DIALLING Performed Instrumentally by our HEMISPHERE in PLANE; Projected and first fitted by Mr William Oughtred, and laid down according to his Method formerly published for this very subject.  
TOGETHER, With twenty one several Diagrams or Schemes, demonstratively showing the reason and groundwork of all Dialling: As also, how to know, distinguish and set down the Hour-lines for both Faces of all Planes at one working.  
By a Practitioner in the same Art.  
PONDERIBUS SONITUM JACOBUS RIMEUS   printer's or publisher's device 
LONDON, Printed by William Dugard; and are to be sold by William Hope, at the sign of the Anchor, on the North-side of the Royal Exchange. 1652.   

TO THE READER.  
HEre I present unto your view a small Tract of Dialling performed by Mr Oughtred's projection of our Hemisphere in Plane, according to his own method in his book formerly published for this very subject, which by often practice & trial I have found so useful and satisfactory in all cases whatsoëver (to this Art belonging) that I have fitted twenty several Diagrams or Schemes for ten several proposed Planes. The greater 10 showing the drawing & setting down the hour-lines Substyler and Poles height for every of them by the Hemisphere or Instrument itself. The other or lesser 10 show how to distinguish and set down the hour-lines which are in use for either face of any of these Planes.  
In which ensuing discourse I assume to myself nothing but the making and fashioning of the Diagrams, and a larger Explanation (demonstratively) of what Mr Oughtred hath in brief set down, wherein I resemble one lighting his Candle at another man's, which as it cannot diminish or take away the lustre from his, so I must acknowledge it hath given a light unto mine.  
Here is further to be noted, for a great encouragement to the Practitioner, that if he but barely know the rules of the Globe as Aequinoctial, Meridian's, Parallels of declination, Poles, etc. He shall most plainly see (as in a glass) what the fundamental reason of Dialling is, neither can he be hoodwinked (in the use of this Instrument) from the true and full knowledge thereof.  
I have likewise for better conveniency fitted or ordered the Diagrams (to this tract belonging) to be stitched or bound up by themselves, that so you may lay open before you what Diagram you pleas, by the particular Explanation whereof according to its number in the Book you shall most easily learn the order of working, which I conceiv to be a readier way to the learner then to bind them together and so be forced to turn from page to page for every letter. Lastly these Diagrams or Schemes (being made only for Demonstrationsake) are not so large as to expect every thing to be exactly performed according to Arithmetical Calculation; yet I presume no man shall find such difference from the truth as may make the least sensible error in Time, for had I made them of a greater circumference the work would have been so much the truer.  
Which as I have found both useful and satisfactory to myself, so if it prove but acceptable to thee, I have my desire.  
R. L.   

A Definition of certain Terms or Words hereafter used for the better understanding of the ensuing Discourse.  

A Plane.  
BY a Plane is meant any thing which is flat or smooth, be it Wall, board, Stone, or any thing else whereon lines may be conveniently drawn; which must be conceived to have length and breadth, but no thickness; that so when the Sun forsake's the one face or side of any Plane, it must presently appear upon the other: for every Plane hath two faces or sides, whereon some hour-lines may be drawn for good use; Except the Horizontal, which can have but one.  
Again, every Plane must cut or divide the Globe of the Univers into two equal halus: As the Horizontal Plane divide's the upper Hemisphere which we see, from the lower which we see not: Or the Equinoctial Plane may be said to divide the Northern half of the Globe from the Southern half thereof, or the Meridian Plane; to divide the Eastern half from the Western half thereof: and so do all other Planes whatsoever, according to their several situations.   

The Horizontal line.  
THe Horizontal line, is a line always level, or parallel to the Horizon; being indeed a level-line.   

A Perpendicular line.  
A Perpendicular line is a line crossing the Horizontal, or any other line at right angles, but most proper, that line which crosseth the Horizontal at right angles, pointing out the Zenith and Nader.   

Declination.  
THe Declination of a Plane is, what a Plane varie's or differs from beholding the due North or South, either to the East or Westwards.   

Inclination.  
THe Inclination of a Plane is the quantity of an Arch in degrees and minutes which any plane is seated, or lifted up above the level of the Horizontal line.   

Pole of the world.  
THe Pole of the world is represented in the Instrument by the letter P, through which all the Meridian's pass.   

Pole of the Plane.  
THe Pole of the Plane must ever be 90: 00″ from the Plane itself, even as the Pole of the world is 90: 00″ from the Equator.   

Pole of the Meridian of the Plane.  
SO must also the Pole of the Meridian of the Plane be 90: 00″ from the Meridian itself; which will be far better understood in every of the several Diagrams following.   

Meridian or Hour-circle.  
BY a Meridian is meant any of the Hour-circles in the Instrument, whereof there are but 12 drawn, thereby pointing out the 24 hours, being 15: 00″ each from other, which is an hour of time, and are sometimes termed Meridian's, and sometimes Hour-circles, as occasion shall be given.   

The proper Meridian of the Plane.  
THe proper Meridian of the Plane is that very Meridian, either real in the Instrument, or pricked down, which passeth through the Pole of the Plane.   

Meridian of the place.  
THe Meridian of the Place is that Meridian in this our Horizon, which makes (the Sun being thereupon) Midday and Midnight with us.   

The Prime Vertical Circle.  
THe Prime Vertical Circle is the Circle or Azumith which passeth through the Zenith and the points of East and West in the Instrument, represented by the line W z E.   

The Scale.  
THe Scale of the Instrument is the Radius or Semidiameter thereof, viz. z S being divided into 90: 00 unequal degrees.   

The limb of the Instrument.  
THe limb of the Instrument is often named, by which you are to understand is meant the innermost Circle of all in the Instrument, which is in every of the Diagrams or Schemes divided into 360: 00″.   

An Hour-line.  
BY an Hour-line is meant any straight line drawn from the Pole of the Plane, through the Intersection or crossing of any hour-circle with the Plane unto the limb of the Instrument.   

The Substiler line.  
THe Substiler line is likewise a straight line drawn from the Pole of the Plane, through the intersection or crossing of the proper Meridian of the Plane, with the Plane itself, unto the limb of the Instrument, over which line the Style or Gnomon is always placed.   

