Mr. DESARGVES. universal way of making all manner of Sun dials▪ Published by Daniel king▪ & Sold by Isaake Pridmore at y, golden falcon in the▪ A● 1659.    

Mr. De Sakgue's UNIVERSAL WAY OF DYALING.  
OR Plain and easy directions for placing the axletree, and marking the hours in sun-dials, after the French, Italian, Babylonian and Jewish manner.  
Together with the manner of drawing the lines of the signs, of finding out the heighr of the Sun above the Horizon, and the East rising of the same, the Elevation of the Pole, and the position of the Meridian.  
All which may be done in any superficies whatsoever, and in what situation soever it be, without any skill at all in Astronomy.  
By DANIEL KING Gent.  
LONDON, Printed by Tho. Leach and are to be sold by Isaac Pridmore at the Golden falcon in the Strand, near the New Exchange, 1659.   

TO THE illustrious GEORGE VILLIERS, Duke, and Marquess of Buckingham, Earl of Coventry, Viscount Villiers, Baron Whaddon, and Ros, Knight of the most noble Order of the Garter, &c.  
Sir,  
HAving had the honour to observe your grace's great affection, and love to Sciences and Arts, and your own excellency being most eminent therein, together with your unparalleled love and inclination to the splendour of your Native Country, in promoting Learning and Ingenuity. These high merits with my own particular obligations and attendance, encourage my endeavours of the patronage to a new birth never presented to the English Nation; presuming by God's assistance to bring forth something of worth that hathnot yet seen light, and if your Grace shall please to pardon my observant presumption, you will hereby more strictly engage him ever to honour your heroic worth, who is,  
The very humblest of Your Servants, Daniel King.   

The Preface.  
Concerning the particulars of this TREATISE.  
WHereas the Superficies or outsides whereon dials may be made, may be either flat, bowed, or crooked, plain or rugged, & situated diversely, the most part of the books treating of this matter, contain severally the manner of making flat dials in all kinds of positions, Horizontal, Vertical, Meridional, Septentrional, Oriental, Occidental; Declining, Inclining; Inclining, and Declining; and accordingly in all other kinds of superficies.  
They may also show, for those that are ignorant of it, the way to find the elevation of the Pole, the Meridian line, the Declinings, Inclinings, and other particularities.  
But Monsieur de Sargues intention being to publish nothing, if it be possible, that is to be found in another Book, and to give you only the general Rule to make, and not to copy out a number of examples all differing one from another; I will give you but one example only in this volume, by this universal manner, the discourse whereof may be applied generally to all kinds of superficies, and in what situation soever they be, without having any knowledge of the Pole, nor of the height of the Sun, nor of the declining or inclining, nor of the Meridian line, nor of any other thing in Astronomy, and without a needle touched, nor of any kind of thing that may give a beginning to that, as you shall see yet better when we shall treat of the practice.  
And in practising this general Rule, you shall find at one and the same time, the elevation of the Pole, and the position of the Meridian, you shall know how to place the needle of your dial, and so you shall come to find the equal hours, which are called hours after the French way, alias Astronomical.  
The rest being more curious than necessary, I thought to set down nothing else but those two things; But I have been persuaded, for the satisfaction os some, to add also the manner of drawing upon the same superficies, the lines of the signs, the hours after the Italian and Babylonian way, and of the ancients, the height of the Sun, and the situation thereof, in respect of the Horizon.  

Of the Practice of Sun-Dyals.  
MAny diverse things are represented in Sun-Dyals by the shadow of the Sun, to divers ends, the hour is shown by them, and serves only for that purpose every day. Other things are represented also by it, as the signs, and other particularities, whereby it may serve sometime for the divertisement of a few  
In ancient time the hours were not counted as they are now a days, and in Italy at this day they are counted otherwise than in France  
The manner that they count their hours in France is according to Astronomy, and here is at length a general way of framing and making dials with hours equal to the Sun, according as the hours are now counted in France, after which way one may come to show if need be, by the shadow of the Sun, and also of the Moon, whatsoever can be shown concerning other circumstances to satisfy curiosity.  
There are two things which together do compose those kinds of dials of equal hours, after the French way, the one is as piece that shoots out or sallies out of the superficies of the dial, the shadow whereof falling upon this superficies, shows what a clock it is, the other are the lines drawn upon the superficies of the dial, each of them representing one of the hours after the French way.  
They make dials after the French way wherein there is only the shadow of one portion alone, as might be a button of the piece that shoots out, that shows what a Clock it is. But in this general way, there is still the whole shadow of all the length, in a direct line of this piece that shoots out, showing continually what a Clock it is, of which piece or length, you may take if you will a button, and mark the hours with that button only, together with all the other particularities that may be added to such dials. Some call by one and the same name these two kinds of pieces the shadow whereof shows the hour, as well that same whose shadow shows continually the hour at length, as that which hath but the shadow of a button to show it.  
But to the end we may distinguish both kinds of pieces one from the other, that same whose shadow shows the hour at length, and is the original spring of all the others, I call it the axletree of the dial.  
This axletree may be made as well with a straight, round, and smooth rod of iron or brass, as with a flat piece, and cut of one side in a straight line.  
There are often other rods in the dials, which serve to bear up the axletree that shoots out, and those kind of rods I call the supporters of the axletree.  
The lines that are drawn upon the superficies of the dial, and that show each of them one of the hours after the French way, I call them lines of the hours after the French way.  
In the innumerable number of such kinds of dials as may be made after the French way, it happens that the superficies of the dial, is either all flat, or is not so altogether.  
When the superficies of the dial is all flat, every line of the hours is a straight line.  
And when the superficies of the dial is not altogether flat, it may be that every line of hours is not all straight also.  
To make one of those dials of equal hours after the French way, by this universal way, there are two things to be done one after another. The first is to place the axletree as it ought to be, that is, shooting out of the superficies of the dial; The second is, to draw the lines of the hours as they must be upon the same superficies.  
And by means of this general way, you shall do those two things without knowing in what day, nor in what time of the year, nor in what Country you are, without knowing what the superficies of the dial is, whether it is plain or rough, nor which way it looks, without knowing any thing concerning the making and the placing of the parts of the world, or without any skill in Astronomy, without any needle touched with the loadstone, or any instrument or figure that may serve for a beginning towards the making of a dial: But by the means only of the Beams of the Sun, by one general rule you shall place the axletree, and draw the lines of those hours upon one of those dials whatsoever the superficies may be, and which way soever it looks, with all the celerity and exactness that is possible in art; and if you are equally exact in every operation, you shall make by this means many dials upon different superficies, and turned towards several parts of the world, which shall agree plainly among themselves, and if you do not do it, you may be sure that the fault is on your part, and not in the rules, since that others do succeed well in it.  
There are some pieces that are requisite for the framing of a dial, and whereof it is composed, such are the axletree rod with its supporters.  
There are other pieces that must be used in the making of a dial, as Rules, Compasses, a Squirt, a lead with his two frames, one to mark with, and the other to level.  
There are some other things that you shall use also, as pegs, and rods, either of iron, or of brass, or of wood, some sharp at both ends, the others sharp at one end only, a table, either of wood, or of slate, or of any other stuff, flat and solid, to draw upon if need be, some straight lines with the rule; and in case the superficies of the dial were plain and even, you must use some fine strings, supple and strong, some mastic, cement, or plaster, or such like stuff fit to seal with, &c. all which things you must have in readiness, whensoever you will go about the making of a dial.  
And though you would learn to make but one of these dials well, it is fit you should have some models of all those pieces, and when you are upon those chapters that concern them, as you shall understand an Article, it will be requisite that with the models of those pieces, you work at the same time an actual model of the thing which that article shall teach you to do, and so you must work from one end to another, till you have at last every way completed an actual model of this kind of dials, and you shall need to make but few of such models of dials upon any superficies, turned towards several parts of the world, to bring you acquainted with the practice of making dials to the life, or after the natural, in what kind or odd situation of superficies soever they may be.  
Lastly, you shall find the precepts and the descriptions to be more troublesome, than the actual making or working, study only to be as exact in every one of these operations of making of these dials, as in the practice of other Arts.    

The Epistle to the READER.  
Courteous Reader,  
THis Treatise being originally written in French, and generally approved of all those that have any skill in the Art of Dyalling, I have thought it my duty to lay hold upon this occasion, to show how desirous I have ever been to procure any good unto my Country. Therefore I have caused it to be carefully translated into English, and have set it forth for the good and utility of all such, as are curious, and true lovers of that Art; Reputing myself most happy to meet with any occasion, whereby I may contribute any thing towards the advancement of learning, and of the public good,— Non enim nobis solum nati sumus; We are not meant to be wholly and solely for ourselves. As for the work itself, I am so confident it will so gain the attentive Readers approbation, as that I shall forbear to say any more in commendation of it, than that it is an expedite, and sure way of obtaining the site of the Axis, and of other requisites in the framing of all sorts of dials, of no less curiosity, than use, performed without the ordinary rules and presupposals of the spiritual calculations and practice; I need premise no more, but advise to follow the directions that are set down through all the book, for effecting that which is promised, and thou shalt see the same plainly and readily performed. Accept then, Courteous Reader, this small labour, the undoubted Testimony of my Love, as kindly▪ as I offer it cordially unto thee, hoping that God will enable me to give thee hereafter some thing of more consequence,  
So Farewell. Vtere & fruere. Thine D. K.   

To all Lovers of Ingenious Practices.  
THe French have excelled all other Nations in the Art of Perspective for this last Age, their many Books and curious Writings so excellently composed do witness for them.  
Dyalling I account one kind of Perspective, for that glorious Body the Sun, the Eye of the world, traceth out the lines and hour-points by his Diurnal Course, and upon the resubjected Plane by the laws of Picture, Scenographically delineates the dial.  
Many have writ upon this subject of several countries, in several Ages, many are the Rules and Practices set down; But among all those of foreign Parts, none hath performed the same with more ease, and less trouble, than Monsieur du Sargues the Author; as wholly laying aside those tedious observations of azimuths, Declination, Reclination, Inclination, Meridian; Substile, &c. and performing the operation only by three observations of the sun's shadow from a Point.  
It will not be amiss to give the Reader a small consideration hereof; the point B of the pin AB, in all the figures is always one part of the Axis, or Gnomon of the dial, and may be used to show the hour: this point B, you must imagine to be the centre of the Earth (for the vast distance to the Sun, maketh the space betwixt the centre and superficies of the Earth to be insensible) and from it at all times of the year (excepting the Aequinoctial day) the Sun in its course forms two Cones, whose Apex is the point B, that next the Sun termed Conus luminosus or the light Cone, the other whereof our Author makes use, termed Conus umbrosus the dark Cone, now in this dark Cone, if by any three points equally distant from the Apex B, the Cone be cut, the Section will be a Circle parallel to the Equinoctial: And thereby, as the Author shows many ways, the position of the Axis or Gnomon may be found out, and the dial easily made.  
Now it rests, courteous Countryman, that we be very grateful, and every way forward to encourage Mr. D. King, one very industrious in the studies of Antiquities and Heraldry; who out of his desire to serve his Country, hath caused this piece speak English, hath been very careful to see the cuts well done, and will (no doubt) proceed to cause some of those rare pieces of perspective in French to be translated.  

Then prosper King, until thy worthy hand,  
The gallic learning make us understand.   
Ionas Moor Mathesios' Professor.   

Books Printed for Isaac Pridmore, and are to be sold at the Golden falcon near the New-Exchange.  
THE Rogue, or the life of Gusman de Alpherache the witty Spaniard, written in Spanish by Matthew Aleman, Servant to his Catholic Majesty; the fifth and last Edition Corrected.  
A Physical discourse, exhibiting the cure of Diseases by signatures, whereunto is annexed a Philosophical discourse, vindicating the soul's prerogative in discerning the truths of Christian Religion with the eye of reason, by R. Bunworth.  
Self-examination or Self-Preparation for the worthy receiving of the Lord▪ Supper; delivered in a Sermon concerning the Sacrament, by Daniel Cawdrey, sometimes Preacher there, with a short Chatechism: the third Edition.  
The Obstinate Lady, a Comedy written by Sir Aston Cockaine.  
Sportive Elegies written by Samuel Holland Gent.  
A New discovery of the French Disease, and running of the Reins, with plain and easy directions for the perfect curing the same, by R. Runworths'.  
The unspotted high Court of justice, erected and discovered in three Sermons Preached in London and other places, by Thomas Baker, Rector of St. Mary the More in Oxon.  
A Chain of Golden poens, embellished with wit, Mirth, and Eloquence, together with two most excellent Comedies, viz. the Obstinate Lady, and Trapolin supposed a Prince, by Sir Aston Cockaine.  
The Ascent to bliss by three steps, viz. Philosophy, History, and Theology, in a brief discourse of man's felicity, with many rem●●keable examples of divers Kings and Princes.  
The Heroical Loves, or Anthcon & Fidelta a poem, by Thomas Bancroft.  
Advice to Balam's ass, or Momus Catechised, in Answer to a certain scurrilous and abusive scribbler by, John Heydon A●●hor of advice to a daughter, by T. P. Gen●.  
The Analysis of all the Epistles of the new Testament, wherein the chief things of every particular chapter are reduced to heads, for the help of the Memory, and many hard places explained, for the help of the understanding, by John Dale Master of Arts, and fellow of Magdal 〈…〉 in Oxford.   


