Mechanic dialing: TEACHING Any Man, though of an Ordinary Capacity and unlearned in the Mathematics, to draw a True SUN-DYAL On any Given Plane, However situated: Only with the help of a strait RULER and a pair of COMPASSES; And without any Arithmetical Calculation. By Joseph Moxon, Hydrographer to the Kings most Excellent Majesty. LONDON. Printed for Joseph Moxon on Ludgate-hill, at the Sign of Atlas. MDCLXVIII MECHANIC dialing. Description of dialing. dialing originally is a Mathematical Science, attained by the Philosophical contemplation of the motion of the Sun, the motion of the Shadow, the Constitution of the Sphere, the Situation of Planes, and the consideration of Lines. Explanation. THE motion of the Sun is regular, it moving equal Space in equal Time; But the motion of the Shadow irregular in all parts of the Earth, unless under the two Poles, and that more or less according to the Constitution of the Sphere and situation of the Plane. And therefore Scientifick Dyalists by the Geometric considerations of Lines, have found out Rules to mark out the irregular motion of the Shadow in all Latitudes, and on all Planes, to comply with the regular motion of the Sun. And these Rules of adjusting the motion of the Shadow to the motion of the Sun may be called Scientifick dialing. But though we may justly account dialing originally a Science, yet such hath been the Generosity of many of its studious Contemplators, that they have communicated their acquired Rules; whereby it is now become to many of the Ingenious no more difficult than an Art, and by many late Authors so entitled: Nay more, by this small Treatise it will scarce be accounted more than a Manual Operation; for, though (hitherto) all the Authors I have met with seem to presuppose their Reader to understand Geometry, and the projecting of the Sphere already, or else endeavour in their Works to make him understand them, as if they were absolutely necessary to be known by every one that would make a Dyal, when as in truth (the contemplative pains of others aforesaid considered) they are not; but indeed are only useful to those that would know the reason of dialing. Thus they do not only discourage young beginners, but also disappoint many Gentlemen and others that would willingly either make them themselves, or set their Workmen about them, if they knew how to make them. This little Piece I have therefore composed for the help of those who understand neither the Projection of the Sphere, or Geometrical Operations: Only, if they know how to draw a strait Line between two Points by the side of a Ruler, describe a Circle with a pair of Compasses, erect a Perpendicular, and draw one Line parallel to another, they may know how to draw a Dyal for any given Plane, however situated in any Latitude. But perhaps these two last little Tricks are not known to all new beginners, therefore I shall show them. First, How to erect a Perpendicular. For Example, in Fig. 1. Upon the Line AB you would erect a Perpendicular to the Point C: Place one Foot of your Compasses upon the point C, and open the other to what distance you please; For Example, to the point A, make there a mark; then keeping the first Foot still in C, turn the other Foot toward B, and make there another mark; then open your Compasses wider, suppose to the length AB, and placing one Foot in the point A, with the other Foot describe a small Arch over the point C, and removing the Foot of your Compasses to the point B, with the other Foot describe another small Arch, to cut the first Arch, as at D. Then lay your strait Ruler to the point where the two small Arches cut each other, and upon the point C, and by the side of the Ruler draw the Line CD, which shall be a Perpendicular to the Line AB. Another way with once opening the Compasses, as by Fig. 2. Draw the Line AB, and place one Foot of your Compasses upon the point you would have the Perpendicular erected, as at the point C, and with the other foot describe the Semicircle A ab B, then placing one Foot in B, extend the other Foot to b, in the Semicircle; and keeping that Foot in b, extend the other Foot to D, and make there a small Arch: Then remove one Foot of your Compasses to A, and extend the other Foot to a in the Semicircle, and keeping that Foot in a, extend the other to D, and make there another small Arch, to cut the first small Arch; and laying a strait Ruler to the point where these two small Arches cut each other, and upon the point C, draw by the side of the Ruler the Line CD, which shall be Perpendicular to the Line AB. To erect a Perpendicular upon the end of a Line, as by Fig. 3. On the point B, at one end of the Line AB, place one Foot of your Compasses in the point B, and extend the other on the Line towards A, as to b, and with it describe the Arch ba C; then placing one Foot in b, extend the other to a in the Arch, and make there a mark; Divide with your Compasses the Arch ba into two equal parts, and keeping the Feet of your Compasses at that distance, measure in the Arch from a to C, then draw a strait Line from the point C to the end of the Line B, and that strait Line shall be Perpendicular to the end of the Line AB. To draw a Line Parallel to another Line, as by Fig. 4. Example. If you would draw a Line Parallel to the Line AB, open your Compasses to the distance you intent the Lines shall stand off each other, and placing one Foot successively near each end, describe with the other Foot the small Arches CD; lay a strait Ruler to the top of these Arches, and draw a Line by the side of it, and that Line shall be Parallel to the Line AB. Definitions. A Dyal Plane is that Flat whereon a Dyal is intended to be projected. Of Dyal Planes some be Direct, other Decliners, others Oblique. Of Direct Planes there are five sorts: 1. The Horizontal whose Plane lies flat, and is parallel to the Horizon, beholding the Zenith. 2. The South Erect, whose Plane stands upright, and directly beholds the South. 3. The North Erect, whose Plane stands upright, and directly beholds the North. 4. The East Erect, whose Plane stands upright, and directly beholds the East. 5. The West Erect, whose Plane stands upright, and directly beholds the West. Of Decliners there are infinite: and yet may be reduced into these two Kind's: 1. The South Erect Plane, declining more or less towards the East or West. 2. The North Erect Plane, declining more or less towards the East or West. Of Oblique Planes some are Direct, others Declining; and are of four sorts: 1. Direct Inclining Planes, which lean towards you, and lie directly in the East, West, North, or South quarters of Heaven. 2. Direct Reclining Planes, which lean from you, and lie directly in the East, West, North, or South quarters of Heaven. 3. Inclining Declining Planes, which lean towards you, but lie not directly in the East, West, North, or South quarters of Heaven: But decline more or less from the North or South, towards the East or West. 4. Reclining Declining Planes, which lean from you, but lie not directly in the East, West, North, or South quarters of Heaven: But Decline more or less from the North or South, towards the East or West. If the Situation of the Plane be not given, you must seek it: For, there are several ways how to know these several kinds of Planes used among Artists; But the readiest and easiest is by an Instrument called a Declinatory, fitted to the variation of your Place: And if it be truly made, you may as safely rely upon it as any other. OPERATION I. The Description of the Clinatory. THE Clinatory is made of a square Board, as ABCD, of a good thickness;, and the larger the better; between two of the side is described on the Centre Aa Quadrant as OF divided into 90 equal parts or degrees, which are figured with 10, 20, 30, to 90; and then back again with the Compliments of the same numbers to 90: between the Limb and the two Semidiameters is made a round Box, into which a Magnetical Needle is fitted; and a Card of the Nautical Compass, divided into four Nineties, beginning their numbers at the East, West, North, and South points of the Compass, from which points the opposite sides of the Clinatory receives their Names of East, West, North and South. But, Note, that the North point of the Card must be placed so many degrees towards the East or West sides of the Clinatory as the Needle varies from the true North point of the world, in the place where you make your Dyal; which your Workman that makes your Clinatory will know how to fit. Upon the Centre A, whereon the Quadrant was described, is fastened a Plumb-line, having a Plumbet of Lead or Brass fastened to the end of it, which Plumb-line is of such length that the Plumbet may fall just into the Grove GH, below the Quadrant, which is for that purpose made of such a depth that the Plumbet may ride freely within it, without stopping at the sides of it. See the Figure annexed. With this Clinatory you may examine the situation of Planes. As if your Plane be Horizontal, it is direct: and then for the true scituating your Dyal you have only the true North and South Line to find: which is done only by setting the Clinatory flat down upon the Plane, and turning it towards the right or left hand, till you can bring the North point of the Needle to hang just over the Flower-de-luce, for than if you draw a Line by either of the sides parallel to the Needle, that Line shall be a North and South Line. If