GNOMONIQUES, OR The Art of Drawing SUN-DIALS On all sorts of PLANES By Different Methods.  
With the Geometrical Demonstrations of all the Operations.  
By Mr. DE LA HIRE of the Royal Academy of Sciences.  
Rendered into English and Illustrated by an Example in Numbers.  
By JOHN LEEK Professor of the Mathematics.  
LONDON, Printed for Rich. Northcott adjoining to St. Peter's Alley in Cornhill, and at the Mariner and Anchor upon Fish-street-hill near London-Bridge. 1685.   

THE PREFACE.  
I Have always considered the Description of Sun-dials' as one of the most ingenious and useful Inventions derived from the Study of the Mathematics. Also there is nothing▪ that draws more Admiration from all Men, than to see Strait-lines drawn on a Plane at Unequal Distances, to measure exactly the equal Divisions of the time of the Continuance of a Day: and although the Sun appears in different places of Heaven according to the different Seasons of the Year, yet the same Strait-lines do still determine the same hour at all these different Seasons.  
But these Hour-lines have nothing in them that deserves to be considered, if we compare them to those which trace forth the way which the Sun makes in his Distances from the Equinoctial Line, which questionless have given place to the more profound Meditations of the Sections of a Cone; which at this day is the most considerable part of our Speculative Geometry.  
I might easily demonstrate that we are beholding to Sun-dials' for the discovery of those admirable Curve Lines whereof we found very great use in all parts of the Mathematics; for we cannot consider the Shadow of the end of any body pointed on a Plane, without perceiving at the same time the Curvature which the way of the Sun traces forth; which is most like to that of the Section of an upright Cone which hath a Circle parallel to the Equinoctial for its Base, on which we may suppose the Sun moves than when he makes that Shadow.  
But although the Properties of these Curve Lines serves as a Foundation for the most part of the Descriptions of Dial's, yet in this Work I do not intent to explain any thing thereof in particular, for that would be to departed too much from my Subject.  
Seeing also that divers other knowing Geometers have largely treated thereof, it seems to me that it will be very useless to re-search into the time of the most ancient Antiquity, and who was the Inventor of this Art: We may only believe very likely that it was perfected by little and little, and that the first Men seeing the necessity which they had to divide the continuance of a day into divers parts, judged that it could not be better done than by the same Sun which limits the continuance thereof.  
We may also be very easily persuaded that the Meridian Line hath been the first which they have drawn, as well because that it divides the apparent day, which is the time during which we may see the Sun, into two equal parts, and that the Shadow of a Strait-line raised perpendicularly upon an Horizontal Plane, was always extended along this Meridian Line when the top of it meets with it, although it be in different places; so that it serves us to know the greatest height of the Sun above the Horizon every day, the which changes during the time of half a Year, it seemed to them one of the most considerable Phenomena's of the Sun.  
The most part of the Ancients divided the space of time (which is from the rising to the going down of the Sun) into twelve equal parts, which they called Hours, and they began their Account from Sun rising; and although the Hours changed their length during the half year to those which did not Inhabit under the Equinoctial Line, where the apparent days are always equal to one another, therefore they have always Midday at the Sixth Hour: but these sorts of Hours have no Circles of the Sphere that represent them.  
The Babylonians began the day at the Sun Rising, and divided the Duration into 24 equal Hours: The Italians began it at Sun Setting, and made also their Hours equal: These two manners of beginning the day have the Horizon for their term; the first did always know how long time they had from the Sun Rising above the Horizon, and the others did always know that which remains to the Sun Setting.  
The Astronomers and the greatest part of the Nations of Europe begin their Day when the Sun comes to the Meridian, these last when the Sun comes to the Meridian under the Horizon; and the first when the Sun comes to the Meridian above the Horizon; this manner of beginning the day has great advantages beyond other.  
I undertake in this Work only the Description of the Astronomick Hours, which have for their Term the Meridian, and of the Italians and Babylonians that begin from the Horizon. In the Construction of these sorts of Dial's there are two principal Operations which we may consider each in particular, which has obliged me to divide this Treatise into Two Parts.  
In the First Part, after I have explained as brief as I can, that which is necessary to be known for the right understanding of the Construction of Sun Dial's, in which I give the definition with that of all its parts: I propose following divers Manners and Practices, to draw their principal Line, with the Points which are necessary for the description of the Hour Lines, to the intent that we may serve ourselves with those which are most fit in the different Rencontres of the Planes proposed.  
Each of these Practices have advantages in particular Cases; I observe by which we may serve ourselves very near following the Exposition of the Plane proposed, to the intent that those that have not sufficient knowledge of the different Rencontres of Planes with the Circles of the Sphere, may not give themselves the trouble to follow a Method from which they cannot draw a great advantage.  
In all these Practices which I propose, I make no use of the Magnetical Needle, for there happens great change to the Variation of the Magnetical Needle; besides we are not assured that there is no Iron hid, or some Stone or Brick which is of the Nature of Iron, which may turn aside the Needle from its true Direction towards the place whither it would go if it were free.  
Also I approve not that Method which many do practice to found the Declination of a Plane, that is to say, the Angle which the Meridian Line makes on a Horizontal Plane with the Horizontal Line, which is the meeting of the Plane of the Dial with that Plane: they draw on a Plane set Parallel to the Horizon (the which we call a Level Plane) a Meridian Line following one of the Practices, which may be seen hereafter; and when the Sun marks Midday on that Horizontal Plane, they mark a Point of Shadow on the proposed Plane; but you must observe that the Errors that one commits in all the Operations, as well in the placing of the Plane levelly, as in the determination of the Meridian, are multiplied and increased in transporting them to another Plane.  
For the same Reasons we aught also to reject all sorts of Instruments, unless they be very plain and very large: Therefore I have thought good to use only the Ruler and Compass, the Plomb Line and the Level, and to draw the Lines and Circles only upon the given Plane.  
Although we may determine the length of the Lines, and also the most part of the Angles by Calculation of Spherical and Strait-lined Triangles, which serves for the drawing of Dial's, Yet I have thought good that I aught not to propose any (of those ways) in this small Treatise, because that the most part of those Calculations are much longer than the Practice, and they are founded but upon the same Angles, and the same Lines which I have used in the Practice; so that those that can use Calculation, shall found no great difficulty to apply them to Numbers and Sins, where I only propose Lines and Angles.  
After I have taught to Draw the Principal Line of Dial's, I proceed to the Second Part, wherein I show to draw the Astronomical Hour Lines, and than to describe the Parallels of the Signs.  
I propound no particular Construction on the Horizontal and Vertical Planes, which only gives particular Rules for each Case, the which in the ordinary way happens very seldom: Therefore the Methods that I teach are for all sorts of Planes indifferently considered. I know well that there are divers Cases where we might found Abridgements, but these Abridgements consists only in certain Lines and Points which come to be united in the general Practices which I give: Also it is to be observed that the Portions of Curve Lines that I describe, are always Conic Sections, that is to say, either Ellipses, Hyperboles, or Paraboles, and sometimes Circles, when the Plane is perpendicular to the Axis of the Cone, the which is always an upright Cone.  
The most part of the Practice that I teach being founded on the Declination of the Sun, I give you a Table thereof Calculated for the days of Four Years following one another, to the end to comprehend the Bissextile or Leap Year; I there also add the Differences of the Declinations for every day, with a Table of Latitudes of the Principal Cities of the World.   

THE CONTENTS.  

The Preface.  
CHap. I Fig. 1. Of the Circles of the Sphere necessary to be known for the drawing of Sun Dial's.  
Chap. II. Of the definition of Sun Dial's, and the principal parts which serves for their Construction.  
Chap. III. Fig. 2. To mark the Points of Shadow.  
Chap. IU. To draw the Horizontal Line.  
Chap. V Fig. 4. To Draw the Substylar Line, Two Points of Shadow being given in a certain Condition. 

To place the Centre and draw the Equinoctial Line, the Declination of the Sun and one Tract of the Shadow being given.    
Chap. VI Fig. 5. To place the Substylar Line, the Equinoctial Line, and the Centre of the Dial, and to determine the Position of the Axis. 

Two Points of Shadow, what you will being given, with the Declination of the Sun at the time of Observation of the Points of Shadow.    
Chap. VII. Fig. 6. To place the Substylar Line, the Centre of the Dial and Equinoctial Line, one only Point of Shadow being given, with the Declination of the Sun and the Altitude of the Pole above the Horizon.  
Chap. VIII. Fig. 7, 8. To found the Centre of the Dial, the Substylar Line and the Equinoctial Line, one only Point of Shadow being given, and the shortest Shadow.  
Chap. IX. Fig. 9 To found the Centre of the Dial, and to draw the Substyle and Equinoctial. 

Two Points of Shadow being given, with the Declination of the Sun at the time when you marked the Points of Shadow.    
Chap. X. Fig. 10, 11. To found the Centre of the Dial, and to draw the Substylar and Equinoctial Lines. Two Points of Shadow, what you please being given, with the Declination of the Sun at the time of taking the Points of Shadow.  
Chap. XI. Fig. 12. To found the Centre of the Dial, and draw the Equinoctial Line, the Substylar being drawn, and one Point of Shadow being given, with the Declination of the Sun. 

And to draw the Equinoctial Line, the Centre of the Dial being placed.  
And to found the Centre of the Dial, the Equinoctial being drawn.    
Chap. XII. Fig. 13. To draw the Equinoctial and Substylar Lines, and to found the Centre of the Dial. 

Two Points of Shadow, what you please being given, with the Declination of the Sun.    
Chap. XIII. Fig. 14. To found the Points of the Hours of Six and of Midday upon the Equinoctial, and to draw the Meridian Line. 

The Equinoctial and Horizontal Lines being placed.    
Chap. XIV. Fig. 15. To draw the Meridian, and to found the Point of the Line of Six Hours on the Horizontal Line. 

One only Point of Shadow being given, the height of the Pole and the Declination of the Sun.    
Chap. XV. Fig. 16. To draw the Meridian Line, Two Points of Shadow being given, in a certain condition.  
Chap. XVI. Fig. 17. To place the Centre of the Dial, or to determine the Inclination of the Axis with the Meridian, to draw the Substyler and Equinoctial, the Meridian being posited, and the Altitude of the Pole being given.  
Chap. XVII. Fig. 18. Remarks and Practices for many Ahridgements in the Operations of the foregoing Chapters.   

The Second Part.  
The Preface. Of the choice we aught to make of the Operations to draw the Substylar, Equinoctial and Meridian Lines; and to place the Centre of the Dial, following the Expositions of the proposed Superficies.  
Chap. I Fig. 21. To Mark the Points of the Astronomick or French Hours on the Equinoctial Line, and by those Points to draw the Hour Lines  
Chap. II. Fig. 22. To Mark upon the Horizonta● Line the Points of the Astronomick and French Hours. 

And to draw the Hour Lines by those Points.    
Chap. III. Fig. 23. Six Intervals of Hours following one another being given, to araw all the other Hours.  
Chap. IU. Fig. 24. To draw the Arches of the Signs  
Chap. V Fig. 25. The Equinoctial being given, t● draw a Parallel to it by a Point given upon an● Hour Line.  
Chap. VI Fig. 26. To draw the Italian and Babylonian Hours upon an Horizontal Surface.  
Chap. VII. Fig. 28, 29. To draw the Italian and Babylonian Hours on a Surface, which is not Horizontal.  
Chap. VIII. Fig. 28. To continued the Description of the Italian and Babylonian Hours, when the Parallel or Equinoctial is not on the Surface.  
Chap. IX. Fig. 30, 31. Four Astronomick Hour-Lines following one another being given, with the Equinoctial Line. 

To found the other Hours.    
Chap. X. Fig. 32. A Drawn Dial being given, to found the foot of the Style which did serve to draw the Dial, and to determine the length thereof.  
Chap. XI. Fig. 33, 34. To place the Axis.  
Chap. XII. To Draw Reflecting Dial's.  
Chap. XIII. Of the Use of the Table of the Sun's Declination, and of the Difference of Meridian's of divers considerable Towns in respect of Paris.    

GNOMONIQVES, OR The ART of DRAWING SUN-DIALS On all Sorts of PLANES.  

THE FIRST PART.  

CHAP. I  
Of the Circles of the Sphere necessary to be known for the Drawing of SUN-DIALS.  
THE Sphere is an Instrument whereby we explain the Daily Motion of the Celestial Bodies, according as they appear to us to move always from East to West, and also the proper Motion of the Sun, which moves from West toward the East, and makes his Revolution thro' the Twelve Celestial Signs in the space of one year.  
This Instrument is composed of divers Circles, of which we only describe those that belong to our present Subject: Those Circles whose Planes pass thro' the Centre of the Earth, are called Great Circles of the Sphere, and all the other are Lesle.  
But before we speak of these Circles, we aught to consider the Axis of the Sphere, which we conceive to be as a straight line about which the Instrument is turned. The two Ends of the Axis are called Poles, the one North and the other South.  
The Earth is placed in the middle of this Instrument, and consequently the Axis passes thro' the Centre thereof. We may understand from Astronomical Observations that the Globe of the Earth is so little, in respect to its distance from the Sun, so that we may consider it as a Point, if we compare it to that distance.  
The Equinoctial or Equator is a Great Circle, and one of the chiefest of the Sphere, the Plane whereof is at Right-angles to the Axis, it divides the Sphere into two equal parts, whereof one is called Septentrional, and the other Meridional.  
Fi● 1. The Ecliptic is another Great Circle whose Plane makes an angle with the Plane of the Equinoctial of 23 degrees 30 minutes; the Sun moves under this Circle, going from the; West toward the East, and makes one entire Revolution in 365 days and near 6 hours. The Inclination of this Circle to the Equinoctial, causes the different Declinations of the Sun in regard to the Equinoctial; it is divided into Twelve equal parts, which are called Signs: And we begin from the Intersection thereof with the Equinoctial, proceeding towards the North. The Names of the Twelve Signs are Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces. Their Characters are ♈, ♉, ♊, ♋, ♌, ♍ ♎, ♏, ♐, ♑, ♒, ♓.  
The Tropics are two Circles parallel to the Equinoctial, which touches the Ecliptic in the point of its greatest distance from the Equinoctial; therefore these Circles are distant from the Equinoctial 23 deg. 30 min. on one side toward the North, and on the other side toward the South.  
So that it is manifest, That when the Sun is in the common Intersection of the Ecliptic and Equator, the Motion of the Sphere about its Axis, which goes from East to West, and is called the Motion of the Primum Mobile, makes him appear to us in the Equinoctial; and also when he is in his greatest distance from the Equinoctial, the same Motion of the Primum Mobile makes him to appear to us to move in the Tropics.  
The Zenith is an imaginary Point in the Sphere, marked by a straight Line coming from the Centre of the Earth, and passing by some place of the Superficies thereof. This Line is called the Vertical Line of that place.  
The Horizon is a Great Circle, whose Plane cuts the Vertical Line at Right-angles. The Horizon of a Place distinguishes the visible part of Heaven of that Place, from that part of Heaven which is not there seen.  
The Meridian is a Great Circle which passes thro' the Poles and Zenith, the Plane whereof is at Right-angles with the Planes of the Equinoctial and Horizon. When the Sun comes to this Circle, he is in the middle of his apparent Course during a day, and is at his greatest height above the Horizon, because this Circle passes thro' the Zenith and Poles.  
If we suppose the Equinoctial to be divided into 24 equal parts, beginning from the Meridian, the 6th and 18th part shall fall on the Intersections of the Horizon and Equinoctial, because the Meridian and Horizon are at Right-angles to one another; and if we imagine other Circles like the Meridian, that is to say, that pass thro' the Poles of the World and Point of Division of the Equinoctial, those Circles which we call Meridian's, shall be the Hour Circles, among which is the Meridian of the Place, whereof all the Planes intersect one another in the Axis. We may also conceive others, which divide each part into two or four, to mark the half-hours and quarter-hours; for if we suppose these Circles to be fixed, than when the Primum Mobile turns the Sun with his Ecliptic about the Axis, the time of his apparent Course shall be divided into hours, halves, and quarters, by these Meridian's.  
Also we number the Declination of the Sun upon the like Meridian's, which do all intersect the Equinoctial at Right Angles, which we make to pass thro' the Centre of the Sun in the Ecliptic: We number this Declination from the Equinoctial towards the Poles: Therefore it is either South or North: The Angles of Declination are measured by Arches of Circles.  
Those Circles that pass by the Vertical Line are called Vertical Circles (or Azimuth) and their Planes are perpendicular to the Plane of the Horizon; they serve to measure the height of the Sun above the Horizon, which is numbered from the Horizon toward the Zenith.  
It is manifest from that which has been said before, that there are infinite Orisons and Meridian's, and that there are only these two great Circles, which may change according to the different places on the Earth, for they are established by the Vertical Line.  
The Amplitude of Rising or Setting is counted on the Horizon, beginning from the Points where the Equinoctial cuts the Horizon, and is numbered toward the South or North.  
If we conceive that in the Revolution of one day the Horizon moves, as being fastened to the Axis, so as it cannot change its Inclination, than when it shall pass by the 24 equal Divisions of the Equinoctial, it shall represent the 24 Circles of the Italian or Babylonian Hours.   

CHAP. II.  
Of the Definition of Sun-dials', and of the principal parts which serves for their Construction.  
THe Distance from the Centre of the Earth to the Superficies thereof not being considerable, in respect of the Distance of the Earth from the Sun, we may take any Point on the Superficies and consider it as its Centre in relation to the Motion of the Sun.  
Therefore if we place a Style which is a Pointed Rod upon any Plain Surface, and than consider the Point of that Style as the Centre of the Earth, the Intersections of that Surface with the Planes of the Hour Circles, of the Equinoctial or Equator, of the Horizon and of the other great Cricles, shall be straight Lines, which retain the Names of the Planes of the Circles from whence they were produced: All these Lines on that Plane Surface with the Style makes a Sundial.  
The Shadow of the Point of the Style, which is one of the Points of the Axis shows the Hours.  
1st Figure  
It is also evident that the Shadow of the top of the Style gives the Hours, and shows when the Sun meets with any one of the Circles of the Sphere; for when the Sun comes to a great Circle, the Shadow of the Axis is extended in the Plane of that Circle, if that Circle passes by the Axis; and if it passes not by the Axis, the shadow of the Point of the Style shall be in the Plane of that Circle; for the Planes of great Circles pass by the Point of the Style.  
And if we conceive a Conical Superficies which has for its Base a lesle Circle of the Sphere, and for its Vertex the Point of the Style, that Conical Superficies shall meet the Surface of the Dial in a Curve Line; so as when the Centre of the Sun shall touch that lesle Circle which is the Base of the Conical Superficies, the Shadow of the Point of the Style shall touch the Curve Line which is the meeting of that Curve Superficies with the Plane of the Dial: for the Point of the Style is on that Superficies whereof it is the Vertex.  
The Foot of the Style is that Point on the Plane of the Dial, which is the meeting of a straight Line drawn perpendicularly to that Plane, and which passes by the Point of the Style.  
If the Plane of the Dial be considered as the Plane of the Horizon of any place, the straight Line that passes by the Point of the Style and by its Foot, shall be the Vertical Line of that Place: and the Plane that passes by that Vertical and by the Axis, shall be the proper Meridian of that Place, considered as the Horizontal of a Place.  
The meeting of the Meridian and Surface of the Dial is called the Substylar Line, or the Meridian of the Plane or Surface of the Dial, which we aught to distinguish from the Meridian of the Place, which is the meeting of the Meridian proper to that Place and of the Surface of the Dial, at lest if they be not coincident, which happens when the Dial does not decline from the East or West.  
We see by the position of these Lines that the Substylar Line is always at Right Angles with the Equinoctial Line.  
We aught to make the Dial so as the Foot of the Style be not encumbered, for that Point serves for many Operations; therefore the Style must be planted a little obliquely upon the Surface.  
I understand by the length or height of the Style the straight Line drawn from the Point to the foot thereof.  
The Arches of the Signs on the Surface of the Dial, are the Descriptions of the Parallels to the Equinoctial, which pass thro' the 12 equal Divisions of the Ecliptic Line, which show the beginning of the Signs.  
2d. Fig  
3d Fig   

CHAP. III.  
To Mark the Points of Shadow.  
Fig. 2. THe Practices which are taught in this Treatise, being founded on the Points of Shadow of the Point of the Style to be very small, it is very necessary to mark them very exactly; but it is very difficult because of the Penumbra: See here two ways by which it may be done.  
The first is to fit a small round Plate to the Point of the Style, which may be parallel to the Plane of the Dial, whereof the Centre may be joined to the Point of the Style; than having drawn the Shadow of the said Plate on the Plane of the Dial, take the middle of that Shadow, which shall be the Shadow of the end of the Style at the same time when we observed the Shadow of the Plate.  
The Second way is to make a small round hole in a little piece of Pasteboard or thin Plate, or other like body, and having applied it to the end of the Style, so as the Centre of the Hole may be joined to the Point of the Style, and that the small Plate may regard the Sun perpendicularly; the light of the Sun shining thro' the Hole shall mark a clear Circle or Oval D E in th● Shadow of the Plate on the Plane of the Dial which we draw on the said Plane; and if it be an Oval, having drawn a straight Line DPE from the Poin● P, which is the foot of the Style, whereof S i● the Point, which may pass thro' the Centre of that Oval and cut it in D and E, or draw D G and E F parallel to one another, and making any Angle with D E, D G being made equal to D S, and E F equal to E S, the Line G F shall cut D E in the Point A, which shall be the Shadow of the Point of the Style S, at that time when the Oval was drawn: But we may take the Centre of the Oval for the Point A, without falling into any sensible Error, as we may see by this Operation.  
But if the Shadow be a Circle, the Centre of that Circle shall be the Shadow of the Point of the Style.  
Here Note, That we aught to make the Hole as small as is possible, for so the Operation shall be more exact. It is sufficient that we may see distinctly the Figure of the Light in the Shadow of the small Plate.   