1   
I Figure, To all sorts of People.  
I come now to the first of those two things that you are to do for to make one of those dials, which is the manner how to find the position, or the placing of the axletree.  
WHen you have a mind to find out the right placing of the axletree of one of those dials by this general way, mark first which way the light of the Sun comes to the place where you will make your dial, and which way it goes out again.  
Then make fast upon the place, as the figure above doth show, with cement, plaster, mastic, or the like, a peg or pin, AB, by the great end A, putting the other small or sharp end B as far out of the superficies of that place as you can, In such sort that while the Sun doth shine upon that place, the shadow of the end of the pin B may fall always upon this superficies, and for the rest it is no matter how this pin or peg be framed, or placed, or turned, you are only to look to the small end thereof, that must be in such a manner, that you may set or apply upon it one of the feet of the Compass.  
Then in a fair Sunshiny day, when the light is very clear, and the shadow very clean, whilst it falls upon this superficies in the figure below, mark in it, as the figure shows, in one and the same day, at three several times as far asunder as you can, three several points CDF, each of them at the end of the shadow of this pin AB that answers to the small end of it B.  
You must nor that there is a certain time and place in which you cannot mark the points of the shadow; That is when this superficies is flat, and situated after such a manner, that the ground plot thereof being stretched at length, answereth and reacheth into the centre of the Sun.  
For in that case how short soever the peg or pin B may be, the shadow hereof cannot go and fall in this superficies, but at the end of an extreme length.  
Therefore when the days are equal with the nights, or very near, you cannot mark in this manner three points of shadow in a flat superficies, which is situated in that manner called parallel to the Equator.  
When you have thus marked three points of shadow, you have no more need of the light of the Sun, and you may make an end of the rest in any other time and season, as well by night as by day, as I shall say three times together for one and the same manner, in three several ways to be expressed, after I have briefly satisfied the Theoriciens that take pleasure to see the reasons of the precepts, or rules of the practice of the Arts, before they see the precepts themselves.  
1   

To the Theoriciens.  
This Resolution will serve you.  
AFter you have conceived that the Sun in his full revolution of a natural day makes a Circle parallel to the Equator, and the rest of this Hypothesis, for dials.  
The three beams of the Sun or straight lines, BC, BD, BF, make in their point or common end B. some angles or corners equal one to another; with an other straight one that makes the fourth, which is the axletree of the dial.  
Now the position or placing of these three straight lines BC, BD, BF, is given out; Therefore the placing of this fourth which is the axletree of the dial is given also.  
You shall have hereafter in the fourth figure an other resolution of this kind, before you have the way to compose some problems, or propositions about it.  
I said to the Theoriciens, because if you were not at all versed in any kind of practice, either of Geometry or Art, you might hardly understand me at first concerning the 2d. & third figures following, because of the short & compendious way whereby I express myself unto those that are skilled in Geometry: but I can assure you, that when you have understood what is written in order for all sorts of people, if you come again to these second and third figures, you shall know at the very first sight what they mean.   

For the Theoriciens,  
And for those that are skilled in Geometry.  
THe I figure is a plate of some thin, flat, smooth, and solid stuff▪ as Iron tinned, or the like, being round, and having a hole just in the centre, greater or lesser, according to the occasion.  
The II figure is a straight rod, round, smooth, and solid, as of Iron, or the like, of the bigness of the hole in the plate.  
The III figure is as it were a whirl made of the plate, and of the rod put thorough the plate, in such sort that it is perpendicular to the said plate, as the squire that turns round about doth represent unto you, and is so fast that it cannot stir or move.  
In the V figure AB is the peg or pin that hath marked unto you the points of the shadow CDF, the rods or sticks BC, BD, BF, are solid and strong, as of wood, or the like, having each of them a slope edge in a direct line all along, going from the point of the peg B to each point of the shadow CDF, and are so turned or ordered, that in applying the whirl unto them, the edge of the plate may go andtouch the three slope edges of the rods all at once; and the rods or sticks are made fast in this situation, in such sort that they cannot move nor stir.  
The rule that crosseth over the three slope edges, BC, BD, BF, toucheth them all three, or else two at the time only, whereby it shows whether those slope edges are all three in one and the same situation, or upon one and the same ground or no, and on which side is their hollowness when there is any.  
The hand applies the whirl unto it, and keeps it there till the axletree BO● come to touch the end of the peg or pin B, and that at the same time the edge of the plate EDH touch the three slope edges of the rods.  
And when the whirl is placed or settled after this manner, the rod is the axletree of the dial, and placed as it ought to be, and there remains nothing else but to make it fast in this situation or position.  
The IV figure doth show, that if you go to make use of thin and supple strings in this practice or working, in pulling those two marked with je, and Ih, to make them fast in direct lines, they would make the two strings marked with bc, bf to bend, so that you can do nothing exactly with them, which is the reason that Monsieur Desargues hath not thought fit to make use of them for the Beams of the Sun, but rather of the slope edges of the rods that are both stiff and strong.  
2  
3   

To the Theoriciens,  
And others that are skilled in Geometry.  
THis foregoing figure shows to the eye that all the pieces of the Instrument are made so strong and firm, that they cannot bend.  
AB is the pin, by whose point B, you have had the points of shadow C, D, F.  
The three sticks or rods BC, BD, BF, have each of them a slope edge in a direct line at length, going from the point of the pin B to the three points of shadow, C, D, F.  
The slope edges of the two longest sticks or rods, BC, BF, have some portions made in them, equal every one to the third and shortest stick BD.  
The three sticks IH▪ ID, IE, are every one longer than BD, and all three made even, than they are joined all by the end to one of the points EDH, of the slope edges of the other sticks, BC, BD, BF, and their other ends I, are brought together in one and the same point, I.  
The rod by▪ is straight, round, smooth and strong as of iron or the like, it hath a straight line by, drawn from one end to an other, and one of the points B, of this line of the said rod toucheth the point of the pin; And with an other point I of the same it toucheth the point I of the three rods or sticks.  
This being so, the rod by comes to be the axletree of the dial rightly placed, there remains nothing else but to make it fast in this position or situation.  
The figure shows in the rods that go from the point of the pin B, to the points of shadow CDE, how one may make fast those rods at one end to the pin, and also all together to one point, by binding them to it; And how they may be made f●●t at the otherend to one point of the superficies of the dial, by fastening them to it with mastic, plaster, cement, or thelike.  
This way is more sure than that with the strings; But yet it is not the easiest▪ nor the least troublesome, in my judgement.   

To the Theoriciens,  
Another resolution of the same kind with the former.  
THe position or placing is given of the four points BC DF, and the placing of the two straight lines BE, BH, that divide in two the angles CBD, and DBF, and of the two ground plots that pass unto those two straight lines BE, and BF, and that are perpendicular to the ground plots of those Angles CBD and DBF, are given out; Therefore the intersection or intercutting of these two ground plots so perpendicular is given.  
But this intercutting is the axletree of the dial, therefore the position of the axletree of the dial is given.  
Any one may frame at his pleasure upon that which is granted concerning this composition, many other resolutions, and divers compositions of problems, and divers general ways of practice.  
In the mean time you shall have here three several ways one after another, to see which is the most advantageous for the actual practising of the Art, and to induce you to seek or try if there is any other shorter.  
4   

For the Theoriciens.  
The Composition of the problem, or Proposition, in Consequence of the Resolution made upon the lowermost figure of the first draught  
THe first figure is the place of the dial, with the pin and the points of shadow, CDF.  
Make a ground plot of it, II upon one straight line BD, and with one point B, three Angles DBN, DBR, DBH, equal to the three Angles of the first figure, that are between the beams of the Sun, DBC, DBF, CBF, every one to his own respectively.  
From the centre B, II figure, and from any space BD, draw a half circle that may meet in the points, DNRH, the straight lines BD, BN, BR, BH.  
Make in the third figure a triangle DGV, with three spaces, equal to the three spaces DH, DR, DN, every one to his respectively, as having the condition necessary for that purpose.  
Find the centre O of the circle eugd, drawn about this triangle VGD.  
Draw two Diametters do, POB, of this circle, perpendicular one unto another.  
Lengthen one POB sufficiently of one side and on the other.  
From one of the ends D, from the other EOD draw as far as that which is lengthened POB one straight line DB, even with the straight line DB of the II figure, for it must reach unto it, viz. In the Equinoctial at the point O, and in an other place at an other time, lengthen sufficiently III figure this straight line BD.  
Make in it the segments or cuttings even with the beams of the Sun of the I figure BD, BF, BC.  
Take in the III figure, in the straight line POB, conveniently a point I other than B.  
Make in the IIII figure three rods or sticks CI, DI, FI, each of them sharp at both ends, and equal with the three spaces CI, DI, FI, of the third figure.  
Draw a straight line along the axletree rod, mark in this line of the axletree conveniently figure IIII, one Cut by equal with the space by of the III figure.  
Set figure IIII one of the ends of the stick CI to the point of shadow, C, one of the ends of the stick DI to the point of shadow, D, and one end● of the stick FI to the point of shadow F.  
Let the ends of those sticks or rods be so well fastened to the points of shadow CDF, that they cannot stir.  
Bring together the other ends I, of those sticks in one point I.  
Put one of the point▪ B of the axletree rod to the point of the pin B, and the other point I with the three ends of the sticks CI, DI, FI, set or joined together. And if you have been very exact in the work, the point I of the pin will go and place itself with the three ends of the sticks set together in the point I, if not, you have not wrought exactly.  
6   

To the Theoriciens.  
It is no matter whether the Figures come right to the Compasses, you are only to take notice what this ensuing discourse ordains you to do.  
MAke figure I with three straight lines CQRD, DIPE, and CF, a Triangle even and like unto the Triangle figure III, of the three points of shadow CDF, upon the straight line CQRD figure I. make a Triangle CBD, both like and equal with the Triangle figure III. of the sunbeams CBD, and upon the straight line FPID figure I. make a Triangle FDB, like to the Triangle figure III. of the sunbeams FDB, make longer if need be figure I. on the side of D the straight lines CQRD and FPID. By the points B and B draw a straight line BRAYH perpendicular to CQRD, and a straight line BIAKL perpendicular to the straight line FPID, find out the end or point A, common to these two straight lines BRAYH, BIAKL, and by this end A draw a straight line AE perpendicular to the straight line BRAYH, and a straight line AG, perpendicular to the straight line BIAKL, from the point R draw as far as the straight line AE a straight line RE even with RB, from the point I draw as far as the straight line AG a straight line IG even with IB.  
From the point E carry to the straight line BRAIH a straight line EH, perpendicular to the straight line RE, from the point G, carry to the straight line BIAKL a straight line GL, perpendicular to the straight line IG, from the points B and B carry a straight line BQ that may divide in half the Angle CBD and a straight line BP that may divide in half the Angle DBF. By the points Q and H draw a straight line QOH, and by the points P and L draw a straight line POL, find the end or the point O common to the two straight lines QOH and POL, and from the point A for centre and space AO draw an half circle that may meet with the straight lines all in K and AH in Y.  
Now make in some other place even or flat, as in the second figure in one and the same line BDFC three cuts BC, BD, BF, even with the Sun beams, figure III. BC, BD, BF, each of them to his own from the point B of this second figure for centre, and from the interval or space ay or GK, of the first figure, draw an half Circle O from the point C figure II for centre, and from the space CO, of the first figure draw an other half Circle O from the point D of the II figure for centre and space DO of the first figure, draw an other half Circle O, and from the point F also of the II figure for centre and space foe of the I figure, draw an other half Circle O, and if you have done right, all these half Circles will meet in the same point O, if not, you have not been exact in working.  
By the points B and O draw a straight line boy, take in this line a point at discretion, first make three rods even with the spaces CI, DI, FI of the second figure, and every one sharp at both ends, make in the length of the axletree rod figure III the space by, even with the space by of the II figure.  
Lastly set these rods to the axletree figure III as I have said at the end of the fifth table, and the axletree of the dial is placed.  
There are some situations of superficies of dials, where practising this manner of drawing one or the other of the points LH or O comes so far from the straight line CF, that you should have need of too great a space to come to it.  
But in what manner soever the superficies of the dial may be situated, and at all times or seasons of the year, I mean, in any strange or odd kind of example that may be found, you may work or practise these kinds of draughts with as much ease as in the most easy pattern.  
5  
And by means of these three angles even with those in the air between the beams of the Sun, you may choose at pleasure within the lines that represent those beams, other points CDF and otherwise disposed between them, than those which the shadow of the point of the pin hath given upon the superficies of the dial, and upon those three points chosen out at pleasure, you may make an other triangle CDF, and practise afterwards this manner of drawing as far as the triangle CBO figure II than in this triangle; and in the straight line BC, make BC, BD, BF, even with the beams in the air, BC, BD, BF, of the third figure, contained from the point of the pin B, to the points of shadow CDF in the superficies of the dial each of them to his own, and after you have taken, as it is said, the point I in the straight line boy, you must make use of the points CDF, last made in the triangle OCB, for to set the rods CI, DI, FI, to the axletree by then to work on as before.  
To make other points instead of those of the superficies of the dial, you need only to make some at the two extremities or furthest ends CF, and make BC, and BC, equal one to the other, and unequal with the middlemost BD but a little bigger, more or less according as the angles DBC, DBF, are more or less unequal among themselves, and instead of making figure I the triangle CDF of the spaces between the points of shadow CDF of the superficies of the dial, you shall make it of the spaces between the points that are set in the place of these points of shadow.  
6   