Your Plane either Recline or Incline, Apply one of the sides of your Clinatory parallel to one of the Semidiameters of the Quadrant to the Plane, in such sort that the Plumb-line hanging at liberty, may fall upon the Circumference of the Quadrant, for then the number of degrees of the Quadrant comprehended between the side of the Quadrant parallel to the Plane, and the Plumb-line shall be the number of degrees for Reclination, if the Centre of the Quadrant points upwards; or Inclination, if the Centre points downwards. If your Reclining or Inclining Plane Decline, Draw upon it a Line parallel to the Horizon, which you may do by applying the backside of the Clinatory, and raising or depressing the Centre of the Quadrant, till the Plumb-line hang just upon one of the Semidiameters, for than you may by the upper side of the Clinatory draw an Horizontal Line if the Plane Incline, or by the under side if it Recline. If it neither Incline or Recline, you may draw a Horizontal Line both by the upper and under sides of the Clinatory. Having drawn the Horizontal Line, apply the North side of the Clinatory to it, and if the North end of the Needle points directly towards the Plane, it is then a South Plane. If the North point of the Needle points directly from the Plane, it is a North Plane: but if it points towards the East, it is an East Plane: if towards the West, a West Plane. If it do not point directly either East, West, North, or South, than so many degrees as the Needle declines from any of these four points to any of the other of these four points, so many degrees is the Declination of the Plane. You may find a Meridian Line another way; thus, If the Sun shine just at Noon, hold up a Plumb-line so as the shadow of it may fall upon your Plane, and that shadow shall be a Meridian Line. OPERAT. II. To describe a Dyal upon a Horizontal Plane. FIrst draw a North and South Line (which is called a Meridian Line) through the middle of the Plane: Thus, Set your Declinatory flat upon the Plane, and turn it to and fro till the Needle hang precisely over the Meridian Line of the Declinatory; then by the side of the Declinatory parallel to its Meridian Line, draw a strait Line on the Plane, and if that strait Line be in the middle of the Plane, it shall be the Meridian Line, without more ado: But if it be not in the middle of the Plane, you must draw a Line parallel to it through the middle of the Plane for the Meridian Line, or twelve a Clock line: And it shall be the Meridian Line, and also be the Substilar Line; then draw another strait Line through the middle of this Line, to cut it at right Angles for the VI a Clock Lines; and where these two Lines cut one another make your Centre, whereon describe a Circle on your Plane as large as you can, which by the Meridian Line, and the Line drawn at right. Angles with it will be divided into four Quadrants; one of the Quadrants divide into 90 degrees thus, Keeping your Compasses at the same width they were at when you described the Quadrant, place one Foot in the twelve a Clock Line, and extend the other in the Quadrant, and make in the Quadrant a mark with it; so shall you have the sixtieth degree marked out: then place one Foot of your Compasses in the six a Clock Line, and extend the other in the Quadrant, and make in the Quadrant another mark with it; so shall that Quadrant be divided into three equal parts; each of these three equal parts contains 30 degrees: Then with your Compasses divide one of these three equal parts into three parts, and transfer that distance to the other two third parts of the Quadrant, so shall the whole Quadrant be divided into nine equal parts. Then divide one of these nine equal parts into two equal parts, and transfer that distance to the other eight equal parts, so shall the Quadrant be divided into eighteen equal parts. Then divide one of these eighteen equal parts into five equal parts, and transfer that distance to the other seventeen equal parts, so shall the whole Quadrant be divided into 90 equal parts. Each of these 90 equal parts are called Degrees. Note, That you may in small Quadrants divide truer and with less trouble with Steel Deviders, (which open or close with a Screw for that purpose,) than you can with Compasses. In this Quadrant (thus divided) count from the Substilar or Meridian Line the Elevation of the Pole, that is, the number of Degrees that the Pole of the World is elevated above the Horizon of your Place, and draw a Line from the Centre through that number of Degrees for the Stilar Line. Then on the Substilar Line choose a point (where you please) and through that point draw a Line at right Angles to the Substilar Line as long as you can, for the Line of Contingence, and from that point in the Substilar Line measure the nearest distance any part of the Stilar Line hath to that point; and keeping one Foot of your Compasses still in that point, set off that distance in the Substilar Line, and at that distance describe against the Line of Contingence a Semicircle, which divide from either side the Meridian or Substilar Line into six equal parts thus; Draw a Line through the Centre of this Semicircle parallel to the Line of Contingence, which shall be the Diametral Line, and shall divide this Semi Circle-into two Quadrants; one on one side the Substilar Line, and the other Quadrant on the other side the Substilar Line: Then keeping your Compasses at the same distance they were at when you described the Semicircle, place one Foot first on one side the Diametral Line at the Intersection of it and the Semicircle, and then on the other side, at the Intersection of it and the Semicircle, and extend the other in the Semicircle, and make marks in the Semicircle on either side the Substilar Line: Then place one Foot of your Compasses at the Intersection of the Semicircle and the Substilar Line, and turn the other Foot about on either side the Semicircle and make marks in the Semicircle, so shall the Semicircle be divided into six equal parts: Divide one of these equal parts into two equal parts, and transfer that distance to the other five equal parts, so shall the whole Semicircle be divided into twelve equal parts. These twelve Devisions are to describe the twelve Hours of the Day, between six a Clock in the Morning, and six a Clock at Night. If you will have half Hours you may divide each of these twelve into two equal parts, as before: If you will have Quarters you may divide each of these twenty four into two equal parts more, as before. For thus proportioning the Devisions in the Semicircle, you may proportion the Devisions and Sub-devisions of Hours upon the Dyal Plane; for a strait Ruler laid upon each of these Devisions, and on the Centre of this Semicircle, shall show on the Line of Contingence the several Distances of all the Hours and parts of Hours on the Dyal Plane: And strait Lines drawn from the Centre of the Dyal Plane, through the several Devisions on the Line of Contingence shall be the several Hour Lines and parts on the Dyal Plane. But an Horizontal Dyal in our Latitude will admit of four Hours more, viz. V, IV, in the Morning, and VII, VIII, in the Evening. Therefore in the Circle described on the Centre of the Dyal Plane transfer the distance between VI and V, and VI and IV, on the other side the six a Clock Line; And transfer the Distances between VI and VII, and VI and VIII on the other side the opposite six a Clock Hour Line, and from the Centre of the Dyal Plane draw Lines through those transferred Distances for the Hour Lines before and after VI Then mark your Hour Lines with their respective numbers. The Substiler Line in this Dyal (as aforesaid) is XII, from thence towards the right hand mark every successive Hour Line with I, TWO, III, etc. and from XII towards the left hand with XI, X, IX, etc. The Style must be erected perpendicularly over the Substilar Line, so as to make an Angle with the Dyal Plane equal to the Elevation of the Pole of your Place. Example. You would draw a Dyal upon a Horizontal Plane here at London; First draw the Meridian (or North and South Line) as XII B, and cross it in the middle with another Line at right Angles, as VI, VI, which is an East and West Line; where these two Lines cut each other as at A, make the Centre, whereon describe the Semicircle B, VI VI; but one of the Quadrants, viz. the Quadrant from XII to VI, towards the right hand you must divide into 90 equal parts (as you were taught in Fol. 12.) and at 51 ½ degrees (which is London's Latitude) make a mark, and laying a strait Ruler to the Centre of the Plane, and to this mark draw a Line by the side of it for the Stiler Line. Then on the Substilar Line choose a point as at C, and through that point draw a Line as long as you can perpendicular to the East and West Line VI, VI, as OF, (which is called the Contingent Line,) where this Contingent Line cuts the Substilar Line place one Foot of your Compasses, and from thence measure the shortest distance between the point C and the Stilar Line. And keeping one Foot of your Compasses still in the point C, set off the shortest distance between the point C and the Stilar Line on the Substilar Line, as at D; which point D shall be a Centre, whereon with your Compasses at the same width you must describe a Semicircle to represent a Semicircle of the Equinoctial. This Semicircle divide into six equal parts (as you were taught Fol. 13.) to