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
Meridian Dials be the height of the Stile Now to describe this Dial Geometrically 't is yet more easily performed for if you draw as in Scheme 24. the Line AB parallel to the Horizon and then take a Point in the middle of it suppose K do but prick on both sides of it the Tangent of 15 30 45 60 and 75 and the several Perpendiculars drawn throu these Pricks will be true Hour-lines which you may figure as you see in the before mention'd 24th Scheme and as for the Stile the Tangent of 45 or distance between the 12 a Clock line and that of 9 or 3 gives you its height which is to be a Pin or Gallowes Stile as before and the 12 a clock line the Substilar OPERATION XX. How to describe a Direct reclining North or South Dial. SUPPOSE then that the Plane lay directly South and that its Reclination were 20 Degrees you have nothing to do but either Geometrically to make on it a direct Vertical South Dial for the Elevation of 71 Degrees and ½ I mean for a Plane 20 Degrees neerer the Pole than your own Zenith or to fix your String on 71 gr and 30 min. in your Meridian that is to say at A in scheme 25th and then to draw your said string over the East or West Points of your Globe for 't will represent this Plane since it Reclines or falls back from the Zenith 20 degrees therefore the Distances between the Hour-Circles that intersect with your String must for the former reasons give you in any blind Circle which shall be equal to a great one on your Globe marks viz. b c d e f g for the corresponding hour-Hour-lines and the Meridian being the Substilar since 't is the Hour Circle that falls on the Plane at right Angles the Height of your Stile must as in all Direct Vertical Dials be the distance from the Pole to A the supposed Point or Place where your String is fixed Now had your Plane Reclin'd 20 Degrees the other way that is to say had it Reclin'd so many Degrees facing the North you must have fixed your String at N viz. 20 Degrees short of the Zenith and consequently your said String would have intersected with the Hour Circles at o p q r s therefore a Direct Vertical North Dial for the Latitude of 31 g. 30 m. will be the required Dial. OPERATION XXI How to make a Declining Reclining Dial by the Globe SUppose your Plane declin'd 40 Degrees Eastward as did the late Declining Vertical and then Reclin'd 20 Degrees with a Southern Aspect and by the way you must remember that I mean in general by a Planes Reclining with a Southern Aspect its looking towards that Quarter tho' it be turned more or less from Direct South towards the East or West in like manner a Declining Reclining Plane with a Northern Aspect turns from direct North towards one of the aforesaid Points Supposing then a Plane thus Reclining Do but describe or place it on your Globe and your Operation will be as easy as any of the former First mount your Bead 71 Degrees and half above the Horizon that is to say fix it to 20 Degrees from the Zenith of the Globe then seeing your Plane has a Southern Aspect and so lies beyond your said Zenith Northward move your String till it cuts in the Horizon 40 Degrees Westward from the Northern Meridian or back part of the 12 a Clock Hour Circle In the next place take a Thred and tying it about your Globe so that it lies not only on your Bead but crosses also the Horizon at 40 Degrees from the East point Northward and 40 Degrees from the West Point Southward the said Thred will represent your Plane Reclining and Declining as aforesaid Or in short fix a small Needle in the Point where the Bead lies which we suppose at A in Sch. 26. and fastning to it a Thred or part of the string draw it over the Horizon at 40 Degrees from the East-Point Northwards and it will give you the Eastern or Morning side of your Plane as it will the Western or Afternoon side if you draw it as in Scheme 27. over 40 Degrees of the Horizon from the West-Point Southwards This being done describe a blind Circle or Semi-Circle equal to a great one on your Globe for Example sake the blind Semi-Circle A. T. C and drawing from O the Center the blind Line OA perpendicular to the Horizontal line H h take the distance with your Compasses between A the station of your Needle or Bead and the point in the 12 a clock hour Circle crost by the Thred or Edge of your Plane and this distance from A in your blind Circle gives you there towards your left hand the Point k to which if you draw a fair Line from the Center it will be the 12 a clock Line of your Dial and the distance from the said station of your Bead or Needle to the intersection of the Thred with the next Hour-Circle will give you l the mark of the 11 a clock Line and in this manner you must run over all other intersections of your Thred and Hour-Circles to the very Horizon on both sides of the Globe I mean on the Morning and Evening side of it represented by Scheme 26 and 27 and placing their distances on your blind Circle on both sides of the aforesaid OI do but draw lines to them from the Center and your Dial is describ'd And here you must observe that I have in Scheme 26. or Eastern Face of the Globe plac't A the Station of the Bead or Needle above the Meridian since its true place cannot be exprest for it ought to have bin on the other side of it I mean on the Western side which Scheme 27 is supposed to represent Now for the Stile and Substilar there is no difference from the Rules of the Declining Vertical since 't is but finding the nearest point on your Thred to the Pole by your Compasses for the distance between the said Point on your Thred and it's intersection with the 12 a clock Hour-Circle is the distance in the blind Circle between k and M for the Substilar and the distance between the said neerest Point and the Pole gives MX the height of the Stile above the Plane Nay if you measure the Distance between each Point and A in any great Circle 't will give you the Degrees or Distances between A and your Stile Substilar and each Hour-line and consequently performs the second way as we have all along mention'd of describing Dials by the Globe As for the Demonstration of this Dial what we have formerly said about the rest proves it also for supposing that the Thred represents truly your Plane and that the Hour lines of a Dial are as I have show'd you all along the several intersections of the Hour-Circles with the Plane this Dial must be true
