will the now Radius RM be to the several Degrees in its Tangent Line As for the Demonstration or Reason of this Dial every body that understands Gnomonics comprehends it I doubt not at the first sight for the Angle O in the Triangle KOA being by construction equal to the Elevation do but place the Base AO on a Meridian Line and if you consider the Side KO as the Indicating Side of the Stile or Cock it necessarily follows that it will represent the Axis of the World for it is evident that its Top K will point directly to the Pole and touch it if produc'd whilst O its other extremity passes throu ' the Center of the Horizontal Plane therefore if a Circle whose Radius is AL were so plac't on this Stile or Axis that its Diameter crost it at right Angles at L the said Circle would represent Circulum maximum semper apparentium for that Circle in the Heavens ever touches the Horizon as this would do at A. This Circle then being parallel to the Aequator is divided by the Hour Circles into twenty four equal parts and consequently each fifteen Degrees in its Tangent Line GH will correspond with its said equal parts or Divisions Now GH is also the Tangent Line of the Horizon as touching it in the Point A but where the Hour Circles cut the Horizon or its Tangent line there the Points will be to which from the Center the Hour Lines in an Horizontal Dial are to be drawn ergo O the Center of your Horizontal Plane and the several fifteen Degrees in the common Tangent GH are the true points of the Hour Lines Besides as the distance between each Hour Line if AL be the Radius is 15 Degrees so if AO be Radius I mean OA the Radius of the Horizontal Plane the said Hour Lines will be distant as many Degrees asunder as they are in the Horizon of the World or as you found them in the Fabrick of the second Horizontal Dial by the Globe Here also you may see that the true place of this Dial is to be in the Center of the Earth and not on its superficies but by reason of the Suns vast distance the Error which thereby happens is not sensible nay because the Error is not sensible we may safely conclude that the Sun is vastly distant from us So much then for Horizontal Dials since there now remains nothing necessary to be known but how to find whether they stand Level or no which is handled in the first Section and how to draw a Meridian Line for their true placing which is learnt by the following Operation But before we go further let me advise you whensoever you make a Dial of consequence of what kind soever it be to describe it first on Paper and thence to mark out the Lines on your real Plane for thereby you will not only keep your said Plane neat and more judiciously chuse the best place for the Center of your Dial but besides the several conveniences which practice will show you the Lines themselves will be more exactly drawn by reason you can manage your Paper draught as you please OPERATION IV. How to draw a true meridian Line on any Horizontal Plane COmpose your Globe on the Plane or Place where your Dial is to stand and making marks or pricks there on each side of the Pedestal at the Letters S and N draw but a Line throu ' those marks and that will be a true Meridian Line and if you do the like under the Letters E and W you will have a true East and West Line OPERATION V. How to Describe a Vertical or an Erect Direct South Dial by your Globe for the Elevation of London The first way THIS Dial is made on the Plane of the Primary Vertical which passes from the Zenith to the Nadir throu the East West points and being therefore erect and facing also directly the South t is commonly called an Erect Direct South Dial so that if you draw but your String from the Zenith to the Nadir thro either of the Intersections of the Horizon with the Equator 't will appear upon the Superficies of the Globe like the emerging edge of a thin Plate and consequently represent the said Plane or at least as much of it as is requisite This being don't open your Compasses at 60 Degrees as before and describe on a sheet of paper the blind Semi-Circle I PC as in 〈◊〉 10 with the Diamiter or Meridian IOT throu ' it then take with your Compasses the distance between the Zenith of your Globe and the Intersection of your String with the nearest Hour Circle and 't will in your Blind Circle on both sides of the Meridian or twelve a Clock Line to wit from I to k and I to h give you marks by which you may draw from the Center O the Hour Lines of 1 and 11 as will the distance from k to l and h to g viz. the distance from the said first Intersection to the second the marks of 2 and 10 and in this manner you must proceed to 6 and 6 as the latest and earliest hours that this kind of Dial shows for since its Sides lye full East and West and that the Sun never comes to the East before 6 in the morning nor is later in the West than 6 at night 't is impossible that the Plane should significantly contain more Hour-Lines And as for the Stile or Cock the distance on your Globe between the Zenith and the Pole being the Complement of the Elevation gives you from I to K the Degrees of its height above the Plane so that you may easily place and erect it the Substile being still the Meridian The Rules in the first Horizontal Dial will show you also both how to contract and enlarge it and how to resolve especially if you consult the 7th 8th and 10th Schemes any difficulty that can possibly arise in the present Operation for Scheme the 7th shows you the Globe it self with the String drawn from the Zenith to the Nadir throu ' the East Intersection of the Aequator with the Horizon and Scheme the 8th the Globe cut into this Plane by the said String and lastly the lower part of Scheme the 10th to wit the Semi-Circle PIC the Dial described by the foregoing Directions Now for the Demonstration it follows in the 8th Operation OPERATION VI. How to make this Vertical South Dial by the Globe for the Elvation of London The second Way DEscribe a Blinde Circle of what bigness you please with a Diameter throu ' it and placing your String on the East or West Poynt of the Globe as before measure by your Bead or Compasses in any great Circle the distance between the Zenith and each Intersection of the said String with the Hour Circles and you will have the Degrees of every Hour from 12 a Clock as the before
