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
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-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
is the Tangent Complement of FS or NR the Stiles height above the Plane And here also you see that the very same Dial the figures only transpos'd will serve both for an Horizontal and this Direct Vertical one to those that live in the Latitude of 45 Degrees since the Elevation of the Pole and Complement of it is there the same OPERATION X. To describe by the Globe Meridian Dials or as others call them East or West Dials for the Elevation of London THese Dials tho' Vertical and Direct as passing thro' our Zenith and facing also two Cardinal Points or Quarters of the World are very different from the former nor has any body I believe taught yet their Description by the Globe To perform therefore this Operation you must by the help of your String or Compasses describe on your Globe with Chalk or the like matter an Arch as in Sch. 11. which having its Pole at K the East-point for examples sake of the Aequinoctial cuts somewhere or other the 11 a Clock Northern hour Circle I mean the 11 a Clock hour Circle on the Northern or black part of the Globe and this Arch by reaching from the point C in the Aequinoctial Colure or 6 a Clock Circle to H in the Horizon on the said Northern side of the Globe will be a piece of a little Circle parallel to the Meridian containing the Degrees of the Elevation of the Pole and cutting all the Hour-Circles also from 6 to 11. But if this be thought too troublesom a work the Globe-maker may avoid it by putting 6 Pricks or Asterisks upon the Globe where the said Arch and Hour-Circles would intersect as may be seen in the said 11 Scheme at C O S T V and Z so that if beyond C he adds one prick more viz. at R to give you from H the Radius or 60 Degrees of the said Arch you need nothing else This being premis'd describe on a sheet of paper HR or 60 degrees of the said Arch being Radius a blind Circle as in Sch. 12 and drawing the Line H h how you please throu ' K its Center to represent the intersection of the Horizon open your Compasses to the said Arches full extent to wit from H to C and putting one foot on the blind Circle at H and the other marking there at C draw the line PC π throu ' the Center K and 't will represent the intersection of the Aequinoctial Colure or 6 a Clock hour Circle with your said blind Circle or Plane so that if you take from off your Globe the distances between the point C and the several Intersections of the Hour Circles with the said Arch CH and place them on your blind Circle on the right hand side of PC π as well below the Horizon H h as above it and draw lines thro' them viz. O ο S σ T τ V υ and Z ζ you will have a compleat East Dial describ'd after you have drawn 2 lines more on the left side of the said C π to wit the Line N ν distant from it as is O ο and the Line M μ as is S σ. As for the figuring each hour line it must be according to the Figures of the corresponding Hour-Circles cut by the aforesaid Arch CH and thus you will find them figured in the forementioned Scheme 12 which shews you too how the Borders or Parallels are drawn for the said Figures to lye in as being only double Lines equidistant at pleasure on both sides of the Horizon H h and here also by the blind Lines and by the fair ones you have before your Eyes what is necessary to be exprest on your fair Plane and what not Nor is there any difference in the Construction of a West-Dial except it be in turning on your draught the hour-Hour-Lines or Parallels the other way to the end they may all point Northwards on their respective Planes for thus in Sch. 11. do the Prick Lines m 8 n 7 c 6 o 5 s 4 t 3 u 2 and z 1. which would truly represent this Dial if they were produced in the said Scheme Now for the Substilar 't is the 6 a Clock Hour Line since that Hour Circle falls on the Plane at right Angles and as for the Cock it may be a Gallows Stile as in Scheme 13 or a Pin as in Scheme 14 so it be plac't on the Substilar and perpendicular to it having its height equal to the Distance between the Pricks or Asterisks C and P in the said 11 Scheme or which is all one to the distance between K and X. viz. the nearest distance between the Substilar and the 9 a Clock hour line in an East-Dial and the Substilar and the 3 a Clock Line in a West Dial. But here you are to remember that when I say that the height of the Stile is to be equal to the distance between C and P. I mean in rigour equal to the Sine and not the Chord of that Arch but seeing the Chord of 10 Degrees differs not sensibly from the Sine and by the way the Arch CP on the Globe will not be above 10 Degrees from the Meridian the interval between C and P will serve the Turn But if you would be more exact take between your Compasses the distance of double CP to wit the interval of suppose 20 degrees and half of it is the required distance for half the Chord of 20 Deg. is equal to the Sine of 10. Or if you please you may erect a needle at C Paralel to P the elevated Pole of the Globe and the distance between them will be the true Height of your Stile To Conclude You may contract and enlarge these Dials as you please by drawing the hour-lines twice or thrice or according to any other proportion nearer or farther asunder and so abateing or heightning in the like manner