To the Theoriciens.  
MAke in one and the same plain, as in the first figure, with three right lines CgkD, CrtF, DieF, a triangle, CDF equal and like to the triangle of the three points of shadow, fig. IV. CDF make upon the said three straight lines CgkD, CrtF, DieF, three other triangles CBD, CbF, DBF, equal and like to the triangles in the air of the beams of the Sun, III. fig. CBD, CBF, DBF every one to his own. By the points B and b I. fig. draw a straight line Bqg that may part in two the Angle DbF. Draw out of the point C at your discretion a straight line aqkty perpendicular to the straight line CgkD, and out of the point F, draw a straight line hPirx perpendicular to the straight line FeiD, make in the triangle Fcb the section or cutting Cl, equal with Ca, of the triangle CBD, and the section Fs, with Fh, of the triangle FbD, from the point t centre, and space tl, draw a bow lm, from the point k, centre and space ka, draw a bow am, that may meet with the bow lm, in m▪ and draw along the straight line km, from the point r centre and space rs, draw a bow sn, from the point i. centre and space i, h, draw a bow in (hn) that may meet with the bow (sn) in (n) draw along the straight line (in): Make in the straight line (km) the section or cutting (ku) equal with kq. By the point (u) bring to the straight line aqkty, a straight line (Vy) perpendicular to the straight line (km); make in the straight line (in) a section (iz) equal with iP; by the point (zx) carry to the straight line (hPirx) a straight line (zx) perpendicular to the straight line (in) find out the butt end (y) common to the two straight lines (aqkty) and vy. And also the butt end x common to the straight lines hPirx and zx, draw the straight lines goy, and eox▪ find the butt end o common to these straight lines gov, eox.  
Make in an other place figure II. a Triangle gqy, of the three straight lines, as, gq, gy, and thou of the first 
7  figure; make in the II. fig, and in the straight lines gy and gq, the section (go) equal to go of the I. fig. And the section gb also equal to go of the I. figure; draw if you will the straight line (bo) of the second figure. Make again in another place fig, 3. a Triangle (cbo) of the three straight lines (bo) of the Triangle (gbo) of the second figure. And of CB and CO of the first figure; and upon bc fig. 3. make the cuts bc, bd, bf, equal to the lines BC, BD, bF, of the first figure, every one to his own respectively. And if you have done rightly, the spaces fo, do, co of the Triangle cbo, fig. 3. are equal with the spaces foe, DO, CO of the first figure, every one to his own respectively. Take fig. 3. in the straight line (bo) according to your discretion the point (i) other then (b) make three sticks sharp at both ends, and equal to the three spaces ci, di, si, of the third figure: mark along upon`the rod or axletree the space (By) equal to the space (By) of the third figure: work as I have said, and as the fourth figure doth show you, and you shall find the axletree of the Dial placed in his right place. You may after this manner, as in others substitute, or bring in other points CDF in stead of those of shadow of the superficies, or face of the dial and work by this mean, everywhere with the like ease.   

Figure 8, For those that have skill in Geometry.  
THe higher figure is the place of the dial with the face unequal to the pin AB, and to the three points of shadow CDF, all marked, as it is said.  
Get a flat and solid thing, as a slate, a board, paceboard, or the like.  
Draw upon it in the lower figure a straight line BDFC, make in that line three cuts BC, BF, BD, equal with the three spaces BC, BF, BD, of the place of the dial, each of them to his own respectively, then from the point B of the lower figure for the centre, and from the spaces BC, BF, BD, draw some circles DH, FE, CG.  
By this means you see whether the spaces BC, BD, BF, of the higher figure or dial, are equal or unequal one unto another, and when these spaces are unequal among themselves, as it happens in this example, you see which is the least, and which is the biggest, as in this example, the space BD, comes to be the shortest of the three.  
Now from the point C of the lower figure for the centre, and from the space between the two points of shadow C and F of the higher figure, draw a circle E, that may meet in one point E, the circle of the space BF, viz. the circle FE, for it must meet with it, then draw the straight line FB, that may go and meet in one point H, the circle of the shortest space BD, viz. the circle DH.  
Again from the point C of the lower figure for centre, and space between the two points of shadow C and D of the higher figure, draw a circle N, that may go and meet in the point N, the circle of the shortest space BD, viz. the circle DH, for it must meet with it.  
From the point F in the lower figure for centre, and from the space between the two points of shadow FD of the higher figure, draw a circle that may meet in the point R, the circle of the shortest space BD, viz. the circle DH, for it must meet with it▪ By this means the three spaces or straight lines DH, DR and DN, of the circle DH, which is that of the shore●t space BD, have the conditions that are requisite for the making of a triangle.   

Figure 9, For those that are skilled in Geometry.  
MAke in another place, as in the lower figure, a triangle DGV of three straight lines, equal with the three spaces, DH, DR, DN, of the higher figure, every one to its own.  
Find in the lower figure the centre O of a circle, the edge whereof may reach to the points VDG, according as the lower figure doth declare.  
Draw a straight line do, through the Diameter or midst of this circle.  
By the point O in the lower figure, draw a straight line POQ, perpendicular to this Diameter do.  
From the point D in the lower figure, for the centre and space BD of the higher figure, draw a circle that may meet as in B, the straight line QOP, for it must meet with it in one or two points, viz. In the times of the Equinoxe in one point only, which is the point O, and all the rest of the year, in two points divided on both sides from the point O.  
And that you may be exact in working, do as much on the other part, and of E for centre.  
Draw in the lower figure the straight line BD, which in the times of the Equinoxe is joined with the straight line odd, and all the rest of the year is divided from it, and draw along this straight line BD beyond the point D.  
Make in the straight line BD of the lower figure two Sections BC and BF, equal to the two Sections BC, & BF of the higher figure, each of them to his own respectively.  
Take in the straight line QOP of the lower figure of one side or other of the point B, at your discretion, a point I, besides the point B, and that may stand as far from the point B, as the occasion may give you leave.   

Figure 10, To those that are skilled in Geometry.  
THen according as the lower figure shows you, cut three sticks CI, FI, DI, sharp at both ends, and equal to the three spaces CI, FI, DI, of the higher figure, each of them to its own space respectively, and upon the rod below, whereunto you mean to make the axletree of the dial, make a Section by, equal with the space by of the higher figure.  

8   

9   

10    


11   
Figure 11, For those that are skilled in Geometry.  
AFterwards as you see in the lower figure set to the points of shadow CDF upon the place of the dial, the three ends CDF of the sticks CI, DI, FI, each of them to his own point; and the point B of the axletree rod to the point of the pin AB, then bring into one point alone in the air I, the three other ends I of these three sticks CI, DI, FI, and set them to the point I of the axletree rod.  
For the ends of these three sticks, and the point I of the axletree ought to meet all four together in one point in the air I, and then you shall find the axletree rod placed as it must be in the dial. So that you need no more but to make it fast afterwards in this position or placing, or else to place an other in an other place that may be a parallel to it.  
If the matter was only about that which is sufficient to show Geometrically the truth of the proposition, it were sufficient to have either the three sticks only without the spaceBI of the axletree rod, or the spaceBI of the axletree rod with two sticks, without a third. But to make the operation sure and effective, you can not be confident that you have done rightly without a fourth stick that may serve for a proof; This is that which Monsieur de Sargues had a mind to impart unto you.   

Figure 8, The same thing over again, but in other Terms.  
To the workmen of many sorts of Arts.  
When you have marked the three points of shadow CDF in the place where you mean to make one of these dials, draw with the rule in some even or flat place, as you see in the lower figure, a line BD, FC, and make in that line a prick or point B where you shall think fit, or at your discretion.  
Then go to the place of the dial in the higher figure, take with the Compasses the space from the point B, of the ●pin●●… B to the point of shadow C. And with that space come back to the line BD, FC of the lower figure, set one of the feet of the Compasses to the point B, and with the other foot go and mark in that same line BD an an other point or prick C, then with the same space, give a stroke with the compass CG about the point B.  
Go back again to the place of the dial above, take with your Compasses the space from the point B, of the pin AB to the point of shadow F, and with this space come back to the lower figure; Set again one of the feet of the compass to the point B, and with the other foot go and mark in that same line BC an other prick or stay F, and draw again about the point B with the same space of the compass this half circle FE.  
8  
Then look in the lower figure which of the three strokes CG, FE, DH, is nearest the point B, and which are the furthest off, as in this example you see that the stroke DH is nearer to the point B than any of the two others FE and CG, and if they were either two or three together it were no matter.  
When you know which of these strokes of the lower figure CG FE and DH, is the nearest to the point B, and which are the farthest, as here, the stroke DH is the nearest, and the two CG, FE are the farthest off.  
Go to the higher figure to the points of shadow C, and F, which are even with the two strokes below CG, and FE, which are the furthest from the point B, and open your Compasses upon the points of shadow C and F, and remember well the two letters or coats upon which you have opened your Compasses, and with this space come back to the lower figure, and set one of the feet of the Compasses to the point C, and with the other foot go and mark a point E, upon the stroke of the compass FE, for it must reach to it.  
Then draw with the rule by the two points E and B, a line ebb, that may go and make a point H upon the stroke DH, which is the nearest to the point B.  
Go back again to the higher figure, and open your Compasses upon the points of shadow C and D, and with this space come back to the lower figure▪ set one foot of the compass upon the point C, and with the other foot go and mark a point N▪ upon the stroke DH, which is the nearest to the point B, for it must reach to it.  
Go back again to the higher figure, and open your Compasses upon the point of shadow F and D, and with this space come again to the lower figure, and set one foot of the compass upon the point F, and with the other foot go and make a point R upon the stroke DH, which is the nearest to the point B, for it must reach to it.  
After that, you have no more to do upon the place of the dial, till you place the axletree as it ought to be, and you have in the lower figure upon the stroke DA, which is the nearest to the point B, four several points or stays DNRH to make three point perdus with, as you shall see, in the mean time remember when you open the compass upon the points of shadow in the place of the dial, to take great notice upon what letters you have opened your compass, that you may apply the same space below upon the two strokes which are equal with the two points of shadow upon which you have opened your compass, and set one foot upon one of the strokes, and the other upon the other stroke. And moreover that the points NR, may well come out from betwixt the points D and H; and that I have caused them to come in so betwixt them, by reason of the smallness of the place, and what way soever they come to be disposed, it is but one and the same thing still.   