the help of the said Rulers was perpendicular to your Wall or Plane is turned thereby from true South as formerly it stood towards the East the above-mentioned number of 90 Degrees but had the shade fallen on the 10th Degree your Plane would for the same Reason have declin'd 40 Degrees towards the West In short therefore the difference of these two Azimuths is the thing that resolves the Question for when they are equal there is no Declension at all Of Reclining Dials THE Horizontal Plane lay open we saw to the whole Hemisphere whilst each Vertical one enjoy'd but half of it for by being Vertical a moiety of the said Hemisphere is before and the other behind it Now the Reclining Plane which is exprest by Sch. 29. instead of being perpendicular to the Horizon bends towards it yet so that its bending has nothing in it of overwhelming or tendency towards those that behold it as it happens to Inclining Planes exprest by Scheme 30 but still receeds according to the Degrees of its Reclination farther and farther from them making thereby an obtuse Angle with the Horizon and consequently faces more than half the apparent Heavens as the Inclining one does less whose Angle is therefore ever Acute As for the kinds of Reclining Planes there are I may say 4 to wit the Aequinoctial the Polar the Direct Reclining and the Declining Reclining Plane for each of these appropriates to it self a particular Fabric or way of making and therefore we will Treat of them in Order OPERATION XVIII How to describe a Dial on an Aequinoctial Plane both by the Globe and Geometrically also THIS Plane is represented by the Globe when 't is Compos'd and cut as in Scheme 20 quite throu ' at the Aequinoctial therefore open your Compasses at 60 Degrees there and describing the Blind Circle ABCD in Scheme 21 divide it as the Hour-Circles cut the said Aequinoctial in Sch. 19th that is to say divide it into 24 equal Divisions and there will rest nothing more to be done but to draw Lines from the Center O through as many of those Divisions as you shall think necessary and then to Figure them successively from Morning to Night As for the Stile seeing the Axis of the World is at right Angles with any Diameter of the Aequator and runs throu ' the Center of it it must needs follow that the Perpendicular Pin OP plac't in the Center of your Dial will perform that Office for when it directly points to the Pole it represents the said Axis as the divided blind Circle does the Aequinoctial and its Divisions therefore since the Shade of the Axis ever falls according to the time of the Day on This or That intersection of the Hour-Circles with the Aequator the Shade of the Pin must fall also on the corresponding Hour-line of the Dial as being in the effect the same thing in case the 12 a Clock Line be plac't on a Meridian line and mounted at A its South side above the Horizon the Complement of the Elevation of the Pole i. e. 38 Degrees and a half for by this means your Plane from an Horizontal one will be perfectly that of the Aequator Nor is it hard to mount thus the said South side of your Dial since 't is but opening your Compasses in any great Circle of your Globe at twice as many Degrees as is the Complement of the Elevation to wit 77 Deg. and they will give you the true length of a Perpendicular to underprop withal the aforesaid A or Southern point of the 12 a clock line of your Dial. And the reason of it is because AC the Diameter of your Dial being by Hypothesis equal to the Diameter of the Globe becomes now C being Center of the new Arch made by the mounting or raising the side of your Plane above the Horizon a Radius double to OA the former Radius Therefore since the Chord of a double Arch is ever the Sine of the single Arch in a Circle whose Radius is double the other it follows that the Chord of 77 Degrees is in respect to the double Radius AC the Sine of 38 g. 30 m. and consequently will perform if erected Perpendicularly the design'd Operation Now for the Geometrical Construction of this Dial since it consists only in dividing a Circle into 24 equal parts with a perpendicular Cock or Stile there is no need of more words about it so that we 'l end here with a Memorandum viz. that as the Reclining face of this Plane shews the Hour from Spring to Autumn so the Inclining Face or other side of it does the same for the remaining half year to wit from Autumn to the Spring OPERATION XIX How to describe a Polar Dial both by the Globe and Geometrically also THE true Plane of this Dial is speculatively the Plane of the Aequinoctial Colure or 6 a Clock Hour-Circle but in practice that of any Circle parallel to it so that the Construction and Demonstration of a Dial on it is mutatis mutandis the same with that on a Meridian Plane of which we have already so fusely treated Make then by your Globe for example sake an East Dial on a Meridian Plane according to any of the former ways and if you alter but the Figures that is to say if having figur'd the Substilar instead of 6 with 12 you mark the Morning 7 a Clock Hour line of the said East Dial with 1 that of 5 with 11 and so on in Order it will be a true Polar Dial showing you exactly the Hour when it directly faces the South and Reclines so that the Apex or uppermost part of the Substiler or 12 a Clock line points just to the North Pole for then the back-part of the Plane makes an Angle with the Horizon equal to that of our Elevation This Operation may be also perform'd of it self without the former consideration since 't is but putting one foot of your Compasses on the Intersection of your Meridian or 12 a Clock hour Circle with the Aequator of your Globe to wit on K in Scheme 22 and so describing with Chalk the Arch CAE I mean an Arch which reaching from the said Meridian cuts the Morning 7 a Clock or if you please the Evening 5 a Clock Hour Circle somewhere or other for then if you draw a blind Circle as in Sch. 23. of the same bigness and take the several distancces between the Pricks or intersections of the Hour-Circles with the said Arch to wit the distances between C and O C and S c. and place them on the blind circle on both sides of PCK π the Substilar or 12 a clock line as well below the line AE ae as about it the lines drawn from the said Pricks will be true Hour lines and the distance between C and P or between K and X will for the reasons mentioned in the Description of the