your Stile The Demonstration is obvious for since the points M N C O S T V and Z in the upper part of the blindCircle or Plane and the Points μ ν π ο σ τ ◠ζ on the lower part of it are by being equal in distance to those on the Arch the intersections of the morning hour Circles of 4 5 6 7 8 9 10 11 with the edges of the said Plane it follows that the Lines drawn from the corresponding Points must be the true hour lines of this Dial since the hour Lines as we said of all Dials are only the Intersections of the respective hour Circles with the Plane Again the shade of the Axis the Axis being a part of all the hour Circles falls ever on the Hour-Line or Interfection of this or that Hour Circle as often as the Sun comes into the Plane of that Hour-Circle therefore the Stile of this Dial representing truly the Axis since 't is above the Plane and distant from it as 't is on the Globe will cast its Shade every hour on the
Hour Circle i. e. that which corresponds with the time of the Day would be just interpos'd between your Eye and the Sun but since the whole Axis is the common Section of the Hour-Circles let your Eye be but in any part of it the same Interposition must happen so that seeing the Suns Ray by reason of the Blew Colour penetrates not the Glass his Picture must needs be on the outside of it where the said Ray would have otherways past Now the Ray that goes from your Eye throu ' the two Holes being the Axis therefore whilst your Eye remains in this Posture it will follow that wheresoever you see the Suns Picture on the Glass there his place must be and consequently his said Picture must show the Hour OPERATION XXXI How to make an Horizontal Concave Dial by the Globe and Geometrically also COmpose so your Globe in the Concavity given suppose BAC in Scheme 37. that A the Center of the said concavity shall concurr with the Center of the said Globe then drawing your String over each necessary hour Circle on the Globe to the sides of the Concavity mark as many Points as shall be convenient for the Describing the corresponding hour Circles and the Pin AD erected in the Nadir at D as high as the said Center A I mean a Pin equal to the Semi-diameter of the Concavity will with its Top always show you the hour Tho the former way be impracticable when the Hole is less than the Globe yet it serves to illustrate and make easy the Geometrical Operation for you have nothing you see to do but to draw hour Circles within as you would without were the said Concavity a whole Sphere and then the Top of its Semi-Diameter i. e. the poynt which lyes in the Center A will perform the Stiles part for since the Sun is every Hour as we have before showd you in the same Plane of the true hour Circle and since A the Top of the Semi-Diameter being in the Center of the Concavity is part of the Axis or Common Section of all the Hour-Circles it follows that its Shadow must fall on the true Hour OPERATION XXXII How to describe Geometrically a Cieling Dial. SEeing the Glass which reflects the Suns Rayes to show us the Hour is commonly fixt in the corners and by-places of Windows the Globe can seldom be so well order'd by reason of its Bulk as to help us in the Construction of this Dial therefore I shall only give you the Geometrical way which is as I take it both short and new and because these Dials have commonly the Windows or inlets for the Sun Southerly for otherwise they will show but very few hours we 'l suppose ours also in the following Example to stand thus and afterwards you shall see the difference between such a Dial and those whose Windows have another Aspect First make on any Past-board Trencher c. an Horizontal Dial as in Scheme 38. and fix in O its Center a Thred of a good Length to wit OP then fasten the said Dial so with a Nail to a Long Masons Ruler that its Fiducial edge KL may lye upon the Meridian or 12 a Clock Line and having cemented and plac't Level a piece of Looking Glass of the bigness of a Three pence in the Window or what convenient place else you please of your Chamber which we 'l suppose to be G find by the Plumet AE the Poynt A in the Cieling WXYZ being the poynt in Scheme 39. directly over the said G and draw throu ' it a Meridian line viz. the Line AL. In the next place fix one end of a piece of Packthred on G the Center of the Glass and the other on some point of your Meridian line in such manner that it make an Angle with it of 51. 30′ i. e. the Angle of the Elevation which may be easily perform'd by the application of the side of a Quadrant to the said extended Packthread and when 't is right let the Point thus found in your Meridian line be called B. Lastly take the distance between the aforesaid Points A and B and marking it suppose at C on the edge of your Ruler from O the Center or fastning of the Horizontal place so the said Rulers Fiducial edge KCL along the Meridian line on the Cieling that the point C may lye just on A and all is done for then if you draw but the Thred OP streight over each Hour-line of the Horizontal it shows you where you are to draw all the fair Lines of the required Dial. Sch. 40. As for the truth of this Dial it appears in Scheme 40. by the right Angle Triangles OGH and GHF where HF is part of HM a suppos'd Meridian line on the Floor under that in the Cieling G the Station of the Glass in the Window H the Point under the said Station as formerly A was