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(2.) As the Sine of 38 d. 29 m. laft found.

Fig.

Is to the Sine of the Half Sum of the Sides, 99 d. 14 m. (or XLV.

89 d. 36 m.)

So is the Sine of the Difference, 27 d. 24 m.

To the Sine of 47 d. 41 m.

To which Sine add the Radius, and half thereof will be the
Sine of 59 d. 19m. Whofe Complement 30 d. 41 m. is
Half the Angle at P.

Thus the Angle at P, being 61 d. 22 m. that reduced into Time,
makes 4 h. 6 m. ferè.

At which Time the Twilight Begins in the Morning. And that fubftracted from 12 h. leaves 7 h. 54 m. for the Time that Twilight will End in the Evening.

PRO B. XXXV.

The Elevation of the Pole, and the Altitude of a Star; whofe Right Afcenfion and Declination are both known: To find out the Moment of Time, and the Azimuth of the Star at that Time.

Rom.

I. By the Cœleftial Globe.

Ectifie the Globe to the Latitude, (fuppofe London) 51 d. 30 m. Bring the Place of the Sum (fuppofe 27 d. 54 m. of Pifces) to the Meridian, and the Hour-Index to 12, and the Quadrant of Altitude to the Zenith: Then turn the Body of the Globe about, till the Star (fuppofe Caput Andromeda) meet with 5 d. 24 m. (the given Altitude thereof) Then will the Hour. Index point at the Time required, viz. 3 h. 47 m. Morn."

II. By Trigonometrical Calculation.

The Sun at Noon is at the higheft. Take away therefore the Fig. Sun's Right Afcenfion, from the Right Afcenfion of the Star; and XLVI. the Remainder being turned into Time, fhews how many Hours XLVII. Afternoon the Star is at the highest. Therefore,

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SFig. XLVI.

Fig. XLVII.

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Fig. XLVI.

XLVII.

Given,

In the Triangle PZ S there is

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1. The Side PZ, the Distance of the Pole of the A-
quator, and the Horizon of London, 38 d. 30 m.
2. The Side Z S, the Complement of the Star's Altitude,
84 d. 36 m.

3. The Side PS, the Complement of the Star's Decli
nation, 62 d. 42 m.

1. The Angle P, and must be turned into Time:

In Fig. XLVI.

Taken from the Time

And

muft be

of Culmi

Required,

In Fig. XLVII.

Added to

nation,

that the Time of Obfervation may appear.

2. The Angle at Z, the Azimuth of the Star, at the
Time of Obfervation.

The Canon for Calculation. By Cafe XI. of O. A. S. T.
1. For the Angle at P.

(1.) As the Radius,

Is to the Sine of PZ, 38 d. 30 m. (the Complement of the
Latitude.)

So is the Sine of ZS (the Complement of the Star's Altitude,

84 d. 36 m.

To the Sine of 38 d. 18 m.

(2.) As the Sine of 38 d. 18 m.

Is to the Sine of the half Sum of the Three Sides of the Triangle,
92 d. 54 m. (or 87 d. 6 m.)

So is the Sine of the Difference (between the Half Sum, and the
Side Z S, oppofite to the Angle at P) 8 d. 18 m.

To the Sine of 61 d. 10m. The Double whereof 122 d. 20 m.
is the Quantity of the Angle at P. Which, turned into Time,
gives 8 h. 10 m.

2. For the Angle at Z.

As the Co-fine of the Star's Altitude ZS, 84 d. 36m.

Is the Sine of the Angle at P, 122 d. 20 m. Or. (comp. to 180d.) 57 d. 40 m.

So is the Sine of the Side SP, (the Comp. of the Star's Decline-
tion) 62 d. 42 m.

To the Sine of the Angle at Z, 48 d. 57 m. And that is the
Star's Azimuth from the North-Eastward.

Example

Example in Caput Andromeda.

255

Fig.

XLVI.

In the Latitude of 51 d. 30 m. the Altitude of the Head of Andromeda, was Obferved in the Eastern Part to have 5 d. 24 m. XLVII. of Altitude above the Horizon: The Sun's Longitude at that time being in 27 d. 54 m. of Pifces . At which Time alfo the Right Afcenfion of the Sun is

D. M. 358 04

The Right Afcenfion of Cap. Andromeda, found as before 357 45
To which add 360 d. it makes

717 45

d.}

From which fubftra&t the Sun's Right Afcenfion 358 d. 4 m. there remains

359 41

Which turned into Time, gives 23 h. and 57 m. which is as
many Hours as the Star is coming to the Meridian after
the Sun.