CHAP. IU.  
To Draw the Horizontal Line.  
A Style the Point whereof is S, being planted upon the Plane of the Dial, we apply a Rule A S, so as one of the Edges thereof A S may be levelly, touching the Point of the Style, and that the end of that Edge of the Ruler may touch the Plane of the Dial at the Point A, which shall be one of the Points of the Horizontal Line.  
By the Point A draw a levelly Line on the Plane of the Dial, which shall be the Horizontal Line.  
There are so many ways to set a Ruler level, that I will prescribe none in particular.   

CHAP. V  
To found the Substylar Line, two Points of Shadow being given in a certain condition. To found the Centre and draw the Equinoctial Line, the Declination of the Sun, and one Tract of Shadow being given.  
Fig. 4. LEt there be a Style, the Point whereof i● S, and P the Foot, having marked the Point A, which let be the Shadow of the Point S, on the Plane of the Dial; on the Point P as a Centre, and at the Distance P A, describe the Circle A B, and when the Shadow of the Point S comes again to the Circle A B on the same day, at the Point B we mark that Point. Than draw the Line A B, and divide it into two equal parts in D, the straight Line P D, shall be the Substylar Line, which aught to be at Right Angles to the Line A B. We may take divers Points, as A, for to found divers Points as B, and if the Operation be made exact, all the Points as D, ought to be in one and the same straight Line with the Point P.  
If the Sun's Declination has changed considerably in the time which has passed between the two Observations of the Points of Shadow A and B; which may fall out when the Sun is near the Equinoctial Points, or when the Style P S is 
4th. Fig  very high; or lastly, when there has passed much time between the Observations, we shall not have the Substylar Line exactly.  

The Demonstration of the former Practice.  
Supposing the Declination of the Sun not to be changed between the two Observations of the Points of Shadow A and B; by construction the two Triangles A P S, B P S, are equal and alike; therefore the Angle P S A is equal to the Angle PSB; the Sun therefore was equally elevated above the Plane of the Dial, than when the two Points of Shadow were observed. Therefore the Sun was equally distant from the Meridian of that Plane at the times of the two Observations: Therefore the Line P D, which divides the Line or Arch A B into two equal parts, shall be the Meridian of the Plane.   

To found the Centre of the Dial, and to draw the Equinoctial Line, knowing the place where the Shadow of the Point of the Style cuts the Substylar Line.  
If between the Points of Shadow A and B, we mark a Succession of Points of Shadow, so that we may have the Point L where the Shadow of the Point of the Style meets with the Substylar Line between the Observations, having erected P F perpendicular to the Substylar Line, and equal to the height of the Style P S, let the straight Line F L be drawn, and having made the Angle L F E equal to the Declination of the Sun at that time when the Point of Shadow L did me with the Substylar Line, so as the Point E, or the meeting of the straight Line F E with the Substylar Line, may be always towards the Convex part of the Curvature of the Line A L B, that Point E shall be the Point where the Equinoctial Line E G cuts the Substylar Line; which Lines shall intersect one another at Right Angles.  
Than having drawn F C perpendicular to F E, the straight Line F C determines the position of the Axis with the Substylar Line; and if it meets with it in the Point C, that Point shall be the Centre of the Dial, and the Line F C shall be the Angle of Inclination of the Axis with the Substylar Line, which serves to place the Axis, and to found the other necessary Points for the Constructton of the Dial.  
It is not necessary that the Point of Shadow L should be taken on the same day when we observed the other Points A and B, it is sufficient that we have the Declination of the Sun than when we make Observation of the Point L, and on which side the convexity of the Tract of the Shadow shall be on that day, to found the Point E.   

Demonstration.  
The Demonstration of this Operation is manifest, if we consider that we have made the Angle L F E, and that the Line F C aught to be extended in the Plane of the Meridian, which is perpendicular to the Plane of the Dial, and that the meeting of the Line D P C and the Point F, aught to be conjoined in the Point of the Style S.   

Another way of finding the Substylar Line by the Amplitude of the Suns Rising and Setting upon the Plane of the Dial.  
When the Sun gins to rise on the Plane of the Dial, mark the Shadow of a Small Thread extended from the Foot of the Style to its Point, and do also the same when the Sun sets on the Plane of the Dial, the Angle comprehended between these two Lines of Shadow, whose Vertex is at the Foot of the Style, being divided into two equal parts, shall give the Substylar Line.  
This is manifest, for that Angle is the Sum of the Ortive and Occasive Amplitudes of the same day, which we suppose to be equal.  
The Substylar Line being placed, we may found the Centre of the Dial by the Practice of the 11th Chapter, using only one Point of Shadow, and knowing the Declination of the Sun at the Hour where the Point of Shadow has been marked; and if the Dial have no Centre, we may have the Inclination of the Axis with the Substylar Line; which shall serve instead of the Centre for the placing of Hours: Also by the same Practice we may have the position of the Equinoctial Line.    

CHAP. VI  
To place the Substylar and Equinoctial Lines, and the Centre of the Dial, and to determine the position of the Axis. Any Two Points of Shadow being given, with the Declination of the Sun at the time of Observation o● the Points of Shadow.  
Fig. 5. A Style being placed on the Plane of the Dial, whereof the Point may be S and P the Foot, and any two Points of Shadow A and B taken at pleasure.  
Upon some certain Plane having made the Angle d s a equal to the Sum or Difference of a Right Angle, and that of the Declination of the Sun on that day on the which the Points of Shadow were marked, according as the Declination is North or South; for you would have a Point o● the Substylar Line as Q which may answer to 〈◊〉 Point of the Axis, which may be more North than the Point of the Style, you must make the Angle d s a equal to the Sum of a Right Angle and Angle of the Declination of the Sun, if the Declination be North, but equal to the Difference of a Right Angle and Angle of the Declination if it be South.  
5th Fig  
Take s d of any length, and make s a and s b equal to the Intervals S A, S B, from the Point of the Style to the Points of the Shadow; from the Point A as a Centre at the distance a d, describe Two Arches of Circles at T and L, and from the Point B, at the distance of the Line b d, intersect the former Arches at the Points L and T, than draw the straight Lines L T and A B, which shall intersect one another at Right Angles in the Point O; and from the Point O as a Centre, at the distance L O or O T, which are equal, describe the Semicircle L D T.  
Than draw the straight Line P G K parallel to A B, intersecting L T in the Point G, and make G K equal to P S the height of the Style: and from the Point P, as a Centre, at the distance of the Line s d, describe the Arch I, cutting the straight Line L T in the Point I  
And from the Point K as a Centre, and at the distance G I, describe the Arch D R, cutting the Semicircle L T in the Point D: Than to the Line L T, let fall the perpendicular D Q and draw the straight Line Q P, which shall be the Substylar Line.  
If the Point Q falls too near to the Point P, we may take s d greater, and begin the work again to determine the Substylar Line more exactly.  
If the Declination of the Sun be considerably changed between the Observations of the Points of Shadow, be it that the Observations be made on the same day or on different days, the Angles P s a, d s b must be made according to the different Declinations at the times of Observation, a●● the Angle d s a being made as has been taught before for the time of the Observation of the Po●●● of Shadow A, make s a equal to S A; also t●● Angle d s b being made for the Observation of t●● Point of Shadow B, make s b equal to S B.  
Than from the Points P and Q of the Substylar Line raise the Perpendiculars P N equal to t●● height of the Style P S, and Q M equal to Q D than if the straight Line M N being drawn me with the Substylar Line in C, the Point C shall 〈◊〉 the Centre of the Dial, and the Line N E perpendicular to N M, meeting with the Substyle Line; the Line V E being drawn perpendicular 〈◊〉 the Substylar Line passing thro' the Point E shall 〈◊〉 the Equinoctial Line of the Dial.  
We may observe that the Point D ought t● be on the side with the Semicircle T L G, whi●● is cut by the Line P G K, and which answers 〈◊〉 the Point M of the Axis, which is supposed 〈◊〉 be more toward the North than the Point S; th●● Point D may indifferently meet with the Circle 〈◊〉 either side of the straight Line T L.  
Also, We may observe that if the straight Li●● L T passes by the Point P, that Line shall be th● Substylar Line, and it shall always pass by th● Point P, if the Points of Shadow A and B a● Points of the Equinoctial Line.  

Demonstration.  
It is clear by this Construction that the Line s d presents the Axis, and s a and s b the Lines of the ●adow from the Point of the Style, which make with the Axis s d an Angle equal to the Sum or Difference of a Right Angle and Angle of the Declination of the Sun, according to that which hath been prescribed in the Practice: Therefore if we conceive that the two Triangles a s d, b s d, each apart have their Points a and b in the Points of Shadow A and B, and their Points s joined together with the Point S of the Style, if the Points d of each of these Trianglec are also joined together, these two Triangles in this Position compose a Pyramid A B S d whereof the Line S d is the Axis of the Dial.  
But to found on the Plane of the Dial a Point which may answer perpendicularly to the Point d, as P answers to the Point S, we must consider in the Pyramid the Triangle A B D ●o move upon the Line A B, so as it is manifest that the Point d describes a Circle such as is T L D, on a Plane which is perpendicular to the Plane of the Dial; and the Intersection of those two Planes is the straight Line T L, which aught to cut the Line A B, which conjoins the Points of Shadow A and B in the Point O, and O T, O L shall be equal: There remains nothing more but to mark the Point d or D upon the Circle L D T couched on the Plane of the Dial.  
The Plane of the Circle L D T being perpendicular to the Plane of the Dial, if we conceive tha● the straight Line s d moveth upon its extreme, s b●ing immovable, and put upon the Point of the Sty● S, and that the other extreme thereof d, may 〈◊〉 always in its moving upon the Plane of the Circ●● L D T, it is evident that that extreme d shall describe a Circle R D upon that Plane which sha●● have for its Centre the Point K, where the Line S 〈◊〉 drawn from the Point of the Style S perpendicular● to the Plane L D T, meets with the same Plane, th●● which Line S K shall be equal to P G: but the S● midiameter of the Circle R D shall be equal to G● if P I be made equal to s d; but the intersection 〈◊〉 the Circle L D T with the Circle R D, which is th● Point D, determines the position of the Point d of th● Axis. So as D Q considered as perpendicular to th● Plane of the Dial, coming from the Point d or D does there mark the Point Q, which is one of th● Points of the Substylar Line, seeing that the Poin● d is one of the Points of the Axis.  
It is also evident that the Plane N M C P bein● supposed to be perpendicular to the Plane of the Dia●● the Line N M does there represent the Axis in i●● position, the Point C the Centre of the Dial, a●● consequently the Line V E ♈ shall be the Equi●noctial.   

Another Practice upon the same Positions and Constructions.  
The same Preparations being made as before, ●●e two small Rods of any firm Matter, as of Wood of a sufficient thickness, or of Iron: and ●ake them pointed at the ends, and equal in length 〈◊〉 the straight Lines a d, b d; it is not material ●hether they be straight or crooked, if the Di●ances between their Points be equal to a d and 〈◊〉 d.  
Put one of the Points of that Rod which is equal ●o a d on the Point of Shadow A, and one of the Points ●f the other Rod to the Point of Shadow B, and ●oyn them together by their other Points; but so as ●he Points that are joined together may approach ●r fall back from the Point of the Style without altering the other Points of the Rods, which are ●et on the Points of Shadow A and B; than we ●ake with the Compasses or otherwise, the distance between the Points a and d, and set that distance between the Point of the Style and the ●oints of the Rods that are joined together, so ●s that Point, it may be more toward the North ●han the Point of the Style: by this means the ●ommon Point of the Rods being fixed, shall be one ●f the Points of the Axis which aught to pass ●y the Point of the Style, therefore the situation ●f the Axis shall be determined.  
By this common Point of the Rods so fixed, whic● I call D, having drawn a Line perpendicular 〈◊〉 the Plane of the Dial, which shall meet it in th● Point Q, the Line P Q shall be the Substyle Line.  
The Point C on the Plane of the Dial whe●● it is met by the Line D S, drawn by the Point 〈◊〉 the Style S, and by the end of the Rod D, sh●● be the Centre of the Dial, and by the Pract●●● of the 11th Chapter we may draw the Equinoctial Line: but if we have not the Centre, we m●● draw it by the Practice of the same Chapter.  
This Practice may serve to make you understa●●● more easily the Demonstration of the Construct●●● which is proposed in that Chapter.  
6st Fig    

CHAP. VII.  
To place the Substylar Line, the Centre of the Dial and the Equinoctial Line. One only Point of Shadow being given, with the Declination of the Sun and height of the Pole above the Horizon.  
Fig. 6. HAving placed a Style upon the Plane of the Dial, whose Point may be S, and P the foot, and A one Point of Shadow, draw a Horizontal Line by the Practice of the 4th Chapter. And by the Point P draw the Lines B P H perpendicular to the Horizontal Line h H, and P Z parallel to H h and equal to the Height of the Style PS. Than from the Point H, where P H meets with the Horizontal Line, draw H Z and Z B perpendicular to Z H, which shall meet with H P at the Point B if the Horizontal Line passes not thro' the Point P: First let it meet at the Point B.  
Upon some Plane make the Angle d s a equal to the Sum or Difference of a Right Angle, and of the Declination of the Sun at the time when the Point of Shadow was observed, according to the Precepts which have been given in the 6th Chapter, and make the Angle d s b equal to the Sum of a Right Angle and the height of the Polls above the Horizon.  
Than taking the Point d at pleasure on the Line s d, make s b equal to Z B and s a equal to the length of the Shadow from the Point of the Style S to the Point of Shadow A, and draw the straight Lines a d, b d.  
And by the Points A and B draw the straight Lin● A B, and from the Point B as a Centre, at the distance b d describe the Arch f L either above obelow the Line A B; and likewise from th● Point A as a Centre, and at the distance a d, describe the Arch g d, cutting the Arch L f at th● Point L, and from the Point L draw the stra●● Line O L perpendicular to A B.  
From the Point O as a Centre, at the distan●● O L describe the Arch D L.  
And from the Point P draw the straight Lin P G K perpendicular to O L; and from the sam● Point P, at the distance d s, describe the Arch I e●ther on the one or the other side of G, cutting th● Line L O at the Point I  
Than make G K equal to P S the height o● the Style, and from the Point K, at the distanc● G I describe the Arch R D, cutting the Arch D L in D, and from the Point D draw the straight Lin● D Q perpendicular to L O, and the Line P Q which passes thro' the Points P and Q shall be the Substylar-Line.  
It the Point Q be too near to P, we may found another, by taking another Point d on the Line s d, as has been said in the Practice of the Precedent Chapter; also we must have a regard to the other Observations which have been made upon the same Practice, by that which is there founded on the same Principles.  
Consequently we place the Equinoctial Line and the Centre of the Dial by the Practice of the 11th Chapter: but we have also here this advantage, that the Line which passes thro' the Point B and thro' the Centre of the Dial, shall be the Meridian Line.  
In the second place, if the Horizontal Line passes thro' the Point P, or if the Point B be too far distant from the Point P, we must fasten another Style upon the Plane of the Dial, whereof the Point may pass by the Line of the Plummet hanged from the Point S of the Style, the Point of that second Style being called B, we perform the Operation as before to found the Lines d a, d b: but we may use the small Rods, as has been taught in the foregoing Chapter, otherwise the Operation would be too long.  

Demonstration.  
This Operation is so like to the precedent; that the Demonstration does not much differ from it; for here instead of a Second Point of Shadow we have the Point B, and the Point b s of the Triangle b s d being applied in B S, that Triangle moving upon B S and meeting the other Triangle a s d whereof the Points a ss are applied in A S, and which moves upon A S, so as the Points d d of the two Triangles may be joined together; than the Axis which aught to pass by the Points S d, shall be stayed in that Position, as we have seen in the precedent Chapter: for in all the different Positions of the Triangle b s d moving upon B S, the Line S d which represents the Axis, remains always elevated above the Horizon, with an Angle equal to the Elevation of the Pole above the Horizon, and it shall not be stayed in that Position, but by the meeting of the other Triangle a ss d, so as the two Points d d of the two Triangles may be joined together.  
The rest of this Operation being altogether like to the precedent, we shall not here again repeat the Demonstration.  
●th Fig    

CHAP. VIII.  
To found the Centre of the Dial, and to situate the Substylar and Equinoctial Lines. One only Point of Shadow being given, and the shortest Shadow.  
Fig. 7. HAving placed a Style on the Plane of the Dial, whose Point let be S and Foot P, and marked a Point of Shadow A, and divers other Shadows following one another, as G B, which may determine the length of the shortest Shadow, which is P B the Semidiameter of a Circle, that has for its Centre the Foot of the Style P, which touches the Tract of the Points of Shadow G B in the Point B, and determines it exactly; which cannot be done by the meeting of the Two Curves, that is of the Circle N B and Tract of Shadow G B:  
Therefore upon some Plane p s equal to th●e height of the Style P S, draw c p b perpendicular to s p and make p b equal to P B, and s b shall be equal to S B; and prolong s b to a, and let s a be equal to the length of the Shadow S A.  
Than make the Angle a s c equal to the Sum or Difference of a Right Angle, and the Angle of the Declination of the Sun at the time when the Points of Shadow were observed, following that which has been noted in the Practice of the 6th Chapter, the Line s c meeting b p in c draw the straight Line a c.  
Than on the Dial from the Point P. at the Distance p c describe the Arch C F, and from the Point of Shadow A, at the Distance a c describe the Arch D C, intersecting the Arch C F in the Point C, the Point C shall be the Centre of the Dial, and the straight Line C P shall be the Substylar, which shall cut the Tract of Shadow in the Point B, where the Circle N B aught to touch it.  
The Substylar Line being placed, we may draw the Equinoctial Line by the Practice of the 11th Chapter.  

Demonstration.  
8th Fig  
Note, That if in this Practice the Lines b p and s c be parallel or make a very acute Angle, the Axis shall not meet with the Plane of the Dial, or else shall meet with it at a great distance from the Point P. Than you must use the following Practice.  
Fig. 8. After you have made the same preparation as before, take s g on the Line s c of any length, and from the Point g draw a straight Line g m parallel to s p, which shall be also perpendicular to b p; from the Point g as a Centre, at the distance g a, describe the Arch a e intersecting b p in the Point e.  
From the Point A as a Centre, and at the distance e m describe the Arch M Q, and from the Point P as a Centre, at the distance p m, describe the Arch M F intersecting the Arch Q M in the Point M, and M P B shall be the Substylar Line.  
Than draw the straight Line P Z perpendicular to the Substylar Line M P B an equal to P S the height of the Style, and also M G perpendicular to the same Substylar Line M P B, and equal to m g, the Line G Z shall determine the Position of the Axis in respect of the Substylar Line, and Z E being drawn perpendicular to Z G meeting the Substylar Line in E, the Equinoctial Line shall be WE perpendicular to the Substylar Line intersecting it in the Point E.   

Demonstratson.  
There is no difficulty in the Demenstration of this Practice after that which has been explained before, we must only conceive here the Triangle e m g to be perpendicular to the Plane of the Dial, that it is Right angled at the Point m, and that it is moved about the Point e which is put upon the Point A, so as the Line e m is always on the Plane of the Dial, and also that the Trapezium p s g m is movable about the Line p s which aught to be applied to the Style P S, so as the Triangle e m g and the Trapezium p s g m may meet in than common Line m g or M G, the Line M G shall be another Style answering to the first, the Point M shall be the Foot thereof, and G shall be the Point and the Line that passes by the Points of these two Styles S and G, shall be the Axis, and consequently the Line M P that passes by the Feet of these correspondent Styles shall be the Substylar Line.  
It is also manifest that the Line Z G determines the Inclination of the Axis in respect of the Substylar Line.  
We may take divers Points of Shadow as A, to confirm this Operation, as we have done in th● Practice of the 5th Chapter, the shortest Shadow P B remaining always the same.  
9th Fig  
It is to be observed, That if the Sun's Declination hath considerably changed berween the Observation of the Point of Shadow A, and the shortest Shadow P B, the Angle p s c must be made as we have taught before for the time of the shortest Shadow, and than draw s a which makes with c s the required Angle for the time of Observation of the Point of Shadow A: for it is not necessary that the Observations of the Point of Shadow A, and the Point of the shortest Shadow B, should be made on the same day.    