corresponding hour Line and as for the reason why the height of the said Axis is equal to the distance between the 3 or 9 a Clock Lines and the Substilar it shall be shown in the Demonstration of the next Operation OPERATION XII How to describe an East or West Dial Geometrically for the Elevation of London DRAW the blind Line H h and cross it from your left hand as in Sch. 13. with AE ae another blind-line to make an Angle at their Intersection K equal to the Complement of the Elevation then pricking in the said Line AE ae on the right side of K the respective Tangents of 15. 30 45. 60. and 75 Degrees as also on the left the Tangents of 15 and 30 Draw but Perpendiculars through the Pricks and you have an East-Dial whereas should you cross as in Sch. 14. H h with AE ae from the right hand and pricking the aforesaid Tangents the other way draw Perpendiculars through them you would have a West-Dial By these Schemes also you may know how each Dial is to be Figur'd the East-Dial containing as you see all the hours from 4 in the morning 'till Noon and the West all the hours from Noon to 8 at Night Now for their Cocks they are as I said in the last Operation to be a Pin or a Gallowes Stile and in height equal to the Tangent of 45. Degrees or distance between the 9 or 3 a Clock hour Lines and that of six which is ever their Substilar These Dials must be true if their Planes lye in or Parallel to the Meridian for since the Line H h by being plac'd according to our Hypothesis horizontal represents the intersection of the Horizon and the line AE ae that of the Aequator by making an Angle with the said H h equal to the complement of the Elevation the substilar must be the Intersection of the Aequinoctial Colure or 6 a Clock hour Circle with the Plane since that Hour-Circle falls on the Plane at right Angles If then a Gallows Stile be set on the said Substilar and Perpendicular to it its Shade must needs constantly cross the Aequator AE ae at right Angles Now when the Sun is in the Plane of the 6 a clock hour Circle his Ray makes no Angle with the said Stile because the Sun and the Stile are in the same Plane and so the shade falls directly along the Substilar but when he gets for examples sake into the next hour Circle his Ray the height of the Stile being Radius makes an Angle of 15 Degrees with the said Stile and consequently the distance of the two shades are in the line AE a the Tangent of those Degrees The like therefore being said of the next Hour Circle and so on it follows as I mention'd in the beginning that the pricking from the intersection K the Tangents of 15 30 45 60 and 75 Degrees in the line AE ae must give you points to draw the perpendiculars or true hour-lines of this Dial by as also that the Tangent of 45 Degrees gives the height of the Stile since the Tangent of those Degrees which you see gives the 3 and 9 a clock lines is equal to the Radius Here also we see not only why these hour-lines are so unequally distant since they are so many Parallels marshall'd according to the Divisions of a Tangent line but why the 12 a Clock hour line can never be really express'd for 't is the Tangent of 90 Degrees which is infinite OPERATION XIII How to describe a Declining Dial by the Globe for the Elevation of London The first Way THIS Plane as passing from the Zenith to the Nadir is still Vertical and should you may suppose be by right the primary Vertical but by its tendency towards the East or West Points its Dial takes the Appellation of a Declining one that is to say of a Dial whose Plane declines so many degrees from facing directly the North and South as is its tendency towards the said East or West points As for the way of making this Dial it differs little from the first Direct Erect one already treated of for supposing your present given Plane declines 40 Degrees from full South towards the East you must draw your String which ever represents the Edges as we have said of your Plane not throu ' the East Point of the Horizon of your Globe as before but throu ' 40 Degrees further towards the North for this makes the String to represent part of a Plane that comes nearer by so many Degrees the facing of the East than it did Then opening your Compasses at 60 Degrees in any of the great Circles and describing as in Sch. 17th the blind one PZW prick in it from its Meridian Line OZ the distance between the Zenith of your Globe and the intersection of your String with the first Hour-Circle to wit between Z and b in Sch. 15. and it will give you a mark for the 11 a Clock line on your Dial and the distance between the Zenith and the Intersection of your String with the next Hour-Circle to wit between Z and c will give you the mark of the 10 a Clock line and thus you must proceed to every Hour-Circle cut thus by your String till it falls on the Horizon that is to say from z to d e f g h letters marking as you see in the said Scheme the 9 8 7 6 5 and 4 a Clock Hour Circles and consequently giving you those Hour-lines on your Dial. Now for the Afternoon hour lines which are no longer equal in distance to the Morning ones you have nothing to do but to draw your String on the West-side of your Globe throu ' 40 Degrees in the Horizon the contrary way viz. from the West towards the South and the distance between the Zenith and the Point in the first Hour-Circle cut by your String to wit from Z to k in Sch. 16. will give you the mark for 1 a Clock and the distance from thence to the next Point or Intersection gives you that of 2 to wit from Z to l and in this Order you are to proceed to n the 4 a Clock Hour Circle that is to say till you come to the intersection of the String with the Horizon on the West-side of your Globe As for your Stile and Substilar they differ also from those of direct North and South Dials for the said Stile or Cock is to be no longer plac'd on the 12 Ã Clock Line nor will its height now be equal to the Complement of your Elevation therefore having drawn your String throu ' the Degrees of Declension in the Horizon as before and putting one foot of your Compasses in the North Pole find with the other the nearest Point on your String to wit S as in Sch. 15. and the distance between S the said nearest Point and the Zenith of your Globe will be ZS
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
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