the Point over it and to facilitate the Demonstration let us imagine GH equal to GA i. e. that the Glass lyes in the middle between the Floor and Cieling This being so suppose that GH instead of representing a Perpendicular Line in the Wall as here we conceive it had been a Perpendicular Stick and that you were to describe an Horizontal Dial on the Floor whose Stile was to be the said Stick I say supposing this you must you know to perform the Operation produce the Meridian Line MH to suppose N and fastning a String on G find in it the Point v. g. O for the Center of the Dial I mean a Point to which a String being extended from G makes with the Meridian OH the Angle of the Elevation and so draw the several Hour-lines from the said O according to their respective Angles and Distances all which is exprest at large in the third Scheme or first Horizontal Dial for there you see GH is a Perpendicular Stile showing the Hour with its top and that O is the Center of the Dial having a Line drawn to it from G making the Angle of the Elevation with the Meridian OH Now since O in our present case is a point without the Chamber and consequently the Line MH cannot be produc'd to it you must draw your Thred from G to the said Meridian Line HM within the Chamber and find in it the Point F to wit the Point where the said Thred GF makes with it an Angle equal to that of the Elevation for thereby you will have the distance of O your true Center from H as being the distance of F from H seeing the side GH is common and the Angles in both Triangles equal This being so if you put out of the Chamber an Horizontal Dial whose Center shall lye on O and its Meridian Line concurr with HF 't is but producing all its Hour-Lines on the Floor and it must necessarily follow that G the Top of the Perpendicular Stile will show you truly the time of
level to the Sun do but move it till the shade of the said Needle or Pin falls directly along the Diurnal Parallel where 't is placed or if it be not placed in any of the said Parallels move the Globe till the shade falls parallel to the next Diurnal Parallel and 't will be as truly Compos'd as before supposing you know as we have already taught you whether it be Forenoon or Afternoon when you operate for as in the Morning the Stiles of Dials cast their shades Westward and in the Afternoon Eastward so must your Needle or Pin do when the Globe is Compos'd But here the Reader must take notice that in case the shade of the Needle or Pin will by no means fall sensibly parallel but as you move the Globe draws nearer and nearer its being so till at last it shortens to nothing then the Sun is exactly South and consequently your Globe is compos'd as soon as the shade thus vanishes Now Because the shadow of the Pin is on the Globe an Arch of a Great Circle this way of Composing the Globe cannot be accounted Mathematically true For as the Sun approaches each Tropick and the Tropicks not Great Circles it will happen Mornings and Evenings when the Pin projects long shadows that the shadow of the Pin will not ly exactly in the Parallel of the Day but will more or less intersect it in the Center or Pin-hole Therefore tho' the aforesaid way of Composing the Globe be true enough for ordinary uses yet I shall give you two other waies without exception Observe the Concentrics between the North Pole and its Polar Circle and first you will find that they are equal in number to the Parallels either from the Equator to the Tropick of Cancer or to those from the said Aequator to Capricorn for to avoid the confusion of too many Parallels there are usually but 8 Northern and 8 Southern described on the Globe 2ly That they are distant from the Pole as the said Parallels are from the Equator And 3ly That they are markt not only with the Daies of the Month of the Northern Parallels but with those of the Southern also The Day of the Month then being for example sake Apr. 10. Move but the Globe when level till the shade of Extuberancy touches the Concentric markt Apr. 10. and 't will be truly Composed supposing that the Eastern face of the Globe looks towards the Forenoon or Eastern parts of Heaven and the Western face towards the Afternoon In like manner If the Day of the Month or Suns Parallel be an imaginary one between any two that are exprest for to avoid as I mention'd the confusion of too many Parallels there are usually but 8 Northern and 8 Southern described I say in like manner If the Day of the Month or Suns Parallel happens thus let the said Shade but touch or fall proportionably between the correspondent Concentrics and the Globe will be Compos'd as before The reason of the Operation is this The Sun illuminating as has been said half the Globe the Shade of Extuberancy or in other terms the Confines between the Obscure and Illuminated parts will be still 90 degrees from the point or place where the Sun is vertical therefore if the Sun be v. g. in the Equator the aforesaid Shade or Illumination must terminate in the Poles of the World and when he is in the Parallel of Ap. 10. the Illumination must fall short of the South Pole and go beyond the North Pole as many degrees as the said Parallel declines from the Equator But the Concentric of Ap. 10. is by Construction just distant from the Pole those degrees Ergo when the said shade of Extuberancy or the Illumination touches this Concentric the Globe must