greatest extuberancy and this Circle being 90 Degrees from the point of the Globe which lies directly under our Zenith it must differ from the Horizon of the Globe as many Degrees as its Zenith differs from that in the Heavens therefore the way prescribed is at least speculatively true Operation II. To find the Suns Almucantar or Height THere are three distinct ways of performing this independent of the following Operations and each of great use for the first gives you the Suns height in an instant if he shines The second if you have the least glimps of him or can guess at his place in a Cloud The third if you know the hour by any good Watch Pendulum or the like whether we see the Heavens or no. I. As for the first way 't is this your Globe being level move it 'till the shade of the Pin in the Zenith falls directly upon the Meridian and then the shade of the Extuberancy i. e. that made by the swelling or bellying out of the Globe will touch the true degree in the Quadrant of Altitude reckoning from the Zenith to it And thus you will find not only the Sun's height sooner perchance than by any ordinary Quadrant but will still have it before your eyes as long as you please nothing being to be further done but to move sometimes the Globe that the shade of the said Pin may still concur with the Meridian But if your Globe be fix'd or that for some particular reason you have no mind to stir it at all draw your string from the Zenith through the shade of its Pin i. e. lay the string in the Plane of the Sun and then if you mount your Bead till it reaches the nearest part of the shade of Extuberancy it will by bringing it to the Meridian or Quadrant of Altitude lye on the true Degree reckoning as before from the Zenith to it The Reason of the Operation is this The Sun when he rises brushes the Zenith and Nadir of the Globe with his Rayes for he illuminates alwayes within some few Minutes just half of it therefore when he gets v. g. a Degree higher he must needs illuminate a Degree beyond the Zenith and so proportionably from time to time or else he would sensibly illuminate more or less of the Globe at one moment than at another which is absurd Now since the Sun in truth illuminates more than an Hemispere the Reader must remember that Ptolomy reckons this excess take one time with another to be about 26 minutes and Tycho something less therefore substract 13 minutes or half the said Excess from what the shade of Extuberancy mark 's and you have his Height with all ordinary Exactness but should you chance at any time to doubt how far the said Shade of Extuberancy which is not so discernable as that made by a Gnomon just reaches erect then a piece of stick straw quill c. or if you please rest your Finger on the Globe between the Sun and the point in dispute and where the shade of your Finger straw stick or quill is lost that will be the true Term of the shade As for the Second Way for both the former we reckon but one turn the Meridian of your Globe to the Sun as before or because we suppose him not to shine out-right direct by your Eye the said Meridian so that it lye in the same Plain with him and this you may do in a manner as well if you have the least glimps of him or can by any accident guess whereabouts he is as if you had the fore-mentioned help of the Pin's shade in the Zenith Having thus done Take your String in both hands and cross with it as exactly as you can at right Angles that part of the Meridian next your body whether it happens to be the Quad. of Alt. or that of Proportion then putting your Face close to it and moving your Ey lower and lower till by reason of the Extuberancy you can but just see the Sun or his supposed place in Heaven do but bring your String held as before to this point viz. bring your String towards you till it just takes away the Sun or his supposed place from your Ey and the degree in the Meridian on which it then lies will be counting from the Zenith the Height required for so far his raies would reach did he shine out-right The third way is when we know the Hour by any Watch Pendulum c. thus Find among the Aequin or Diurnal Parallels that belonging to the present Day which we will suppose Apr. 10. and drawing your string from the Zenith over that Point in the said Parallel where 't is cut by the Hour given i. e. by the morning 9 a Clock Circle move your Bead to the said Point and the distance from the Bead to the Horizon will be the required Height viz. about 36 degrees as you 'l find if you bring the Bead to the Meridian and count the degrees between it and the Horizon The Suns Height may be also known by its Azimuth as by Operat 5. Having therefore by any of the aforesaid waies his Height 't will upon any doubt soon appear whether it be Fore or Afternoon for as long as ever he increases in Degrees i. e. mounts higher and higher above the Horizon it wants of Noon whereas if he falls or declines 't is after Noon OPERAT. III. To Compose the Globe either by a Meridian Line or without it to the site of the World IF you have a Merid. line drawn viz. a Line lying exactly North and South place the Globe level with its Merid. directly over it i. e. place so the little Notch in the Pedestal markt S that it cover the Southern extremity of the said line and the Notch N the Northern and then the Poles and Circles on the Globe will without sensible error correspond with those in Heaven and each painted Region or Countrey on it will be turn'd towards the real one which it represents But if you have no line drawn Know the day of the Moneth and you have two quick waies to do this Operation without any forreign helps The Globe having in it smal pin-holes on the several intersections of the Merid. with the aforesaid Diurnal Parallels or to be exacter on each point of the Merid. which an imaginary Parallel of each fifth day would cut for tho' we are to suppose Parallels for every day throughout the year yet there being no sensible difference in the Sun from 5 daies to 5 days such holes will be abundantly sufficient nay the aforesaid ones from ten Dayes to ten Days may very well serve the turn in any ordinary Operation I say the Globe having holes in its Meridian at this distance put the Zenith Pin or if you think better a Needle in the Hole which most agrees with the true day of the Month and then exposing your Globe