mentioned Seventh Scheme shows you so that by the help of your Sector or of any Line of Chords or Quadrant you may mark them successively in your Blind Circle on both sides of the Diameter and then if you draw from the Center Lines throu ' those marks your Dial is finish't for as to the Stile and Substilar you need no other Instruction than what you had in the last Operation which also directs you to the Demonstration since the same serves both OPERATION VII How to draw a Line Parallel to the Horizon together with two ways how to place truly all paper Draughts on their respective Plane HAving lately advised you To Delineate all Dials on Paper before you draw them on your designed Plane and having show'd you how to describe this Dial 't is now time to teach you how to draw an Horizontal Line on this Plane that you may thereby truly place your Draughts Slip therefore out your two Rulers which are under the the Pedestal as I already mentioned and placing the end of one on a convenient Center chosen by you in your Plane you 'l have by the end of the other when the Plummet falls on the Asterisk or little Star a cond Point and consequently marks to draw the required line by so that if you then place the Center of your said Draught on the Center of the Plane and its 6 a Clock Hour Line on your Horizontal Line all the other Lines will fall on their true places and thereby show you where with a Cole or the like to mark out points for the perfect and final drawing of them The Cock also of the Paper Dial will direct you in the placing of the other for they are both to be of the same height above their respective Planes with their Tops pointing the same way viz. downwards to the Horizon in all these South Dials But if you will have yet a more easy way of placing a Paper Draught not only on this but on any Plane for which 't is made look what a Clock 't is by your Globe and moving your said Draught on its Plane 'till it shows exactly the true Hour do but fix it there and you may mark out the Points for your fair Lines with all the ease imaginable OPERATION VIII How to make a Vertical or Erect Direct North Dial for the Elevation of London THERE is no difference between the Fabrick of this Dial and the former unless it be in figuring it for a South Dial reverst is a North Dial the After-noon Hour Lines being mark't with the Morning Figures and the Morning ones with those of the Afternoon So that the Top of the Stile points now upwards as may be seen by Scheme 9th and by the upper part of Scheme 10th to wit by the Semi Circle PTC therefore when you chuse a Center in your design'd or real Plane for this Dial let it be in the lower part of it to have Room for the Hour Lines to run upwards And by the way you must here remember that tho' I bad you in the making of this your Vretical South Dial to take the distance between the Zenith and the Intersection of the String with the next Hour Circle for the 1 and 11 a Clock Hour Lines c yet that Section of your Globe by your String from the Zenith as aforesaid gives in truth a North Dial and therefore in strictness you ought to have taken the Distance between the Nadir and the several Intersections of the Plane with the Hour-Circles but since both Dials are as I told you alike 't is best always to operate thus from the Zenith as being more at hand than the Nadir and consequently more convenient The Demonstration or reason why these Dials show the Hour differs even at first Conception but little and at the second not at all from that already given for the Horizontal Dial. By the first Conception I mean our considering these Planes as Vertical and Erect for since the hour-Hour-lines of all Dials are as I show'd you in the former Demonstration the Intersections only of the respective Hour-Circles with the Planes and since the hourly indicating Shade is the Shade of the Axis or of the Hour-Circle which then lies in the Plane of the Sun it must follow that the Mark made for example sake by the 4 a Clock Morning Hour Circle on the String and the Center of the said Plane which is the common passage of all the Hour-Circles will be two true Marks or Points for you to draw that hour-Line by and consequently that the Shade of the Axis will still fall on the said hour-line as often as the Sun comes into the Plane of that Hour-Circle Now your blind Circle is by construction equal to the Circle made by the String on the Globe and the Marks on its Limb are equal to the Marks on the said String therefore the Dial must be truly drawn and the Stile plac't on the 12 a Clock line to wit on the intersection of that Hour-Circle which falls on the Plane at right Angles must truly cast its shade from time to time seeing by its Site and Angle it corresponds with the Axis of the World As for our second Conception in reference to these Dials we shall find by it that their Planes are real Horizontal ones to some People or other for this Section of the Globe being a great Circle will be the Horizon to those that live in the Pole of it viz. to those under our Meridian 90 Degrees from our Zenith which being a point in our Horizon makes their Horizontal Dials always our Direct Vertical ones and their Direct Vertical Dials our Horizontal ones 'T is plain then that the present Dials are exactly describ'd if our former Directions and Proof of an Horizontal one be true for all the Hour Lines are here drawn from the Center to the several intersections of the Hour-Circles and Horizon which as we are to suppose the String represents Nor do's the Cock of these Dials differ from the former Rules for having the Meridian or 12 a Clock line for Substilar for the former reason and being 38 Degrees and a half above it it makes an Angle equal to the Elevation of the People who have the said Plane for Horizon OPERATION IX To make the aforesaid North and South Dials Geometrically for the Elevation of London THere is no need of a Scheme for this Operation since 't is a Corollary from what we have now said for make but an Horizontal Dial Geometrically as we formerly show'd you in Scheme the 6th according to the Complement of the Elevation of your Place and that will serve the figuring only consider'd for either Dial. Here then you may see that OS or ON the Basis or Foot of the Stile of these Dials that is to say the distance between its Center and its Horizontal edge or side is ever the Tangent of the Elevation for 't
the Horizontal Draught with the Figure 12 as the Meridian Line and 2 and 10 min. with Figure◠and so on all along whereas if the Declination were Westward then 11 and 10 minutes will be the said Meridian Line 10 and 10 minutes your 1 a Clock Line for thus you must operate in all other Cases that is to say you must still allow by the new figures the difference of Longitude that chances to be between you and them to whom the Declining Plane is Horizontal But because this manner of Dialling may seem to some troublesom and confus'd especially when the said Difference of Longitude happens to be a Fraction and not even Hours I shall here adjoin a second Geometrical Way OPERATION XVI How to describe Geometrically a Dial declining 40 Degrees Eastward for the Elevation of London The second way HAving made an Horizontal Dial for this Elevation in the lower part of your Paper Plane as 't is exprest by the prick lines in Scheme 18 and drawn from the Center A the several Hour-Lines upward as far as you think fit and Figur'd them to show what Hour-Lines they are chuse in AC the 12 a clock line any Point suppose P and draw throu ' it the blind Line GD making with the said AC an Angle of 50 Degrees or