if its Eastern face looks towards the Fore-noon part of Heaven or the Western the Afternoon be illuminated as the Earth is and consequently Compos'd for its corresponding with the Earth in its site and position is all we mean by Composing As for the reason why I mark each Concentric with the 4 opposite Months whereas the Parallels are markt only with 2 of them 't is that the Globe may be Composed by the help of the Northern Concentrics even when the Sun is in his Southern Declension it being more convenient and ready for one to cast his Ey on the North Pole than to stoop to the South Pole about which otherwise there must have been the like number of Concentrics and markt as the Southern Parallels are I say this is the reason of thus marking the Concentrics for since the Sun in its Northern declension illuminates beyond this Pole he must in his Southern fall proportionably short of it therefore move the Globe as before let it be Summer or Winter or any other time of the Year till the said Illumination or Shade touch the Concentric markt with the day of the Month and 't will be still Composed The second way I shall defer to Operat 10. because the intermediate ones conduce much to the facilitating it as you 'l see OPERATION IV. To find the Day of the Month. THis Operation is also perform'd two ways as being the Converse of the former therefore since that requires the knowledge of the Day of the Month this must require the Globe Compos'd Having then Compos'd it by a Meridian line or otherwise Consider upon what Excentric or between which of them the said Shade of Extuberancy or Illumination falls and that will shew the Day of the Month. As for the second way you shall have it when we come to Operat X. which treats as we said of the Second way of Composing the Globe OPERATION V. To find the Sun's Azimuth THe Sun's Azimuth is an Arch of a great Circle which passeth through the Zenith and Nadir over his body so that his Mornings or Afternoons distance reckon'd by the Degrees of the Horizon from the Meridian or Southern Cardinal section of the Globe is the thing requir'd and for performing the Operation there are four several ways Compose your Globe Then standing on the illuminated side or side next the Sun and fixing your String by its nooze in the Zenith hold it up by the plummet-Plummet-end and move it along till its Shade falls on the middle of the Fulcrum or supporting Pillar or to be more exact till it covers the Center of the Projection being the point you see directly answering the Nadir for then the Degree in the Horizon which the said Shade falls upon gives from the above mentioned Meridian the requir'd Azimuth Or else guide your String by winking or by any other convenient means which practice will show you till it concur with the Shade of the Zenith-pin that is to say till they both ly in the same Plane for then the Shade of the String it self if it hangs strit along the Globe will cut the Horizon as before In case you have
their Fame among the Moderns they were Dia-Meroes Dia-Syenes Dia-Alexandrias Dia-Rhodou Dia-Romes Dia-Boristheneos and Dia-Riphoeon being all names made by the Addition of the Greek Preposition 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 i. e. per to some remarkable Town River or Place thro' which the middle of each Clime past so that the middle of the first went thro' Meroe an Ethiopian City on the Nile where according to some Queen Candace Reigned the second thro' Syene in Egypt lying just under the Tropic the third thro' Alexandria the fourth thro' the Isle of Rhodes the fifth thro' Rome the sixth thro' the mouth of Boristhenes now called Nieper by the Cossacks and the other Inhabitants and the seventh and last thro' the Riphoean Hills part of which lay according to their account in or about the Latitude of 50 Degrees and consequently corresponded with the Cimerians 'T was here then that Alfraganus and other Arabians ended Northwards who besides several smal particulars err'd not a little in making Rome and the Boristhenes only a Clime asunder when as their longest days differ at least an hour And as for the Southern Climes to wit those on the other side of the Aequinoctial they thought fit to consider them but not knowing what to call them as being ignorant for the most part of the Places they went through they added 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 i. e. Contra to the former Denominations so that making Anti-dia Meroes serve for the first Clime Anti-dia Sienes for the second they proceeded in the same order with the Rest But now before I end I shall endeavour to solve a difficulty which startles not a few viz. how it comes to pass seeing the Climes are assigned as we mentioned by the Antients to know the length of the Summer Solstitial day in every Country that the middle of the first Clime which in rigour should lye no further from the Aequator than to encrease the day a quarter of an hour runs over Meroe where the Excess is at least an hour I answer the Antients deeming it more equal that the middle of the Clime and not the end of it should be the Point where the half hourly increment was to begin fixt the Terme à quo not in the Aequator but a quarter of an hour further and therefore Taprobane which some now think Sumatra was the place where Ptolemy commences all his Climes making thereby the middle of his first to pass per Sinum