The English globe being a stabil and immobil one, performing what the ordinary globes do, and much more / invented and described by the Right Honorable, the Earl of Castlemaine ; and now publish't by Joseph Moxon ...

SECT. V. Of Dialling. (Book 5)

THE Performance of this Section no less appertains than the* 1.1 rest to our Globe; since it not only represents to our view, what Dials are in themselves, and why they shew us the time of the Day, but afford's us also so natural and so easy a way of describing them, that no person can be hardly so ingnorant, who will not (with the least Instruction) in three hours time be able to make one on most Planes, unless perchance▪ for want of a little Digestion and private practice, the Rules shove one ano∣ther out of his Memory. Nay all the accidental Requisits to Dialling (being these that follow) are here performed without the help of any forrein Instrument.

• 1. To draw a Meridian Line.
• 2. To draw a Line Parallel to the Horizon on a Plane not Horizontal.
• 3. To find the Declension of a Plane.
• 4. To find how much a Plane Reclines.
• 5. To find how much a Plane Inclines.
• 6. To find how long the Sun shines on a Plane; that is to say, when he'l come on, and when He'l go off it, at any time of the year.

Besides we have already taught you, how to know when a Plane is * 1.2 Level▪ how to find the † 1.3 Height▪ of the Sun, and how his * 1.4 Azimuth, all which properly appertain to this Science.

These then are the Operations, I mean by Requisites, which shall be all treated of either immediately before, or immediate∣ly after each Dial that requires them. But tho' I intend to show you, as I now mentioned, this easy and natural way of Dialling, even in what bigness you please without need of Qua∣drant or Scale of Reduction, without your knowing the least Pro∣blem in the Mathematics, or my requiring any thing else of you, let that you can draw a Line from Point to Point, and describe a Circle, when the Radius is given; I say, tho' I intend this,

yet that you may not be wholly Ignorant of their Geometrical Construction, I shall teach you that also, and therein perchance something both expedite and new; but then I must suppose (for I have not time to instruct you,) that you can Erect a Perpendicular, draw a true Parallel, and know so much of the * 1.5 Sector as how (besides the use of the Line of Lines) to find the Chord and Sine of any Number of Degrees by it. I men∣tion the Sector, not that any Ordinary Ruler, with a Line of Sines, Line of Chords, &c. is unsufficient, but because you have under the Pedestal of your Globe a Sector to your hand, which I have order'd there for two Reasons▪

First it will be necessary, the Globe standing erect and at right Angles with the Horizon, to bring the edge of its Pedestal to your Plane in several Operations; Now seeing this cannot be done unless the Pedestal be greater than the Globes Diameter (which would, make it perchance cumbersome, and to take up too much room) there are therefore under it two small wooden Rulers about half an Inch Broad, to slip in, or out to a determin'd length, so that it will be better to have one of them a Sector (which is a very necessary Instrument) than not, especial∣ly since 'tis no hindrance to the use, we chiefly intend it for. Nor would I have any thing more on it (to avoid the Confu∣sion of Devisions) than the Line of Sines on the one side, and the Line of Lines on the other▪ and as for its length, if it be six inches, it will make an even Foot at full extension▪ which ren∣ders it a very convenient standing measure. My second Rea∣son for the said Sector is, because I will show you, how to find by it the TANGENT and SECANT of any degree even ac∣cording to any Radius, and how usefull an Operation this is (and particularly in Dialing,) everybody well knows that has the least insight in it.

Suppose then you desire the Tangent and Secant of 50 De∣grees;* 1.6 open your Sector, and taking (GR) the given Radius between your Compasses, put it over in the Line of Sines, at the Sine of 40, viz. at the Complement of the Angle required, and the distance between the Figures of 50 and 50, will be the Tangent sought for; and the distance between the Figures of 90 and 90 will be the Secant of the said 50 Degrees.

These two Problems I recommend to you because they are extremly ingenious and useful, found out it seems a while ago

by Mr. Line (the Author of that admirable Horological Pile in White-Hall Garden) as, since his death, one of his Scholars (that hinted them to me) informed me, and upon consideration I found them exactly true, having thus de∣monstrated them.

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Sch. 1.

'Tis evident by Sch. 1. that* 1.7 as the sine Complement AB is to the Sine CB, so the Radius AD is to the Tangent ED. Again 'tis evident by Scheme 2. that as KC: KS :: CR: ST, but by Constructi∣on KC is on the Sector the Sine of 4 or Sine comple∣ment of 50, and CS the Sine of 50 it self, therefore CR being the Radius given, ST must be the Tangent re∣quir'd, viz. the Tangent of 50, according to that Radius. On the other side we see by the said first Scheme that as the Sine Complement AB is to the Radius AD, so is the Radius AC. (or AD) to the Secant AE. Again you see by Scheme second, that KC: KD :: CR: DP; but by construction KC is the Sine of 40 or Sine Complement of 50, and KD is Sine of 90 or Radius of the Sector, and CR is the Radius given, ergo PD must be the Secant required.

Having thus done with the Preliminaries, we'l now come to the business it self, after I have told you there are five principal Planes on which a Dial may be made; to wit the Horizontal, the Vertical, the Declining, the Reclining and the Inclining Plane; and first of the Horizontal, since that Dial is the Basis and Foundation of all the Rest.

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Sch. 3.

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Sch. 4.

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Sch. 5.

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Sch. 6

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Sch. 10.

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Sch. 7.

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Sch. 9.

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Sch. 8.

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Sch. 11.

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Sch. 12.

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Sch. 13

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Sch. 14.

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Sch. 15.

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Sch. 16.

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Sch. 17

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Sch. 18.

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Sch. 19.

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Sch. 20.

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Sch. 21.

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Sch. 22.

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Sch. 23.

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Sch. 25.

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Sch. 26.

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Sch. 27

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Sch. 24

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Sch. 28

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Sch. 31

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Sch. 32

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Sch. 33

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Sch. 34

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Sch. 35.

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Sch. 36.

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Sch. 37.

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Sch. 39

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Sch. 38

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Sch. 41

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Sch. 42

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Sch. 34

OPERATION I.

BEfore you proceed further, you must know Reader, that the Printer* 1.8 (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 Di∣rect Reclining Plane, the Direct Inclining Plane, the Reclining Declining Plane, and the Inclining Declining Plane. First, then of the Horizon∣tal, that Dial being (as is said) the Foundation of this Science, and af∣terwards 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* 1.9 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* 1.10 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 far∣thest 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 de∣scribe a larger Circle, suppose, NESW; or if a lesser Di∣al, 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 de∣signed 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* 1.11 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 (pas∣sing

throu' that Mark and Center) will give you in the Bor∣der 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 ei∣ther 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 (fal∣ling from the Indicating side O π) they stand Perpendicular to it; that is to say, the Pin AB (being part of the Trian∣gle or Stile IO π) must be erected at B, the Pin GH at H, and π I at I; and the reason, why they perform this Office, as well as the whole Triangle IO π is, because their Tops are parts of the Line O π, which is the only side of the said Triangle, that shows the Hour, as we mention'd before. Now for the De∣monstration of all, it follows in the next Operation.

OPERATION II. How to describe an Horizontal Dial by the Globe for the Elevation of London.

The second way.

DEscribe a Circle of what bigness you please, and draw a* 1.12 Meridian, or 12 a Clock line throu' it, as before; then count in the Horizon of your Globe how many Degrees there are between the Hour-Circles of 12 and 1, or, (which is the same thing) between 12 and 11, and you will find their number to be about 11. 40′. These place on both sides of your said Meridian Line by the help of a Quadrant, or Line of Chords, and they'l give you (if you lay your Ruler as before on the Center) the 11 and 1′ a Clock Hour Lines of your Dial, to wit, the distance from I to k, and from I to h, as may be seen in the aforesaid third Scheme. Proceed then in this manner as to the rest of the Hour lines, and for your Stile and Substilar, the former Directions are sufficient.

The Demonstration or Reason why these Dials show the* 1.13 Hour is not difficult; for if: you consider your Globe, you will see that all its. Hour Circles are equally distant from each other, and that the Axis of the World (of which the two Poles are the extremities) lies in the middle of them, and is in truth a part of each, as being the common Section of them all; therefore when the Sun comes into the Plane of any Hour Circle (for example to that of 4 in the morning) the shade of that Hour-Circle will fall there, where the said Hour Circle cuts the Horizon on the Opposite or Western side, and consequently the Axis being in that Plane, as a part of it, its Shade must needs fall there also. Now since the Blind Circle or Limb of the Dial described is a Circle representing the Horizon, and having by Construction its Hour-lines di∣stant from each other as the Hour Circles of the Globe or World are distant in their Horizons, and since the Hour-lines of This (and consequently of all other Dials) are only the

intersections of the Hour-Circles with their respective Planes, it must needs follow, if we place in the middle of the said Dial a Cock or Stile, making an Angle of 51, 30, with its Meridian line or Substilar (to wit, the Angle which the Axis of the World makes with the intersection of the Meridian and Plane of the Horizon) 'twill cast a Shade directly on the Hour line corresponding to the Hour Circle in whose Plane the Sun then lies, in case the Meridian or 12 a Clock line of the Dial be plac't North and South, like the Meridian of the Globe when compos'd; for the Globe it self without it be compos'd will not (as we have formerly mention'd) shew the Hour, because its Hour-Circles do not then correspond with the Heavenly ones. And as for the reason why the 12 a Clock line is the Substilar, 'tis because the true Height of the Axis above the Plane (which the Stile or Cock, as I showd you, represents) is to be measured in the Hour Circle that falls on the Plane at right Angles, which being the Meri∣dian or ordinary 12 a Clock Hour Circle, it follows that its In∣tersection with the Plane must be the Substilar, or Line with which the Stile is to make the Angle of the Elevation.

All that we have then said of this Dial may be clearly seen by Sch. 5. which represents your Globe cut into an Horizon∣tal Plane, with its Dial on it, as Sch. 4. does the Globe en∣tire, when you consider it in the description of the said Dial; for there you have before your eyes (by the Letters I k, l, &c.) not only how to open your Compasses from Hour-Circle to Hour Circle for the true placing the Distances of each Hour-Line on your blind Circle, but also the number of Degrees in the Horizon between every Hour Circle and the Meridian. Besides, by the Horizons oblique cutting the Hour Circles, you may see how that (notwithstanding the equality of the Suns Horary motion) the Hour-lines of this Dial must be unequal, and consequently that they are of different distances in diffe∣rent Latitudes.

OPERATION III. To describe an Horizontal Dial Geometrically, for the Elevation of London.