Complement of your Declension then erect the Perpendicular PB on the said blind line at P and taking with your Compasses AP being your Radius the Tangent of 5◠Degrees and ½ or true Elevation of the Pole put one foot on P and where the other marks on the said Perpendicular suppose at F there will be the Center of your Declining Dial so that having bordred your Plane with fitting Parallels to contain the standing Figures of each hour you have nothing more to do but to draw fair Lines from the said Center F to your Border throu ' the Intersections of the Line GD with the several Hour-lines of the Horizontal Dial that is to say you have nothing more to do but to draw fair Lines throu ' the Points KLMNO PQR which give the hours of 7 8 9 10 11 12 1 and 2 and by the way you may have as many other Morning or Evening hours as you please if you draw the said GD long enough for the other hour-lines of the Horizontal to meet with it Nor is there more difficulty here about the Stile and Substilar than in any of the former Dials for AP being Radius 't is but taking the Sine of 40 Degrees or Declination of the Plane with your Compasses from the Sector and putting one foot on your 12 a clock Line at P the other foot will in the line GD to wit at M give you the Point for to draw the Substilar FM and the Sine Complement of the Declension or Sine of 50 Degrees will be XM the Stiles height Nay if for want of a Sector or the like you cannot conveniently find the Sine of the said Declension do but observe where a Perpendicular from A falls on GD suppose at M and PM will be the distance in the said GD between the 12 a Clock line of this Dial and its Substilar and AM equal to XM the height of the Stile above it Thus then we see that the Fabrique of a Declining Dial which is wont to terrify young Students is in a manner as quick and easy as that of the Horizontal since two ordinary Lines more viz. GD and BP give us all the Points necessary for its Description The Demonstration and Reason of this Dial is evident for the Horizontal being by construction true any Erect Plane facing the South that crosses its Meridian or 12 a clock line AC at right Angle◠will represent a Primary Vertical or Direct South Plane and then the Center of the Dial described on it will be distant from P the intersection of the two Planes on the said AC the Tangent of the Elevation as I shew'd you before Now since GD is by Hypothesis the Edge of a Vertical Declining Plane and since as we show'd you in the before cited place that the 12 a Clock line as well in a Declining as in a Primary Vertical Dial is Perpendicular to the Horizon containing in it the Centers of the said Dials it follows that FP being the Tangent of the Elevation and Perpendicular also to the said DG where it cuts the 12 a Clock line of the Horizontal must be the 12 a Clock line and F the Center of our present Dial whose Declension is 40 Degrees Eastward since FP declines so many Degrees from CP toward the morning Hours for the said CP and FP represent the 12 a Clock lines of a Direct and of our thus Declining Vertical Plane if you consider them flatted down and lying in the Horizon This being so 't is evident that the Lines drawn from F to KLMN c. are the true Hour lines of our Dial as falling from its Center to the several Points made on its Horizontal edge by the Hour Circles or which is all one by their Intersections with the Horizontal Dial. As for the Stile and Substilar let us but consider the Triangle AMP and we shall find that P is by construction the Angle of 50 Degrees and A that of 40 as substended by the Sine of the Declension so that A being a right Angle AM must be a perpendicular therefore the Hour Circle whose intersection the said AM happens to be falls at right Angles on our present Plane and consequently gives the Substilar Now since the Axis of the World passes through F and A the Centers of the two Dials when they are joyned as we now suppose them at GD the common Section of their Planes I say since the Axis passes throu ' their Centers its Elevation or Height above our Plane must be AM as being the only Perpendicular that can fall from it upon the said Plane and consequently its Measure but AM you see is the Sine Complement of 40 since PM is the Sine of 40 Therefore in all Declining Dials The Sine of the Declension from their 12 a Clock Line gives in their Horizontal Edge their Substilar and the Sine Complement their Stile Q. E. D. OPERATION XVII To take the Declension of a Plane COmpose your Globe and find exactly the Azimuth i. e. what Degree of the Horizon is cut by the String 's shade when it passes throu ' the Zenith and Nadir which wee 'l suppose to be the 50th from the South towards the West then having slipt out to an equal length the two Rulers from under your Pedestal Hold your Globe level and apply the said Rulers as soon as you can to your Plane as you did when you drew an Horizontal Line and find again the Azimuth which now being for example 90 Degrees shows your Plane declines 40 towards the East because the Azimuth being now increast so many Degrees the Meridian which by
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-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
Species there are two sorts here viz. one of plain and simple Pricks the other of small Astricks alternatively plac'd so that 't is but observing of what Species the Prick next a Star is as suppose an Astrisk and then following with your Eye a File or Arch of Astrisks 'till you come to the Horizon for the Figures at their termination there give you the requir'd Azimuth Thus then the confusion which the several Almucantars and Azimuths would make were they all describ'd on the Plane is avoided seeing the Plane is now less fill'd than if the Almucantars were only exprest on it for disjoyn'd Pricks circularly plac'd occupy not the room of a continued Circle and yet each Row or Circle of the said Pricks perform both the forementioned Offices How to operate by the Projection or Pedestal FIRST the Reader must remember that I call Rectifying the first Plane the placing and adjusting it so that all the Stars may appear above and below the Horizon as they then really do in the Heavens themselves which Operation being a main and principal matter for all the other are in Truth but so many Deductions or Corollaries I will now begin with it nor is there any thing here requir'd but the height of some Star in view as the Lion's Heart or the like which you may find by the Globe as you do the Sun 's or Moons height as I mentioned before Now for cleerness sake let us suppose this Star to be about 45 Degrees high Westwardly and then if you move your Plane till the said Star lyes thus under a Prick of this height you have without ever moving more the Plane the several