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Spherical Trigonometry applied to Dialling; fhewing the Grounds and Reafon, as well as the Practical Method, of drawing Hour - Lines on all Sorts of Planes, in all Places.

T

HIS moft ingenious and furprising of all the Mathematical Arts can never be well understood by any Perfon, who hath not very well studied the Doctrine of the Projection of the Sphere, and the Nature of Spherical Triangles, in the Manner as delivered in the former Part of this Book. For on these two Hinges turns all the Mystery of the Art of Shadows.

The Planes on which Dials are defcribed, receive their Denominations according to their Situations and Pofitions, with regard to the Horizon, and the Cardinal Points thereof. For all Planes are either

1. Horizontal, or parallel to the Horizon.
2. Erect, or perpendicular to the Horizon.
3. Reclining, or Oblique to the Horizon.
4. Inclining Planes.

1. Of Horizontal Planes there is but one Kind.
2. Of Erect Planes there are two principal Kinds;
1. Direct, or those which directly behold the East,

Weft, North, and South Points of the Horizon.
2. Declining, or those which, declining from those
four Cardinal Points, do behold fome intermedi-
ate Point between them.
VOL. II.

Tt 2

Of

Of Erect Direct Planes there are thefe fubdivifions.

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3. Of Reclining Planes, or thofe which are pofited oblique to the Horizon, like the Roofs of Houses, &c. there are these Kinds.

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and E.

N and E.

N and W.

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If the Planes behold none of thefe Points direct, but fome other between them, they are thus denominated.

1. South Recl. Decl. Eaft) 2. South Recl. Decl. Weft,

Planes whofe

(S and E.

S and W.

3. North Recl. Decl. Eaft Poles lie be-N and E. 4. North Recl. Decl. Weftj

tween

N and W.

4. Of Inclining Planes there are the fame Divifions and Denominations as are juft enumerated of Recliners; for an Inclining and Reclining Plane are the fame, only the Reclining beholds the Zenith, the Inclining the Horizon of the Place; and one Dial ferves for both.

These

These are all the Varieties of Planes on which Dials may be defcribed, and it behoves every one who pretends to this Art to have a perfect Notion thereof.

I fhall now fhew the Geometrical Construction of Dials on all the foregoing Planes, from the Principles of Spherical Trigonometry, in the enfuing Problems.

PROBLEM I.

To find the Sun's Altitude by a Quadrant.
Practice.

When you are to make a Dial, you must first find the Sun's Altitude by a Quadrant, thus; hold your Quadrant against the Sun in fuch manner as that the Ray of the Sun Beams which paffeth through the Hole of the upper Light may fall precifely on the Hole of the lower Light; then will the Thread cut the Limb of the Quadrant in the Degree and Minute of the Sun's Altitude.

PROBLEM II.

To find the Horizontal Distance of the Sun from the Pole of the Plane.

Practice.

'Draw an Horizontal Line on the Plane or Wall, and apply the Edge of the Quadrant thereto at Right Angles, taking Care that the Limb of the Quadrant, be towards the Sun. Having thus applied the Quadrant, and in a Pofition as level as poffible, hold up a Thread and Plummet against the Limb of the Quadrant, at full liberty, moving this way and that 'till the Shadow of the Thread falls juft on the Center of the Quadrant; and then obferve what Number of De

grees

grees are contained between the Shadow of the Thread and that Side of the Quadrant standing perpendicular to the Plane, for they are the Horizontal Distance required.

PROBLEM III.

To find the Plane's Declination.

Practice.

The Declination of the Plane is the Distance or Quantity of the Arch of the Horizon contained between the Pole of the Plane, or a Line drawn perpendicular thereto, and the true Point of North and South.

Having found the Altitude of the Sun, and Horizontal Distance of the Plane, proceed for the Declination by these Rules.

1. First, find the Sun's Azimuth by Prob. 17 Of Aftronomy. Or if it be Collin's, or Sutton's Quadrant, you may find the Azimuth near enough the truth by it.

2. When you make your Obfervation of the Horizontal Distance, mind whether the Thread's Shadow doth fall between the South and that Side of the Quadrant which was perpendicular to the Plane.

3. If the Shadow fell between them, then the Sun's Azimuth from the South, and the Horizontal Distance added together do give the Declination of the Plane. And in this Cafe the Declination is towards the fame Coaft on which the Sun's Azimuth is.

4. If the Shadow fall not between them, then the Difference of the Sun's Azimuth and Horizontal Dif tance, is the Declination of the Plane; and in this Cafe, if the Azimuth be the greater of the two, the Plane declines towards the Coaft on which the Sun is; but if

the

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