Describe a fair Circle as ABCD, and if you would have your Dial of another Shape, you may afterwards de∣scribe about it what Figure you please; I say, describe the fair Circle ABCD, and draw throu' its Center O the Line AOC for your Meridian or 12 a Clock hour line, and cros∣sing it at right angles with BD for the Morning and Even∣ing 6 a Clock hour lines, mark in it (by the help of your Line of Sines or any way else) from A the value of 51. 30. or Latitude of your dwelling, which happening to reach, (for example sake) to K, draw the blind line OK; then throu' any point of AO (suppose A) draw GH, another blind line, paral∣lel to BD, or at right Angles with the said AO, and taking with your Compasses the nearest distance between A and OK, which being (suppose) the point L, let AL, by the help of your Sector (according to our former * 1.14 directions,) be the Ra∣dius to the Tangent Line GH, so that marking in it on both sides of A, the Tangents of 15, 30, 45, 60, and 75 Degrees, the said Center O and the point 15 will give you the Hour-lines of 1 and 11, the Center O and 30, those of 2 and 10, and in this manner proceed to 75, which will give you the Hour∣lines of 5 and 7; and as for those beyond the 6 a Clock lines, do but produce 8 in the Morning, and 'twill give you 8 at Night, and 7 in the Morning 7 at Night, as will 4 and 5 in the Evening, the like forenoon Hours.

Thus then you have not only an Horizontal Dial Geo∣metrically described, almost as soon as the former, (and this without embroyling the Plane with multiplicity of blind Circles and Lines) but a way also (in case you have no Sector) how to make any Tangent Line serve your turn; for, 'tis but ta∣king between the Compasses 45 Degrees of it (i. e. a distance equal to its Radius) and finding out (by a trial or two) the Point

(suppose) R in the line OA, where one foot of your Compasses being placed, the other just touches M (the suppos'd nearest point or distance in OK from the said R) draw throu' R a line at right Angles with the Meridian, and noting in it, as we show'd you before, the Degrees of each hour (according to this new Tangent line) the Center O and these Degrees will give you the points of each hour line; for as the former Radius AL was to the several Degrees in its Tangent Line, so will the now Ra∣dius RM be to the several Degrees in its Tangent Line.

As for the Demonstration or Reason of this Dial, every body* 1.15 that understands Gnomonics comprehends it, I doubt not, at the first sight; for the Angle O in the Triangle KOA, being by con∣struction equal to the Elevation, do but place the Base AO on a Meridian Line, and if you consider the Side KO as the Indica∣ting Side of the Stile or Cock, it necessarily follows, that it will re∣present the Axis of the World; for it is evident that its Top K will point directly to the Pole, and touch it, if produc'd, whilst O its other extremity passes throu' the Center of the Horizontal Plane; therefore if a Circle (whose Radius is AL) were so plac't on this Stile or Axis, that its Diameter crost it at right Angles at L, the said Circle would represent Circulum maximum sem∣per apparentium, for that Circle in the Heavens ever touches the Horizon, as this would do at A. This Circle then being parallel to the Aequator, is divided by the Hour Circles into twenty four equal parts, and consequently each fifteen Degrees in its Tan∣gent Line GH, will correspond with its said equal parts or Divi∣sions. Now GH is also the Tangent Line of the Horizon, as touching it in the Point A, but where the Hour Circles cut the Horizon, or its Tangent line, there the Points will be, to which (from the Center) the Hour Lines in an Horizontal Dial are to be drawn; ergo O the Center of your Horizontal Plane, and the several fifteen Degrees in the common Tangent GH are the true points of the Hour Lines. Besides as the distance▪ between each Hour Line (if AL, be the Radius) is 15 Degrees, so if AO be Radius (I mean OA the Radius of the Horizon∣tal Plane) the said Hour Lines will be distant as many Degrees asunder, as they are in the Horizon of the World, or as you found them in the Fabrick of the second Horizontal Dial by the Globe. Here also you may see, that the true place of this Dial is to be in the Center of the Earth, and

not on its superficies, but by reason of the Suns vast distance, the Error, which thereby happens) is not sensible; nay, because the Error is not sensible, we may safely conclude, that the Sun is vastly distant from us.

So much then for Horizontal Dials, since there now remains nothing necessary to be known, but how to find whether they stand Level or no (which is handled in the first * 1.16 Section) and how to draw a Meridian Line for their true placing, which is learnt by the following Operation. But before we go further let me advise you (whensoever you make a Dial of consequence, of what kind soever it be) to describe it first on Paper, and thence* 1.17 to mark out the Lines on your real Plane, for thereby you will not only keep your said Plane neat, and more judiciously chuse the best place for the Center of your Dial, but (besides the several conveniences which practice will show you) the Lines themselves will be more exactly drawn, by reason you can ma∣nage your Paper draught as you please.

OPERATION. IV. How to draw a true meridian Line on any Horizontal Plane.

* 1.18COmpose your Globe on the Plane, or Place where your Dial is to stand, and making marks or pricks there (on each side of the Pedestal) at the Letters S and N, draw but a Line throu' those marks, and that will be a true Meridian Line, and if you do the like under the Letters E and W, you will have a true East and West Line.

OPERATION. V. How to Describe a Vertical, or an Erect Direct South Dial by your Globe for the Elevation of London.

The first way.

THIS Dial is made on the Plane of the Primary Vertical, which passes from the Zenith to the Nadir throu the East

West points, and being therefore erect, and facing also directly the South, tis commonly called an Erect Direct South Dial; so that if you draw but your String from the Zenith to the Nadir thro either of the Intersections of the Horizon with the Equator, 'twill appear upon the Superficies of the Globe, like the emerging edge of a thin Plate, and consequently repre∣sent the said Plane, or at least as much of it as is requisite.

This being don't open your Compasses at 60 Degrees, as * 1.19 be∣fore,* 1.20 and describe on a sheet of paper the blind Semi-Circle I PC (as in 〈◊〉〈◊〉 10) with the Diamiter or Meridian IOT throu' it, then take with your Compasses the distance between the Zenith of your Globe, and the Intersection of your String with the nearest Hour Circle, and 'twill in your Blind Circle on both sides of the Meridian or twelve a Clock Line, (to wit from I to k, and I to h) give you marks, by which you may draw from the Center O the Hour Lines of 1 and 11; as will the distance from k to l, and h to g (viz. the distance from the said first Intersection to the second) the marks of 2 and 10; and in this manner you must proceed to 6 and 6, as the latest and ear∣liest hours, that this kind of Dial shows; for since its Sides lye full East and West, and that the Sun never comes to the East before 6 in the morning, nor is later in the West than 6 at night, 'tis impossible that the Plane should significantly contain more Hour-Lines. And as for the Stile or Cock▪ the distance on your Globe between the Zenith and the Pole (being the Com∣plement of the Elevation) gives you from I to K the Degrees of its height above the Plane, so that you may easily place and erect it, the Substile being still the Meridian. The Rules in the first Horizontal Dial will show you also both how to contract and enlarge it, and how to resolve (especially if you consult the 7th. 8th. and 10th. Schemes) any difficulty that can possibly arise in the present Operation; for Scheme the 7th. shows you the Globe it self with the String drawn from the Zenith to the Nadir throu' the East Intersection of the Aequator with the Horizon; and Scheme the 8th. the Globe cut into this Plane by the said String, and lastly the lower part of Scheme the 10th. (to wit, the Semi-Circle PIC) the Dial described by the foregoing Directi∣ons. Now for the Demonstration it follows in the 8th. Opera∣tion.

OPERATION VI. How to make this Vertical South Dial by the Globe for the Elvation of London.

The second Way.

DEscribe a Blinde Circle of what bigness you please with a* 1.21 Diameter throu' it, and placing your String on the East or West Poynt of the Globe as before, measure (by your Bead or Compasses in any great Circle) the distance between the Zenith and each Intersection of the said String with the Hour Circles, and you will have the Degrees of every Hour from 12 a Clock, as the before 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 Demon∣stration, 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 * 1.22 Pa∣per, before you draw them on your designed Plane, and ha∣ving show'd you how to describe this Dial, 'tis now time to teach* 1.23 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* 1.24 Draught not only on this, but on any Plane for which 'tis made, look what a Clock 'tis 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 Di∣al 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; there∣fore 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* 1.25 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, 'tis best always to operate thus from the Zenith, as being more at hand than the Nadir, and consequently more conve∣nient.

The Demonstration or reason why these Dials show the* 1.26 Hour, differs even at first Conception but little, and at the second not at all from that already given for the Horizon∣tal 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 Demonstrati∣on) 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 re∣ference 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. 'Tis plain then, that the present Dials are exactly describ'd, if our for∣mer 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 Meri∣dian 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 Geome∣trically, for the Elevation of London.

THere is no need of a Scheme for this Operation, since 'tis 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 Elevati∣on of your Place, and that will serve (the figuring only con∣sider'd) for either Dial.

Here then you may see that OS, or ON the Basis or Foot* 1.27 of the Stile of these Dials, (that is to say, the distance between its Center and its Horizontal edge or side) is ever the Tan∣gent of the Elevation; for 'tis 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* 1.28

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 Nor∣thern 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* 1.29 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 'twill represent the intersection of the Ae∣quinoctial 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 Intersecti∣ons 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 com∣pleat 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 Ho∣rizon 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,* 1.30 except it be in turning on your draught the Hour-Lines or Parallels the other way, to the end they may all point North∣wards 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 'tis the 6 a Clock Hour Line, since that* 1.31 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 be∣tween 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 draw∣ing 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,* 1.32 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 re∣presenting truly the Axis▪ (since 'tis above the Plane, and di∣stant from it as 'tis on the Globe) will cast its Shade every hour on the corresponding hour Line▪ and as for the reason, why the height of the said Axis is equal to the distance between the 3 or 9 a Clock Lines and the Substilar, it shall be shown in the Demonstration of the next Operation.

OPERATION XII. How to describe an East or West Dial Geometrically for the Elevation of London.

DRAW the blind Line H h and cross it from your left hand* 1.33 (as in Sch. 13.) with AE ae another blind-line to make an Angle at their Intersection K equal to the Complement of the Elevation, then pricking in the said Line AE ae on the right side of K, the respective Tangents of 15. 30 45. 60. and 75 De∣grees, as also on the left the Tangents of 15 and 30, Draw but Perpendiculars through the Pricks, and you have an East-Dial; whereas should you cross (as in Sch. 14.) H h with AE ae from the right hand, and pricking the aforesaid Tangents the other way, draw Perpendiculars through them, you would have a West-Dial. By these Schemes also you may know how each Dial is to be Figur'd, the East-Dial containing (as you see) all the hours from 4 in the morning 'till Noon: and the West all the hours from Noon to 8 at Night. Now for their Cocks, they are (as I said, in the last Operation) to be a Pin, or a Gal∣lowes Stile, and in height equal to the Tangent of 45. Degrees, or distance between the 9 or 3 a Clock hour Lines and that of six, which is ever their Substilar.