following Operations at a time First You see all the Stars that are then above the Horizon and below it for all the painted ones within the Circle HRST on the second Plane represent the real ones then in sight and the rest those that are below the Horizon Secondly You see what Stars are Rising what are Setting what are Culminating and what are in their Lowest Depression Thirdly If you look after any particular Star suppose the Lion's Heart by seeing him on the West-side of PS the Meridian of the said second Plane you are sure he is not only in a Declining state but also by following the Prick next him to the Horizon according to its Species that his Azimuth is 45 Degrees Fourthly You will see his Bearing to be about S. W. if you follow the Azimuthal Arch to the Nautical Characters there Fifthly You see that the Hour of the Night is 10 by observing under what Hour-Line the 10th of April i. e. the day of the Month the Suns place in the Ecliptick lyes Sixthly By any real or imaginary Hour Line that runs over the said Star you find his Right Ascension to be near 148 Degrees for thus the said Hour Line cuts the Limb. Seventhly By his being behind the Sun about 8 hours as appears by the Hour Lines that pass over the Star and the Suns place you have the difference of their Right Ascensions which amounts to about 120 Degrees Eighthly Which is the most surprising and not performable even by a Coelestial Globe you no sooner see these things in relation to this or any other particular Star but at the same time also even without touching your Projection you have them in relation to all the Stars in general for when the First Plane is rectify'd we have besides the Hour the Heighths Azimuths Bearings Right Ascensions c. of all the other Stars above the Horizon Concerning the other Operations they are more restrain'd as being peculiar to the Star you enquire after for if you would know when the Lions Heart Sets which for continuation's sake we will call the ninth Operation do but move your first Plane till the said Star touches the Horizon and the imaginary Hour Line that passeth over the Sun's place in the Ecliptic show's you that 't will be then about 3 and a quarter in the morning 10ly By the Figures about the Horizon you will see at the same time that his Occasive Amplitude is near 23 Degrees Northward and his then Bearing by the Nautecal Caracters to be WNW or thereabouts 11. By the imaginary Hour-line that then passes over the said Star viz. that of about 7 and a quarter you have half the time of his constant aboad above the Horizon and consequently know that from his Rising to his Setting there are about 14 hours and an half 12. By reason that the imaginary hour-Hour-line of about 7 and a quarter passes over the Star as we said at his Setting it follows that it 's Ascensional difference i. e. the difference between its Right and Oblique Ascension is about an Hour and a quarter or 18 Degrees 13. By the Degree of the Ecliptic that Sets with the Star which is the 26 of ♌ and by the opposite Degree which then Rises viz. the 26. of ♒ you see that on the 8th of August he Sets Achronically and on the 2. of February Cosmically 14. Remove the said Plane till the said Star brushes the Horizon on the East-side and by the precedent method mutatis mutandis you will find when he Rises what his Ortive Amplitude is how he then Bears how long he is under the Horizon when he Rises Cosmically and when Achronically 15. By placing the point of a Pin or Needle on the Class over the Lions Heart and then moving the first Plane till the divided 6 a Clock Hour-line PE lyes just under the said point the Divisions there will show its Declination to be about 13 Degrees and 33 Minutes The like you may do with your Compasses for if you take the Distance between the Pole and Star and measure it on PE you have what you seek for Many other Operations are performable by the Projection touching the Stars but since these are the most material ones and since I have not time to treat more fusely I leave the rest to be found out by my Reader himself who may easily do it if he understands either the Caelestial Globe or any Instrument belonging to the Stars And here he is to remember that knowing but the Hour at any time let him put the Suns place or day of the Month under the Hour-line that corresponds with it and the Projection will be rectified and consequently having a true view of the then posture of the Heavens he may opperate as before In the next place if he knows but the Suns place in the Ecliptic of the first Plane and opperates with the said place as if it were a Star he may find out the former Operations in relation to the Sun it self that is to say he may at that moment know his Height Azimuth Bearing Amplitude c. 16. If you would know the Stars in the Heavens you may also do it by the help of this Projection for your first Plane being rectified it gives you as I said the 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 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
Clock Thirdly when at 4½ Fifthly when at 5 and in the like Proportion go on till the Days come to their greatest Decrease and putting the said days of the Month in Order as they are in the Scheme under the corresponding Hours on the morning side of your Dial for his Rising do the like for his Setting on the Evening side of it and you may perform the Operation with sufficient Exactness In like manner you are to proceed for the Quarters half Quarters c. if you would have them exprest 6ly To avoid also the trouble of deviding the Circle 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 according to the Suns Diurnal Increment and Decrement in Amplitude you need only find by your Globe what the said Amplitude amounts to on every of the aforementioned Days which are markt on your Dial for the Suns Rising and Setting and then put it in Figures under each Day as the Scheme shows you 7ly Open your Compasses at the Tangent of 28 Degrees AB being the Radius and putting one Foot on B describe the Circle XYZ afterwards describe another according to the Tangent of 35 Degrees then a third according to that of 40 and so on in the same Proportion as far as your Plane permits Now if you mark these Circles with the Figures of the Complement of their Degrees that is to say the Circle of 28 Degrees with the Figure 62 that of 35 with 55 that of 40 with 50 c. you will always know the height of the Sun for what Circle soever the Shade of AB touches with its Top that will be the requir'd Height and if it falls between 2 Circles 't is but considering which of them it comes nearest to and then you may guess at the Height with sufficient exactness 8ly and 9ly Devide one of these Circles viz. SEWN into Degrees and under each 11 Degree and ¼ place the several Points of