each of which equal parts, and to the Centre of the Equinoctial Semicircle lay a strait Ruler, and where the strait Ruler cuts the Line of Contingence make marks in the Line of Contingence. Then lay the strait Ruler to the Semicircle of the Dyal Plane, and to each of the marks in the Line of Contingence, and by the side of it draw twelve strait Lines for the twelve Fore and Afternoon Hour Lines, viz. from VI in the Morning to VI in the Evening. Then in the Quadrant VI B, measure the distance between the VI a Clock Hour Line, and the V a Clock Hour Line, and transfer the same distances from the VI a Clock Line to VII, and V on both sides the VI a Clock Hour Lines, and through those distances draw from the Centre of the Plane the VII and V a Clock Hour Lines, and measure the distance between the VI a Clock Hour Line and the IV a Clock Hour Line, and transfer the same distance from the VI a Clock Line to VIII and IV, and through those distances draw from the Centre of the Plane the VIII a Clock and IV a Clock Hour Lines. If you will have the half Hours and quarter Hours, or any other division of hours, you must divide each six devisions of the Equinoctial into so many parts as you intent, and by a strait Ruler laid to the Centre of the Equinoctial, and those devisions in the Equinoctial Circle make marks in the Line of Contingence, as you did before for the whole Hour Lines; and Lines drawn from the Centre of the Plane through those marks shall be the sub-devisions of the Hours: But you must remember to make all sub-devisions short Lines, and near the verge of the Dyal Plane, that you may the easier distinguish between the whole Hours and the parts of Hours; as you may see in the Figure. Having drawn the Hour Lines, set the number of each Hour Line under it, as you see in the Figure. Last of all sit a Triangular Iron, whose angular point being laid to the Centre of the Dyal Plane, one side must agree with the Substilar Line, and its other side with the Stilar Line; so is the Style made. And this Style you must erect perpendicularly over the Substilar Line on the Dyal Plane, and there fix it. Then is your Dyal finished. OPERAT. III. To describe an Erect Direct South Dyal. YOU may know an Erect Direct South Plane by applying the North side of the Declinatory to it; For then if the North end of the Needle hang directly over the North point of the Card in the bottom of the Box, it is a South Plane; but if it hang not directly over the North point of the Card, it is not a Direct South Plane, but Declines either East or West, and that contrary to the pointing of the Needle Easterly or Westerly from the North point of the Card: for if the North point of the Needle points Easterly, the Plane Declines from the South towards the West: if it point Westerly, the Plane Declines from the South towards the East. You may know if the Plane be truly Erect or upright, by applying one of the sides AB or AD to it; for then by holding the Centre A upwards, so as the Plumb-line play free in the Grove, if the Line falls upon 0, or 90, the Plane is upright; but if it hang upon any of the intermediate Degrees, it is not upright, but Inclines or Reclines. If you find it Incline, apply the side AB to it, and see what number of Degrees the Plumb-line falls on, for that number of Degrees counted from the side AB, is the number of Degrees of Inclination. If you find the Plane Reclines, apply the side AD to it, and see what number of Degrees the Plumb-line falls on, for that number of Degrees counted from the side AD is the number of Degrees of Reclination. These Rules being well understood, may serve you to find the situation of all other sorts of Planes. But for the making a Dyal on this Plane, you must first draw a Meridian Line through the middle of the Plane, by applying a Plumb-line to the middle of it, till the Plumbet hang quietly before it: for than if the Plumb-line be blacked (for a white Ground, or chalked for a dark Ground) and strained as Carpenters do their Lines, you may with one stroke of the string on the Plane describe the Meridian Line, as A XII: This Meridian is also the Substilar Line. Then on the top of this Meridian Line, as at A, draw another Line athwart it to cut it at right Angles, as VI, VI, for an East and West Line. At the meeting of these two Lines on the top, make your Centre, whereon describe a Semicircle on your Plane, as large as you can, which by the Meridian Line and the East and West Line will be divided into two Quadrants. One of these Quadrants divide into 90 Degrees (as you were taught Fol. 12.) and from the Substilar Line count the Compliment of the Poles Elevation, which (here at London where the Pole is elevated 51 ½ Degrees, its Compliment to 90) is 38 ½ Degrees, and make there a mark, as at E. Then on the Substilar Line choose a point (where you please) as at F, for the Line of Contingence to pass through: which Line of Contingence draw as long as you can, so as it may cut the Substilar Line at right Angles, and from the point F in the Substilar Line measure the shortest distance between it and the Stilar Line, and keeping one Foot of your Compasses still in the point F, transfer that distance into the Substilar Line, as at G; then on the point G describe a Semicircle of the Equinoctial against the Line of Contingence, which Semicircle divide into twelve equal parts, (as you were taught by the Example in the Horizontal Dyal, Fol. 13.) and by a strait Ruler laid to each of these Devisions, and to the Centre of the Semicircle make marks in the Line of Contingence by the side of the Ruler: For strait Lines drawn from the Centre of the Dyal Plane through these marks in the Contingent Line shall be the 12 Hour Lines before and after Noon. Then mark your Hour Lines with their respective Numbers: The Substilar or Meridian Line is XII, from thence towards the right hand with ay, TWO, III, etc. and from thence towards the left hand with XI, X, IX, etc. The Style must be erected perpendicularly over the Substilar Line, so as to make an Angle with the Dyal Plane equal to the Compliment of the Poles Elevation, viz. 38 ½ Degrees. OPERAT. IV. To make an Erect Direct North Dyal. THE Erect Direct North Dyal, Stile and all, is made by the same Rules, changing upwards for downwards, and the left side for the right, the Erect Direct South Dyal is made: for if the Erect Direct South Dyal be drawn on any transparent Plane, as on Glass, Horn, or an oiled Paper, and the Horizontal Line VI, VI, turned downwards, and the Line VII marked with V, the Line VIII with IIII, the Line V with VII, and the Line IIII with VIII, then have you of it a North Erect Direct Dyal. All the other Hour Lines in this Dyal are useless, because the Sun in our Latitude shines on a North Face the longest Day only before VI in the Morning, and after VI at Night. OPERAT. V. To describe an Erect direct East Dyal. Hung a Plumb-line a little above the place on the Wall where you intent to make your Dyal, and wait till it hang quietly before the Wall: Then if the Line be rubbed with Chalk (like a Carpenter's Line) you may by holding the Plumbet end close to the Wall, and straining it pretty stiff, strike with it a strait Line, as Carpenters do: This Line shall be a perpendicular, as AB. Then choose a convenient point in this Perpendicular, as at C, for a Centre, whereon describe an occult Arch, as DE; This Arch must contain the number of Degrees of the Elevation of the Equinoctial, counted between D and E, which in our Latitude is 38 ½, or (which is all one) the Compliment of the Poles Elevation. Therefore in a Quadrant of the same Radius with the occult Arch measure 38 ½ Degrees, and set them off in the Plane from E to D: Then from D to the Centre C in the Perpendicular draw the pricked Line DC; this pricked Line shall represent the Axis of the World. Then cross this Line at right Angles with the Line CF, and draw it from C to F, so long as possibly you can: This Line shall be the Contingent Line. Then choose a point in this Contingent Line, as at VI, draw a Line through that point at right Angles for the Substilar Line, as G VI H for the Substilar Line; then open your Compasses to a convenient width, (as to VIG) and pitching one Foot in the point G, with the other Foot describe a Semicircle of the Equinoctial against the Line of Contingence, which Semicircle divide from VI both ways into six equal parts, as you were taught by the Example in the Horizontal Dyal: and laying a strait Ruler on the Centre of this Semicircle of the Equinoctial, and to each of those equal parts mark on the Contingent Line where the Ruler cuts it, for those marks shall be the several points from whence Lines drawn parallel to the Line CD shall be the respective Hour Lines. The reason why the Contingent Line is drawn from VI to F, so much longer than from VI to C is; because the Hour Lines from VI towards XII are more in number towards Noon, than they are from VI backward towards IIII: for this Dyal will only show the Hours from a little before IV in the Morning to almost Noon: For just at Noon the Shadow goes off the Plane; as you may see if you apply a strait Ruler to the Centre of the Equinoctial Semicircle G, and lay it to the point 12 in the Semicircle; for the strait Ruler will then never cut the Line of Contingence, because the Line of Contingence is parallel to the Line G XII on the Equinoctial Circle, and Lines parallel, though