Diagram.  
BY the word Diagram is meant a Geometrical figure, whereof for every proposed plane (in the ensuing Tract) there are two of the same number from nᵒ A to ndeg; K, (both included).  
 hemisphere in plane for the latitude 51: 30″   

An explanation of the Instrument.  
THis Instrument is our Hemisphere in Plane for the Latitude 51: 30 first projected by Mr William Oughtred, and by him (as also by Mr Gunter) a Method is set down for the making thereof, wherein the Circle  

N E S W is the Horizontal Circle of the Plane.  
N z S the Meridian, or North and South line.  
W z E the Prime Vertical Circle, or East and West line.  
z represents the Zenith, and sometimes the centre of the Instrument.  
P the Pole of the world.  
W ♈ E the Equator.  
The Arch ♋ represents the Summer Tropic.  
The Arch ♑ represents the Winter Tropic.  
z S the Semidiameter is divided into 90: 00″ unequal degrees.   
Next there are drawn through the Pole in the the Instrumen P 12 Meridian's which divide the whole face of the Plane into 24 parts, which are 15: 00″ each from other (being an hour of time) therefore they are taken and deemed for hour-circles also, and figured accordingly, which will be sufficient for our present matter. These several Diagrams following, being made only for demonstrationsake; But the Instrument would be much more useful and exact if it were larger, and every Meridian, or every 5th Meridian of the 360: 00″ were therein drawn, as hereafter will appear.  
Further you are to note, that this Instrument is most proper for all dials with centres, in which the hour-lines, substiler and stile do meet, whereof their natures, qualities and number follow:   

First, The Horizontal Plane.  
THe Instrument itself is an Horizontal dial: for if straight lines be drawn from the centre z to the several intersections of the hour-circles with the limb (which represents the Horizontal Plane) and in z erect a Style or Gnomon 51: 30″ above the Plane, equal to N P right over the Meridian z P n It shall be a perfect Horizontal Dial.   

An upright Plane fully beholding the North or South.  
THis Plane is likewise represented by the line W z E in the Instrument, being the Prime Vertical Circle, or East and West line thereof, being right to the Meridian N z S, therefore to the Southwards a due South face, and to the Northwards a due North face.   

The Vertical Inclining Plane.  
ALL Planes which are not levelly to, or with the Horizon, nor yet perpendicular thereto, but lie sloping betwixt the Zenith and Horizon, are called Inclining Planes: And if their Horizontal lines shall be parallel to the Prime Vertical W z E, that is, fully beholding the North or South, they are then called Vertical Inclining Planes: These Planes may Incline either to the North-part of the Horizon, or to the South; and each Plane hath two faces whereon some hour-lines may be drawn for use; The upper face lying open to the Zenith, and the lower to the Nader, wherein you are to consider the height of the Pole above the Plane by comparing the Inclination of the Plane to the Horizon, with the Latitude of the place: and will be better understood by the demonstration following, wherein A c A represent the Horizontal line.  

z c n the Perpendicular line.  
N S the two Poles of the world, whereof N is elevated 51: 30″ above A according to our Latitude.  
A c A the Equinoctial line or Plane.   
The other lines are several Planes proposed according to their several Inclinations to the Horizon.  
I will first propose a Plane whose Inclination is 20: 00″ in the North part of the Horizon, viz. the line B c B; now hereby you may plainly perceiv that the North-pole N is elevated above the upper face of this Plane but 31: 30″; for if I take the angle of the Plane's Inclination A c B 20: 00″ out of A c N 51: 30″, there will rest the angle B c N 31: 30″ for the North-Pole's height above the upper face, and therefore the South-pole S must be elevated so much above the lower face of the said Plane as the angle B c S shows, being the same with the angle B c N.  
If the Plane proposed were D c D, Northerly likewise, you may perceiv by the situation of the Plane that the North-pole N will be elevated above the lower face of this plane 20: 00″; for if I take the angle A c N 51: 30″ out of A c D 71: 30″ there will rest the angle N c D 20: 00″ for the North-pole's height above the lower face, and and therefore so much for the South-pole above the upper face thereof as the angle D c S shows.  
Again, if the Plane proposed were F c F 18: 30 Inclination in the Southern part of the Horizon. Here, because F in the South-part of the Horizon is lifted up above the Horizontal line 18: 30″ according to the Inclination thereof; therefore F being the other end of the Plane, must be so much depressed under it, so that if I add the angle AcF 18: 30″ to the angle A c N 51: 30″, I shall have the angle F c N 70: 00″ for the North-pole's height above the upper face, and so much must the South-pole's height be above the lower face of this Plane, as the angles F c N and F c S show.  
If the Plane proposed were the line N c S, it must of necessity be a Polar Plane, because it is the same with the Axis of the world, so that neither Pole will be elevated above this Plane.  
If the Plane proposed were the line A c A it must needs be an Equinoctial Plane, and so the Poles will be elevated 90: 00″ above either face of the said Plane.   

The upright declining Plane.  
THese Planes, because they do not directly behold the South or North, nor yet the East or West fully, but are Planes which lie betwixt both, (and upright withal) are called upright declining Planes, which have likewise two faces whereon some hour-lines for use may be drawn, the one face lying open to the Southwards, the other face to the Northwards.   

The Declining Inclining Plane.  
IF a Plane shall Decline any way from the due North or South either to the East or Westwards, and is not withal an upright plane, but is such a plane as with its Declination inclineth to the Horizon, it is then called a Declining Inclining Plane.   

The Meridian Plane.  
THose Planes whose Horizontal lines are the same, or parallel to the Meridian, and withal are upright Planes, are generally called Meridian Planes.   

The Meridian Inclining Plane.  
BUt those Planes whose Horizontal lines are the same, or parallel to the Meridian, and withal the Plane shall Incline to the Horizon, than it is called a Meridian Inclining Plane.   

The Polar Plane.  
THose Planes wherein a line may be drawn parallel to the Axis of the world are called Polar Planes (although with some distinction) because that line pointeth unto the two Poles; and these Planes are always parallel to some one of the hour-circles in the Instrument; For the direct Polar Plane is parallel to the hour-circle of 6, and the Meridian plane is parallel to the hour-circle of 12.   

The Polar declining Plane.  
BUt the Polar declining Plane is a Plane which is parallel to some one of the hour-circles in the Instrument betwixt 12 and 6, because of its Inclination to the Pole, and Declination from the prime Vertical, or, East and West line in the Instrument.  
Next the Substiler and hour-lines in a Meridian Plane make an Arch with the Horizontal line of the plane, equal to the Latitude of the place: In direct Polar planes they make an Arch of 90: 00″, but in Polar declining planes they make an Arch with their Horizontal line of their planes more than the latitude of the place, and less than 90: 00″.  
Again, in direct Polar planes the Substiler line is the same with the Meridian or 12 a clock line: In a Meridian plane it is the same with the hour of 6, but in a Polar declining plane it is betwixt 12 and 6, according to the inclination or distance of the proper Meridian of the plane with or from the Meridian of the place.  
I should now proceed to the setting down those several planes upon the Instrument in their order; but because some of them are plain regular dials, falling under every man's apprehension, and some comprehended in what shall be spoken of the others, I will omit these several following.  
The first therefore which I shall omit or pass over, is the Horizontal plane, whereof the Instrument itself is a perfect pattern, as in folio 5 is declared.  
The next following is the due South and North plane, which is likewise represented in the Instrument by the line W z E, and is fully comprehended and taught in any of the Diagrams nᵒ A B or C, of the upright declining planes.  
The next following is the Vertical inclining plane, which is likewise sufficiently explained in any of the Diagrams of number D E F G, of the declining inclining planes, there being but the same method and order of working for the one as for the other.  
The last which I pass over are the direct Polar and Equinoctial planes, being very plain regular dials, and indeed, not so proper for the Instrument as aforesaid.    