the Pixidis Nauticae or Mariners Compass in the Order as they are express'd in our said Scheme and you will not only have by the Shade of AB the Suns Azimuth at all times but see also how he bears from you according to the Points of the Compass and if the Shade be at any time too short lay on it but a Ruler Label of Paper or the like and that will truly lengthen the said Shade and resolve your Question 10thly Devide AF the Northern half of the Meridian as many times as you can by the Stile or Radius AB and then each Devision into ten equal parts as you see it done in the said Scheme and by it you will know at all times the Proportion between any Perpendicular and its Shade and consequently besides many other things the height of any Tower Tree or the like for having found the Sun to be suppose 25 Degrees high and that the Circle of Altitude cuts the Linc AF in the 22 Devision if therefore you measure the Shade of your Tower and finding it for Examples sake to be 66 Yards long you have what you seek for as the said 22 is to 10 the Stiles height so is 66 the length of the Shade to 30 the height of the Tower So much then for the Construction of Dials And now let me desire all those that are pleased to follow this Geometrical way which perchance is as expedite a one and as free from blind Lines as can be not to rest satisfy'd till they fully comprehend what they do for the Mechanical way of Dialling is as soon lost as learnt it being impossible without continual Practice not to forget the Rules especially if one can make many Dials when as a man that understands the reason of the Operations by having in his Head a true Idea of the Sphere and its Projection will 20 years after without Memorandums or Notes be able reflecting but a little to make not only all Dials he formerly knew but new ones also at first fight To Conclude I here present my Reader with the Globe in a new Dress for being painted or stain'd on Marble according to Sch. 43. 't will be fit for any Garden or open Portico and least it might appear too plain the corners of its Base or Pedestal may be adorned with handsom well turn'd Branches which not only embellish the whole Machin by their Make But hold out Bowls of Glass and Wyar for use also For on the First Corner to wit That markt with A there is placed as a Rarity The blind man's Dial. On the Second markt with B. The Dial that shows the Hour when the Sun shines not which will be often very useful On the third mark't with C there is an Armillary Wyer Sphere having a Vane on the Top that continually shows on the brass Plane within graduated and Nautically Character'd from what Quarter the Wind exactly blows as also if you turn the said Vane into the Plane of the Sun his Azimuth and Bearing Besides the Sphere being an Horizontal Concave Dial shows the Hour too for the Shade of the Pin's top in the Center ever fall's on the true Hour-Circle as I show'd in the Construction of such a Dial. And by the way you must know this Branch stands not in it's true place in the Scheme I mean on the third Corner of the Base because in Perspective 't will fall on the Globe it self and consequently not appear well to the Eye in a Picture Lastly on the fourth Corner markt with D there is another Glass Bowl of the former Dimension containing orderly all the Constellations and remarkable Stars and therefore if you know the Hour it will compose the said Bowl or Globe and so represent the then position of the Heavens but tho you are Ignorant of the Hour if you see a known Star and move the Bowl on its Axis till the painted star on it lyes just between your Eye and the Real one you have the Hour and consequently may know the Globe being now Compos'd any Star or Constellation above the Horizon for the Axis of this Bowl having one end pointing directly to the North Pole and the other fixt in the Center of a Rundle containing on its Limb the Days of each Month fitted to the right Ascension of the Stars and moving also on a Plane divided into 24 equal parts figured with the hours of a Natural Day 't will follow that the Day of the Month when the Globe is Compos'd must lye on the true Hour as the true Hour move'd to the Day of the Month must Compose the Globe as is before hinted These short directions are sufficient for any Mathematician or Instrument-Maker and as for the Branch it self 't is as you see not in its true Place for the above mentioned Reason J. Moxon To the Reader HAving Courteous Reader engaged to show you the Problems and Operations on the Sector which the Noble Author supposes every one that studies the Geometrical way of Dialling to know I
shall here begin I. Upon a Line given AB to erect CD a Perpendicular IF there be a Point as C given in AB the Line on which the Perpendicular is to fall Mark on both sides of the said Point with your Compass the equidistant Points M and N then opening them at pleasure put one foot on M and describe the blind Arch EF and putting the other Foot in N describe the blind Arch GH and the fair line from D their Intersection to the Point C will be the Perpendicular requir'd Now if you have no Point assign'd in the said Line AB to terminate your Perpendicular by take two Points there at pleasure as suppose M and N and opening how you will your Compasses describe the blind Arches EF and GH above your Line and OP and QR below it and the Intersections of these Arches to wit D and S will be two points to draw your Perpendicular by II. Upon C the end of AC a given Line to draw DC a Perpendicular OPEN your Compasses at a convenient width and putting one Foot on C let the other within reach of AC mark any where as at F then touching or cutting from thence the said AC with the moving Foot of your Compasses at suppose E and describing on the other side of F the blind Arch GH lay your Ruler on FE and it will cut the said Arch at suppose D so that DC will be the requir'd Perpendicular III. A Line AB being given how to draw DG a Parallel to it HAVING taken two points in the said Line as suppose A and B open your Compasses at what width you please and putting one foot on A describe the blind Arch CDE and putting one foot on B describe the blind Arch FGH then if you lay your Ruler on the highest part or greatest Extuberancy of the said Arches to wit on the Points D and G the Line so drawn will be the requir'd Parallel IV. To describe a true Square AB being a Line as long as the side of the Square you design erect on the end A the Perpendicular DA of the former length then taking between your Compasses the said AB put one foot on D and describe the blind arch EF and again putting one foot on B describe the blind arch GH to cut EF and if from their Intersection C you draw the fair lines CB and CD you have a true Square V. To draw an Oblong or as they commonby call it