CHAP. IX.  
To found the Centre of the Dial, and to draw the Substylar and Equinoctial Lines. Two Points of Shadow being given, with the Declination of the Sun at the times when the Points of Shadow were observed.  
Fig. 9 A Style being placed on the Plane of the Dial whereof the Point may be S and the Foot P, and two Points of Shadow A and B being observed.  
Make the Angle e s a equal to the Sum or Difference of a Right Angle and the Declination 〈◊〉 the Sun at the hour when we observed the Poi●● of Shadow A, following the Rules that we ha●● given in the Practice of the 6th Chapter.  
Let several straight Lines be drawn, as A E, ●●ward the place where we suppose something ne●● they aught to meet the Centre of the Dial, a● from the Centre A at any distance describe th● Arch of a Circle D R, intersecting A E in th● Point D. And having made s a equal to S A from the Centre Q at the distance a r, equal 〈◊〉 A R describe the Arch d r.  
Than from the Point s as a Centre, at the d●stance S D, describe the Arch d n, cutting the Ar●● r d in d, and draw the straight Line a d e to me●● with the Line s e in the Point e, and make A equal to a e, and having found several Points 〈◊〉 E, draw the Curve Line E C by all those Point● If we perform the same Operation by the Poi●● of Shadow B, we shall found another Curv●● Line F C, which intersecting the first Curve Li●● in the Point C, determines the Centre of th● Dial.  
And C P shall be the Substylar Line.  
We may draw the Equinoctial by the Practi●● of the 11th Chapter.  
The Demonstration of this Practice is so easy that it needs no Explication.  
10.th Fig.   

CHAP. X.  
To found the Centre of the Dial, and to draw the Substylar and Equinoctial Lines.  
Any Two Points of Shadow being given, with the Declination of the Sun at the time of Observation of the Points of Shadow.  
●ig. 10. LEt P S be a Style, whereof P may be the Foot and S the Point, and 〈◊〉 and B two Points of Shadow. From the ●oint A, at the Distance A S, describe the Arch Q Z: Also from the Point B, at the Distance 〈◊〉 S, describe the Arch R Z, intersecting the ●rch Q Z in the Point Z, and draw the straight Lines, Z B and Z A, and the Line Z P forth at length, than make Z H equal to Z B, and draw the ●●tait Line B H, which being divided into two equal parts in E, let Z E be drawn to meet with A F in the Point F.  
Than erect P N perpendicular to Z L, an● equal to the height of the Style P S, and dra● G O and O L at Right Angles, and O L shall mee● with Z P L at the Point L; the Line L F shall b● one Line which aught to pass by the Centre of the Dial.  
If the Centre of the Dial that is to be found, b● more North than the Point of the Style; and 〈◊〉 the Declination be more North, having made th● Angle c s b equal to the Sum of a Right Angle and of the Declination of the Sun at the tim● when we have marked the Points of Shadow; bu● if the Declination be South, having made the Ang●●● c s b equal to the difference of those Two Angle  
And if the Centre of the Dial required be mo●● South than the Point of the Style, and the Declination be North, having made the Angle c s b ●●qual to the Difference of a Right Angle and D●●clination of the Sun, but if the Declination be South having made that Angle equal to the Sum of the two Angles.  
Let s b be made equal to S B, and from th● Point b having drawn b d perpendicular to 〈◊〉 prolonged, if it be necessary: from the Point B, and at the Distance b d describe the Arch D cutting Z F in D, then making d e equal to D E draw the straight Line s e.  
From the Point P draw P I N perpendicular to L, and make I N equal to I S, and draw F N, ●en draw N C, making an Angle with F N equal the Angle e s c, the Line N C shall meet with ●e Line F L, at the Point C, the Centre of the ●ial if it have a Centre, and the same Line C N ●etermines the inclination of the Axis upon the ●ine C F on a Plane which is inclined to the ●lane of the Dial towards P, with an Angle equal 〈◊〉 S I P.  
And C P shall be the Substylar Line, the Equinoctial Line may be found by the Practice of the 11th Chapter.  
But if the Dial have no Centre, which is known by the Line N C being parallel to the Line F L; or if the Centre be at a great Distance from the Point P: Than draw the Line T V X parallel to P N, and make V X equal to V T, and draw ● X cutting V I in M, and draw P M meeting T V in Y, and make V y equal to V Y, and y P shall be the Substylar Line. The Equinoctial Line is drawn by the Practice of the 11th Chapter.  

To found the Centre by three Points of Shadow.  
Having marked three Points of Shadow, we ●ay found the Line F L by two of those Points ●hich shall pass thro' the Centre, and by one of ●ose Points and the third, we may found also another Line which shall pass thro' the Centre; and the Point where these two Lines do meet, shall be th● Centre of the Dial.  
This Practice may be performed without knowing the Declination of the Sun.  
The Centre being found, and consequently th● Substylar Line which passes thro' the Centre an● by the Point P, the Equinoctial Line is draw● by the Practice of the 11th Chapter.   

Demonstration.  
Fig. 11. To demonstrate this Practice, w● must suppose that the Triangle A S B in th● 11th Figure to be the same with the Triang A Z B of the 10th Figure, so as the Point Z joined to the Point of the Style, and that th● Line A B remains in its position.  
It is evident that all the Planes that are perpe● dicular to the Plane of the Triangle S A B an● which passes by the Point of the Style S, interse●● one another in the Line S L, which is raised perpendicular to the Point S upon the Plane of th● Triangle: Therefore the Plane L S F is perpendicular to the Plane S A B, and I say that the A● is of the Dial, and consequently the Centre, shal● be in that Plane L S F.  
For the Line S F divides the Angle A S B i● to two equal parts; and S B being equal to S ● the two Triangles S B D, S H D, being equ● each and like to the Triangle s b d of the 10● Figure, and being joined together by their common side S D, that Line S D shall be the Ax●● in its true position, and the Arch of a Circle B ● ●●scribed from the Centre D at the Distance D B, ●●all be the parallel of the Sun, for the Angle B D is equal to the Sun's Declination.  
But in these two Equicrural Triangles H S B, 〈◊〉 D B, because that the common Base B H is equally divided in the Point E, the Plane that passes by the Lines E S, E D, shall be perpendicular to the Planes of the two Triangles, therefore the Plane which is perpendicular to the Planes of the two Triangles, and that passes by their Vertical Points S D, shall carry the Axis: but the Plane that passes by the Point S, and by the Point E, and which shall be perpendicular to the Plane of the Triangle H S B, shall be also perpendicular to the Plane of the other Triangle.  
Therefore the Axis shall be in the Plane F L S that is perpendicular to the Plane of the Triangle A S B, and which passes by the straight Line S E F, and consequently the Axis meets ●ith the straight Line F L in some Point, or is parallel to it, and the Point C where the Axis meets with the Line F L shall be the Centre of the Dial, which was to be demonstrated.  
Than to demonstrate the Practice which serves to found the Centre of the Dial, we must consider that B H being the Chord of the Arch of the parallel of the Sun's Declination contained between the two Lines of Shadow S A, S B, and the Line B D being equal to the Line b d, and d e being equal to D E, the Triangle s d e is the same with the Triangle S D E of the 11th Figure.  
Therefore the Angle c s e is that which the Axis aught to make with the Line F L on a Plane which passes by that Line F L, and by S the Point of the Style.  
Therefore in the 10th Figure the Triangle F C N having the Line I N equal to I S of the Triangle F S C, and being both of them perpendicular to the common Line I L, and by the same Point I, and the Angle F N C being equal to the Angle e s c, the Point c shall be the Centre of the Dial, if the Line N C meet with the Line I L, which was that which was a●● last to be demonstrated. We suppose that i● this Practice the Difference of Declination between the Observations is not considerabl● changed.  
As concerning the Practice which serves t● found the Substylar Line, when the Point C is a● a great Distance from the Point P, or when th● Dial has no Centre; it is evident by Construction that T V being to V Y as N I is to I P, th● Lines N T and P y aught to meet one another in the same Point C, which shall be the Centre or they shall be parallel to one another, and the the Dial shall have no Centre.  
12.th Fig.    

CHAP. XI.  
To found the Centre of the Dial, and to draw the Equinoctial Line.  
The Substylar Line being drawn, and one Point of Shadow being given, with the Declination of the Sun. AND To draw the Equinoctial Line the Centre of the Dial being found. AND To found the Centre of the Dial, the Equinoctial Line being drawn.  
●ig. 12. LEt a Style be placed on the Plane of the Dial, whose Foot let be P, and ●●int S, and Point of Shadow A, and Substy● Line C P.  
Make the Angle d s a equal to the Sum or ●fference of a Right Angle, and of the Suns ●clination, as we have done in the Practice of ●e 6th Chapter, s a being made equal to S A, ●●ke any Point as d upon the Line s d, and draw ●●e straight Line a d.  
From the Point A draw the straight Line A R perpendicular to the Substylar Line C P, and from the same Point A as a Centre, and at the Distance a d, describe the Arch N cutting the Substylar Line in N.  
From the Point R as a Centre, and at the Distance R N, describe the Arch N D; erect the perpendicular P Z at Right Angles to the Substylar Line, and equal to P S the height of the Style, than from the Point Z as a Centre at the Distance s d, describe the Arch G D cutting the Arch N D in D: The Line Z D determines the situation of the Axis in respect of the Substylar Line, and if it meets with the Substylar Line as at the Point C, that Point C shall be the Centre of the Dial.  
Than from the Point Z draw the straight Line Z E perpendicular to Z D, meeting the Substylar Line in E, the Line V E perpendicular to the Substylar Line drawn through the Point E, shall be the Equinoctial Line.  
If the Centre of the Dial were given, the Line C P shall be the Substylar Line, and P Z being perpendicular to the Substylar Line, and equal to the height of the Style, having drawn C Z and Z E perpendicular to C Z, meeting the Substylar Line in E, the Line V E being drawn perpendicular to the Substylar thro' the Point E, shall be the Equinoctial Line.  
If the Equinoctial Line were given, draw the straight Line P E by the Foot of the Style perpendicular to the Equinoctial Line V E, and P E shall be the Substylar Line; and having made P Z perpendicular to the Substylar Line P E drawn by P the Foot of the Style, and equal to the height of the Style, having drawn Z E and Z D perpendicular to Z E, the Line Z D determines the inclination or position of the Axis with the Substylar Line; and if Z D meets the Substylar Line as at C, that Point C shall be the Centre of the Dial.  
The Demonstration of these Three Practices has nothing in it that deserves to be explained after the foregoing Demonstrations; for we may sufficiently know that the Triangle E Z C is the same with E S C which was upon the Plane, which was drawn by the Point of the Style, and by the Substylar Line.   

CHAP. XII.  
To draw the Equinoctial Line, and the Substylar Line, and to found the Centre of the Dial.  
Any Two Points of Shadow being given with the Declination of the Sun.  
Fig. 13. LEt a Style be set on the Plane of the Dial, whose Foot may be P, and Point S, and let the Two Points of Shadow be A and B.  
Make the Angle as f equal to the Declination of the Sun at the time when we have marked the Point of Shadow A, and make s a equal to S A, and on the Point A as a Centre describe the Arch M L at any Distance; and on the Point a describe the Arch l m equal to the Arch L M.  
Than by the Point A draw several strait Line● as A L, A M, toward the place where the Equinoctial aught to pass, which may be known by the Declination being North or South, and knowing near the position of the Wall in regard of the Axis.  
13.th Fig.  
Than from the Point s as a Centre, and to 〈◊〉 Distance S M, describe the Arch m cutting ●e Circle l m in m; than having drawn a meeting ●eting s f in f, make A F equal to a f: Also make ss l equal to S L, having drawn a l cutting s f in d, make A D equal to a d, and so of all the other Lines which we have drawn by the Point A, by which we found the Points as F D, and by all those Points as F D we draw a Curve Line F D.  
We do the same thing for the Point of Shadow B, about which we draw a Curve Line I G, and the straight Line D C, which touches the two Curves, shall be the Equinoctial.  
By the Practice of the aforegoing Chapter we draw the Substyle, and found the Centre of the Dial (if it has a Centre) with the inclination of the Axis to the Substylar Line.  

Demonstration.  
The Demonstration of this Practice is not difficult to those who understand the Conique Sections; they will presently conceive that these Curve ●ines, which are traced about by the Points of ●badow, are the Sections of an upright Cone, whereof the Angle at the Vertex of the Triangle ●●awn by the Axis is double the Declination of the San.  
The Axis of the Cone is a straight Line drawn from the Point of the Style to the Point of Shadow.  
Than if we conceive that there is a Plane which touches these two Cones both together, and which passes by their Vertex, as is the Point of the Style; that Plane shall be the Equinoctial; for the Plane that passes by the Axis of the Cone, and by the Line, where the touching Plane meets the Cone, shall be perpendicular to the touching Plane, as may easily be seen; and the Angle contained between the Axis of the Come and the Tangent Line, being the Declination of the Sun by Construction, the touching Plane shall necessarily be the Plane of the Equinoctial.  
14. Fig.    

CHAP. XIII.  
〈◊〉 found the Points of the Hours of 6 and 12 on the Equinoctial Line, and to draw the Meridian Line.  
The Equinoctial and Horizontal Line being given. Or to draw the Meridian, the Centre of the Dial being given.  
Fig. 14. LEt P S be the height of the Style, whereof P is the Foot and S the Point, and let N L be the Horizontal Line, and M N the Equinoctial Line; the Point N where the Equinoctial Line meets with the Horizontal Line, is the Point where the Line of the Hour 〈◊〉 6 intersects the Equinoctial Line.  
From the Centre N, and at the Distance N S, equal to the height of the Style, describe the Arch K H, and taking any Point as O in the Equinoctial Line for a Centre, at the Distance O 〈◊〉 describe the Arch I H intersecting the Arch ● H in H, than draw the straight Line N H, and H M perpendicular to N H; the Point M where H M meets the Equinoctial Line, is the Point where the Meridian Line aught to intersect the Equinoctial Line.  
Than having hanged up a Line with a Plummet F, so as the Line may pass by S the Poin of the Style, mark any Point as C, on th● Plane of the Dial, so as you may see with o● Eye the Points M and C both hid together b● the Line of the Plummet, and this is that whic● we call bourning, and the Line MC shall b● the Meridian Line.  
But if the Centre of the Dial were given, an● that it were the Point C, we must mark some Poin● as Mon the Plane of the Dial, which we may se● to pass by the Line of the Plummet with the Poin● C, and the Line C M shall be the Meridian.  
We may also draw this Meridian Line in th● Night with a Candle, in holding it at a d●stance from the Line of the Plummet, so as th● Shadow thereof may pass by M or by the Poin● C which of them is given: For the Shadow 〈◊〉 that Line shall be the Meridian Line.  

Demonstration.  
The Horizontal Circle cuts the Equinoctial Circle into two equal parts and equally distant from t●● meeting of the Meridian with those Circles; for t●● Equinoctial is perpendicular to the Axis, and t●● Horizontal is perpendicular to the Vertical Lin● and the Meridian passes by the Axis and by th● Vertical Line; Therefore the Horizontal Line cu●● the Equinoctial Line in the same Point where th● Hour Circle of 6 meets with them, which i● 90 degrees from the Meridian on the same Equinoctial 
15.th Fig.  Circle; therefore the Line and Plummet ●●ing a part of the Vertical, the Plane that passes 〈◊〉 that Line, and by a Point of the Meridian or of the Axis, shall be the Plane of the Meridian, the Intersection whereof with the Plane of the Dial is the Meridian Line.    

CHAP. XIV.  
To draw the Meridian Line, and to found the Point of the Hour Line of six on the Horizontal Line.  
Only one Point of Shadow being given, with the height of the Pole, and the Declination of the Sun.  
Fig. 15. A Style being given, whereof let S be the Point, P the Foot, being placed on the Plane of the Dial, draw the Horizontal Line H h, and from the Point P draw P H perpendicular to H h, and draw P Z parallel to H h, and equal to P S the height of the Style, and make H E equal to H Z.  
Than having marked the Point of Shadow A, as far from Noon as it is possible, hung a Plummet T, so as the Line thereof may pass by S the ●oint of the Style, and bourn it (as we did in the foregoing Practice for the Meridian) mark the Point h on the Horizontal Line, by which you see the Line pass, than when it also passe● by the Point of Shadow A, and than draw th● straight Line h E.  
Than draw an Arch of a Circle z oh f on the Centre c at any Distance, and make z o equal 〈◊〉 the height of the Pole above the Horizon, an● draw the Lines o c, z c, to c the Centre of th● Circle.  
Let the Arch oh m be made equal to the Declination of the Sun at the time when the Poin● of Shadow was made toward Z, if the Sun b● in North Signs, and toward f if it be in South Signs, for those that have their Zenith in the North part of the Sphere; but on the contra●● for those that have it in the South part; and draw● a m parallel to oh c.  
Draw c f perpendicular to c z from the Centre c, and make the Angle d c f equal to the Angle h S A, and draw the Lined d e parallel to f c, meeting c z in e and a m in a.  
On the Point E as a Centre, at the Distance de● describe the Circle B D, meeting E h (prolonged i● it be necessary) at the Point B make B M equa● to d a, and from M raise D M perpendicular to E B, intersecting the Circle BD in D, then draw E D (prolonged if it be necessary) and the Point F where it intersects the Horizontal Line, shall be the Point of the Meridian upon the Horizon.  
And E G being drawn perpendicular to E D, giveth the Point G where the Hour Line of six meets with the Horizontal Line.  
It must be observed that the Line M D which ●s drawn perpendicular to B E, may meet with the Circle B D on either side of the Point B; but you must take care that if the Point of Shadow A, ●s marked before Noon, to make use of the Point D, which is on the right hand of the Point B ●s in the Example, and if the Point A were marked after Noon, you must take the Point D where M D meets the Circle on the left hand of B ●o have the position of the Meridian Line; if D E meets not the Horizontal Line, but than when it 〈◊〉 prolonged toward E, the Point F shall appertain to the Line of Midnight: All this aught to ●●e understood concerning those that have their Zenith on the North side of the Equinoctial, for 〈◊〉 is contrary with those which have their Zenith ●n the Southern Hemisphere.  
If the Line E D meets not the Horizontal Line ●eing likewise prolonged towards E, than the Dial ●●all have no Line of Midday nor of Midnight, ●nd the Plane of the Dial shall be either Oriental ●r Occidental.  
It is also to be observed that the Angle H E F made by the Line E D with the Horizontal Line E H, is the Angle of the Declination of the Plane.  
By the foregoing Practice the Meridian Line or the Line of Midnight may be drawn by the Point F.  

Demonstration.  
The Figure f m z c is a Projection of the Sphere on a Plane by Lines parallel to one another, which Projection is called the Analemma; the Circle f m 〈◊〉 represents the Meridian, and the Point Z the Zenith● o c the Equinoctial Line, a m the parallel of th● Sun's Declination at the time when the Point of Shadow A was marked, c f the Horizontal Line▪ d a e the parallel of the Sun's Altitude at the sam● time, and the Point a determines in the Sphere th● place where than the Sun was.  
If a Verticle Circle z a g be conceived to p●● through the Point ato the Horizon in g, it is ●●nifest that the Line f g in the Circle whose Radius 〈◊〉 f c, shall be the Versed Sine of the Angle whi● the Verticle Circle z a g maketh with the Merid●● Circle a d f, and that d a shall be the same Verse Sine in the Circle, whose Radius is d e.  
Therefore if we conceived the Plane h E F D 〈◊〉 be the Plane of the Horizontal Circle, we may s●● by Construction that the Line E F D is the Meridian on that Plane, and that the Point F is in 〈◊〉 Plane of the Dial, the Point of the Meridian wh●●● it meets with the Horizon, and that the Point 〈◊〉 is the Point where the Hour of six cuts the Horizon seeing that F E G is a Right Angle; for B M 〈◊〉 qual to d a is the Versed Sine of the Angle B E D which aught by consequence to be equal to the Ang●● which is made by the Meridian with the Virtue Circle.  
16.th Fig.    

