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
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
represent the Illumination and the other the Obscurity you may perform this Operation at any time whether the aforesaid Luminaries shine or no. How easy therefore is it to conceive the whole Mistery of the Moons four principal Changes and what men mean by them For first we see that as She is call'd New by an Astronomer from her being with the Sun i. e. as fully between our Eye and the Sun as her then Course permits so no sooner has he found by their several motions that she is gotten 90 Degrees or six hours from the Sun but he says she is in her first Quarter and when they are asunder 180 Degrees or 12 Hours to wit as far as ever they can be that she is Full and lastly as soon as they are distant 270 Degrees or 18 hours on the same side and 90 Degrees or six hours on the other that she is in her last Quarter so that at their next meeting she becomes New again OPERATION XIV How to find how long the Moon wants of any Change or Cardinal Point and consequently how old she is I Propose not this Operation as a thing exact but seeing it is a Corollary of the former I thought fit to hint it therefore pray take it for better for worse and make of it what you can To resolve then these Questions by the Globe you are to expose it as before to the Moon when she shines and move about it till you can there just describe her shape and by the way you will come nearer the mark if you only consider the Lunular or lesser Portion whether it happen to be the obscure or the illuminated part of her whole Discus or Orbe I say describe her Shape on the Globe as neer as you can and observe how many Degrees the breadth of the Horn'd or Lunular Portion will be in any great Circle that crosses it in the middle at Right Angles and that will give you taliter qualiter what you seek for as appears more clearly by the ensuing Example Having observ'd suppose the illuminated Portion of the Moon to be Lunular expose your Globe and move about it 'till you perceive on it an illuminated Lunula proportionable to the Real one then finding its measure by some great Circle that crosses it at right Angles to be 40 Degrees these consequences will follow First if the Moon be in her Encrease she is past being New 40 Degrees i. e. three days and about seven hours seeing her hourly Elongation from the Sun is one time with another about half a Degree and half a minute but if she be in her Decrease she wants so many days and hours from being again New In the next Place it will happen that the obscure part of the Globe is 140 Degrees broad for both parts or portions making up the apparent Hemisphere the said obscur'd Part becomes the supplement of the former 40 Degrees so that 140′ amounting to about 279 hours or 11 days and 15 hours you may conclude that if she be Encreasing she wants so much of being Full as also that she is 50 Degrees or almost 100 hours i. e. four Days and almost four hours past her first Quarter whereas if she be Decreasing she will want eleven Days and fifteen hours from her next Conjunction and be four days and almost 4 hours beyond her last Quarter As for knowing the Moons state in relation to her Waxing and Waining you need only observe on what side of her Discus her illuminated Part stands for if it be on the West-side of it she is in a Waxing Condition if on the East-side in a Waining or Declining one And here also remember that as to the measuring the aforesaid Portions of the Moons Discus represented on your Globe you may do it by the Horizon if she illuminates not much beyond the Zenith or by the Aequator when the illumination reaches to the Pole or neer it or by the Ecliptic when it extends it self a good way further for the said Portion of the Moons Discus is measur'd at first sight by that great Circle which lies equally distant from each Horn of the Lunula on the Globe i. e. by that great Circle which crosses it as we said in the middle at Right Angles and when no great Circle does so you had best measure it exactly with your Compasses seeing that on the knowledge of its breadth the Resolution of all the former Questions depend Many things of great use may be drawn from knowing the true proportion of the illuminated and obscure parts of the Moons Orb but this I leave to them that have exacter Instruments than the Globe and more time to make Deductions The END of the Third Section SECT IV. Shewing the Proportion between Perpendiculars and their Shades SEeing there is the same proportion between all Shades and their Perpendiculars at least to sense and seeing the several Almucantars of the two great Luminaries are the chief Cause of the lengthning or shortning of them I have here adjoin'd a few by Operations even in Altimetry it self as belonging naturally to our Globe since it not only shows us several ways of finding from time to time the said Almucantars but gives us also at the same instant without trouble as appears by the ensuing Operations the above-mentioned Proportion and consequently the height of all things Perpendicular to the Horizon OPERATION I. How to find the Proportion between the Perpendicular and its Shade COnsider the Northern or back part of the Globes Meridian which we will call hereafter the Quadrant of Proportion and which is not only devided like the Southern or fore-part into Degrees but markt also in relation to the affair in hand with several Figures of which that next the Zenith is 17 and the remotest 188. And by the way you must take notice that when you see a Cross behind any Figure it signifies half an Integer more so that 17 + is 17 Degrees and a half 26 + is 26 and a half c. When you would therefore Operate Turn the Southern or fore-part of the Meridian towards the Sun 'till they be both in the same Plane i. e. 'till the shade of the Pin in the Zenith falls directly upon the Quadrant of Proportion and what Figure soever suppose 25 the shade of Extuberancy cuts that will be the then Proportion between Perpendiculars and their Shades for here you may take notice that we ever suppose the Shade to be 100. Nay if finding by any of the former ways the Sun's height to be suppose 14 Degrees you rectify your Bead to 76 Degrees or the Complement of it you need only clap back your String that is to say draw it from the Zenith over the Devisions of the afore-mention'd Quadrant and then the Figures under the Bead to wit 25 will shew you the required Proportion In short take but the Suns Height any how and reckon from the Zenith as many Degrees on