Fig. 9, To the workmen of many sorts of Arts.  
SEt your compass upon the points D, and H, of the higher figure, and with that space go to some flat or even place in the lower figure, and make two points D, and V, so that the space DV below, may be even with the space DH above.  
Then go to the figure above, and set your Compass upon the points D and R, and with this space come back to the figure below, and set one foot of the compass to the point V, and with the other foot draw a line from the point D, to the point G, so that the space VG below, may be even with the space DR above.  
Go back again to the figure above, and set your Compass to the points D & N, and with this space come back to the lower figure, & set one foot of the Compass to the point D, and with the other foot draw a line from the point V, to the point G, so that the space DG below, may be even with the space DN above, and may meet in G the other circular line that you have drawn about the point V, for it must meet with it.  

i.e. That are made without any aim, or heed.  And so you have made in the lower figure three points VGD that will be perdus or lost. Now find a centre O, upon which having set one of the feet of the Compass, and the other upon D, let this foot in turning the Compass about, go and pass by those three points perdus VGD, then draw with the rule by the points, as it were O and D, a line do, and setting again one foot of the Compass to the point O, and turning the other foot to E, make in the line do, the side OE, even with OD.  
Then by the point O, draw a line QOP, that may cut the line do, in two equal parts; again set your compass to the points B, and D, of the figure above, and with this space go to the figure below, set one foot of the compass to the point D, & with the other foot draw from the point E, a line B, that may meet as it were in the point B, the line QOP, and make with this other foot of the compass a point B, in the line QOP, for it must meet with it, if you have done exactly.  
9  
9  
When the days and the nights are equal, it meets with it in one point alone, viz. O, and at some other times it meets in two points, one of one side of the O, and the other on the other side, as in the point E. Then remove your Compass out of his place, and with the same space of the points B and D of the figure above, set one foot of the Compass to the point E, and with the other foot draw from the point D, with your Compass another line B, that may go and meet the line▪ QOP, with the line that you have traced with the Compass about the point D, and both of them in one and the same point B, for it must do it if you have been exact: And that serves to mark more exactly this point B, in the line QOP, how near soever it is to the Point O. After that, Whether the Point B of the lower figure meets with the Point O or not, draw with the rule by the points B and D, the line BD, and draw this line BD, as you see beyond the point D.  
That being done, open your Compass upon the points B and C, of the higher figure, and carry this space to the line BD of the lower figure, and from B into C.  
Set your Compass again upon the points B and F of the higher figure, and bring this space to the line BD of the lower figure, and from B into F.  
And finally make in the line QOP, a point I, at your discretion of one side or other of the point B, and let it be as far distant from the point B, as occasion will give you leave. And so you have in this lower figure from the point I to every one of the points BDFC, all the measures that are necessary for the placing of the axletree or needle in your dial, in the manner hereafter following.   

Figure 10,  
To the workmen of many sorts of Arts.  
CUt three rods or sticks sharp at both ends as you see below, one CI of the length that is betwixt the point C, and the point I of the figure● above▪ the other FI of the length that is betwixt the point F, and the same point I of the higher figure, the other DI of the length that is betwixt the point D, to the same point I of the higher figure, then open your compass upon the points B and I of the higher figure, and bring down this space upon the axletree rod, and make in the same as you see, two points B and I with this same space by of the figure above.  

10    


11   
Figure 11,  
To the workmen of many sorts of Arts.  
Go to the place of the dial below which I have expressed again, a purpose to avoid the confusion of lines, and put the end of the rod CI to the point of shadow C, the end F of the rod FI to the point of shadow F, and the end D of the rod DI to the point of shadow D, and set the point B of the axletree rod to the point B of the pin AB.  
Then bring together into one point in the Air I, the three other ends of the three rods CI, DI, FI, for they must come in there together, and bring the point I, of the axletree rod, to the same point I in the Air together with the three other ends of the rods I, for these four things must come altogether into one and the same point in the Air I, if so be you have been exact in working. And when these three ends of the rods and the point I of the axletree rod, are all four gathered together into one and the same point in the Air I, the axletree rod will come to be placed directly as it must be in the dial, and so you need no more but to make it fast in that place, or to fasten an other either near it or far from it, that may be even with it, or parallel to, or equally distant from it.  
If the four points I should go and meet in the body of the dial, you must but take in its figure the point I nearer, or in the other side of the point B, and make an end of the rest as I have said.   


8   
Figure 8, I will say the same thing over again, but more at large.  
To all sorts of People that have neither skill in Geometry nor in Arts; but are apt and sit to learn them both▪  
BEfore you undertook to make this dial, you had nothing about you, nor knew nothing wherewith to further you in it, and going about it, you have made use of the pin AB, as it were at a venture.  
Now you must consider that having placed the pin AB, in this manner, you have given out of yourself in the end thereof a point alone unmovable and fixed in the air.  
Then by means, of this fixed point in the air B, and of the sunbeams, you have found out three other unmovable and fixed points of shadow CDF, on the outward face of the place where you have a mind to make your dial▪ So you see that by means of this end of pin B, and of the sunbeams, you have established upon the place where you intend to make your dial four points fixed and divided one from another, viz, one in the Air, which is the point or the end B of the pin AB, and three in the superficies of the dial which are the three points of shadow CDF.  
Whereby you have found also six spaces, that is to say, the lengths of six straight lines unmovable, fixed, distinct, and divided one from the other.  
For if you consider well, you shall see that you have found out by this means the spaces, or lengths, or distances that are from the point B of the pin AB, to every one of the three points of shadow CDF, viz. the space from the point of the pin B to the point of shadow C, the space from the same point of the pin B to the point of shadow D, and the space again from the same point of the pin B to the point of shadow F.  
And for your better instruction, if you will make these three lines visible to the eye, set unto every one of them either a ruler or a string stretched out in a direct line from the point of the pin B, to every one of the points of shadow CDF, as the points do show it unto you; And so you may see the three lines BC, BD, BF, which otherwise are invisible in the Air.  
And besides these three spaces or lengths, you have also found out the three spaces or lengths that are from every one of the three points of shadow CDF, unto the other, viz. the space from the point of shadow C, to the point of shadow F, the space from the point of shadow C, to the point of shadow D, and the spaces from the point of shadow F to the point of shadow D, as you may see by the points that are there. So you have six spaces or lengths BC, BD, BF, CF, CD, DF, which you have already found unmovable, and fixed to the place wherein you intend to make your dial, which are so great a furtherance unto your work, that there remains nothing else to do, but by the help and means of the said six spaces or lengths, to find also three or four more, that you may have all that is requisite for the placing of the axletree rod of your dial as it ought to be.  
You must know that there are several ways whereby these six spaces which you have found already, viz, BC, BD, BF, CF, CD, DF, are made use of to find out those three or four more, which you must have to enable you to place the axletree rod of your dial as it must be.  
And that of all those several ways a man may have a liking to one for one reason, and another man to an other for some other reason, and of those several ways Monsieur de Sargues hath showed me three or four at the most, viz. that which he hath set down in the figure of his model or project page, and of the others for which you must know how to make sometimes somekind of alteration, and which I have set down in short, there is one in the sixth figure, and another in the seventh.  
As for this it is such, that there is no occasion but you may practise it in effectually, and with the like ease everywhere, without you need either to add or alter any thing, as you shall see presently.  
Draw with the rule, as you see in the figure below in some flat or even place a straight line BD, FC, then go to the figure above, and open your compass, and set one of the feet to the point B of the pin AB, and the other foot to the point of shadow C, and by that means you shall take with your compass the space or the lengths that are from the point of the pin B, to the point of shadow C, whereof you will be pleased to remember, to the end that when I shall bid you for brevity sake, take after the same manner with your compass such a space, you may be able to do with your compass even as I told you just now of the space BC in the figure above.  
Now with this space BC of the figure above, come back to the figure below, and set at your discretion one of the feet of the compass upon the straight line that you have drawn there, as for example set it to the point B, then turning the compass about upon this point B, draw with the other foot a circular line CG, which circle by this means shall have a space BC equal with the space BC of the higher figure, and will meet the line BD for example in the point C.  
Go back again to the figure above, take there after the same manner the space from the end or point of the pin B to the point of shadow D, and with this space come back to the figure below, and set again one of the feet of your compass to the point B, and holding it still upon this point B draw with the other foot a second circular line DH, that will be equal with the space above BD, and that may meet the line BC, for example in the point D.  
Go back again to the figure above, and take with your compass the space betwixt the point of the pin B and the point of shadow F, and with this space come back to the figure below set one of the feet of the compass to the point B, and draw with the other foot a third circular line FE, with the space BF of the figure above, and that may meet the line BD for example in the point F.  
By this means you have set away and transported the three spaces BC, BD, BF, from the rise or place which they had in the place of the dial above in a flat and even place below; and all of them united together in one single line BDFC, in which you may see whether those spaces be equal amongst themselves, as they may be in some occasions which is indifferent, or whether they be unequal, by seeing whether the points CDF, are united together two or three in one single point, or whether they are disunited or divided one from another, and when these three points CDF, are disunited or divided one from the other which happens most commonly, and that these three spaces BC, BD, BF, are unequal amongst themselves, as it falls out in this example, you see which of these spaces are the greatest and which is the least, considering which of these three points CDF is the nearest, and which is the furthest from the point B, that is to say also that by this means you see which of these circular lines CG, EF, DH, is nearest to the point B, and which are furthest as in this example you see that of the three spaces BC, BD, BF, those two BC and BF, are the greatest, and BD is the least; and of the three half circles CG, FE, DH, you see that DH is the nearest to the point B, and that the circle FB is nearer to it than the circle CG.  
When you have thus found out which of the three spaces BC, BD, BF is the least, and which of the three circles CG, FE DH is the nearest of the point B. Go back again to the higher figure to the points of shadow CDF, and take with the compass the space betwixt the points of shadow C and F, which are at the ends of the two greatest spaces BC and BF, and with this space CF of the higher figure, go to the lower figure to the same point CF, and set one foot of the compass upon either of those points C and F, that is the furthest from the point B, viz. C, and holding still the point of the compass upon this point C, go and mark with the other foot a point for example E, in the circle of the other of these two points C and F, viz. in the circle of the point 〈◊〉 which is the circle FE, for this other foot of the compass must reach to this circle of the point F, as for example in the point E; This being done▪ draw by this point E to the point B a straight line ebb, that may go and meet, in one point H the circle of the point D which is the nearest to the point B, and mark in it this point H.  
Then go to the figure above, and take with the compass the space betwixt the two points of shadow C and D, and with this space CD of the figure above go to the figure below, to the same points C and D set one foot of the compass to that point of these two C and D, which is the furthest from the point B, viz. C, and keeping this foot of the compass upon this point C, go and mark with the other a point, for example N in the circle of the other of the two points C and D, viz. in the circle of the point▪ D with the circle DH, for this other foot of the compass must reach to the circle of the point D, for example in the point N.  
Go back to the figure above, take with your compasses the space betwixt the two points of shadow F and D, and with this space FD of the figure above, come back to the figure below to the like points F and D set one foot of the compass upon that point of these two F and D, which is furthest from the point B, viz. F and holding still the point of the compass upon this point F, go and mark with the other foot a point for example R, in the circle of the other of these two points F and D to wit, in the circle of the point D which is the circle DH, for this other foot of the compass must reach to the circle of the point D, for example in the point R. This being done, you have no more to do in the place of the dial, till you go and place the axletree rod in it as it must be, and in the figure below in the circle of the point D, which is nearest to the point B, you have found by this means four points DN, RH, different and divided one from the other, and when the two points N and R should be found united together, it were no matter. Now by the means of these four points you have three spaces amongst others from the point D, to every one of the three points HR and N to wit, the space from D to H, the space from D to R, and the space from D to N, of which spaces you see which is the greatest and which is the least, when they are all three unequal, as in this example, for it may happen that there will be two found equal amongst them.  
And with the help of these spaces DH, DR, DN, you shall find presently those four that you must have for the placing of the axletree rod of your dial, as it ought to be.   

Figure 9, To all sorts of People▪  
TAke with the compass in the figure above, of these three spaces DH, DR DN, that same that is the greatest of all, as in this example here, the space DH and with this space DH of the higher figure, go to some place that is flat, as in the figure below, and set at the same time the two feet of the compass upon it, as for example, to the two points D and V▪ and mark these two points as it were D and V, which by this means will be distant one from the other, the length of the space DH of the figure above.  
Go back again to the figure above, take there with the compass the space from D to R, and with this space go to the figure below, set one of the feet of the compass to the point V, and from thence draw with the other foot towards the point D a circle G, which by this means shall be made of the space DR of the figure above.  
Go back again to the figure above, take with the compass the space from D to H, and with this space go to the figure below, set one of the feet of the compass to the point D, and from thence with the other foot draw towards the point V another circular line that may meet, for example, in G the other circle, which you have drawn about the point V, for this other foot of the compass must meet the other circle in one or two points, and for example, in the point G for one.   