in the blind Circle of Scheme the 17th to wit the distance between the Meridian Line of your Dial and your Substilar which in this our Example lyes from the Moridian towards your left hand or Morning hours and the distance from the said Point in the String to the Pole being from S to P will in the said blind Circle be the height of your Stile so that if you erect and place your said Stile from the Center all along the Substilar OS it will continually show you the Hour But if you fancy that the Extension of your Compasses from the Pole to the String will not give you precisely this Point since your said Compasses may seem to touch it in several Points I say if you doubt or fancy this fasten a Thred on the Pole and drawing it streight over the Horizon at 40 Degrees from the Meridian of your Globe Eastwardly i. e. till it passes thron ' the Pole of the Plane see where the said Thred crosses your String or edge of the Plane and there the true requir'd Point will be The Demonstration of this Dial is in the following Operation OPERATION XIV How to describe by the Globe a Declining Dial for the Elevation of London The second way DRaw your String over at 40 Degrees in the Horizon from the East Northwardly and from the West Southwardly as before and the respective distances between the Zenith and the Intersection of your String with the Hour-Circles will give you in any great Circle of the Globe the Degrees of their respective distances as well for the Morning as Afternoon and the proportionable Degrees in any Circle will give you the Points for the Drawing of your Hour-lines as I showd you in the Construction of the former Vertical North and South Dials and as for the Stile and Sub-stilar you must operate as directed in the foregoing Operation that is to say the number of Degrees between Z and S gives you the Sub-stilar and those from P to S the height of your Stile As for the Demonstration or Reason why Dials thus made show the Hour it is this First you see that the String by being on one side removed 40 Degrees from the East point Northward and on the other side 40 Degrees from the West Point Southward represents on the Globe the requir'd Plane and therefore wheresoever the Hour Circles cut it there the Shade of the Axis will fall as we show'd you before in the former Dials Now two Points made by the intersections of each Hour Circle with the Plane being given you to wit the Center where they all meet and their respective marks on the String or supposed Edges of your Plane it must needs follow that if you draw Lines throu ' those Points they will be true Hour Lines for as we have often said the Hour-Lines of all Dials are only the intersections of the Plane with the hour Circles In the next place since PS by construction is the nearest distance from the Pole to your String or Plane it appears that the Hour-circle which cuts the said Plane at S falls on it at right Angles and consequently that as PS the height of the Pole or Axis above the String or Plane gives the true height of the Stile of this Dial so the intersection of the Plane with the said Hour-Circle must be the true Substilar for the Substilar as we already mention'd is only the intersection of the Plane with the Hour-Circle which falls at right Angles on it Ergo The distance between Z and S gives in your blind Circle the distance from your 12 a Clock line to the Substilan and PS the height of the Stile And by the way here it appears not only why the 12 a Clock Lines of Declining Dials continue perpendicular but also why their Centers keep the same distance from the Horizontal Edges of their Planes as do the Centers of the primary Vertical or direct North and South Dials I say here all this appears for the 12 a Clock Line which is ever the intersection of your Meridian with these Planes being a Perpendicular in the primary Vertical Plane becomes the Axis of the Horizon and all Vertical Dia's by their Declension more only about it so that both the Center and the said 12 a Clock Line remain the same in all therefore the Tangent of the Elevation being as I formerly show'd you the length of the foot of the Stile or distance between the Center of a Primary Vertical Dial and its Horizontal Edge is that of a Declining one also But to proceed with the Demonstration you must remember that this Dial is an Horizontal one as we show'd you before to those that dwell in the Pole of the Circle describ'd by the String i. e. to those in our Horizon 40 Degrees Eastward from the Meridian or which is all one to those that dwell where the Thred cuts the Horizon but all the Hour-Lines are truly drawn according to the former Rules of an Horizontal Dial to wit from the Center to the Points where the respective Hour Circles cut the String or Limb of the Plane therefore it must truly shew the hour OPERATION XV. How to describe Geometrically a Declining Dial for the Elevation of London The first way THIS Dial being as I said an Horizontal one to those in our Horizon 40 Degrees Eastward from the Meridian Find as we show'd you in the Geographical or 20 Section what Elevation or Latitude they have and describe Geometrically an Horizontal Dial on paper for the said Elevation In the next place consider the difference between both Longitudes to wit how many Hours the Sun comes sooner to their Meridian than yours so that if he comes suppose 3 hours 't will follow that the 3 a clock hour line is to be the true 12 a clock line of this Plane because 't is really so late with those People when 't is but Noon with you and consequently that their 4 will be your 1 a clock and their 2 your 11. and in the like manner you are to mark the rest having nothing more to do but to draw on your fair Plane a Line Parallel to the Horizon and to place on it at right Angles the true 3 a Clock Line that is to say the 12 a clock line according to your now alteration or present figuring the Hour-Lines for you will have all the requisite Marks or Points not only to draw the other Hour Lines but also plainly to see where the Substilar will fall and how high the Cock it self is to be for they are all to correspond with those in the said Horizontal or Paper draught Now in case the difference of Longitude between these 2 Places happens to be a Fraction as suppose one hour and 10 minutes then if the Declination of your Plane be still Eastward as in the former example 10 minutes past 1 must be markt in