a Long Square AB being the longest side of this Square erect on the end A the Perpedicular DA of the length of the shortest then taking between your Compasses the line AB put one foot on D and describe the blind arch EF and taking between your Compasses the line AD describe the blind arch GH to cut the said EF and if from their Intersection C you draw the fair lines CB and CD you have the Square you design VI. To Describe an equilateral Triangle or an Isosceles OPEN your Compasses at AB being the side of the Triangle you design and putting one foot on A describe the blind Arch EF and again putting one foot on B describe the blind Arch GH to cut the said EF and if from their Intersection C you draw the fair lines CA and CB you have a true equilateral Triangle Nor is there any difference in the Description of the Isosceles ASB for the only difference between them is that the sides AS and BS of the Isosceles are longer or if you please they may be shorter than the Base AB whenas all three sides are equal in the equilateral Triangle VII To make a Triangle of three given Lines SUPPOSE the first line given be AB the second AC the third BC and that you are to make a Triangle of them let AB be the Base and taking the given line AC between your Compasses put one foot on the Base at A and describe the Blind Arch EF then taking the given line BC between your compasses put one foot on the Base at B and describe the Blind Arch GH to cut the said Arch EF and if you draw lines from their Intersection at C to A and B on the aforesaid Base you have your intent VIII To describe an Oval CROSS RP at right Angles with IM and taking with your Compasses on the said lines from the intersection O equal distances to wit OA OB OC and OD and draw through the point C the lines AK and BH each equal to twice AC as also throu ' D the lines AN and BL each equal to twice BC then A and B being Centers describe the Arches KPM and HLR in like manner C and D being Centers describe the Arches HIK and LMN and the figure thus drawn will be a perfect Oval So much for the Geometrical Problems necessary for Dialling and as for the Instrumental ones i. e. those performed by the Sector they are as I may say of two sorts some belonging to one side of it and some to the other for the side marked with L is divided into 100 equal parts and called the LINE of LINES and the side mark'd with S the LINE of SINES First then of the LINE of LINES which by the way tho' it be divided as I said but into 100 parts may yet stand for 1000 if you fancy every 10 Divisions a Line of 100 parts and in like manner it will stand for 10000 parts if every division be deemed 100 therefore a Line v. g. of 75 equal parts may be exprest by 75 of those Divisions or by 7½ or by ¾ The Use of the LINE of LINES marked with L. I. To divide a Line into any number of equal parts SUPPOSE your Line were to be divided in 23 equal parts take it between your Compasses and opening your Sector place one foot of your said Compasses on the 23 division of the Sector and the other foot on the 23 over against it and the distance between the Figures 1 and 1 on the said Sector will give you one equal Division of your Line and the distance between 2 and 2 will give you two equal Divisions of it and in this manner proceed till you quite run over it as you design II. To find the proportion between any two Lines SET over the greater Line at 100 and 100 on the Sector then taking the lesser between your Compasses find where it will be just set over also or lye parallel to the former which hapning suppose at 50 and 50 you may conclude that the Proportion required is as 100 to 50. III. To divide a Line as any other Line proposed is divided that is to say according to any Proportion SUppose you saw a Line containing 65 equal parts of the Sector devided into three pieces the first containing five equal parts of the Sector the other fifteen so that the last must be 45 then suppose you would divide after this proportion another Line containing but thirteen equal
Word is Rendred the Nature of Things signified Discussed and where Need requires Illustrated with apt Figures and Diagrams With an Appendix exactly containing the Quantities of all sorts of Weights and Measures The Characters and meaning of the Marks Symbols or Abbreviations commonly used in Algebra And sundry other Observables By Joseph Moxon Price 2s 6d The English Globe invented by the Right Honourable the Earl of Castlemain and of which this Book shews the use containing about a Foot in Diameter are made by Joseph Moxon Price ordinary made up 40s and with the Projection described in Section 6. of this Book Price 50s At the place aforesaid you may also have all manner of Maps Sea-Plats Drafts Mathematical Books Instruments c. at the lowest Prizes FINIS * pag. 24. * p. 73. †p. 80. * p. 82. †p. 85. Of the Circles describ'd on the Globe The 4 Cardinal points of the Globe * vid. Oper 2. 5. in Sect. 2. What the Operations of the Globe are perform'd with A Memorandum How the Treatise is divided The first way A Memorandum The second way The Reason and Demonstration of the Operation The first way The Reason and Domonstration of the operation How much the Sun illuminates more than half the Earth How to know the terms of the shade of Extuberancy when the sun shines faintly The second Way The Third way To know at any time whether it be Forenoon or Afternoon * Operat 1. pag. 4. A way to Compose the Globe by the Sun * Operat 2. pag. 7. A Memorandum The first way of Composing the Globe The Demonstration The 2d way vid. Op. 10. The 1. way The Second way vid. Op. 10. The first way * Operat 3. pag. 8. The second way The Third way * vid. Oper. 2. pag. 5. A Memorandum The 4th way The first way The 2. way * Op. 2. pag. 6. †Op. 5. pag. 10 A Memorandum * Op. 3. pag. 8. The 1st way The 2. way The 3d. way * 2. pag. 6. The 4th way A Memorandum * vid. the particulars in the conclusion or last Chapter The 2d way of composing the Globe by the shade Demonstration * Op. 3. pag. 8 The 3d. way of finding the day of the month * Op. 2. pag. 6 Op. 5. pag. 10. To find when and at what declension the Sun rises or sets earlier or later accord * Op. 6. pag. 11. * Vid. Oper. 13. Sect. 2. Preliminary Considerations The grand Divisions of the Earth The Boundary between Europ and Asia The Division of each modern Country from the other The Ancient Limits of several Nations A Table of reducing Degrees into Miles What the Latitude of a Place is and how to find it What the Longitude is Of the Grand Meridian Of the most noted Places where Author 's have plac't the grand Meridian Where we fix our Grand Meridian How to find the Longitude of any place A Memorandum A preliminary Discourse of Climes What a Clime is What a Parallel is Of the Antiquity and number of Climes Of