Avalitum or Mouth of the Red Sea and the middle of his second per Meroen But the Arabians thinking that for several Degrees from the Aequator all was either Sea or by reason of the Heats scarce Habitable or else judging it for their Honour to have their own Country in the first Clime began half an hour beyond Taprobane and so Dia Meroes tho the Days are there 13 hours long leads the Van in their Catalogue These few things premis'd I shall now shew you the way I take therein which I think in all respects clear and ready First I make the primary Circle of Longitude to be the Circle particularly appropriated to this use being devided and mark't according to the true distance of each Clime from the other and as to the place where they commence on our Globe I rather follow Ptolomies Astronomical than Geographical Method for besides the aforementioned excess of the Arabians should we begin but a quarter of an hour from the Aequator it makes a great space of the Earth viz. from Taprobane to the Aequator to be in no Clime at all and which is more it causes a little confusion when the length of the day is greater in every Clime than what the said Clime can justly challenge according to its Rank and Number I say as for the place where the Climes commence I rather follow Ptolomies Astronomical than Geographical way and therefore beginning at the very Aequator my first Parallel or middle of my first Clime is supposed to run over the places that enjoy 12. hours and a quarter of Day and the end of it noted on the primary Circle of Longitude or 2 a Clock Hour Circle with the Figure I. over the places that have 12. and 1 2 and thus we proceed to the Polar Circles to wit where the 24th Clime or 48th Parallel terminates so that from thence we come to the Devisions on the said Circle of Longitude which show where the days are as long as an ordinary Week where as long as a Month and where as two arriving at last at the Poles themselves where there is a constant half year of light and as much of Darkness And to give you a Remembrance of the Names of the aforesaid old Climes and that you may also see without Calculation or Trouble where the Ancients plac'd them I have set down the first Syllable of their names as Mer. Sy. Al. c. according to their respective Latitudes To find then in what Clime any place is v. g. Constantinople you are only to draw your String from the Pole over that City and mounting up the Bead thither to move it to the said Primary Circle of Longitude and 't will lye on the Clime or Paralel required But if you would know what places are suppose under the 4th Clime throu'out the World i. e. what places have their longest day just 14. hours Fix the Bead on the 4th Clime and moving it on its Noose from the Pole round the Globe you may conclude that every place it passes over has the Sun exactly so long above the Horizon when the days are at the longest and in the same manner you must proceed on the South of the Aequator to find the Countrys that lye under the 4th Southern Clime In short here we have besides what has been already said a view not onely of the strange inequallity of the Climes especially between the first and last but also of their exact distance in Degrees and consequently in Miles by help of our Table of Reduction mentioned in the first Operation of this Section But seeing we are a little fallen into Speculation 't will not be perchance improper to proceed yet further and to consider here as in a natural and fit place the Bounds and Terms of the five Zones so called from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Cingulum as enclosing the whole World within their respective Districts 'T is with the Torrid one we 'l then begin whose Bounds are the two Tropics so that the Diurnal Parallels not only remarkably distinguish it from the other Zones but shew why the several Inhabitants within this space were called by the Ancients AMPHISCII i. e. Vtrinque umbrati or men that had two shadows from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 utrinque 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Vmbra nay by the said Parallels you may find when the shade will change and be different For since by these Paths or Traces the Sun as we often hinted
must know our selves and so are all other People and Places of the World that are in neither of the two former ones for take any point not under the Poles or the Aequator for your Zenith and 't will be impossible to describe an Horizon or Circle 90 Degrees from it which cuts not the Aequator and all its Parallels obliquely 'T is this Obliquity then that gives name to the Position and 't is this that makes the great inequalities in days and nights for if the Horizon has a greater portion of one Diurnal Parallel above it than of another as it must needs have by its slanting 't will follow when the Sun is in such a Parallel that the Day will be longer than when the portion was less and consequently since more of one Parallel is under the Horizon than of another that one Night is shorter than another and seeing the nearer the Pole is to the Horizon the more equally it cuts the said Parallels and the further it is from it the greater the inequality happens to be 't is no wonder that by how much the greater the Elevation is by so much the longer the Days are and when the whole Horizon falls below some of the Parallels that then during