is not then able tho' up to shine upon it 't were needless as we said to express more Hour lines 'T is the Describing also of the Plane with your String that brings us to the knowledge of the second part of this Operation I mean the knowing at all times when the Sun comes on and goes off any Plane for having describ'd one Declining v. g. 20 Degrees Eastward do but observe what Diurnal Parallels and Hour-circles intersect on the Edges of your Plane and you have your Intent for you will by this means see that tho' the Sun rises for example sake on the 11 of June before 4 the first hour circle which intersects with this Parallel on the Edges of the Plane is that of a Quarter before six whereas about the beginning of May he is there at half an hour past five and on the 10 of April at or near 5. Now if you consider in the same manner the West-side of the Globe you will see from time to time at what hour he goes off it and thus you may do let the Plane be what it will Here therefore it evidently appears if you should erect at any time suppose about the 10th of April a Perpendiculur stile on an Horizontal Plane and draw every Hour a Line along the Shade of the said stile why such a Dial will be false as only telling you the true Hour twice in the year to wit on the 10th of April and about the 10th of August viz. on the days on which the Sun run's in the same Diurnal Parallel I say all this now evidently appears since every Line thus drawn on an Horizontal Plane except the Meridian or 12 a clock line is no Hour line but an Azimuthal Section I mean the Section of the said Plane with a Circle that then passes over your head throu ' the body of the Sun so that if one of these Lines should Bear suppose almost SE and be figur'd with 10 in the morning Draw but your String from the Zenith over that Bearing or Point of the Compass in the Horizon of your Globe and it will truly represent the said shade or Line on your Plane for it show's it to be 10 of the Clock on the Parallel belonging to the said 10th of April But since your String cuts also on your Globe v. g. the Tropic of ♑ at a little before 9 and the Tropic of ♋ at almost half an hour past 10 you may conclude that this will be the true time of the Day on the 11 of December and 11 of June tho' the shade of the Perpendicular stile still show's 10 a clock at the aforesaid Bearing let the Season of the year be what it will therefore a Dial thus made must be false Of several ingenious and humersome Dials HAving thus run throu ' all Planes I shall at present show you how to make use of the former Principles as to the ready Describing of several ingenious and humersome Dials for all are now in a manner but Corollaries from what we have already said and consequently easy both in Speculation and Practice OPERATION XXVI How to make a Dial on any Plane whose stile shall be an Arrow fixt casually on it EXamine what the Plane is and having found it to be suppose a Vertical one Declining 40 Degrees East-ward describe by your former Rules such a Dial on Paper with the Paper stile F x M. as in Scheme 31. exactly set and mounted then draw on the Plane an Horizontal Line H h and place on it your said Paper draught so that the 12 a clock Line FP may fall at right Angles on the said Horizontal line Lastly move your Draught along it till some part of F x or Indicating side of the stile suppose the Point A just touches the Top or most prominent Part of the Arrow and fixing there the said Draught if you draw fair Lines on your Plane under those on the Paper the said Arrow will always show you the Hour with its Top. The Reason is plain for you see by the said Top's just touching the Edge or Indicating side of the Paper-stile it has the effect of the Top of AB I mean the Top of a Perpendicular falling from the said side on the Sub-stile so that X the Top of XM both in the present Scheme and also in Scheme 18. or Example of a Declining Plane has this Effect also Now since the Top of AB or XM or of any other Perpendicular that falls from the Indicating side XF on the substile FM will perform the Office of the stile as we show'd you at large in Demonstration of the first Horizontal Dial or first Example it must necessarily follow that A the Arrow's Top do's the like OPERATION XXVII How to make a Dial to show the Hour without a stile on any Plane DEscribe as in Scheme 32. a Dial on P the given Plane and erect for the present a true stile as FAB of Paper or the like then fixing a Glass or any other transparent matter suppose G at what distance you please before the said given Plane and Parallel to it mark where A the Top of the Stile just touches the said Glass and if there you paint a little Asterisk or spot it will as often as the Sun shines describe such another Figure at suppose D by its shade on the said Plane P and move also from Hour Line to Hour Line according to the true time of the day The reason of this is also Evident for if A the top of the real Stile show's the Hour by casting a Shade as we show'd you all along on the Hour Lines then the Asterisk being there painted where the said Top touches the Glass must do the like for it is you see the Stile 's Apex or Top and consequently casts a true shade to know the Hour by This Dial serves not only for all double Windows or for Cavities that have over them any Glass or Transparent matter but shows us how to make one for any Plane that is illuminated by a Ray coming throu ' a Hole since if you describe the Planes proper Dial on Paper and move it duly as before on the said Plane 'till the Stile or if that be too short 'till a Thred drawn along its Indicating side touches the Hole it will give you marks for the drawing the fair and standing Hour-lines of your Plane which the said Ray will dayly run over in order and consequently show you from time to time the Hour for the Ray passing as you see throu ' the Hole v. g. at A and falling on the true Hour Line at D performs what A the Apex of the true Stile FAB would do OPERATION XXVIII How to describe a Dial having a Picture of a Man in it that shall Point to the Hour from time to time with his Finger THIS Dial is on several Planes of Mr. Lines his forementioned Pile
