesser by the whole Sine, part the product by the Sine of the greater: and thereof shall come a Sine, the Complement of whose ark shall be the distance of the Substile from the Meridian.  
But mark here the diversity which doth arise by reason of the compounded ark: for hereby you shall have three sundry ways, both in finding out the distances between the Meridian and the Horizon, the Style and Substile, etc. And likewise in the delineation of the figure. Whereof ariseth three sundry kinds of dials: the first of them being already taught.   

The second kind.  
If the compounded ark be just 90. degrees: then the distance of the Style from the Substile shall be the Complement repeated: and the distance of the Substile from the Meridian shall be also 90. degrees.  
Note that in the working hereof, there is no difference from the first kind. For here as in the other before, you shall first ●inde out the complement repeated: then the distance of the Meridian from the Horizon: Afterward the Elevation of the Meridian, which being added to the Elevation of the Pole, if the total number be just 90.ᵈ. you shall not need to proceed forward, for this (as before I made mention) shall be the distance of the Meridian from the Substile.  
In this kind you shall have no intersection of the Meridian and the Contingent line: therefore you shall begin the division of the Equator at the line of the Substile, which shall show the 6. hour, either in the morning, or evening, according as the plat doth decline.   

The third kind.  
But if the compounded ark be greater than 90 degrees, subtract it from 180, and that which remaineth shallbe called the difference kept. Then multiply the Sine of the Complement repeated, by the Sine of the difference kept, dividing the product by the whole Sine, hereof shall come a Sine, whose ark shall be the distance of the Style from the Substile.  
Then compare the Complement of this distance, with the Compliment of the difference kept, increasing the Sine of the lesser by the whole Sine, and parting the product by the Sine of the greater: whereof shall come a Sine, the Compliment of whose ark, being taken from 180 degrees, the remainder shall show how much the line of the Substile must be distant (upward by the Horizon) from the Meridian. For the better understanding of these three kinds, mark these three examples following, with the delineation of their figures.  

Example of a North Dial, whose  
Declination is, 45. d.  
Reclination 45. d.  
Elevation of the Pole 52. d.   
First I work this altogether like to the South reclining 45.ᵈ. declining 45.ᵈ. until I have found out the Elevation of the Meridian.  
Then I add the Elevation of the Meridian which is, 35.ᵈ. 16.ᵐ. to the Elevation of the Pole 52.ᵈ. the total number is 87.ᵈ. 16.ᵐ. which is the compounded ark: and being less than 90.ᵈ. I proceed with this ark, etc.  
Now therefore I multiply the Sine of this compounded ark, which is 99884, by the Sine of the complement to be repeated being 86602, and the product 8650154168 I part by the whole Sine. The quotient thereof is 86501, whose ark 59 54.ᵐ. is the distance between the Style and the Substile.  
Then I compare 30.ᵈ. 6.ᵐ. being the Complement of this distance, with▪ the Complement of the compounded ark, which is 2.ᵈ. 44.ᵐ. and finding this least, I increase 4768 the Sine thereof, by the whole Sine, dividing the product, which is 476800000, by 50151 the Sine of the greater: the quotient Sine yieldeth an ark 5.ᵈ. 27.ᵐ. the Complement hereof 84.ᵈ. 33.ᵐ. is the distance of the Substile from the Meridian.  
In drawing of this Dial, first as before, make a line Parallel to the Horizon A B. Then extend the line C. D. cutting the other squirewise. Mark the intersection or centre with E. draw thereupon a circle A. B. C. D. whose two quadrants at the least, toward A. (if the plat decline Westward, or else toward B. if it decline Eastward) being divided into 90.ᵈ. each of them, number therein from A. towards D. the distance of the Meridian from the Horizon, which is 54.ᵈ. 44.ᵐ. and at the end hereof from the centre ●. draw the line F. for the Meridian, or 12. hour. Account from this upward toward C. the distance of the Substile from the Meridian, bring 84.ᵈ. 33.ᵐ. make likewise at the end of this number from E. the line for the Substile. Again number from hence upward (if there be so much space, or else downward) 59 54.ᵐ. which is the distance between the Style and the Substile, and at the end hereof draw the line H. for the Style. Let the Contingent line cut the Substile squirewise in the point G. which you may take in any place of the Substile. Then with your compasses the least distance of this point G. and the Style being taken, with that wideness extend them forth in the line of the Substile toward E. making there the point I. upon which draw the Equator or Equinoctial circle. This done place the ruler upon the centre I. and the intersection of the Meridian and the Contingent line, and where it shall touch the Equator there begin to divide it into 24. equal parts. Finish all other things, as in those which went before.  
A North reclining declining.   dialing diagram  

The second kind of North Dial, reclining 45. degrees 14. minutes, declining 38. degrees.  
IT shall not be necessary to show an example of this second kind, because the Complement repeated, the distance of the Meridian from the Substile, and between the Style and Substile, etc. is found out altogether like to the other before, and that which followeth. Therefore it sufficeth to set down the number of these distances.  
First the Complement repeated is, 64.ᵈ. 18.ᵐ.  
The distance of the Meridian from the Horizon is, 60.ᵈ. 59  
The Elevation of the Meridian is 38.ᵈ. which I add to the Elevation of the Pole being 52.ᵈ. and the total 90.ᵈ. is the compounded ark.  
Draw therefore in this kind as before, a line Parallel to the Horizon A B. make another squirewise, C. D. Draw the two quadrants from C. by A. to D. dividing each of them into 90.ᵈ. Then number herein the distance of the Meridian from the Horizon, which is 60.ᵈ. 59 drawing from E. at the end here of, the line F. for the Meridian. From this account 90.ᵈ. being here the distance of the Substile from the Meridian: and like wise at the end of this number from E. draw the line for the Substile. Again from this line downward, (because there is no space upward,) account the Complement repeated 64.ᵈ. 18.ᵐ. which here we take for the distance of the Style from the Substile: make likewise by this number the line E. H. for the Style. Let the Contingent line cut the Substile squirewise in any place. Then taking the least distance with the compasses of the point G and of the Style, with that widensse in the Substile, make the point I. upon which draw the Equinoctial circle by G And because here, (as in all other Dial's, where the compounded ark is just 90.ᵈ.) there is no intersection of the Meridian, and the Contingent line: therefore begin the division of the Equator into 24 equal parts at the Substile, which in this kind shall show the sixth hour, either in the evening or morning, according to the declination of the plat. Place the ruler upon the centre I. and so many of these parts as you can, making marks in the line of Contingence as before, draw the hour lines by them, etc. finish the rest as in the other.  
A North Declining Reclining.   dialing diagram 

Example of the third kind of North Dial before mentioned.  
Reclining 21.d. 30.ᵐ.  
Declining 30. d.   
Here first, as before I multiply the Sine of the Declination being 50000, by 93041 which is the Sine of the complement of the Reclination and the product 4652070000 I part by the whole Sine: the quotient 46520 yieldeth an ark 27.ᵈ. 43.ᵐ. The complement whereof is 62.ᵈ. 17.ᵐ. being the Complement to be repeated.  
Then I increase 86602 the Sine of the complement of the Declination by the whole Sine, dividing the product, by 88525 the Sine of the Complement repeated: The ark of the quotient Sine 97827, is 78.ᵈ. 2.ᵐ., which is the distance of the Meridian from the Horizon.  
This done, I augment this quotient Sine 97827, by the Sine of the Complement of the reclination, which is 93041: the product 9101921907 I distribute by the whole Sine: and the quotient 91019 giveth an ark 65.ᵈ. 32.ᵐ. the Elevation of the Meridian.  
Now I add 65.ᵈ. 32.ᵐ. being the Elevation of the Meridian, to 52.ᵈ. the Elevation of the Pole, and the total number 117.ᵈ. 32.ᵐ. being the compounded ark, because it is greater than 90.ᵈ. I subtract it from 180, and there remaineth 62.ᵈ. 28.ᵐ. which is the Difference kept.  
Afterward I multiply 88525 the Sine of the Complement repeated, by 88674 the Sine of the difference kept, parting the product 7849865850 by the whole Sine. The quotient is 78498 whose ark is 51.ᵈ. 43.ᵐ. which is the distance between the Style and the Substile.  
Then comparing the Complement of this distance, being 38.ᵈ. 17.ᵐ. with the complement of the difference kept, which is 27.ᵈ. 32.ᵐ. and finding this to be least, I amplify 46226 the Sine thereof, by the whole Sine, dividing the product 4622500000 by the Sine of the greater, which is 61955, and 74612 the quotient coming hereof yieldeth an ark 48.ᵈ. 15.ᵐ. whose Complement being 41.ᵈ. 45.ᵐ. I take out of 180, and the remainder is 138.ᵈ. 15.ᵐ. the distance of the Substile from the Meridian.   

The delineation of the Figure.  
First as before, draw a line A. B. parallel to the Horizon. Make another C. D. squirewise to the former. Draw two quadrants at the least C. A. D. dividing each of them into 90. degrees, number therein from A. toward D. the distance of the Meridian from the Horizon, which in our example is found to be 78.ᵈ. 2.ᵐ. draw at the end hereof from E. the line F. for the Meridian. Then account from this upward 138.ᵈ. 15.ᵐ. being the distance of the Substile from the Meridian, drawing likewise at the end of this number from E. the line for the Substile. Afterward account from this line (on which side you will) 51. 43ᵐ. the distance between the Style and the Substile, extending the line E H. by the end of this number for the Style. Let the Contingent line cut the Substile squirewise in any point where you will. Afterward taking with your compasses the least distance between the point G. and the Style, with that wideness in the Substile toward E. make the point I. upon which draw the Equinoctial circle by G. And because in this kind, where the Compounded ark is greater than 90.ᵈ. there is none intersection of the Meridian and the contingent line beneath the ceutre E. therefore draw forth the Meridian beyond the centre, then place your ruler on the intersection of the. Meridian and Contingent line above the centre, and the point ●. and where it cutteth the Equator, there begin to divide it into 24. equal parts. Finish that which remaineth as before.  

A North reclining declining.   dialing diagram    

The making of a South Inclining direct Dial. CAP. 16.  
IF the inclination of the plat be less than the Elevation of the pole, take that from this, accounting the remainder for the Elevation of the Pole. And with this number as if it were the Elevafion of the Pole make a Dial like to the South erect direct, and it shall be fit for your plat.  
But if the Inclination of the plat be equal with the Elevation of the Pole, make your Dial altogether like to the North reclining direct, whose reclination likewise is equal to the Elevation of the Pole. They differ only herein, that this is placed downward to the earth, and that upward.  
But if the Inclination of the plat be greater than the Elevation of the Pole, subtract this from that, and with the number remaining, as if it were the Elevation of the Pole, make your Dial like to the North erect direct.  

Example of this kind incling 65. d.  
Subtract 52.ᵈ. the Elevation of the Pole, out of 65.ᵈ. being the Inclination of the plat, and the remainder 13.ᵈ. account the Elevation of the Pole. For the delineation hereof, resort to the North erect Dial, Cap. 5.  
A South inclining direct.   dialing diagram 
You see in this Figure, that all the lines for the hours are drawn opposite from the Contingent line, beyond the centre E. In like manner must you do in the delineation of all such, which incline more than the elevation of the Pole.  
Let the Style in this kind be placed directly over the Meridian, with so great an angle as D. E. F. is, etc.    