These Dials must be true, if their Planes lye in or Parallel* 1.34 to the Meridian; for since the Line H h, by being plac'd ac∣cording

to our Hypothesis horizontal, represents the intersecti∣on of the Horizon, and the line AE ae that of the Aequator, by ma∣king an Angle with the said H h equal to the complement of the Elevation, the substilar must be the Intersection of the Aequinocti∣al Colure (or 6 a Clock hour Circle) with the Plane, since that Hour-Circle falls on the Plane at right Angles. If then a Gallows Stile be set on the said Substilar and Perpendicular to it, its Shade must needs constantly cross the Aequator AE ae at right Angles. Now when the Sun is in the Plane of the 6 a clock hour Circle, his Ray makes no Angle with the said Stile, because the Sun, and the Stile are in the same Plane, and so the shade falls directly along the Substilar; but when he gets (for examples sake) into the next hour Circle, his Ray (the height of the Stile being Radius) makes an Angle of 15 Degrees with the said Stile, and conse∣quently the distance of the two shades are in the line AE a the Tangent of those Degrees. The like therefore being said of the next Hour Circle and so on, it follows (as I mention'd in the beginning) that the pricking from the intersection K, the Tangents of 15, 30, 45, 60 and 75 Degrees in the line AE ae, must give you points to draw the perpendiculars or true hour-lines of this Dial by, as also, that the Tangent of 45 Degrees gives the height of the Stile, since the Tangent of those Degrees, (which you see gives the 3 and 9 a clock lines) is equal to the Radius.

Here also we see not only why these hour-lines are* 1.35 so unequally distant, since they are so many Parallels mar∣shall'd according to the Divisions of a Tangent line, but why the 12 a Clock hour line can never be really express'd, for 'tis the Tangent of 90 Degrees which is infinite.

OPERATION XIII. How to describe a Declining Dial by the Globe for the Elevation of London.

The first Way.

THIS Plane (as passing from the Zenith to the Nadir) is* 1.36 still Vertical, and should (you may suppose) be by right the primary Vertical, but by its tendency towards the East or West Points, its Dial takes the Appellation of a Declining one,

that is to say, of a Dial, whose Plane declines so many degrees from facing directly the North and South, as is its tendency to∣wards the said East or West points.

As for the way of making this Dial it differs little from the* 1.37 first Direct Erect one, already * 1.38 treated of; for supposing your present given Plane declines 40 Degrees from full South towards the East, you must draw your String (which ever represents the Edges, as we have said, of your Plane) not throu' the East Point of the Horizon of your Globe, as before, but throu' 40 Degrees further towards the North, for this makes the String to represent part of a Plane that comes nearer (by so many De∣grees) the facing of the East than it did. Then opening your Compasses at 60 Degrees in any of the great Circles, and descri∣bing (as in Sch. 17th.) the blind one PZW, prick in it from its Meridian Line OZ, the distance between the Zenith of your Globe and the intersection of your String with the first Hour-Circle (to wit between Z and b in Sch. 15.) and it will give you a mark for the 11 a Clock line on your Dial; and the di∣stance between the Zenith and the Intersection of your String with the next Hour-Circle (to wit between Z and c) will give you the mark of the 10 a Clock line, and thus you must proceed to e∣very Hour-Circle cut thus by your String, till it falls on the Ho∣rizon, that is to say from z to d, e, f, g, h, letters marking (as you see in the said Scheme) the 9, 8, 7, 6, 5 and 4 a Clock Hour Circles▪ and consequently giving you those Hour-lines on your Dial.

Now for the Afternoon hour lines (which are no longer equal* 1.39 in distance to the Morning ones,) you have nothing to do but to draw your String, on the West-side of your Globe, throu' 40 De∣grees in the Horizon the contrary way (viz. from the West to∣wards the South) and the distance between the Zenith and the Point in the first Hour-Circle cut by your String (to wit from Z to k in Sch. 16.) will give you the mark for 1 a Clock, and the distance from thence to the next Point or Intersection gives you that of 2, to wit, from Z to l, and in this Order you are to pro∣ceed to n, the▪ 4 a Clock Hour Circle, that is to say, till you come to the intersection of the String with the Horizon on the West∣side of your Globe.

As for your Stile and Substilar they differ also from those of* 1.40 direct North and South Dials; for the said Stile or Cock is to be no longer plac'd on the 12 à Clock Line, nor will its height now

be equal to the Complement of your Elevation, therefore having drawn your String throu' the Degrees of Declension in the Ho∣rizon as before, and putting one foot of your Compasses in the North Pole, find with the other the nearest Point on your String, to wit S (as in Sch. 15.) and the distance between S the said nearest Point and the Zenith of your Globe will be ZS in the blind Circle of Scheme the 17th, to wit the distance between the Meridian Line of your Dial and your Substilar, which in this our Example lyes from the Moridian towards your left hand or Morning hours, and the distance from the said Point in the String to the Pole (being from S to P) will in the said blind Circle be the height of your Stile; so that if you erect and place your said Stile from the Center all along the Substilar OS it will continually show you the Hour.

But if you fancy that the Extension of your Compasses from* 1.41 the Pole to the String will not give you precisely this Point, since your said Compasses may seem to touch it in several Points; I say, if you doubt or fancy this, fasten a Thred on the Pole, and drawing it streight over the Horizon at 40 Degrees from the Meridian of your Globe Eastwardly (i. e. till it passes thron' the Pole of the Plane) see where the said Thred crosses your String (or edge of the Plane) and there the true requir'd Point will be. The Demonstration of this Dial is in the follow∣ing Operation.

OPERATION XIV. How to describe by the Globe a Declining Dial for the Elevation of London.

The second way.

DRaw your String over at 40 Degrees in the Horizon from* 1.42 the East Northwardly, and from the West Southwardly, as before, and the respective distances between the Zenith and the Intersection of your String with the Hour-Circles will give you in any great Circle of the Globe the Degrees of their respective distances as well for the Morning as Afternoon, and the pro∣portionable Degrees in any Circle will give you the Points for the Drawing of your Hour-lines, as I showd you in the Con∣struction of the former Vertical North and South Dials; and

as for the Stile and Sub-stilar, you must operate as directed in the foregoing Operation, that is to say, the number of Degrees between Z and S gives you the Sub-stilar, and those from P to S the height of your Stile.

As for the Demonstration or Reason why Dials thus made* 1.43 show the Hour, it is this; First you see that the String, by be∣ing on one side removed 40 Degrees from the East point North∣ward, and on the other side 40 Degrees from the West Point Southward, represents on the Globe the requir'd Plane, and therefore wheresoever the Hour Circles cut it, there the Shade of the Axis will fall, as we show'd you before in the former Dials; Now two Points made by the intersections of each Hour Circle with the Plane being given you (to wit, the Center where they all meet, and their respective marks on the String, or sup∣posed Edges of your Plane) it must needs follow, that if you draw Lines throu' those Points, they will be true Hour Lines; for (as we have often said) the Hour-Lines of all Dials are on∣ly the intersections of the Plane with the hour Circles. In the next place, since PS by construction is the nearest distance from the Pole to your String or Plane, it appears that the Hour-cir∣cle which cuts the said Plane at S, falls on it at right Angles, and consequently that as PS (the height of the Pole or Axis above the String or Plane) gives the true height of the Stile of this Dial, so the intersection of the Plane with the said Hour-Circle must be the true Substilar; for the Substilar (as we already mention'd) is only the intersection of the Plane with the Hour-Circle, which falls at right Angles on it; Ergo The distance between Z and S gives in your blind Circle the distance from your 12 a Clock line to the Substilan, and PS the height of the Stile.

And by the way, here it appears not only why the 12 a Clock* 1.44 Lines of Declining Dials continue perpendicular, but also why their Centers keep the same distance from the Horizontal Edges of their Planes, as do the Centers of the primary Vertical or direct North and South Dials; I say, here all this appears; for the 12 a Clock Line (which is ever the intersection of your Meridian with these Planes) being a Perpendicular in the pri∣mary Vertical Plane, becomes the Axis of the Horizon, and all Vertical Dia's▪ by their Declension more only about it, so that both the Center and the said 12 a Clock Line remain the same in all; therefore the Tangent of the Elevation, being (as I

* 1.45 formerly show'd you,) the length of the foot of the Stile, or distance between the Center of a Primary Vertical Dial and its Horizontal Edge is that of a Declining one also.

But to proceed with the Demonstration; you must remem∣ber* 1.46 that this Dial is an Horizontal one (as we show'd you * 1.47 be∣fore) to those that dwell in the Pole of the Circle describ'd by the String, i. e. to those in our Horizon 40 Degrees Eastward from the Meridian, or (which is all one) to those that dwell where the Thred cuts the Horizon; but all the Hour-Lines are truly drawn according to the former Rules of an * 1.48 Horizontal Dial, to wit from the Center to the Points where the respective Hour Circles cut the String or Limb of the Plane, therefore it must truly shew the hour.

OPERATION. XV. How to describe Geometrically a Declining Dial for the Elevation of London.

The first way.

THIS Dial being (as I said) an Horizontal one to those in* 1.49 our Horizon 40 Degrees Eastward from the Meridian, Find (as we show'd you in the * 1.50 Geographical or 20 Section) what Elevation or Latitude they have, and describe Geometrically an Horizontal Dial on paper for the said Elevation. In the next place consider the difference between both Longitudes, to wit how many Hours the Sun comes sooner to their Meridian than yours, so that if he comes, suppose, 3 hours, 'twill follow, that the 3 a clock hour line is to be the true 12 a clock line of this Plane, because 'tis really so late with those People, when 'tis but Noon with you, and consequently that their 4 will be your 1 a clock, and their 2 your 11. and in the like manner you are to mark the rest, having nothing more to do but to draw on your fair Plane a * 1.51 Line Parallel to the Horizon, and to place on it at right An∣gles the true 3 a Clock Line, (that is to say the 12 a clock line ac∣cording to your now alteration or present figuring the Hour-Lines,) for you will have all the requisite Marks or Points, not only to draw the other Hour Lines, but also plainly to see, where the Substilar will fall, and how high the Cock it self is to be; for they are all to correspond with those in the said Ho∣rizontal

or Paper draught. Now in case the difference of Lon∣gitude between these 2 Places happens to be a Fraction, as (sup∣pose) one hour and 10 minutes, then (if the Declination of your Plane be still Eastward as in the former example) 10 mi∣nutes past 1 must be markt in the Horizontal Draught with the Figure 12, as the Meridian Line, and 2 and 10 min. with Figure ••. and so on all along; whereas if the Declination were Westward, then 11 and 10 minutes will be the said Meridian Line, 10 and 10 minutes your 1 a Clock Line, &c; for thus you must operate in all other Cases, that is to say, you must still allow by the new figures the difference of Longitude, that chances to be between you and them, to whom the Declining Plane is Ho∣rizontal. But because this manner of Dialling may seem to some troublesom and confus'd (especially when the said Diffe∣rence of Longitude happens to be a Fraction, and not even Hours) I shall here adjoin a second Geometrical Way.

OPERATION. XVI. How to describe Geometrically a Dial declining 40 Degrees Eastward, for the Elevation of London.

The second way.