in Whitehal Garden and as no Dial can be more useful so perchance none ever struck the Fancy both of the Ignorant and Learned with a more sudden Admiration than this as I have often found by Experience both in England and elsewhere Nor truly can it but surprize one at first to think that a Picture without a Machine or Movement should have his Finger ever on the Hour and as duly attend the Sun's motion as if he were alive I say this cannot but surprize one and yet this very Dial is as easy to be made as any of the former Suppose then as in Scheme 33 that the Plane given you were that of the Vertical Cavity a b c d lying directly South describe therefore on the Glass ABCD the contrary Dial i. e. a Direct North Dial with a Paper Style truly mounted and placing the said Glass over the Plane and Paralel to it see where the Stile just touches the said Plane and at that point suppose E let the top of the Pictures Finger be painted then throwing away your Paper Stile and now by the Help of a handsome Frame or the like fixing there your Glass all its painted Hour Lines by hindring the Sun's Passage or Light will project so many Dark Lines on you Plane whilst the then true one falls directly on the Mans Finger and consequently shows you what a Clock it is For if there were a Hole that passed at E the Top of the Mans Fingers throu ' the Center of the World to our Antipodes it necessary follows by the Reasons in our former Operation that at 10 of the Clock suppose at night the Sun being then Northward must cast its Rays throu ' the said Hole or top of the Finger on the 10 a Clock Line of this North Dial on the Glass but since at 10 a Clock in the morning the Sun is in the same Plane as he was at 10 at night only his Station is contrary therefore he must now cast the Shade of the Hour Line the contrary way i. e. on the Mans Finger for in the day time the Hour-line is between the Sun and the Finger whereas in the night time the Finger or Hole is between him and the Hour-Line This Dial needs not always be made on a Glass for 't is sufficient if you raise a thin Frame aaaa in Scheme 34. on the Pillars bbbb above P your Plane as high as the Glasse's true Station or Place for then you may cross the said Frame with small Strings or Wyars which will by their interposition cast the same shade as the Hour-lines of the Glass would have done so that if the Figures belonging to the said Lines be put on the Frame at the end of each corresponding Wyar and then pierc'd the Sun Beams passing throu ' their Cavities will distinguish each very perfectly on the Plane Tho I have not time to show you all the particulars of this Learned Man's rare Invention in Dialling for most of the Dials on the aforesaid Pile may be naturally and expeditely describ'd by the help of this Globe yet I will give you two more viz. the two following ones because besides their prettiness we may have use of them as you shall see by and by OPERATION XXIX To make a Dial by which a Blind man may constantly know the Hour YOU must first get made in Brass the Armillary Hemisphere ABCDE as in Scheme 35 8 Inches suppose in Diameter representing your Globe cut throu ' the Horizon but the said Hemisphere is not to have any thing solid remaining besides the Horizon ABCE with the Pieces of the Hour Circles 1234 c that reach to it from the Nadir or rather from the Tropic of Capricorn AFC on the Northernside for the Southerly Circles are superfluous Then having plac'd the said Hemisphere directly North and South as your Globe stands when Compos'd fix G a Glass Bowl of clear water 4 Inches in Diameter i. e. half the former in the midst or center of it for the Sun's Beames passing throu ' the Water will contract in a Point and ever burn at suppose H the true Hour-Circle so that if a Blind-man puts but his Hand on the said Brazen Hour Circles he will soon find by the Heat where the Sun marks and consequently tell you the Hour for he may easily feel how far it is from the middlemost Hour Circle I mean the 12 a Clock Circle or Meridian As for the Reason of this Operation 't is presently conceiv'd for when the Sun is over against suppose the 5 a Clock Hour Circle on the South-side of the Dial he must needs be over against the same Hour on the north-North-side both hours making but one Circle Now since the Center of the Bowl by being in the Center of the Hemisphere is in the Plane of all the Hour Circles and since according to the nature of Refraction all Parallel Rays of the Sun passing throu ' a Sphere of Water are where they meet with the Direct Ray that passes throu ' the said Center contracted into a point viz. on the opposite side at the distance of half its Diameter or two Inches according to our present Example I say seeing this it must needs follow that at 5 of the Clock the Sun will burn on the corresponding Hour-Circle and if so then a Blind-man by feeling the Heat and finding its distance from 12 must needs be able to tell you the true time of the Day OPERATION XXX To make a Dial to show the Hour when the Sun shines not PRepare a Blew Glass Bowl as in Scheme 36th and describe on it with their Respective Figures all the Hour-Circles of the Globe or as many as you think fit then fixing it where you intend and composing it truly by your Globe if you place your self so at some Distance that a little Hole being made at each Pole to wit at P p you may see quite throu ' the Bowl 't will follow that the Hour-Circle suppose A which the Sun's Picture appears on will be the true time of the Day I call this to know what a Clock it is when the Sun shines not because now the least faint Appearance of him serves the turn tho' it be not enough to cast any shadow nay let the Sun be quite cover'd and if you can but guess by the Adjacent Brightness whereabout he is you will be able to guess the Hour without any sensible Error for the said Brightness appearing on the Bowl will be proportionably distant from the Sun 's true place there as 't is from the Sun in the Heavens 'T is clear that the Suns Picture must fall if any where on the true Hour-Circle because by Composing the Bowl according to the true Position of the Heavens the Hour-Circles of the one concur with the other and fall exactly in the same Plane therefore were your Eye in the Center of the Bowl its true