9   
Figure 9, To the workmen of many sorts of Arts.  
SEt your compass upon the points D and H of the higher figure, and with that space go to some flat or even place in the lower figure, and make two points D and V, so that the space DV below, maybe even with the space DH above.  
Then go to the figure above, and set your compass upon the points D and R, and with this space come back to the figure below, and set one foot of the compass to the point V, and with the other foot, draw a line from the point D to the point G, so that the space VG below, may be even with the space DR above.  
Go back again to the figure above, and set your compass to the points D and N, and with this space come back to the lower figure, and set one foot of the compass to the point D, and with the other foot draw a line from the point V to the point G, so that the space DG below, may be even with the space DN above, and may meet in G the other circular line that you have drawn about the point V, for it must meet with it.  
And so you have made in the lower figure three points VGD, that will be perdus or lost.  
Now find a centre O, upon which having set one of the feet of the compass, and the other upon D, let this foot in turning the compass about go and pass by those three points perdus VGD, then draw with the rule by the points, as it were, O and D, a line do, and setting again one foot of the compass to the point O, and turning the other foot to E, make in the line do, the side OE, even with OD.  
They by the point O, draw a line QOP, that may cut the line do in two equal, parts. Again, set your compass to the points B and D of the figure above, and with this space go to the figure below, set one foot of the compass to the point D, and with the other foot draw from the point E a line B, that may meet as it were in the point B the line QOP, and make with this other foot of the compass a point B in the line QOP, for it must meet with it, if you have done exactly.  
When the days and the nights are equal, it meets with it in one point alone, viz. O, and at some other times it meets in two points, one of one side of the O, and the other of the other side, as in the point E, then remove your compass out of his place, and with the same space of the points B and D, of the figure above set one foot of the compass to the point E, and with the other foot draw from the point D with your compass another line B, that may go and meet the line QOP, with the line that you have traced with the compass about the point D, and both of them in one and the same point B, for it must do it if you have been exact, And that serves to mark more exactly this point B in the line QOP, how near soever it is to the point O. After that, whether the point B of the lower figure meets with the point O or not, draw with the rule by the points B and D the line BD, and draw this line BD, is you see, beyond the point D.  
That being done, open your compass upon the points B and C of the higher figure, and carry this space to the line BD of the lower figure, and from B into C.  
Set your compass again upon the points B and F of the higher figure, and bring this space to the line BD of the lower figure, and from B into F.  
And finally make in the line QOP a point I, at your discretion, of one side or other of the point B, and let it be as far distant from the point B, as occasion will give you leave; And so you have in this lower figure from the point I, to every one of the points BD, FC, all the measures that are necessary for the placing of the axletree or needle in the dial, in the manner hereafter following.   

Figure 9, To all sorts of People.  
THat being done open your compass at your discretion, and the more that the occasion will permit you to open it, it will be so much the better, and with this opening, set one of the feet of the compass to the point G of the figure above, then turning about this point of the compass upon this point G, draw with the other foot four circles H, L, M, S, about the point G. viz. two H, and L from the point D, and two M and S from the point V, then remove your compass, and set one of the feet upon the point D, and with the other foot draw from the point G, two circles that may meet in two points, viz. Land G, those two circles that you have drawn about the point G, from the point D, if this same foot of the compass could not meet with these two circles HL, that you have drawn about the point G & from the point D, it is because you had not opened your compass enough, before you did set it upon the point G, and in such a case you shall open it more, and set it again upon this point G, and when this same foot meets these two first circles, for example, in H and in L, matk these two points L, and H.  
Afterwards remove the compass and keeping still the same open, set one foot to the point V, and turning the compass upon the point V, draw with the other foot from the point G, two circles that may meet likewise in two points as S, and M, the two circles that you have drawn about the point G, from the point V, and mark those two points S and M, in which those two circles meet with the other two: And therefore note that before you make your compass to turn upon the point G▪ you must open it in such a manner, that when you shall set it afterwards upon the points D and V, the other foot may meet with the circles that you have drawn about the point G.  
9  
Then draw by the two points H and L, a long straight line H, L, O, and by the two points S and M, draw an other long line and as straight, M, S, O, and these two lines HL and MS, being sufficiently drawn at length, will meet in one and the same point O.  
By the two points D and O, draw a straight line DO, and draw it at length, as you see from the point O, then set one of the feet of the compass upon the point O, and the other foot upon the point D, and turn that foot of the compass which is upon the point O. The other foot which is upon the point D must go and touch every one of the points G and V, and when this other foot hath gone over the point D, G, V, go and mark at the very same time with it a point, as E in the line do, and so you shall make the portion OE of the line odd, equal with the portion OD.  
Then open your compass at discretion more than from the space odd, and as much as the occasion will give you leave, the more the better, and the compass being so open at discretion, set one of the feet to the point D of the lower figure, and turning that foot of the compass upon the point D, draw with the other foot from the point O, two circles, as P and Q then remove the compass, keeping still the same distance, set one foot upon the point E, and turning it about, draw with the other foot from the point O, two other circles, that may meet in two points with the two circles that you have drawn about the point D, and as for example in the two points Q and P, and draw with the rule by these two points, as Q and P, a long straight line QP, that must reach to the point O, if you have been very exact in the working; if it doth not reach to it you have not been very exact, and I advise you to begin it again: If it reaches to it, go back to the figure below, and take with the compass the distance between B and D, then with this space go to the figure below, set one foot of the compass upon the point D, and turning it about, draw with the other foot from the point O, a circle that may meet the line QOP, as for example in the point B, for this other foot of the compass must go and meet that straight line POQ either in one or in two points, because the space from B to D of the higher figure ought never to be smaller or lesser than the space DO of the figure above.  
It is true that twice in the year, viz. in autumn and in the Spring, when the days and nights are equal, that space BD of the figure above comes to be equal with the space DO, of the figure below, and in those times that other foot of the compass that tutns about the point D of the figure below, meets the line QOP just in the point O. But at all other times the space BD of the figure above is somewhat bigger than the space DO of the lower figure; And then the other foot of the compass that turns about the point D, meets the line QOP, in two points, one of each side of the point O, as for example in B, for one.  
And that you may be the more exact, remove the compass from one part of the straight line boy unto the other, and with the same opening of the space BD of the higher figure, set one of the feet upon the point E of the figure below, and turning this foot upon this point E, draw with the other and from the point D an other circle that will meet (if you have been exact in the working) the straight line QOP, and the circle also that you have drawn about the point D, and both in one point; as for example in the point B, which will enable you to discern well the point B in the straight line POQ; mark this point B in the line POQ, whether you find it united with the point O, and so both of them making but one and the same point, as it falls out, when the days and nights are equal, or whether you find it divided from the point O, as it falls out in other seasons, and as you see in this example▪ Then draw with the rule by these two points B and D a straight line BD, which you shall stretch out sufficiently beyond the point D. When the days and nights are equal, as in autumn and in the Spring, and that the point B is found to be united with the point O, the line BD comes likewise to be united with the line odd, and both together make but one line; But at any other time, as the two points B and O are two several points and divided one from the other, so the two lines BD and odd, are two several lines▪ and divided one from the other,  
This being done, go to the figure above, mea●ure with your compass the space from B to C, and with this space go to the figure below, set one foot of the compass upon the line BD, to the point B, and set the other foot in any place of the same line BD where it may light upon; as for example in the point C, by this means you shall make the portion BC, of the line BD of the lower figure, equal with the portion BC, of the line BD of the higher figure, make after the same manner with the compass, the portion BF, of the line BD of the lower figure, equal to the Portion BF, of the line BD of the figure above.  
Finally, in the same figure below, and in the line QOP, mark at your discretion another point, I, of one side or other of the point B, according as you shall find it most convenient for the place of the dial, and as far from this point B, as occasion will permit, the further the better, and so you have found the four spaces that you wanted for the perfect placing of the axletree of your dial.  
For in so doing, you have found in this figure below the distances that are from every one of the four points BDF C, to one and the same point I, that is to say the space from B to I, the space from D to I, the space from F to I, and the space from C to I, which distances by, DI, FI, and CI, will serve you to place the axletree of the dial in the manner following.   

Figure 10, To all sorts of People.  
CUt (as you see in the lower figure,) three sticks sharp it both ends, one CI of the length of the point C, to the point I, otherwise of the space CI of the figure above: The other FI, of the length of the space FI of the higher figure, and take with your Compasses the space by of the figure above, and being so open, see both feet at once upon a straight line, along the axletree rod of the lower figure, for example, in two points as B and I, and mark these two points B, I, in the axletree rod.  

10    


11   
Figure 11, To all sorts of People.  
THat being done, go to the place of the dial, the which, to avoid the confusion or multiplicity of lines, I have set below in the lower figure, set in this lower figure one of the ends of the stick CI, to the point of shadow C, one of the ends F, of the stick FI, to the point of shadow F, and one of the ends D, of the stick DI to the point of shadow D, and one of the points B of the axletree rod, set it to the point B, of the pin AB.  
And holding thus the three ends CDF, of the three sticks to the points of shadow CDF, every one respectively to his own, and the point B of the axletree rod, to the point of the pin B, bring together the three other ends I, of the three sticks or rods CI, DI, FI, into one point in the air I, for they must meet there▪ then bring the point I of the axletree rod, also to the point in the air I, with the three ends I of the sticks, for it must come and meet there exactly, if you have done right, or if the straightness of the place hath not hindered you.  
If the straightness of the place of the dial hinders the three ends I, of the sticks from meeting together in one point in the air I, take the point I in the figure below in the ninth cut, or that above in the tenth figure in an other place, than in that where you had taken it and according to the occasion, then bring the sticks to it as before (for you may take it anywhere, or in any place of the line POQ, of one side or other of the point B;) but the further you can take it from the point B, will be better, and take it in so many-places, that having set the sticks of the points CDF, to this point I, and mark'ed the space by upon the axletree rod, the four points I, may at last meet together in one point in the air I.  
And when the point B of the axletree rod, is at the point B, of the pin AB, and when the three ends I, of the sticks, and the point I of the axletree rod are met, as you see in the lower figure, all four together in one and the same point in the air I.  
The axletree rod will come then to be placed, just as it ought to be in the dial.  
That if you do not care to be sure that your dial must be as just, as it is possible for art to do, in such a case, you may spare one of the four lengths CI, DI, FI, by, and content yourself with three only, as being sufficient for the theory: But the fourth will serve you for a proof, to see whether or no you have been very exact in working, and will justify the three others.   

Figure 12, To all sorts of People.  
THe figure above shows you how that which you have done with three sticks, may be done either with many Compasses, with the help of some body, or else with other kinds of branches tied or fastened one with another.  
The same figure above, as also the figures below, show how every one of those branches may be of two several pieces, which go in by couples into one hoop or ring, and slide along one by another, and are made fast with a screw to the measure where you will have them to stand upon, and these pieces may be made of tinned iron, or of iron, if you are afraid that their points will grow dull by often using them.  
Or otherwise they show you that instead of one stick, you may have two, both sharp at one end, which you shall fasten and bind together at the other end, of what length or measure you please.  
The same figures do show you also, that two divers branches, viz. CI, and FI, may be fastened together in the place where you will have them to stand together, with a press and a screw to fasten them with.  
The higher figure shows you besides, that you may ●●●●en or bind with strings or threads, the axletree rod with the point B, of the pin AB, and the two branches CI, FI, with the axletree rod, to make them stand fast of themselves in their place.  
When you have found thus the placing of the axletree rod, it is in your choice, either to seal it and fasten it in that place, or to place another instead of it, that may go the same way, and that may be every way equally distant from it; But that you may be the more exact, it will be as good to seal or fasten that in the place, where the practice of the Draught hath caused it to meet, than to place another, unless there was some occasion or necessity for it.  
12   

