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I. What a Great Circle is.
A Great Circle of the Sphere is fuch a Circle, as divideth the XLIV. whole Sphere into Two Equal Parts or Hemispheres: Of which, there are generally accounted Six, viz.
Equinoctial 3 Colure.
Befides thefe Six Principal Great Circles of the Sphere there are other Circles, which are Great Circles alfo. As are all Azimuth or Vertical Circles. Alfo all intermediate Meridians or Home Circles. And these are fuch as we have most occafion to make use of in this Place: And here note, That all Great Circles projected upon a Plain, Steriographically (or Circular,) are either, (1.) Perfect Circles as the outward or Primitive Circle is. Or, (2) SemiGreat Circles, projected within the Primitive Circle. Or, (3) They are Streight Lines, paffing through the Centre of the Primitive Circle, as afterwards will appear.
II. How to Project the Circles of the Sphere upon a Plain, and fwerable to a Prefixed Time and Place.
ET it be required to Projet fuch Circles of the Sphere in Plano, upon the Plain of the Meridian, in the Latitude North, 40 Deg. Upon the 10th of June, at the time of the Sun's Rifing or Setting, and alfo at 10 in the Morning, or 2 in the Afternoon, the fame Day: The Sun then having 23 Deg. 30 Min. of North Declination.
First, With 60 Deg. of a Scale of Chords, upon the Point A, defcribe the Primitive Circle Z H NO, reprefenting the Meridian of the Place.
Secondly, Draw the Right Line HA O for the Horizon of the Place; and at Right Angles thereto the Line Z A N, for the Equinoctial Colure, Z being the Zenith, and N the Nadir Points.
Thirdly, Take 40 Deg. (the Latitude given) out of your Scale of Chords, and fet them from O to P, from Z to Æ, from H to S, and from N to a; and draw the Line P AS for the Axis of the World, and Hour-Circle of Six; and the Line A a, for the Equinoctial.
Fourthly, Becaufe the Sun's Declination at the time given is 23 Deg. 30 Min. take 23 Deg. 30 Min. and fet them from to, and from a to: And if you lay a Ruler from E or a
to, it will cross the Axis of the World P A S, in the Point F; XLIV. fo have you Three Points, F, through which you may draw
the Circle F; which will cut the Horizon HA O," in the Point, the Place where the Sun will rife that Day.
Fifthly, This Point being found, you have Three other Points given, viz. P, O and S; through which you may defcribe the Circle POS which is the Hour at which the Sun Rifeth.. And thus have you Projected, fo far of the Problem, asconcerns the Time of the Sun's Rifing. Now for his Place of being at Ten in the Morning, or Two in the Afternoon.
Sixthly, Ten in the Morning, or Two in the Afternoon, are (either of them) Two Hours, or 30 Deg. diftant from the Meridian: Wherefore, take 30 Deg from your Scale of Chords, and fet them upon the Meridian, from A to G, then a Ruler laid from P to G, will cross the Equinoctial Circle A a, in the Point B, fo have you Three Points, S, B and P, by which you may Project the Circle S B P, for the Hour Circle of Ten in the Morning, or Two in the Afternoon: And the Circle S BP, will cut the Tropick of Cancer A(the Line which the Sun traces that Day,) in the Point C; in which Point the Sun will be at Ten in the Morning, or at Two in the Afternoon. Seventhly, And now have you Three other Points, N. C and Z, through which you may draw the Azimuth Circle ZCN, for the Azimuth, that the Sun will be upon at Ten and. Two a Clock.
III. Concerning the Spherical Triangles that are made by the Interfections of thefe Great Circles thus projected.
Many are the Triangles, that are made by the Interfe&ions of thefe few Circles upon this Projection, but I fhall exemplifie only in Two of them; namely, the Triangle P O'O, Right-angled at O, which is compofed of Three Arches of Great Circles of the Sphere, viz. (1.) Of . O, an Arch of the Horizon, comprehended between O, the North Point of the Horizon, and, the Point in the Horizon, whereon the Sun Rifeth that Day, or his Amplitude from the North. (2.) Of P 0,an Arch of a Meridian or Hour-Circle, paffing through P, the North Pole of the World; and O, the Place of the Sun's Rifing that Day, equal to 66 Deg. 30 Min. the Com
plement of the Sun's Declination that Day. And, (3.) Of P O, Fig. an Arch of the Meridian of the Place, comprehended between XLIV. O, the North Point of the Horizon, and P, the North Pole, equal to the Latitude given, 40 Deg.
•The other Triangle fhall be the Oblique-angled Triangle Z CP, conftituted by the Interfection of Three other Great Circles of the Sphere; namely, Of (1.) Z P, an Arch of the Meridian of the Place, comprehended between Z the Zenith, and P, the North Pole, equal to 50 Deg. the Complement of the Latitude of the Place. (2.) Of PC, an Arch of a Meridian, or HourCircle of Ten or Two a Clock, equal to 66 Deg. 30 Min. : the Complement of the Sun's Declination; or, the Sun's Diftance from the elevated Pole P. And (3.) Of Z C, an Arch of an Azimuth or Vertical Circle, paffing through the Hour-Circle SC P, in the Point P, where the Sun is at Ten or Two a Clock that Day, and is the Complement of the Sun's Altitude at that time..
IV. To find the Poles of thefe Great Circles, they being thus
The Pole of any Great Circle of the Sphere, is, always, 90 Deg. diftant in all Places, from the fame Circle, upon a Right Line, or Circle, which cuts it at Right Angles: So,
1. The Pole of the General Meridian, or Primitive Circle ZHN O, is A, the Center thereof, in all Places distanc 90 Deg.
2. The Poles of the Horizon H O, are Z and N, the Zenith and Nadir Points.
3. The Poles of the Equinodial 'e, are P and S, the Poles: of the World.
The Poles of the Equinoctial Colure, or Axis of the World
5. The Poles of the Prime Vertical Circle, or Azimuth of Eaft
And thus you fee, that all Great Circles of the Sphere, which, when Projected, become Strait Lines, their Poles fall all in the Periferie of the Primitive Circle: But for all other Great Circles, which confift of Circular Arches, as P OS, PBS, and Z DN, the Poles of them fall within the Primitive Circle ZHNO, upon those Great Circles, which cut them at Right Angles : S>,