the Rest Sch. 3. Sch. 4. Sch. 5. Sch. 6 Sch. 10. Sch. 7. Sch. 9. Sch. 8. Sch. 11. Sch. 12. Sch. 13 Sch. 14. Sch. 15. Sch. 16. Sch. 17 Sch. 18. Sch. 19. Sch. 20. Sch. 21. Sch. 22. Sch. 23. Sch. 25. Sch. 26. Sch. 27 Sch. 24 Sch. 28 Sch. 31 Sch. 32 Sch. 33 Sch. 34 Sch. 35. Sch. 36. Sch. 37. Sch. 39 Sch. 38 Sch. 41 Sch. 42 Sch. 34 OPERATION I. BEfore you proceed further you must know Reader that the Printer skipping a line in the last Paragraph and then adjusting the number of Planes to those he found exprest has left out two so that the before mentioned principal Planes are 7 viz. the Horizontal Plane the Direct Vertical Plane the Declining Vertical Plane the Direct Reclining Plane the Direct Inclining Plane the Reclining Declining Plane and the Inclining Declining Plane First then of the Horizontal that Dial being as is said the Foundation of this Science and afterwards of the rest in Order for the Author treats of all Dials that are to be described on the aforesaid Planes J. M. How to describe an Horizontal Dial by the Globe for the Elevation of London The first way OPen your Compasses at 60 Degrees in any great Circle of your Globe and draw on a sheet of Paper a blind Circle with a fair Diameter throu ' it for the Meridian or 12 a Clock hour line of your Dial Then take with your Compasses in the Horizon of your said Globe the several Distances between the next 8 morning or evening hour Circles and its Meridian or ordinary 12 a clock hour Circle and marking these Distances successively in the blind Circle on both sides of its Diameter they and the Center will be the Points by which you may draw all hour Lines from 4 in the Morning till 8 at Night and if you would have a Dial bigger than the blind Circle draw about it a bigger Circle if a lesser a less nay if you describe any other Figure as an Oval Square Oblong c. the said Points will as well guide your Ruler as when the blind Circle it self was the Extremity or border of your Plane But least this Direction should be too obscure for a Beginner I will here adjoyn an Example Having opened your Compasses as I said at 60 Degrees in any great Circle of your Globe and describ'd a blind Circle to wit I p T c as in Scheme third draw a fair line IT any how throu ' the Center O for your Meridian or 12 a clock hour Line and by the way remember that in the Fabrique of this Dial you place the point I ever towards you and T farthest from you to the end you mistake not when directed to this or that hand Having then proceeded thus far put one foot of your Compasses on the Meridian or according to the Polar Figures the 12 a clock Circle of your Globe where it cuts the Horizon and the other foot on the 1 a Clock Circle and mark this distance in the blind Circle from I towards the left hand to wit from I to k and it will give you a point or mark for your 1 a Clock hour line and from I to h towards your right hand the mark for your 11 a Clock hour Line In the next place take in the said Horizon the distance between the 1 and 2 a Clock Circles and place it from k onwards to l for a mark for your 2 a Clock Line and from h to g for the 10 a Clock line and so on till you come to r 8 at night and to a 4 in the morning which are the latest and earliest Summer hours If then you would have a larger Dial describe a larger Circle suppose NESW or if a lesser Dial a lesser Circle as MPQR and laying your Ruler on the Center O and on each of the former Marks or Letters in the said blind Circle successively draw but a fair line to the designed Limb or Border whether it be a Circle or the square VXYZ or any other Figure and your Dial wants nothing bdt a Cock but remember that you need not draw your hour lines quite from the Center O because meeting all there they will be apt to blur therefore describe about the said Center at what distance you please a little Circle like γ ♌ λ and your lines will terminate there with more neatness and convenience Now if you have a mind to put on half hours and quarters you will not much err if you divide each hour into four parts but to be exact you must make use of your String thus You know that the distance between each hour-Circle in the Aequator is 15 Degrees Draw therefore your String from the Pole throu ' the Aequator of your Globe over 7 degrees and 30 minutes or half the distance between each Hour Circle and where the String cuts the Horizon there will be the true half Hour of that Hour so that if you mark with your Compasses the said distance on the blind Circle between the corresponding hour lines the Ruler passing throu ' that Mark and Center will give you in the Border the place of that half hour and in like manner you are to proceed in marking out the rest as also the Quarters and all other Subdivisions As for the Stile or Cock of this Dial it must always at the Center make an Angle with the Meridian or 12 a Clock Line OI equal to the Distance between the Pole and the nearest part of the Horizon of the Globe that is to say an Angle equal to the Elevation or Latitude of the Place therefore your Dial being made suppose for London open your Compasses at the aforesaid distance or at 51 Degrees and ½ and placing one foot on I the other will fall on K in the said blind Circle so that drawing the blind line OK to π you will have the Triangle IO π which if you so erect that the Point O lyes just on the Center and the Base IO on your 12 a Clock line or Substile your Dial is finish'd And here you may take notice that tho' this Stile be the Triangle IO π yet you may fashion it into what shape you please in case the side π O which indicates or shows the Hour makes still an Angle of 51 Degrees and 1 2 with the Meridian IO nay you may make it a Pin or upright Stile as appears by the Perpendiculars AB GH and π I for either of them will serve the turn by marking the hour with the shade of its Apex or Top but then they must not be plac'd in the Center O but thereon the Substilar where falling from the Indicating side O π they stand Perpendicular to it that is to say the Pin AB being part of the Triangle or Stile IO π must be erected at B the Pin GH at H and π I at I and the reason why they perform this Office as well as
