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2. Set 30 deg. the Plains Declination, from N to c, if the Plain Fig. Decline Eastward, as in this Example, or from N to e, if Weftward: LVI. Then lay a Ruler from Z to c, and it will cut the Horizontal Line of the Plain in K, fo have you three Points Z, K and N, by which to defcribe the Arch ZK N, reprefenting the Meridian of the Place. And to find the Pole thereof, fet 90 de. from c, to d, and then a Ruler laid from Z to d, will cut the Horizontal Line HO, in W, which is the Pole of the Meridian Circle Z K N and the Weft-point of the Horiz.

3. Set 51 deg. 30 m. the Latitude of the Place, from O to a, and from N to b: Then a Ruler laid from W to a, will cross the Meridian in P, the Pole of the World: And laid from W to b, it will cross the Meridian in E, the point through which the Equinoctial Circle is to pafs: And now you have two points W and E, through which the Equinoctial Circle æ Æe may be defcribed (by the XXI Geometrical Problem, Lib. 1.)

4. Through P, the Pole of the World, and Q, the Pole of the Plain, draw the right Line PQ, for the Axis of the World, and Sub-filar Line of your Dial: And in this Line (extended) will the Center of the Equinoctial Circle æ æ be found.

5. From P, the Pole of the World, lay a Ruler to A, the interfection of the Meridian and Equinoctial, and it will cut the Plain B: At this point B, begin to divide the Semicircle H NO of the Plain, into 12 equal Parts, at the points o☺ ☺, &'c.

6. From Q, (the Pole of the Plain) lay a Ruler to every of the points o, &c. and it will cross the Equinoctial Circle æ Eæ, in the the Points ***, &c.

7. Lay a Ruler to P (the Pole of the World) and every of the points ***, &c. and it will cut the Primitive Circle reprefenting the Dial Plain, in the points N, 9, 10, 11, &c. on the Weft fide, and N, 1, 2, 3, &c, on the Eaft fide of the Meridian.

8. Laftly, Lines drawn from the Centre Q through thefe points, fhall be the true Hour-lines of an Erect Plain Declining from the South-Eastward 30 de. in the Latitude 51 de. 30 m. And now,

Concerning the other Requifites belonging to this Ered Decliing Plain.

These are all of them reprefented to the Eye in the Scheme of the Projection of the Plain: Wherein, by the interfection of the feveral Circles there is conftituted a Right-angle Spherical Triangle ZTP, in which,


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CZ T is the distance of the Sub-file from the Meridian.

LVI. The fideZ P the Complement of the Latitude of the Place.
TP the height of the Pole above the Plain.

SPZ T, the Complement of the Plain's Declination."
The Angle Z P T, the Plain's difference of Longitude.
ZTP, is a Right Angle.

And all these Sides and Angles may be meafured upon the Proje
ction it felf, by the Precepts deliver'd in the firft Book, Se&t. in.
of Spherical Trigonometry Geometrically performed by Projection. And
they will be found to be as here expreffed, viz.

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de. m.


ZTP. -90---00

Side Z P-38----30>Angle Z P T-36----25

2T P

T P−32...- 36

And now I fhall fhew how all these may be found

By Trigonometrical Calculation.

Before any Ered Declining Dial can be made, there are Three things which must be found, befides the Hour-diftances, and those are thefe :

1. The height of the Pole (or Stile) above the Plain.
2. The diftance of the Subftile from the Meridian.
3. The Plain's difference of Longitude.

All which may be found in the Triangle ZT P: In which there is given (befides the Right Angle at T,)

(1.) The Angle TZ P, the Complement of the Plain's Declination, 60 deg.

(2.) The Side Z P, the Complement of the Latitude of the Place 38 de. 30 m.

By which may be found,

I. The Side T P: The Height of the Pole, or Stile, above the Plain:

By CASE I. of R. A. S. T.

As the Radius, S. 90 deg.

Is to the Side ZP, 38 d. 30 m. the Co-Latitude.

So is the Sine of T Z P: the Co-declination 60 de.

To the Sine of T P, 32 deg. 26 m.

Which is the height of the Pole (or Stile) above the Plain.

-II. The


II. The Side Z T, The Distance of the Sub-file from the Meridian: LVI..

By CASE II of R. S. T.

As the Radius Sine 90 de.

Is to the Tangent of Ź P, the Co Latitude 38 d. 30 m.

So is the Co-fine of T Z P, 30 deg.

To the Tangent of Z T, 21 de. 40 m.

Which is the Diftance of the Sub-file from the Meridian.

III. The Angle ZP T, The Plains difference of Longitude: By:

Cafe III. of R. A. S. T..

As the Radius,

Is to the Tangent of the Declination T Z P, 60 deg.

