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
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
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
I. II. III. c. as well where they fall upon the Polar Circles as the Aequator and tho all the Circles that thus cut the Aequator and Polar Circles at a Roman Figure be Circles of Longitude yet they now serve for true Hour Circles also since they are not here express'd and drawn according to the usual manner of Terrestrial Globes at the distance of 10 Degrees but of 15 asunder And here be pleas'd to remember that since there is a difference between the Roman Figures which belong to the same Circle for if it cuts for example at IIII. on the Polar Circles 't wil cut you see at six hours difference viz. at X in the Aequator the reason of it will appear by and by very plainly when we come to the Operations that concern these Circles of which the broadest passing through the Zenith and Nadir has two Quadrants gradually divided on one half of it the first called the Quadrant of Altitude reaching from the Pin Z or Zenith to the Horizon H the second called the Quadrant of Depression reaching from thence to the Nadir whereas on the other half or back part of the said Circle there is a single Quadrant only viz. from the Zenith to the Horizon which we shall for the future term the Quadrant of Proportion This Circle is also markt on the Polar Circles with the Figure XII representing thereby not only the 12 a Clock hour Circle or Meridian of the place for which the Globe is particularly design'd but the Colurus Solstitiorum also so that the Colorus Aequinoctiorum must be the 6 a Clock Circle whose half is as you see divided for several uses into Degrees from Pole to Pele By these two Circles then you have readily presented to you the 4 Cardinal Sections or Points of the Globe for as the Graduated half of the said Meridian shews the Globes Southern part or face and the opposite its Northern so the graduated half of the six a Clock Circle gives its Eastern and the plain side of it its Western Now for the Aequinoctial Parallels or Sun 's Track for every 10th day throughout the year for to avoid Confusion of Circles I describe no more they are distinguish'd by the Days of the Month when the Sun comes to them the uttermost of which are the two Tropics markt not only with the 11th of June and 11th of December but with ♋ and ♑ the usual Characters of Cancer and Capricorn Lastly for the Meridian of the World or first Circle of Longitude 't is markt with the Letter L and prickt also and tho in the present Longitude i. e. that of London it stands for the 2 a Clock Hour-Circle yet in its self 't is changeable as shall be shewn hereafter when we treat of its Properties and Divisions These are then the Circles here describ'd either common as I said to all Globes or particular to this and being well observed and remembred will much facilitate the ensuing Operations which are all naturally performed either by the shadow of the Sun and Moon alone or by the help of a small String hanging sometimes from the Pin P representing the North-Pole sometimes from the Pin Z representing as I said the Zenith and garnisht with a little Bead and Plumet according to its Figure in the Scheme aforesaid And here you are to take notice that tho the one end of the String be absolutely fastned to the Pole to prevent the loosing of it yet 't will serve for the Zenith as commodiously as if it always hung from thence for there is made at a convenient distance from the said fastned end a little Noose or Ring which as occasion requires is now to be over this Pin and now over that Nay if you give your String but half a turn about either of the Pins you will with a little Allowance as exactly perform your Operation as if you used the said Noose it self To conclude the whole Treatise is divided into six Sections The first solving several Questions that relate to the Sun in our Elevation The second many Geographical ones together with some that concern the Sun not only where we live but all the World over The third is of the Moon The 4th of the Proportion of Perpendiculars to their shades with some useful Corollaries thence arising The 5th of Dialling and the 6th of the Stars SECT I. Solving many questions relating to the Sun in our Elevation Operation I. To set the Globe level or parallel to the Horizon I Begin here because 't is what we first suppose done in most Operations especially in the nice ones nor is the performance difficult for we have nothing to do but to place the String and Plumet exactly upon the South side of the Meridian or 12 a Clock hour Circle and if it hangs just over the little Star on the Pedestal then the Plane where the Globe stands is Horizontal and Level otherwise 't is faulty as much as the Plummet varies from being Perpendicular to the said Star for the Star you must suppose is engraved by the Globe-maker there where he found the Plumet to hang upon his Placing the Globe truly level Let therefore the String and Plumet be always long enough to touch almost the Pedestal for thereby you may better perceive any Error and remember also that in case the said Pedestal to be less cumbersom be not as big as the Diameter of the Globe then there is to be under it a little wooden Ruler which being drawn out and markt with a Star will serve for this and several other uses as you will see anon There is another way speculatively true tho perchance not so exact in practice which is thus perform'd Place your Globe on your Plane with the String lying on the Meridian as before and if the Extuberancy or swelling of the Globe just touches and bears up the String at the Horizontal Circle then the Plane is Level or Parallel to the Horizon otherwise it differs as many degrees as are between the point where the said String touches the Globe and its Horizon The reason of this is That seeing the greatest and most extuberant Circle on a Globe is that which lies 90 degrees from its Pole the Horizon becomes here the greatest and most extuberant one that can be described from the Zenith therefore the Globe being on a Level which makes its Zenith to correspond with the Zenith in the Heavens the String cannot fall short of the Horizon because it must rest on the most extuberant Circle that occurs nor can it touch below it because the Plummet drawing the said String perpendicular from the greatest extuberancy hinders its bending and consequently its inclination to any part of the Globe beneath the Horizon Now if the Plane be not level then the Zenith of the Globe and Heavens not corresponding another Circle or part of the Globe instead of the Horizon must have the
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
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
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