the Sun's aboad there the Inhabitants have no night at all therefore it follows that if a Star be neerer the Pole than is the Latitude of a Place it can never set in that Place Yet notwithstanding this strange inequality and disproportion of Day and Night all People in all Positions by that time the Sun finishes his annual Course make them even and thereby enjoy an equal share of both for if under the Pole the Sun be six months above the Horizon he is as long under it and if we and the Rest that live in the Oblique Sphere have Summer Days of a mighty length our Winter Nights are of the same Dimension therefore it follows that at the long Run the Inhabitants under the Aequator or in the Right Sphere who have always 12 hours of Day and as much of Night cannot boast of having more of the Suns Company than they that live in the two other and consequently that the assertion is true 'T is in the Oblique Sphere then that the above-mentioned Brazen Horizon is chiefly intended but as I said in the beginning 't is forty to one so many Universal Operations being perform'd by the Globe in its set Posture that in 7 years a man lights on a Question that could invite him to change it were it moveable as other Globes are so that having show'd you that in case of Necessity it may be in effect altered even without stirring it from its Pedestal I shall proceed OPERATION XVI How to take the Elevation of the Pole in any place whatsoever SUppose you were in a strange Place and that your Globe being one that had bin fitted for London you desire to know the present Elevation Expose your Globe to the Sun on a Meridian Line with the Pin or Needle in the Hole on the Parallel of the 10 of April or true day of the Moneth and observing at 12 a clock when the Sun comes into the Plain of the Globes Meridian that the shade of the said Needle or Pin loses not it self as it would do were the Sun directly opposite to it for so it had hapn'd at London or in any place in the Latitude of 51 e 30′ I say having thus expos'd your Globe and observing this move your Pin or Needle from Hole to Hole or from one Degree of the Meridian to the other 'till it's shade be wholly lost and finding the said Needle or Pin on the Parallel suppose of June 11th which is about 11. 30′ higher then it 's proper place to wit the Parallel of the 10th of April you may conclude that your present Elevation is 63 degrees i. e. 11. 30′ higher than the Globe's whereas had you bin oblig'd to move your Needle or Pin so many Degrees lower than the 10th of April your Elevation had bin but 40. The Demonstration is obvious for since the Earth is round as nothing perchance proves it better than the Experience we have that as so many miles suppose 60 elevates or depresses the Pole one Degree so just 60 Miles more elevates or depresses it another I say since the World is round and that the Degrees of the Globe answer to its Degrees it must follow that the difference between the Pins situation now on the Globe and where it would have stood on it at London is the true difference of the two Elevations OPERATION XVII How to know in what Elevation the Sun Rises or Sets an hour or any other space of time earlier or later than he do's in the Globes Elevation IF the Sun rising at London on the 10th of April about 5 and setting about 7 you would know in what Elevation or Latitude he then rises for examples sake at 4 and sets at 8 take the distance of 90 Degrees with your String or Compasses in any great Circle and placing one end of your String or one foot of your Compasses where the Parallel of the day intersects with the Hour-Circle of either 4 in the morning or 8 at night observe where or at what point the other end of your said String or other foot of your said Compasses touches in the Meridian or 12 a Clock Circle of the Globe and you will find it to be at or about 8 Degrees and 30 Minutes beyond the Zenith towards the North Pole so that the Elevation required is greater than your own by those 8 Degrees and 30 minutes that is to say the Elevation is that of 60 or thereabout whereas had your String or Compasses touch't 8. ° 30 ′ on the other side of your Zenith the required Elevation would have been less than your own so many Degrees i. e. it would have been that of 43 Degrees or thereabout This appears true by placing your Brazen Horizon or by describing an imaginary one over the two points made by the Intersection of the Parallel of the Day and Hour-Circles of 4 in the morning and 8 in the evening for in the Elevation belonging to such an Horizon 't is evident that the Sun rises at 4 and sets at 8. Now the Pole of every Circle being 90 Degrees from it and the Point in the Meridian being 90 Degrees from the aforementioned Intersection it follows that the said Point in the Meridian is the Zenith or Pole of this new Horizon and consequently by being distant from the Aequator 60 Degrees that so many Degrees is the Latitude or Elevation required The END of the second Section SECT III. Of the Moon HAving now finish'd with the Sun wee 'l make a step if you please to the Moon and show you how to resolve all the useful ordinary Questions concerning her whether we see her by Night or by Day for 't is equal to us whether
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
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-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