CHAP. XV.  
To draw the Meridian Line. Two Points of Shadow being given in a certain condition.  
Fig. 16. HAving fixed a Style on the Plane of the Dial, whereof let P be the Foot and S the Point, draw the Horizontal Line H h by the Fourth Chapter, and from the Point P draw P Z parallel to h H, and equal to P S the height of the Style, and having drawn the straight Line Z H, draw Z C perpendicular to it.  
Than the Point of Shadow A being marked 3 or 4 hours before Noon, to the intent that the Operation may be more exact, let A B be drawn perpendicular to Z C.  
Than make B E equal to B C, and C D equal to E A, and draw Z D.  
By the place where you judge something near that the Shadow of the Point S will arrive when the Sun shall be so far from Noon as it was when the Point of Shadow A was marked, draw many straight Lines as Q F perpendicular to H P, and by the Points F where they intersect H P, draw I F R perpendicular to Z C, and making F G equal to F R, from the Point G as a Centre, and at the Distance R I describe an Arch Q, which shall cut F Q in the Point Q, and so we may found as many Points Q as we will; and by all these Points drawing a Curve Line Q Q, we mark the second Point of Shadow O, than when the Shadow of the Point of the Style S meets with the Curve Q Q.  
Than draw the Line A O, and on the Point A as a Centre, and at the distance A S describe the Arch X Y, also from the Point O at the distance O S describe the Arch X N, cutting the Arch X Y in X, and draw X A, X O; than divide the Angle A X O into two equal parts by the Line X M, which shall meet with A O in M, the Point M shall be one of the Points of the Meridian Line, and by the Practice of the 13th Chapter we may draw the Meridian Line, which aught to pass by the Point M.  
We suppose in this Practice that the Sun's Declination has not changed considerably between the two Observations of the Points of Shadow A and O.  

Demonstration.  
The Curve Line O Q Q which aught to pass by the Point V, as is easy to understand by the Description, shall also pass by the Point A, if we make the Description thereof on the other side of the Line P H; and it is the Section of a Right Cone, whereof the Vertex is the Point S, and D Z C the half of the Triangle which passes by the Axis, couched upon the Plane of the Dial, and whereof Z C represents the Axis, the Angle D Z C is the Compliment of the height of the Sun above the Horizon at the time when the Point A was marked, and the Lines as I R are the Sections of 〈◊〉 Planes parallel to the Base of the Cone with its triangle by the Axis, and upon which the Secti●● of the Cone is a Circle, and the Intersections 〈◊〉 those Circles with the Plane of the Dial are the Points Q. which are in the Superficies of the Cone and consequently in the Conique Section A V Q Q whereof H V P is the Axis: But this Conique Section represents the parallel to the Horizon where the Sun was found at the time of the two Observations of the Shadow in A and O; Therefore the Line X M which divides the Angle A X O into two equal parts, is the Intersection of the Meridian with the Plane A O S, for that Line shall divide 〈◊〉 Chord of that Angle; the Point M is there●●● one of the Points of the Meridian Line on the ●●me of the Dial, in supposing the Declina●●● of the Sun to be the same for the two Points of ●●adow A and O.    

CHAP. XVI.  
To found the C●nter of the Dial or determine the Inclination the Axis with the Meridian, to draw the Substylar and Equinoctial Lines.  
The Meridian being found, and the height of the Pole being given.  
Fig. 17. LEt the Point P be the Foot of the Style whereof S is the Point, and let M T be the Meridian. Draw the Honzontal Line H h intersecting the Meridian Lin● in T; and from the Point T as a Centre at th● distance T S describe the Arch B A, and feom an● Point of the Meridian as M, at the Distance M 〈◊〉 describe the Arch D A intersecting the Arch B 〈◊〉 in the Point A, and draw the straight Line A T, than make the Angle T A C equal to the heigh of the Pole above the Horizon which is he●● given; if the Line A C meets with the Meridian, as at the Point C, then C shall be the Centre of the Dial, and the same Line A C in th● Position determines the Inclination of the Axi● with the Meridian Line; And c P shall be the Substylar Line.  
17.th Fig.  
From the Point A draw A E perpendicular 〈◊〉 A C, intersecting the Meridian Line in E, and the Line E V drawn thro' the Point E perpendicular to the Substylar Line C P, shall be the Equinoctial Line.  
But if the Dial have no Centre, draw the Line R P G at pleasure thro' the Point P meeting with the Meridian in G, and with the Line A C in R, than draw another Line O N parallel to R G, meeting the same Lines at the Points N and O; and draw the straight Lines R N and G O intersecting one another in the Point 〈◊〉, and make R p equal to G P, and draw p F to meet with the Line O N in Q; and P Q shall be the Substylar Line, and the Equinoctial Line is found as before.  

Demonstration.  
By the Construction it is manifest that the Line A T represents the Intersection of the Horizontal Plane on the Plane of the Meridian, therefore the ●ngle T A C being equal to the height of the ●●le above the Horizon, it is manifest that the ●●ne A C determines the Inclination of the Axis ●ith the Meridian M T; therefore consequently the Point C shall be the Centre of the Dial if the Line A C meets with the Meridian Line: But the Line A E which is perpendicular to A C which represents the Axis, shall be the Intersection of the Plane of the Equator and Meridian; therefore the Point E shall be one of the Points of the Equinoctial Line on the Meridian Line; but the Equinoctial Line aught always to be perpendicular to the Substylar Line, wherefore E V shall be the Equinoctial Line.  
If we have not the Centre of the Dial, we have drawn R P G and O Q N parallel to one another, and R N, G O, intersecting in F, and p F Q passing thro' the same Point F, than by reason of the like Triangles we have R G to O N as R p or G P to Q N; therefore P Q, M T, RO aught to meet in one and the same Point C, which shall be the Centre of the Dial, or else they shall be all three parallel to one another, and than the Dial shall have no Centre.  
18.th Fig.    

CHAP. XVII.  
Remarks and Practices for divers Abridgements in the Operations of the foregoing Chapters.  

I.  
Fig. 18. HAving found the Substylar Line p e and the Equinoctial Line e u for the Style s p, if you would remove the Substylar and Equinoctial Lines to another place of the Plane of the Dial; the Line P E parallel to p e shall be another Substylar Line, and V E parallel to e u or perpendicular to P E passing thro' any Point of the Substylar Line P E, shall be the Equinoctial Line, and we determine by the following Method the Position of a Style for the two Lines P E, E V, whereof the height shall be given of any length, or we will determine the height of a Style whereof the Position shall be given upon the Substylar Line P E.  
First, Let the Line A R be given for the height of the Style, which aught to be set for the Substylar and Equinoctial Lines P E, E V  
Make E P equal to e p, and set it the same way (that is the Point P must be set above the Point E if the Point p be above the Point e, and below it, if it be below it) and make E Z equal to p s, and E Z equal to R A given, and draw z P and Z R parallel to Z P, meeting E P in R; and the Point R shall be the Foot of the Style, the height whereof R A is given.  
Therefore if you fix a Style, whereof the Foot may be R, and the Distance between the Point thereof A and Foot R, may be the height equal to the Line Z E, the Proposition is satisfied.  
But if the Point R were given for the Foot of the Style, and the height were required: dra● P Z as before, and by the Point R draw R Z parallel to P z, and E Z shall be the height of the Style whose Foot is the given Point R.   

II.  
The Substylar Line C E, the Equinoctial Line E V, and the Meridian Line C M, answerable to the Style S P being given, we may take what Point we will in the Substylar Line as K, to the Centre of the Dial, without altering the Substylar Line or Equinoctial, and the Line K m drawn parallel to C M shall be the Meridian Line, but the height and position of the Style must be changed by the following Method.  
Make E z equal to P S, and draw M P and P Z, and draw m R and R Z parallels to M P ●nd P Z, and the Point R shall be on the Substylar Line, which is the Foot of the Style, whereof the height is R Z perpendicular to the Point R on the Plane of the Dial.   

III.  
If the Substylar Line C E were given with the Meridian Line C M answering to the Style P S, we may take any Point as p to be the Foot of a Style, whereof the height is to be determined; or the Style being given of any height to determine the Position of the Foot p without changing either the Meridian or the Centre of the Dial.  
If the Foot of the Style p be given, and we are to determine its height, by the Foot of the Style P, put for the finding of the Meridian and Substylar Line, let there be drawn P S perpendicular to the Substylar Line and equal to the height of the same Style, and let C S be drawn by the Centre of the Dial; and from the given Point p let p s be drawn parallel to P S, till it meet C S in the Point s, and p s shall be the length of the height of the Style, which aught to be placed at the Point p, and the Meridian C M and the Centre of the Dial C are not changed.  
But if p ss were given for the height of the Style, it must be put upon P S prolonged if it be necessary in P z, and z s must be drawn parallel to C P to meet with the Line C S in m ss, and s p being drawn parallel to S P, shall give the Point p on the Substylar Line for the Foot of the Style required, whereof the height is given.   

iv  
Fig. 19 The Meridian C M being given, with the Equinoctial Line E M, we may found another Equinoctial as e m, without changing the Meridian, the which Equinoctial e m shall make the Angle e m C with the Meridian C M equal to the Angle E M C, but we must found another Style by the following Method.  
If the Centre of the Dial be given at the Point C, by the Foot of the Style P which have served to found the Meridian and the Centre C, having drawn P S perpendicular to the Substylar Line CP, and equal in length to the same Style, let e s be drawn parallel to E S, and from the Point s draw s p parallel to S P, meeting the Substylar Line in the Point p, which shall be the Foot of the Style required, whereof p ss shall be the height.  
20th Fig.   

V  
Fig. 20. If after you have drawn the Meridian Line C M and the Substylar Line B P, we can●●t have the Equinoctial, because the Style has ●een put too long, we may diminish it as much 〈◊〉 we please, without changing either the Foot ●●ereof or the Substylar Line; but we must found another Meridian and another Horizontal Line which may answer to that Style, and these Meridian and Horizontal Lines shall be parallel to the first Meridian and Horizontal Lines.  
Therefore draw the Line e m M by any Point 〈◊〉 the Substylar Line as e, which may be perpendicular to it, that Line may be the Equinocti●● Line; but the height of the Style must be ●●anged in drawing e s perpendicular to the Line 〈◊〉 ss which determines the inclination of the Axis ●ith the Substyle, and that Line e s meeting SPARKE ●hich is a perpendicular to the Substylar Line by ●●e Foot of the Style, and which is his height, 〈◊〉 that for the Equinoctial Line e m, P z shall be the height of the Style required; but there must ●e another Meridian found, whether the Centre of the Dial be found or not found.  
Take on the Substyle the length e B equal to e draw B M, and also take e b on the Substyla Line equal to e z, and by the Point b draw b 〈◊〉 parallel to B M meeting the Equinoctial Lin● e m in the Point m, and let c m be drawn parallel to C M, the Line c m shall be the Meridian for the Style, whereof the Foot is in P, the heigh is P z, and the Equinoctial Line e m.  
And we may yet change if we will the Foo● of the Style or its height, according as necessity requires by the former Observarions.  
But for the Horizontal Line D H which ha● been found for the Style whose Foot was P and P S the height, the which meets the Substyle a●● the Point H, if you would found another for the Style whose height is P z, without placing effectually that Style to make use of the Practic● taught in the 4th Chapter; you have no more t● do but to draw S H, and by the Point z the Lin● Z h parallel to S H meeting the Substyle in h 〈◊〉 and the Line d h parallel to D H, shall be th● Horizontal Line required.   

VI  
Lastly, a Dial being drawn on a Plane, we may transfer it in what other place we will on the same Plane, by drawing of parallel Lines to those that are drawn, so that we keep the same order and the same proportion between them in their meetings, but the Style aught to be put at the Poi●● which answers to the Point of the first, which for its Foot.     

THE SECOND PART.  

THE PREFACE.  
Concerning the choice which we aught to make of the Practices to draw the Substylar, Equinoctial, and Meridian Lines; and to place the Centre of the Dial, according to the Expositions of the Planes proposed.  
HAving taught in the first Part of this Treatise different Practices to found the principal Lines of Sun Dial's, I think it will not be amiss to give some Instructions concerning the Use which we may make of these Practices, following the different Expositions of the Planes proposed on which the Sun Dial's are to be drawn.  
Therefore it is necessary to know something near, the Position of the Plane in regard of North or South before we begin any thing; which may be done by a small Declinatory, which presently shows on what side the North, South, East or West is; which those that are used to observe the Sun, may know by seeing in what manner i● shines upon the Plane according to the Hour and Season of the Year.  
Than we may well conceive after what manner the Axis shall meet with that Surface, an consequently we may judge of the Position 〈◊〉 the Substylar Line, of the Equinoctial, and also of the whole Dial.  
But considering a Dial wholly made, it is not difficult to know among divers manners which we may use, that may be most fit and most easy for the Construction of the Dial.  
Therefore we may presently see that it will be useless to found the Centre of a Dial or the Meridian on a Plane which comes near either to the East or West, and that the Equinoctial Line being set on such a Plane, we need not found the Point of Midday, and that we must use the Point of the Sixth Hour to begin the Divisions of the Horary Intervals on that Line, or on the Horizontal Line.  
Also we may see that on these sorts of Planes we cannot use the Practices where we aught to have the Points of Shadows after Midday, which may be answerable to others taken in the Morning; for that which is of those which give the Position of the Meridian of the Plane, the Substylar Line, by the correspondent Points, we can never fitly make use of them on these sorts of Planes: For if the first Point hath been marked a little too far from that Meridian, we can never have it's correspondent Point.  
Also we must not use the Practices of correspondent Points of Shadows, or the Tract of the Shadow, if the Circle that is described from the Foot of the Style as a Centre, meets that Tract in Angles too acute; for we cannot determine exactly that meeting, and this inconvenience may hap to the Practices on all sorts of Planes in any Season of the Year.  
If the Dial be large, or if the Declination o● the Sun has changed considerably between the Observations of the Points of Shadow, we have not exactly the Lines which we seek by those Practices where we suppose that it hath not been changed between the Observations.  
In the Practices of this Second Part we suppose always that the Equinoctial Line or Horizontal Line is drawn, and that we have marked on that Line the Point where the Hour of 12 or 6 meets with it; at which Points we begin the Division of the Hours on those Lines; but to draw them we must have the Centre of the Dial, or at lest the Inclination of the Axis to the Substylar Line, which has been taught in the First Part.   

CHAP. I  
To mark the Points of the Astronomique Hours on the Equinoctial Line and by those Points to draw the Hou● Lines.  
Fig. 21. LEt S P be a Style, whereof S is the Point and P the Foot, E XII, i● the Equinoctial Line; on which the Point XII is the meeting of the Equinoctial with the Meridian, and the Point VI is the meeting thereof with the Hour of Six and with the Horizon; P E● is the Substylar Line which meets with the Equinoctial in E.  
21. st Fig.  
Than by the Centre of the Dial, and by the Points of the Hours which are marked upon the Equinoctial Line, draw straight Lines, which shall be the Hour Lines.  
But if we have not the Centre of the Dial, and we have only the Inclination of the Axis Z z to the Substylar Line e E, you must take any Point, as e, on the Substylar Line, and by that Point e having drawn a straight Line e 12, parallel to the Equinoctial Line E XII; and drawing e z perpendicular to Z z, make e a equal to e z, and by the Point a draw the straight Lines a 11, a 12, a 1, etc. parallel to the Lines A XI, A XII, A I, etc. and by these Points, where these Lines meets with the Line e 12, and by those which are correspondent to them on the Equinoctial Line, draw the Hour Lines 11 XI, 12 XII, 1 I, 2 TWO, etc.  

Demonstration.  
The Plane S XII E is the Plane of the Equinoctial by the Constructions of the First Part, and the Triangle A XII E being the same with the Triangle s XII E, we aught to put upon the Plane of the Triangle A XII E the Hour Lines 15 Degrees the one from the other about the Point A, in beginning from the Line A XII or A VI, which are the Intersections of the Plane of the Meridian, or of the Plane of the Hour Circle with the Plane of the Equinoctial; but all the Planes of the Hour Circles intersect one another in the Axis, and do also pass by the Centre of the Dial which is the meeting of the Axis with the Plane of the Dial: Therefore the straight Lines drawn from the Centre of the Dial to the Points of the Hours marked on the Equinoctial Line shall be the Hour Lines of the Dial, if the Dial has no Centre; or if the Centre be at a great distance from the Equinoctial Line, the Practice that is here taught for that Case gives the Point● on the Line e 12, which is parallel to the Equinoctial Line; so as all the distances of the Hour Lines on that Line e 12, are in the same proportion with their Distances on the Equinoctial Line, and the each of these Distances on the Line e 12 have the same proportion to their correspondent Distance on the Equinoctial Line, as the Line e z has too the Line E Z; therefore it follows that if the Dial have no Centre all the Hour Lines are parallel to one another and to the Axis. And if it have a Centre, all the Lines shall meet in that Centre: which is very manifest, because of the like Triangles which are made by the Lines E XII, e 12, which are parallel to one another.  

22. ᵈ Fig.     

CHAP. II.  
To mark the Points of the Astronomique Hours on the Horizontal Line.  
And to draw the Hour Line by those Points.  
Fig. 22. S Is the Point of a Style given, whereof P is the Foot, M H D is the Horizontal Line, P H is drawn by the Poine P (the Foot of the Style) perpendicular to the Horizontal Line; M is that Point where the Meridian Line intersects the Horizontal Line, upon the Line H P set H s equal to H S, and draw the straight Line s M a, and make the Angle M s A equal to the Angle of the elevation of the Pole above the Horizon, than from any Point as A, taken on the Line s A, raise a perpendicular from 12 to s A, till it meet with s M in 12, and draw the Line 9, 12, 4, perpendicular to s a, and make 12 a equal to 12 A; and from the Point a, as a Centre, describe a Circle at any Distance, and divide it into equal parts from 15 Degrees to 15 Degrees, beginning the Division where the Line a 12 intersects the Circle; and draw Lines from the Point a, to the Divisions of the Circle, to meet with the Line 9.4 at the Points 9, 10, 11, 12, 1, 2, 3, 4, etc. And by the same Points and the Point s, draw straight Lines which shall meet the Horizontal Line in the Points of the Hours required, which must be marked according to the Diurnal Motion of the Sun, of which the Point M shall be Noon, and D the Point of the Hour of Six.  
If the Line s D drawn perpendicular to ss M meet with the Horizontal Line at the Point D, that Point shall be the Hour of Six on the Horizontal Line, which is the same Point where the Equinoctial Line aught to meet with the Horizontal Line, as hath been taught in the First Part.  
If we have not the Point of Midday on the Horizontal Line, and we have but D the Point of the Hour of Six, than draw s D and s M perpendicular to s D, the which s M meets the Horizontal Line, or not in M, for it is indifferent; than we do the same as we did before to found the Points of the Hours on the Horizontal Line. We may see by this Practice that it is not necessary that the Line s M a should meet with the Horizontal Line.  
The Hour Lines are to be drawn from C the Centre of the Dial, and by the Points of the Hours which have been found on the Horizontal Line.  
If the Dial have no Centre, it is better to follow the Method of the precedent Chapter than this, seeing that we have always an Equinoctial Line in that Case.  

Demonstration.  
The Plane s M D is the same with the Plane S M D by Construction, which is the Plane of the Horizon; therefore all this Operation must be considered as made on the Plane of the Horizon which passes by the Point of the Style S, whereof s M is the Meridian given, or the Substylar Line, or s D is the Line of the Hour of Six, the Point s is the Centre of the Dial, the Line s A is the Inclination of the Axis to the Substylar Line s M; therefore it is that taking the Point A at pleasure for the Point of the Style, by the Practice of the 11th Chapter the Line 9, 4, shall be the Equinoctial answering to the Point of the Style, and by the Practice the former Chapter the Lines s 12, s 1, s 2, etc. shall be the Hour Lines on that Plane, which meets the Horizontal Line M H D, which is also on the same Plane by Construction with the Points of the Hours which they design; therefore the Lines which are drawn thro' those Points of the Hours on the Horizontal Line, and by C the Centre of the Dial, shall be the Hour Lines of the proposed Plane.    

CHAP. III.  
Six Intervals of Hours following one another being given, to draw all the other Hours.  
Fig. 23. AS let the Six Intervals of Hours from C A to C F be given, draw E e parallel to c 5, cutting c A in the Point A, C B in the Point B, C D in the Point D, etc. And make A b equal to A B, A d equal to A D, etc. And from the Centre C, and through the Points b d e, etc. draw the Lines of the Hours that follow the precedent Hours.  
If also you would have other Hours following the first or last found, you must repeat the Operation in drawing another Line as E e parallel to that which is the last of the Six Intervals of Hours.  
If the Dial has no Centre, you must draw another Line as s t parallel to E e, on which found the Points of the Hours as you found them on the Line E e, and in joining the Horary Points of the two parallel Lines E e and s t, you shall have the Hour Lines required.  
23. ᵈ Fig.  