The making of a North inclining direct Dial. CAP. 17.  
IF the Inclination of the plat be less than the complement of the Elevation of the pole, add the inclination to the Elevation: and with that number, as if it were the Elevation of the Pole, make a North erect direct dial, for your plat.  
But if the inclination be equal with the complement of the Elevation of the Pole, then make a Dial as you were taught in the second kind of South reclining direct: for as that plat so reclining in the South is placed upward: so is this placed down ward. Note that this North dial showeth but only 4. hours, the two first in the morning, and the two last in the evening, and that only in the midst of Summer.  
But if the inclination be greater than the Complement of the Elevation of the Pole, subtract this lesser out of the other being greater: and with the number remaining, as if it were the Elevation of the Pole, make your dial like to an horizontal. This kind likewise showeth very few hours.  

Example of this last kind, inclining 45. d.  
First I take 38.ᵈ. being the complement of the Elevation of the Pale, out of 45.ᵈ. the inclination of the plat, and there remaineth 7.ᵈ. which I keep for the Elevation of the Pole. For the delineation of this dial, resort to the horizontal, Cap. 3.  

A North inclining Direct.   dialing diagram    

The making of the East and West inclining Dial's. CAP. 18.  
MUltiply the Sine of the Elevation of the Pole by the Sine of the Inclination of the plat, dividing the product by the whole Sine: the quotient ark shall be the distance of the Style from the Substile.  
Then compare the complement of this distance with the complement of the Elevation of the Pole, increasing the Sine of the lesser by the whole Sine, part the product by the Sine of the greater, where of shall come a Sine, the Complement of whose ark shallbe the distance of the Substile from the Meridian.  
It shall not be necessary to show any example hereof, because the East and West reclining be altogether like to this. Wherefore resort thither, for the working of your example, for those distances in drawing this figure shall be here observed.  

The delineation of an East Dial, inclining 45. d.  
First draw a line Parallel to the Horizon, A. B. make the perpendicular line C. D. squirewise to the other. Describe the quadrant from A. to D. number from A. toward D. 33.ᵈ. 52.ᵐ. the distance of the Substile from the Meridian, and at the end hereof from C. draw the line E. for the Substile, account from this 42.ᵈ. 9 drawing likewise at the end of this number the line G. for the Style. Let the Contingent line cut the Substile squirewise. Finish all things else, as in the East reclining.  

An East and West inclining.   dialing diagram    

The making of a South inclining declining Dial. CAP. 19  
FIrst multiply the Sine of the Declination, by the Sine of the complement of the inclination, parting the product by the whole Sine. The quotient Sine shall yield an ark, whose complement shall be named the complement repeated.  
Then increase the Sine of the Complement of the Declination by the whole Sine, and the product divide by the Sine of the Complement repeated: whereof shall come a Sine, whose ark is the distance of the Meridian from the Horizon.  
Afterward, this Sine being multiplied by the Sine of the Complement of the inclination, and the product parted by the whole Sine: the ark of the quotient Sine shall be the Elevation of the Meridian. Which ark you must add to the Elevation of the Pole. And if the total number be less than 90.ᵈ. it shall be named the Doubtful ark. But if it be greater than 90.ᵈ. take it from 180, and let the remainder be called the Doubtful ark.  
This done, augment the Sine of the Complement repeated, by the Sine of the Doubtful ark: & the product arising thereof being divided by the whole Sine, the quotient ark shall be the distance of the Style from the Substile.  
Now compare the Complement of this distance, with the Complement of the doubtful ark, multiplying the Sine of the lesser, by the whole Sine, dividing the product by the Sine of the greater: the ark of the quotient Sine coming thereof shall be the distance of the Substile from the Meridian.  
But note, if the doubtful ark be found without subtraction from 180, (which is if it be less than 90ᵈ.) than you must subtract the distance of the Substile from the Meridian out of 180, and the number remaining shall be the true distance of the Meridian from the Substile.  
Mark this likewise, if the doubtful ark be equal 90 degrees, let the Complement repeated be the distance of the Style from the Substile: then shall there be just 90ᵈ. for the space between the Substile and the Meridian, as before is taught in the North declining reclining, whither you may resort for the working hereof. Here it shall be sufficient to show two examples, with the delineation of their figures: the one, where the doubtful ark is subtracted from 180ᵈ. and the other, where there is no subtraction hereof, because it is less than 90 degrees.  

Example of a South Dial  
Inclining 45 degrees.  
Declining 45 degrees.   
Forasmuch as this is altogether like to the North reclining 45ᵈ. declining 45ᵈ. whither you may resort: I will omit the working of the former part of this example, until I come to the finding out of the distance of the Substile from the Meridian. The former part of this example you may find to be thus wrought in the North reclining declining:  

1 First the complement repeated 60ᵈ.  
2 Then the distance of the Meridian from the Horizon 54ᵈ. 44ᵐ.  
3 Next the Elevation of the Meridian 35ᵈ. 16ᵐ.  
4 The compounded ark there, which we call the doubtful ark here 87ᵈ. 16  
5ᵐ. The distance of the Style from the Substile 59ᵈ. 54ᵐ.  
6 Last, the distance of the Substile from the Meridian 84ᵈ. 33ᵐ.   
But here (as before is taught) you must subtract this distance from 180ᵈ. because the doubtful ark was less than 90ᵈ. and then the remainder 95ᵈ. 27ᵐ. shall be the true distance of the Substile from the Meridian.  
In the delineation of this dial, first, as before, make a line paralele to the Horizon A. B. Draw another perpendicular C. D. cutting the former squirewise. Make two quadrants from A. by D▪ to B. If the plat decline westward, number the distance of the Meridian from the Horizon from A. toward D. etc. But if the declination be Eastward (as in this example it is) then account this distance of the Meridian from the Horizon, which is 54ᵈ. 44ᵐ. from B. toward D. drawing at the end hereof from the centre C. the Meridian line which shall show the 12 hour. Account likewise from that toward A. 95ᵈ. 27ᵐ. the distance of the Substile from the Meridian: making in like manner the line E. C. for the Substile. From this line on which side you will, number the distance of the Style from the Substile, being 59ᵈ. 54ᵐ. draw by this the line G. representing the Style. Let the line of Contingence cut the Substile squirewise in any point where you will. Make the Equator, begin the division thereof, and draw the hour lines in all respects as in the North reclining 21 degrees 30 minutes, declining 30 degrees. Cap. 15.  
A South declining inclining Dial.   dialing diagram 

The second example of a South Dial  
Inclining 33.d. 40.m. Where the doubtful ark is subtracted from 180.  
Declining 31.d. Where the doubtful ark is subtracted from 180.   
First 51503 the Sine of the Declination being multiplied, by 83227 the Sine of the Complement of the inclination: and the product 4286440181 divided by the whole Sine: the ark of the quotient is 25ᵈ. 23.ᵐ. The complement whereof being 64ᵈ. 37ᵐ. is the complement repeated.  
Then I increase the Sine of the Complement of the declination which is 85716, by the whole Sine, and I part the product 8571600000, by 90346 the Sine of the Complement repeated. The quotient 94875 yieldeth an ark 71ᵈ. 35ᵐ. which is the distance of the Meridian from the Horizon.  
Again, I multiply this Sine 94875, by the Sine of the Complement of the Inclination being 83227, and the product 7895961425 arising thereof, I divide by the whole Sine. The ark of the quotient is 52ᵈ. 9ᵐ which is the Elevation of the Meridian.  
Now this being added to the Elevation of the Pole 52ᵈ. and the total number 104ᵈ. 9ᵐ. taken from 180, there remaineth 75ᵈ. 51ᵐ. which is the doubtful ark.  
Therefore I augment 90346 the Sine of the Complement repeated, by 96930 the Sine of the doubtful ark, and the product 8760399890 I part by the whole Sine. The ark of the quotient 61ᵈ. 10ᵐ. is the distance of the Substile from the Style.  
This done, I compare 28.ᵈ. 50.ᵐ. the Complement of this distance, with 14.ᵈ. 9 the Complement of the doubtful ark, multiplying the Sine of the lesser, which is 24446, by the whole Sine, dividing the product 2444600000, by 48226 the Sine of the greater. The quotient Sine 50741 yieldeth an ark 30ᵈ. 30.ᵐ. whose Complement 59ᵈ. 30ᵐ. is the distance of the Substile from the Meridian.  
First draw a line A. B. Parallel to the Horizon: draw another C. D. squirewise to the line A. B. make the two quadrants 
The delineation of the figure. 71.  from A. by D. to B. And because this Declination is on the left hand, number from A. towards D. 7ᵈ. 35ᵐ. the distance of the Meridian from the Horizon, drawing by this from C. the Meridian line. From that toward B. account 59ᵈ. 30ᵐ. the distance of the Substile from the Meridian, making in like manner at the end of this number from C. the Substile line E. Again from this forward (if you can, or else backward) number the distance of the Style from the Substile, which is 61ᵈ. 10ᵐ. Draw likewise by this from C. the line G. for the Style. Let the Contingent cut the Substile squirewise in any place, where you will, etc. The making of the reclining declining Dial's before, shall teach you the finishing of this likewise.  
A South declining inclining.   dialing diagram  

The making of a North inclining declining Dial. CAP. 20.  
THe working of this kind is altogether like the South reclining declining dial.  
First therefore multiply the Sine of the Declination, by the Sine of the Complement of the inclination, dividing the product by the whole Sine. The quotient shalt give an ark: whose Complement shall be named the Complement repeated.  
Then augment the Sine of the Complement of the Declination by the whole Sine, & the product part by the Sine of the Complement repeated. Whereof will come a quotient Sine: whose ark shall be the distance of that which is as it were the Meridian from the Horizon.  
For the finding out of the Elevation of the Meridian, because it may be done two sundry ways, I will set them down both, referring the choice to yourself.  
Compare the Complement of the ark last found out, with the Declination of the plat, multiplying the Sine of the lesser, by the whole Sine: and dividing the product by the Sine of the greater. The quotient Sine shall yield an ark: whose Complement shall be the Elevation of the Meridian. Or multiply the Sine of the ark last found out, by the Sine of the Compliment of the inclination, part the product by the whole Sine: and the ark of the quotient Sine shallbe the Elevation of the Meridian.  
Now comparing the Eluation of the Pole with the Elevation of the Meridian, subtract the lesser from the greater, retaining the number remaining, which shall be called the difference kept.  
Afterward increase the Sine of the difference kept, by the Sine of the Complement repeated, and divide the product by the whole Sine. The quotient arising of this division shall give an ark, which shall be the distance between the Style and the Substile.  
To conclude, the Complement of this distance being compared with the Complement of the difference kept, multiply the Sine of the lesser by the whole Sine, and part the product by the Sine of the greater. The Complement of the ark of the quotient Sine, shall be the distance of the Substile from the Meridian.  
But note, if the Elevation of the Meridian, & the Elevation of the Pole be found equal, make an Equinoctial dial as before you were taught in the Meridional reclining declining: for there is no difference, but only that this is placed downward, and the South upward.  
There be of this north, as of the South reclining declining, three sundry kinds of Dial's, as by the examples and figures following you may perceive.  