continued to never so great a length never meet. To these Hour Lines, set Figures as may be seen in the Scheme. The Style IK of this Dyal as well as of all others must stand parallel to the Axis of the World; and also parallel to the Face of the Plane, and parallel to all the Hour Lines, and stand directly over the Substilar or VI a Clock Hour-Line, and that so high as is the distance of the Centre of the Equinoctial Semicircle from the Contingent Line. OPERAT. VI To describe a Dyal on an Erect Direct West Plane. AN Erect Direct West Dyal, is the same in all respects with an Erect Direct East Dyal: Only as the East Dyal shows the Forenoon Hours, so the West shows the Afternoon Hours. Thus if you should draw the East Dyal on any transparent Plane, as on Glass, Horn, or oiled Paper, on the one side will appear an East Dyal, on the other side a West: Only the numbers to the Hour Lines (as was said before in the North Dyal) must be changed; for that which in the East Dyal is XI, in the West must be I; that which in the East Dyal is X, in the West must be TWO; that which in the East Dyal is IX, in the West must be III, etc. The Style is the same. OPERAT. VII. To describe a Dyal on an Erect North, or Erect South Plane Declining Eastwards or Westwards. THese four Dial's, viz. the Erect North Declining Eastwards, the Erect North Declining Westwards, the Erect South Declining Eastwards, and the Erect South Declining Westwards, are all projected by the same Rules; and therefore are in effect but one Dyal differently placed, as you shall see hereafter. First draw on your Plane a strait Line to represent the Horizon of your place, and mark one end of it W for West, and the other end E for East. Choose a point in this Horizontal Line for a Centre, as at A, whereon you may describe a Circle to comprehend all these four Dial's: Draw a Line as MAM perpendicular to the Horizontal Line WE, through the Centre A for a Meridian Line, and on that Centre describe a Circle, which by the two Lines WAE, and MAM will be divided into four Quadrants, which will comprehend the four Dial's aforesaid: for if it be a North declining West you are to draw, the upper Quadrant to the left hand serves your purpose: If a South Declining West, the same Lines continued through the Centre A into the lower Quadrant to the right Hand serves your turn; if a North Declining East, the upper Quadrant to the right Hand serves your turn; or if a South Declining East, the same Lines continued through the Centre A into the lower Quadrant to the left hand serves your turn; and you must draw the Declination, Compliment of the Poles Altitude, Substile, Stile and Hour Lines in it; but the Hour Lines must be differently marked as you shall see hereafter. I shall only give you an Example of one of these Dial's; viz. A South Declining East. We will suppose you are to draw a Dyal that declines from the South 50 Degrees towards the East; here being but one Dyal, you need describe but one Quadrant of a Circle. Set off in the lower Quadrant WAM 50 degrees from the Meridian Line M towards W, and from the Centre A draw a strait Line through that mark in the Quadrant as DA, which may be called the Line of Declination; then set off from the Meridian Line the Compliment of the Poles Elevation, which in our Latitude is 38 ½ degrees, and there draw another Line from the Centre as AP, which we will call the Polar Line. Then take in the Horizontal Line a convenient portion of the Quadrant, as AB, and from the point B draw a Line parallel to the Meridian Line AM, and continue that Line till it intersect the Polar Line, as at P, from which Point P draw a Line parallel to WA, as PC: Then measure the distance of AB in the Horizontal Line, and set off that distance in the Line of Declination, as from A to D, and from that point of distance draw a Line parallel to the Meridian AM through the Horizontal Line at R, and through the Point D, and continue it through the Line PC, as at S; then laying a strait Ruler to the Centre A, and the Intersection of the Line PC, at S draw the Line AS for the Substile: Then upon the Point S erect a Line perpendicularly as ST; Then measure the distance between R and D, and set that distance off from S to T, and from the Centre to the point T draw the Line AT for the Style or Gnomon; and the Triangle SAT made of Iron or Brass and erected perpendicularly over the Substile SA shall by its upper side TA cast a shadow upon the Hour of the day. But you will say the Hour Lines must be drawn first: It is true; Therefore to draw them you must choose a point in the Substile Line where you think good, and through it draw the Line FF as long as you can for the the Line of Contingence: then with your Compasses take the shortest distance between this point and the Style, and transfer that distance below the Line of Contingence on the Substile as at A, and with your Compasses at that distance describe on the Centre A a Circle to represent the Equinoctial; Then (as you were taught in the Example of the Horizontal Dyal) divide the Semicircle of the Equinoctial into twelve equal parts, beginning at the point in the Equinoctial Circle, where a strait Line drawn from the Centre of it to the Intersection of the Line of Contingence with the Meridian Line cuts the Equinoctial Line, as here at the Point G; Then lay a strait Ruler to the Centre of the Equinoctial Circle, and to every one of the Devisions in the Semicircle, and mark where the strait Ruler cuts the Contingent Line; for strait Lines drawn from the Centre A of the Dyal to those several marks on the Contingent Line shall be the Hour Lines; and must be numbered from the Noon Line or Meridian A M backwards, as XII, XI, X, IX. etc. towards the left hand. So is your Dyal finished. This Dyal drawn on any transparent matter as Horn, Glass, or an oiled Paper, shall on the other side the transparent matter become a South Declining West, (Style and all) but then the I a Clock Hour Line must be marked TWO, the XII XII, the XI a Clock Hour Line I, X, TWO, IX, III, etc. If you project it anew, you must describe the Quadrant MW on the other side the Meridian Line, on the Centre A from M to E, and then count, (as before) the Declination, Altitude of the Pole, Substile, and Style in the Quadrant, beginning at M towards E, and work in all respects as with the South Declining East; only number this South Declining West as in the foregoing Paragraph. If you project a North Declining East, you must describe the Quadrant above the Horizontal Line from M upwards, towards E on your right hand, and count (as before) the Declination, Altitude, Compliment of the Pole, Substile, and Style from the Meridian Line, and work as with the South Declining East: It must be numbered from the Meridian Line M towards the right hand with XI, X, IX, VIII, etc. If this Dyal were drawn on transparent matter, the other▪ side would show a North Declining West: But if you will project it anew, you must describe the Quadrant above the Horizontal Line, from M upwards towards W, and count from the Meridian Line AM the Declination, Compliment, Altitude of the Pole, Substile and Style, and work with them (in all respects) as with the South Declining East; but then the XI a Clock Hour Line must be marked I, the X, TWO; the IX, III, etc. OPERAT. VIII. To draw a Dyal on an East or West Plane Reclining, or Inclining. DRaw a strait Line parallel to the Horizon, to represent a Meridian, or XII a Clock Line, and mark one end N, the other S; Choose a point in this Line, as at A for a Centre: then if your Plane be an East or a West Incliner, let fall a Perpendicular upon this Centre, (that is, the Perpendicular must stand above the Meridian Line NS.) as A, and upon the Centre A describe a Semicircle above the Meridian Line NS;) But if your Plane be an East Incliner, or a West Recliner, let fall a Perpendicular from the Centre A under the Meridian Line, and upon the Centre A describe a Semicircle under the Meridian Line. If your Plane be a West Incliner, work (as shall be taught) in the Quadrant on the left hand above the Meridian Line. If an East Recliner, in the Quadrant on the right hand above the Meridian Line. If it be a West Recliner, work in the Quadrant on the left hand under the Meridian. If an East Incliner, in the Quadrant under the Meridian Line the right hand. For Example, An East Dyal Reclining 45 Degrees. You would draw a Dyal on an East Plane Reclining 45 Degrees: Therefore in the Quadrant on the right hand above the Meridian Line, set off from the Perpendicular A 45 Degrees on the Quadrant, for the Reclination of the Plane; and set off also in the Quadrant 38 ½ Degrees from the Perpendicular for the Compliment of the Poles Elevation, and at these settings off make marks in the Quadrant: Then lay a strait Ruler to the Centre A, and to the marks in the Quadrant, and draw strait Lines through them from the Centre. Then choose in the Meridian Line NS a convenient point, as at B, and through that point draw a Line parallel to the Perpendicular A, which will intersect the Line drawn for the Compliment of the Poles Elevation AP in P; from which point P, draw a Line parallel to the Meridian Line NS, to cut the Perpendicular A in C, and also the Line of Obliquity AO in O. Then measure the length AO, and set off that length in the Perpendicular ACE from A to E, and draw the Line EGLANTINE parallel to the Meridian Line NS, which will cut the