Page images
PDF
EPUB

6. Set 24°20', the Plane's Declination, from the Scale of Chords, on the Primitive Circle from H to c, from D to e, and from 0 to f.

7. Then a Ruler laid from Z, to the Point c, e, f, will give the Points W, B, E for the Weft, South, and Eaft Points of the Horizon of London.

8. Lay a Ruler from W to Z, and it will cut the Primitive in b; then lay off 38° 28′ (the Co-latitude of London) from the Chords from b to d.

9. Take the Tangent of 54° 00' and fet from 2 to K, and draw thro' Ka Line parallel to HO, infinitely extended, and thereon fet off the Half-Tangent of 24° 20', from K to M; then with the Tangent of 65° 40′ ( fet the other way from K, on the fame Line extended) you find the Center whereon to draw the Circle Z BM, which fhall be the Meridian of London.

10. A Ruler laid from W to d will cut the Meridian in P for the North Pole of the World.

II. Thro' P and draw the Right Line NQT, which fhall be the Meridian of the Place fought after, and the Axis of the World; and, being produced, will meet the Meridian of London in S, the South Pole of the World.

12. Cross NT at Right Angles with V X, and a Ruler laid from 0 to V will give the Point Æ, where the Equinoctial interfects the Meridian of the enquired Place.

13. Thro' V, Æ, and X, draw the Equinoctial Circle, as hath been taught; which alfo will pafs thro' W and E, the Eaft and West Points of the Horizon of London.

14. Divide P S into two equal Parts in E, and draw E G at Right Angles thro' the fame; extending it infinitely make P E Radius, and lay off the Tangents of 15°, 30°, 45°, &c. both ways from E on

the

the Line E G; and thus you will have the Centers of the Hour Circles to be projected; which when done your Projection is finished.

15. Now to find the Requifites, take QP in your Compaffes, and measure it on a Scale of Half-Tangents, and you will find it equal to 72° 34' whose Complement PN, or 2, is 17° 26'; and that is the Latitude in which this Reclining Plane will be an Horizontal Dial.

16. Lastly, lay a Ruler from P to a, and it will cut the Primitive in b; then Th measured on the Chords will be found 14° 41', equal E a, the Dif ference of Longitude required.

Therefore a Plane declining 24° 20', and Reclining North 36° oo' in the Latitude of London, will be an Horizontal Plane in the Latitude 17° 26', and Difference of Longitude from London 14° 41', which is about Caffena in Negroland.

The

G

The I

CHAP. VII.

Theorems for the Explication of the Doctrine of Spherical Triangles, and the Manner of their Solution.

[blocks in formation]

in common with the other Great Circle

therefore the common Section

ABC,

D,

AFC;

A C,

fhall be a Diameter of each; and fo will cut them into two equal Parts. Q, E. D.

THEOREM II.

If from the Pole B, of any Great Circle AFC, be drawn a Right Line B D, to the Center thereof, the faid Line will be perpendicular to the Plane of that Circle.

Demonftration.

Let be drawn the Diameters

n the Circle

EF, GH,
AECF;

Then

BDF, BDE,

then because in the Triangles

the Sides

are equal to the Sides

and the Base

therefore fhall the Angle

[blocks in formation]

and fo each is a right one; thus it is demonftrated that

Angles

BDG, BDH,
BD,

ЛЕСЕ,

Q, E. D.

are right ones; and fo is the Line perpendicular to the Plane of the Circle by Euclid 11. Prop. 4.

THEOREM III.

If a great Circle EB F, be defcribed about the Pole A then the Arch BF, intercepted between AB, AF, is the Measure of the Angle BAF, or BDF.

Demonftration.

By Theorem 2, the Angles

are right ones; because the Arches

AD B, ADE
AB, AF,

are Quadrants; Confequently the Angle (whofe Measure is the Arch

BDF,
BF,)

is equal to the Inclination of the Planes ABC, AFC; and fo equal to the Spherical Angle BAF, or BCF,

COROLLARY I.

2. E, D.

Because AB, A F, are Quadrants, A fhall be the Pole of the Circle paffing thro' the Points B, F; for A is at Right Angles to the Plane FD B, by Euclid 11. Prob. 14.

COROLLARY II.

The Vertical Angles are equal; being each equal to the Inclination of the Circles; alfo the adjoining VOL. II.

[blocks in formation]
« PreviousContinue »