Prob. 92. To project the Circles of the Sphere on the Plain of the Meridian for the Latitude of London 51°. 32′ North.

With the Chord of 60°, draw the primitive Circle H ZRN for the Meridian of the Place; draw the right Circle HOR for the Horizon, and ZON at right Angles for the Prime Vertical, or Azimuth of Eaft and Weft, in which Z will be the Zenith and N the Nadir; take 51°. 32', the Latitude of the Place, and fet it from H to P, and from R to S; then will P reprefent the North Pole of the World elevated above the Horizon, and S the South Pole of the World depreffed below the Horizon, equal to the Latitude of the Place; from P to S draw a Diameter POS, which represents the Axis of the World; at right Angles to POS draw EOQ, which will be the Aquinoctial; take from the Line of Chords 23°29', and lay it, both Ways, from A to a, and from A to ; lay it alfo from Qto, and from Q to b; draw the Ecliptic Ov; take the Half-tangent of 23°. 29′, and lay it, both Ways, on the Axis of the World to c and d; then, thro' the three Points a, c,, draw, by Prob. 17, the Parallel ac for the Tropic of Cancer; and, by the fame Problem, draw db, thro' the three Points v db, for the Tropic of Capricorn; or, if you take from the Scale of Tangents

Tangents 66°. 31', the Complement of 23°. 29, and fet one Foot of the Compaffes in cor d, letting the other fall on the Axis of the World produced to a fufficient Length, that will be the Center, on which the Tropic may be drawn, as before.

The Center O, where the Ecliptic interfects the Equator, is the Beginning of V; 30° being taken from the Scale of Half-Tangents, and laid on the Ecliptic from O towards, is the Beginning of 8; 60° being taken from the fame Scale, and laid from the Center towards, is the Beginning of II; and the Beginning of is at the Interfection of the Tropic with the Meridian: Now, reckoning back 30° from towards O, we shall come to the Beginning of N, the fame Point II is; reckoning back 60° from towards O, we fhall come to the Beginning of m, the fame Point where began; and, reckoning back 90°, we come to the Center, or Beginning of, the firft of the Southern Signs: Then take 30° from the Scale of Half-Tangents, and fet from to m for the Beginning of that Sign; take 60° from the fame Scale, and fet from to for the Beginning of that Sign; the Beginning of is at W, the Interfection of the Tropic with the Meridian; reckoning back from 30° will be the Beginning of; and reckoning back 60° for the Beginning of will be the Beginning of M.

Take from the Line of Chords 23°. 29', and lay it, both Ways, from the North Pole P to A and T; then, by Prob. 42, Cafe 2, lay 23°.29′ from P on the Axis of the World tor: Now take the Tangent of 23°. 29, and, placing one Foot of the Compaffes in r, let the other fall on the Axis of the World extended beyond P, this will be the Center, on which may be defcribed the Arctic Circle ArT about the North Pole P: And in the fame Manner may the Antarctic Circle Bn D be defcribed about S, the South Pole.

If one Foot of the Compaffes be placed on the Equator, either within, or continued without the primitive Circle, and oblique Circles are drawn, paffing thro' the Poles of the World, as PS, they will be Meridians, or Hour-Circles. In the fame Manner, placing one Foot of the Compaffes in the Horizon, either within the primitive Circle, or produced beyond it, and drawing oblique Circles, Zt N, thro' the Zenith and Nadir, they are Azimuth-Circles.

If fmall Circles, fuch as x cz, are drawn parallel to the Horizon HOR, by Prob. 61, at any Distance less than 90°, they are Parallels of Altitude.

Q92

Having

Having explained the Manner of projecting the principal Circles of the Sphere, I fhall now fhew the young Aftronomer their Ufe; and, as the Sun's Place in the Ecliptic is the Foundation of most of the other Problems in Aftronomy, I fhall begin with that.

Prob. 93. To find the Sun's Place in the Ecliptic.

Rule. If the given Day of the Month is more than the Day of the Sun's Entrance in that Month, fubtract the Day of the Sun's Entrance from the given Day, and the Remainder will be the Degrees of the Sun's Place in the Sign he enters in that Month.