Demonstration.  
If we imagine a Plane to pass by the Line E A e, and which is parallel to the Plane of the Hour Circle of the Line C 5, it is manifest that the Plane of the Hour Circle which passes by the Line C A, shall be perpendicular to the Plane that passes by E A e, because that the Planes of the Hour Circles C A, C 5, are inclined to one another by a Right Angle, and by consequence the Axis meets not with the Plane drawn by E A e, and it shall ●e parallel to it, and all the Lines which are the meetings of the Planes of the Hour Circles with the Plane that passes by E A e, shall be all parallel to one another and to the Axis.  
But because the Plane that passes by C A is perpendicular to the Plane by E A e, the meetings of the Hour Lines upon that last Plane shall be equally distant by order from the meeting of the Plane drawn by C A; that is to say, that the first ●●n one side shall be as far distant as the first on the other side, and that the second hour on one side shall be as much as the second hour on the other side, and so following in order; therefore the Line E A e which is on that Plane, meets the parallel Hour Line at equal distance by order, both on the one and the other side of the Point A, which was to be demonstrated.  
We may conceive the Plane that passes by E A e, as the Plane of a Dial which has no Centre, upon which all the Hour Lines are parallels to one another and to the Axis; and that the Hour Line which passes by the Point A is the Meridian or the Substylar Line of the Plane; than we may see manifestly that the Hour Lines that are on one and the other side of that which passes thro' A, are equally distant from it in order, and by consequence if we draw a straight Line any way upon that Plane, the meetings of the Hour Lines both on the one side and on the other of that which passes by A, shall be at an equal Distance from A.    

CHAP. IU.  
To draw the Parallels of the Twelve Signs.  
24.th Fig.  
If the Centre of the Dial be towards the North in regard to the Point of the Style, make s c equal to S C of the Dial, which is the distance between the Point of the Style and the Centre, but if the Centre be towards the South, in respect of the Point of the Style, make s c upon c s prolonged on the other side of the Point s.  
Than to found the Points of the Parallels of the Signs upon the Hour Lines; as for Example, On the Line of Midday, you must take the distance S XII from the Point of the Style S to the Point XII, which is the Intersection of the Line of Midday with the Equinoctial, and set it from ss to 12 ●pon the Line s a, and having drawn the Line 12 which cuts the Lines of the Signs in the Points d, f, g, h, i, k, then we transport ●he Intervals 12 h, 12 i, 12 k, 12 g, 12 f, ●2 d, in XII H, XII I, XII K, XII G, XII F, XII D, on the one and other side of the Equinoctial Line, as they are on the two sides of the Line s a.  
And after the same manner having found the other Points upon each Hour Line, and likewise on the halves and quarters, or other Lines coming from the Centre, we draw by all the Points which belong to the same Sign, the Line of the Parallel of the Sign, and so for each of them in particular.  
But if we have not the Intersection of the Equinoctial Line upon the Hour Line, on which you would have the Points of the Signs, in that case you may have always the Centre of the Dial; therefore it it is, that having taken any Point, as R on the Hour Line, and having made the Triangle c s r on the Base c s equal, and like to the Triangle C R S on the Base C S, and the Line c r being continued if it be necessary, shall meet with the Lines of the Signs in those Points, which being set on the Dial, in observing from the Point C which is the Centre, the same Distances which they have from the Point c.  
25.th Fig.  
The Construction of this Practice is so plain, that it needs no Demonstration, for it is easy to see that the Plane c s 12 is the same with the Plane of the Hour Circle C S XII, and so of the rest.  
A part of an Arch of a Sign being given, which is Parallel to the Equator, we may describe that Arch by the Practice of the following Chapter.   

CHAP. V  
The Equinoctial Line being given, we may draw a Parallel to it by a Point given on an Hour Line.  
Fig. 25. THe Equinoctial Line A G is given, and the Hour Lines a A, P B, c C, d D, etc. which intersect the Equinoctial in the Points A B C D, etc. and the Point P on one of the Hour Lines is also given, by which it is required to draw a Parallel to the Equator.  
By the Point P, and by the Point A, on the Equinoctial, the Point of the Hour next to B, on which is the given Point P, draw A P meeting with the next Hour to P B at the Point c, then draw c E, so as the Points E and A may be equally distant from the Hour c C; the intersection d of the Line c E with the Hour Line D d, shall be one of the Points of the required Parallel; after the same manner we found another Point f, and so of the rest.  
But here it is to be observed, that if there be only whole Hours on the Dial, we shall not have the Points of the Parallel but for every other Hour; and if we have the Lines of the half Hours, we shall have the Points of the Parallel from Hour to Hour, and if we have the Lines of the Quarter of Hours, we shall have the Points of the Parallel from Half Hour to Half Hour, and so of the rest; for if A P meet the Line of the half Hour s b at the Point s, having drawn s D so as b D and b A may be equal, s D shall meet the Hour Line c C at the Point K, which is as far from s b as P B, the Point K shall be one of the Points of the Parallel.  
If instead of the Equinoctial Line we have one of its Parallels, that parallel being given, we do the same thing as if it were the Equinoctial Line.  
26.th Fig.  

Demonstration.  
The straight Lines on the Plane of the Dial represent great Circles of the Sphere, therefore it is that the two great Circles represented by C A and by C E, which are equally inclined to the Hour Circle c C, because that C A and C E are equal, meeting P B and D d, which are equally distant and equally inclined to C c, in the Points P and d, equally distant from the Equinoctial Line B D, therefore P and d shall be in one and the same parallel to the Equator, and so of the other Points.    

CHAP. VI  
To draw the Italian and Babylonian Hours upon an Horizontal Plane.  
Fig. 26. THe Astronomique Hours being drawn on the Dial whose Centre is C, and the Meridian C A and V A the Equinoctial Line, c E being divided into two equal parts in A, let the Points b, c, d, e, f, g, h, etc. of a Parallel to the Equator, be found on the Hour Lines by the Practice of the foregoing Chapter, which Parallel passes by the Point A.  
The Line A 12 parallel to the Equinoctial shall be the Line of the 12th Italian Hour.  
The straight Line b VII, which passes by th● Point of the Seventh Hour in the Morning of th● Equinoctial, and by the Point of the First Hou● after Noon of the Parallel, shall be the Line 〈◊〉 the 13th Italian Hour.  
The straight Line c VIII which passes by th● Point of the Eighth Hour in the Morning on th● Equinoctial Line, and by the Point of the second Hour after Noon of the Parallel shall be th● 14th Italian Hour, the straight Line d IX, which passes by the Point of Nine in the Forenoon on the Equinoctial, and by the Point of Three in the Afternoon on the Parallel, shall be the 15th Italian Hour, and so of the rest; there being always Six Hours distance between the Hour of the Equinoctial and that of the Parallel.  
The Babylonian Hours are marked after the same manner, but only that which is done on one side of the Meridian for the Italian Hours, is made on the other side of the Meridian for the Babylonian Hours, and they are counted after another manner; as for Example, the straight Line that passes by the Point of Midday of the Equator, and by the Point of the Sixth Hour in the Morning of the Parallel, is the Sixth Babylonian Hour; that which passes by the first Hour Afternoon on the Equinoctial, and by the Point of the Seventh Hour in the Morning on the Parallel, shall be the Seventh Babylonian Hour, and  
27.th Fig.  
so following, so as A 12 parallel to the Equator shall be the 12th Babylonian Hour for the Horizontal Dial.  

Demonstration.  
Fig. 27. If we imagine a Parallel to the Equator, which passes thro' the Intersection of the Meridian and Equator, it is evident that the Horizontal Circle shall touch that parallel at the Point where the Meridian Circle cuts it; and if we conceive that the Horizontal Circle be fastened with the Parallel and the Meridian, than when the Meridian hath advanced 15 Degrees on the Equator in going according to the Motion from the East to the West, the Plane of the Horizontal Circle in that Position, shall mark the first Italian Hour on the West part, and the first Babylonian Hour on the East part; and than when it hath advanced 15 degrees more, the Plane of the Horizontal Circle shall mark the second Italian and Babylonian Hour, and so on; so as than when it is arrived to the 12th Hour, the Meridian Circle shall be found in its first Position, for it has made half a Revolution; but the Horizon that cuts the Plane of the Meridian at Right Angles, shall be found in a Position opposite to that which it had at first; that is to say, that if S H be the meeting of the Horizon with the Meridian, S C the Axis, C E the Plane of the Dial parallel to the Horizon; after that the Meridian hath passed over 12 Hours of the Equator, or that S H has puffed over 12 Hours of the Parallel; the Line S H shall fall in 12 S A: So as the Angle 12 S C shall be equal to the Angle H S C, therefore C S A shall be an Equicrural Triangle; and so also the Triangle S A E shall be Equicrural, S E being the meeting of the Equator and Meridian; therefore S A is equal to A C and A E both together; therefore C A and A E are equal: But C E represents the Meridian of the Horizontal Dial, wherein C is the Centre, E the Intersection of the Equinoctial, and A the Intersection of the 12th Italian or Babylonian, which must be perpendicular to the Meridian; for in that Position the Plane of the Meridian is returned to its place, and the Horizon which has not changed its Inclination which it has with the Meridian remains always at Right Angles to it. Therefore it shall intersect the Plane of the Dial in a Line perpendicular to the Meridian, the Plane of the Dial being parallel to the Horizon in that first Position, and that Line aught to pass thro' the Point A of the Meridian, which divides the Line C E into two equal parts.  
If we conceive a Cone which has for its Base the Parallel to the Equator H 12, and for its Vertex the Centre of the Sphere, the Section of that Cone with the Plane of the Dial, shall be the representation of that Parallel, but the Plane of the Dial is parallel to one of the Planes, which touches the Cone; therefore the Section or the Representation of that Parallel shall be a Parabola.  
28.th Fig.  
And seeing that all the Planes of the Italian and Babylonian Hours are also the Planes which touch that Cone, the Lines of those Hours shall touch the Parabola which is in that Section.    

CHAP. VII.  
To draw the Italian and Babylonian Hours, on a Plane which is not Horizontal.  
Fig. 28. THe Astronomique Hours being described, and the Horizon R H which is one of the Hours required, being drawn on the Plane of the Dial with the Equinoctial Line, we draw a Parallel to the Equator d, b, R, e, f, g, which passes by R the Intersection of the Horizon, with any Hour Line, as in this Example, with the Hour Line of TWO, by the Practice of the 5th Chapter. And seeing that the Horizon, which is the Line of the 24th Italian Hour, intersects the Parallel in R, at the Point of the Second Hour after Noon, and the Equinoctial Line at the Point of the Sixth hour after noon the Line of the first Italian Hour shall pass by the Point e of the Parallel, which is the Third Hour after Noon, and by the Point of the Equinoctial; the Line of the Second Italian Hour shall pass by the Point f of the Parallel, which is the Fourth Hour, and by the Point of the vl Hour on the Equinoctial Line; and so of the rest we found all the Points by which the Italian Hours aught to pass, so as the Eighteenth Italian Hour passes always by the Point of Midday of the Equinoctial Line, and by a Point of the Hour of a Parallel, which shall be so far from the Point of Midday as the Point R of the same Parallel, which is the Intersection of it with the Equinoctial Line, is the Point of the Sixth Hour after Noon, which is in this Example at Four Hours distance, seeing the Point R is an Hour of the Afternoon.  
Fig. 29. But if the Point R, by which the Parallel to the Equator is described, were the Intersection of an Hour before Noon, we must consider that that Parallel aught to meet also the Horizon in a Point of an Hour, which is so far from Noon, as that is by which we have described it; for Example, If the Point R were the Intersection of Nine in the Morning with the Horizontal Line, the Parallel to the Equator described by the Point R, aught to meet the Horizontal Line in the Point H, which is upon an Hour Line, so far distant from Noon as is the Point R; that is to say, that the Point H shall be the meeting of the Third Hour after Noon with the Horizontal Line; and the Line of the 24th Italian Hour, which is an Occidental Portion of the Horizon, aught to be taken from the Point of the Third Hour of  
29.th Fig.  
the Parallel with the Point of the Sixth Hour after Noon of the Equator, and in counting as we have done before, we shall found that the First Italian Hour shall pass by the Point of the Fourth Hour on the Parallel, and by the Point of the Seventh Hour after Noon on the Equinoctial; and that the Line of the Second Italian Hour shall pass by the Point of the Fifth Hour of the Parallel, and by the Point of the Eighth Hour of the Equinoctial, and so on; and we draw only those that are visible, for the others are of no use, and serve only to count and to place those which are of use.  
These Rules are for the Italian Hours, but for the Babylonian Hours, which have for the Twenty fourth Hour the Oriental Part of the Horizon, if the Parallel which is described by the Point R of the Horizon, were the meeting of the Horizon with the Line of the Ninth Hour before Noon, the First Babylonian Hour shall pass by the Point of the 10th Hour in the Morning of the Parallel, and by the Point of the 7th Hour in the Morning on the Equinoctial; the Line of the Second Babylonian Hour shall pass by the Point of Eleven before Noon on the Parallel, and by the Point of Eight on the Equinoctial, and so of the rest; and if the Point R of the Parallel were the Point of any Afternoon Hour, we must take its correspondent before Noon to begin to count the Babylonian Hours, which is the contrary of that which we have done for the Italian Hours.  
We must observe, that if we describe the Lines by the Intersections of the Astronomique Hour-Lines with the Italian or Babylonian, in taking those Points of Intersection in order from the Equinoctial Line, those Lines shall be the Parallels to the Equator, which shall meet the Horizontal Line in the same Points where the Hour Lines meet.  

Demonstration.  
After that which hath been demonstrated in the aforegoing Chapter, concerning the generation of the Italian and Babylonian Hours, it suffices here to explain after what manner the Horizontal Line meets the Astronomique and Italian Hour Lines in one and the same Point.  
It is easy to see that when the Horizon, which we have supposed to be fastened to the Meridian without changing its Inclination that it has with it, is moved by the motion of the Meridian, that if the Meridian advanceth 15 degrees on the Equator; also that Point of the Horizon, let it be what it will be, shall advance 15 degrees on the Parallel which that Point describes: Therefore if we describe a Parallel to the Equator by one Point of the Horizon, by which Point passes an Astronomique Hour Line, it is manifest that all the same Hour Lines shall divide that Parallel from 15 to 15 degrees; Therefore it is that all the Italian Hour Lines which aught to pass by the Points of the Hours of the Equinoctial, shall also pass by the Points of the Hours of the Parallel; and it is also manifest that we may draw Six Parallels to the Equator by the six Intersections of the Horizon with the six Hour Lines 12, 1, 2, 3, 4, 5, and that the Italian Hours pass in order by the Points of the Hours of those Paralles; and it also follows that all the Lines of the Italian and Babylonian Hours shall touch the Parallel described by the Point of meeting of the Horizon and Meridian, and that all the Points of touching shall be the meeting of the Astronomique Hours with the Parallel, and that there will be always six Hours Interval between the touching Point upon the Parallel, and the Point of meeting of the same Hour upon the Equator; for the Plane of the Horizon touches that Parallel at the Point of Midday, the which Point describes the Parallel.    

CHAP. VIII.  
To continued the Description of the Italian and Babylonian Hours, when the Parallel or the Equator is wanting on the Plane of the Dial.  
Fig. 28. IF the Point h be the last that is found o● the Parallel by means of the Equinoctial Line, following the Practice of the 5th Chapter of this Second Part, and that the Line h III be the last Italian Hour which we can mark by help of that Parallel, that Line h III shall meet with some Astronomical Hour in some Point as m, if by the Practice of the Fifth Chapter of this Part we found the Points l n o of the Parallel which passes by m, and if they be on the Hour before or after that, on which is the Point m; we continued to draw the Lines of the Italian or Babylonian Hours by the Points of the Hours of the Parallel m n oh, and by the Points of the Hours of the Equator, in following the same order as before, and if the Equator be wanting in using the Practice taught at the end of the 5th Chapter, we shall found the Points of another Parallel by the Parallel that is  
30th Fig.  
given, and than we may join the Points of the Hours on the two Parallels in following the former Order.  
There is nothing in all this that deserves any long Explication, nor any other Demonstration, than that we have given in the Two foregoing Chapters.   

CHAP. IX.  
Four Astronomique Hours being given, following one another in order, with the Equinoctial Line.  
To found the other Hours.  
Fig. 30. LEt the four Hour Lines following one another be A a, B b, C c, D d, with the Equinoctial Line E F; from a Point a, taken at pleasure in one of the last Lines A a, having drawn a D which cuts B b in B, and C c in g; also by the same Point a, having drawn a C which cuts B b in h, let A h be drawn, which meets C c in c, and B g which meets D d in d: let c b, c d be drawn prolonged to the Equinoctial Line to the Points E, F, the Hour Lines E e, F f, drawn by the Points E F, shall be the Hour Lines required, whereof E e shall be distant fro● A a one Hour, and F f shall be two Hours fro● D d: Therefore B D being prolonged to f in th● Line F f, and F d to l, in the Line B b, having drawn C f which cuts D d in i, l i prolonge● shall meet the Equinoctial in M, by which th● Hour Line M m shall be between the two Hou● Lines D d and F f, and these Seven Hour Line being found, we may have all the rest by the Practice of the Third Chapter.  
Fig. 31. There are many Cases where Three Hour Lines are sufficient with the Equinoctial and Horizon; for Example, If we have three Hour Lines a 2, b 3, and c 4, and the Equinoctial Line 24, and Horizontal Line a c, having drawn a 4, which cuts b 3 in d, and having drawn 2d which cuts c 4 in f, draw 2 c, which cuts b 3 in e, and 4 e, which cuts a 2 in h; there aught a straight Line to pass by the three Points f b h, which shall meet the Equinoctial in the Point g, which is one Point of the Hour as far from the Hour b 3, as is the Hour Line of Six: Therefore if the Hour Line b e be the fourth Hour, a h shall be the third, and g i shall be the second; but in this Example, b e being the third Hour, g i shall be the 12th Hour.  
The first Hour between 12 and 2 is found by drawing g c, which cuts a 2 in k, and 4 k which cuts the Hour Line g i, which was drawn by the Point g to the Point i, and in drawing 2 〈◊〉  
31th Fig.  
which cuts g h in n, the Hour Line by the Point 〈◊〉 shall be the first Hour.  

Demonstration.  
By that which hath been demonstrated in the 5th Chapter it is manifest that the Points a and c are in the same Parallel; also that the Points b and d are also in the same, therefore the great Circles that are represented by the Lines c b E, c d F, are equally inclined to the Equator, the one on one side, and the other on the other, but c b F has the same Inclination as a b D, because of the equal parts of their Parallels a c, b d; and there is a space of three Hours between c C and F, and between c C and E; therefore the Hour which passes by the Point E is the next to that which passes by the Point A, but there are two between the Points D and F.  
By the same Construction the Points l and f are in the same Parallel, but f i c and l i M represent great Circles equally inclined to the Equator, because of the Points l and f which are in the same Parallel, and equally distant from the Point i; therefore D C and D M represent equal parts of the Equator, which are each of them an Hours distance from one another.  
As concerning that which is of three Hour Lines, seeing that by Construction the Points a and f are in the same Parallel; and likewise h and c, the Lines f h and a c which is the Horizon, represent Circles equally inclined to the Equator; therefore there is as much distance between the Point of the Ho●● of Six upon the Equinoctial and the Hour Line b●● as there is between the same Hour Line b e an● the Point g. We demonstrate as before that th● Point n divides the interval between the Hour Line● g i and a k.    

CHAP. X.  
A Dial being given which is already drawn, to found the foot of the Style which did serve to draw it, and to determine the height thereof.  
Fig. 32. THe Line A B is the Equinoctial Line, and the Distance A B on that Line is the Interval of any six Hours; having divided A B into two equal parts in the Point G, from the Point G as a Centre, on the Diameter A B describe the Circle A, S, B, d, f, and mark the Points d and f which divide the Semicircle into three equal parts; A F, F D and D B are each the Interval of two Hours on the Equinoctial Line: the Lines d D, f F aught to meet the Circumference of the Circle at the Point S, and the Line S E P drawn perpendicular to the Equinoctial Line shall be the Substyle.  
32th Fig.  
If we have C the Centre of the Dial, on the Diameter C E, having described the Semicircle C Z E, and having applied in it the Line E Z equal to E S: Z P being drawn perpendicular to the Substylar Line E P, and meeting it at the Point P, that Point shall be the Foot of the Style, whereof P Z shall be the height.  
But if we have not the Centre of the Dial, having drawn a e parallel to the Equinoctial, and from the Point a having drawn a ss parallel to A S which meets the Substylar Line in ss, from the Point E as a Centre, and Semi-diameter E S, describe the Arch z, and from the Point e as a Centre and Semi-diameter e s describe the Arch x, and draw ●he straight Line x z which shall touch the two Arches x and z, and that Line x z shall determine the Inclination of the Axis to the Substylar Line, and having drawn E z perpendicular to x z from the Point E, and from the Point z the straight Line z P perpendicular to the Substylar Line E P, the Point P shall be the foot of the Style, whereof P Z shall be the height.  