Example of a North Dial 
First example.   
Inclining 20 d.  
Declining 30 d.   
First I multiply 50000 the Sine of the declination, by 93969 the Sine of the Complement of the inclination, and the product thereof 4698450000, I divide by the whole Sine 100000, & the quotient Sine 46984 yieldeth an ark 28ᵈ. 2ᵐ. whose Complement 61ᵈ. 58ᵐ. is the Complement repeated.  
Then I augment the Sine of the Complement of the declination, which is 86602 by the whole Sine, parting the product 8660200000, by 88267 the Sine of the Complement repeated: the quotient is 98113. The ark thereof being 78ᵈ. 51ᵐ. is the distance of the Meridian from the horizon.  
Now in comparing the Complement 
Two ways in working the example, for the two precepts, of the finding out the Elevation of the Meridian.  of the ark last found out, with the declination of the plat: I find the Complement of the ark least. Wherefore I multiply the Sine thereof being 19337 by the whole Sine: and the product 1933700000, I divide by 50000 the Sine of the greater. The quotient 38673 giveth an 
Both these ways tend to one end.  ark 22ᵈ. 46ᵐ. whose Complement 67ᵈ. 14ᵐ. is the Elevation of the Meridian. Or multiply 98112 the Sine of the ark last found out, by the Sine of the Complement of the inclination, which is 93969: and the product 9219566568 part by the whole Sine. The quotient shall be 92195, whose ark 67 degrees 14 minutes is the Elevation of the Meridian.  
This done, I compare the Elevation of the Meridian 67ᵈ. 14ᵐ. with the Elevation of the Pole 52ᵈ. subtracting the lesser from the greater, and there remaineth 15ᵈ. 14ᵐ. which is the difference kept.  
Then I multiply 26275 the Sine of the difference kept, by 88267 the Sine of the Complement repeated: parting the product 2318455682, by the whole Sine. The ark 11.ᵈ. 47.ᵐ. of the quotient Sine 23184, is the distance of the Style from the Substile.  
Now the Complement of this distance being 76.ᵈ. 36ᵐ. compared with 74.ᵈ. 47.ᵐ. the Complement of the difference kept, I increase 96494 the Sine of the lesser, by the whole sine, and the product thereof 9649400000, I distribute by 97277 the Sine of the greater. And the quotient is 99191. whose ark is 82.ᵈ. 43.ᵐ. The complement whereof being 7.ᵈ. 17.ᵐ. is the distance of the Substile from the Meridian.  

The delineation of the Dial.  
If the Equator of the Meridian be greater than the Elenation of the Pole, draw a line A. B. parallel to the Horizen, make another C. D. squirewise to the former, describe the quadrant on the left hand from A. to C. (because the Declination is on the right hand) numbering therein from A. upward 76.ᵈ. 32.ᵐ. the distance of the Meridian from the Horizon, draw at the end thereof from D. the line G. representing the Meridian. From this backward, toward A account the distance of the Substile from the Meridian, which is 7.ᵈ. 41.ᵐ. making likewise at the end of this number from D. the line E. for the Substile. From hence account 11.ᵈ. 47.ᵐ. being the distance of the Style from the Substile, drawing in like manner from D. the line F. for the Style. Let the line of contingence cut the Substile squirewise as before, make the Equator in this, as in the other before. Place the ruler upon the centre H. and the intersection of the Contingent line, and the Meridian, beginning the division thereof into 24. equal parts, where the ruler shall touch the Equator. Finish all things remaining, as before.  
Remember that in this kind of North inclining declining Dial, the Meridian line rather representeth the 12. hour at midnight, then in the day: wherefore in accounting the hour lines, let the meridian be always for the 12. hour in the night. And then if your plat decline Westward, account backward as it were 11. 10. 9 which hours with the 12. you may omit in the delineation of your dial, because they have no use in our Elevation. But account forward 8, 7, 6, 5, 4. etc. so many as you think shall be necessary, for some declination will receive more, than other will: the greater the more, the lesser the sewer hours. But if the declination of your plat be Eastward then account from the Meridian line forward 1, 2, 3, 4. etc. so many as shall be needful, omitting likewise the first three, etc.  
A North inclining declining.   dialing diagram  

The second kind.  
But if the Elevation of the Meridian, and the Elevation of the Pole be equal, the making of the Dial differeth from the other before. For an example hereof, and the drawing of the Figure, resort to the South reclining 33ᵈ. 30.ᵐ. declining 32.ᵈ. for these are both alike, only remember here, that if this decline Eastward, make the quadrant likewise on the East side, etc. But if the plat decline toward the West, make the quadrant Westward (as in the figure following) finishing all things else as here you see done. Observe the same order in naming the hour lines which I taught you before.  
A North inclining declining.   dialing diagram 

The third Example of a North Dial.  
Inclining 45. d.  
Declining 45. d.   
If the Elevation of the Meridian be lesser than the Elevation of the Pole, make your Dial thus,  
For the working of the example, and the delineation of the figure, resort to the South reclining 45.ᵈ. declining 45.ᵈ. for that agreeth with this, except only that in this, the Style is placed downward, and showeth but few hours, and those likewise are drawn downward: but in the other the Style standeth upward, showing many hours, and those likewise for the most part are drawn upward. Mark also in this kind, if your plat decline westward, draw the Quadraut on the West side, but if the declination be Eastward, make the Quadrant on the East side, etc.  
The making of these, are more pleasant to know, then profitable to use: unless the declination be the greater, for so it may show the more hours.  
For the naming of the hours, do as you were taught before in the first of these three kinds.  
A North Declining Inclining.   dialing diagram   

The delineation of those Dial's, whose plat lieth nigh to the Elevation of the Pole. CAP. 21.  
WHen as the declination and reclination, or declination and inclination, shall cause the plat of your Dial to lie nigh to the Elevation of the Pole, than the hour lines must be drawn very long, to make the distances somewhat large, so that they may be discerned one from another. And commonly these following be such as here we mean.  

A South  
Reclining direct.  
Declining erect.  
Declining reclining.   

An Eastor West  
Reclining.  
Inclining.   

A North  
Inclining direct.  
Declining erect.  
Declining inclining.   
In all these kinds oftentimes the distance between the Style and the Substile is so small that it is not possible to draw your Dial, except the plat be very large: which at all times you shall not have.  
Therefore the best way is this. First draw your Dial very true (as before hath been taught) upon a large paper, making your quadrant so great as may be. Draw likewise the hour lines very long, which will cause the greater distance between the Style and the Substile, and in like manner between the hour lines. It will also procure the greater Equator, which will be a great help in the drawing of your Dial, as by the figure following you may perceive.  
When you have thus drawn your Dial, then at the end of the paper where the lines be of greatest distance, draw the line I K. parallel to the horizontal line, make likewise the other line L M. (according to the greatness of your plat) of equal distance from that other. This done, find out (or rather draw a line upon your plat which may represent) the true Horizon of the plat: cutting the paper in sunder by the line I K. and the line L. M. placing it upon your plat very plain, so that one of the edges may be parallel to the Horizon. Then make marks upon the plat at both ends of all the hour lines drawing by those marks, the lines for the hours. Draw likewise light lines upon the plat for the Style & the Substile, easily to be extinguished, as you did upon the paper.  
Let the Style N. be fastened over the Substile as you have been taught in the East, & West, or Equinoctial dials.  
But here note, that the Style must not be of equal height at both ends. But let it be so high, as the distance between the Style & the Substile is in the place where it standeth: as in the examples following you may plainly perceive.  

A North Dial.  
Declining Westward 65. d. Inclining 22.d. 20.ᵐ. Distance of the Meridian from the horizon 50.d. 50.ᵐ. Distance of the Substile and the Meridian 6.d. 35.ᵐ. Distance of the Style and Substile 4.d. 15.ᵐ. Elevation of the Pole 52. d.  

hitherto we have taught the delineation of all kinds of Dial's, which are to be made upon any plain plat or superficies: now followeth the garnishing of them, with the 12. Signs, and the hours unequal.   dialing diagram    

How to draw the 12. Signs in all kinds of Dial's before mentioned. CAP. 22.  
FOr somuch as sometime the 12 signs of the Zodiac are placed in Sun dials, know in what sign the Sun is at any time (which albeit any kind will receive, yet most commonly the vertical directly opposite to the South are garnished therewith.) Therefore in drawing the 12. Signs, in these South, and all other kinds of dials before mentioned, do thus.  
Prepare a little Table of Iron, Brass, or close grained wood, in breadth 3. or 4. in length 5. or 6. inches. Chose for the Diameter one of the shortest sides, wherein draw upon the centre A. half a circle to be divided into two quadrants distinguished with a line drawn from the centre A. to the ark B. part each quadrant into 90. degrees. The line A. B. is here in place of the Equator, showing in the ark the beginning of Aries and Libra. Moreover how to find the other Signs on both sides, seek in the Table of the declination of the Sun from the Equinoctial circle, which is afterward expressed. First find out the degrees of the distance (in this Table) of the beginning of Taurus from the Equator being 11. degrees 30. minutes. Account this distance in the quadrant from the letter B. towards the left hand, and at the end of this number draw a line from the centre A. which shall show the beginning of Taurus and Virgo. Account like wise the same distance on the other side from the line A. B. and make a line for the beginning of Pisces and Scorpius. Again in the 20.ᵈ. 12.ᵐ. from the line A. B. you shall have the beginning of Gemeni & Leo: and so many degrees and minutes on the other side, the beginning of Aquarius and Sagittarius. To conclude on both sides in the 23 degrees 30 minutes there must be on the one part Cancer, and on the other Capricorn.  
Afterward draw lines from the centre A by every mark so long as your Table will receive▪ and at the end of these lines let the characters of the 12 signs of the Zodiac be fixed: as in the figure following you may see.   dialing diagram 

The use of this trigonal instrument in writing the 12 signs in dials.  
YOur Dial being made and the Style placed therein: take your trigonal instrument, and set it upon the Style, so that the whole Diameter thereof may stand plain upon the edge or upper part, the centre A. of your instrument always remaining in one place of the Style. Let the line of ♋ bend upward to the centre of your dial, and the line of ♑ down ward. Then fasten a thread at the uppermost end of your trigonal in every line of the signs, so that you may direct it downward by the centre A. to the plat of your Dial: and where the end of the thread shall touch the Dial, make a mark. Then moving the Trigonal on the right hand, so that the thread may be stretched on the left hand, make there likewise a prick upon the plat. When you have thus done 4 or 5 times upon the one side of the dial: (that is on the left hand) do so likewise on the other side on the right hand. These marks being thus made, draw (with your compasses or rather with a crooked ruler made according to those marks) a line by them. Do thus with each one of the 7 lines of your Trigonal. When you have drawn all these lines, (being crooked, except the middle line, which is right, and representeth the Equator) then at the ends of them write the characters of the 12 signs on this wise.  
First on the right hand toward the East, at the end of the upper line, writ or place ♑. At the second line downward ♒. At the third ♓. At the fourth ♈. At the fift ♉ at the sixth ♊, at the seventh ♋. Then on the other side on the left hand toward the West, at the end of the nethermost line place ♋. At the second ♌, at the third ♍, at the fourth ♎, at the fift ♏, at the sixth ♐. And finally again at the uppermost set ♑.  
This done, remember that in what place of the Style the centre A. of your trigonal was placed, there fasten a small piece of iron or any such like thing, which may show with the shadow thereof, the Sign which the Sun shall be in at any time: unless the hour lines be so short, that where the small piece of iron should be placed, you may cut the Style that the end thereof shall show it. But the first is the most convenient way for the use thereof.   dialing diagram 
You may in like manner with this Instrument draw or place the 12 signs in Horizontal, East, West, and all other kinds of Dial's, which before are recited: because the difference is nothing but in placing the 12 signs at the ends of the lines: the true doing where of, you may easily perceive by the course of the Sun. For in the South Erect, when the Sun occupieth Cancer, then is the line of ♋ furthest distant from the centre of the Dial. But in the horizontal Dial, the line of ♋ is next to the centre. Few words shall suffice for this matter: experience shall easily teach you herein.    