How to set down upon the Instrument these several Planes following, viz.  

First, The upright declining Plane.  
THe first therefore which I will begin withal is, how to set down upon the Instrument any upright, or upright declining Plane: First you must be very careful which way your declination tends; for if you mistake herein, your work must be extreme falls.  
This upright declining Plane is proposed to decline 30: 00″ to the Eastwards of the South; therefore I reckon in the limb of the Instrument 30: 00″ to the Northwards of E, and 30: 00 to the Southwards of W, in the greater Diagram, number A, and so draw the line A z A for the Horizontal line of the Plane. Next I cross it at right angles with the line C z E for the perpendicular line thereof; for you must know that the Horizontal line of the Plane, and the Perpendicular line thereto, in all upright planes must be the two first lines: and in all incliners the Horizontal line must be the first, and a line crossing the Horizontal at right angles must be the second (which upon the matter is all one) before you can proceed any further.  
This being done, you perceiv the Southern face of this plane to behold the Southeast quarter of the Instrument, and the Northern face thereof to behold the Northwest quarter of the Instrument, which is according to the declination proposed.   

How to set upon the Instrument any Vertical Incliner, or any Declining Inclining Plane.  
FOr all sorts of Incliners you must have recours unto the Semidiameter or graduated scale in the Instrument z S.  
Now this Plane Decline 24: 20″ Westerly from S. is proposed to Incline 36: 00″ Northerly.  
First, because this plane declines to the Southwestwarde, therefore in the greater Diagram number D I reckon from W in the limb 24: 20″ Northerly, and so much from E southerly, and so I draw the Horizontal line of the Plane A z A, which I cross at right angles with the line B z B.  
Next, I consider the Inclination proposed 36: 00″ northerly, which I take out of the Scale from S to e and put it over in the northern part of the line B z B from B to T, or, (which is much the better way) I take the compliment to 36: 00″ being 54: 00″ out of the Scale from z to e and put it over in the Northern part of the line from z to T, because the Inclination proposed is northerly, of which you must have as much care as to the Declination; which point T will be the highest point in this plane, being nearest the Zenith z. Now from the two ends of the Horizontal line A and A, and through this point T; having found a centre, I draw the Arch of a circle A T A, which Arch is the Declining Inclining plane proposed: And these two foregoing examples will be sufficient for setting down or placing upon the Instrument all Planes whatsoëver.   

To find the place of the Pole of every upright Plane.  
NOw after your Plane is justly set upon your Instrument with its horizontal and perpendicular lines, the next thing requisite to know, is the place of the Pole of your Plane, which must be carefully placed; for from the Pole of every Plane (having a centre) the Substyler and all the hour-lines must be drawn to the limb of the Instrument.  
You are likewise to know, that the Pole of every Plane is always 90: 00″ from the Plane itself, and ever, in all upright planes, at that end of the perpendicular line in the limb of the Instrument, to which the face of your Plane is opposite, for which you intent your Dial, because it is the Semidiameter, and therefore the Radius or Sine of 90: 00″ which is general.  
Thus in the greater Diagram nᵒ A the Southeasterlie face of the plane being proposed, therefore at the Southern end of the perpendicular line, viz. at C in the limb is the place of the Pole of that face of the plane. If the Northwesterlie face (which is the contrary face) were intended for principal, than the Pole thereof would be at E in the Northern end thereof.   

To find the place of the Pole of every Inclining, or of every Declining Inclining Plane.  
IF the Plane be a Vertical Incliner, or a Declining inclining Plane, as in the greater Diagram nᵒ D is proposed, than the Pole of the Plane must be always in the line which crosseth the Horizontal at right angles, betwixt the limb and the centre z, so that you know z T being the compliment of Inclination 54: 00″ if I take 36: 00″ out of the Scale from z downwards, it will extend to d, which being put over in the same line B z B from z, it will reach unto m, which must be the place of the Pole of this Plane, because z T is 54: 00″ of the Scale from z, and z m is 36: 00″ from z, which makes the line T z m to be 90: 00″ betwixt the Plane at T, and the Pole thereof at m; and this is general for all Planes of this nature.   

To draw the Hour-lines upon all Planes with Centres.  
YOu are to mark diligently where the Hour-circles in the Instrument do cross or cut your Plane; for if a straight Ruler be laid to the Pole of the Plane, and to those several intersections or cross of the Hour-circles, with the Plane and straight lines drawn from those several intersections or cross to the limb of the Instrument, those straight lines shall be the Hour-lines for your Plane, and by reckoning the degrees in the limb of the Instrument you shall know, how much every hour-line is distant either from the Horizontal line of the Plane, or from the Perpendicular line thereof, or from the Substiler, or any one hour-line from another; by which you may transfer the same lines upon your Plane by a table made thereby as you pleas.   

To draw the Substiler line upon all Planes with Centres.  
YOu are to mark diligently where the Pole of your Plane falls to be either upon an hour-circle, or meridian, or betwixt any two; for if the Pole of the Plane fall upon a Meridian, as in the Diagram number D it doth fall just upon the Meridian or hour-circles of one in the point m, than this hour-circle of one must be the proper Meridian of the Plane, because it passeth through the Pole of the Plane at m. Now if you lay a ruler to the Pole of the Plane m, and to the intersection or crossing of this Meridian with the plane itself, in the point L, and draw the straight line m L K to the limb of the Instrument, this line shall be the Substiler line.  
Furthermore, because the proper Meridian of the Plane, and the hour-circle of one, in this Diagram nᵒ D, be one and the same, therefore the Substiler and hour-line of one, must be one and the same line also, and serve for both.  
But if the Pole of the Plane fall betwixt two meridians or hour-circles in the Instrument, as in the Diagram nᵒ A it doth, viz. betwixt the hour circles of 9 and 10 in the Instrument, at the point C: in the limb (because it is an upright Plane) then for better demonstration sake, I will in the Diagrams following, (where need shall require) make or prick one down, which in this Diagram nᵒ A is C L P E, which is the proper meridian of this Plane, because it passeth through C the Pole thereof. Now laying a ruler to the Pole C, and to the Intersection of this proper meridian with the Plane itself in the point L, and draw the line C L K, it must be the Substiler line for this Plane.   