Figure 3, To those that have understood what hath been said before.  
HAving understood what I have said before, concerning those many ways of finding the position of the axletree of the dial, you may compose others besides, making use partly of that of one figure, and partly of that of an other.  
For example, here is one way composed of two of those that are afore.  
Out of the third or fifth figure, you shall take in the sunbeams or sticks BC, BD, BF, three spaces equal each unto the other; And out of the 5 and 6 figures, you shall make a triangle of three lines equal to the three spaces HE, DE, DH of the third figure, and you shall find the centre O of the circle, circumscribed about this triangle.  
You shall find also within the ground plot of the points HDE, the points like to A & O of the 6 figure or cut, which in this case come to be united together in one and the same point O. That is to say, having found one of these two points A & O, you have found also the other, because they are united or gathered together into one.  
So you have in the second figure of the third cut, the spaces DO, and DI, for two sides of a triangle with straight angles or corners ODI, whose side DI, holds up the straight angle, and the sides DO, and DI, do contain or comprehend it.  
Make this triangle ODI, with three sticks, or with any other thing that may be strong and small as you will, so that you may at your need lengthen the side IO, from the right angle O.  
3  
Set the point D, of this triangle DIO, to the point of shadow D, and holding this point of triangle to this point of shadow D, make the side IO, of this triangle (drawn at length if need be) come and touch the point B, of the pin AB for if you have been very exact in working, it must touch it.  
Take a stick HI, of the length of DI, set one of the ends to the point H, and bring the other end to the point I, of the triangle ODI, without the side IO, leave the end B of the pin AB, for it must be so, if you have wrought exactly as you ought.  
You may have also an other stick EI, of the length DI, and set one end to the point E, and bring the other likewise to the point I, of the triangle DIO, without the side OI, leave the end B of the pin AB.  
That being done the rod by, comes to be the axletree of the dial, and placed as it ought to be, and so of all the other ways that you find besides.  
You may, if you will make use of a triangle rectangular EOI. and of the stick HI, content yourself only with the three equal lengths EI, DI, HI, to find out the point thereby, that you may draw from thence a line to the point B, without making use of any thing else to know if you have done exactly or no, you can not be sure whether you have done well or ill.  
But when you have together with that, either a fourth length by▪ or the straight angle DOI, that will serve you to try whether or no you have been exact in your operations, for as concerning an effectual execution, unless you have from time to time such a kind of proof, to show whether you have wrought exactly as you ought, you cannot assure yourself that your work is as well as it can be done.  
One thing I must tell you, that in some certain occasions according to the times and the placing, or according as the superficies of your dial is, the shadow of the pin comes to be of such a length, and the extremity or end thereof so weakened, and so diminished in strength, and so confuse in the superficies of the dial, that it is very hard to find out   

Figure 13, To all sorts of People.  
I come now to the next and second thing that you are to do, which is to trace out the lines of the hours.  
IN this example I suppose that the axletree rod doth not meet the superficies of the dial, about the place that you work in; and therefore I represent it suspended in the air, with two or three supporters as you see, I suppose also that the superficies of the dial is not smooth, but rough and uneven as I have said.  
When you have placed the Rod by, which is the axletree of the dial, as you see both in the higher and lower figure, you have made an end then of the first of those two things that you were to do, for the making of your dial: Now there remains but the second to be done, which is the finding and the tracing out of the lines of the hours in the dial; and for that purpose.  
Consider in your higher figure, that the superficies and the axletree of your dial are two divers things, and differing one from an other, and there is no such communication from the one to the other, as that with them alone you may find out directly the place of the lines of the hours, without making use of a third thing that may be a means betwixt those two.  
The meanest and the least thing that you can have to be a means betwixt the superficies and the axletree of the dial, is a Ruler.  
13  
To the end that this middle rule may serve you alike in all occasions, it must have all the conditions that you see represented in the figure below. First it must be as long as the place will give you leave, and it must cross over if need be the whole superficies of the dial, and reach over on both sides if it be possible.  
Secondly it must be in the air, and suspended between the superficies and the axletree of the dial.  
Thirdly it must be placed as far from the axletree rod, as possible may be.  
Fourthly it must be placed like a cross, in regard of the same axletree rod.   

Figure 14, To all sorts of People.  
TO place this middle rule well, and as it ought to be betwixt the superficies and the axletree of the dial.  
Choose along the axletree rod by, of the higher figure, some fit or convenient place, as in the point O, and make a round and fixed stay in that place, by winding, or tying some strong thing about this axletree rod, as the figure doth show.  
Tie a string to the axletree rod by, by the means of a ring that may be so big, that you may turn the string with it about the axletree rod easily, as the lower figure shows you; Then with the corner of a squire ED, in the lower figure, thrust on the ring where the string is, and put it close to this stay O, and holding the string fast between the stay O, and the squire ED, set the back of one of the sides OE of this squire's length, to the axletree rod by, and by this means, the other side DO, of this squire, will shoot out into the air like a wing from the axletree rod by, then stretch out the string in a straight line from the stay O of the axletree, along the back of the other side odd of the squire. And holding still in this manner the ring close to the stay of the axletree, by means of the squire, and the back of one of the sides joined at length to the axletree rod, and the other side of the squire like a wing, and the string stretched out in a straight line along this wing, turn both the squire and the string altogether still in this same manner about the axletree rod, as the lower figure doth show.  
14  
When you have found out those two places that are farthest one from another, in which this string turning in this manner with the squire along the side like a wing, may go and meet the superficies of the dial, as here the places G and H.  
Make with mastic, or plaster, or cement, or such like stuff, a little knob flat at the top in each of these places, as for example, one in G, and another as in H, which two knobs may shoot out of the superficies of the dial, in such sort, that you may lay a Ruler on the top of them, going from one of the knobs to the other, as you see here in the lower figure.   


15   
Figure 15, To all sorts of People.  
When you have thus made those two knobbs G, H, in the lower figure, take the squire again and the string, and set them again close to the stay O, of the axletree rod, as you know they were.  
And make them go about again as before, both together about the axletree by, and while you are turning about, the string will fall right over against the two knobs, shorten or lengthen it so, that it may go and touch a point, at the top of each one of the two knobs, one after another▪ viz. a point as P, at the top of the knob G, and a point as Q, at the top of the knob H, and mark these two points Q and P, upon these two knobs.  
When you have marked two points in this manner, set a Ruler in the lower figure upon these two knobs, and place it so, that it may pass from one to the other, by those two points Q and P, and make the Ruler fast in this place with cement or plaster, or the like, in such wise that it may not stir any way.  
And this rule so placed, is the third and middle piece between the superficies, and the axletree of the dial, by means whereof you shall cause, as I shall say hereafter, this superficies and this axletree to have what communication soever you please one with another.  
After you have placed this middle rule in this manner, between the superficies and the axletree of the dial.  
Consider that in France now they reckon 34 hours, for one day and a night, and that these 24 hours, are divided in twice twelve hours, and that every one of these 12 hours is subdivided in twice 6 hours.  
So that in the 24 hours of one day and one night, as they are now reckoned in France, there are two hours that are each of them of 12, that is to say, one hour of 12 in the midst of the night, and another hour of twelve in the midst of the day, these two hours of 12, are called midnight and midday, than there are two other hours, each of them of 6. viz. an hour of 6 in the evening, and another of 6 in the morning.  
Where you must note, that both the two hours of 12, and the two hours of 6, come always to meet together in one and the same line, though it may be lengthened if need be, viz. the two of twelve in one line, and the two of 6 in an other.  
And you shall know, that it is an infallible thing, that within the compass of the superficies of the dial where you work in, if you have placed the axletree pretty near it, there must needs be either one of the hours of 12, or one of the hours of 6, and sometimes they meet there both at once, that is to say, one of the hours of 12, and one of the hours of 6.  
There be many situations of superficies of dials, in which, within the compass, where one may trace in the hours, there is only the line of the hours of 12, and there is found not any one of the hours of 6, and others in which there is found only the line of the hours of 6, and not any one of the hours of 12. But there is no dial in which, within the compass where it is traced in, but the hours of 12, and any one of the hours of 6 are found in it, I mean that one may find in it, either one or other of the hours of 12, and of the hours of 6, by setting the axletree near enough to the superficies of the dial.  
And now since you are sure, that there is without doubt in your dial, either one of the hours of 12, or one of 6.  
You shall begin to seek in it, first the place of that sort of hours of twelve or of six, that may be in it.  
And when you have found the point, either of one of the hours of six, or of one of twelve; You shall find afterwards the points of the other hours, that meet with it in the dial it is in your choice to begin to seek the point, of which of the two sorts of hours of 6, or of 12, you will, and I will show you two ways of seeking them out, both one after another, that when they come to be both in the dial, you may find them out both there if you will, for they serve for a proof one unto another, if you have been exact in your operation.  
That you may finish your dial as you ought, seek in the middle rule the point which is there to be found, either of the hours of twelve, or of the hours of six, for it is set there on purpose to serve for that chiefly.  
For example, seek out first in it the point of one of the hours of six, as I am going to show, than I will show you the way to find out the point of one of the hours of 12, and afterwards I will show you the way to find out the points of all the other hours of the Day.   


16   
Figure 16, To all sorts of People.  
TO seek out in the middle rule, whether the point of one of the hours of 6 be there.  
Take the string that is made fast to the axletree, set it very close again to the point or stay O, as it was when you made it turn about the axletree, then stretch it out in a straight line from the point O of the stay of the axletree, to the middle rule PQ, and holding still this string so stretched out, make it turn about the point O, carrying it from one to the other of the points Q and P▪ along the rule QP, and making it longer or shorter if need be, and set a Carpenters level or triangle over it, to see, while it turns thus, stretched out in a straight line about this point O, keeping along the middle rule, whether there is any place, wherein it comes to be found level, as for example, you see in the higher figure (over the leaf) and when you find it to be level, you must make it fast there.  
And that your level may be more fast, you may set it by the middle upon the axletree rod close to the stay, or you may set close to the stay a ruler notched at one end, as you see the ruler N, is notched, then guide it with the string, And it will serve to fasten the level upon it.  
Or to say it again otherwise, you must, as you know, cause to go about the axletree rod the squire and the string stretched out, as I have said, in a straight line, and made longer if need be, this string will go and pass all along the ruler PQ.  
And if it happens that the string, so guided or carried along the middle rule PQ, come to be found level, as the higher figure shows, mark in this middle rule the point 6, where this string toucheth or reacheth unto when it is so level, and remember that this point 6, is the point of one of the hours of 6, either of the evening, or of the morning. This is that which concerns one point of one of the hours of 6, that if the string in turning thus, comes to pass from one end of the middle ●●…e to the other, without falling to be level it is a sign, that not one of the hours of 6, comes to be found in this dial, to frame it from the point that you have taken for a rest or stay.  
Now that you may seek out the point of the hours of 12, in the lower figure.  
Fasten the centre of a hanging plummet, with a string S, to the middle of the body of the axletree rod, above it or under, as the lower figure shows, it matters not, and as the occasion will permit or require, and set this plummet so, that it may come and fall as near the middle rule as you can.  
Then tie to the axletree rod, as far as you can from the superficies of the dial, another string I with a loose knot, and when the plummet of the first string S, comes to be at rest, make it so fast that it may not stir, then stretch out this second string I in a straight line, in such sort, that coming from the axletree rod, it may go and touch the string of the hanging plummet S, without breaking (or stirring) the string nor the lead, and so holding this second string I, stretched out close to the string with the plummet on S, see whether this second string stretched out in this manner can, being made shorter or longer, if need be, go and meet the middle rule, in a point, or not.  
And when this second string so stretched out comes to meet the middle rule, in one point, as 12, mark in the middle rule this point 12, in which this string so stretched out doth meet with it, and note that that point 12, is the point of one of the hours of 12.  
When you have found out and marked in the middle rule the point of one or other of the hours, either of 6, as you see in the figure above, or of 12, as you see in the figure below, if you have them both, they shall serve for a proof one to another, if you have but one, you may make use of that alone.  
Let us suppose first, that it is the point of one of the hours of 6, as the point, 6▪ 〈◊〉 you shall go on in finding out the points of the other hours, which may be found in your dial, in this manner following.   