HAving made an Horizontal Dial for this Elevation in the* 1.52 lower part of your Paper Plane, (as 'tis exprest by the prick lines in Scheme 18) and drawn from the Center A the several Hour-Lines upward as far as you think fit, and Figur'd them to show what Hour-Lines they are, chuse in AC (the 12 a clock line) any Point, suppose P, and draw throu' it the blind Line GD making with the said AC an Angle of 50 De∣grees or Complement of your Declension; then erect the Perpen∣dicular PB on the said blind line at P, and taking with your Compasses (AP being your Radius) the Tangent of 5•• Degrees and ½, or true Elevation of the Pole, put one foot on P, and where the other marks on the said Perpendicular (suppose at F) there will be the Center of your Declining Dial; so that having bordred your Plane with fitting Parallels, to contain the standing Figures of each hour, you have nothing more to do,

but to draw fair Lines from the said Center F, to your Border, throu' the Intersections of the Line GD with the several Hour∣lines of the Horizontal Dial; that is to say, you have nothing more to do, but to draw fair Lines throu' the Points KLMNO PQR which give the hours of 7, 8, 9, 10, 11, 12, 1 and 2; and by the* 1.53 way you may have as many other Morning or Evening hours as you please if you draw the said GD long enough for the other hour-lines of the Horizontal to meet with it. Nor is there more difficulty here about the Stile and Substilar than in any of the former Dials; for (AP being Radius) 'tis but taking the Sine of 40 Degrees (or Declination of the Plane) with your Compasses from the Sector, and putting one foot on your 12 a clock Line at P, the other foot will in the line GD (to wit, at M) give you the Point for to draw the Substilar FM, and the Sine Complement of the Declension, or Sine of 50 Degrees, will be XM the Stiles height. Nay, if (for want of a Sector or the like) you cannot conveniently find the Sine of the said Declen∣sion, do but observe where a Perpendicular from A falls on GD suppose at M, and PM will be the distance in the said GD be∣tween the 12 a Clock line of this Dial and its Substilar, and AM (equal to XM) the height of the Stile above it. Thus then we see that the Fabrique of a Declining Dial (which is wont to terrify young Students) is in a manner as quick and easy as that of the Horizontal, since two ordinary Lines more, viz. GD and BP give us all the Points necessary for its Description.

The Demonstration and Reason of this Dial is evident; for,* 1.54 the Horizontal being by construction true, any Erect Plane fa∣cing the South, that crosses its Meridian (or 12 a clock line AC) at right Angle•• will represent a Primary Vertical or Direct South Plane, and then the Center of the Dial described on it will be distant from P the intersection of the two Planes on the said AC) the Tangent of the Elevation, as I shew'd you * 1.55 before. Now since GD is (by Hypothesis) the Edge of a Vertical De∣clining Plane, and since (as we show'd you in the before cited place) that the 12 a Clock line, as well in a Declining as in a Primary Vertical Dial, is Perpendicular to the Horizon, con∣taining in it the Centers of the said Dials, it follows that FP (being the Tangent of the Elevation, and Perpendicular also to the said DG where it cuts the 12 a Clock line of the Horizon∣tal) must be the 12 a Clock line, and F the Center of our present

Dial, whose Declension is 40 Degrees Eastward, since FP de∣clines so many Degrees from CP toward the morning Hours; for the said CP and FP represent the 12 a Clock lines of a Direct, and of our thus Declining Vertical Plane, if you consider them flatted down, and lying in the Horizon. This being so, 'tis evident that the Lines drawn from F to KLMN, &c. are the true Hour lines of our Dial, as falling from its Center to the several Points made on its Horizontal edge, by the Hour Circles or (which is all one) by their Intersections with the Horizontal Dial. As for* 1.56 the Stile and Substilar, let us but consider the Triangle AMP, and we shall find that P is by construction the Angle of 50 Degrees, and A that of 40, as substended by the Sine of the Declension, so that A being a right Angle, AM must be a perpendicular; therefore the Hour Circle, whose intersection the said AM happens to be, falls at right Angles on our present Plane, and consequently gives the Substilar; Now since the Axis of the World passes through F and A, the Centers of the two Dials, when they are joyned (as we now suppose them) at GD the common Section of their Planes; I say, since the Axis passes throu' their Centers, its Elevation or Height above our Plane must be AM, as being the only Perpendicular that can fall from it upon the said Plane, and consequently its Mea∣sure; but AM you see is the Sine Complement of 40, since PM is the Sine of 40, Therefore in all Declining Dials, The Sine of the Declension (from their 12 a Clock Line) gives in their Horizontal Edge their Substilar, and the Sine Complement their Stile. Q. E. D.

OPERATION XVII. To take the Declension of a Plane.

COmpose your Globe and find exactly the Azimuth, i. e. what Degree of the Horizon is cut by the String's shade, when it passes throu' the Zenith and Nadir, which wee'l suppose to be the 50th from the South towards the West; then having slipt out (to an equal length) the two Rulers from under your Pedestal, Hold your Globe level, and apply the said Rulers, as soon as you can, to your Plane, (as you did when you drew an * 1.57 Horizontal Line) and find again the Azimuth, which now being (for ex∣ample)

90 Degrees shows your Plane declines 40 towards the East, because, the Azimuth being now increast so many De∣grees, the Meridian (which by the help of the said Rulers was perpendicular to your Wall or Plane) is turned thereby from true South (as formerly it stood) towards the East the above∣mentioned number of 90 Degrees; but had the shade fallen on the 10th. Degree, your Plane would (for the same Reason) have declin'd 40 Degrees towards the West. In short therefore, the difference of these two Azimuths is the thing that resolves the Question; for when they are equal there is no Declension at all.

Of Reclining Dials.

THE Horizontal Plane lay open (we saw) to the whole He∣misphere, whilst each Vertical one enjoy'd but half of it; for, by being Vertical, a moiety of the said Hemisphere is be∣fore, and the other behind it. Now the Reclining Plane (which is exprest by Sch. 29.) instead of being perpendicular to the Horizon bends towards it, yet so, that its bending has nothing in it of overwhelming or tendency towards those, that behold it (as it happens to Inclining Planes exprest by * 1.58 Scheme 30) but still receeds, according to the Degrees of its Reclination, farther* 1.59 and farther from them, making thereby an obtuse Angle with the Horizon, and consequently faces more than half the appa∣rent Heavens, as the Inclining one does less, whose Angle is therefore ever Acute.

As for the kinds of Reclining Planes, there are (I may say) 4, to wit, the Aequinoctial, the Polar, the Direct Reclining, and the Declining Reclining Plane; for each of these appropriates to it self a particular Fabric, or way of making, and therefore we will Treat of them in Order.

OPERATION XVIII. How to describe a Dial on an Aequinoctial Plane, both by the Globe, and Geometrically also.

THIS Plane is represented by the Globe, when 'tis Compos'd* 1.60 and cut (as in Scheme 20) quite throu' at the Aequino∣ctial, therefore open your Compasses at 60 Degrees there, and

describing the Blind Circle ABCD in Scheme 21, divide it as the Hour-Circles cut the said Aequinoctial (in Sch. 19th.) that is to say, divide it into 24 equal Divisions, and there will rest nothing more to be done, but to draw Lines from the Center O, through as many of those Divisions as you shall think ne∣cessary, and then to Figure them successively from Morning to Night. As for the Stile (seeing the Axis of the World is at right Angles with any Diameter of the Aequator, and runs throu' the Center of it) it must needs follow that the Perpen∣dicular Pin OP plac't in the Center of your Dial, will per∣form that Office; for when it directly points to the Pole it represents the said Axis, as the divided blind Circle does the Aequinoctial, and its Divisions; therefore since the Shade of the Axis ever falls (according to the time of the Day) on This or That intersection of the Hour-Circles with the Aequator, the Shade of the Pin must fall also on the correspon∣ding Hour-line of the Dial, as being (in the effect) the same thing, in case the 12 a Clock Line be plac't on a Meridian line, and mounted at A (its South side) above the Horizon, the Com∣plement of the Elevation of the Pole, i. e. 38 Degrees and a half for by this means your Plane, from an Horizontal one, will be perfectly that of the Aequator.

Nor is it hard to mount thus the said South side of your* 1.61 Dial, since 'tis but opening your Compasses, in any great Circle of your Globe at twice as many Degrees as is the Complement of the Elevation, to wit 77 Deg. and they will give you the true length of a Perpendicular to underprop withal the aforesaid A▪ or Southern point of the 12 a clock line of your Dial. And* 1.62 the reason of it is, because AC the Diameter of your Dial being (by Hypothesis) equal to the Diameter of the Globe, becomes now (C being Center of the new Arch, made by the mounting or raising the side of your Plane above the Horizon) a Radius double to OA the former Radius. Therefore since the Chord of a double Arch is ever the Sine of the single Arch in a Circle, whose Radius is double the other, it follows that the Chord of 77 Degrees is (in respect to the double Radius AC) the Sine of 38 g. 30 m. and consequently will perform (if erected Perpendicular∣ly) the design'd Operation.

Now for the Geometrical Construction of this Dial, (since it* 1.63 consists only in dividing a Circle into 24 equal parts, with a

perpendicular Cock or Stile,) there is no need of more words about it; so that we'l end here with a Memorandum, viz. that* 1.64 as the Reclining face of this Plane, shews the Hour from Spring to Autumn, so the Inclining Face, or other side of it does the same, for the remaining half year, to wit, from Autumn to the Spring.

OPERATION XIX. How to describe a Polar Dial, both by the Globe, and Geo∣metrically also.

THE true Plane of this Dial is speculatively the Plane of the Aequinoctial Colure or 6 a Clock Hour-Circle, but in practice that of any Circle parallel to it, so that the Construction and Demonstration of a Dial on it, is (mutatis mutandis) the same with that on a Meridian Plane, of which we have already so fusely * 1.65 treated.

Make then by your Globe (for example sake) an East Di∣al on a Meridian Plane, according to any of the former ways, and if you alter but the Figures, that is to say, if having fi∣gur'd the Substilar instead of 6 with 12, you mark the Morn∣ing 7 a Clock Hour line of the said East Dial with 1, that of 5 with 11, and so on in Order, it will be a true Polar Dial, showing you exactly the Hour, when it directly faces the South, and Reclines so, that the Apex or uppermost part of the Substiler or 12 a Clock line points just to the North Pole; for then the back-part of the Plane makes an Angle with the Horizon equal to that of our Elevation.

This Operation may be also perform'd of it self without the former consideration, since 'tis but putting one foot of your Compasses on the Intersection of your Meridian or 12 a Clock hour Circle with the Aequator of your Globe (to wit, on K in Scheme 22) and so describing with Chalk the Arch CAE, I mean an Arch which reaching from the said Meridian, cuts the Morning 7 a Clock, or if (you please) the Evening 5 a Clock Hour Circle somewhere or other; for then if you draw a blind Circle (as in Sch. 23.) of the same bigness, and take the seve∣ral

distancces between the Pricks or intersections of the Hour-Circles with the said Arch, to wit, the distances between C and O, C and S, &c. and place them on the blind circle, on both sides of PCK π the Substilar or 12 a clock line, as well below the line AE ae, as about it) the lines drawn from the said Pricks will be true Hour lines, and the distance between C and P or be∣tween K and X will (for the reasons mentioned in the Descripti∣on of the Meridian Dials) be the height of the Stile.