To find the height of the Pole above all Planes having Centres.  
THe next and last thing to know, is, how much the Pole is elevated above your Plane: Now the height of the Pole above any Plane, is the measure of the arch of the proper meridian of the Plane, which is intercepted or included betwixt the Pole in the Instrument and the Plane: To know the quantity of which arch in degrees and minutes, first reckon in the limb of the Instrument, from either end of the proper meridian 90: 00″, mark where this acount end's, and from thence through the centre z draw an obscure, or occult line to the opposite point in the limb, then take the distance betwixt the proper meridian and the centre z in the occult line, and see how many degrees and minutes it is in the Scale from z downward; for the compliment thereof being taken out of the Scale from z, and extended in the occult line on the other side the centre, shall point out the pole of this proper meridian; which had, lay a ruler to the Pole of this proper meridian, and to the Pole in the Instrument P, and make a mark in the limb where the ruler cut's. Again lay the ruler to the Poles of the proper meridian, and to the intersection or crossing of this meridian with the Plane, and make a second mark in the limb where the ruler cut's; for the degrees and minutes reckoned in the limb betwixt these two marks, will be the exact measure of the arch of the proper meridian of the plane, which is intercepted or included betwixt the Pole and the Plane, which is the height of the Pole above all planes having centres.  
Thus in the Diagram nᵒ A, from either end of the proper meridian of the Plane, viz. from E or C in the limb, I reckon 90: 00″ which end's at A or A, being in this Diagram the ends of the horizantal line, and therefore the same line A z A serves for the occult line before spoken of (and will be always so in all upright planes). Next I find the distance betwixt the proper meridian and the centre z, viz. z L to be in the Scale 21: 00″; so that taking 69: 00″ the compliment thereof out of the Scale from z downwards, and extending it in the occult line on the other side the centre, it will point out the place of the Pole of this proper meridian to be at h.  
Now laying a ruler to the Pole of this proper meridian at h, and to the Pole in the Instrument P make a mark in the limb where the ruler cut's, viz. n; or (which is all one) draw the obscure line h P n; then keeping the ruler at h, turn it to the point L, where the proper Meridian cut's the Plane, and make the mark o in the limb, or the line h L oh, and you shall find the degrees and minutes betwixt n and o in the limb to be 32: 40″, which is the measure of the arch of the proper meridian betwixt P and L, which is the North-pole's height above the Northern face of the Plane, and therefore the South-pole must be elevated so much above the Southern face thereof.  
Again, in the Diagram nᵒ D, the proper meridian of the Plane which passeth through the Pole of the Plane, is the meridian or hour-circle of one, viz q m i L q: from q therefore, either end of this hour-circle, which is (now the proper meridian of the Plane) I reckon 90: 00 in the limb, which end's at D, from which point D through the centre z I draw the occult line D z D; Next I find the distance betwixt the centre z and the proper meridian in the obscure line to be z i 9: 00″ of the Scale; therefore taking the compliment thereof 81: 00″ out of the Scale from z downwards, it will extend on the other side the centre, in the occult line from z to h, which will be the place of the Pole of this proper meridian; now laying a ruler to h the Pole of the proper meridian, and to P the Pole in the Instrument, draw the obscure line h P n, then laying the ruler to h, and to the point L, where the proper meridian crosseth the Plane, and draw the line h L oh, and you shall find the degrees and minutes betwixt n and o in the limb to be 17: 00″, which is the measure of the arch of the proper meridian betwixt P and L, which is the North-pole's height above the upper face of this declining inclining Plane, and therefore the South-pole must be elevated so much above the lower face thereof.   

To place the Style or Gnomon.  
NOw the height of the Pole above the Plane being found for right placing the Style or Gnomon you must know it must ever point directly to the pole, which is elevated above that face of your plane for which you intent your Dial.   

To know what hour-lines must be in use for either face of any Plane whatsoever.  
Because you shall not encumber nor pester your Dial with useless or unnecessary hour-lines, you shall most plainly perceiv, after your Plane is truly placed upon your Instrument, what hour-lines will be in use for either face thereof, only by observing the intersections, or cross of the Tropiques and Parallels of the Sun's Declination with the Plane, which is very plain and of excellent use.  
For example in the Diagram no A▪ you perceiv the Southern face of the Plane to be A C A in the Instrument, and the Northern face thereof to be A E A; now imagining the Sun's diurnal arch to be in the Summer Tropic, you may perceiv that from its rising, which is about ¼ before 4 till it come to the point d, where the Tropic crosseth the plane; it will shine upon the Northern face thereof, which is until about 40″ past 4 in the morn, from this point d: it immediately appears upon the Southern face thereof, and so continues till the same Tropic cut's the Plane again in the point c, which you see will be amongst the hour-circles about ½ an hour past 2, and then forsaking the Southern face, it appears again upon the Northern face, and so continues till Sunset, which is about ¼ past 8 in the Evening.  
So that for the Northern face, the hour-lines in use will be first 4 in the morn, and then from 3 to 8, (both included); And for the Southern face, all the hour-lines from 5 in the morn, to 4 afternoon, (both included); because you see that when the Sun is in the parallel of 20: 00″ South Declination, the Sun will shine upon that Southern face from its rising till its setting, being about a little past 4 afternoon, because that parallel 8, 20, 4, crosseth not the plane at all.  
Again, in Diagram nᵒ D, the Sun being in the Summer Tropic, it will shine upon the Northern or lower face of that plane, from its rising to the crossing of that Tropic, with the Plane in the point a, which is about 25 past 6 in the morn, from thence in the same moment of time, it will appear upon the Southern or upper face thereof, and continues thereupon till it cross the Plane again in the point b. which is about 30″ past 7 in in the Evening, and then departing from that face, shine's upon the Northern face of the said Plane till Sunset; But when the Sun is in 10: 00″ South Declination, viz. the parallel g, i, 10, f, it will shine upon the Northern face from its rising at g, which is a little before 7 in the morn, till it cross the Plane in the point i, which is about ¼ past 7 in the morn; So that for the Northern, or lower face of the Plane, the hour-lines in use will be from 4 to 7 in the morn, (both included) and only 8 in the evening; and for the Southern or upper face, all the hour-lines from 7 in the morn to 7 at evening, both included.  
Further you must know, that by the Instrument you shall always have 12 of the 24 hour-lines pointed out unto you by the intersections or cross of the hour-circles with the Plane.  
It will fall out in all southerly Inclining, or in southerly Inclining Declining Planes, that although the Northern face of a Plane be the face proposed to find out the hour-lines and Substiler for, yet in these planes the hour-lines will be drawn in and by the Instrument; for the lower or southerly face thereof, which is the contrary face to the intended, which you shall fully understand in the Diagrams nᵒ E & G, following.  
You are to note, that the Tables following for every Diagram, are the several distances in degrees and minutes taken out of the limb of the Instrument, which the Substiler and hour-lines are distant from the Perpendicular line of the plane in all upright Planes, and in all Incliners, from the line crossing the Horizontal at right angles, (which upon the matter is all one line).  
Except the two Polar Declining Planes, whose hour-lines in the Tables take their several distances from the Substiler lines.  
Lastly, in the 10 lesser Diagrams you have three whole circles, or rundles, in each whereof the greater (which represents the Plane) is divided into 24 hour-lines by the Instrument, in which you have two scrolls, whereof the outermost is the Dial for the face proposed, and in the innermost scroll are contained the hour-lines which are in use for the contrary face; and the two lesser circles or rundles are the dials for both faces of the proposed planes, whereof, in all Inclining planes, the rundle Nᵒ V is for the upper, and the rundle Nᵒ L stands for the lower face of any of them.    