Figure 17, To all sorts of People.  
MArk at your discretion in the rule PQ. two several points MN, and consider the point in the middle of the body of the axletree, close by the stay O, that is the Point about which you have turned the string with the corner of the squire. You see there three several points unmovable, and fixed, viz. the point M, and the point N, in the middle rule, and the point O, in the middle of the body of the axletree rod, close to the stay. And so having those three points fixed, M, N, O, you have by this means the three several distances, viz. the measures of the distances that are from one of these three points, to the two others, viz. the space or distance from the point M, to the point N, the distance from the point M to the point O and the space from the point N, to the point O.  
Remember two things, one is, that the point O, is in the middle of the body, that is to say of the bigness, and not in the out side of the axletree rod. The other is, that these two points MN, that you have marked at discretion in the middle rule, are not for all that certainly the points of hour, and that they are to serve you to find out the points of hour, and perhaps they may chance to be some of them; and may be not, and perhaps you must blot them out after you have found out the points of hour.  
This being done so, take with your compass upon the middle rule, the distance from the point M, to the point N, and with this space go to some place that is flat or smooth, and set both the feet of your compass therein at once, as in the figure below in the points M and N, and by these two points, draw a straight line MN, as long at either end as the rule PQ.  
Then go back to the dial above, take with the compass the distance which is from the posnt M, to the middle of the bigness of the axletree close by the stay O, or else otherwise, take the distance which is from the point M, to the axletree towards the stay O, and add unto it half of the bigness of the axletree; and with this space MO, come back to the figure below, set one of the feet of the compass to the point M, and turning this foot about upon this point M, trace with the other foot a line crooked like a bow O, go back to the figure above, take again with your Compass the distance which is from the point N, to the middle of the bigness of the axletree close by the stay O, and with this space come back to the lower figure, set one of the feet of the Compass to the point N, and turning this foot upon this point N, trace with the other foot another crooked line that may meet with the other in one point, as O, for it must meet with it.  
Then open your Compass at discretion, rather more than less, and set one of the feet of the Compass so open at discretion to the point O, and turning this foot of the Compass upon this point O, trace with the other foot a round RGSH. Go back to the dial in the figure above, take with your Compass upon the rule QP, the distance which is from one of the points M or N, to the point of 6 hours, and with this space, for example of M6, come back to the lower figure, set one of the points or feet of the Compass upon this point M, go and mark with the other foot in the line M, a point as 6, of the same side upon the rule.  
And so you have in the line MN, one and the same thing as you have in the dial in the middle rule, viz. the three points MN and 6, of the same distance, in each of these two straight lines.  
This being done, draw in the figure below by the two points O and 6, a straight line O, 6, which may divide the round RGSH, in two halfs RGS, and RHS.  
Open the Compass at your discretion, and as much as the space will give you leave, and keeping your Compass so open at discretion, set one of the feet to the point S, and turning this foot upon this point S, trace with the other foot, two crooked lines L and D, then with the same space, remove your Compass out of his place, and set one of the feet to the point R, and turning this foot about upon this point R, trace with the other foot two other crooked lines, that may meet in two points L and D, the two crooked lines that you have drawn about the point S, and mark those two points L and D, and draw by those two points a straight line LD, which may pass by the point O, if you have been exact in your operations.  
So you have divided this round, into four quarters of a round, with the two straight lines For, LOD, and if the straight line LOD, drawn in length comes to meet the line MN, in a point as 12, it shows that there is also the point of the hours of 12, in your dial, viz. in the middle rule between the superficies and the axletree; now divide with your Compass every one of these quarters of the round, into six parts equal, as you see in the points that are upon the brim of the round RGSH, and by the centre or middle point of this round O, and by every one of the points of these divisions of the edge or brim of the round, draw some lines or beams, as you see some drawn already, that may go and meet the straight line MN, as in the points, 5, 4, 3, 2, 1, 11. and these points are the points of the other hours, that are to be found in your dial.  
18   


17   
Figure 18, To all sorts of People.  
NOw take with the Compass in the figure above, the space from 6 to 5, and with this space go to the dial in the figure below, set one of the feet of the Compass to the point 6, and keeping this foot of the Compass upon this point 6, go and mark with the other foot in the middle rule another point 5, and by this means you shall transport with your Compass the space 6, 5, ●●rom the line of the figure above, which represents your table, or the flat place in the dial of the figure below, upon the middle rule MN, & so accordingly take with your Compass every one of the other spaces, 5, 4, 4, 3, 3, 2, 2, 1, 1●, 12, 11, from the higher figure, and bring them in this manner to the dial upon the middle rule, in the lower figure, and so you have done in this middle rule in the dial of the lower figure, all and the same spaces as those are, that are upon the table of the lower figure: And those points of the middle rule of the lower figure, are as many points of hours that will be in the dial, among which you know that the point 6, is the point of one of the hours of six, either of the evening, or of the morning, whereby you shall come to know which are the other hours, whereof you have the points so marked in the rule of the dial.  
As for to let you know, whether this point of 6 hours, is either of those in the morning, or of those of the evening, I will not trouble this paper with it, because that is plain enough of itself: And you see well enough, whether the shadow of the axletree rod will fall upon this point, either in the morning about the beginning of the day, or else in the evening about the shutting up of the day. By this means you may see well enough, whether the hours of your dial are of those of the forenoon, or of the afternoon, for to mark them accordingly, without speaking any further about it. If you have found upon the middle rule MN, the point of one of the two hours of 12, and not the point of one of the hours of 6, you are but to do with that point of hour of twelve, the same thing that I said you should do with the point of one of the hours of 6.  
When you have so transported the points of the hours from the Table of the figure above, to the dial in the lower figure upon the middle rule▪ That which remains to do, is to transport those points of hours from the middle rule, into the superficies of the dial in the manner following, and to trace in it afterwards the line of the hours, as I will show you.  
You see, there are two strings tied to the axletree rod, in the figure below, set the rings of these two strings, as far as you can one from another, and as from R in S, then take one of these two strings, as that which comes from the point R, carry it stretched out in a direct line from the axletree rod, to one of the points of hours that are marked in the middle rule, for example in the point of hour 12, and cause this string coming so from the axletree, to pass to this point of hour 12, of the middle rule, and to go altogether in a straight line as far as the superficies of the dial; and mark in the superficies of the dial the point in which this string so carried, meets it; By this means you shall transport this point or hour 12, from the middle rule into the superficies of the dial to the point XII.  
And after the same manner, you shall transport one after another the points of hour 11, 12. 1, 2, 3, 4, 5, 6, from the middle rule, into the superficies of the dial to the points XI, XII▪ I, II, III, IV, V, VI.   

Figure 19, To all sorts of People.  
How to trace the lines of the hours upon the superficies of the dial.  
OF the two strings, fig. above, that are made fast to the axletree rod, stretch out one in a straight line, from the axletree rod, as from the point R to a Point of hour of the middle rule, as to the point of hour I▪ and holding this string so stretched, take the other or second string coming from the point S, and stretching it likewise in a straight line, make it cross over the first string by▪ and let it touch it without breaking his straight line, and let it go in a straight line from thence, to the superficies of the dial as to the point D, and mark the point D, in the superficies of the dial, wherein this second string so carried, comes to touch it; then make this second string to go and touch again the first in another place, and with this second string go and touch in the same manner, another point E, in the superficies of the dial E, And so remove this second string along the first string, as many times as you shall have need to mark any several points, as D, E, L, in the superficies of the dial, to trace the line of that hour there, then draw a line as fine and delicate as you can by all those points deal, in the superficies of the dial, that line shall be the line of that hour I.  
And after this manner, you shall trace in the superficies of the dial, the lines of all the other hours that are in the middle rule, and your dial will be finished.  
The lower figure shows you to the eye, how that after you have transported as above said, all the points of hour from the middle rule into the superficies of the dial, you may take away this middle rule, and the two knobs that hold it up, and make an end of tracing the rest of the lines in the superficies of the dial, as I have said, with the strings coming from R and I, and by means of the points of hours XI, XII, I, II, III, IV, V, VI.   

Figure 20, To all sorts of People.  
AFter that you have placed the axletree of the dial as it must be, if you desire to find the points of the hours in the superficies, with some extraordinary instruments, that which is the plainest of all, viz. a round flat plate, and stiff, as of tinned iron, or the like, and divided into 24 parts equal one unto another, and set up in the manner of a rotunda or whirl, by the squire, or with right angles about the axletree of the dial, as the figure below doth show, is the most common and the shortest way of all.  
The figure H, shows you this round alone, and how it is open of one side, that the centre thereof may be placed with the centre of the axletree.  
The 1. figure shows the neck that may be applied unto this round about the centre, to the end that one may with this neck, set the round to the axletree of the dial by the squire, or with angles straight between themselves, as you see in the second figure.  
When you have thus set this round to the axletree of the dial, the lower figure shows you how you must place the string of the plummet, hanging upon the axletree by a point of one of the divisions of the edge of this round, that it may give you the points of the hours in the superficies of the dial.  
The strings which coming from the axletree pass afterwards to the points of the division of this round in its 24 parts, show you, how you must afterwards bring the strings from the axeltre, by the points of the division of this round in 24 parts, equal to the superficies of the dial, that you may have the points of the hours in this superficies.  
The string LS, XII, that passeth to the string with the plummet RS, gives the point of the hours of 12. The string LVI, that passeth to one of the points of this division in 24, and is found to be level, gives the point of the hours of 6.  
The other strings show you, that the way of tracing the points of the other hours is the same as above.  
20   

 
Figure 21, To all sorts of People.  
When you have brought, as I have said, by means of this rotunda and the strings, all the points of the hours, into the superficies of the dial; You may take away the rotunda if you will, and make an end of tracing the lines of the hours as before with the strings, and by means of the points of hours, which you have brought into the superficies of the dial, as you see in the figure below, the line DELKPQSYIZG.  
And for this purpose by means of the said strings, carry a string in a straight line from the axletree to the point of hour, for example I, and holding it there stretched in a straight line, carry of one side or other according to the occasion, an other string coming also from the axletree, as from I, or from B, that may go in a straight line as far as the superficies of the dial, and let it go and touch, and cross over the string IR, several times in several places, and at every time go with this second string to touch and mark a point in the superficies of the dial, until you have enough, as you see the points D, E, L, K, P, Q, Y, ay, Z, G, and carry by these points a line sweetened, that shall be a line of hour, do the like for the lines of the other hours, and you have done.  
When you have marked in the superficies of the dial, a point of every one of the hours that are to be found in it: If you desire to trace the lines of the hours every one at once, without making use of the strings, as in the figure above, you may do it when it is dark, as by night, with the light of a candle, in that manner as it is expressed in the lower figure.  
Set a light behind the axletree rod of the dial, and turn the same lightabout this axletree, until the shadow of this axletree come to one of the points of hour I, and trace in the superficies of the dial a line DELKPQSYIZG, all along this shadow of the axletree, that line shall be a line of hour, do the like for every one of the other points of hour, and you have finished your dial.   


22  
Instruments to work with all,   

21   
Figure 22, Several Instruments to work withal in these occasions hereafter specified.  
I Did not intend to burden my memory with any thing in this matter, but with Monsieur de Sargues universal rules for the placing of the axletree, and for the tracing in a dial the hours after the French way, without meddling with the rest, which is more curious than useful.  
But to follow the advice of many considerable persons whom I do honour, I have set down also the way to mark that which is commonly called the signs: The hours after the Italian or Babylonian way: The hours after the manner of the Ancients: The elevations of the Sun above the Horizon; and the rising of the same.  
And for as much as none can do any of these things universally, without using these instruments more or less; This table following shows to the eye all the pieces that are used in those several occasions. These instruments are first a circle, a half circle, or the quarter of a circle, which is all one which is made to turn about its Diameter set fast in it's due and convenient place, or down right, as in the fourth figure, or level, as in the second or third figure, or else inclining or hanging downward, as in the first figure.  
The way to make this circle to move in all kinds of positions, is to set two rings in its Diameter, through which one may put in a stick straight, round, and smooth, about which this circle may turn round like a weather cock about his needle or spear, as in the second figure, and there must be within those hoop rings, a screw to fasten this circle in that place, or which way soever you will have it to stand.  
The sticks or rods are represented by the 7th. figure, with a fork at the end of every one, bored in the cheek, to put a pin through, as you see, that one may be set plum or down right, and the other level, being made fast at one end to the axletree rod as in the 5th. figure. And for this purpose also the axletree rod is bored in O.  
The 8th. figure shows the axletree rod by itself bored with O, and the pin Q, put through the hole, to show more plainly that which the 5th figure represents, viz. all the pieces set or joined together being marked with qo.  
You see that the hoop rings are near the edge or brim of the circle, a purpose to leave the centre O, and a space about it free, having commonly a piece taken off, that this circle may turn freely about the forked end, that is to say about his centre, without any let or hindrance at all.   