the 7 common Northern Climes Of the 7 Southern Climes Why the middle of the first Clime has 13. hours of day How the first Circle of Longitude is divided as to the Climes To find in what Clime any Place lies Of the inequality of the Climes * pag. 23. Of the 5 Zones Of the bounds of the Torrid Zone which contains the Amphiscii To find when the shade changes side here Of the bounds of the frozen Zones which contain the Periscii Of the bounds of the Temperate Zones which contain the Heteroscij First way Second way Third way Of the Periaeci Of the Antaeci Where they have no Night and where no Day When 't will be perpetual Day or Night at any Place * Op. 6. pag. 11. Where 't is Dinner-time all the World over Where 't is the time of Rising all the World over Where 't is Supper time all the World over Where 't is Bed-time all the World over The Reason or Demonstration of the Operation * Oper. 10 sec 1. pag. 14. To find the Sun's height in any place The Reason of the Operation To find the Sun's Depression To find all the Places that have the Sun at the same height How Astronomers begin their Computation of Time How the Italians How the Babilonians To find the Babilonish and Italian hour when the sun is in the Aequator * Op. 10. sec 1. pag. 14. * Op. 10. sec 1. pag. 14. To find the Italian Hour when the Sun is in the Aequator To find the hour both the said ways at any time * Op. 18. p. 19 A most ready way of finding at any time the Babilonian and Italian Hour all the world over Of the Judaic way of Computing time A most ready way to find the Judaic Hour Why the days of the Week being called by the Names of the Planets follow not each other after the order of the Planets * ♄ Saturn ♃ Jupiter ♂ Mars ☉ Sol ♀ Venus ☿ Mercury ☽ Luna The Advantage in reckoning the Italian way The Advantage in reckoning the Babylonian way Of the Parallel Sphere Oblique Sphere All Positions taking the year round enjoy an equal share of the Sun's presence * Vid. Op. 3. sec 1. pag. 8. The Demonstration How the Earth is prov'd Round The Demonstration * pag. 5. * pag 10 * pag. 11 * vid. pag. 8. 15 A Memoran * p. 12. How you are to operate A Memorandum An Example Two Memorandums The reason or demonstration of the Operation A memorandum Why 6 hours must be added sometime to the Tables * p. 49. * p. 16. * pag. 49. * Op. 7. sec 2. pag. 33. * Vid. Op. 1. 2. pag. 49. A Corollary An Example A memorandum * Sect. 1. Op. 2. p. 5. *  1. 2. 3. 4. 6.  12. 11. 10. 9. 8. 7. 6. 87. 93. 110. 140. 200. 300. 625. * Op. 2. Sect. 1. pag. 6. A preliminary Discourse * pag. 4. †pag. 5. * pag. 10 * ☞ Because every body that desires to know these and the following Problems has not perchance at hand Mr. Gunter's Book I shall add them to this Treatise as the Reader will find at the end of it J. Moxon How to find the Tangent and Secant of any degree Demonstration A Memorandum The Construction An Example How to draw the half hours quarters c. The Construction Demonstration * pag. 71. The Demonstration * pag. 4. A Memorandum * pag. 8. * pag. 73. The Construction The construction * pag. 79. How to draw a Line Paralel to the Horizon and how to place truly the draught on its Plane An easier way how to place any paper draught on its Plane A Memorandum Demonstration A Memorandum Some few things to be premis'd The Construction of an East Dial. Of a West Dial. Of the Stile and Substilar The Demonstration The Construction The Demonstration The reason of the
unequal distance of these hour-hour-lines What a Declining Plane is The Construction * pag. 80. To describe the Morning hours of a Declining Dial. To describe the Afternoon Hours How to make the Stile and Substilar of a Declining Dial. A Memorandum The Construction The Demonstration of these 2 declining Dials A Memorandum * pag. 84. Another Demonstration * pag. 83. * pag. 73. The construction * pag. 23. * pag 8. The Construction A ready way to find the Stile and Substile of a declining Dial. Demonstration * pag. 92. The Demonstration of the Stile and Substilar * pag. 82. * vid. pag 104. * vid. pag 105. The Construction and Demonstration How to make an Horizontal Plane an Aequinoctial one The Demonstration The Geometrical Construction A Memorandum * pag. 8. The Construction The Construction and Demonstration of a declining direct Dial facing the South A Direct North reclining Plane * pag. 89. How to describe the Plane of this Reclining Dial on the Globe The Construction A Memorandum Of the Stile and Substilar * pag. 82. Another Demonstration The construction * pag. 2. First way * pag. 4. * pag. 4. The second way The first part of the Operation * pag. 89. The second part of the Operation Why every Erect Stile or perpendicular show's not always the true Hour The Construction * pag. 89. The Demonstration * pag. 94. * pag. 75. Demonstration The Advantage or use of this Dial. The Construction Demonstration Another Dial of the same nature The construction The Demonstration The Construction Demonstration The Construction The Geometrical way The Demonstration The Construction The Demonstration * pag. 73. How this Dial is to be made when the windows lye not Southward The Construction of it as to the Hour at home * pag. 73. The Construction of it as to the Hour in other places To find the Suns place and day of the Month. To find the Rising and Setting of the Sun To find the Suns Amplitude To find the Height of the Sun To find the Suns Azimuth and Bearing To find the Proportion of Perpendiculars to their Shades The description of the Branches or Embellishments in Sheme 43. * p. 111. †p. 112. * vid. p. 113. * p. 71. Of the Sector * Op. 17. Sect. 1. pag. 19. A Memorandum * p. 6. * p. 68. * Op. 5. way 2. p. 10. A Memorandum Of the first Plane and its bigness Of the Circles and Stars on it and how they are placed Of the second Plane and it's bigness * p. 6. †p. 68. * p. 132. Two Memorandums The Principle on which this Projection depends Of the Concentric Circles The general Rule for projecting the great Oblique Circles Of projecting the Ecliptic Of projecting the Horizon Of projecting the primary Vertical Of projecting the rest of the Azimuths An expedite way of finding the said Centers and Radius Lemma 1. * Eucl. 27. 1 †6. 1. Lemma 2. †Eucl. 27. 1. * 6. 1. The Ecliptic truly projected A Memorandum The way of describing G. Frisius's Meridians The way of describing G. F. hia Parallels How to describe the Circles of Altitude on the 2 Plane * p. 138 and 139. * p. ◠First kind * pag. 4. ‡ p 8. 15. * pag. 13. ‡ p. 31. * pag. 36. ‡ pag. 39. * p. 40. * p. 35. * p. 5. ‡ p. 37. * p. 10. ‡ p. 11. * p. 21. ‡ p. 51. * p. 65. * p. 70. A Memorandum * p. 8. 2d kind * p. 18. * p. 9. 3 kind * p. 14. * p. 15. * p. 35. * p. 13. * p. 11. * p. 10. 12 * p. 16. * p. 17. * p. 19. * p. 17. * p. 33. * p. 33. * p. 34. * p. 36. * p. 31. 4th kind