Here followeth a particular Explanation of the several Diagrams following, viz.  

Diagrams Nᵒ A & A.  
Being the first example of an Upright Declining Plane, proposed to decline 30: 00″ easterly from South.  
According to what hath been formerly taught I draw the line  

A z A for the Horizontal lines of this Plane.  
C z E for the Perpendicular lines of this Plane.  
C the Pole of the Plane because a southerly face is proposed.  
C L P E the proper Meridian of the Plane.  
L the point of this Meridian's cutting or crossing the Plane.  
C L K the Substiler line of this Plane.  
A z A serves for the occult line, being 90: 00 from C or E.  
z L 21: 00″ the Meridian's distance from z in the occult line.  
z h 69: 00 the Pole of this proper Meridian at h.  
n o in the limb, being the measure of the arch of this proper Meridian, betwixt P and L is 32: 40″.   
The Pole's height above this Plane 32: 40″.  
The hour-lines which are in use for the Northerly face of this Plane, are four in the morn, and all the hour-lines from 3 to 8 (both included).  
And for the Southern face thereof, all the hour-lines from 5 in the morn to 4 afternoon, both included.  
Here in this Diagram nᵒ A, because the South-easterly face of the Plane is the face proposed to make a Dial for, therefore the hour-lines and substiler, as they are drawn in the semicircle of the Instrument A E A shall be the perfect Dial, (without alteration) to be transferred upon the Southern face thereof, which may easily be done by making a Table out of the limb of the Instrument of the several distances of the hour-lines and substiler from the Perpendicular line of the Plane. Thus you shall find  


The Table.  
The hour of— 5: 75: 00″— 11: 10: 30″  
The hour of— 6: 57: 20″— 12 Perpendicular  
The hour of— 7: 45: 00″— 1: 13: 20″  
The hour of— 8: 34: 00″— 2: 29: 45  
The hour of— 9: 26: 00″— 3: 52: 30″  
The hour of— Substiler line— 22: 20″— 4: 79: 40″  
The hour of— 10: 18: 00″    
Now if you would know how to set off the hour-lines which are in use for the North-westerly face, which is the back or contrary face to the proposed upon that face of your Plane, which hour lines really by the Instrument you have not.  
Sweep a circle as I have done in the lesser Diagram of this nᵒ A, viz. A B A E, and set therein by help of the Table above, all the hour-lines from 5 in morn, to 4 afternoon, both included; in the lower semicircle A E A just as they lie in the Instrument, which will be the perfect Dial for the Southeasterlie face proposed.  
Now if you continue these 12 hour-lines through the centre to their opposite quadrants in the upper semicircle A B A, you shall divide the whole circle into 24 hours, that so you may take thereout the hour-lines which are in use for the Northwesterlie face. As I have done and distinguished by the two Scrolls; whereof the greater or outermost Scroll is the Dial for the face proposed, and in the lesser Scroll are the hour-lines for the contrary face of the same Plane.  
And when you pleas to transfer these hour lines in the lesser scroll which are for the contrary face, upon that face of your Plane, you must place them contrary to what they are in the lesser scroll, because they are the hour-lines for the contrary face to the proposed, as I have done in the little rundle nᵒ 2, being the perfect Dial for the Northwesterlie, or contrary face, as the little rundle nᵒ 1 is the Dial for the proposed.  
The better to conceiv a reason hereof, imagine the two little rundles nᵒ 1 nᵒ 2 were both cut out of the Paper, and their backsides clapped together, their Horizontal and Perpendicular lines agreeing aright, and centre opposite to centre, than you will plainly perceive, that the Substiler line K z m in the rundle number 2 will be one line with the Substiler line L z n in the rundle nᵒ 1.  
As also the hour-lines will meet, and make one line with each hour-line of its opposite quadrant, and this is general for all Planes whatsoever.   

An explanation of the Diagrams Nᵒ B & B.  
THe second example of an Upright Declining Plane proposed to decline 63, 0″0 Westerly from South.  
Here you have  

A z A, the Horizontal lines of this Plane.  
B z B the perpendicular lines of this Plane.  
C, the place of the Pole, because the Southwesterly face is proposed, being in the South-west quarter.  
C L P E, the proper Meridian of the Plane.  
L, the point of this Meridian's crossing, or cutting the Plane.  
C, L, K, the subtler line.  
A z A, servs for the occult line, being 90, 0″0, from C, or E.  
z L, 36, 0″0, the Meridian distance from z in the occult line.  
z h, 54, 00 the Pole of this proper Meridian at h.  
n, o, in the limb, being the measure of the arch of this proper Meridian, betwixt P & L, is 16. 30″,  
The Poles height above this Plane 16. 30″.   
The hour-lines in use for the Southwesterly face, are from a little past 10, or indeed from 11 to 8 (both included) because 11, is the first perfect, or real hour-line. And for the Northeasterly or contrary face, all the hour-lines from 4 in the morn, to 11 before noon, both included.  
Here you have the hour-line of 9 at night given you by the Instrument for one of the 12 which for this Southwesterly face is useless, in respect the Sun continues not so long above our Horizon, yet it must not be left out of either the Table, or greater rundle of the lesser Diagram nᵒ B, because without it you cannot divide the whole rundle (which represents the Plane) into 24 hours, which being done, you will be enabled thereby to take thereout, such hour-lines as are in use for the Northeasterly or contrary face; as the lesser scroll in the lesser Diagram nᵒ B shows, but they are to be placed contrary to what they are in the lesser scroll, as the little rundle, for the Northeasterly face shows, by the reason formerly given.  