Now to describe this Dial Geometrically, 'tis yet more easily performed, for if you draw (as in Scheme 24.) the Line AB parallel to the Horizon, and then take a Point in the middle of it (suppose K) do but prick on both sides of it the Tangent of 15, 30, 45, 60, and 75, and the several Perpendiculars drawn throu these Pricks will be true Hour-lines, which you may figure as you see in the before mention'd 24th Scheme; and as for the Stile the Tangent of 45, (or distance between the 12 a Clock line, and that of 9 or 3) gives you its height, which is to be a Pin or Gallowes Stile as before, and the 12 a clock line the Substilar.

OPERATION XX. How to describe a Direct reclining North or South Dial.

SUPPOSE then that the Plane lay directly South, and that* 1.66 its Reclination were 20 Degrees, you have nothing to do, but either Geometrically to make on it a direct Vertical South Dial for the Elevation of 71 Degrees and ½ (I mean for a Plane 20 Degrees neerer the Pole than your own Zenith) or to fix your String on 71 gr. and 30 min. in your Meridian (that is to say at A in scheme 25th. and then to draw your said string over the East or West Points of your Globe, for 'twill represent this Plane, since it Reclines or falls back from the Zenith 20 de∣grees; therefore the Distances between the Hour-Circles that intersect with your String, must (for the former reasons) give you in any blind Circle (which shall be equal to a great one on your Globe) marks (viz. b, c, d, e, f, g,) for the corresponding Hour-lines; and the Meridian being the Substilar (since 'tis

the Hour Circle that falls on the Plane at right Angles) the Height of your Stile must (as in all Direct Vertical Dials) be* 1.67 the distance from the Pole to A, the supposed Point, or Place where your String is fixed. Now had your Plane Reclin'd 20 Degrees the other way, that is to say▪ had it Reclin'd so many Degrees facing the North, you must have fixed your String at N, viz. 20 Degrees short of the Zenith, and consequently your said String would have intersected with the Hour Circles at o, p, q, r, s; therefore a Direct Vertical North Dial for the Lati∣tude of 31 g. 30 m. will be the required Dial.

OPERATION. XXI. How to make a Declining Reclining Dial by the Globe.

SUppose your Plane declin'd 40 Degrees Eastward (as did the late Declining * 1.68 Vertical) and then Reclin'd 20 Degrees with a Southern Aspect, and by the way you must remember, that I mean in general by a Planes Reclining with a Southern Aspect, its looking towards that Quarter, tho' it be turned more or less from Direct South towards the East or West; in like man∣ner a Declining Reclining Plane with a Northern Aspect turns from direct North towards one of the aforesaid Points. Sup∣posing then a Plane thus Reclining, Do but describe or place it on your Globe, and your Operation will be as easy as any of the former.

First mount your Bead 71 Degrees and half above the Hori∣zon,* 1.69 that is to say fix it to 20 Degrees from the Zenith of the Globe; then seeing your Plane has a Southern Aspect, (and so lies beyond your said Zenith Northward) move your String till it cuts in the Horizon 40 Degrees Westward from the Northern Meridian, or back part of the 12 a Clock Hour Circle. In the next place take a Thred and tying it about your Globe so, that it lies not only on your Bead, but crosses also the Horizon at 40 Degrees from the East point Northward, and 40 Degrees from the West Point Southward, the said Thred will represent your Plane Reclining and Declining, as aforesaid. Or, in short fix, a small Needle in the Point where the Bead lies (which we suppose at A in Sch. 26.) and fastning to it a Thred or part of

the string, draw it over the Horizon at 40 Degrees from the East-Point Northwards, and it will give you the Eastern or Morning side of your Plane, as it will the Western or Afternoon side; if you draw it (as in Scheme 27.) over 40 Degrees of the Horizon from the West-Point Southwards.

This being done, describe a blind Circle or Semi-Circle e∣qual* 1.70 to a great one on your Globe, for Example sake, the blind Semi-Circle A. T. C, and drawing from (O) the Center the blind Line OA perpendicular to the Horizontal line H h, take the distance with your Compasses between A the station of your Needle or Bead, and the point in the 12 a clock hour Circle crost by the Thred or Edge of your Plane, and this di∣stance from A in your blind Circle, gives you there towards your left hand the Point k, to which if you draw a fair Line from the Center it will be the 12 a clock Line of your Dial, and the distance from the said station of your Bead or Needle to the intersection of the Thred with the next Hour-Circle will give you l, the mark of the 11 a clock Line; and in this man∣ner you must run over all other intersections of your Thred and Hour-Circles to the very Horizon on both sides of the Globe (I mean on the Morning and Evening side of it, represented by Scheme 26 and 27) and placing their distances on your blind Circle, on both sides of the aforesaid OI, do but draw lines to them from the Center, and your Dial is describ'd.

And here you must observe that I have (in Scheme 26. or* 1.71 Eastern Face of the Globe) plac't A (the Station of the Bead or Needle) above the Meridian, since its true place cannot be ex∣prest; for it ought to have bin on the other side of it, I mean on the Western side, which Scheme 27 is supposed to repre∣sent.

Now for the Stile and Substilar there is no difference from* 1.72 the Rules of the Declining Vertical, since 'tis but finding the nearest point on your Thred to the Pole by your Compasses; for the distance between the said Point on your Thred and it's in∣tersection with the 12 a clock Hour-Circle is the distance in the blind Circle between, k and M for the Substilar and the di∣stance between the said neerest Point and the Pole, gives MX the height of the Stile above the Plane. Nay, if you measure the Distance between each Point and A in any great Circle,* 1.73 'twill give you the Degrees or Distances between A and your Stile, Substilar, and each Hour-line, and consequently performs

the second way (as we have all along mention'd) of describing Dials by the Globe.

As for the Demonstration of this Dial, what we have for∣merly* 1.74 said about the rest proves it also; for supposing that the Thred represents truly your Plane, and that the Hour lines of a Dial, are (as I have show'd you all along) the several in∣tersections of the Hour-Circles with the Plane, this Dial must be true, since all the Lines on it are the said intersections, as drawn from its Center to the Points made by the Hour-Cir∣cles on its Edges: Nor can there be any error in the Substilar or Stile, the first being the intersection of the Plane with it's true Meridian of the Plane, I mean with that Hour Circle which falls on it at right Angles, and the other being the real Height (as you see) of the Pole above the Plane, ergo, the whole must be true.

OPERATION XXII. How to describe by the Globe a Dial Declining and Reclin∣ing as the former, with a Northward Aspect.

THere is no need here of a Scheme, the Construction of this* 1.75 Dial being in a manner the same as the former, only now you must draw your String and Bead (fitted to the Reclination) the contrary way, that is to say, over the South or forepart of the Globe throu' the 40th Degree in the Horizon East-ward from the Meridian or 12 a clock hour circle, then fixing a Needle (as * 1.76 I show'd you) on your Globe, or else tying a thred round it so, that it crosses still your Bead▪ and the aforesaid two Points in the Horizon, you have there the Plane repre∣sented, and may consequently (by the help of the former In∣structions) describe this Dial, whose Stile is to point upward, because of its Northern Aspect.

OPERATION. XXIII. How to describe all Inclining Dials, whether Direct or Declining.

AN Inclining Dial (of what sort soever it be) is the back or hinder part of a Reclining one of the contrary Aspect so that its hour-lines must be mark with the opposite Figures, and▪

the Stile must point the other way; therefore if you desire a Dial Declining East-ward 40 Degrees, and Inclining 20 with a Sou∣thern Aspect, describe only the last Dial, (which has, you see, the▪ same Declination and Reclination with a Northern Aspect) and then if you mark the Morning hour lines with the Evening Figures, and place the Paper draught the contrary way, that is to say, let the Apex of the stile point downwards, you will per∣form the Operation.

As for the Geometrical Description of Reclining or Inclining Dials since 'tis very intricate, I shall not now trouble you with it, especially having already show'd you so facil a way by the Globe.

OPERATION. XXIV. How to find the Degrees of the Reclination or Inclination of any Plane by the Globe.

THere are two ways to perform this Operation; for first* 1.77 (as I show'd you in taking the * 1.78 Level of a Plane,) the String rests just on the Horizon of the Globe, when it stands on an Horizontal Plane, or one 90 Degrees from being Erect and Vertical.

Draw therefore on the Reclining Face or side of the Plane (represented by Scheme 29.) a Line parallel to the Horizon (suppose AB) and let fall the Perpendicular CD, then place the Notches of the Pedestal of the Globe (mark't with SN) on the said Perpendicular, and consider what Degree in the Meri∣dian (counting from the Zenith) the String just lyes or rests upon, and that will be as well the Inclination, if the Plane in∣clines, as the Reclination if it reclines; for the Complement of this (I mean the distance between the Point, or Resting place of the String and the Horizon) showing always how much the Plane want's of being * 1.79 Level or Horizontal, the Degrees from the Zenith, must needs show how much it wants of being Erect or Vertical.

As for the second way, Draw a Perpendicular on the Recli∣ning* 1.80 side of your Plane▪ as I now show'd you, and placing on it (after the same manner) the Notches of the Pedestal, expect 'till the Shade of the Pin in the Zenith falls upon the Meridian of your Globe; for this show's the Sun to be at that moment in the Plane of the said Meridian; then observing on what Degree of it the Shade of Extuberancy falls, place but your Globe Level or Horizontal with your Meridian in the Plane of the Sun as be∣fore, and as the difference of these Degrees shows how much your Plane wants of being Horizontal, so that the Complement show's what it wants of being Erect, and consequently the value of it's Reclination if it reclines, or Inclination if it in∣clines.

[illustration]

Sch 29.

[illustration]

Sch 30

You may also if you please draw your Perpendicular on the Inclining side of your Plane (as in Scheme 30^th) but then the requir'd Inclination, if it inclines, or Reclination if it reclines, will be the difference in Degrees between the aforesaid shades of Extuberancy, after you cast away 90; for by how much the Inclination happens to be, by so much the shade of Extube∣rancy exceed's 90, since 90 is the difference between an Hori∣zontal, and an Erect Plane.

OPERATION XXV. How to find how long the Sun can possibly shine on a Plane, as also (from time to time) when we may expect him af∣ter his Rising to come on, or before his Setting to go off the said Plane.

I Defer'd this Operation till we had treated of all Planes, be∣cause the applying of it would then be better understood. 'Tis (tho' obvious and easy, of great Use) as not only showing us what Hour lines are absolutely necessary on all Dials, and what not, but telling us also at what a clock (all the year long) we may expect the Sun on our Plane, and at what a clock he must go off it; for (notwithstanding He be above the Horizon, He will not always so long shine on a Plane not Horizontal, as by the Ear∣liest and Latest hour lines (that may be justly exprest on it) one might expect.