23  
for the signs.   
Figure 23, How to mark the Signs.  
GEt, figure 2 below, a half circle both thin and stiff ctsrd, draw there a beam OZS, perpendicular to the Diameter CPOQD, take on both sides of this beam OS, 23 degrees and half, for example 23 degrees and half from s towards t, and as much from s towards r, draw the straight line r, t, make upon Diameter t, r, a half, circle tzr, divide the edge or brim of that half circle in six equal parts, as in the points that you see there; draw by those points as far as the half circle CtsrU, some straight lines that may be perpendicular to the straight line r, t, bring from the centre O, by the points that those perpendicular have made upon the edge of the half circle CtsrD, some straight lines, as you see that the strings show you, and with those lines drawn out sufficiently, you shall mark the signs in the dial, as I shall say.  
You see that the half circle is cut thorough, or made hollow from the point P, to the point Q, round about the centre O, according to the circumference PZQ, which is notched also in the points that you see in it, which are betwixt every degree of the half circle; And the centre O, and these notches, are there a purpose to fasten a string upon them, instead of bringing it from the centre O.  
The two figures s32tgez, s45rbcz, both on the right and on the left side of the half circle CtsrD, show as you may judge by their letters or coats, each of them one half of the figure tsrz, of the half circle CtsrD, which I have made thus bigger than each half of this figure, that one may set in the letters ge, cb, and some figures, 2345, about the edges of the two half circles, and also the signs, as you see, which I could not do in the middle figure without confusion.  
The lines coming from the points t23s, s45r, towards the lowest past of the figure or plate, coming near one an other, go and seek the centre of the half circle t23s45r, Every one of the three faces between those straight lines is to hold two signs, mark them there in the same order that you see them, close by these straight lines, one of one side, and an other on th'other side.  
And by this means the straight line of the half circle t23s45r, which from the centre of the half circle passeth to the points, is that of the signs of Aries, and of Libra; that which passeth to the point 3, is that of Taurus, and of Virgo; that which passeth to the point 2, is that of Gemini, or the twins, and of Leo; that which passeth to the point t, is that of Cancer; that which passeth to the point r, is that of Capricornus; that which passeth to the point 5, is that of Sagittarius and Aquarius; that which passeth to the point 4, is that of the Scorpion, and of the Fishes.  
The two figures s32cgez, s45rbcz, show you also that with one quarter of circe marked on both sides with 6 signs in one part, and 6 others in the other part, you may do the same thing as well as with the half circle, by turning this quarter of circle, as you see in the said figures, once of one side, and once of th'other.  
I shall for all that speak to you always as if you had the half circle in your hand.  
Therefore when you will mark the lines of the signs in the Superficies of the dial, The first figure shows you how you must set up your half circle with the axletree, for to turn it about the same, without going up or down along the axletree.  
The first figure above shows how you must make your half circle, viz. about the axletree, set then the half circle to the axletree of the dial, as you see in the figure that is under the first.  
Tie a string with a loose knot, just in the centre of the circle.  
Turn the half circle about the axletree, cause at the same time the string coming from the centre, to pass by one of the lines of the signs, holding it longer or shorter as need shall require, go and touch with the string many several points in the superficies of the dial one after an other.  
Draw a line sweetened, by all those points, and it is the line of the signs that are marked along the straight line of the half circle which the string doth cover, in turning with it about the axletree.  
Do the like for every line of the signs, mark the signs in the dial by the lines so drawn according to the situation, in regard of the Country and of the place of the dial, and as the figure shows, you have marked the signs in the dial.  
And if the strings could not come from the centre, fasten them with a knot to the beams, coming from the centre, in the notches or clefts of the circumference PZQ.  
Set a button or an other mark in the axletree, in the place where the centre of the half circle CtsrD hath been, and the shadow of the button will go and mark the sign that the Sun is in.   

Figure 24. To mark the hours after the Italian or Babylonian way.  
THE first figure shows how you must set on your half circle, and how to make it turn about the axletree.  
Moreover, the line NO (Shows you what kind of line coming from the centre O of this half circle, you must make use of, in making the half circle to turn about.  
When you have drawn the lines of the hours after the French way at length, in the superficies of the dial, as the figure below doth show.  
Set on, as the same figure shows you also, the half circle O t rto the exeltree of the dial with a string ON in its centre.  
Let this half circle hang down right or Plum, and when this half circle is just down and at rest, draw the string ON, in a straight line coming from the centre O, and closing with the half circle, in such sort as it may go, and touch it all at length, turn this string as a beam of the half circle about the centre O, till it be very level, as the figure shows by the setting on of the Carpenters level A.  
When the string ON is stretched out very level close to the half circle, mark exactly upon the edge of this half circle, the point wherein the string ON toucheth it, as doth the letter E.  
Then let the half circle turn about the axletree.  
Make in the mean time the string ON, to pass by the point t, which you have marked upon the edge of the half circle, and making it shorter or longer according as need shall require, go and touch with this string many several points one after an other in the superficies of the dial in divers places, 1, 2, 3, 4, 5, 6.  
Draw an obscure line by those points, as you see the line bowed or crooked, 23, 24, 1, 2, 3, 4, 5, 6, and which reacheth beyond the axletree towards h.  
This line crosseth over the superficies of the dial, out of the Equinoctial line PQ, and meets by the way all the lines of the hours after the French way, as you see it doth in 23, 24, 1, 2, 3, 4, 5, 6.  
There remains to trace the lines of these hours after the Italian or Babylonian way in the superficies of the dial and when you know how to trace one, you shall be able also to trace the other.  
Therefore to trace a line of those kinds of hours, It is no matter which you begin to trace first; Count upon the equinoctial line PQ fixes paces of hours equal, after the French way, one after an other, as from XII. to VI. Afterwards follow the lines of the hours, after the French way, that pass by the points XII. & VI the two extremities of these six spaces every one to the aforefaid line of which you have found the place, by turning the string with the half circle about the axletree by the point t, as you see, as far as the points 24. And 6.  
Take conveniently in these two lines of hours after the French way, In each of them one of the points wherein it meets with the Equinoctial line PQ, or else the said line found with the string 1, 2, 3, 4, 5, 6, that is to say in one, the point that the line placed with the string makes in it, and in the other; the point that the Equinoctial line makes in it, for example, In the line, XII, 24, take in it the point 24. Wherein it meets with the line found with the string: and in the other, VI, take in it the point VI. wherein it meets with the equinoctial line.  
Set either a string or a rule by these two points so taken 24. and VI, as you see the line, 24. 6.  
Then with the string coming from the centre of the half circle Q go razing or laying even the string 24. VI by making it longer or shorter as need requires; as you see in O, g, mark many several points in the superficies of the dial one after an other, as for example 24, g, VI, more or less, according as the superficies of the dial is more or less uneven.  
Draw an obscure line by the points 24, g, VI, and it will be a line of hours after the Italian or Babylonian way, and so of all the rest.  
The string h, OH shows that you may if need be, do the like both of one and of the other side of the centre O, to go and place of one part or other according to the occasion, the line, as 2, 3, 4, 5.  
And if you have a straight line, as might be O, q, which may turn about the centre O, and be perpendicular to the axletree by, and you hold the half circle with this straight line, set one at a convenient or reasonable distance from the other: And let it be always exactly of the distance of six hours after the French way: First of all, this string describes the Equinoctial line in the superficies of the dial, Secondly when one of the two, either the half circle or the straight line O q, is found in one of the points of the hours of the Equator, th'other is likewise found in it, in an other point of hour, then drawing with a string coming from the centre O, a straight line that may go from the point, as t, to the end of the straight line O q which you shall go drawing with this string made shorter or longer, as need requires, and mark some points of line of hour, after the Italian or Babylonian way in the superficies of the dial.  
And for this purpose there is nothing so easy as to have a circle of Equator, that may be fitted to the half circle, and where you may have always a space ready made for its hour.  
25  
To mark the hours after the Manner of the Jews.   


24  
for the hours after the Italian way.   
Figure 25, To mark the hours after the manner of the Ancients or the Jews.  
YOu must know first that it would be very troublesome to draw in the superficies of the dial, the lines of this kind of hours in such a manner as that they might be always just and right in theory, all the year long.  
And therefore it is sufficient to draw them just by demonstration in three points only, viz, in their points of both ends, and of the middle, which are the points of those circles that appear the greatest above the Horizon being parallel to the Equator, and of the Equator itself.  
The rest goes as it may, and therefore it may be said, that the lines of such hours traced in this manner are false in the rest of their length, yet curiosity makes them pass for current.  
Wherefore to mark this kind of lines of hours.  
The higher figure 4 shows which way you must make this half Circle to turn about, viz. about a straight axletree line placed level in the centre of the axletree of the dial.  
And to be short, set up and make very fast a rod in a straight line passing to the centre O, and let it be first within the joint of the axletree rod, secondly let it be level, as the figures do show of a plummet P. and of a level A, this being done, tie some strings with a loose knot to this rod so leveled NL as you see NR, and LT.  
Take the string from about the centre O, stretch it out in a direct or straight line from the centre O to one of the points of hour, after the French way, of the Equinoctial line of the dial, for example to the point of 1 hour, as you see the string OI.  
This string being thus stretched out, take the other strings of one or th'other end NL, and cross over this string OI with them, and so go and mark many points in the superficies of the dial, as TIR.  
Draw an obscure line by those points as TIR, it is a line of hours after the manner of the Ancients or the Jews, do the like with the other hours and half hours of the Equinoctial line.  
If you leave a rod in the dial, as NOL, the shadow thereof will go and show these hours continually at length, if you will not leave it in, the button or centre O of the axletree of the dial will show them.   


26  
for the height of the sun.   
Figure 26. How to mark the Elevation of the Sun above the Horizon.  
THE higher fig. 3. shows which way you must turn the half circle, viz. about a straight axle-line hanging down right.  
Set up your half circle, so that it may turn like a weathercock about a rod hanging down right, or plum, above or below the axletree of the dial, it matters not which.  
Whilst you turn it thus as it is above said, cause in the mean time the string coming from the centre O to pass by one of the degrees of the edge of the circle; and make the string shorter or longer as need shall require, mark with it many several points in the superficies of the dial, according as you see them ranked one by an other, in four places.  
Draw a small or obscure line through all these points, and this will be one of the lines of the Elevation of the Sun.  
Count the Degrees in the edge of the circle, beginning at the first of the beam which is level, and ending at the 90. Beam which is down right or plum.  
Mark in the line of the dial, the number of the Degrees of the border of the circle, where the string passes that hath marked the points of that line, and so of all the others, and the shadow of the button of the axletree which is in the centre of the circle, will show the Elevation of the Sum above the Horizon.   

Figure 27: How to mark the Sun rising, or East rising of the Sun.  
THE figure 2 above shows how you must place the half circle, viz. parallel unto the Horizon, I would not put a level to it to avoid confusion. It shows also that one of the Diameters of the circle must be set within the centre of the dial, that is to say, that it must go directly from the south to the North, and accordingly the Diameter which is perpendicular to it, will go from East to West.  
When your circle is set fast in this position, let a plummet op in the lower figure hang from the centre O.  
This being done, from each point of Degree of the edge of the circle, as from x and from z. mark with a string XT or ZR. many points in the superficies of the dial.  
Draw a small or obscure line through these points, as tie or SR. it is a line of the Suns Eastrising-  
Mark in it the number of Degrees of the point of the circle from whence the string comes, according as you will count them, to begin either from the East, or from the South. And so of all the other Degrees accordingly.  
And the shadow of the button O will show which way the sight of the Sun comes upon the dial.  
I will take here occasion to tell you, that if for some reason or other, you could observe, in one and the same day, but two shadows of the Sun in stead of three, as we have said in the placing of the axletree in the dial, the declining of the Sun in that day, will serve you for a third shadow, or else two other shadows observed in an other day. I mean you may find equally the placing of the axletree by one or other of those ways above mentioned, and with 3 shadows; and with 2 shadows, and the declining of the Sun in that day, and with 4 shadows, two of one day, and two of an other, which are three ways that come all to one.  
27  
for the Eastrising of the sun.   


28   
Figure 28.  
I do not specify in this volume these kinds of flat dials, wherein you may work without knobs or middle rule: And where you may draw the Equinoctial line; trace out and divide the circle Equator: in a word, where you may do all: yea and in the very superficies of the dial, you may easily come to know them you self, by putting this universal way into practice.  
Here is only a way how to trace out all the twelve lines of the hours equal, after the French way, in the flat dials where the axletree meets the superficies athwart in the space that you work in, so that you shall have no need of a greater place. And what I have already said, and what I am now going to say again, will serve to find out the way to do the like in all kinds of dials universally.  
When you have drawn upon your dial the Equinoctial line M 12 M, drawn conveniently and divided the circle Equator Q 12 Q bring to the Equinoctial line, the beam of the 12 hours' Q, 12. Draw of both sides of the Equator, and from the Equinoctial line a straight line MQ parallel to the beam of 12. hours O 12. bring the beams of the other hours to the first, which they shall find of the Equinoctial in rt: and of MQ in c, d●g, Q, Bring in the dial the line of the twelve hours B. 12. draw by the point M, of the Equinoctial line, and from the centre of the dial B a straight line ML parallel to the line of twelve hours B 12; make upon this line ML and upon the point M a triangle LMN like to the triangle in the air OB 12. and let the angles of these triangles in the points L and B be equal one unto an other, Carry the spaces Mq, Mg, Md, Mc, from the straight line MQ into the straight line MN, viz. from M into N, into u, into i, into o, bring by the points N, u, i, o, some straight lines NL, ub, if, oh, parallel to the side, NL of the triangle LMN: Carry from the centre of the dial B by the points r, t, h, f, o, L, Some straight lines, BL, Bh, Bf, Bh, Bt, Br; These are such lines of hours as you may continue beyond the centre B, and mark them according to their orders.  
THE END.