Line BP prolonged in G. Measure also the length of CO, and set that length off from A to Q on the Line of Obliquity AO, and draw the Line QR parallel to the Perpendicular ACE. Then measure the distance of ARE, and upon the Line GPB set it off from G to S; and laying a strait Ruler to the point S and the Centre A, draw by the side of it the Line AS; for the Substile Line. Then measure the length of QR, and from S raise a Perpendicular, and in that Perpendicular set that length off from S to T; and laying a strait Ruler to the Centre A and the point T, draw the Line AT for the Stilar Line, which Stilar Line being perpendicularly erected over the Substilar Line AS, will stand parallel to the Axis of the World, and cast its shadow on the Hour of the Day. To draw the Hour Lines on this Plane, you must (as you have several times before been directed) choose a point in the Substilar Line, and through that point draw at right Angles with the Substilar Line the Line of Contingence so long as you can: Then measure the shortest distance between that Point and the Stilar Line, and transfer that distance below the Line of Contingence in the Substilar Line, as at A, and with your Compasses at that distance describe against the Line of Contingence the Equinoctial Circle; Then divide the Semicircle of the Equinoctial next the Line of Contingence into twelve equal parts, (as you have formerly been taught) beginning at the Point in the Equinoctial Circle, where a strait Line drawn from the Centre of it to the Intersection of the Line of Contingence with the Meridian Line NS cuts the Equinoctial Circle, as here at the point D: Then lay a strait Ruler to the Centre of the Equinoctial Circle, and to every one of the Devisions in the Equinoctial Semicircle, and mark where the strait Ruler cuts the Contingent Line: for strait Lines drawn from the Centre A of the Dyal through these several marks in the Contingent Line shall be the Hour Lines, and must be numbered from the Meridian or Noon-Line NS which is the XII a Clock Line upwards, with XI, X, IX, VIII, etc. The Centre of this Dyal must stand downward. If this Dyal were turned with its Centre upwards, it would show a West Inclining 45 degrees, only the numbers to the Hour Lines must be changed; for to XI you must set I, to X, TWO; to IX, III, etc. and the Substile over which the Style must stand, must be placed in the Semicircle (at first described) as much to the right hand the Perpendicular A, as it doth on the left hand. If this Dyal were drawn on Glass, Horn, or an oiled Paper, and you turn the Meridian Line NS upwards, the backside shall be an East Inclining 45 degrees, and the Hour Lines must be numbered as they are on the East Reclining: But the Substile over which the Style must stand, must be placed, in the Semicircle (at first described) as much to the left hand the Perpendicular A, as it is on the oiled Paper to the right hand. If you turn the Meridian Line NS downwards, the backside shall be a West Recliner 45 degrees, and the Hour Lines must be numbered from the XII a Clock Line upwards, with I, TWO, III, etc. You must note that all the Hour Lines of the Day will not be described in this single Quadrant, nor does the Quadrant at all relate to the Hour Lines; but is described only for setting off the Compliment of the Poles Elevation and Reclination of the Plane, that by working (as hath been shown) you may find the place of the Substilar Line, and the Angle the Style makes with it: For having the Substilar Line, you know how to draw the Line of Contingence, and to describe the Equinoctial Circle, by which all the Hours are described on the Plane. To draw a Dyal on a Direct South or North Plane Inclining or Reclining. Direct Reclining or Inclining Dial's are the same with Erect Direct Dial's that are made for the Latitude of some other Places; the Latitude of which Places are either more than the Latitude of your Place, if the Plane Recline; or less, if the Plane Incline: and that in such a proportion as the Arch of Reclination or Inclination is. Thus a Direct South Dyal Reclining 10 degrees in London's Latitude, (viz. 51 ½ degrees) is an Erect Direct South Dyal made for the Latitude of 61 ½ degrees. And a Direct South Dyal Inclining 10 in the Latitude of 51 ½ is an Erect Direct South Dyal in the Latitude of 41 ½ degrees: and is to be made according to the Direction given in Operat. III. OPERAT. IX. To draw a Dyal on a South or North Inclining Declining, or Reclining Declining Plane. THese four sorts of Dial's viz. the South Inclining Declining, and South Reclining Declining, and North Inclining Declining, and South Reclining Declining, are all projected by the same Rules; and therefore are in effect but one Dyal differently placed, as you shall see hereafter. First draw on your Plane a strait Line parallel to the Horizon, and mark one end W for West, and the other E for East. On South Incliners and Recliners, E on the right hand, and W on the left: on North Incliners and Recliners E on the left hand and W on the right. Choose a point in this Horizontal Line for a Centre, as at A; Through this point A draw a Line Perpendicular to the Horizon, and on this point (as on a Centre) describe a Semicircle, one Quadrant above, and another below the Horizontal Line. (though for this Example I describe but one.) Then if the Plane respect the South, set off in the lower Quadrant from the Perpendicular the Declination, the Inclination, or the Reclination, and the Compliment of the Altitude of the Pole; and through these several settings off in the Quadrant, draw strait Lines from the Centre A; then take in the Horizontal Line towards the Semicircle, a convenient distance from the Centre A, as B, and through the point B draw a strait Line parallel to the Perpendicular, and prolong it through the Polar Line, as BP: Through the point P, draw a Line parallel to the Horizontal Line, as PC; this Line will cut the Line of Obliquity in the point O: Then measure the distance of AO, and set off that distance on the Perpendicular from A to F, and through the point F draw a strait Line parallel to the Horizontal Line, as FG, for the Horizontal Intersection. Then measure the distance of CO, and set off that distance on the Perpendicular from A to I; from the point I, draw the Line ID parallel to the Horizontal Line, to cut the Line of Declination in the point D. Then measure the distance of AB, and set off that distance in the Line of Declination from A to E; and from the point E draw a strait Line parallel to the Horizontal Line WE, to cut the Perpendicular in the point K. Measure the distance of EKE, and set off that distance on the other side the Perpendicular in the Horizontal Intersection, from F to H, and from the point H draw HN parallel to the Perpendicular to cut the Horizontal Line in the point N. Then to find the Meridian Line, Substile and Style, do thus. If your Plane be a Southern Incliner, or a Northern Recliner, measure the distance of LD, and set off that distance in the Horizontal Intersection from F to M, and through the point M draw the Line AM for the Meridian Line. Then add the distance of ALL to AK, thus: measure the distance of ALL, and place one Foot of your Compasses in the point K in the Perpendicular Line, and extend the other to X, and measuring the distance of AXE, set it off in the Line of Obliquity from A to Q; and from the point Q draw the Line QR parallel to the Perpendicular, and cutting the Horizontal Line in the point R. Then measure the distance of ARE, and set off that distance from H in the Horizontal Intersection to S on the Line HN, and to the point S draw the Line AS for the Substile. Then measure the distance of QR, and set off that distance perpendicularly from the point S to T; and lastly, from the point A, draw the strait Line AT for the Stilar Line, which Stilar Line being perpendicularly erected over the Substilar Line AS, will stand parallel to the Axis of the World, and cast its shadow on the Hour of the Day. But if the Plane be a Southern Recliner, or Northern Incliner, measure (as before) the distance of LD, and (as before you were directed) to set it off from F in the Horizontal Intersection on the right hand the perpendicular Line; So now, set that distance from F to m in the Horizontal Intersection on the left hand in the Perpendicular Line, and draw the Line A m for the Meridian Line. Then as before you were directed to add ALL to AK: So now, subtract the distance of ALL from AK, and the remainder will be LK: Set therefore the distance of IK from A to q in the same Line of Obliquity, and from the point q, draw the Line qr parallel to the perpendicular. Measure then the distance of A r, and set off that distance in the Line HN, from H to s for the Substilar Line: Then erect on the point s a Perpendicular, and on that Perpendicular set off from s to t the distance of qr: And lastly, from A draw the Line A t for the Stilar Line. If K falls upon L the Plane is parallel to the Axis of the World, and the Dyal drawn upon it will have no Centre: But s will fall upon H, and AH (or A s) will be the Substile. I shall give you two Examples of these Rules: One of a Dyal with a Centre, and the other of a Dyal without a Centre. And first, OPERAT. X. How to draw a Dyal with a Centre, Declining 20 Degrees, and Inclining 30 Degrees. HAving by the foregoing Precepts of the last Operat. found the Substile, Stile, and