Thus the Angle at P, being found to be 122 d. 20 m. which (the Star being in the Eastern Part) must be substracted from the Time of Culmination, 23 h. 57 m. and then there will remain 15 h. 47 m. for the Time of Obfervation Afternoon, which is 3 h. 47 m. in the Morning.

PROB. XXXVI.

The Time of Obfervation, and Latitude of the Place, being given:
To find out the Altitude and Azimuth of a Star given.

R

I. By the Cœleftial Globe.

Ectifie the Globe to the Latitude, the Quadrant of Altitude to the Zenith, the Place of the Sun to the Meridian, and the Index to 12. Then turn the Body of the Globe about, till the Index points at the Time given, and there keep it: Then the Quadrant of Altitude laid over the Star will give you upon it the Star's Altitude; and when it cuts the Horizon, the Azimuth thereof alfo.

H. By Trigonometrical Calculation.

This is but the Converse of the former Problem: And the Trianglé made upon the Globe the fame, viz. the Triangle Z PS, in Fig. XLVI, and XLVII. In which Triangle there is

Given,

Fig. XLVI. XLVIL

Given,

Required,

1. The Side Z P (the Diftance of the Poles of the Equa-
tor, and Horizon of London) 38 d. 30 m.
2. The Side P S, the Complement of the Star's Declina-
tion (Cap. Andromeda) 62 d. 42 m.

3. The Angle S PZ (the Time between the Time of Ob.
Jervation, and the Time of the Stars next Culmination,
viz. 8h. 10m) which is 122 d. 20m.

1. The Side ZS, (the Complement of the Star's Obfer. ved Altitude).

2. The Angle at Z (the Azimuth of the Star, from the North Part of the Meridian.)

The Canons for Calculation. By Cafe IX. and Cafe I. ofO. A.S.T. 1. For the Angle at Z.

(1.) As the Sine of half the Sum of the Sides ZP and ZS, 50d. 36 m.

Is to the Sine of their half Difference, 12 d. 6 m.

So is the Co-tangent of half the Given Angle, Z PS, 61 d. 10m.
To the Tangent of 8 d. 30m. the half Difference of the Two
Angles, Z and S.

(2.) As the Co-fine of half the Sum of the Given Sides, 50 d.

36 m.

Is to the Co-fine of their balf Difference, 12 d. 6 m.

So is the Co-tangent of half the given Angle Z, 61 d. 10 m.

To the Tangent of 40 d. 22 m. Which is the half Sum of the
Two Angles.
Added to

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2D. M.D. M. For the SZ.

Then 8 d.

30 m.

Subftracted from

40 2248 52.

gives (3152)

Angle S.

So that the Azimuth of the Star is 48 d. 52 m. from the North
Part of the Meridian.

2. For the Side Z S.

(3.) As the Sine of the Angle at Z, (laft found) 48 d. 52 m.
Is to the Sine of the Side SP (the Complement of the Star's
Declination) 62 d. 42 m.

So is the Sine of the Angle at P, 122 d. 20m. (or 57 d. 40 m.)
To the Sine of 84 d. 36 m. for the Side ZS. Whofe Com-
plement, 5 d. 24 m. is the Altitude of the Star at the Time
of Obfervation.

Aftrono

Aftronomical Problems

Relating to

ASTROLOGY

INTRODUCTION.

Astrology confifteth principally of Two Parts, viz. the one

Mathematical, as is the Aftronomical Part; the other Judiciary, as is the Aftrological Part.

The Mathematical Part teacheth how in a Scheme or Figure (as they call it) to reprefent the Face of the Heavens in Plano, for any Hour of the Day or Night, at all Times of the Year, and in all, Parts of the World.

The Aftrological Part teacheth how (from the Sight of the faid Pofition of the Scheme or Figure of the Heavens at the Time of itsErection) to give a determinate Judgment of what was demanded upon that Erection of the Scheme or Figure; as of Annual Revolutions, Elections, the Nativity of a Perfon, &c.

The principal Authors that have given their Opinions concerning the dividing of the Heavens into 12 Manfions or Houses, are, 1. Prolomy, 2. Alcabitius, 3. Campanus, and 4. Regiomontanus: Which laft Way is now generally received and practiced among the Aftrologers of these Times, and by them termed the Rational Way of Regiomontanus.

Now, becaufe (as I faid before) that the Erection of a Figure of the Heavens is the Mathematical Part of Aftrology, I fhall therefore fhew how by the Globes to erect a Figure of the Heavens according to the Rational Way of Regiomontanus.

PROB

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