If then you would find (suppose on a Declining Plane) what* 1.81 hour lines may be justly and necessarily drawn on it, I mean what the earliest and latest hour lines ought to be, you are only to draw you String from the Zenith (according to the Declen∣sion) on both sides of the Meridian (or 12 a clock hour circle) to the very Horizon; that is to say, you must operate in the same manner as you do, when you describe the Plane in the Fa∣brick of this kind of Dial; for the Hour circles cut by your said String in the Horizon show you exactly how early he can come on, and how late he can stay on it; so that to express fur∣ther Lines were needless. This then makes you stop at 4 in the Afternoon in your late * 1.82 Example, where the Plane declines 40 Degrees Eastward, whereas had it declin'd but 20 your ear∣liest hour (as you may see if you try) would have bin five in the Morning, and the latest five at Night. In short, describe your Plane (let it be what it will) on your Globe with your String, and your Hour circles, (as we said) that intersect with it in the Horizon answer the Question, since it clearly appear's (your String ever representing the Edges of the Plane) that if the Sun lyes Easterly in the Morning, and Westerly in the Evening

of the Hour-circles, that meet your String in the Horizon, He must be behind your Plane; therefore since he is not then able (tho' up) to shine upon it, 'twere needless (as we said) to express more Hour lines.

'Tis the Describing also of the Plane with your String that* 1.83 brings us to the knowledge of the second part of this Operation, I mean the knowing at all times when the Sun comes on, and goes off any Plane; for having describ'd one (Declining, v. g. 20 De∣grees Eastward) do but observe what Diurnal Parallels and Hour-circles intersect on the Edges of your Plane, and you have your Intent; for you will by this means see, that, (tho' the Sun rises (for example sake) on the 11 of June before 4) the first hour circle, which intersects with this Parallel on the Edges of the Plane, is that of a Quarter before six, whereas a∣bout the beginning of May, he is there at half an hour past five, and on the 10 of April at or near 5. Now if you consider in the same manner the West-side of the Globe, you will see from time to time at what hour he goes off it; and thus you may do, let the Plane be what it will.

Here therefore it evidently appears, if you should erect at a∣ny* 1.84 time (suppose about the 10th of April) a Perpendiculur stile on an Horizontal Plane, and draw every Hour a Line along the Shade of the said stile, why such a Dial will be false, as only tel∣ling you the true Hour twice in the year, to wit on the 10th of April, and about the 10th of August, viz. on the days on which the Sun run's in the same Diurnal Parallel; I say, all this now evidently appears, since every Line thus drawn on an Horizon∣tal Plane (except the Meridian, or 12 a clock line) is no Hour line but an Azimuthal Section; I mean the Section of the said Plane, with a Circle that then passes over your head throu' the body of the Sun; so that if one of these Lines should Bear (sup∣pose) almost SE, and be figur'd with 10 in the morning, Draw but your String from the Zenith, over that Bearing, or Point of the Compass in the Horizon of your Globe, and it will truly re∣present the said shade or Line on your Plane; for it show's it to be 10 of the Clock on the Parallel belonging to the said 10th of April: But since your String cuts also on your Globe (v. g.) the Tropic of ♑ at a little before 9, and the Tropic of ♋ at almost half an hour past 10, you may conclude that this will be the true time of the Day on the 11 of December, and 11 of June,

tho' the shade of the Perpendicular stile still show's 10 a clock at the aforesaid Bearing, let the Season of the year be what it will; therefore a Dial thus made must be false.

Of several ingenious and humersome Dials.

HAving thus run throu' all Planes, I shall at present show you how to make use of the former Principles, as to the ready Describing of several ingenious and humersome Dials, for all are now in a manner but Corollaries from what we have already said, and consequently easy both in Speculation and Pra∣ctice.

OPERATION XXVI▪ How to make a Dial on any Plane whose stile shall be an Arrow fixt casually on it.

EXamine what the Plane is, and having found it to be, sup∣pose,* 1.85 a Vertical one Declining 40 Degrees East-ward, de∣scribe by your* 1.86 former Rules: such a Dial on Paper with the Pa∣per stile F x, M. (as in Scheme 31.) exactly set, and mounted; then draw on the Plane an Horizontal Line H h, and place on it your said Paper draught so, that the 12 a clock Line FP may fall at right Angles on the said Horizontal line. Lastly, move your Draught along it, till some part of F x or Indicating side of the stile, (suppose the Point A) just touches the Top or most promi∣nent Part of the Arrow, and fixing there the said Draught, if you draw fair Lines on your Plane under those on the Paper, the said Arrow will always show you the Hour with its Top.

The Reason is plain: for you see by the said Top's just touch∣ing* 1.87 the Edge, or Indicating side of the Paper-stile, it has the ef∣fect of the Top of AB, I mean the Top of a Perpendicular fall∣ing from the said side on the Sub-stile, so that X the Top of XM (both in the present Scheme and also in Scheme.* 1.88 18. or Example of a Declining Plane) has this Effect also. Now since the Top of AB or XM or of any other Perpendicular, that falls

from the Indicating side XF on the substile FM will perform the Office of the stile (as we show'd you at large in Demonstra∣tion of the * 1.89 first Horizontal Dial or first Example,) it must necessarily follow, that A the Arrow's Top do's the like.

OPERATION XXVII. How to make a Dial to show the Hour without a stile on a∣ny Plane.

DEscribe (as in Scheme 32.) a Dial on P the given Plane, and erect for the present a true stile (as FAB) of Paper or the like, then fixing a Glass or any other transparent mat∣ter (suppose G) at what distance you please, before the said given Plane and Parallel to it, mark where A the Top of the Stile just touches the said Glass; and if there you paint a little Asterisk or spot, it will (as often as the Sun shines) describe such another Figure (at suppose D) by its shade on the said Plane P, and move also from Hour Line to Hour Line, according to the true time of the day.

The reason of this is also Evident; for, if A the top of the real* 1.90 Stile show's the Hour by casting a Shade (as we show'd you all along) on the Hour Lines, then the Asterisk being there painted where the said Top touches the Glass, must do the like; for it is, you see, the Stile's Apex or Top, and consequently casts a true shade to know the Hour by.

This Dial serves not only for all double Windows, or for Cavi∣ties* 1.91 that have over them any Glass or Transparent matter, but shows us how to make one for any Plane, that is illuminated by a Ray coming throu' a Hole, since if you describe the Planes pro∣per Dial on Paper, and move it duly (as before) on the said Plane, 'till the Stile, or (if that be too short) 'till a Thred drawn along its Indicating side, touches the Hole, it will give you marks for the drawing the fair and standing Hour-lines of your Plane, which the said Ray will dayly run over in order, and consequently show you from time to time the Hour; for the Ray passing (as you see) throu' the Hole (v. g.) at A, and falling on the true Hour Line at D, performs what A, the Apex of the true Stile (FAB) would do.

OPERATION XXVIII. How to describe a Dial, having a Picture of a Man in it, that shall Point to the Hour from time to time with his Finger.

THIS Dial is on several Planes of Mr. Lines his foremen∣tioned Pile in Whitehal Garden; and as no Dial can be more useful, so perchance none ever struck the Fancy, both of the Ignorant and Learned, with a more sudden Admi∣ration 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 sur∣prize 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* 1.92 were that of the Vertical Cavity, a b c d, lying directly South; describe therefore on the Glass (ABCD) the contra∣ry Dial, i. e. a Direct North Dial, with a Paper Style truly mounted; and placing the said Glass over the Plane, and Para∣lel 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 pain∣ted; 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 con∣sequently shows you what a Clock it is.

For if there were a Hole that passed at E (the Top of the* 1.93 Mans Fingers) throu' the Center of the World to our Antipo∣des, it necessary follows (by the Reasons in our former Operati∣on) 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 contra∣ry)

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 'tis suffi∣cient* 1.94 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 Sta∣tion or Place, for then you may cross the said Frame with small Strings or Wyars, which will by their interposition cast the same shade as the Hour-lines of the Glass would have done; so that if the Figures belonging to the said Lines be put on the Frame, at the end of each corresponding Wyar, and then pierc'd, the Sun Beams passing throu' their Cavities, will distinguish each very perfectly on the Plane.

Tho I have not time to show you all the particulars of this Learned Man's rare Invention in Dialling; (for most of the Di∣als on the aforesaid Pile may be naturally and expeditely de∣scrib'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 Hemis∣phere* 1.95 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 Cir∣cles (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, 'tis presently con∣ceiv'd;* 1.96 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 Hemi∣sphere) 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 Cen∣ter) 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 correspond∣ing Hour-Circle, and if so, then a Blind-man (by feel∣ing 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* 1.97 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, 'twill follow that the Hour-Circle (suppose A, which the Sun's Pi∣cture 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,) where∣about he is, you will be able to guess the Hour without any sen∣sible Error; for the said Brightness appearing on the Bowl will be proportionably distant from the Sun's true place there, as 'tis from the Sun in the Heavens.

'Tis clear that the Suns Picture must fall (if any where) on the* 1.98 true Hour-Circle, because (by Composing the Bowl according to the true Position of the Heavens) the Hour-Circles of the one con∣cur with the other, and fall exactly in the same Plane; therefore were your Eye in the Center of the Bowl, its true 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 hap∣pen; so that seeing the Suns Ray (by reason of the Blew Colour) penetrates not the Glass, his Picture must needs be on the out∣side 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 Pi∣cture 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* 1.99 BAC in Scheme 37.) that A the Center of the said conca∣vity 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 conve∣nient 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* 1.100

than the Globe, yet it serves to illustrate and make easy the Geo∣metrical 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* 1.101 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, there∣fore 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* 1.102 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 'tis 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.

[illustration]

Sch. 40.

As for the truth of this* 1.103 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 An∣gles and Distances; all which is exprest at large in the third * 1.104 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 ma∣king 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 Hori∣zontal Dial whose Center shall lye on O, and its Meridian Line concurr with HF, 'tis 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 the Day; But by Construction all the hour-lines are thus drawn on the Cieling, and consequently are exactly over the supposed ones on the Floor, Ergo, the Reflext Ray from G must as truly show you the Hour above, as the Direct Ray below; for both Rayes are ever in the same Plane.

Nor is there to be any real Difference in the Operation tho'* 1.105 the Chamber-window should look another way; for you are only to remember, that whilst it enjoys the least Point of South, the Center of your Dial is without the Chamber, when it looks full East or West 'tis in the side or edges of it, and when it ver∣ges Northward, 'tis altogether within; so that in a full Southern Aspect, the said Center will be most abroad, and in a full Northern one the Contrary; all which plainly appears to any one, that will consider an Horizontal Dial truly plac'd (having a Perpendicular for its Stile) if he draws over the Hour-lines, a Line that shall represent the aforesaid side of your Chamber according to its Position and Site.