How to place the unequal hours in a horizontal Dial. CAP. 23.  
FIrst you shall understand, that by an unequal hour is meant the 12 part of the day whether it be short or long. For when the Sun is in the Equinoctial circle, the day hath 12 equal, and like wise 12 unequal hours. But in winter solstice, when the days be at the shortest, and containeth with us (where the Pole is elevated above the Horizon 52 degrees) only 7 hours, 36 minutes: then the unequal hours be less. For if you divide these 7 hours 36 minutes by fractions into 12: you shall find only 38 minutes of an equal hour, to make one unequal hour. But in the summer solstice, when the days be at the longest, and hath 16 hours, 24 minutes: then one hour with 24 minutes maketh one hour unequal. These be like wise called the hours of the Planets, and are placed in dials on this manner.  
Your Dial being made and prepared, draw upon it the 12 signs of the Zodiac (as you were taught before) so large as your plat will give leave, etc.  
Note that always the Meridian line, or 12 equal hour, is the 6 unequal hour. Mark likewise, that when the Sun entereth into the beginning of ♈ and ♎, both the equal and the unequal hours be of like quantity. For the 7 equal hour in the morning is the first unequal, and the 8 equal the 2 planetare hour, etc.  
But in the tropic of ♋ it is otherwise: for then the unequal hours be greater than the equal. Therefore account how many hours and minutes is in the longest day for your Elevation, changing all the minutes of those hours into one total sum. Then divide this number by 12: and the quotient shall show how many minutes of an equal maketh one unequal hour. Example hereof: with us the longest day is 16 hours 24 minutes. Wherefore I multiply 60 which is the minutes of an equal hour, by 16, which is the sum of hours of the longest day, and the product ariseth to 960, whereunto I add 24 the minutes remaining, than the number shall be 984, which being divided by 12, the quotient is 82. Wherefore I conclude that 82 minutes of an equal, maketh one unequal or planetare hour. Then to draw them in dials work thus.  
First in the tropic of ♋ divide the space between each hour into 60 equal parts, but because those distances be so small, for the most part, that this cannot be done: therefore it shall suffice to divide every one of them into three equal parts. Then shall every part contain 20ᵐ. and three of them 60ᵐ. which is one equal hour.  
When as therefore you will place the 7 unequal hour, account from the Meridian line, or 12 hour, 4 parts and 2 minutes (which is 82 minutes) and make there a mark. Then place your ruler by this point or mark, and the intersection of the line of ♈ and ♎, and the first hour after noon: draw a line from one tropic to another, that is from ♋ to ♑.  
Then for the placing of the 8 unequal hour, account from the 7 last made, 4 of the foresaid parts and two minutes make there a mark, by which and the intersection of the line of ♈ and ♎ and the line of the second hour after noon, the ruler being placed, draw a line, as before.  
For the 9, account from the 8 last made, likewise 4 parts and two minutes, drawing a line as before. Do in like manner for the drawing of all the other remaining: that is, for the 10, and 11. The 12 being the Sun sitting shall need no line.  
As you have finished the unequal hours on this side for the afternoon, do in like manner for those in the forenoon: accounting from the Meridian 4 parts and 2ᵐ. of an equal hour, to each unequal, drawing, as before, lines from one tropic to another.  
You may place the figures for the unequal or planetare hours, at the end of the lines under the tropic of Capricorn, or above the line of Cancer as you will.  
Note that the same part of the Style which showeth the 12 signs, shall like wise show the unequal hours. For better understanding hereof:  
Behold the Figure following.  
An horizontal Dial.   dialing diagram  

How to place the hours unequal in a South erect direct Dial. CAP. 24.  
THere is almost the same manner of placing the unequal hours in this kind, as in the horizontal, differing only herein, that whereas you divided the spaces between the hour lines, in the tropic of ♋, (which is next to the centre of your Dial) and accounted 4 parts 2 minutes of an equal, to one unequal hour: in this you must divide the tropic of ♑, (which is likewise next to the centre of the Dial) taking only 2 parts wanting 2 minutes, (that is 38m.) of an equal hour to each unequal.  
Therefore your Dial being prepared, and the 12 signs of the Zodiac placed therein, etc. account how many hours and minutes be in the shortest day in your country, changing all the hours into minutes (giving to each hour 60 minutes.) Divide the product by 12, and the quotient shall show how many minutes of an equal, will be answerable to each unequal hour. Example, the shortest day with us is 7 hours, 36 minutes, which being multiplied into one gross number of minutes, the product is 456, which I divide by 12, and the quotient is 38. Therefore I affirm that 38 minutes of a vulgar or equal, maketh one planetare or unequal hour.  
Then divide the tropic of ♑ here, as in the horizontal you did the tropic of ♋. Afterward account from the Meridian line 38m. (or 2 parts wanting, 2m.) of an equal hour to each unequal, etc.  
Finish all things else in this Meridional erect direct, as you were taught in the Horizontal. The figure ensueth.  
A South Dial.   dialing diagram  

The placing of the hours unequal in East and West Dial's. CAP. 25.  
FIrst draw your Dial true, then place the 12 Signs of the Zodiac in it: which you may do on this wise. The Style of your Dial being fixed therein, as before hath been taught, take your trigonal (before prescribed) and place it upon the edge of the Style: then according to the direction of the thread (as in the South Dial before) make marks upon the plat, as in the Figure following you may see. Afterward by these marks draw lines as before you did, placing at the ends of them the Characters of the 12 Signs.  
Moreover note that where the centre A. of your trigonal doth touch the edge of the Style, there fasten a small piece of Iron or such like thing, which may show she place of the Sun in every Sign, and likewise the planetare or unequal hours.   dialing diagram 
When as therefore you would place the hours unequal in those kinds of dials, account the member of hours and minutes, which the half of the longest day in your Country doth contain.  

Example.  
With us the longest day containeth 16 hours, 24ᵐ. the half thereof is 8 hours 12ᵐ. wherefore numbering in the circumference of the circle from A. to D. 6 hours, (for that quadrant containeth 6 equal hours) and from D. toward C. the other 2. hours and 12 minutes, so that you may have from A. by D. toward C. the just half of the longest day, at the end thereof make a point X. Then divide the ark A. X. into 6. equal parts. Afterward placing the ruler upon the centre E. and each one of these parts, where it shall touch the Contingent line D. F. make marks. Likewise in the other line of Contingence G. B. make these marks, being of equal distance in this line from the point B. as the other be from the point D. Then place the ruler upon two of these marks (being equidistant from the points B. D.) in both the Contingent lines, and where it shall touch the tropic of ♋ make marks. This done, lay the ruler upon the first mark in the Tropic of ♋ next unto X, and the intersection of the line of the 6 hour, with the line of the beginning of ♈ and ♎, drawing a manifest line from one tropic to another, which (if you have wrought all things true) will be parallel to the Horizon, and it shall be the 12. unequal hour line in the West Dial, and the Horizon or Sun rising in the East Dial. Again, place your ruler upon the second mark in the tropic of Cancer, and the intersection of the line of the 5. hour with the Equator, or line of Aries and Libra, drawing like wise a line, which shall show the first hour in the East, and the 11. hour in the West Dial, make on this wise all the other unequal hours, namely the 10, 9, 8, and 7, in the West Dial. And the 2, 3, 4, 5, in the East Dial.  
 dialing diagram    

The making of an horizontal Spherical or hollow Dial. CAP. 26.  
FIrst prepare your Sphere or plat perfectly hollow, of what quantity you wil Then with your compasses divide the upper brim thereof, into 4 equal parts, marking it with the letters A. B. C. D. Afterward open your Compasses to the wideness of one Quadrant, either A. C. or A. D. and with that wideness, one foot being placed in the point D. draw an obscure or light line from A. to B. This done, set one foot of your Compasses in the point C and with the other you shall try whether the plat be perfectly Spherical or not: for if it be, your Compasses will fall in the same line which you made before, but if it be not perfect you may amend it. And then draw this line manifest, so that it may continue. For it shall be the Meridian line, showing the 12 hour. Again, place one foot in the point A. drawing with the other a light line from C. to D. then one foot being set in B. you may try with the other, as before, the truth of the line last made. Now where these two lines cut each other in the bottom of the plat, place the letter E. Then dividing the quadrant A. E. into 90.ᵈ. and accounting therein the Elevation of the Pole from E. toward A. make the point F. by which the ark line of ♈ and ♎, representing the equinoctial circle, shall be drawn. Afterward seek out the greatest declination of the Sun from the Equator, (which you shall find in the table of the declination of the Sun) which is 23.ᵈ. 30.ᵐ. Then account 23.ᵈ. 30.ᵐ. from F. toward E. making there a mark for the ark of Cancer. Likewise from F. toward A. number the same distance, for the ark line of Capricorn. This done, account from F. toward E. 20.ᵈ. 12.ᵐ. make there also a mark for the line of ♊ and ♌. And the like space from F. towards A. for the ark of ♒ and ♐. To conclude, number from F. towards E. 11.ᵈ. 30.ᵐ. making there also a mark for the ●ine of ♉ and ♍: and so much from F. toward A. for ♓ and ♏.  
When you have thus made marks for all the arks of the 12 Signs, open your compasses to the Quadrant of the Sphere, that is, from A. to C. which wideness of the compasses remaining, place one foot in the point F. in the ark of ♈ and ♎ in the Meridian: and where the other foot shall touch the same Meridian towards B make the point G. which shall represent the Pole antarctic, by which as it were from a centre, draw a line from the point D. by F. to C. which shall be the line of Aries and Libra. Then one foot of your compasses remaining in the point G. with the other draw lines from one side of the plat to the other, by every mark before made for the 12. Signs of the Zodiac.  
The 12. Signs being thus finished, proceed to the division of the equal hours on this manner. Divide the ark of the Equator into 12. equal parts, beginning at D. by F. ending in C. Then open your Compasses to the quadrant of the plat, and the same wideness of them remaining, place one foot on the first point of the division next C. in the Equator, and if you have divided the Equator equally, the other foot will touch the first part beyond F. towards D. by which from the centre G to the edge of the plat, draw a line, which shall show the first hour afternoon. This done, remove your compasses (that wideness remaining) placing one foot in the second part from C. towards F. and the other foot touching the second part from F towards D. make a line as before, from the centre G. to the brim of the plat, to show the 2. hour afternoon. In like manner, finish all the other hour lines, namely the 3, 4, 5, 6 7, 8, for the afternoon. Then do likewise on the other side for the hours in the forenoon, namely the 11, 10, 9, 8, 7, 6, 5, and 4 drawing lines from the centre G. by every division, to the brim of the plat, etc.  