Meridian, you must (as you have often been directed) choose a point in the Substilar Line, through which, at right Angles to the Substilar Line draw the Line of Contingence as long as you can: Then measure the shortest distance between the point of Intersection and the Stilar Line, and transfer that distance on one side the Line of Contingence upon the Substilar Line, and so describe the Equinoctial Semicircle against the Line of Contingence: Then lay a strait Ruler to the Centre of the Equinoctial Circle, as at A, and to the point where the Line of Contingence cuts the Meridian Line, as at Z, and mark where the strait Ruler cuts the Equinoctial Circle, and from that mark begin to divide the Semicircle into twelve equal parts, and by a strait Ruler laid to those devisions and the Centre of the Equinoctial, make marks in the Line of Contingence. Then shall strait Lines drawn from the Centre A of the Dyal through every one of those marks in the Contingent Line be the Hour Lines of the Dyal, and must be numbered from the XII a Clock Line towards the right Hand with I, TWO, III, IV, etc. And the other way with XI, X, IX, etc. OPERAT. XI. How to draw a Dyal without a Centre, on a South Plane; Declining East 30 Degrees, Reclining 34 Degrees 32 Minutes. HAving by the Precepts of Operat. IX found the Substile, you must find the Meridian Line otherwise than you were there taught: For, having drawn the Lines of Latitude, Declination and Reclination, and found the Substile, measure the distance of BP, and set it off on the Line of Declination from A to K, and draw from the Perpendicular OF the Line KQ parallel to AB: Then measure the length of KQ, and set it off on the Polar Line AP, from A to V; then take the nearest distance between the point V and the Line AB, and set it off on the Line QK from Q to M; through which point M, draw a Line from the Centre A: Then measure with your Compasses in the Semicircle WNE (which in this Dyal may represent the Equinoctial) the distance of the Arch N m, and set off that distance from the Intersection of the Substile with the Semicircle at S to T in the Semicircle, which point T shall be the point in the Equinoctial that you must begin to divide the Hours at, for the finding their distances on the Line of Contingence. Then consider (according to the bigness of your Plane) what height your Style shall stand above the Substile, and there make a mark in the Substile: For the distance between the Centre A and that mark must be the height of the Style perpendicularly erected over the Substile, as at I. Draw through this point I a Line of Contingence, as long as you can to cut the Substile at right Angles, and then laying a Ruler to the Centre A, and successively to each Division of the Equinoctial make marks in the Line of Contingence, and through those marks draw strait Lines parallel to the Substile, which shall be the Hour Lines; and must be numbered from the left hand towards the right, beginning at the XII a Clock Line with I, TWO, III, etc. and from the right hand towards the left on the XII a Clock Line with XI, X, IX, etc. The Style to this Dyal may be either a strait Pin of the length of AI, or else a Square of the same height, erected perpendicularly upon the point I, in the Substile Line. OPERAT. XII. To make a Dyal on the Ceiling of a Room, where the Direct Beams of the Sun never come. Found some convenient place in the Transum of a Window to place a small round piece of Looking-Glass, about the bigness of a Groat, or less, so as it may lie exactly Horizontal. The point in the middle of this Glass we will mark A, and for distinction sake call it Nodus. Through this Nodus you must draw a Meridian Line on the Floor, Thus, Hang a Plumb-line in the Window exactly over Nodus, and the Shadow that the Plumb-line casts on the Floor just at Noon will be a Meridian Line; or you may find a Meridian Line otherwise by the Clinatory. Having drawn the Meridan Line on the Floor, find a Meridian Line on the Ceiling, thus, Hold a Plumb-line to the Ceiling, over that end of the Meridian Line next the Window; If the Plumbet hang not exactly on the Meridian Line on the Floor, remove your hand on the Ceiling one way or other, as you see cause, till it do hang quietly just over it, and at the point where the Plumb-line touches the Ceiling make a mark, as at B; that mark B shall be directly over the Meridian Line on the Floor: then remove your Plumb-line to the other end of the Meridian Line on the Floor, and find a point on the Ceiling directly over it, as you did the former point, as at C, and through these two points B and C on the Ceiling, strain and strike a Line blacked with Smal-Coal or any other Colour (as Carpenters do) and that Line BC on the Ceiling shall be the Meridian Line, as well as that on the Floor: Then fasten a string just on the Nodus, and remove that string, forwards or backwards, in the Meridian Line on the Ceiling, till it have the same Elevation in the Quadrant on the Clinatory above the Horizon that the Equinoctial hath in your Habitation, and through the point where the string touches the Meridian Line in the Ceiling shall a line be drawn at right Angles with the Meridian, to represent the Equinoctial Line. Thus in our Latitude the Elevation of the Equator being 38 ½ degrees; I remove the string fastened to the Nodus forwards or backwards in the Meridian Line of the Ceiling, till the Plumb-line of the Quadrant on the Clinatory, when one of the sides are applied to the string, falls upon 38 ½ degrees: and then I find it touch the Meridian Line at D in the Ceiling: therefore at DIEGO make a mark, and through this mark strike the line DE (as before I did in the Meridian Line) to cut the Meridian Line at right Angles: This Line shall be the Equinoctial Line, and serve to denote the Hour Distances, as the Contingent Line does on other Dial's, as you have often seen· Then I place the Centre of the Quadrant on the Clinatory upon Nodus, so as the Arch of the Quadrant may be on the East side the Meridian Line, and underprop it so, that the flat side of the Quadrant may lie parallel to the string, when it is strained between the Nodus and the Equinoctial, and also so as the string may lie on the Semi-diameter of the Quadrant, when it is held up to the Meridian Line on the Ceiling. Then removing the string the space of 15 degrees in the Quadrant, and extending it to the Equator on the Ceiling, where the string touches the Equator, there shall be a point through which the I a Clock Hour line shall be drawn: and removing the string yet 15 degrees further to the Eastwards in the Semicircle of Position, and extending it also to the Equator, where it touches the Equator, there shall be a point through which the TWO a Clock Hour Line shall be drawn. Removing the string yet 15 degrees f●rther, to the Eastwards in the Semicircle of Position, and extending it to the Equator, there shall be a point through which the III a Clock Hour Line shall be drawn: The like for all the other Afternoon Hour Lines. So oft as the string is removed through 15 degrees on the Quadrant, so oft shall it point out the Afternoon distances in the Meridian Line on the Ceiling. Having thus found out the points in the Equator through which the Afternoon Hour Lines are to be drawn, I may find the Forenoon Hour distances also the same way, viz. by removing the Arch of the Quadrant to the West side the Meridian, as before it was placed on the East, and bringing the string to the several 15 degrees on the West side the Quadrant; or else I need only measure the distances of each Hours distance found in the Equator from the Meridian Line on the Ceiling; for the same number of Hours from XII, have the same distance in the Equinoctial Line on the other side the Meridian, both before and afternoon: The XI a Clock Hour distance is the same from the Meridian Line, with the I a Clock distance on the other side the Meridian; the X a Clock distance, the same with the TWO a Clock distance; the IX with the III, etc. And thus the distances of all the Hour lines are found out on the Equator. Now if the Centre of this Dyal lay within doors, you might draw lines from the Centre through these pricks in the Equator, and those Lines should be the Hour lines, as in other Dial's: But the Centre of this Dyal lies without doors in the Air, and therefore not convenient for this purpose: So that for drawing the Hour Lines, you must consider what Angle every Hour Line in an Horizontal Dyal makes with the Meridian; that is, at what distance in Degrees and Minutes the Hour Lines of an Horizontal Dyal cut the Meridian; which you may examine, as by Operat. II. For an Angle equal to the Compliment of the same Angle, must each respective Hour Line with the Equator on the Ceiling have. Thus upon the point marked for each Hour distance in the Equinoctial Line on the Ceiling, I describe the Arches I, TWO, III, IV, as in the Figure, and finding the distance from the Meridian of the Hour Lines of an Horizontal Dyal to be according to the Operat. II. Thus, The 1 a clock Hour line 11.40 whose Compliment to 90 is 78.20 The 2 a clock Hour line 24.15 whose Compliment to 90 is 65.45 The 3 a clock Hour line 38.14 whose Compliment to 90 is 51.56 The 4 a clock Hour line 53.36 whose Compliment to 90 is 36.24 I measure in a Quadrant of the same Radius with those Arches already drawn from the Equinoctial Line for the 1 a Clock Hour 78.20 for the 