Demonstration.  
If we suppose the Circle A S B d to be on the Plane of the Equinoctial, it is manifest that the Point of the Style aught to be one of the Points of the Circumference of the Semicircle A S B, because the six Intervals of Hours that are on the Equinoctial from A to B, for the Angle A S B, in what place soever the Point S is, aught to be a Right Angle: And also the Lines drawn from that Point S to the Points of the Hours of the Equinoctial Line, aught to comprehend equal Angles each of 15 degrees in the same Point S; therefore the Arches A f, f d, and d B being each of 60 degrees, shall subtend Angles of 30 degrees, which are each in value two Hours, and consequently the Lines f F and d D shall be Hour Lines on the Equinoctial; therefore they meet in the same Point S upon the Circumference of the Circle which determines the Point of the Style in respect of the Equinoctial, and the Line S E drawn perpendicular to the Equinoctial Line, doth there give the Point E which is one of the Points of the Substylar Line; but the Substylar Line aught to be perpendicular to the Equinoctial Line; therefore the same straight Line S E P shall be the Substylar.  
The Axis aught to make a Right Angle with the Plane of the Equinoctial, therefore the Line E S which is equal to that which aught to meet the Axis on the Plane of the Equinoctial, aught to make a Right Angle with the Axis.  
Therefore if we have c the Centre of the Dial, and if we describe the Semicircle C Z E, applying E Z equal to E S in that Semicircle, it is manifest that C Z E shall be a Right Angle, and that the Line C Z represents the Inclination of the Axis to the Substylar Line C E, and that the Point Z represents the Point of the Style which has served to draw the Dial: Therefore Z P which is drawn perpendicular to C E, shall be the height of the Style, whereof the Point P shall be the foot.  
But if we have not the Centre of the Dial, it is manifest by Construction that the two Lines E S e s having one of their extremes in the Points E e, the other will be had in the Axis, if the Line that joins the two last extremes be perpendicular to those two Lines which aught to be parallel; but it is also evident that the Line x z which touches the two Arches of Circles which have for their Semi diameters E S, e s, shall be perpendicular to the two straight Lines E z, e x, which come from the Centre to the touching Points; therefore the Line x z gives the Inclination of the Axis to the Substylar Line, and the Line z P drawn perpendicular to the Substyle, shall determine the height of the Style, whereof P shall be the foot.    

CHAP. XI.  
To place the Axis.  
IF we would have the Hours shown only by the shadow of the Point of the Style, we aught to make and to place the Style after such a manner as may serve without changing of it.  
We may give it divers forms, but one of the best is, to make it waved to the end, that the Shadow thereof may not unite with the Hour Lines in any place, and that we may always know that it is but only the Shadow of the Point that serves to show the Hours.  
But if you would have a portion of the Axis to show the Hours, and that the Axis be represented by an Iron Rod, the Style we have placed aught to have the Point very small, that it may enter into a little hole made in the Rod, so as the Point of the Style may exactly answer to the middle of the thickness of the Rod; the Style may remain, if you would have it, to sup port the Axis; but if the Axis be not very long, and if it be strong enough to sustain itself alone being fastened at one end, we may take away the Style when the Axis is fixed on the Surface of the Dial. We may do the same, if we fasten to the end of the Style a Point of an Iron Wyer, which there may be very small, and may take away but the half of the thickness of the Rod, so as the Dial being drawn to that Point, there is nothing to be done but to take it away to place the Axis, whereof the middle of the thickness aught to answer to that Point; therefore whether the Style remains to uphold the Axis, or whether we take it away when the Axis is fixed in its place, we must fasten it to the end of the Style to stay the end thereof, which aught to answer to the Centre of the Dial if it have any.  
Fig. 33. We may make the Rod which serves for the Axis as it is marked in the Figure, so as the hole signified by A may be made to lodge the Point of the Style, and that it be let in as far as the middle of the thickness of the Rod, the Point B which answers also to the middle of the Rod, aught to be applied exactly to the Centre of the Dial, this Rod being so stayed at the Point B and at the Point A, we must fasten the Foot C on the Plane of the Dial.  
But if you would not have a Foot to the Axis as G, and that you would only fix the Rod to the Centre of the Dial, you must draw divers Lines which may pass by the Centre of the Dial, and stay the Rod on the Point of the Style A, and by any other place, so as the end may enter in a hole made in the Plane of the Dial at the place of the Centre, may be divided by the middle of its thickness by each Line that passes by the Centre.  
Fig. 34. Also we may make use of a thin Plate which must be cut according to the Inclination of the Axis with the Substylar Line; it must be set perpendicularly on the Plane of the Dial, in applying ●ne of its sides to the Substylar Line, and the other side passing by the Point of the Style shall serve for the Axis. If you would draw on the Plate a perpendicular Line to that side of it which is applied to the Substylar Line, and equal to the height of the Style, the which shall represent the Style, so as the end of that Line meet with the other side of the Plate, that Line when the Plate shall be set in its place, shall answer to the Style. See here divers Figures of these sorts of Plates with their feet to fasten them on the Plane of the Dial.  
33th Fig:   

CHAP. XII.  
To Draw Dial's by Reflection.  
TO make a Dial that may show the Hours by the Reflection of the Light of the Sun, you must make use of a small piece of polished Metal very even and flat; of a round form, and of abont an Eighth part of an Inch in Diameter; and having placed it, and fastened it in a place very stable and immovable, we mark the Points of Light on the Plane where we intent to draw the Dial, which serve instead of the Points of Shadow, the middle of the Mirror or Glass aught to be considered as the Point of a Style, whereof we found the foot in drawing from the middle of the Glass a Line perpendicular to the Plane of the Dial, the Point where this Line meets with the Plane of the Dial, shall be the Foot of the Style.  
We may found the Substylar Line, the Equinoctial Line, the Centre of the Dial and Meridian by the practices, where we make no use of the Horizontal Line, nor of the height of the Pole.  
Having found the Equinoctial Line, and the Point where the Meridian Line intersects it, we draw the Hours, following the Methods of the Second Part of this Treatise.  
Here it is to observed that if the Inclination of the Glass be never so little changed, all the Dial will be considerably changed. Therefore these sorts of Dial's do very hardly remain many years in a good condition; for there always happens some alterations to the Wall on which they are fixed.  
But if in place of the Glass we fill some small Vessel either of Glass or Potter's Earth, of about an Inch in Diameter, with Water or Quick Silver, that Vessel being put upon a place marked on some Transum of a Window or the like, so as you may always set it in the same place again, if you would take it away, the Reflection of the Light from the Water or Quick Silver, shall give the Hours on the Dial of which we must draw the Lines, as has been taught in the First and Second Part of this Work, in observing only that the middle of the Superficies of the Water or Quick Silver, serves for the Point of the Style, and that the Operations which are made on the Horizon below for the Dial's, which gives the Hour by the Shadow of a Point, aught to be made on the Horizon above, and that which is made above in those Dial's, in these Dial's to be made below.   

CHAP. XIII.  
Concerning the Table of the Sun's Declination, and of those of the Difference of Meridian's of divers considerable places in respect of Paris.  
THe Tables of the Declination of the Sun which are at the end of this Work, is calculated for the Meridian of Paris, and for each day at Noon, on the side are the Differences between the Declination of one day and that of the next day following; they are made for Four Years following one another. The First Year gins in 1681, which is the First after the Bissextile or Leap Year, the Second is the following Year 1682, the Third is 1683, and the Fourth Year 1684, is Bissextile or Leap Year.  
Than afterwards there follows a Table of the Differences of Meridian's of the principal places of the Earth in respect of Paris; it is calculated ●n Hours and Minutes, and it serves to found the Declination of the Sun in all those places at any Hour proposed. There is also another Table which is joined to that, in which we may found the Latitude or Height of the Pole of the same places.  
We begin the Days in these Tables and in the following Calculations, from each Day at Noon, and continued it to the next following Day at Noon, and we count the Hours to 24, soas the Third Hour in the Morning of any day proposed, is the Fifteenth Hour after Noon of the preceding day.  

RULE I. To continued the Tables of the Sun's Declination.  
IF you would have the Declination of the Sun for the Years following those that are calculated, you must add to the Declination increasing of each day, and subtract from the Declination decreasing after Four Years fully passed, one Minute for 32 Minutes of Difference, half a Minute for 16 Minutes of Difference, a quarter of a Minute for 8 Minutes of Difference, and so of the rest in proportion.  

Example.  
If you would know the Declination of the Sun for the 5th of January at Noon for the Year 1685, which is the First Year after the Bissextile or Leap Year, and is the Fourth that follows to the Year 1681, whereof we have the Calculation in the Table. I found that the Difference between the 5th and 6th of January 1681, is 12 Minutes, and the Declination is decreasing; therefore I see that there must be substracted about ⅓ of a Minute from the Declination of the 5th of January 1681, the which is 20 Degrees 58 Minutes; and we shall have 20 degrees 57 Minutes ⅓ for the Declination of the Sun the 5th of January at Noon in the Year 1685; if there be Eight Years passed between the Year in which you would have the Sun's Declination, and that which answers to it in this Table, that is to say, that which is equally distant from the Leap Year, you must double this Correction; if it be 12 Years, you must triple the Correction, and so forth.    

RULE II. To found the Declination of the Sun at all Hours of the Day.  
HAving found in the Table of differences of the Sun's Declination, the difference of the Declination of the Sun between the given Day and the next following Day, take the part proportional of that Difference answerable to the given Hours, which we add to the Declination of the same Day if it increase, but subtract it if it decrease.  

Example.  
As to know the Declination of the Sun on the 15th day of March at 4 a Clock in the Afternoon for the Year 1683, I found in the Table that the Difference between the 15th and 16th day is 23 minutes and ½ (for there is sometimes 24 minutes and sometimes 23 minutes) and the Declination of the Sun for the 15th of March at Noon in the Year proposed 1683, being 1 degree 59 minutes, to which here must be added the proportional part of the Difference for 4 Hours, which is the sixth part of 24 Hours; therefore you must take the sixth part of 23 minutes and ½, which is about 4 minutes, which being added to the Declination found 1 degree 59 minutes, because the Declination increases, and we shall have 2 degrees 3 minutes, the Declination increasing on that Day at the required Hour; but if the Declination were decreasing, we must subtract the part proportional of the Declination found in the Table.  
This aught to be understood of the Declinations of the Sun for the Meridian of Paris, as we found them in this Table, but for other places on the Earth, they are to be reduced according to the following Rule.    

RULE III. To know the Declination of the Sun at a given Hour in any place that is set down in the Table.  
THe Table of Differences of Meridian's of places which are here set down in respect of Paris, with the word add or subtract shows how much later or sooner it is Noon at Paris than at that place; the chief use whereof is to know the Declination of the Sun at a given Hour in any place set down in the Table.  
Therefore if you would know the Declination of the Sun at a given Hour in any place that is in the Table, you must found the Difference of its Meridian in respect of Paris, and join it to the given Hour, if there be add after it, and if the Sum exceed 24 Hours, we take the overplus in the next following day to that as was given; but if there be found subtract, we take that Difference of Meridian's from the Hour proposed; but if the Hour be too little, add 24 Hours to it, and than subtract, and the remainder shall be attributed to the foregoing Day.  
Having therefore found this Sum or Difference of Hours, we found the Declination of the Sun for that Hour at Paris by the Second Rule, and you shall have that which you require for the given place.  

Example.  
If we would know for Rome the Declination of the Sun for the 22th day of August 1682, at 4 a Clock in the Morning.  
First, Because the Hour proposed is in the Morning, I reduce it to 16 Hours after Noon on the 21th day of August, and having found in the Table of Differences of Meridian's, that for Rome I must subtract 47 minutes, therefore I take 47 minutes from 16 hours, and there remains 15 hours and 13 minutes after Midday on the 21th of August 1682; and by the Second Rule I found that the Declination for that day and that Hour is at Paris 8 degrees and 15 minutes, which is that which was required for 4 a Clock in the Morning the 22th day of August 1682 at Rome.   

Another Example.  
If you would know for Peking in China the Declination of the Sun at 2 a Clock in the Morning on the 25th of July 1684.  
First, because the Hour proposed is before Noon, I reduce it to the foregoing day, which will be the 24th of July at 14 Hours after Noon, and I found in the Table of Differences of Meridian's, that for Peking I must subtract 7 hours 45 minutes, the which being substracted from 14 hours, there remains 6 hours 15 minutes; therefore I search by the Second Rule the Declination of the Sun for Paris on the 24th of July, at 6 a Clock 15 minutes after Noon, in the Year 1684, and I found 17 degrees, and about 13 minutes which is the Declination of the Sun for Peking on the day and hour proposed.   

Another Example.  
I would know at Quebec the Declination of the Sun on the 25th of March 1683, at 10 a Clock in the Morning, which being reduced, will be 22 Hours after Noon on the 24th of March, and I found in the Table that I must add for Quebec 4 Hours 36 minutes, therefore we have the 24th of March at 26 Hours 36 minutes, or the 25th of March at 2 a Clock 36 minutes after Noon. For which time in the Year 1683, I found the Sun's Declination to be 5 degrees and about 30 minutes, which is that for Quebec on the day and hour required.       

An ADVERTISEMENT concerning the Figures.  
HEre you must observe that the Dial's drawn in these Figures, are not made expressly for any place; for it is impossible to make the Magnitude of the Lines which we must use equal to those that are here drawn: we aught only to follow the Precepts, and not to measure with the Compasses the length of the Lines, to see if they agreed: For Example, although we say make the Line a ss equal to A S, yet these two Lines are not equal in the Figure, because that sometimes one of the ends of the Line A S as S being the Point of the Style which is not in the Plane, the apparent Magnitude of that Line is not the true Magnitude, and it also happens oftentimes that the Lines of the Figure answers not among themselves according to the Discourse which was done to order the place of the Figures: it is sufficient to observe well after what Manner, of what Magnitude, and in what Angles we prescribe to draw the Lines without depending on the Figure which serves but to help the Imagination, and to guide you in the Operations, seeing that it is almost impossible to meet with two like Dial's among a great number which we make on the Planes proposed, as it is found ordinarily.   


A TABLE Of the Sun's Declination For the Year 1681, Which was the First after Bissextile or Leap-year.  
For the year 1681. Days January Diff. M D. M. South. 1 21 41 10 2 21 31 11 3 21 20 11 4 21 9 11 5 20 58 11 6 20 46 12 7 20 34 12 8 20 22 12 9 20 9 13 10 19 56 14 11 19 42 14 12 19 28 14 13 19 14 15 14 18 59 15 15 18 44 15 16 18 29 16 17 18 13 16 18 17 57 16 19 17 41 17 20 17 24 17 21 17 7 17 22 16 50 18 23 16 32 18 24 16 14 18 25 15 56 18 26 15 38 19 27 15 19 19 28 15 0 19 29 14 41 19 30 14 22 20 31 14 2  Days Februar. Diff. M D. M. South. 1 13 42 20 2 13 22 20 3 13 2 20 4 12 42 21 5 12 21 21 6 12 0 21 7 11 39 21 8 11 18 21 9 10 57 22 10 10 35 22 11 10 13 22 12 9 51 22 13 9 29 22 14 9 7 22 15 8 45 23 16 8 22 23 17 7 59 23 18 7 36 23 19 7 13 23 20 6 50 23 21 6 27 23 22 6 4 23 23 5 41 23 24 5 18 24 25 4 54 23 26 4 31 23 27 4 8 24 28 3 44 24 Days March Diff. M. D. M. So. No. 1 3 20 23 2 2 57 24 3 2 33 23 4 2 10 23 5 1 47 24 6 1 23 24 7 0 59 24 8 0 35 24 9 0 11 23 10 0 No. 12 24 11 0 36 24 12 1 0 23 13 1 23 24 14 1 47 24 15 2 11 23 16 2 34 23 17 2 57 24 18 3 21 23 19 3 44 23 20 4 7 24 21 4 31 23 22 4 54 23 23 5 17 23 24 5 40 23 25 6 3 23 26 6 26 22 27 6 48 22 28 7 10 22 29 7 32 23 30 7 55 22 31 8 17  Days. April Diff M D. M. North 1 8 39 22 2 9 1 22 3 9 23 21 4 9 44 21 5 10 5 21 6 10 26 21 7 10 47 21 8 11 8 21 9 11 29 21 10 11 50 20 11 12 10 20 12 12 30 20 13 12 50 20 14 13 10 19 15 13 29 19 16 13 48 19 17 14 7 19 18 14 26 19 19 14 45 18 20 15 3 18 21 15 21 18 22 15 39 17 23 15 36 17 24 16 13 17 25 16 30 17 26 16 47 17 27 17 4 16 28 17 20 16 29 17 36 16 30 17 52 15 Days May Diff. M. D M. North 1 18 7 15 2 18 22 14 3 18 36 15 4 18 51 14 5 19 5 14 6 19 19 13 7 19 32 13 8 19 45 13 9 19 58 12 10 20 10 12 11 20 22 12 12 20 34 11 13 20 45 11 14 20 56 11 15 21 7 10 16 21 17 10 17 21 27 10 18 21 37 9 19 21 46 9 20 21 55 9 21 22 4 8 22 22 12 8 23 22 20 7 24 22 27 7 25 22 34 6 26 22 40 6 27 22 46 6 28 22 52 5 29 22 57 5 30 23 2 5 31 23 7 Days June Diff. M. D. M. North 1 23 11 4 2 23 15 3 3 23 18 3 4 23 21 2 5 23 23 2 6 23 25 2 7 23 27 1 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 2 15 23 25 2 16 23 23 2 17 23 21 3 18 23 18 3 19 23 15 4 20 23 11 4 21 23 7 5 22 23 2 5 23 22 57 5 24 22 52 6 25 22 46 6 26 22 40 6 27 22 34 7 28 22 27 7 29 22 20 8 30 22 12  Days July Diff. M. D. M. North. 1 22 4 8 2 21 56 9 3 21 47 9 4 21 38 10 5 21 28 10 6 21 18 10 7 21 8 11 8 20 57 11 9 20 46 11 10 20 35 12 11 20 23 12 12 20 11 12 13 19 59 13 14 19 46 13 15 19 33 13 16 19 20 14 17 19 6 14 18 18 52 14 19 18 38 15 20 18 23 15 21 18 8 15 22 17 53 15 23 17 38 16 24 17 22 16 25 17 6 16 26 16 50 17 27 16 33 17 28 16 16 17 29 15 59 17 30 15 42 18 31 15 24  Days August Diff. M D. M. North. 1 15 6 18 2 14 48 18 3 14 29 18 4 14 11 19 5 13 52 19 6 13 33 19 7 13 14 20 8 12 54 20 9 12 34 20 10 12 14 20 11 11 54 20 12 11 34 20 13 11 14 21 14 10 53 21 15 10 32 21 16 10 11 21 17 9 50 21 18 9 29 21 19 9 8 22 20 8 46 22 21 8 24 22 22 8 2 22 23 7 40 22 24 7 18 22 25 6 56 22 26 6 34 23 27 6 11 23 28 6 48 23 29 5 25 23 30 5 2 23 31 4 39  Days September Diff. M. D. M. No. So. 1 4 16 23 2 3 53 23 3 3 30 23 4 3 7 23 5 2 44 23 6 2 21 23 7 1 58 24 8 1 34 23 9 1 11 24 10 0 47 23 11 0 24 24 12 0 South 0 23 13 0 23 24 14 0 47 23 15 1 10 24 16 1 34 23 17 1 57 24 18 2 21 23 19 2 44 23 20 3 7 24 21 3 31 23 22 3 54 23 23 4 17 24 24 4 41 23 25 5 4 23 26 5 27 23 27 5 50 23 28 6 13 23 29 6 36 23 30 6 59 23 Days October Difi. M D. M. South 1 7 22 22 2 7 44 23 3 8 7 22 4 8 29 22 5 8 51 22 6 9 13 22 7 9 35 22 8 9 57 22 9 10 19 22 10 10 41 21 11 11 2 21 12 11 23 21 13 11 44 21 14 12 5 21 15 12 26 21 16 12 47 20 17 13 7 20 18 13 27 20 19 13 47 20 20 14 7 19 21 14 26 19 22 14 45 19 23 15 4 19 24 15 23 19 25 15 42 18 26 16 0 18 27 16 18 18 28 16 36 17 29 16 53 17 30 17 10 17 31 17 27 16 Days November Diff. M D. M. South 1 17 43 16 2 17 59 16 3 18 15 16 4 18 31 15 5 18 46 15 6 19 1 15 7 19 16 14 8 19 30 14 9 19 44 13 10 19 57 13 11 20 10 13 12 20 23 12 13 20 35 12 14 20 47 12 15 20 59 11 16 21 10 11 17 21 21 10 18 21 31 10 19 21 41 10 20 21 51 9 21 22 0 9 22 22 9 8 23 22 17 8 24 22 25 7 25 22 32 7 26 22 39 7 27 22 46 6 28 22 52 6 29 22 58 5 30 23 3 5 Days December Diff. M. M. D. South 1 23 8 4 2 23 12 4 3 23 16 3 4 23 19 3 5 23 22 2 6 23 24 2 7 23 26 2 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 2 15 23 25 2 16 23 23 3 17 23 20 3 18 23 17 4 19 23 13 4 20 23 9 5 21 23 4 5 22 22 59 6 23 22 53 6 24 22 47 6 25 22 41 7 26 22 34 7 27 22 27 8 28 22 19 8 29 22 11 9 30 22 2 9 31 21 53 10   