The Table.  
The hour of— 10: 83: 00″— 35: 40″ Substiler.  
The hour of— 11: 33: 30″— 5: 38: 00″  
The hour of— Perpendicular 12: 00; 00″— 6: 41: 45  
The hour of— 1: 15: 15— 7: 47: 30″  
The hour of— 2: 23: 00″— 8: 55: 20″  
The hour of— 3: 29: 00″— 9: 69: 00″  
The hour of— 4: 33: 30″—     

An Explanation of the Diagrams Nᵒ C & C.  
THe third example of an Upright Declining Plane proposed to decline 63: 00″ easterly from North.  
Here you have  

A z A the Horizontal lines of this Plane.  
B z B the Perpendicular lines of this Plane.  
C the Pole of the Plane in the North-east quarter of the Instrument, because the North-east face is set for principal.  
C L P E the proper meridian of this Plane.  
L the point of this Meridian's crossing the Plane.  
C L K the Substiler line.  
A z A serves for the occult line 90: 00″ from C or E.  
z L 36: 00″ the Meridian's distance from z in the occult line.  
z h: 54, 00″ the Pole of this proper Meridian at h.  
n o in the limb, being the measure of the Arch of this proper Meridian, betwixt P and L is 16: 30″.  
The Pole's height above this Plane— 16: 30″.   
The hour-lines in use for the North-easterly face, are from 4 in the morn, to 11 before noon (both included); and for the Southwesterly, or contrary face; all the hour-lines from a little after 10, or indeed from 11 to 8 (both included) because the hour-line of 11 is the first real or perfect hour-line.  
Note that these two Planes nᵒ B & nᵒ C are both one and the same Plane, only here is all the difference, that in the Diagram nᵒ B the South-westerly face is proposed for principal; and in the Diagram nᵒ C the North-easterly face is set for principal, as by the Poles of each face doth appear; by which you may plainly perceiv, that what face soever of any Plane is set for principal, the hour-lines and Substiler thereof, being taken out of the Instrument just as they lie therein, shall be the perfect dial, without any alteration for the face proposed; but the hour-lines for the contrary face must be contrarily placed, etc.  
All which doth most plainly appear in the two lesser Diagrams, nᵒ B nᵒ C.  
Furthermore, because these two Planes have the same Declination, therefore at one working you have the hour-lines, Substilers and and Poles height for both, as by the Tables and hour-lines in use for each do appear, all agreeing in every respect the one with the other.  


The Table.  
The hour of— 10: 83: 00″—: 35: 40″ Substiler.  
The hour of— 11: 33: 30″— 5: 38: 00″  
The hour of— Perpendicular 12. 00: 00″— 6: 41: 45  
The hour of— 1: 15: 15— 7: 47: 30″  
The hour of— 2: 23: 00″— 8: 55: 20″  
The hour of— 3: 29: 00″— 9: 69: 00″  
The hour of— 4: 33: 30″—     

An explanation of the Diagrams Nᵒ D & D.  
THe first example of a Declining Inclining Plane, falling betwixt the Pole and Horizon:  
Proposed to Decline 24: 20″ westerly from the South.  
Proposed to Incline 36: 00″ Northerly.  
Here you have likewise  
A z A the Horizontal line of the Plane.  
B z B the line crossing the Horizontal at right angles.  
z T 54: 00″ from z being the compliment of Inclination.  
A T A the Declining Inclining Plane proposed.  
z m 36: 00″ from z being the place of the Pole of the Plane at m.  
q m i L q the proper Meridian of the Plane, being in this Diagram the same with the hour-circle of one.  
D z D the occult line 90: 00″ from either q in the limb.  
z i 9: 00″ the Meridian's distance from z in the occult line.  
z h 81: 00″ from z the Place of the Pole of this Meridian at h.  
L the point of this Meridian's crossing the Plane.  
m L K the Substiler line, and hour-line of one also.  
n o in the limb, being the measure of the arch of this proper Meridian betwixt P and L is 17: 00″.  
The North-pole's height above the upper or South-westerlie face is 17: 00″, so much also must the South-pole be elevated above the lower or Northeasterlie face, as by the situation of the Plane in the Diagram doth appear.  
The hour-lines in use for the upper face are from 7 in the morn, to 7 in the evening (both included).  
And for the lower face from 4 in the morn to 7 before noon (both included) and only 8 at evening, when the Sun is in the Summer Tropic.  


The Table.  
The hour of— 8: 63: 00″— 2: 10: 40″  
The hour of— 9: 42: 20″— 3: 5: 30″  
The hour of— 10: 32: 00″— Perpendicular  
The hour of— 11: 25: 00″— 4: 1: 40″  
The hour of— 12: 20: 00″— 5: 12: 40″  
The hour of— Substiler— 1: 15: 40— 6: 33: 50″  
——— 7: 75: 30″     

An explanation of the Diagrams Nᵒ E & E.  
THe second Example of a Declining Inclining Plane falling to the Southwards of the Zenith.  
Proposed to Decline 45: 00″ Westerly from South.  
Proposed to Incline 38: 30″ southerly.  
Here you have  
A z A the Horizontal line of this Plane.  
B z B the line crossing the Horizontal at right angles.  
z T 51: 30″ from z, being the compliment of Inclination.  
A T A the Declining Inclining Plane proposed.  
z m 38: 30″ from z, being the place of the Pole of the Plane at m.  
q m i L q the proper Meridian of this Plane.  
D z D the occult line 90: 00″ from either q in the limb.  
z i 36: 00″ the Meridian's distance from z in the occult line.  
z h 54: 00″ from z the Plane of the Pole of this Meridian at h.  
L the point of this Meridian's crossing the Plane.  
m L K the Substiler line.  
n o in the limb, being the measure of the arch of this proper Meridian betwixt P and L is 62: 40″.  
The North-Pole's height above the upper face 62: 40″.  
Here in this Diagram nᵒ E, when the Sun is in the Summer Tropic, it will begin to appear upon the upper, or Northwesterlie face of this Plane, just at 7 in the morn, but I have made the hour-lines in use for this upper face, from 8 in the morn, to 8 at evening (both included) because the hour-line of 8 in the morn is the first real or perfect hour-line.  
And the hour-lines in use for the lower or Southeasterlie face are from 4 in the morn, when the Sun is in the Summer Tropic to the point f in the Winter Tropic, which is about half an hour past 2, therefore from 4 to 2, (both included).  
Here, as I have formerly said, (although in this Diagram) the Northwest face is the face proposed and set down, as the Pole of its Plane at m, in the Northwest quarter of the Instrument plainly shows, yet the hour-lines in and by the Instrument are all drawn for the Southeast, or lower face, as the lesser scroll in the lesser Diagram nᵒ E shows, being the same with the hour-lines and Substiler in the Instrument, viz. from 4 in the morn, to 3 afternoon: Now if by these you divide your plane into 24 hours, all the hour-lines which shall be in use for the upper face, viz. from 8 in the morn to 8 in the evening (both included) shall be the perfect dial for this upper face, because it is the face set in the Instrument for principal, as the greater scroll and little rundle nᵒ V shows, being all one.  
But the hour-lines which are in use for the lower face, from 4 to 2, must be contrarily placed, etc. as the little rundle nᵒ L shows, and this is general for all southerly Incliners whatsoever.  