The placing of the unequal hours in this Dial.  
When you will place the hours unequal in this Spherical Dial, divide the Tropic of Cancer and of Capricorn each of them into 12. equal parts, as before you did the Equator for the equal hours. Then with the compasses join each three points, of these three arks, answerable into one line or ark, until you have made 12. lines to show the 12. unequal hours, as in the figure following you may see. Whereof the Meridian, or 12. unequal hour shall always be the 6. unequal hour.  
In placing the figures to the equal & unequal hours, and the Characters to the 12. Signs of the Zodiac, the figure may sufficiently show, notwithstanding you may place them where you will, giving to each sign his proper Character, and every hour equal and unequal their proper figures or names.  
Fire the Style in the centre G. standing up so high as the brim of the plat, so that the upper end may appear as the centre of the circumference, which you may try, by placing a ruler upon the points A. B. And again upon C. D. In like manner you may examine it with your Compasses, but the former, as practise will teach you, is the more convenient way.  
You may, if you will, have the Style stand above the plat, so that it may show the equal hours above the edge of the Sphere, and then fasten a knot of equal height with the plat, which shall show the motion of the Sun in the 12. Signs and the unequal hours, which otherwise the end of the Style should do: as in the figure following you may perceive.  
The Figure of an hollow horizontal Dial.   dialing diagram   

The making of a South Spherical erect direct Dial. CAP. 27.  
YOu shall understand that the making of this kind differeth almost nothing from the horizontal: except only in the accounting the beginning of the Elevation of the Pole, & drawing the hours unequal.  
First your Sphere or plat being prepared, and parted into four Quadrants, as before in the horizontal you were taught, divide the quadrant E. A. into 90 degrees. Then number in it the Elevation of the Pole from A. toward E. marking the end of the number with F. Afterward open your compasses to the quadrant of the plat, and the one foot placed in F. extend the other in the Meridian line towards B. making the point G. where it shall touch the Meridian line: from which point as a centre you shall draw the lines for the 12 signs, as you were taught in the horizontal. Divide likewise, and draw the lines for the equal hours, as before in the horizontal.  
The placing of the hours unequal in this kind is done, as in the South Erect Direct upon a plain superficies, except only that whereas there you did draw them with a ruler, here you shall do it with compasses on this manner. Open your compasses to the Quadrant of the plat, that is, from A. to C. accounsing from the Meridian in the tropic of Capricorn, two parts wanting two minutes of an equal hour, making there a mark: And that wideness of your compasses remaining, draw a line from that mark made in the Tropike ♑, by the intersection of the line of Aries and Libra, & the line of the first hour in the afternoon, unto the tropic of Cancer. Draw all the other lines for the unequal hours in like manner, as in the figure you may plainly see,  
Fix the Style in the centre G. as you did in the horizontal, placing the figures for the equal and unequal hours, and the characters to the 12 signs, as in the example following, or otherwise as you will.   dialing diagram  

The making of a Dial upon a Quadrant, or the Table described in the beginning of the Book. CAP. 28.  
IF you will make this Dial upon a Quadrant alone, prepare and divide it altogether like one of the Quadrants of the table, whose making is prescribed in the first Chapter: or if you will make it upon that instrument already prepared, work thus.  
Divide the diameter E. D. into four equal parts; then leaving out the fourth part next E. describe upon the middle point between C. and S. an obscure half circle: which divide into six equal parts. From each point of which division draw light lines unto the Semidiameter D. E. touching it squire wise. Then draw nigh to the line D. E. two other lines parallel thereto: which with the cross lines drawn from the half circle may make 12 unequal quadrangles, wherein the 12 signs of the Zodiac may be figured: the North towards D. the South toward S. And to the intent you may proportionally part these Quadrangles into five or ten degrees: divide those six parts of the half circle equally into so many degrees. Then draw again from those divisions right lives to the Dimetient D. E. squirewise, as before.  
This done upon the point E. from S. that is to say, from the end of ♐ and beginning of ♑, draw an ark to the Semidiamer D. E. which shall be for the winter Tropike. Again upon the same centre, draw another from the beginning of ♈ and ♎ for the Equator. To conclude, make another from the angle D. that is, from the end of ♊ and beginning of ♋, which shall be for the Summer Tropike. By the help of these arks, you shall draw the hour lines, so that you first know the height of the Sun above the Horizon, at every hour, when it occupieth the beginning of ♑, ♈ and ♋. Whose altitude is thus found out.  
Take the Elevation of the Pole, and the Complement thereof: also the declination of the Sun from the Equinoctial, and the distance of the Sun from the Meridian, accounting 15 degrees for every hour, with the Complement of this distance. Then if you desire to know the altitude of the Sun at six of the clock either in the morning or at evening (at which hour only in Summer it is above the Horizon) multiply the Sine of the Elevation of the Pole, by the Sine of the declaration of the Sun, dividing the product by the whole Sine, and you shall have your desire.  
But if the Sun shall be distant from the Meridian fewer hours than six, multiply the Sine of this distance (giving to every hour 15 degrees) by the Sine of the Complement of the altitude of the Pole, and the product hereof divide by the whole Sine. Then take the ark of the quotient from 90 degrees, and the first number found out shall remain, which must be kept. Then compare the Sine of this number found out, with the Sine of the Elevation of the Pole, augment the lesser by the whole Sine, dividing the product by the greater: Whereof shall come a quotient Sine, to the complement of whose ark add the declination of the Sun, if it shall be in the North signs, or subtract it from that, if it occupieth the South signs. And if the number which cometh hereof shall be greater than 90, take it from 180, and you shall have the second number found out. The Sine of this number found out being multiplied, by the Sine of the first number found out, shall yield a product, which part by the whole Sine, and the ark of the quotient Sine shall show the desired altitude of the Sun for the hour propounded.  
But if the distance of the Sun from the Meridian shall exceed six hours (that is 90 degrees) take the degrees of that distance from 180, multiplying the Sine of the remainder by the Sine of the Compliment of the altitude of the Pole, dividing the product by the whole Sine, and subtract the ark of the quotient from 90, the remainder shall be called the First number found out. The Sine of which number found out compared with the Sine of the Elevation of the Pole, increase the lesser by the whole Sine, and distribute the product by the greater. Then take the Complement of the Declination of the Sun, from the ark of the quotient, and you shall have the second number found out.  
Finally the Sine of the first number found out, and the Sine of the second being multiplied by themselves, and the product parted by the whole Sine; the quotient Sine shall yield an ark, which shall be the altitude of the Sun.  
But when the Sun occupieth the beginning of ♈ or ♎ you shall find the altitude thereof every hour, only by multiplying the Sine of the Complement of the distance of the Sun from the Meridian, by the Sine of the Complement of the Elevation of the Pole, dividing the product by the whole Sine, the quotient which cometh hereof shall yield the ark of your desire. At both the six hours, because the one six is the rising, and the other the sitting, there is then no altitude of the Sun above the Horizon.  
Moreover, to know how much the Meridian altitude is, of the Sun entering into ♋, add the greatest Declination thereof, to the Complement of the Elevation of the Pole: and by subtracting the greatest Declination of the Sun, from the complement of the Elevation of the Pole, you shall likewise have the Meridian altitude of the Sun entering into ♑.  
You shall likewise seek the height of the Sun, being in the 10 degree of ♉ above the Horizon, at 8 of the clock before noon, and at 4 after noon: at 7 in the morning, and 5 in the evening: at both 6. and the 5. hour in the morning, and 7. at night. Also at 5. in the morning and 7. in the evening, the Sun being in the beginning of ♊.  
But because the working of this, to find out these altitudes, requireth much time and labour, we will set down these distances ready found out, calculated for the Elevation of the Pole 50.ᵈ. and 52ᵈ. which you may likewise use without any notable difference, where the Pole is Elevated 49.ᵈ. 51.ᵐ. and 53ᵈ.  

The first table calculated for 50 degrees.  Hour. Hour. ♋ ♈ ♎ ♑ 10 deg. ♉ 10 deg. ♊   D. M. D. M. D M. D. M. D. M.  12 63 30 40 0 16 30     11 1 61 2 38 23 15 19     10 2 54 42 33 50 11 51     9 3 46 15 27 2 6 24     8 4 36 53 18 45   30 23   7 5 27 15 9 35   20 55   6 6 17 47 0 0   11 19   5 7 8 48     2 3 7 14 4 8 0 36       0 0  

The second table calculated for 52 degrees of latitude.  Hour Hour ♋ ♈ ♎ ♑ 10 deg. ♉ 10 deg ♊   D. M. D. M. D M. D. M. D. M.  12 61 30 38 0 14 30     11 1 59 13 36 28 13 21     10 2 53 15 32 7 10 0     9 3 45 12 25 33 4 45     8 4 36 12 17 32   29 36   7 5 26 53 8 46   19 22   6 6 17 42 0 0   11 4   5 7 8 58     2 4 7 46 4 8 1 1       0 0  
If therefore you will make your Quadrant horological for the Elevation of the Pole 50.ᵈ. extend a thread, or lay a ruler, from the centre E. by the 63.ᵈ. 30.ᵐ. of the limb of the quadrant, beginning at F and where the thread so placed shall touch the tropic of ♋, there make a point or mark. Again, let the thread be placed upon the 40.ᵈ. of the limb, and where it shall touch the Equator, there also make a mark. Thirdly let the thread be drawn by the 16.ᵈ. 30.ᵐ. of the quadrant: and where it shall touch the tropic of ♑, there make likewise a mark. This done, search out the centre (by the 5. Proposstion 4. Euclid) and join these three marks into one ark, which shall be the line for the 12 hour. Afterward place the thread upon the 61.ᵈ. 2.ᵐ. of the limb, and where it shall cut the tropic of ♋ note it. Draw likewise the thread upon the 38.ᵈ. 23.ᵐ. of the limb, mark where it cutteth the Equator. Then by the 15.ᵈ 19 of the limb extend the thread, making a mark in the section thereof and the tropic of ♑. Draw into one ark these three marks (as you did before) finding out the common centre: so shall you have the line of the 11. hour before noon, and 1. after noon. In like manner are the rest of the hour lines drawn by 3. points accounted and found out by the altitude of the sun. But because before 9 and after 3. in this Elevation of the Pole it will not serve for this delineation, there must another be made by the table of the 12. Signs or Quadrangles. Therefore draw that from the 10.ᵈ. of ♉, so shall you have a line or place wherein the third points may be marked. Then to prepare the points of the ark for the hours of 8. and 4. of 7. and 5. and both the 6. besides the two marks made in the tropic of ♋ and the Equator, let the third be noted in the ark drawn from the 10. degree of ♉. 〈◊〉 draw the line of the 5. and 7. hours, use besides the tropic of Cancer and this drawn from the 10. degree of ♉, the third which shall come from the beginning of ♊. The line of 4. in the morning, and 8. in the evening is very short, included in the angle next unto I. Thus you have finished 9 arks or lines for the hours, of which each one of them doth show two hours: except one which is only for the ●2 hour.  
And to the intent that there should be no space left void and unprofitable between the tropic of ♑ and the centre E. you may draw in that place the unequal hours, which you shall very easily do on this manner. Describe upon the centre E. an ark very little distant from the tropic of ♑, to be divided into 6 equal parts, and take the middle point between the beginning of that ark at S. and between E. Then upon this point▪ as it were a centre, make a half circle from S. to E. which shall be the line of the 6. unequal hour, that is of midday. Afterward one foot of the compasses being placed in the same Semidiameter E. D. and removed each way as occasion shall require: and the other extended in the mean time so far that it may touch the second point of the ark divided into 6. parts, and the centre E. draw an ark or line from the centre to that point, which shall show the 5. and 7. hours. Inlike manner are the other 4. drawn, one foot of the compasses being removed, as necessity shall require, in the Semidiameter E. D (which may be prolonged if it shall be too short) & the compasses so opened, that the other foot may touch the point of the divided ark and the centre E. and join them all into one ark line: then fix a thread in the centre E. well waxed, having two small beads upon it to move with a plummet of iron or lead. Last of all prepare two sights of brass or other metal with joints, which you must place in one right line upon the side or edge A. B. so that you may direct them, and again turn them down upon that side A. B. at your pleasure: that (if you make this Dial upon the Table or instrument prescribed in the beginning of this book) they be no hindrance to you in trying or examining of your plaits. Thus is your dial prepared: but for further instruction behold the Figure.   dialing diagram 

The use of this Dial or quadrant horological.  
TO know the hour of the day when the Sun shineth, work thus. First seek out in some Calendar in what Sign and degree the Sun is at that day: then extend the thread with the beads upon the Semidiameter E. D. and place the nethermost bead upon that degree, in the Table of the 12. Signs, which the Sun than occupieth. Afterward lifting up your quadrant, the thread with the plummet hanging at liberty, let the Sun beams pass thorough both the sights, and then the nethermost bead shall show the hour of the day either before, or after noon: for each line as you see, hath two hours, but you may easily discern what hour of the day it is.  
The unequal hour of the day is thus found out. The neither bead being placed upon his just degree in the Zodiac, lay it upon the ark of the 12. hour, the thread being extended: thus holding this immovable, place the uppermost bead upon the line of the 6. hour unequal. This done the Sun beams entering both the sights, the highest bead shall show the unequal, and the nethermost the equal hour.  
The rising and sitting of the Sun, whereby the length of the day and night is known, you shall find out on this manner. Place the neither bead upon the just degree of the Sun in the Zodiac, then extend the thread upon the semidiameter E. F. of the quadrant, and the bead shall show at or between what hours the Sun riseth and falleth.  
The height of the Sun. above the Horizon at any time of the day is thus taken. Receive the Sun beams by the holes in your sights, and the thread with the plunmet having liberty to move, shall cut in the limb of the Quadrant the just degree of the altitude of the Sun above the Horizon. In like manner is the Elevation of the Stars above the Horizon searched out.  
You shall find out the elevation of the Pole by the help of your Quadrant, thus. Take the height of the Sun at 12 of the clock, when the days and nights be of equal length, which being subtracted from 90ᵈ. the elevation remaineth. But if you assay that upon any other day than the Equinoct. you must consider whether the Sun occupieth the North, or the South signs, and then use the table of the Declination of the Sun on this wise. When the Sun is in the North signs, subtract his declination from the Meridian height thereof. But if it be in the South signs add the declination to his Meridian altitude. The remainder or total sum being taken from 90.ᵈ. the Elevation of the Pole shall remain.    