A TABLE Of the Sun's Declination For the Year 1682. Being the Second after Bissextile or Leapyear.  
For the Year 1682. Days January Diff. M D. M. South 1 21 43 10 2 21 33 10 3 21 23 11 4 21 12 11 5 21 1 12 6 20 49 12 7 20 37 12 8 20 25 13 9 20 12 13 10 19 59 14 11 19 45 14 12 19 31 14 13 19 17 14 14 19 3 15 15 18 48 15 16 18 33 16 17 18 17 16 18 18 1 16 19 17 45 17 20 17 28 17 21 17 11 17 22 16 54 17 23 16 37 18 24 16 19 18 25 16 1 18 26 15 43 19 27 15 24 19 28 15 5 19 29 14 46 19 30 14 27 20 31 14 7  Days February Diff. M D M. South 1 13 47 20 2 13 27 20 4 13 7 21 4 12 46 20 5 12 26 21 6 12 5 21 7 11 44 21 8 11 23 22 9 11 1 21 10 10 40 22 11 10 18 22 12 9 56 22 13 9 34 22 14 9 12 22 15 8 50 23 16 8 27 22 17 8 5 23 18 7 42 23 19 7 19 23 20 6 56 23 21 6 33 23 22 6 10 23 23 5 47 23 24 5 24 24 25 5 0 23 26 4 37 24 27 4 13 23 28 3 50 24 Days March Diff. M. D. M. So. No. 1 3 26 23 2 3 3 24 3 2 39 24 4 2 15 23 5 1 52 24 6 1 28 24 7 1 4 23 8 0 41 24 9 0 17 24 10 0 No. 7 23 11 0 30 24 12 0 54 24 13 1 18 23 14 1 41 24 15 2 5 23 16 2 28 24 17 2 52 23 18 3 15 23 19 3 38 24 20 4 2 23 21 4 25 23 22 4 48 23 23 5 11 23 24 5 34 23 25 5 57 23 26 6 20 22 27 6 42 23 28 7 5 22 29 7 27 22 30 7 49 22 31 8 11 22 Days April Diff. M. D. M. North 1 8 33 22 2 8 55 22 3 9 17 22 4 9 39 21 5 10 0 21 6 10 21 21 7 10 42 21 8 11 3 21 9 11 24 21 10 11 45 20 11 12 5 20 12 12 25 20 13 12 45 20 14 13 5 19 15 13 24 19 16 13 43 19 17 14 2 19 18 14 21 19 19 14 40 18 20 14 58 18 21 15 16 18 22 15 34 18 23 15 52 17 24 16 9 17 25 16 26 17 26 16 43 17 27 17 0 16 28 17 16 16 29 17 32 16 30 17 48 15 Days May Diff. M. D. M. North. 1 18 3 15 2 18 18 15 3 18 33 14 4 18 47 14 5 19 1 14 6 19 15 14 7 19 29 13 8 19 42 13 9 19 55 12 10 20 7 12 11 20 19 12 12 20 31 12 13 20 43 11 14 20 54 11 15 21 5 10 16 21 15 10 17 21 25 10 18 21 35 9 19 21 44 9 20 21 53 9 21 21 2 8 22 22 10 8 23 22 18 7 24 22 25 7 25 22 32 7 26 22 39 6 27 22 45 6 28 22 51 5 29 22 56 5 30 23 1 5 31 23 6  Days June Diff. M. D. M. North 1 23 10 4 2 23 14 3 3 23 17 3 4 23 20 3 5 23 23 2 6 23 25 2 7 23 27 1 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 1 15 23 26 2 16 23 14 2 17 23 22 3 18 23 19 3 19 23 16 4 20 23 12 4 21 23 8 4 22 23 4 5 23 22 59 5 24 22 54 6 25 22 48 6 26 22 42 6 27 22 36 7 28 22 29 7 29 22 22 8 30 22 14 8 Days July Diff. M. D. M. North. 1 22 6  2 21 58 8 3 21 49 9 4 21 40 9 5 21 30 10 6 21 20 10 7 21 10 10 8 21 0 10 9 20 49 11 10 20 38 11 11 20 26 12 12 20 14 12 13 20 2 12 14 19 49 13 15 19 36 13 16 19 23 13 17 19 9 14 18 18 55 14 19 18 41 14 20 18 27 14 21 18 12 15 22 17 57 15 23 17 41 16 24 17 25 16 25 17 9 16 26 16 53 16 27 16 37 16 28 16 20 17 29 16 3 17 30 15 46 17 31 15 28 18 Days August Diff. M D. M. North. 1 15 10 18 2 14 52 18 3 14 34 19 4 14 15 19 5 13 56 19 6 13 37 19 7 13 18 19 8 12 59 20 9 12 39 20 10 12 19 20 11 11 59 20 12 11 39 20 13 11 19 21 14 10 58 21 15 10 37 21 16 10 16 21 17 9 55 21 18 9 34 21 19 9 13 22 20 8 51 22 21 8 29 22 22 8 7 22 23 7 45 22 24 7 23 22 25 7 1 23 26 6 38 22 27 6 16 22 28 5 54 23 29 5 31 23 30 5 8 23 31 4 45  Days September Diff. M. D. M. No. So 1 4 22 23 2 3 59 23 3 3 36 23 4 3 13 23 5 2 50 24 6 2 26 23 7 2 3 23 8 1 40 24 9 1 16 23 10 0 53 23 11 0 30 24 12 0 South 6 23 13 0 17 24 14 0 41 23 15 1 4 24 16 1 28 24 17 1 52 23 18 2 15 23 19 2 38 23 20 3 2 23 21 3 25 23 22 3 48 24 23 4 12 23 24 4 35 23 25 4 58 24 26 5 22 23 27 5 45 23 28 6 8 23 29 6 31 22 30 6 53 23 Days October Diff. M. D. M. South 1 7 16 23 2 7 39 22 3 8 1 23 4 8 24 22 5 8 46 22 6 9 8 22 7 9 30 22 8 9 52 22 9 10 14 22 10 10 36 21 11 10 57 21 12 11 18 21 13 11 39 21 14 12 0 21 15 12 21 21 16 12 42 20 17 13 2 20 18 13 22 20 19 13 42 20 20 14 2 20 21 14 22 19 22 14 41 19 23 15 0 19 24 15 19 18 25 15 37 18 26 15 55 18 27 16 13 18 28 16 31 18 29 16 49 17 30 17 6 17 31 17 23  Days November Diff. M D. M. South 1 17 40 16 2 17 56 16 3 18 12 15 4 18 27 15 5 18 42 15 6 18 57 15 7 19 12 14 8 19 26 14 9 19 40 14 10 19 54 13 11 20 7 13 12 20 20 12 13 20 32 12 14 20 44 12 15 20 56 11 16 21 7 11 17 21 18 11 18 21 29 10 19 21 39 10 20 21 49 9 21 21 58 9 22 22 7 8 23 22 15 8 24 22 23 8 25 22 31 7 26 22 38 7 27 22 45 6 28 22 51 6 29 22 57 5 30 23 2 5 Days December Diff. M. M. D. South 1 23 7 4 2 23 11 4 3 23 15 3 4 23 18 3 5 23 21 3 6 23 24 2 7 23 26 2 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 2 15 23 25 2 16 23 23 3 17 23 20 3 18 23 17 3 19 23 14 4 20 23 10 5 21 23 5 5 22 23 0 5 23 22 55 6 24 22 49 7 25 22 42 7 26 22 35 7 27 22 28 8 28 22 20 8 29 22 12 8 30 22 4 9 31 21 55    

A TABLE Of the Sun's Declination For the Year 1683. Being the Third after Bissextile or Leapyear.  
For the Year 1683. Days January Diff. M D. M. South 1 21 46 10 2 21 36 10 3 21 26 11 4 21 15 11 5 21 4 12 6 20 52 12 7 20 40 12 8 20 28 13 9 20 15 13 10 20 2 13 11 19 49 14 12 19 35 14 13 19 21 15 14 19 6 15 15 18 51 15 16 18 36 15 17 18 21 16 18 18 5 16 19 17 49 17 20 17 32 17 21 17 15 17 22 16 58 17 23 16 41 18 24 16 23 18 25 16 5 18 26 15 47 18 27 15 29 19 28 15 10 19 29 14 51 19 30 14 32 20 31 14 12  Days February Diff. M D. M. South 1 13 52 20 2 13 32 20 4 13 12 20 4 12 52 21 5 12 31 21 6 12 10 21 7 11 49 21 8 11 28 21 9 11 7 22 10 10 45 22 11 10 23 22 12 10 1 22 13 9 39 22 14 9 17 22 15 8 55 22 16 8 33 23 17 8 10 23 18 7 47 23 19 7 24 23 20 7 1 23 21 6 38 23 22 6 15 23 23 5 52 23 24 5 29 23 25 5 6 24 26 4 42 23 27 4 19 23 28 3 56 24 Days March Diff. M. D. M. So. No. 1 3 32 24 2 3 8 23 3 2 45 23 4 2 21 24 5 1 57 24 6 1 34 23 7 1 10 24 8 0 46 23 9 0 23 24 10 0 No. 1 24 11 0 25 24 12 0 49 23 13 1 12 24 14 1 36 23 15 1 59 24 16 2 23 23 17 2 46 24 18 3 10 23 19 3 33 23 20 3 56 23 21 4 19 23 22 4 42 23 23 5 5 23 24 5 28 23 25 5 51 23 26 6 14 22 27 6 37 23 28 6 59 23 29 7 22 22 30 7 44 22 31 8 6 22 Days April Diff M. D. M. North 1 8 28 22 2 8 50 22 3 9 12 22 4 9 34 21 5 9 55 21 6 10 16 21 7 10 37 21 8 10 58 21 9 11 19 21 10 11 40 20 11 12 0 20 12 12 20 20 13 12 40 20 14 13 0 20 15 13 20 19 16 13 39 19 17 13 58 19 18 14 17 19 19 14 36 18 20 14 54 18 21 15 12 18 22 15 30 18 23 15 48 17 24 16 5 17 25 16 22 17 26 16 39 17 27 16 56 16 28 17 12 16 29 17 28 16 30 17 44 15 Days May Diff. M. D. M. North 1 17 59 15 2 18 14 15 3 18 29 14 4 18 44 14 5 18 58 14 6 19 12 14 7 19 26 13 8 19 39 13 9 19 52 12 10 19 5 12 11 20 17 12 12 20 29 12 13 20 40 11 14 20 51 11 15 20 2 10 16 21 13 10 17 21 23 10 18 21 33 9 19 21 42 9 20 21 51 9 21 22 0 8 22 22 8 8 23 22 16 7 24 22 23 7 25 22 30 7 26 22 37 6 27 22 43 6 28 22 49 6 29 22 55 5 30 23 0 5 31 23 5 4 Days. June Diff. M D. M. North 1 23 9 4 2 23 13 3 3 23 16 3 4 23 19 3 5 23 22 2 6 23 24 2 7 23 26 1 8 23 27 1 9 23 28 1 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 1 15 23 26 2 16 23 24 2 17 23 22 3 18 23 19 3 19 23 16 3 20 23 13 4 21 23 9 4 22 23 5 5 23 23 0 5 24 22 55 6 25 22 49 6 26 22 43 6 27 22 37 7 28 22 30 7 29 22 23 7 30 22 16 8 Days July Diff. M. D. M. North. 1 22 8 8 2 22 0 9 3 21 51 9 4 21 42 9 5 21 33 10 6 21 23 10 7 21 13 11 8 21 2 11 9 21 51 11 10 20 40 11 11 20 29 12 12 20 17 12 13 20 5 13 14 19 52 13 15 19 39 13 16 19 26 13 17 19 13 14 18 18 59 14 19 18 45 15 20 18 30 15 21 18 15 15 22 18 0 15 23 17 45 16 24 17 29 16 25 17 13 16 26 16 57 16 27 16 41 17 28 16 24 17 29 16 7 17 30 15 50 18 31 15 32  Days August Diff. M D. M North. 1 15 14 18 2 14 56 18 3 14 38 18 4 14 20 19 5 14 1 19 6 13 42 19 7 13 23 19 8 13 4 20 9 12 44 20 10 12 24 20 11 12 4 20 12 11 44 20 13 11 24 21 14 11 3 21 15 10 42 21 16 10 21 21 17 10 0 21 18 9 39 21 19 9 18 22 20 8 56 22 21 8 34 22 22 8 12 22 23 7 50 22 24 7 28 22 25 7 6 22 26 6 44 22 27 6 22 23 28 5 59 23 29 5 36 23 30 5 13 22 31 4 51 23 Days September Diff. M. D. M. No. So. 1 4 28 23 2 4 5 23 3 3 42 23 4 3 19 24 5 2 55 23 6 2 32 23 7 2 9 24 8 1 45 23 9 1 22 23 10 0 59 24 11 0 35 23 12 0 South 12 24 13 0 12 23 14 0 35 24 15 0 59 23 16 1 22 24 17 1 46 23 18 2 9 24 19 2 33 23 20 2 56 23 21 3 19 24 22 3 43 23 23 4 6 23 24 4 29 24 25 4 53 23 26 5 16 23 27 5 39 23 28 6 2 23 29 6 25 23 30 6 48 23 Days October Diff. M. D. M. South 1 7 11 22 2 7 33 23 3 7 56 22 4 8 18 23 5 8 41 22 6 9 3 22 7 9 25 22 8 9 47 22 9 10 9 21 10 10 30 22 11 10 52 21 12 11 13 21 13 11 34 21 14 11 55 21 15 12 16 21 16 12 37 20 17 12 57 20 18 13 17 20 19 13 37 20 20 13 57 20 21 14 7 19 22 14 36 19 23 14 55 19 24 15 14 19 25 15 33 18 26 15 51 18 27 16 9 18 28 16 27 17 29 16 44 17 30 17 1 17 31 17 18 17 Days November Diff. M. D. M. South 1 17 35 16 2 17 51 16 3 18 7 16 4 18 23 16 5 18 39 15 6 18 54 15 7 19 9 14 8 19 23 14 9 19 37 14 10 19 51 13 11 20 4 13 12 20 17 13 13 20 30 12 14 20 42 12 15 20 54 11 16 21 5 11 17 21 16 11 18 21 27 10 19 21 37 10 20 21 47 9 21 21 56 9 22 22 5 8 23 22 13 8 24 22 21 8 25 22 29 7 26 22 36 7 27 22 43 6 28 22 49 6 29 22 55 6 30 23 1 5 Days December Diff M. M. D. South 1 23 6 4 2 23 10 4 3 23 14 4 4 23 18 3 5 23 21 2 6 23 23 2 7 23 25 2 8 23 27 1 9 23 28 1 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 1 15 23 26 2 16 23 24 3 17 23 21 3 18 23 18 3 19 23 15 4 20 23 11 5 21 23 6 5 22 23 1 5 23 22 56 6 24 22 50 6 25 22 44 7 26 22 37 7 27 22 30 8 28 22 22 8 29 22 14 8 30 22 6 9 31 21 57    

A TABLE Of the Sun's Declination For the Year 1684. Being Bissextile or Leapyear.  
For the Year 1684. Days January Diff. M. D. M. South 1 21 48 10 2 21 38 10 3 21 28 11 4 21 17 11 5 21 6 11 6 20 55 12 7 20 43 12 8 20 31 13 9 20 18 13 10 20 5 13 11 19 52 13 12 19 39 14 13 19 25 15 14 19 10 15 15 18 55 15 16 18 40 15 17 18 25 16 18 18 9 16 19 17 53 16 20 17 37 17 21 17 20 17 22 17 3 17 23 16 46 18 24 16 28 18 25 16 10 18 26 15 52 19 27 15 33 19 28 15 14 19 29 14 55 19 30 14 36 19 31 14 17  Days February Diff. M D. M. South 1 13 57 20 2 13 37 20 4 13 17 20 4 12 57 21 5 12 36 21 6 12 15 21 7 11 54 21 8 11 33 21 9 11 12 21 10 10 51 22 11 10 29 22 12 10 7 22 13 9 45 22 14 9 23 22 15 9 1 23 16 8 38 22 17 8 16 23 18 7 53 23 19 7 30 23 20 7 7 23 21 6 44 23 22 6 21 23 23 5 58 23 24 5 35 23 25 5 12 23 26 4 49 24 27 4 25 23 28 4 2 24  3 38 23 Days March Diff. M. D. M. So. No. 1 3 15 24 2 2 51 24 3 2 27 23 4 2 4 24 5 1 40 24 6 1 16 24 7 0 52 23 8 0 29 24 9 0 5 24 10 0 No. 19 23 11 0 42 24 12 1 6 23 13 1 29 24 14 1 53 24 15 2 17 23 16 2 40 24 17 3 4 23 18 3 27 23 19 3 50 23 20 4 13 24 21 4 37 23 22 5 0 23 23 5 23 23 24 5 45 22 25 6 8 23 26 6 31 22 27 6 53 23 28 7 16 22 29 7 38 22 30 8 0 22 31 8 22  Days April Diff. M. D. M. North 1 8 44 22 2 9 6 22 3 9 28 21 4 9 49 22 5 10 11 21 6 10 32 21 7 10 53 21 8 11 14 20 9 11 34 21 10 11 55 20 11 12 15 20 12 12 35 20 13 12 55 20 14 13 15 19 15 13 34 19 16 13 53 19 17 14 12 19 18 14 31 18 19 14 49 18 20 15 7 18 21 15 25 18 22 15 43 18 23 16 1 17 24 16 18 17 25 16 35 17 26 16 52 17 27 17 8 16 28 17 24 16 29 17 40 16 30 17 55 15    15 Days May Diff. M. D. M. North. 1 18 10 15 2 18 25 15 3 18 40 14 4 18 54 14 5 19 8 14 6 19 22 13 7 19 35 13 8 19 48 13 9 20 1 12 10 20 13 12 11 20 25 12 12 20 37 11 13 20 48 11 14 20 59 11 15 21 10 10 16 21 20 10 17 21 30 10 18 21 40 9 19 21 49 9 20 21 58 8 21 22 6 8 22 22 14 7 23 22 21 7 24 22 28 7 25 22 35 7 26 22 42 6 27 22 48 5 28 22 53 5 29 22 58 5 30 23 3 5 31 23 8 4 Days June Diff. M. D. M. North 1 23 12 4 2 23 16 3 3 23 19 3 4 23 21 3 5 23 24 2 6 23 26 1 7 23 27 1 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 1 14 23 27 2 15 23 25 2 16 23 23 3 17 23 20 3 18 23 17 3 19 23 14 4 20 23 10 4 21 23 6 5 22 23 1 5 23 22 56 5 24 22 51 6 25 22 45 6 26 22 39 7 27 22 32 7 28 22 25 7 29 22 18 8 30 22 10 8 Days July Diff. M. D. M. North 1 22 2 9 2 21 53 9 3 21 44 9 4 21 35 9 5 21 26 10 6 21 16 11 7 21 5 11 8 20 54 11 9 20 43 11 10 20 32 12 11 20 20 12 12 20 8 12 13 19 56 13 14 19 43 13 15 19 30 14 16 19 16 14 17 19 2 14 18 18 48 14 19 18 34 15 20 18 19 15 21 18 4 15 22 17 49 16 23 17 33 16 24 17 17 16 25 17 1 16 26 16 45 17 27 16 28 17 28 16 11 17 29 15 54 17 30 15 37 18 31 15 19  Day August Diff. M. D. M. North 1 15 1 18 2 14 43 18 3 14 25 19 4 14 6 19 5 13 47 19 6 13 28 19 7 13 9 20 8 12 49 20 9 12 29 20 10 12 9 20 11 11 49 20 12 11 29 21 13 11 8 20 14 10 48 21 15 10 27 21 16 10 6 21 17 9 45 22 18 9 23 21 19 9 2 22 20 8 40 22 21 8 18 22 22 7 56 22 23 7 34 22 24 7 12 22 25 6 50 23 26 6 27 22 27 6 5 23 28 5 42 23 29 5 19 22 30 4 57 23 31 4 34  Days September Diff. M. D. M. No. So. 1 4 11 23 2 4 48 23 3 3 25 24 4 3 1 23 5 2 38 23 6 2 15 23 7 1 52 24 8 1 28 23 9 1 5 24 10 0 41 23 11 0 18 24 12 0 South 6 23 13 0 29 24 14 0 53 23 15 1 16 24 16 1 40 23 17 2 3 24 18 2 27 23 19 2 50 24 20 3 14 23 21 3 37 23 22 4 0 24 23 4 24 23 24 4 47 23 25 5 10 23 26 5 33 23 27 5 56 23 28 6 19 23 29 6 42 23 30 7 5 23 Days. October Diff. M. D. M. South 1 7 28 22 2 7 50 23 3 8 13 22 4 8 35 22 5 8 57 22 6 9 19 22 7 9 41 22 8 10 3 22 9 10 25 21 10 10 46 22 11 11 8 21 12 11 29 21 13 11 50 21 14 12 11 21 15 12 32 20 16 12 52 20 17 13 12 20 18 13 32 20 19 13 52 20 20 14 12 19 21 14 31 19 22 14 50 19 23 15 9 19 24 15 28 18 25 15 46 18 26 16 4 18 27 16 22 18 28 16 40 17 29 16 57 17 30 17 14 17 31 17 31  Days November Diff. M D. M. South 1 17 47 16 2 18 3 16 3 18 19 16 4 18 35 15 5 18 50 15 6 19 5 14 7 19 19 14 8 19 33 14 9 19 47 14 10 20 1 13 11 20 14 12 12 20 26 13 13 20 39 12 14 20 51 11 15 21 2 11 16 21 13 11 17 21 24 10 18 21 34 10 19 21 44 10 20 21 54 9 21 22 3 8 22 22 11 8 23 22 19 8 24 22 27 7 25 22 34 7 26 22 41 7 27 22 48 6 28 22 54 5 29 22 59 5 30 23 4  Days December Diff M. M. D. South 1 23 9 4 2 23 13 4 3 23 17 3 4 23 20 3 5 23 23 2 6 23 25 2 7 23 27 1 8 23 28 1 9 23 29 0 10 23 29 0 11 23 29 0 12 23 29 1 13 23 28 2 14 23 26 2 15 23 24 2 16 23 22 3 17 23 19 3 18 23 16 4 19 23 12 4 20 23 8 5 21 23 3 5 22 22 57 5 23 22 52 6 24 22 46 7 25 22 39 7 26 22 32 8 27 22 24 8 28 22 16 8 29 22 8 9 30 21 59 9 31 21 50    