The Table.  
The hour of— 4: 83: 00″— Perpendicular,  
The hour of— Substiler line— 72: 40″— 10: 4: 30″  
The hour of— 5: 69: 30″— 11: 21: 00  
The hour of— 6: 56: 00″— 12: 38: 00  
The hour of— 7: 42: 00″— 1: 54: 00″  
The hour of— 8: 27: 40″— 2: 69: 30″  
The hour of— 9: 12: 00″— 3: 83: 30″     

An explanation of the Diagrams Nᵒ F & F.  
THe third Example of a Declining Inclining Plane, falling betwixt the Pole and Zenith.  
Proposed to Decline 24: 20″ westerly from South.  
Proposed to Incline 75: 00″ Northerly.  
Here you have  
A z A the Horizontal line of the Plane.  
B z B the line crossing the Horizontal at right angles.  
z T 15: 00″ being the compliment of Inclination.  
A T A the Declining Inclining Plane proposed.  
z m 75: 00″ from z, being the place of the Pole of the Plane at m.  
q m i L q the proper Meridian of the Plane.  
D z D the occult line 90: 00″ from either q in the limb.  
z i 15: 00″ the Meridians' distance from z.  
z h 75: 00″ from z the place of the Pole of this Meridian at h.  
L the point of this Meridian's crossing the Plane.  
m L K the Substiler line.  
n o in the limb, being the measure of the arch of the proper Meridian, betwixt P and L is— 20 00″.  
The North Pole's height above the lower or Northeasterlie face 20: 00″; therefore the South Pole must be elevated so much above the upper or South-westerlie face thereof.  
The hour-lines in use for the upper face, are from 8 in the morn to 7 in the evening (both included).  
And for the lower face, from 4 in the morn to 8 afore noon (both included) and only 8 in the evening, when the Sun is in the Summer Tropic.  


The Table.  
The hour of— 8: 60: 00″— 16: 20 Substiler.  
The hour of— 9: 27: 00″— 2: 17: 30″  
The hour of— 10: 10: 20″— 3: 22: 30″.  
The hour of— Perpendicular— 4: 29: 40″.  
The hour of— 11: 00: 40″— 5: 38: 20″  
The hour of— 12: 07: 00″— 6: 52: 30″.  
The hour of— 1: 12: 30″— 7: 79: 30″.     

An explanation of the Diagrams Nᵒ G & G.  
THe fourth example of a Declining Inclining Plane, falling near unto the Winter Tropic.  
Proposed to Decline 24: 20″ easterly from North.  
Proposed to Incline 14: 20″ southerly.  
Here you have  
A z A the Horizontal line of the Plane.  
B z B the line crossing the Horizontal at right angles.  
z T 75: 40″ from z, being the compliment of Inclination.  
A T A the Declining Inclining Plane proposed:—  
z m 14: 20 from z, being the place of the Pole of the Plane at m.  
q m i L q the proper Meridian of the Plane.  
D z D the occult line 90: 00″ from either q in the limb.  
zi 8: 00″ being the Meridian's distance from z.  
z h 82: 00″ from z, being the place of the Pole of this Meridian at h.  
L the point of the Meridian's crossing the Plane.  
m L K the Substiler line.  
n o in the limb, being the measure of the arch of this proper Meridian, betwixt P and L is 116: 00″ southerly, from P to the edge of the Plane at L.  
Here you must remember what was said at first, that every Plane cut's or divide's the globe of the Univers into two equal halus, which considered, if you take 116 the distance of the North-pole from the Southern edge of the Plane at L out of 180: 00″ the semicircle, there will remain 64: 00″ for the elevation of the same Pole above the Northern edge of the same face, being the upper face of the Plane. North-pole's height above the upper face 64: 00″.  
Now the situation of this Plane in the instrument you see, that all the while the Sun hath above 15: 00″ North Declination, it shine's upon the upper face of this Plane, from its rising to its setting (when the days are at longest with us) for you see the parallel of 10: 00″ North Declination, crosseth the Plane but a very small time before the Sun setting, when it is therein.  
So that for the upper or Northeasterlie face you must have all the hour-lines from 4 in the morn to 8 at evening (both included).  
And for the lower or South-westerlie face, first 8 and 9 in the morn, and then from 2 to 7 in the evening, which hour-lines will be of very small use, because the Plane incline's so much Southerly.  
Note likewise, that this Plane is a southerly Declining Inclining Plane, as the Plane in the Diagram nᵒ E is, therefore the same method is to be observed, as is there taught, which is also most plainly demonstrated in the lesser Diagram of this number G.  


The Table.  
The hour of— 8: 79: 10″— Perpendicular.  
The hour of— 9: 64: 30″— 2: 4: 40″  
The hour of— 10: 50: 30— 3: 20: 00″  
The hour of— 11: 37: 20″— 4: 35: 40″  
The hour of— Substiler— 34: 20″— 5: 52: 00″  
The hour of— 12: 24: 00″— 6: 69: 00″  
The hour of— 1: 9: 30″— 7: 85: 10″     

An explanation of the Diagrams Nᵒ H & H.  
Being an example of a Meridian Inclining Plane, proposed to Incline 53: 00″ westerly.  
I have said before, that for all Planes of this nature, the Meridian line, or North and South line in the Instrument N z S must be one and the same line with, and serve for the Horizontal line of the Plane; And therefore the Prime Vertical, or East and West line in the Instrument W z E stands for the line which must cross the Horizontal line of the Plane at right angles: The rest of the work differs nothing from the former Declining Inclining Planes.  
So that here the Meridian  
N z S is the Horizontal line of this Plane.  
W z E stands for the line crossing the Horizontal at right angles.  
z T 37: 00″ from z, being the compliment of Inclination.  
N T S the Meridian Inclining Plane proposed.  
z m 53: 00″ from z, being the place of the Pole of this Plane at m.  
q m i L q the proper Meridian of this Plane.  
D z D the occult line 90: 00″ from either q in the limb.  
z i 35: 00″ this Meridian's distance from z in the occult line.  
z h 55: 00″ from z the place of the Pole of this Meridian at h.  
L the point of this Meridian's crossing the Plane.  
m L K the Substiler line.  
n o in the limb, being the measure of the arch of this proper Meridian betwixt P and L is 28: 30″.  
The North-pole's height above the upper face 28: 30″.  
The hour-lines in use for the Eastern or upper face, are from 4 in the morn to 2 afternoon (both included).  
And for the Western or lower face, all the hour-lines from 1 to 8 in the evening (both included).  


The Table.  
The hour of— 12: 90: 00″— 6: 32: 40″  
The hour of— 9: 65: 00″— 7: 40: 00″  
The hour of— 2: 34: 45—: 44: 30″ Substiler  
The hour of— 3: 7: 50″— 8: 47: 20″  
The hour of— Perpendicular.— 00: 00″— 9: 54: 20″  
The hour of— 4: 10: 45— 10: 63: 00″  
The hour of— 5: 23: 20″— 11: 74: 15  
If the Plane shall incline easterly as this doth westerly, the same order in working is to be observed in every respect.     

An Explanation of the Diagrams Nᵒ I & I.  
THe first example of a Polar Declining Plane,  
Proposed to Decline 52: 00″ westerly from South.  
Proposed to Incline 64: 00″ Northerly.  
In these Polar declining Planes, you are to find out by the Instrument two things in chief, and then the rest of the work will differ nothing from the making of an ordinary Meridian, or direct Polar Dial.  
The first is the Arch betwixt the Substiler and the Horizontal line of the Plane in degrees and minutes.  
The second is the Inclination or distance betwixt the proper Meridian of the Plane, and the Meridian of the place; both which will be easily given you by the Instrument.  
Thus I set down upon the Instrument according to the Declination and Inclination proposed.  