2 a Clock Hour 65.45 for the 3 a Clock Hour 51.56 for the 4 a Clock Hour 36.24 and transfer these distances to the Arches drawn on the Ceiling: For then strait Lines drawn through the mark in the Arch, and through the mark in the Equator, and prolonged both ways to a convenient length, shall be the several Hour Lines (aforesaid;) And when the Sun shines upon the Glass at Nodus, its Beams shall reflect upon the Hour of the Day. Some helps to a young Dyalist for his more orderly and quick making of Dial's. IT may prove somewhat difficult to those that are unpractised in Mathematical Projections, to divide a Circle into 360 Degrees (or which is all one) a Semicircle into 180, or a Quadrant into 90 degrees; and though I have taught you in the projecting the Horizontal Dyal the original way of doing this, yet you may do it a speedier way by a Line of Chords, which if you will be curious in your Practice, you may make yourself; or if you account it not worth your while, you may buy it already made on Box or Brass of most Mathematical Instrument-makers'. This Instrument is by them called a Plain Scale, which does not only accommodate you with the devisions of a Quadrant, but also serves for a Ruler to draw strait Lines with: the manner of making it is as follows. Describe upon a smooth flat even-grained Board a quarter of an whole Circle, as BC, whose Radius AB or AC may be four inches, if you intent to make large Dial's, or two inches if small; but if you will, you may have several Lines of Chords on your Scale or Rule. Divide this Quadrant into 90 equal parts as you were taught in the making the Horizontal Dyal. Then draw close by the edge of your strait Ruler a Line parallel to the edge, and at about 1/20 part of an Inch a second Line parallel to that, and at about ⅛ of an Inch a third Line parallel to both. Then place one Foot of your Compasses at the beginning of the first degree on the Quadrant descibed on the Board, as at B, and open the other Foot to the end of the first degree, and transfer that distance upon your Rule, from B to the first mark or division, between the two first drawn Lines. Then place one Foot of your Compasses again at the beginning of the first degree on the Quadrant described on the Board, as at B, and open the other Foot to the end of the second Degree, and transfer that distance upon your Rule from B to the second mark or division between the two first drawn Lines; And thus measure the distance of every Degree from the first Degree described on the Quadrant, and transfer it to the Rule. But for distinction sake, you may draw every tenth division from the first Line parallel to the edge of the third Line, and mark them in succession from the beginning with 10, 20, 30, to 90: and the fifth Devisions you may draw half way between the second and the third parallel Lines; the single Devisions only between the two first parallel Lines. So is your Line of Chords made. The Use of the Line of Chords. AS its use is very easy, so its convenience is very great; for placing one Foot of your Compasses at the first Division on the Scale, and opening the other to the 60th Degree, you may with the points of your Compasses (so extended) describe a Circle, and the several Devisions, on the Scale shall be the Degrees of the four Quadrants of that Circle, as you may try by working backwards, to what you were just now taught in the Making the Scale: For as before you measured the distance of the Degrees of the Quadrant, and transferred them to the Scale, so now you only measure the Divisions on the Scale, and transfer them to the Quadrant, Semicircle, or whole Circle described on your Paper. For Example: If you would measure 30 Degrees in your described Circle, place one Foot of your Compasses at the beginning of Devisions on the Scale, as at A, and extend the other Foot to the Divisions marked 30, and that distance transferred to the Circle, shall be the distance of 30 degrees in that Circle. Do the like for any other number of Degrees. You may draw your Dyal first on a large sheet of Paper, if your Dyal Plane be so large, if it be not so large, draw it on a smaller piece of Paper; Then rub the backside of your Paper-Dyal with Smal-coal, till it be well blacked; and laying your Paper Dyal on your Dyal Plane, so that the East, West, North, or South Lines of your Paper agree exactly with the East, West, North, or South situation of your Dyal Plane. Then with Wax or Pitch fasten the Corners of the Paper on the Plane, and laying a strait Ruler on the Hour-Lines of your Dyal, draw with the blunted point of a Needle by the side of the Ruler, and the Smal-coal rubbed on the backside the Paper will leave a mark of the Lines on the Plane. If you will have the Lines drawn Red, you may rub the backside of your Paper with Vermilion; if Blue, with Verditer; if Yellow, with Orpment, etc. Then draw upon these marked Lines with Oil Colours, as you please. An Explanation of some Words of Art used in this BOOK. ANgle. The meeting or joining of two Lines. Arch. A part of a Circle. Axis. The strait Line that runs through the Centre of a Sphere, and both ways through the Circumference: though in dialing it is all one with the Diameter of a Circle. Clinatory. See Fol. 8, 9, 10. Chord. See Fol. 44, 45, 46. Compliment. The number that is wanting to make up another number 90 Degr. or 180 Degr. or 360 Degrees. Contingent. A Line crossing the Substile at right Angles. Degree. See Fol. 12. Diameter. The longest strait Line that can be contained within a Circle, viz. the Line that passes through the Centre to the Circumference both ways. Dyal Plane. See Fol. 7. Elevation of the Pole. So many degrees as the Pole is elevated above the Horizon. Equinoctial. The Equinoctial is a great Circle that runs evenly between the two Poles of the World. But when we name the Equinoctial in this Book, we mean a small Circle which represents it, and is the Circle or Arch of a Circle which is divided into equal parts to find thereby the unequal parts on the Line of Contingence. In the Horizontal Dyal it is that Arch of a Circle marked GCH. Horizon. Is a great Circle encompassing the place we stand upon; but in dialing it is represented by a strait Line, as in Operat. III. In the South Dyal the Line VI A VI is the Horizontal Line. Latitude. The Latitude of a Place is the number of Degrees contained between the Equinoctial and the place inquired after. Line of Contingence. See Contingent. Magnetic Needle. The Needle touched with the Loadstone, to make it point to the North. Meridian, is a great Circle of Heaven passing through the North and South points of the Horizon; but in dialing it is represented by a strait Line, as in Operat. II. in the Horizontal Dyal the Line XII A is a Meridian Line. Nadir. The point directly under our Feet. Nautial Compass, Is the Compass used by Navigators, whereon is marked out all the 32 Winds or Points of the Compass. Oblique Plane. See Fol. 7. Parallel. See Fol. 6 Perpendicular. See Fol. 5. Pole. The North or South Points on the Globe of the Earth, are called North or South Pole. Quadrant. The fourth part of a Circle. Radius. Half the Diameter of a Circle. Right Angle. A strait Line that falls perpendicularly upon another strait Line, makes at the meeting of those two Lines a Right Angle. Semicircle. Half a Circle. Semi-Diameter, The same Radius is. Sphere. The highest Heaven with all its imagined Circles is called the Sphere. Style. The Gnomon or Cock of a Dyal. Substile. The Line the Style stands on upon a Dyal Plane. Triangle. A figure consisting of 3 Sides and 3 Angles. Zenith. The Point directly over our Head. FINIS. A Catalogue of GLOBES Celestial and Terrestrial, Spheres, Maps, Sea-Platts, Mathematical Instruments, and Books, made and sold by Joseph Moxon, on Ludgate-Hill, at the Sign of Atlas. GLOBES 26 Inches Diameter. The price 20 l. the pair. GLOBES, near 15 Inches Diameter. The price 4 l. GLOBES, 8 Inches Diameter. The price 2 l. GLOBES, 6 Inches Diameter. The price 1 l. 10 s. CONCAVE HEMISPHERES of the Starry Orb; which serves for a Case to a Terrestrial Globe of 3 Inches Diameter, made portable for the Pocket. Price 15 s. SPHERES, according to the Copernican Hypothesis, both General and Particular, 20 Inches Diameter. Price of the General 5 l. Of the Particular 6 l. Of both together 10. SPHERES, according to the Ptolomaick Systeme, 14 Inches Diameter. Price 3 l. SPHERES, according to the Ptolomaick Systeme, 8 Inches Diameter. Price 1 l. 10 s. Gunter's Quadrant, 13 Inches Radius, printed on Paper, and pasted on a Board, with a Nocturnal on the backside. Price 5 s. Gunter's Quadrant, 4 Inches Radius, printed on Paper, and pasted on Brass, with a Nocturnal on the backside, and a Wooden Case covered with Leather fit for it: A new invention contrived for the Pocket. Price 6 s. A large Map of the World, 10 Foot long, and 7 Foot deep, pasted on Cloth and coloured. Price 2 l. A Map of all the World, 4 Foot long, and 3 Foot deep, pasted on Cloth and coloured. Price 10 s. In sheets 2 s. 6 d. A Map of the English Empire in America, describing all places inhabited there by the English Nation, as well on the Islands as on the Continent. Price 15 s. Six Scriptural Maps, 1. Of all the Earth: And how after the Flood it was divided among the Sons of Noah. 2. Of Paradise, or the Garden of Eden; with the Country's circumjacent inhabited by the Patriarches. 3. The 40 years' travel of the Children of Israel through the Wilderness. 