How to make an instrument, whereby you may know the just hour of the night by the Stars. CAP. 27.  
FIrst prepare a plat or Table of brass or firm and solid wood, which will not change or bend, somewhat thin, let the form of it be round, three inches broad, or more or less as you will. Draw a circle nigh unto the edge, and divide it into 12. equal parts: wherein shall be placed the 12. Signs of the Zodiac, then divide each of these parts into 30. equal parts, which shall show the number of the days that the Sun moveth in every Sign. Make a second circle, wherein you may number the days, and a third circle wherein writ the Characters of the 12. Signs, as you see in the Figure following.  
Draw likewise another circle, wherein you may write the days of every month in the year, which you may do by any Calendar, but for more easiness behold the Table following, whereby you may likewise do it.  
Month. Days. Deg. M. Characters of the 12. Signs. Signs. january 1 20 13 ♑ Capricorn.  15 5 33 ♒ Aquarius.  31 21 44 ♒  February 14 5 52 ♓ Pisces.  28 19 48 ♓  March 15 4 36 ♈ Aries.  31 20 18 ♈  April 15 4 51 ♉ Taurus.  30 19 13 ♉  May 15 4 38 ♊ Gemini.  31 18 50 ♊  june 15 3 3 ♋ Cancer.  30 17 22 ♋  july 15 1 40 ♌ Leo.  31 16 58 ♌  August. 15 1 24 ♍ Virgo.  31 16 58 ♍  September 15 1 40 ♎ Libra.  30 16 30 ♎  October 15 1 30 ♏ Scorpius.  31 17 43 ♏  November 15 2 30 ♐ Sagittarius.  30 18 13 ♐  December. 15 3 38 ♑ Capricorn.  
Enter this Table, and you shall find that the first day of January must be placed against the 20.ᵈ. 13.ᵐ. of Capricorn, and the 15. day against the 5.ᵈ. 33.ᵐ. of Aquarius, etc.  
Therefore lay your ruler upon the centre A. of your plat, and upon the 20.ᵈ. 13.ᵐ. of ♑, and where it shall touch the circle which you made for the days of every month, there make a mark which shall show the first day of January. Then lay it upon the 5.ᵈ. 33.ᵐ. of ♒ and the centre A. and where it shall touch the circle, make a mark for the 15. day of January. Then lay your ruler upon A. and the 21.ᵈ. 44.ᵐ. of ♒, and make likewise a mark for the 31. day of January. Again place the ruler upon the centre A. and the 5.ᵈ. 52.ᵐ. of ♓, for the 14. day of February: work thus with all the rest, until you have set down the beginning and midst of every month, as the Table doth direct you, then having found out the beginning & midst of every month, you may at ease divide every space into so many parts, as there be days in the month, which it serveth for, according as you see in this figure following. It shall be also necessary to make one circle to write the number of the days of the month, and another for the names of the months: you must make also a hole in the centre of this plat, of such bigness, as you may see a Star thorough it: as in the figure you may perceive.   dialing diagram 
Having prepared this plat with the circles upon it, for the 12. Signs and months, you must prepare an other wheel full of teeth, which you shall make on this manner.  
Take a thime Table or plat of the same matter your other was, draw a circle upon it so great as the inward circle of the first plat, and divide it into 24. equal parts. Then draw lines from the centre to every one of those parts, until you have so many as there be hours in the longest night in your Country. Then cut teeth by these lines for the hours, and write the number of them upon the teeth as you see in this figure. Let the tooth for the 12. hour be so long from the centre to the end, as is between the centre A. of the other plat and the circle of the 12. Signs. You must likewise make a hole in the centre of this wheel, of the same bigness of that, in the other plat.  
The thothed wheel.   dialing diagram 
This done, you must prepare a ruler, which shallbe placed upon this wheel. Take therefore a thin piece of wood or brass, and draw upon it a light line: at y● one end thereof in the midst of the line make a hole of the same bigness which that is in the great plat and tothed wheel. Then draw a circle half an inch wide or more if you will, afterward cut the ruler round at the end, cutting off likewise the one half of the breadth of the ruler by the line.  
Let the length of the ruler from th●●●ntre to the end be of such quantity, as is from the centre of the great plat to the outward edge, and an inch or more if y 〈◊〉 〈◊〉 〈◊〉.   dialing diagram 
The foreside of your instrument being finished: the backside must have two small holes, and a handle, which may turn and move about. Prepare them thus.  
Upon your great plat on the backside, right against the fourth degree of ♍, nigh the edge make a small hole, wherein the little pin of the handle may be pla●ed, when need requireth: and writ by this the greater Bear. Again, right against the 28ᵈ. 1ᵐ. of ♎ on the backside make another hole likewise, so that the handle being: removed to it, the pin may enter, as in the other. And by this, writ the lesser Bear. etc.   dialing diagram 
Prepare for your handle a thin plate of iron, brass, or wood, about six inches in length or more if you wil Draw in the midst of it a line: at the one end thereof make a hole in the midst of the line, of such bigness as the hole in the greater plat is. Upon this handle you must fasten a little short pin right in the line, so that the handle being fastened with the pin toward the backside of the great plat, and being moved the pin may fall directly into the hole.  
Behold the Figure following.   dialing diagram 
Prepare a hollow pin of iron or brass, according to this figure above, so great as it may easily enter into the hole of the handle, so that it may turn upon it: but let it go close into the great plat, that it may remain immovable. Let the Tothed wheel and the ruler move likewise upon it. Then place the Tothed wheel and the ruler on the former side of the plat upon the pin, and on the backside the handle with the little pin towards the back of the plat: fasten them so together, that the handle, the Tothed wheel, and the ruler may turn upon the plat severally, or each by themselves at your pleasure, and as need shall require. Notwithstanding you must be careful, that the hole in your pin may be so great, as you may perfectly see the North Star in the night thorough it. Then your instrument shall be finished, whose use followeth afterward.  
For better instruction behold the figure.   dialing diagram 

The use of this instrument.  
FIrst it shall be expedient to find out the Stars serving to this purpose, which you may do by the figure following.   dialing diagram 
You see in this figure the stars which serve to our use. The Polare Star is a bright Star in the North, placed in the tail of the lesser bear, and moveth but a small course, as in the figure the letter A. showeth. There is also another Star of the lesser bear greater and brighter than the other, and is noted with the letter B. and serveth to this purpose also. There is likewise in the greater bear called also Charles wain, and of country men the plough, two great bright stars, and are placed almost in a right line with the Polare Star; & are noted with the letters C. D. And these two Stars have 5 other bright Stars by them, but not so great as the other be. These Stars which you see in the figure, as all other, move equally above the Pole, and finish their course in 24 hours.  
Therefore when you would know the hour of the night by this instrument, do thus. Place the right line of the long tooth of the 12 hour directly over the day of the month, and turn the handle on the backside to the hole of the greater bear, and your instrument shall be prepared.  
Then lift up your instrument by the handle perpendicularly, so that it declineth on neither sides: and beholding the Polare Star thorough the hole in the centre, move the ruler above, until the right line there of be directly against, or seemeth to touch the two Stars of the greater bear, and under the line you shall have the just hour of the night: which you may find out by the number of the teeth with your finger in the night. But if you cannot see the two Stars of the greater bear because of clouds: and yet you may see the Polare Star, and the Star of the lesser bear, noted with the letter B. remove the handle on the backside to the hole of the lesser bear. Then lift up your instrument as before, and behold the Polare Star at the hole, and turn the ruler to the foresaid Star of the lesser bear, and you shall find the true hour of the night, as before is taught.    

The making of a Dial, to know the hour by the Moon. CAP. 30.  
PRepare a square piece of wood or metal 3 or 4 inches over, draw thereon a circle so great as you can, draw also another within that. Now because the Moon finisheth her course in 29 days, 12 hours, and 44 minutes, part the inward into 30 parts in this manner; let 29 be equally divided, and the 30 which is the last must not be so great by a third part: wherefore divide one of those 29 equal parts into three parts, and take two of them for the 30 and last part.   dialing diagram 
Cut out that which is within the inward limb, and prepare another of the same or like wood or metal somewhat thicker, and work it so that it may go into the former so much of the thickness as it may be equal on the backside, and that which remaineth of the thickness, let it hang over the foreside of the first plate to keep it from falling through: draw a circle upon it, and divide it into 24 equal parts: draw from these parts hour lines so many as shall suffice for the longest night: cut that which overhangeth with teeth at every hour, but especially at the 12 hour make a long tooth, fix a wire in the centre for the Style equally distant from the circle on each side. Let the Style hang so much beneath the plat, as it is above, because you shall have as much use on the backside as before.   dialing diagram 
Cut out all that which is within the inward circle, except a little portion to support the Style, so that this shall be but as a hoop. Draw the hour lines on the backside of this plat, as on the foreside, so that the one may be right against another: then draw lines on the inside from every one of these lines on the foreside to the lines on the backside. This plat or wheel must be movable and turn within the former.   dialing diagram 

The use of this Dial.  
PLace this Dial that each side may behold one quarter of the world precisely: the South side the South, the North side the North, etc. Moreover, it must recline according to the elevation of the Equinoctial, so that the Moon being in the Equinoctial circle, shall give light both above and beneath this Dial. You may place it perfectly with your instrument.  
Your Dial being thus placed, when you would know the hour of the night, first learn the age of the Moon by an almanac, and especially the hour of the change, then turn the great over-hanging tooth, to the day of the Moon on the first limb, and to the hour of the change, accounting from 12 of the clock.  
The shadow of the Style shall show the hour of the night either beneath your Dial or else above.  
This Dial will serve also for the Sun, if you turn the great tooth of the movable wheel to the little strike A. which is right opposite to the beginning of the first, and end of the last day of the Moon, near the 15. day, and then it is fit for the Sun. For it differeth nothing from the North reclining direct, where the Reclination is equal to the Elevation of the Pole.   

The use of the Table of the declination of the Sun.  
TAke what degree you will of any sign, and by this table you may know his declination from the Equinoctial circle. The Signs are written partly on the head of the Table, and partly on the foot of the same. The Degrees in the first column do serve for the Signs that be on the head of the Table, and the Degrees in the last column do serve for the Signs in the foot of the Table. And the common Area or angle against the Sign and the Degree which you seek for, doth contain the Degrees and Minutes of the Declination due to the same.   

Example.  
I would know how much the tenth degree of Leo doth decline from the Equinoctial: I must look in the column over Leo, right against the number of 10. in the last column, where I find 17.ᵈ. 46.ᵐ. the declination thereof, etc.  