A z A the Horizontal line of the Plane.  
B z B the line crossing the Horizontal at right angles.  
z T 26: 00″ from z, being the compliment of Inclination.  
A P T A is the Plane proposed, which in this Diagram falls to be the same with the Meridian or hour-cirlce of 9, cutting or crossing the Pole of the Instrument P; and therefore must be a Polar Plane, and neither Pole elevated above either face.  
z m 64: 00″ from z, being the place of the Pole of this Plane at m, which falls just upon the hour-circle of three in the afternoon, and therefore the Substiler, and hour-line of 3 must be one and the same line, and serve for both; as formerly in the Diagram nᵒ D was demonstrated.  
q P m q the proper Meridian of this Plane.   
Now (as in every of the former Diagrams) if from the Pole of the Plane m, through the point of this Meridian's crossing the Plane, which is at P the Pole of the Instrument, you draw a right line to the limb, it must be the Substiler line, and hour line of three also, which is represented by the line m P K, whose Horizontal Arch betwixt A and K in the limb, is 61: 00″ which is the first thing required.  
The next is to observe where the Pole of your Plane falls to be either upon an hour-circle (as here it doth) or betwixt any two, for the distance thereof from the Meridian of the Plane (which in the Instrument is ♣ z S) being taken and converted into degrees and minutes, shall give the Inclination or distance betwixt the proper Meridian of the Plane, and the Meridian of the place, which is here represented by the Arch of the Equator ♈ m in the South-west quarter of the Instrument, and therefore 3 hours of time, or 45: 00″ from the North and South line in the Instrument, which is the Meridian of the place; And this is the second thing required.  
A K 61: The Horizontal Arch.  
♈ m 45: 00″ Inclination or distance betwixt the two Meridian's.  
Here the South-west face of this Plane is proposed for principal or upperface.  
The hour-lines in use for which face, are from 10 before noon, to 8 in the evening (both included).  
And for the lower face from 4 in the morn, to 8 aforenoon (both included).  
Now to take the Table out of the Instrument, observe the place of the Pole of your Plane (which in these planes will always fall upon the Equator) observe I say, the place of the Pole of your Plane, amongst the Meridian's or hourcircles, which justly reckoned either way to the next hour-circles one by one, by allowing 14: 00″ of the Equator for every hour of time, (and so proportionably for any part of an hour, if the Pole fall betwixt two meridian or hour-circles) you may make a Table thereby, to transfer your hour-lines upon your Plane, which must be the several distances of the hour-lines from the Substiler line of the Plane.  


The Table.  
The Substiler & hour of— 3: 00″: 00″  
The Substiler & hour of— 4 & 2: 15: 00″  
The Substiler & hour of— 5 & 1: 30: 00″  
The Substiler & hour of— 6 & 12: 45: 00″  
The Substiler & hour of— 7 & 11: 60: 00″  
The Substiler & hour of— 8 & 10: 75: 00″    
Now for delineating your dial upon your Plane, (the Table being made) first, as in the lesser Diagram of this number I, I draw the Horizontal line A A, next at any convenient point therein, viz. at B; I sweep the Arch C D. and finding out a chord of 61: 00″ for the arch betwixt the Horizontal line and Substiler, I put it over from C to E, through which point E, I draw the Substiler line BHEF, which must be the hour-line of 3 also, to which all the rest of the hour-lines must be parallel. Next at some convenient point in the Substiler line, viz. at H, I sweep a semicircle, whose semidiameter H I, I proportion to be the height of the Style above the Plane; then through this point I, at right angles to the Substiler line, I draw the line G I G, & another line parallel to this A H L, by which two parallel lines, you shall set down your hour-lines upon your Plane, by the Table and this Canon.   

The Canon.  
As H I Radius to I K the Tangent of 75: 00″, being the distance of the hour-lines of 8 & 10 from the Substiler.  
So is H I 50/100 of an inch to I K 1 87/100 of the same measure to be set off from either-side the substiler, in the two parallel lines and so draw the lines through the Plane, as the line 8 K 8 shows. And so for all the rest the same method holds.   

An explanation of the Diagrams K & K.  
THe second Example of a Polar Declining Plane,  
Proposed to Decline 62: 30″ Easterly from South.  
Proposed to Incline 70: 00″ Northerly.  
According to which proposal I draw  
A z A for the Horizontal line of the Plane.  
B z B for the line crossing the Horizontal at right angle.  
z T 20: 00″ from z, being the compliment of Inclination.  
A T P A is the Polar Declining Plane proposed, because it passeth through the Pole of the Instrument P.  
z m 70: 00″ from z, the place of the Pole of this Plane at m.  
q P m q the proper Meridian of this Plane.  
P the point of this Meridian's crossing the Plane.  
m P K the Substiler line, whose Horizontal Arch from A to K is 56: 00″ in the limb of the Instrument.  
The Inclination or distance betwixt the proper Meridian of the plane, and the Meridian of the place is the Arch of the Equator v m, being in time about three hours 48: 00″ before noon, which is 12: 00″ after 8 in the morn, or 57: 00″ in measure, as the Pole of the plane at m shows.  
The Horizontal Arch 56: 00″.  
Distance betwixt the two Meridian's 57: 00″.  
The hour-lines in use for the upper or Southeasterlie face are, from 4 in the morn, to one after noon (and no more) except the plane be very large. And for the lower face from 3 to 8 at evening (both included).  


The Table of the distances of the several hour-lines from the Substiler.  
The hour of— 3: 78: 00″— Substiler line.  
The hour of— 4: 63: 00″— 9: 12: 00″  
The hour of— 5: 48: 00″— 10: 27: 00″  
The hour of— 6: 33: 00″— 11: 42: 00″  
The hour of— 7: 18: 00″— 12: 57: 00″  
The hour of— 8: 03: 00″— 1: 72: 00″  
— 2: 87: 00″    
Thus the Table being made, the delineation or drawing of your lines upon the Plane, differ nothing from that of the former Diagram.   
FINIS.   

If any shall desire to have this Instrument ready printed off in paper for his use or practice, he may be furnished with what quantity he desire's by William Hope Bookseller at the Blue Anchor on the backside the Exchange, London.  
If in brass, he may have this or any other for the Mathematical practice, made by Walter Hayes, at the Cross-Daggers in moorfield's, near Bethlem-gate.   

ERRATA.  
In the Epistle line 20, read circles for rules. Page 9, line 13, for those read these. Pag. 24, line 13, for plane read place. Page 28, line 5, for now, the read now by the. Page 32, line 2, for 〈◊〉: 00″ read 15: 00″.