Exam. 1. The Sun's Place is required on the twenty-feventh of May 1753:

From May

Subtract the Day of Entrance into II

27

21

6

Therefore, on the twenty-feventh of May, the Sun is in fix

Degrees of II.

And, if the given Day is lefs than the Day of Entrance for that Month, add thirty to the given Day, and fubtracting the Day of Entrance in the preceding Month from that Sum, the Remainder will be the Degrees of the Sun's Place in the Sign of the preceding Month:

Prob. 94. The Sun's Place II 7°. 13 given, and his greatest Declination 23°. 29', (always known) to find his prefent Deelination and right Afcenfion.

Con

plement of the Sun's Place to 90°, place one Foot of the Compaffes in O, and let the other fall on the Equinoctial (continued, if neceffary) for a Center, and draw the oblique Circle PORS, by which will be formed the right-angled Triangle TRO, right-angled at R; in which we have given the Angle at 23°. 29, the Sun's greatest Declination, and the Hypothenufe 67.13, to find the Perpendicular R, the prefent Declination; this is an Extreme disjunct, and the Perpendicular is the middle Part; whence, by Lord Nepier's third Rule, and its Note,

As the Radius

Is to the Sine of OY R, 23°. 29',
So is the Sine of YO, 67°.13,

10.

9.600409 9.964720

To the Sine of RO, 21°.33', the Sun's Declinat. 9.565129

The Perpendicular RO, being a Part of the oblique Circle POS, may be measured by Prob. 46: Then, for the right Afcenfion R, having the Angle at T 23°. 29', and the Hypothenufe O 67°. 13', to find the Bafe R, is an Extreme conjunct, and one of the Extremes is required; whence, by Lord Nepier's fecond Rule, and its Note,

As the Co-tangent of V O, 67°. 13′,
Is to the Radius,

So is the Co-fine of OTR, 23°.29',

Co. Ar. 0.376731

10.

9.962453

To the Tang, of Y R, 65°.24', Sun's right Afcenf. 10.339184

The

The Bafe R may be meafured by Prob. 43. If the Declination were North, and increafing, then the right Afcenfion of the Sun is 65°. 24'; but, if it were North, decreasing, that is, if it were between the twentieth of June and the twenty-fecond of September, then the Number found muft be fubtracted from 180°, and the Remainder 114°. 36' is the right Afcenfion; because, as was observed in the twentieth Definition, the right Afcenfion is reckoned from the Point where the Equinoctial interfects the Ecliptic in the first Point of V, 'till it returns to the fame Point again: Therefore, when the Declination is South, increafing, that is, from the twenty-fecond of September to the twenty-first of December, 180° must be added to the Number found by this Problem; and, when the Declination is South, decreafing, that is, from the twenty-firft of December to the twentieth of March, then the Number found by the Calculation must be fubtracted from 360°, and the Remainder is the right Afcenfion fought.

I shall now endeavour to give the young Aftronomer fome Idea of the Caufe of the Inequality of natural Days, arifing from the Obliquity of the Ecliptic to the Equinoctial; by which the Sun's right Afcenfion not increafing exactly the fame with his Place in the Ecliptic, it produces a Variation in the Length of the natural Days: For those Meridians that divide the Equinoctial into equal Portions, will not divide the Ecliptic in the fame Manner, but into unequal Portions; by which the Length of the natural Day will be increafed or diminished; the Length of a natural Day being the Time between the Sun's leaving the Meridian of any Place and coming to it again; which, in different Revolutions of the Earth on its Axis, will be longer or fhorter, as longer or fhorter Portions of the Ecliptic pafs the Meridian. If the Reader finds any Difficulty in understanding the following Explanation, let him pafs it by, and review it when he has made a farther Progrefs.

Let it be required to find the Sun's Place in the Ecliptic when his right Afcenfion is 15°.

With the Chord of 60°, draw EP QS the Meridian; then draw YQ the Equinoctial, and PS the Axis, or Hour-Circle of Six; from A to W, and from Q to 5, fet off 23°. 29', the Sun's greatest Declination, and draw the Ecliptic; then, from the Scale of Half-Tangents, lay 15° from Y to a on the Equinoctial, and draw the Meridian Pa S, which will interfect the Ecliptic in O,

the