The table of the declination of the Sun, from the Equinoctial circle.   Aries. Libra. Taurus. Scorpius. Gemin●. Sagittarius.  Deg. Deg. M. Deg. M. Deg. M. Deg. 1 0 24 11 30 20 12 29 2 0 48 12 11 20 35 28 3 1 12 12 32 20 47 27 4 1 36 12 52 20 58 26 5 1 59 13 12 21 9 25 6 2 23 13 32 21 20 24 7 2 47 13 52 21 30 23 8 3 11 14 12 21 40 22 9 3 34 14 31 21 49 21 10 3 58 14 50 21 58 20 11 4 21 15 9 22 7 19 12 4 45 15 27 22 15 18 13 5 8 15 55 22 23 17 14 5 32 16 3 22 30 16 15 5 55 16 21 22 37 15 16 6 18 16 39 22 44 14 17 6 41 16 56 22 50 13 18 7 4 17 13 22 55 12 19 7 27 17 29 23 1 11 20 7 50 17 46 23 5 10 21 8 12 18 2 23 10 9 22 8 35 18 17 23 13 8 23 8 57 18 33 23 17 7 24 9 19 18 48 23 20 6 25 9 41 19 2 23 22 5 26 10 3 19 17 23 24 4 27 10 25 19 31 23 26 3 28 10 47 19 44 23 28 2 29 11 8 19 58 23 29 1 30 11 29 20 12 23 30 0 Deg. Deg. M. Deg. M. Deg. M. Deg.  Virgo. Pisces.  Leo. Aquarius.  Cancer Capricorn.      

The Table of Sines. The whole Sine containeth 100000. parts.  

The use of the Table of Sines.  
WHereas the making of some Dial's, and this Table of Sines may seem obscure and hard to them who are not acquainted with Cynical computation, it shallbe expedient to declare the use hereof, so much as pertaineth to the understanding of this book, omitting all other uses as impertinent to our present purpose. Wherefore know that the Grades or Degrees are found in the upper head of this Table, and the Minutes pertaining to the Degrees on the left side, & in the Area or common meeting of them both, a number which is called the Sine, answerable to each Degree and Minute, offereth itself. Again the Sine being found out, you may easily know the ark, that is, the Degree and Minutes thereof, these being on the left hand, the other on the head or upper part. Understand by the Complement that which remaineth of any number being taken or substracted from 90. Degrees. If at any time you enter the Table with just Degrees without any Minutes, resolve one Degree into 60. Minutes, and then seek out his Sine. And whereas you shall find some numbers imperfect, you must remember to supply their want with those which be perfect immediately going before.  

Example.  
The Elevation of the Pole at Cambr. is 52. Degrees, whose Sine I desire to know, therefore resolving one of the Degrees into Minutes, I enter the Table with 51. degrees, 60. Minutes, and in the common Area I find the Sine to be 78801. If then you desire to know the complement of this Elevation, subtract 52. out of 90. and the remainder shallbe 38. the complement thereof, and entering the Table with 37.ᵈ. 60.ᵐ. you shall find in the Area 66. which number because it is imperfect, you must supply the want thereof by adding the 3. former figures in that which is next before perfect, to wit, 615. and then the whole number shallbe 61566. the Sine of 38.ᵈ. which is the Complement of 52.ᵈ. the Elevation of the Pole.  
These things well considered, there is nothing in the book so obscure, but it shall seem plain and easy.  
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.     
FINIS.  
Note that some of the letters in the Instrument page 2. are omitted in the printing: place A. at the uppermost corner towards the left hand, B. at the nethermost, C. by 90. beneath, and D. at 90. above.   

D 0 1 2 3 4 5 M Parts Parts Parts Parts Parts Parts  1 29 1774 3519 5262 7004 8744  2 58 1803 48 91 33 73  3 87 32 77 5320 62 8803  4 116 61 3606 49 91 31  5 45 90 35 78 7120 60  6 74 1919 64 5407 49 89  7 203 48 93 36 78 8918  8 32 77 3722 65 7207 47  9 61 2007 51 95 36 76  10 90 36 80 5524 65 9005  11 319 65 3809 53 94 34  12 49 94 38 82 7323 63  13 78 2123 67 5611 52 ●92  14 407 52 96 40 81 9121  15 36 81 3925 69 7410 50  16 65 2210 55 98 39 79  17 94 39 84 5727 68 9208  18 523 68 4013 56 97 37  19 52 97 42 85 7526 66  20 81 2326 71 5814 55 94  21 610 55 4100 43 84 9323  22 39 85 29 72 7613 52  23 69 2414 58 5901 42 81  24 98 43 87 30 71 9410  25 727 72 4216 59 7700 39  26 56 2501 45 88 29 68  27 85 30 74 6017 58 97  28 814 59 4303 46 87 9526  29 43 88 32 75 7816 55  30 73 2617 61 6104 45 84 D 0 1 2 3 4 5 M Parts Parts Parts Parts Parts Parts  31 901 2646 4391 6133 7874 9613  32 30 75 4420 62 7903 42  33 59 2704 49 91 32 71  34 89 34 78 6220 61 9700  35 1018 63 4507 50 90 29  36 47 92 36 79 8019 58  37 76 2821 65 6308 48 87  38 1105 50 94 37 77 9816  39 34 79 4622 66 8106 45  40 63 2908 52 95 35 74  41 92 37 81 6424 64 9903  42 1221 66 4710 53 93 31  43 50 95 39 82 8222 60  44 79 3034 68 6511 51 89  45 1208 53 97 40 80 10018  46 38 82 4826 69 8309 47  47 67 3112 55 98 38 76  48 96 41 84 6627 67 10105  49 1425 70 4914 56 96 34  50 54 99 43 85 8425 65  51 83 3228 72 6714 54 92  52 1512 57 5001 43 83 10221  53 41 86 30 72 8512 50  54 70 3315 59 6801 31 79  55 99 44 88 30 70 10308  56 1628. 73 5117 59 99 37  57 57 3402 46 88 8628 66  58 87 31 75 6917 57 94  59 1716 60 5204 46 86 10423  60 45 89 33 75 8715 52   

D 6 7 8 9 10 11 M Parts Parts Parts Parts Parts Parts  1 10481 12215 13946 15672 17393 19109  2 10510 44 74 15700 17422 38  3 39 73 14003 29 50 66  4 68 12302 32 58 97 95  5 97 31 61 87 17508 19223  6 10626 60 90 15815 36 52  7 55 89 14118 44 65 80  8 84 12417 47 73 93 19309  9 10713 46 76 15901 17622 37  10 42 75 14205 30 51 66  11 71 12504 34 59 79 94  12 99 33 62 88 17708 19423  13 10828 62 91 16016 37 51  14 57 91 14320 45 65 80  15 86 12619 49 74 94 19500  16 10915 48 78 16102 17822 37  17 44 77 14406 31 51 66  18 73 12706 35 60 80 94  19 11002 35 64 89 17908 19623  20 31 64 93 16217 37 51  21 60 93 14521 46 66 80  22 89 12821 50 75 94 19708  23 11119 50 79 16303 18023 37  24 46 79 14608 32 51 65  25 75 12908 37 61 80 95  26 11204 37 65 89 18109 19822  27 33 66 94 16418 37 51  28 62 94 14723 47 66 79  29 91 13023 52 76 94 19908  30 11320 52 80 16504 18223 36 D 6 7 8 9 10 11 M Parts Parts Parts Parts Parts Parts  31 11349 13081 14809 16533 18252 19965  32 78 13110 38 62 80 93  33 11407 39 67 90 18309 20022  34 35 67 96 16619 37 50  35 64 96 14924 48 66 79  36 93 13225 53 76 95 20107  37 11522 54 82 16705 18423 36  38 51 83 15011 34 52 64  39 80 13312 39 62 80 93  40 11609 40 68 91 18509 20221  41 38 69 97 16820 38 50  42 67 98 15126 48 66 78  43 95 13427 54 77 95 20307  44 11724 56 83 16906 18623 35  45 53 85 15212 34 52 64  46 82 13513 41 63 80 92  47 11811 42 69 92 18709 20421  48 40 71 98 17020 38 49  49 69 13600 15327 49 66 78  50 98 29 56 78 95 20506  51 11927 58 84 17106 18823 35  52 55 86 15413 35 52 63  53 84 13715 42 64 80 93  54 12013 44 71 92 18909 20620  55 42 73 99 17221 38 48  56 71 13802 15528 50 66 77  57 12100 30 57 78 95 20705  58 29 59 85 17307 19023 34  59 58 88 15614 36 52 62  60 86 13917 43 64 80 91   

D 12 13 14 15 16 17 M Parts Parts Parts Parts Parts Parts  1 20819 22523 24220 25910 27591 29264  2 48 51 48 38 27619 92  3 76 80 76 66 47 29320  4 20904 22608 24305 94 75 48  5 33 36 33 26022 27703 76  6 61 65 61 50 31 29404  7 90 93 89 78 59 31  8 21018 22721 24417 26106 87 59  9 47 50 46 34 27●15 87  10 75 78 74 62 43 29515  11 21104 22806 24502 90 71 43  12 32 35 30 26218 99 70  13 60 63 58 46 27927 98  14 89 91 87 75 54 29626  15 21217 22920 24615 26303 82 54  16 46 48 43 31 28010 81  17 74 76 71 59 38 29709  18 21303 23004 99 87 66 37  19 31 33 24728 26415 94 65  20 59 61 56 43 28122 93  21 88 89 84 71 50 29820  22 21416 13118 24812 99 78 48  23 45 46 40 26527 28206 76  24 73 74 69 55 34 29904  25 21501 23203 97 83 62 31  26 30 31 24925 26611 89 59  27 58 59 53 39 38317 87  28 87 87 81 67 47 30015  29 21615 23316 25009 95 73 42  30 43 44 38 26723 28401 70 D 12 13 14 15 16 17 M Parts Parts Parts Parts Parts Parts  31 21672 23372 25066 26751 28429 30098  32 21700 23401 94 79 57 30126  33 29 29 25122 26807 85 53  34 57 57 50 35 28513 81  35 85 85 78 63 40 30209  36 21814 23514 25206 91 68 36  37 42 42 35 26920 96 64  38 71 70 63 48 28624 92  39 99 99 94 76 52 30320  40 21927 23627 25319 27004 80 47  41 56 55 47 32 28708 75  42 84 83 75 60 36 30402  43 22013 23712 25403 88 63 31  44 41 40 32 27116 91 58  45 69 68 60 44 28819 86  46 98 96 88 72 47 30514  47 22126 23825 25516 27200 75 41  48 54 53 44 28 28903 69  49 83 81 72 56 31 97  50 22211 23909 25600 84 58 30624  51 39 38 28 27311 86 52  52 68 66 57 39 29014 80  53 96 94 85 67 42 30707  54 22325 24022 25713 95 70 35  55 53 51 41 27423 98 63  56 81 79 69 51 29125 91  57 22410 24107 97 79 53 30818  58 38 35 25825 27507 81 46  59 66 63 53 35 29209 74  60 95 92 81 63 37 30901   