OPERATION XXXIII. To make a compound Dial to wit, one containing several useful Operations.

INnumerable are the ingenious Dials that may be invented, but since we have been long enough on this Subject, either for my Reader's Speculation or Curiosity, I will now conclude, and that with a Recapitulation or summing up of much of what we have already said, by showing the Fabrick of a Com∣pound Dial; that is to say, one that contains many useful Ope∣rations, besides the Hour; for nothing rubs up the Memory more efficatiously, or makes us more Masters of our Rules, than a Practical Example.

The said Dial shows as follows.

• 1. The Hour with us at all times.
• 2. The Hour in what other Countries you please.
• 3. The Sun's Place in each Sign.
• 4. The Day of the Month.
• 5. The time of the Sun's Rising and Setting.
• 6. The Sun's Amplitude.
• 7. The Sun's Height.
• 8. The Sun's Azimuth.
• 9. The Sun's Bearing according to the Points of the Com∣pass.
• 10. The Proportion between Perpendiculars and their Sha∣dows, and consequently the height of any Tower or the like.

To make then this Dial, you must first describe an Hori∣zontal* 1.106 (as in Sch. 41.) about a Foot in Diameter, and let B the Center of the Plane be the Point, where an Erect, or Ʋpright Stile (according to our Directions in the * 1.107 first Horizontal;) shews you with its Top the Hour. Now because the Shade of an Ʋpright Stile, unless it be very short, will presently fall out of the Plane, as well in the Morning as toward Night,

therefore it will be convenient to have your Cock or Stile made so, that AB the Perpendicular or fore-part of it (as in Scheme 42.) should stand at B the said Center of the Plane, to repre∣sent this upright Stile, and its Angle AOB at O the Cen∣ter of the Dial, or Point from whence all the Hour-lines are drawn; for thus the side OA (making with the Meridian line at O, the Angle of the Elevation) represents the Axis of the World, and consequently casts its shadow on the Hour-lines, as the usual Cocks of all Horizontal Dials do.* 1.108

2. Having chosen all the Places, which you desire from time to time to know what a Clock it is at, consider well your Globe, and find under what Hour-Circles the said Places lye; as for Example, suppose Rome lies under the 11 a Clock Hour-Cir∣cle, Constantinople under that of 10, Aleppo 9, &c. Place therefore the said Towns towards the Limb of your Dial, un∣der the corresponding Hour-lines, and you will constantly know the time of the Day in the said Places; for calling it always Noon at each Place you seek after, you have nothing to do but to count the Hours from thence to the shade of the Stile; as v. g. If it be 4 a Clock with you in the afternoon, and you would know the Hour at Aleppo, let Aleppo be 12, and counting from thence (1. 2. 3. &c.) 'till you come to the Hour of the Day, (I mean the Hour then shown you by the Shade,) you will find it to be 7 a Clock there; for Aleppo is (you see) three hours Eastward of you; now had the Hour with you been 4 in the mor∣ning, you must have counted backwards, as 11, 10, 9, 8, and consequently you would have found it there 8 in the morning. In this manner then you must operate all along.

3ly. and 4ly, Find by your Globe exactly the Sun's height e∣very* 1.109 hour at his Entrance into each Sign, then take by the help of your Sector (AB, the Erect Stile in Scheme 42. being Radi∣us) the Tangent Complements of the Heights, and putting one Foot of your Compasses on your Dial at B, make Pricks or Marks in each corresponding Hour-line accordingly; that is to say, if the Sun be high (suppose) 50 Degrees at 12 of the Clock, when he enters ♉ or ♍, then take the Tangent of 40 and prick that distance in the Meridian line, viz. From B to f; and if his height at 1 and 11 a Clock be (v. g.) 48 degrees, take the Tangent of 42, and prick that distance in the 11 and 1 a Clock lines, viz. from B to h and g, and when you have gone thus o∣ver

all the Hour-lines, no sooner will the Sun come into ♉ or ♍ but the Shade of the Point or Apex of the Stile AB will fall e∣very hour on the aforesaid Pricks, and consequently show you the Suns place in the Ecliptic. In like manner you must do with the rest of the Signs, and then with the 10th Degree of eve∣ry Sign, placing still the Character of each Sign about the Limb of your Dial, near the last mark or Prick belonging to it. This being done, see by your Globe what day of the month corresponds with each Sign, and what with their Subdivisions, and if you mark this (as the said 41th Scheme shows you) on both sides of the Meridian, then the said Pricks will (by the help of the Shade of the top of AB) show you also the day of the month. I mention here Pricks not only as an easier way, but a better way than Lines; for besides the great difficulty of draw∣ing them, they embarras and confound a Dial very much, es∣pecially if there be many of them; whereas the said Pricks are never out of an Hour-line, and consequently take up no new room. Now to avoid Confusion and Mistakes, I would have the said Pricks of 3 sorts at least, for if one Row were (v. g.) Astericks and another Crosses, and a 3d Plain Pricks, you would then know at first sight, to what Sign or Day of the month any of them belongs.

5ly. Instead of troubling you with deviding the Circle* 1.110 GKLT (the upper part of the Border of the Dial) for the finding out the time of the Suns Rising and Setting, you need only consult the Days of the Month on your Globe, first, when He rises earliest, Secondly, when He rises at 4 a Clock, Third∣ly, when at 4½; Fifthly, when at 5; and in the like Proportion go on, till the Days come to their greatest Decrease, and put∣ting the said days of the Month in Order (as they are in the Scheme) under the corresponding Hours on the morning side of your Dial for his Rising, do the like for his Setting on the Evening side of it, and you may perform the Operation with sufficient Exactness. In like manner you are to proceed for the Quarters, half Quarters, &c. if you would have them exprest.

6ly. To avoid also the trouble of deviding the Circle* 1.111 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 according to the Suns Diurnal Increment and Decre∣ment in Amplitude, you need only find by your Globe, what the said Amplitude amounts to on every of the aforementioned Days (which are markt on your Dial for the Suns Rising and

Setting) and then put it in Figures under each Day, as the Scheme shows you.

7ly. Open your Compasses at the Tangent of 28 Degrees (AB* 1.112 being the Radius) and putting one Foot on B describe the Cir∣cle XYZ, afterwards describe another according to the Tan∣gent of 35 Degrees, then a third, according to that of 40, and so on in the same Proportion as far as your Plane permits. Now if you mark these Circles with the Figures of the Comple∣ment of their Degrees, that is to say the Circle of 28 Degrees with the Figure 62, that of 35 with 55, that of 40 with 50, &c. you will always know the height of the Sun; for what Circle so∣ever the Shade of AB touches with its Top, that will be the requir'd Height; and if it falls between 2 Circles, 'tis but con∣sidering which of them it comes nearest to, and then you may guess at the Height with sufficient exactness.

8ly, and 9ly. Devide one of these Circles viz. SEWN into* 1.113 Degrees, and under each 11 Degree and ¼, place the several Points of the Pixidis Nauticae, or Mariners Compass in the Or∣der as they are express'd in our said Scheme, and you will not only have (by the Shade of AB) the Suns Azimuth at all times, but see also how he bears from you according to the Points of the Compass; and if the Shade be at any time too short, lay on it but a Ruler, Label of Paper or the like, and that will truly lengthen the said Shade, and resolve your Question.

10thly. Devide AF the Northern half of the Meridian, as* 1.114 many times as you can by the Stile or Radius AB, and then each Devision into ten equal parts (as you see it done in the said Scheme) and by it you will know at all times the Propor∣tion between any Perpendicular and its Shade, and consequent∣ly, (besides many other things) the height of any Tower, Tree or the like, for having found the Sun to be (suppose) 25 De∣grees high, and that the Circle of Altitude cuts the Linc AF in the 22 Devision, if therefore you measure the Shade of your Tower, and finding it (for Examples sake) to be 66 Yards long, you have what you seek; for as the said 22 is to 10 (the Stiles height) so is 66 the length of the Shade to 30 the height of the Tower.

So much then for the Construction of Dials. And now let me desire all those that are pleased to follow this Geometrical way (which perchance is as expedite a one, and as free from blind

Lines as can be,) not to rest satisfy'd till they fully comprehend what they do; for the Mechanical way of Dialling is as soon lost as learnt, it being impossible (without continual Practice) not to forget the Rules, especially if one can make many Dials; when as a man that understands the reason of the Operations (by having in his Head a true Idea of the Sphere and its Proje∣ction) will 20 years after without Memorandums or Notes, be able (reflecting but a little) to make not only all Dials he for∣merly knew, but new ones also at first fight.

To Conclude, I here present my Reader with the Globe in a new Dress, for being painted or stain'd on Marble (according to Sch. 43.) 'twill be fit for any Garden or open Portico; and least it might appear too plain, the corners of its Base or Pedestal may be adorned with handsom well turn'd Branches, which not only embellish the whole Machin by their Make, But hold out Bowls of Glass and Wyar for use also.

For on the First Corner, to wit, That markt with A, there is* 1.115 placed (as a Rarity.) The blind man's * 1.116 Dial. On the Second markt with B. The † 1.117 Dial that shows the Hour, when the Sun shines not, which will be often very useful. On the third, mark't with C, there is an Armillary Wyer Sphere having a Vane on the Top, that continually shows on the brass Plane within (graduated and Nautically Character'd) from what Quarter the Wind exactly blows; as also, (if you turn the said Vane into the Plane of the Sun) his Azimuth and Bearing. Besides, the Sphere (being an Horizontal Concave Dial) shows the Hour too; for the Shade of the Pin's top in the Center ever fall's on the true Hour-Circle, as I show'd in the * 1.118 Construction of such a Dial. And by the way you must know this Branch stands not in it's true place in the Scheme; I mean on the third Corner of the Base, because in Perspective 'twill fall on the Globe it self, and conse∣quently not appear well to the Eye in a Picture. Lastly, on the fourth Corner markt with D there is another Glass Bowl of the former Dimension, containing orderly all the Constellations, and remarkable Stars, and therefore, if you know the Hour, it will compose the said Bowl or Globe, and so represent the then position of the Heavens; but (tho you are Ignorant of the Hour) if you see a known Star, and move the Bowl on its Axis, till the painted star on it lyes just between your Eye and the Real one, you have the Hour, and consequently may know (the Globe being

now Compos'd) any Star or Constellation above the Horizon; for the Axis of this Bowl having one end pointing directly to the North Pole, and the other fixt in the Center of a Rundle containing on its Limb the Days of each Month, fitted to the right Ascension of the Stars, and moving also on a Plane divided into 24 equal parts, figured with the hours of a Natural Day, 'twill follow that the Day of the Month (when the Globe is Compos'd) must lye on the true Hour, as the true Hour move'd to the Day of the Month must Compose the Globe, as is before hinted. These short directions are sufficient for any Mathematician, or In∣strument-Maker; and as for the Branch it self, 'tis (as you see) not in its true Place for the above mentioned Reason.