the whole Triangle IO π is because their Tops are parts of the Line O π which is the only side of the said Triangle that shows the Hour as we mention'd before Now for the Demonstration of all it follows in the next Operation OPERATION II. How to describe an Horizontal Dial by the Globe for the Elevation of London The second way DEscribe a Circle of what bigness you please and draw a Meridian or 12 a Clock line throu ' it as before then count in the Horizon of your Globe how many Degrees there are between the Hour-Circles of 12 and 1 or which is the same thing between 12 and 11 and you will find their number to be about 11. 40′ These place on both sides of your said Meridian Line by the help of a Quadrant or Line of Chords and they 'l give you if you lay your Ruler as before on the Center the 11 and 1′ a Clock Hour Lines of your Dial to wit the distance from I to k and from I to h as may be seen in the aforesaid third Scheme Proceed then in this manner as to the rest of the Hour lines and for your Stile and Substilar the former Directions are sufficient The Demonstration or Reason why these Dials show the Hour is not difficult for if you consider your Globe you will see that all its Hour Circles are equally distant from each other and that the Axis of the World of which the two Poles are the extremities lies in the middle of them and is in truth a part of each as being the common Section of them all therefore when the Sun comes into the Plane of any Hour Circle for example to that of 4 in the morning the shade of that Hour-Circle will fall there where the said Hour Circle cuts the Horizon on the Opposite or Western side and consequently the Axis being in that Plane as a part of it its Shade must needs fall there also Now since the Blind Circle or Limb of the Dial described is a Circle representing the Horizon and having by Construction its Hour-lines distant from each other as the Hour Circles of the Globe or World are distant in their Horizons and since the Hour-lines of This and consequently of all other Dials are only the intersections of the Hour-Circles with their respective Planes it must needs follow if we place in the middle of the said Dial a Cock or Stile making an Angle of 51 30 with its Meridian line or Substilar to wit the Angle which the Axis of the World makes with the intersection of the Meridian and Plane of the Horizon 't will cast a Shade directly on the Hour line corresponding to the Hour Circle in whose Plane the Sun then lies in case the Meridian or 12 a Clock line of the Dial be plac't North and South like the Meridian of the Globe when compos'd for the Globe it self without it be compos'd will not as we have formerly mention'd shew the Hour because its Hour-Circles do not then correspond with the Heavenly ones And as for the reason why the 12 a Clock line is the Substilar 't is because the true Height of the Axis above the Plane which the Stile or Cock as I showd you represents is to be measured in the Hour Circle that falls on the Plane at right Angles which being the Meridian or ordinary 12 a Clock Hour Circle it follows that its Intersection with the Plane must be the Substilar or Line with which the Stile is to make the Angle of the Elevation All that we have then said of this Dial may be clearly seen by Sch. 5. which represents your Globe cut into an Horizontal Plane with its Dial on it as Sch. 4. does the Globe entire when you consider it in the description of the said Dial for there you have before your eyes by the Letters I k l c. not only how to open your Compasses from Hour-Circle to Hour Circle for the true placing the Distances of each Hour-Line on your blind Circle but also the number of Degrees in the Horizon between every Hour Circle and the Meridian Besides by the Horizons oblique cutting the Hour Circles you may see how that notwithstanding the equality of the Suns Horary motion the Hour-lines of this Dial must be unequal and consequently that they are of different distances in different Latitudes OPERATION III. To describe an Horizontal Dial Geometrically for the Elevation of London Describe a fair Circle as ABCD and if you would have your Dial of another Shape you may afterwards describe about it what Figure you please I say describe the fair Circle ABCD and draw throu ' its Center O the Line AOC for your Meridian or 12 a Clock hour line and crossing it at right angles with BD for the Morning and Evening 6 a Clock hour lines mark in it by the help of your Line of Sines or any way else from A the value of 51. 30. or Latitude of your dwelling which happening to reach for example sake to K draw the blind line OK then throu ' any point of AO suppose A draw GH another blind line parallel to BD or at right Angles with the said AO and taking with your Compasses the nearest distance between A and OK which being suppose the point L let AL by the help of your Sector according to our former directions be the Radius to the Tangent Line GH so that marking in it on both sides of A the Tangents of 15 30 45 60 and 75 Degrees the said Center O and the point 15 will give you the Hour-lines of 1 and 11 the Center O and 30 those of 2 and 10 and in this manner proceed to 75 which will give you the Hour-lines of 5 and 7 and as for those beyond the 6 a Clock lines do but produce 8 in the Morning and 't will give you 8 at Night and 7 in the Morning 7 at Night as will 4 and 5 in the Evening the like forenoon Hours Thus then you have not only an Horizontal Dial Geometrically described almost as soon as the former and this without embroyling the Plane with multiplicity of blind Circles and Lines but a way also in case you have no Sector how to make any Tangent Line serve your turn for 't is but taking between the Compasses 45 Degrees of it i. e. a distance equal to its Radius and finding out by a trial or two the Point suppose R in the line OA where one foot of your Compasses being placed the other just touches M the suppos'd nearest point or distance in OK from the said R draw throu ' R a line at right Angles with the Meridian and noting in it as we show'd you before the Degrees of each hour according to this new Tangent line the Center O and these Degrees will give you the points of each hour line for as the former Radius AL was to the several Degrees in its Tangent Line so