4. Of Canaan, or the Holy Land: and how it was divided among the twelve Tribes of Israel, and traveled through by our Saviour and his Apostles. 5. The Travels of St. Paul, and others of the Apostles, in their propagating the Gospel. 6. Jerusalem, as it stood in our Saviour's time; with a Book of Explanations to these Maps▪ entitled Sacred Geography Price 6 s. Useful to be bound up with Bibles. A Sea-Platt, or Map of all the World, according to Mercator, in two large Royal Sheets of Paper; set forth by Mr. Edward Wright, and newly corrected by Joseph Moxon Hydrogr. etc. Price 2 s. Sea plats for sailing to all parts of the World. Price 6 d. the sheet. The famous City of Batavia in the East-Indies, built and inhabited by the Duth; curiously engraved, and printed on four large Sheets of Royal Paper. Price 2 s. 6 d. A small Map of all the World, with Descriptions, on one Sheet. Price 6 d. BOOKS. A Tutor to Astronomy and Geography, or the Use of both the GLOBES Celestial and Terrestrial; by Joseph Moxon Hydrographer to the Kings most Excellent Majesty. Price 5 s. The Use of the Copernican Spheres, teaching to salve the Phaenomena by them, as easily as by the Ptolomaick Spheres; by Joseph Moxon Hydrographer etc. Price 4 s. Wright's Correction of Errors in the Art of Navigation. Price 8 s. New and rare Inventions of Water-works. Teaching how to raise Water higher than the Spring. By which Invention the perpetual Motion is proposed, many hard labours performed, and varieties of Motion and Sounds produced. By Isaac de Caus, Engineer to King Charles the First. Price 8 s. Practical Perspective, or Perspective made easy. Teaching by the Optics how to delineate all Bodies, Buildings and Landscapes, etc. By the Catropticks, how to delineate confused Appearances▪ so, as when seen in a Mirror or Polished Body of any intended shape, the Reflection shall show a design. By the Dioptrics, how to draw part of many Figures into one, when seen through a Glass or Crystal cut into many Faces. By Joseph Moxon Hydrographer, etc. Price 7 s. An exact Survey of the Microcosine. Being an Antaomy of the Bodies of Man and Woman; wherein the Skin, Veins, Nerves, Muscles, Bones, Sinews and Ligaments are accurately delineated. Engraven on large Copper Plates, Printed and curiously pasted together, so as at first sight you may behold all the parts of Man and Woman; and by turning up the several Dissections of the Papers, take a view of all their Inwards: with Alphabetical referrences to the Names of every Member and part of the Body. Set forth in Latin by Remelinus, and Michael Spaher of Tyrol: and Englished by John Ireton Chirurgeon: and lastly, perused and corrected by several rare Anatomists. Price 14 s. Vignola, or the Complete Architect. Showing in a plain and easy way, the Rules of the five Orders in Architecture, viz. Tuscan, Doric, jonick, Corinthian and Composite: whereby any that can but read and understand English, may readily learn the proportions that all Members in Building have to one another: set forth by Mr. James Barrozzio of Vignola, and translated into English by Joseph Moxon Hydrographer, etc. Price 3 s. 6 d. Christiologia, or a brief, but true Account of the certain year, Month, Day and Minute of the Birth of Jesus Christ. By John Butler B. D. and Chaplain to his Grace James Duke of Ormond, etc. and Rector of Lichborough, in the Diocese of Peterburgh. Price 3 s. 6 d. A Tutor to Astrology, or Astrology made easy; being a plain Introduction to the whole Art of Astrology. Whereby the meanest Apprehension may learn to erect a Figure, and by the same give a determinate Judgement upon any Question of Nativity whatsoever. Also new Tables of Houses, calculated for the Latitude of 51 deg. 32 min. Also Tables of Right and Oblique Ascensions to 6 deg. of Latitude. Whereunto is added an Ephemeris for three years; with all other necessary Tables that belong to the Art of Astrology. Also how to erect a Figure the Rational way by the Tables of Triangles, more methodically than hath yet been published; digested into a small Pocket Volume, for the conveniency of those that erect Figures abroad. By W. Eland. Price 2 s. The Use of a Mathematical Instrument called a Quadrant, showing very plainly and easily to know the exact height and distance of any Steeple, Tree, or House, etc. Also to know the Hour of the Day by it; the height of the Sun, Moon or Stars; and to know the time of the Sunrising and setting, and the length of every day in the year, the place of the Sun in the Ecliptic, the Azimuth, right Ascension, and Declination of the Sun: with many other necessary and delightful Conclusions, performed very readily. Also the use of a Nocturnal, whereby you may learn to know the Stars in Heaven, and the hour of the Night by them. With many other delightful Operations. Price 6 d. A brief Discourse of a passage by the North-pole to Japan, China, etc. Pleaded by three Experiments, and Answers to all objections that can be urged against a passage that way. As 1. By a Navigation into the North-pole, and two degrees beyond it. 2. By a Navigation from Japan towards the North-pole. 3. By an Experiment made by the Czar of Muscovy: whereby it appears that to the Northward of Nova Zembla is a free and open Sea as far as Japan, China, etc. With a Map of all the discovered Land nearest to the Pole. By Joseph Moxon Hydrographer etc. Price 6 d. Regulae Trium Ordinum Literarum Typographicarum: Or the Rules of the three Orders of Print-Letters, viz. The Roman, Italic, English, Capaitals and Small. Showing how they are compounded of Geometric Figures, and mostly made by Rule and Compass. Useful for Writing-Masters, Painters, Carvers, Masons, and others that are lovers of Curiosity. By Joseph Moxon Hydrographer etc. Price 5 s. The Use of the Astronomical Playing Cards. Teaching an ordinary Capacity by them to be acquainted with all the Stars in Heaven: to know their Places, Colours, Natures and Bignesses. Also the Poetical Reasons for every Constellation; very useful, pleasant and delightful for all lovers of Ingeniety. By Joseph Moxon Hydrogr. etc. Price 6 d. The Astronomical Cards. By Joseph Moxon Hydrographer, etc. Price plain 1 s. Coloured 1 s. 6 d. Best coloured and the Stars gilded 5 s. The Genteel Housekeepers Pastime: Or, the Mode of Carving at the Table represented in a Pack of Playing Cards. By w●ich, together with the Instructions in this Book, any ordinary Capacity may easily learn how to Cut up, or Carve in M●de all the most usual Dishes of Flesh, Fish, Fowl, and Baked M●●●●; and how to make the several Services of the same at the Table; with the several Sauces and Garnishes proper to each Dish of Meat. Set forth by several of the best Masters in the Faculty of Carving, and published for public Use. Price 6 d. Carving Cards. By the best Carvers at the Lord Mayor's Table. Price 1 s. Compendium Euclidis Curiosi: Or Geometrical Operations. Showing how with one single opening of the Compasses, and a strait Ruler all the Propositions of Euclids first Five Books are performed. Translated out of Dutch into English. By Joseph Moxon. Hydrogr. etc. Price 1 s. An Introduction to the Art of Species. By Sir Ionas Moor. Price 6 d. Two Tables of Ranges, according to degrees of Mounture. By Henry Bond, Senior. Price 6. d. Mechanic Exercises: Or the Doctrine of Handiworks, in six Monthly Exercises; began January 1. 1677. and monthly continued till June 1678. The first three, viz. the Numb. I. Numb. II. Numb. III. teaching the Art of Smithing. The other three, viz. Numb. FOUR Numb. V. Numb. VI teaching the Art of faynery. Accommodated with suitable engraved Figures. By Joseph Moxon Hydrographer, etc. price 6 d. each Exercise. At the place abovesaid, you may also have all manner of maps, Sea-Platts Drafts, Mathematical Books, Instruments, etc. At the lowest prizes. ADVERTISEMENT. THere is invented by the Right Honourable the Farl of Castlemain, a new kind of Globe, called (for distinction sake) the English Globe; being a fixed and immovable one, performing what the Ordinary ones do, and much more, even without their usual Appendancies; as Wooden Orisons, Brazen Meridian's, Vertical Circles, Horary Circles, etc. For it Composes itself to the site and Position of the World without the Mariners Compass, or the like foreign Help; and besides other useful and surprising Operations (relating both to the Sun and Moon, and performed by the Shade alone) we have by it not only the constant proportion of Perpendiculars to their Shades, with several Corollaries thence arising, but also an easy, new, and most compendious way of describing Dial's on all Planes, as well Geometrically, as Mechanically: most of which may be taught any one in few Hours, though never so unacquainted with Mathematics. To this is added on the Pedestal a Projection of all the appearing Constellations in this Horizon, with their Figures and Shapes. And besides, several new things in it differing from the common Astrolabe, (tending to a clearer and quicker way of Operating) the very Principles of all Steriographical Projections are laid down, and Mathematically demonstrated; as is every thing else of Moment throughout the whole Treatise. These Globes will be made and exposed to Sale about August next, (God willing:) against which time the Book for its use will also be Printed, and sold by Joseph Moxon, on Ludgate-Hill, at the Sign of Atlas.