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D 24 25 26 27 28 29 M Parts Parts Parts Parts Parts Parts 1 40700 42288 43863 45424 46972 48506 2 26 42314 89 50 98 31 3 53 40 43915 76 47024 57 4 79 67 41 45502 49 82 5 40806 93 67 28 75 48608 6 33 42419 93 54 47101 33 7 59 46 44020 80 26 58 8 86 72 46 45606 52 84 9 40912 98 72 32 78 48709 10 39 42525 98 58 47203 35 11 65 51 44124 83 29 60 12 92 77 50 45709 55 85 13 41018 42604 76 35 80 48811 14 45 30 44202 61 47306 36 15 71 56 28 87 31 62 16 98 83 54 45813 57 87 17 41124 42709 81 39 83 48912 18 51 35 44307 64 47408 38 19 77 62 33 90 34 63 20 41204 88 59 45916 60 88 21 30 42814 85 42 85 49014 22 57 40 44411 68 47511 39 23 83 67 37 94 36 65 24 41310 93 63 46019 62 90 25 36 42919 89 45 88 49115 26 63 46 44515 71 47613 41 27 89 72 41 97 39 66 28 41416 98 67 46123 64 91 29 42 43024 93 49 90 49217 30 69 51 44619 74 47715 42 D 24 25 26 27 28 29 M Parts Parts Parts Parts Parts Parts 31 41495 43077 44645 46200 47741 49267 32 41522 43103 71 26 67 92 33 48 29 97 52 92 49318 34 75 56 44723 78 47818 43 35 41601 82 49 46303 43 68 36 28 4●208 75 29 69 94 37 54 34 44801 55 49 49419 38 80 61 27 81 47920 44 39 41707 87 53 46406 45 70 40 33 43313 79 32 71 95 41 60 39 44905 58 96 49520 42 86 65 31 84 48022 45 43 41●13 92 57 46509 47 71 44 39 43418 83 35 73 96 45 65 44 45009 61 98 49621 46 92 70 35 87 48124 46 47 41918 96 61 46612 49 72 48 45 43523 87 38 75 97 49 71 49 45113 64 48200 49822 50 98 75 39 90 26 47 51 42024 43601 65 46715 51 73 52 50 27 91 41 77 98 53 77 54 45317 67 48302 49823 54 42103 80 43 92 28 48 55 29 43706 69 46818 53 73 56 56 32 95 44 79 99 57 82 58 45321 70 48404 49924 58 42209 84 47 95 30 49 59 35 43810 73 46921 55 74 60 61 37 99 47 80 50000   

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D 42 43 44 45 46 47 M Parts Parts Parts Parts Parts Parts 1 66934 68221 69486 70731 71954 73155 2 56 42 69507 51 74 75 3 77 63 28 72 94 94 4 99 84 49 92 72014 73214 5 67021 68306 70 70813 34 34 6 42 27 91 33 55 54 7 64 48 69612 54 75 74 8 85 70 33 75 95 93 9 67107 91 53 95 72115 73313 10 28 68412 74 70916 35 33 11 50 33 95 36 55 53 12 72 45 69716 57 75 72 13 93 75 37 77 96 92 14 67215 97 58 98 72216 73412 15 36 68518 79 71018 36 32 16 58 39 99 39 56 51 17 79 60 69820 59 76 71 18 67301 81 41 79 96 91 19 22 68603 62 71100 72316 73501 20 44 24 83 20 36 30 21 65 45 69903 41 56 50 22 87 66 24 61 77 70 23 67408 87 45 81 97 90 24 30 68708 66 71202 72417 73609 25 51 29 87 23 37 29 26 73 51 70007 43 57 49 27 94 72 28 63 77 68 28 67516 93 49 84 97 88 29 37 68814 70 71304 72517 73708 30 59 35 90 25 37 27 D 42 43 44 45 46 47 M Parts Parts Parts Parts Parts Parts 31 67580 68856 70111 71345 72557 73747 32 67601 77 32 65 77 67 33 23 98 53 86 97 86 34 44 68919 73 71406 72617 73806 35 66 40 94 26 37 25 36 87 61 70215 47 57 45 37 67709 83 36 67 77 65 38 30 69004 56 87 97 84 39 51 25 77 71508 72717 73904 40 73 46 98 28 37 23 41 94 67 70318 48 57 43 42 67815 88 39 69 77 63 43 37 69109 60 89 97 82 44 58 30 80 71609 72817 74002 45 80 51 70401 30 37 21 46 67901 72 22 50 57 41 47 22 93 42 70 76 60 48 44 69214 63 91 96 80 49 65 35 84 71711 72916 74100 50 86 56 70504 31 36 19 51 68008 77 25 51 56 39 52 29 98 45 72 76 58 53 50 69319 66 92 96 78 54 73 40 87 71812 73016 97 55 94 61 70607 32 36 74217 56 68114 82 28 53 55 36 57 35 69403 48 73 75 56 58 56 23 66 93 95 75 59 78 44 90 71913 73115 95 60 99 65 70710 33 35 74314   

D 48 49 50 51 52 53 M Parts Parts Parts Parts Parts Parts 1 74333 75490 76623 77732 78818 79881 2 53 75509 41 51 36 98 3 72 28 60 69 54 79916 4 92 47 79 87 72 33 5 74411 66 97 77806 90 51 6 31 85 76716 24 78908 68 7 50 75604 35 42 26 85 8 70 23 53 60 44 80003 9 89 42 72 79 61 20 10 74508 61 91 97 79 38 11 28 80 76809 77915 97 55 12 47 99 28 33 79015 73 13 66 75718 46 52 33 90 14 86 37 65 70 51 80107 15 74605 56 84 88 68 25 16 25 75 76902 78006 86 42 17 44 94 21 24 79104 60 18 63 75813 39 43 22 77 19 83 32 58 61 40 94 20 74702 51 77 79 57 80212 21 21 70 95 97 75 29 22 41 89 77014 78115 93 47 23 60 75908 32 33 79211 64 24 79 27 51 52 28 81 25 99 46 69 70 46 99 26 74818 64 88 88 64 80326 27 37 83 77106 78206 82 33 28 57 76002 25 24 99 51 29 76 21 43 42 79317 68 30 95 40 62 60 35 85 D 48 49 50 51 52 53 M Parts Parts Parts Parts Parts Parts 31 74914 76059 77180 78278 79353 80402 32 34 78 99 97 70 20 33 53 97 77217 78315 88 37 34 72 76116 36 33 79406 54 35 91 34 54 51 23 72 36 75011 53 73 69 41 89 37 30 72 91 87 59 80506 38 49 91 77310 78405 76 23 39 68 76210 28 23 94 41 40 88 29 47 41 79512 58 41 75107 48 65 59 29 75 42 26 66 84 77 47 92 43 45 85 77402 95 64 80610 44 64 76304 20 78513 82 27 45 83 23 39 31 79600 44 46 75203 42 57 49 17 61 47 22 60 76 67 35 78 48 41 79 94 85 52 96 49 60 98 77512 78603 70 80713 50 79 76417 31 21 88 30 51 99 35 49 39 79705 47 52 75318 44 67 57 23 64 53 37 73 86 75 40 81 54 56 92 77604 93 85 98 55 75 76510 22 78711 75 80816 56 94 29 41 29 93 33 57 75413 48 59 47 79811 50 58 32 67 77 65 28 67 59 51 85 96 83 46 84 60 70 76604 77714 78801 63 80901   

D 54 55 56 57 58 59 M Parts Parts Parts Parts Parts Parts 1 80918 81931 82920 83882 84820 85731 2 35 48 36 93 35 46 3 52 65 52 83914 51 61 4 70 81 68 30 66 76 5 87 98 85 46 81 91 6 81004 82015 83001 61 97 85806 7 21 31 17 77 84912 21 8 38 48 33 93 27 36 9 55 65 49 84009 43 51 10 72 81 66 25 58 66 11 89 98 82 40 73 81 12 81106 82114 98 56 89 96 13 23 31 83115 72 85004 85910 14 40 48 30 88 19 25 15 57 64 46 84103 35 40 16 74 81 63 19 50 55 17 91 97 79 35 65 70 18 81208 82214 95 51 81 85 19 25 30 83211 66 96 86000 20 42 47 27 82 85111 14 21 59 64 43 98 26 29 22 76 80 59 84213 42 44 23 93 97 76 29 57 59 24 81310 82313 92 45 72 74 25 27 30 83308 60 87 89 26 43 46 24 76 85203 86103 27 60 63 40 92 18 18 28 77 79 56 84370 33 33 29 94 95 72 23 48 48 30 81411 82412 88 39 64 62 D 54 55 56 57 58 59 M Parts Parts Parts Parts Parts Parts 31 81428 82429 83404 84354 85279 86177 32 45 45 20 70 94 92 33 62 62 36 86 85309 86207 34 79 78 52 84401 24 21 35 95 94 68 17 39 36 36 81512 82511 84 32 55 51 37 29 27 83500 48 70 66 38 46 44 16 63 85 80 39 63 60 32 79 85400 95 40 80 77 48 95 15 86310 41 96 93 64 84510 30 24 42 81613 82609 80 26 45 39 43 30 26 96 41 60 54 44 47 42 83612 57 76 68 45 64 58 28 72 91 83 46 80 75 44 88 85506 98 47 97 91 60 84603 21 86412 48 81714 82708 76 19 36 27 49 31 24 92 34 51 42 50 48 40 83708 50 66 56 51 64 57 24 65 81 71 52 81 73 40 81 96 85 53 98 89 55 96 85611 86500 54 81814 82806 71 84712 26 15 55 31 22 87 27 41 29 56 48 38 83803 43 56 44 57 65 54 19 58 71 58 58 81 71 35 73 86 73 59 98 87 51 89 85701 87 60 81915 82903 67 84804 16 86602   

D 60 61 62 63 64 65 M Parts Parts Parts Parts Parts Parts 1 86617 87476 88308 89113 89892 90643 2 31 90 22 27 89904 55 3 46 87504 35 40 17 67 4 60 18 88349 53 30 79 5 75 32 62 89166 43 92 6 89 87546 76 79 89955 90704 7 86404 60 90 92 68 16 8 18 74 88403 89206 81 28 9 33 88 17 19 93 41 10 47 87602 30 32 90006 90753 11 62 16 44 45 19 65 12 76 30 88458 89258 31 77 13 91 44 71 71 44 89 14 86805 87658 85 84 90057 90802 15 19 72 98 97 69 14 16 34 86 88512 89310 82 26 17 48 87700 25 24 95 38 18 63 14 39 37 90107 90850 19 77 28 52 50 20 62 20 91 42 88566 89363 32 75 21 86906 87756 79 76 45 87 22 20 70 93 89 90158 99 23 35 84 88606 89402 70 90911 24 49 98 20 15 83 23 25 63 87812 33 28 95 35 26 78 26 47 41 90208 47 27 92 40 88660 89454 20 90959 28 87006 53 74 67 33 71 29 21 67 87 80 46 84 30 35 87881 88701 89493 90258 96 D 60 61 62 63 64 65 M Parts Parts Parts Parts Parts Parts 31 87049 87895 88714 89506 90271 91008 32 64 87909 27 19 83 20 33 78 23 41 32 96 32 34 92 37 54 45 90308 44 35 87107 51 88768 89558 21 91056 36 21 87964 81 71 33 68 37 35 78 94 84 46 80 38 49 92 88808 97 90358 92 39 64 88006 21 89609 70 91104 40 78 20 35 22 83 16 41 92 33 48 35 95 28 42 87206 47 88861 48 90408 40 43 21 88061 75 89661 20 91162 44 35 75 88 74 33 64 45 49 89 88901 87 45 76 46 63 88102 15 89700 90457 88 47 78 16 28 12 70 91200 48 92 30 41 25 82 12 49 87306 44 88954 38 95 23 50 20 88157 68 89751 90507 35 51 34 71 81 64 19 91247 52 48 85 94 77 32 59 53 63 98 89008 89 44 71 54 77 88212 21 89802 90556 83 55 91 26 34 15 69 95 56 87405 40 47 28 81 91307 57 19 53 89060 41 93 19 58 33 88267 74 89853 90606 30 59 47 81 87 66 18 42 60 61 94 89100 79 30 91354   

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