So is the Co-fine of Z P, the Co-Latit. 51 de. 30 m. To the Co-Tangent of T PZ, 35 de 25 m.

Which is the Plain's difference of Longitude.. |

Thefe three Requifites being thus found by Trigonometrical Calculation; The Plain's Difference of Longitude 36 de. 25 m. falling between 30 and 45 deg., (which are the Second and Third Equinoctial Hour-diftances) there will be contained therein two compleat Hours, and 6 deg. 25 m. over which fhews that the Sub-tilar Line of the Dial will fall between the Hour-Lines of IX and X in the Morning (in this Eaft Dial) but between the Hour of II and III in the Afternoon if the Plain had Declined Weftward:

Having proceeded thus far. Prepare a Table of Hours fit for the Plain, fuch as is here done..

A Ta-.

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Then againft XII, fet the Plain's Difference of Longitude 26 de. 25 min. (in the fecond Column) and from it main 21 d. 25 m. which fet against fubftract 15 deg. and there will re

XI. and I: And from 21 d. 25 m. fub. ftract 15 de. and there will remain 6 d. 25 m. which fet againft X and II. And (because it is lefs than 15 de.) write the Word Sub-ftile over it, and fubftract it from 15 deg. and there will remain 8 deg. 35 m. which write over Sub-ftile, and againft the Hours of IX and III. Then to thefe 8 d. 35 m. add 15 deg. and and it makes 23 de. 25 m. which fet against VIII and IV: And thus, by the continual Addition of 15 deg. you fhall have fuch Equinoctial Ditances as in the Table: Which Table, thus prepared, the next thing

will be

To find the true Hour-diftances upon the Plain, from the Sub-file. For which, This is

The Canon for Calculation.

As the Radius

To the Sine of the Stiles height 32 d.

36 m.

So is the Tangent of the Aquino

dial Distance 6 d 25 m. To the Tangent of 3 deg. 28 min. Which is the Distance of the Hour-lines of X and II. upon the Plain from the Sub-stile. And fo will the Tangent of the next Equinoctial Diftance 21 d. 25 m. be, to the Tangent of 11 de. 56 min. for the Distance of the Hour-lines of XI and I, from the Sub-ftile: And fo for all the reft, as in the Table. And fo you will find them to be in the Figure alfo.

Fig. And in the making of this Dial, you have made four ; as in LVII. Fig. LVII.


For, if you hold the Paper upon which the South-Eaft declining Fig. Dial is drawn, againft the Light, then fhall you difcover the Stile LVII. to ftand on the Right-hand of the Plain; whereas it now ftands on the Left-hand; fo the fame Hour-lines, Sub-file, Stile and all, being drawn on the back-fide of the Paper, and thofe that are the Forenoon-hours in the Eaft-decliner numbred as the Afternoon-hours in the Weft-decliner; that is, call 11, 1, and 1o, 2, and 9, 3, &c. as in the Tables; fofhall the South-Dial declining Eaft 30 degrees, become a South Dial declining Weftward 30 degrees.

And if you turn the South-Eaft-Dyal upfide-down, fo that the Stile may point upwards towards the North Pole, (and leave out the Hours about 12, as 9, 10, 11, and 1, 2, and 3, which in North Dials reprefent 9, 10, and 11 at Night, and 1, 2, and 3 in the Morning; all which time (in those middle Latitudes) the Sun is under the Horizon) it will become a North-Dial declining Eastward 30 degrees.

Alfo if you turn the South declining Weft-Dial upfide-down, and leave out the hours about Midnight, as 9, 10, 11, 12, 1, 2, and 3, it will then become a North-Dial declining Weftward 30 degrees.

Now for fuch South or North Dials as do decline far towards Eaft or Weft, as 60, 70, 80, or 85 degrees, there you fhall find that the Hour-diftances will fall fo near together, that they will be of no competent diftance one from another, except they be extended very far from the Centre; and therefore the old way hath been (in fuch Cafes) to draw the Dial upon the Floor of a Room, extending the Sub-file, Stile and Hour-lines till they appear of a competent diftance from each other, and then according to the bigness of your Dial-plain, to cut off the Hour-lines, Stile and Sub-file; and to transfer them from the Floor to the Plain upon which the Dial is to be made: but this way being to Mechanical for an Artift to exercife, I fhall therefore here infert a more artificial way of performing this work Geometrically, by which (although the Dial fhould decline 80 or 88 degrees) upon a quarter of a fheet of Paper you may draw your Dial, and have the Stile of a competent height, and all the Hour-lines at a convenient diftance one from another. And fo let this fuffice to be faid in this place concerning Upright declining Dials; for I intend not here to teach the Art of Dialling, but fhew the Ufe of the Globes: and from thence to Calculate the Requifites from them Trigonometrically.



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