J. Moxon To the Reader.

HAving Courteous Reader * 1.119 engaged to show you the Pro∣blems and Operations on the Sector, which the Noble Au∣thor supposes every one (that studies the Geometrical way of Dialling) to know, I shall here begin.

I. Upon a Line given (AB) to erect (CD) a Perpen∣dicular.

[illustration]

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 Perpen∣dicular 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 In∣tersections

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.

[illustration]

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 cut∣ting 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 Pa∣rallel to it.

[illustration]

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.

[illustration]

AB being a Line as long as the side of the Square you design, erect on the end A, the Perpendicular DA of the for∣mer 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.

[illustration]

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 Com∣passes 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 Isos∣celes.

[illustration]

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 In∣tersection 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 be∣tween 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.

[illustration]

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.

[illustration]

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.

[illustration]

* 1.120So much for the Geometrical Problems necessary for Dial∣ling, 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 man∣ner 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 re∣quired, is as 100 to 50.

III. To divide a Line as any other Line proposed is di∣vided; 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 parts of the Sector; Open your Compasses at 13, or length of the Line to be devi∣ded, and putting it over at 65, and 65 on the Sector, the Pa∣rallel at 5 and 5 will be the first division of the Line to be divi∣ded, and one equal part of the Sector in value; the Parallel at 15 and 15 will be the second, and three equal parts in value; and the remainder (being 9 in value,) will be the third; and thus you may do in all other cases.

IV. To encrease or diminish a Line in any Proportion.

SUppose the Proportion were as 4 to 7, take the Line given between your Compasses, and setting it over on your Sector, at the Figures 4 and 4, the distance from 7 to 7 will be a Line encreast (in respect of the given one) as is the Proportion of 4 to 7; In like manner you must do, if any other Proportion were requir'd.

Now if you would diminish a Line as is 7 to 4, put over the Line given at 7 and 7, and the distance of 4 and 4 is the re∣quir'd proportion.

V. Two Lines being given, to find a Third Proportional.

FIND by your Compasses how many parts of the equal Di∣visions of your Sector will measure both your given Lines, so that supposing the one to contain 10 parts, and the other 20, set the second Line (i. e. the Line 20) over at 10, and 10 on the Sector, and the distance or Parallel at 20, and 20 on the Sector, will be 40, the requir'd Proportional.

VI. Three Lines being given, to find a Fourth Proporti∣onal.

THE value of the Lines being found as before, and supposing the first to be 10, the second 20, the third 30, put over the second Line (to wit, 20) at 10 and 10. (the value of the first Line) and the distance or Parallel at 30 and 30. (or value of the third Line) will be 60, the required Proportional.

Of the Use of the LINE of SINES, markt with S.

[illustration]

I. How to find the Sine of any Angle, according to any Radius.

SUppose the Sine of the Angle you require be 50, take the Radius between your Compasses, and put it over at the extremity of the Sector, that is to say, at 90 and 90, and the Parallel at 50 and 50 will be the Sine of 50 Degrees, accord∣ing to that Radius.

II. How to find the Chord of any Arch.

SUppose you would have the Chord of an Arch of 50 Degrees, open your Compasses at the length of the given Radius, and put it over at 90 and 90, then take with your said Com∣passes the Parallel, at the Figures 25 and 25 on the Sector, (i. e.

at the Figures of half the Degrees given) and prick or measure it twice upon any streight Line, and that will be the required Chord.

III. How to make an Angle of any value, as also how to find the value of any Angle already drawn.

[illustration]

SUppose you are to make an Angle of 50 Degrees, draw a Line, as (for Example sake) AD, and taking any Point in it, as B, open your Compasses to a convenient Radius, and put one Foot on B, and describe the blind Arch CF, then taking between your Compasses the Chord of 50, according to the Radius of the said Arch, put one Foot on C, and the other marking at suppose E, draw the Line BE, and you have the required Angle, to wit, the Angle EBC. But if the Angle you would make be above 90 Degrees, as suppose 130, make the Angle of its Supplement, viz. the Angle of 50 as before, and the Angle on the other side, viz. EBA will be the Angle you look for:

Here therefore you see how to find the value of any Angle already drawn, as suppose the Angle EBC, since 'tis but de∣scribing a blind Arch, as CF, and setting over (on the Sector) the Radius of the said Arch at 90 and 90; for if you observe where the measure of this Arch (viz. C. E) marks a Parallel on the Sector, as before, the Figures there (to wit, 25) being doubled (and amounting consequently to 50) will be the requir'd An∣gle.

As for the Tangent and Secant of any number of Degrees, the Noble Author has himself showd you how to find them at pag. 71. and so gentle Reader having finish'd my promise, I bid you Farewel.

Notes

• * 1.1

  A prelimina∣ry Discourse.
• * 1.2

  pag. 4.
• † 1.3

  pag. 5.
• * 1.4

  pag. 10▪
• * 1.5

  ☞ Because every body (that desires to know these and the fol∣lowing Pro∣blems) has not perchance at hand Mr. Gun∣ter's Book, I shall add them to this Trea∣tise as the Rea∣der will find at the end of it: J. Moxon.
• * 1.6

  How to find the Tangent and Secant of any degree.
• * 1.7

  Demonstra∣tion.
• * 1.8

  A Memoran∣dum.
• * 1.9

  The Constru∣ction.
• * 1.10

  An Example.
• * 1.11

  How to draw the half hours, quarters, &c.
• * 1.12

  The Constru∣ction.
• * 1.13

  Demonstrati∣on.
• * 1.14

  pag. 71.
• * 1.15

  The Demon∣stration.
• * 1.16

  pag. 4.
• * 1.17

  A Memoran∣dum.
• * 1.18

  pag. 8.
• * 1.19

  pag. 73.
• * 1.20

  The Constru∣ction.
• * 1.21

  The constru∣ction.
• * 1.22

  pag. 79.
• * 1.23

  How to draw a Line Para∣lel to the Ho∣rizon, and how to place truly the draught on its Plane
• * 1.24

  An easier way how to place any paper draught on its Plane.
• * 1.25

  A Memoran∣dum.
• * 1.26

  Demonstrati∣on.
• * 1.27

  A Memoran∣dum.
• * 1.28

  Some few things to be premis'd.
• * 1.29

  The Constru∣ction of an East Dial.
• * 1.30

  Of a West Dial.
• * 1.31

  Of the Stile and Substilar.
• * 1.32

  The Demon∣stration.
• * 1.33

  The Constru∣ction.
• * 1.34

  The Demon∣stration.
• * 1.35

  The reason of the unequal distance of these hour∣lines.
• * 1.36

  What a De∣clining Plane is.
• * 1.37

  The Constru∣ction.
• * 1.38

  pag. 80. To describe the Morning hours of a Declining Dial.
• * 1.39

  To describe the Afternoon Hours.
• * 1.40

  How to make the Stile and Substilar of a Declining Dial.
• * 1.41

  A Memoran∣dum.
• * 1.42

  The Constru∣ction.
• * 1.43

  The Demon∣stration of these 2 decli∣ning Dials.
• * 1.44

  A Memoran∣dum.
• * 1.45

  pag. 84.
• * 1.46

  Another De∣monstration.
• * 1.47

  pag. 83.
• * 1.48

  pag. 73.
• * 1.49

  The constru∣ction.
• * 1.50

  pag. 23.
• * 1.51

  pag 8.
• * 1.52

  The Constru∣ction.
• * 1.53

  A ready way to find the Stile and Sub∣stile of a de∣clining Dial.
• * 1.54

  Demonstrati∣on.
• * 1.55

  pag. 92.
• * 1.56

  The Demon∣stration of the Stile and Sub∣stilar.
• * 1.57

  pag. 82.
• * 1.58

  vid. pag▪ 104.
• * 1.59

  vid. pag▪ 105.
• * 1.60

  The Constru∣ction and De∣monstration.
• * 1.61

  How to make an Horizon∣tal Plane an Aequinoctial one.
• * 1.62

  The Demon∣stration.
• * 1.63

  The Geome∣trical Con∣struction.
• * 1.64

  A Memoran∣dum.
• * 1.65

  pag. 8. The Constru∣ction.
• * 1.66

  The Constru∣ction and De∣monstration of a declining direct Dial facing the South.
• * 1.67

  A Direct North recli∣ning Plane.
• * 1.68

  pag. 89.
• * 1.69

  How to de∣scribe the Plane of this Reclining Di∣al on the Globe.
• * 1.70

  The Constru¦ction.
• * 1.71

  A Memoran∣dum.
• * 1.72

  Of the Stile and Substilar
• * 1.73

  pag. 82.
• * 1.74

  Another De∣monstration:
• * 1.75

  The constru∣ction.
• * 1.76

  pag. 2.
• * 1.77

  First way.
• * 1.78

  pag. 4.
• * 1.79

  pag. 4.
• * 1.80

  The second way.
• * 1.81

  The first part of the Ope∣ration.
• * 1.82

  pag. 89.
• * 1.83

  The second part of the Operation.
• * 1.84

  Why every E∣rect Stile or perpendicular show's not al∣ways the true Hour.
• * 1.85

  The Constru∣ction:
• * 1.86

  pag. 89.
• * 1.87

  The Demon∣stration.
• * 1.88

  pag. 94.
• * 1.89

  pag. 75.
• * 1.90

  Demonstra∣tion.
• * 1.91

  The Advan∣tage or use of this Dial.
• * 1.92

  The Construction.
• * 1.93

  Demonstra∣tion.
• * 1.94

  Another Dial of the same nature.
• * 1.95

  The constru∣ction.
• * 1.96

  The Demon∣stration.
• * 1.97

  The Constru∣ction.
• * 1.98

  Demonstra∣tion.
• * 1.99

  The Constru∣ction.
• * 1.100

  The Geome∣trical way.
• * 1.101

  The Demon∣stration.
• * 1.102

  The Constru∣ction.
• * 1.103

  The Demon∣stration.
• * 1.104

  pag. 73.
• * 1.105

  How this Di∣al is to be made when the windows lye not South∣ward.
• * 1.106

  The Constru∣ction of it, as to the Hour at home.
• * 1.107

  pag. 73.
• * 1.108

  The Constru∣ction of it as to the Hour in other pla∣ces.
• * 1.109

  To find the Suns place and day of the Month.
• * 1.110

  To find the Rising and Setting of the Sun.
• * 1.111

  To find the Suns Ampli∣tude.
• * 1.112

  To find the Height of the Sun.
• * 1.113

  To find the Suns Azi∣muth and Bearing.
• * 1.114

  To find the Proportion of Perpendicu∣lars to their Shades.
• * 1.115

  The descripti∣on of the Bran∣ches or Embel∣lishments in Sheme 43.
• * 1.116

  p. 111.
• † 1.117

  p. 112.
• * 1.118

  vid. p. 113.
• * 1.119

  p. 71.
• * 1.120

  Of the Sector.