A TABLE Of the Differences of Meridian's Of the most considerable Places In the whole World, In respect of PARIS, With the Height of the Pole or Latitude of the same places.  
The Differences of Meridian's. Names of places. Differences of Meridian's Height of the Pole or Latitude H. M. D. M. Abbeville 0 1 add 50 0 Nor. Agra 5 26 subst. 26 0 Aiz Prov. 0 14 43 33 Aleppo 2 50 36 46 Alexandria Ae. 2 13 30 58 Amiens 0 0 49 55 Amsterdam 0 14 52 21 Angers 0 11 add 47 32 Antwerp 0 15 subst. 51 12 Arles 0 11 43 36 Arras 0 2 50 9 Archangello 2 49 65 30 Athens 1 35 7 40 Avignon 0 11 43 52 Bayonne 0 15 add 43 30 Blois 0 3 47 46 Bourdeaux 0 10 44 50 Bourges 0 0 47 20 Boulogne It. 0 40 subst. 44 20 Breast 0 28 add 48 23 Brussels 0 15 subst. 50 48 Names of Places Differences of Meridian's Height of the Pole or Latitude H, M. D. M. Cadis 0 37 add 36 16 Caen 0 9 49 11 Callais 0 2 50 57 Camboge 6 53 subst. 11 20 Cambray 0 3 50 5 Cap. of b. Es. 1 11 34 32 South Cap-Verd 1 34 add 14 27 North Chartres 0 4 48 30 Constantinop. 2 0 subst. 42 56 Copenhagen 0 42 55 43 Cracovi 1 14 50 10 Dantzick 1 12 subst. 54 22 Dieppe 0 4 add 49 56 Dijon 0 12 subst. 47 28 Dole 0 13 47 20 Douai 0 3 50 15 Dunkirk 0 0 51 1 ½ Ferare 0 41 subst. 44 54 Isle de Fer. 1 38 add 28 10 Fez 0 31 33 10 Lafoy Fleche 0 10 47 42 Names of Places. Differences of Meridian's Height of the Pole or Latitude H. M. D. M. Florence 0 40½ subst. 43 41 Franc●ort M. 0 27 50 4 Fribourg 0 2● 48 16 Gaunt 0 6 51 2 Geneva 0 17 46 20 Goa 4 58 15 30 Grenoble 0 15 45 11 Hambourgh 0 34 53 43 Haure de Gr. 0 8 add 49 7 Jerusalem 2 36 subst. 32 0 Hispaam 4 16 36 14 R●chelle 0 13 add 46 11 Leiden 0 12 subst. 52 12 Lima Per. 5 31 add 12 20 South Lisbon 0 50 ●8 40 North London 0 8 51 32 Loudun 0 8 48 0 Louvain 0 13 subst. 50 50 Lucques 0 37 43 40 Luxembourg 0 20 49 38 Lions 0 11 45 46 Names of Places. Differences of Meridian's Height of the Pole or Latitude D. M. D. M. Macon 0 11 subst. 46 20 Madrid 0 25 add 40 10 Malta 0 53 subst. 35 40 Malaca 6 45 2 20 S. Malo 0 18 add 48 39 Mantove 0 37 subst. 45 1 Marseilles 0 14 43 15 Martinique 4 15 add 14 14 Meaux 0 2 subst. 48 56 Messina 0 58 38 21 Metz 0 19 49 10 Messique 7 8 add 20 30 Milan 0 31 subst. 45 15 Modene 0 38 44 39 Monaco 0 23 43 39 Montpellier 0 6 43 37 Munick 0 50 48 58 Namur 0 13 subst. 50 26 Nancy 0 19 48 39 Nantes 0 16 46 13 Naples 0 56 41 5 Narbonne 0 3 43 6 Nevers 0 2 47 19 Names of Places. Differences of Meridian's Height of the Pole or Latitude H. M. D. M. Ostend 0 3 subst. 51 16 Orleans 0 2 add 47 54 Ormus 4 0 subst. 27 35 Paris 0 0 48 51 Parma 0 36 subst. 44 45 Pavia 0 31 44 58 Padova 0 42 45 31 Pexing Chi. 7 45 40 0 Perigeux 0 9 add 45 34 Perpignan 0 5 subst. 45 52 Pernanibouc. 2 57 add 7 40 South Perouge 0 44 subst. 42 56 North Pisa 0 38 43 9 Plaisance 0 33 44 53 Poitiers 0 6 add 47 7 Portobelo. 5 41 9 55 Prague 0 58½ subst. 50 40½ Quebec 4 36 add 47 0 S. Quentin. 0 6 subst. 49 46 Rennes 0 16 add 47 58 Rheims 0 9 subst. 49 12 Riga 1 31 56 52 Names of Places. Differences of Meridian's Height of the Pole or Latitude H. M. D. M. Rome 0 47 subst. 41 54 Roven 0 4 add 49 27½ Saints 0 11½ add 45 38 Savone 0 28 subst. 44 18 Sienne 0 41 43 11 Sirrah 3 42 34 14 Smirna 2 56 38 22 Strasbourgh 0 24 48 31 Stettin 0 54 53 34 Stockholm 1 7 59 30 Tangier 0 56 add 35 25 Tholouse 0 2 43 29 Toleda 0 26 39 52 Tours 0 6 47 35 Tourney 0 5 subst. 50 32 Tutin 0 25 44 9 Toulon 0 17 43 Valencienne 0 4 subst. 50 20 Verdun 0 38 45 33 Vienna 1 2 48 22    

ADVERTISEMENT.  
I Have begun to travel for the Impression of an entire Work of the Conique Sections, where you shall found not only all the most excellent things has been discovered in this patt of the Mathematics, but also a great number of new Properties which I have discovered, and whereof I have published every one upon different occasions.  
The Method of the Principal Demonstration is particular to myself, and I have used it to abridge very much this whole Work; I have given an Essay of this New Method, which I caused to be Printed, concerning the Conique Sections in the Year 1675.  
See here in few words the order of these Books and what they contain.  
The First Book contains the Lemmas that are necessary for this Method.  
The Second contains the original of the Three Conique Sections, with the Properties of their Diameters, and all that which depends on a Line cut harmonically or into two equal parts, the original of the Asymptotes, and in the end divers Problems depending on these Principles.  
The Third explains the relation of the Ordinate Lines to their Diameters, with the Rectangles of the parts of the Diameters, Parameters, and that which concerns the Tangents.  
The Fourth makes that appear which is most considerable touching the Asymptotes.  
The Fifth Book is filled with near▪ 40 most curious Propositions upon the Conique Sections.  
In the Sixth Book is treated of equal and like Sections.  
The Seventh is continued all along on that which is called the lest and greatest.  
The Eighth treats of the Foci of the Three Sections.  
In the Ninth Book is taught certain, most plain, and most useful Methods for the Description of these Sections.  
Than I show you in an Appendix after what manner we aught to resolve the Conique Sections that have for their Bases, Paraboles, Hyperboles and Elipses, and also all Cylinders of the same Species.  
Afterwards I demonstrate that by the same Method, and without difficulty, we may demonstrate all the Properties of the Conique Sections of whatsoever compounded kind they are, I draw them from their Cones or Pyramids, which I distinguished by Cones of the first kind, of the second kind, of the third kind, etc. These sorts of Cones have for their Bases Circles of all these kinds, and than having explained after what manner we may tender all the Lemmas of the first Book universal, I explain in the same manner that which is the Second Book, and in one part of the following the Sections of all Cones of any kind.  
The Demonstrations are the same as for the Sections of the first Cone, which is that which has the Circle of the first kind for its base: After that I see not any thing that we can desire more universal, nor more easy on this same Atiere.   

The Approbation.  
THis Treatise of the Gnomiques, or the Art of Drawing of Sun Dial's, composed by Mr. De la Hire, hath been read to the Assembly of the Academy Royal of Sciences, made the 9th of May 1682.  
J. B. Du Hamel Secretary of the Academy of Sciences.   

THe Approbation seen, permitted to be Imprinted, made the 23th of May 1682.  
DE LA REYNIE.   

THis Geometrical way of Mr. De la Hire for Drawing Sun- Dial's on Fixed Planes is universal. The Instruments that are used to perform the Practice thereof are only a Plain Scale and a Pair of Compasses for finding the Substylar and Equinoctial Lines on the given Plane, with the situation of the Axis of the Style in respect of the said Plane, the Observations being made by the Sun from several Shadows of the Point of a Fixed Style set upon the said Plane. And by a Level and Plumb Line for finding the situation of the Meridian and Horizontal Lines of the Place on the Plane of the Dial; which Practice of Dialling he chief demonstrates from the Intersections of Planes, the Doctrine whereof is contained in the 11th Book of Euclid's Elements. And because he has given no particular Example of this general way of Dialling in this Treatise, I have thought good to add this Example following:  
A particular Example of this General Method of Dialling Illustrated by Numbers.  
There is a Style fixed on a Fixed Plane, whose perpendicular height is S P 1 inch 64 parts, or 164 parts. Now, the Sun shining, if there be taken Two Points of Shadow of the top of the Style S on the said Plane, as A and B, on that day when the Sun's Declination is 21 deg. 40 min. South, and that P B the Distance of the Point B, from P the foot of the Style may be 55 parts, and the Distance of the Point A from P may be P A 366 parts, and that the Distance of the Point A from B may be B A 415 parts.  
Now from these things given, it is required to found the situation of the Substylar and Equinoctial Lines, and to found the Centre of the Dial on the said Plane, and to determine the position of the Axis of the Dial by Calculation.  
[The Geometrical Operation of this is to be found in the Sixth Chapter, page 16th of this Treatise, and in the 5th Figure.]  

First in the Right-angled Triangle S P B.  
There is given P S 164 parts, the perpendicular height of the Style, and P B 55 parts the distance of the Point of Shadow B from P the foot of the Style;  
To found the Vertical Angle P S B, and the Hypotenuse S B, which is the length of the Ray of the Sun from the top of the Style S to the Point of Shadow B,  
The Proportion is as S P 164 parts, Is to P B 55 parts:  
So is the Radius, To the Tang. of the Angle PSB 18 d. 32 m.  
And as the Sine of the Angle PSB 18 d. 32 min.  
Is to PB 55 parts,  
So is the Sine of 90 deg.  
To S B 173 parts.   

Secondly, In the Right-angled Triangle S P A,  
There is given the perpendicular P S 164 parts  
And the Base P A 366 parts,  
To found the Vertical Angle PSA and the Hypotenuse S A.  
The Proportion is, As S P 164 parts.  
Is to P A 366 parts.  
So is the Radius  
To the Tangent of the Angle PSA 65 d. 52 m.  
And as the Sine of the Angle P SA 65 d. 52 m.  
Is to P A 366 parts,  
So is the Sine of 90 degrees,  
To S A 401 parts.   

Thirdly in the Obliquangular Triangle s d b.  
There is given the two sides s b 173 parts and ss. d 100 parts, taken at pleasure, and the contained Angle d s b 68 deg. 20 min. the Compliment of the Sun's Declination.  
To found the other two Angles ss d b and s b d, and the third side d b.  
From— 180 deg. 00 min.  
Take the given Angle d s b 68 20  
And there shall remain— 111 deg. 40 min.  
The half whereof is— 55 deg. 50 min.  
Than the greater side is s b 173 parts,  
And the lesle side is s d 100 parts.  
Than as the Sum of the sides 273 parts,  
Is to the difference of the sides 73 parts:  
So is the Tangent of 55 deg. 50 min.  
To the Tangent of 21 deg. 30 min.  
The Sum is s d b 77 deg. 20 min.  
The Difference is s b d 34 deg. 20 min.  
Than as the Sine of the Angle s b d 34 d. 20 m.  
Is to the Sine of the Angle d s b 68 d. 20 m.  
So is s d 100 parts,  
To d b 164 parts.   

Fourthly, In the Obliquangular Triangle s d a▪  
There is given the two sides s a 401 parts, and s d 100 parts, and the contained Angle d s a,  
To found the other two Angles s d a and s a d, and the third side d a.  
Than the greater side is s a 401 parts,  
And the lesle side is s d 100  
Than as the Sum of the sides 501  
Is to the Difference of the sides 301  
So is the Tangent of 55 deg. 50 min.  
To the Tangent of 41 31  
The Angle s d a— 97 21  
The Angle s a d— 14 19  
Than as the Sine of the Angle s a d 14 d. 19m.  
Is to the Sine of the Angle a s d 68 20  
So is s d 100 parts,  
To d a 376 parts.   

Fifthly, In the Obliquangular Triangle BAT.  
There is given the Base B A 415 parts, and the two sides A T 376 parts, and B T 164 parts,  
To found the Segments of the Base O A and O B, and the perpendicular O T.  
The greater side is A T 376 parts,  
The lesle side is B T 164 parts.  
The Sum of the sides— 540 parts  
The Difference of the sides 212  
Therefore as the Base B A— 415 parts,  
Is to the Sum of the sides— 540  
So is the difference of the sides— 212  
To the difference of the Seg. of the Base 276  
Therefore to B A— 415 parts  
Add— 276  
The Sum is 691  
The half Sum is A O— 345 ½.  
And from B A 415 parts,  
Subtract— 276  
The Difference is 139  
The half is B O— 69 ½.  
The side B T is 164 parts,  
And B O is— 69 ½  
The Sum is 233 ½ parts, Log. 2.367356  
The Difference is 95 parts, Log. 1.977724  
Sum Log. 4.345080  
Half Sum Log. 2.172540  
Log. of O T 149 parts.   

Sixthly, In the Obliquangular Triangle BPA,  
There is given the Base B A 415 parts, and the two sides P A 366 parts and P B 55 parts,  
To found B x and A x the Segments of the Base and the perpendicular P x.  
The greater side P A 366 parts,  
The lesle side P B— 55  
Their Sum is— 421  
Their Difference is 311  
Therefore as the Base B A 415 parts,  
Is to the Sum of the sides 421 parts:  
So is the difference of the sides 311  
To— 315  
Therefore to B A— 415 parts  
Add— 315  
The Sum is— 730  
The ½ is the greater Segment A x. 365  
And from B A— 415 parts  
Subtract— 315  
The Difference is— 100  
The ½ is the lesle Segment B x— 50  
The lesle side B P 55  
The Segment B x 50  
The Sum is— 105 Log. 2.021189  
The difference is 5 Log. 0.698970  
The Sum of the Log.— 2.720159  
The ½ Sum is the Log. of P x 23, 2. 360079½  
B O 69 parts  
B x 50  
O x 19   

Seventhly, In the Right-angled Triangle P G I,  
There is given the Hypotenuse PI 1000 parts, and the side P G 19 parts,  
To found the side G I  
The greater side P I 100 parts  
The lesle side P G 19  
Their Sum— 119 parts, Log. 2.075547  
Their difference 81 parts, Log. 1.903485  
The Sum of the Log. 3.984032  
The ½ Sum is the Log. of GI' 98 parts, 1.992016   

Eightly, In the Right-angled Triangle K G O,  
There is given KG 164 parts and O G 23 parts, To found the Angle G K G, and the side O K.  
The Proportion is as G K 164 parts,  
Is to G O 23 parts:  
So is the Radius, To the Tangent of the Angle G K O 7 d. 59 m.  
And as the Sine of 7 deg. 59 min.  
Is to G O 23 parts:  
So is the Radius,  
To O K 166 parts.   

Ninthly, In the Obliquangular Triangle OKD,  
There is given the Base O K 166 parts, and O D 149 parts, equal to the Radius of the Semicircle L D T and K D 98 parts,  
To found the Angle O K D.  
O D 149 parts  
K D 98 parts  
Their Sum 247  
Their Difference 51  
Than as the Base O K 166 parts  
Is to the Sum— 247  
So is the difference— 51  
To— 76  
Therefore to O K 166 parts  
Add 76  
The Sum is 242  
The ½ Sum is O y 121  
And from O K 166 parts  
Subtract— 76  
The difference is— 90  
The ½ difference is K 745  
Than as D K 98  
Is to K y 45  
So is the Sine of 90 degrees,  
To the Sine of the Angle K D y 27 d. 20 m.  
The Compliment is the Angle DK y 62 d. 40m.  
Subtract the Angle G K O 7 d. 59 m.  
Remains the Angle D K z 54 d. 41. m.   

Tenthly, In the Right-angled ●●●●ngle D K z,  
There is given the Hypotenuie D K 98 parts and the Angle D K z 54 degrees 41 minutes,  
To found the perpendicular D z and the Base K z:  
For as the Sine of D z K 90 deg.  
Is to the Hypotenuse D K 98 parts  
So is the Sine of the Angle D K z 54 d. 41 m.  
To the perpendicular D z 57 parts.   

Eleventhly, In the Right-angled Triangle P G Q,  
There is given the Base P G 19 parts,  
And the perpendicular G Q 80 parts,  
To found the Hypotenuse P Q.  
The proportion is as G Q 80 parts,  
Is to P G 19 parts:  
So is the Radius,  
To the Tangent of the Angle P G Q 13 deg. 20 min.  
And as the Sine of the Angle P Q G 13 d. 20 m.  
Is to P G 19 parts:  
So is the Sine of 90 degrees,  
To P Q 82 parts.  
Than from N P 164  
Take— N r 57  
Remains— r P 107   

Twelfthly, To found P C;  
There is given N r 57, and r M 82, and N P 164. The Proportion is,  
As N r 57  
Is to r M 82  
So is NP 164  
To P C— 236  
last, To found P E,  
The Proportion is,  
As P C 236  
Is to P N 164  
So is P N 164  
To P E 114  
Now if the Plane of the Dial be on an upright Wall, the Meridian Line may be found as in the 13th Chapter of the First Part, Figure 14th, and the Hours may be set on the Equinoctial Line by the First Chapter of the Second Part, Fig. 21.  
And Six Hours being given, the rest may be found by the Third Chapter of the Second Part.  
Or Four Hours and the Equinoctial Line being given, the rest may  
35 Figure  
be found by the Ninth Chapter of the Second Part.  
And than the Dial may be drawn as in the 35th Figure.   
FINIS.