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And then,

51.32 deg.

0.244

51.076 whofe Com

128.924 for the An

In the Triangle BNS, there is Given, (1) The Side BN 99996. (2) The Side B S, 1411, And (3) The included Angle NBS 128.92 deg. From whence may be found the Angle at the Sun BSN and the Side S N. ;

Thus,

As the Sum of the Sides N B and SB, 101407.

5.006082

4.993811

Is to the Difference of thofe Sides 98585.
So Tang. of half the Angles BSN and BNS, 25.54 d. 9.679276

To the Tangent of 24.92 deg.

14.673087
9.667005

Which added to 25.54 de. gives, 51.08 deg for the Angle BSN, and fubftracted, leaves 0.62 de. for the Angle B NS.

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Fig.

LXXII.

CHAP.

344

Fig.

CHAP. VII.

To find the Place of a Planet in the Ecliptick, &c.

OR that a Planets Place in Longitude in the Ecliptick, differeth

LXXIII. F fomething from the Longitude in its own Orb, I fhall there

fore fhew (by Trigonometry,) How to find the Reduction, and the Ecliptick Place, with the Curtation and Parallax of the Orb, and confequently, the Geocentrick Place and Latitude.

In order whereunto, in the Scheme; Let S reprefent the Sun, H the Place of Mars in his Orbit; V his place in the Ecliptick; E the Earth: A the North Node of Mars; B the South Node; (the two Interfections of the Planets Orb with the Ecliptick; A DB, a Semicircle of the Ecliptick: DR, the limit of the Planets Greatest North Latitude. These things premifed, I proceed to

The Trigonometrical Calculation.

1. In the Spherical Triangle BH V, Right-angled at V, there is given. (1) The Side B H 26.65 de. (the Complement of the Argument of Latitude. )-(2) The Angle HB V, the Greatest Inclination of Mars, 1.85 de. By which you may find the Side BV (By CASE I. of R. A. S. T. Thus

As Radius, 90 de.

To Co-fine H BV, 88. 15 de.

So is the Tangent of BH, 26.63 de.

10.

9-999773

9-700577

9.700350

To the Tangent of BV 26.64 de.

The Difference between BH and BV, is 0.01 de. is to be fubftracted from 201.43 d. the place of Mars, because the Arch BV is leffer then the Arch BH.

Heliocen. Place of Mars, in his Ellipfis. 201.43 deg.

Reduction

Heliocen: Place Corrected

Place of the Sun

Difference

Subft.

.01

201.42

215.85

14.43

Which is the Anomaly of Commutation.

2. In the fame Triangle, there is given as before, whereby the Inclination of the Orbit from the Plain of the Ecliptick H V, may be found (by CASE II. of R. A. S. T. Thus

As

As Radius, 90.00 de.

10:

Fig

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Unto which, the Angle HSV is equal.

9-651800 LXXIII.

8.508974
8.160774

3. For the Curtated Distance S.V. In the Right-angled Plain Triangle HSV, there is given, (1) The Angle V SH, 0.83 d. (2) The Side SH 15204, the distance of Mars in his Orbit, from the Sun By which you may find the Side S V (By CASE IV.. of R. A. S. T.) Thus

As Radius, Sine 90.00 de HV S,

Is to the Side HS, 15204.

So is the Sine of V H S, 89.17 d.

To the Side SV, 15202.

TO...

4.181958

9.999954

14.181912

The Ecliptic Place of the Planet Mars, with his Inclination and Curtation thus attained: The next thing to be enquired after is, the Parallax of the Earths Orb; and his Geocentrick Place, in Longitude and Latitude.

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4. In order whereunto, in Figure LXXIII. the Circle OXEZ, Fig. reprefenting the Earths Orbit; Number the quantity of the Anoma-LXXIII. ly of Commitation 14.43 de. from X (the oppofite Place of Mars from the Sum) to E, drawing the Line EV; which will conftitute an Oblique-angled Plain Triangle SV E: in which there is Given, (1) The Side SV 15202. (2) The Side S E, 10000, (3) The included Angle V SE 165.57 de. (the Complement Angle of Commitation to 180 deg.) By which you may find (1.). the Angles SVE and V E S. And (2) the Side V E.

As the Sum of V S and SE 25 202.

Is to their Difference $202.

4.401435

3.716170

So is the Tang. of half the Angles at E and V, 8.22 d. 9.159743

To the Tangent of 1.71 deg. their difference.

12.875913
8.474478

Which added to the half Sum of the Angles at V and E (8.22) d.) gives 9.03 des for the Angle V ES: And fubftracted, therefrom, leaves 6.51 de. for the Angle SVE: Which is the Parallax of the Orb..

"Then

Fig. LXXIII.

Then for the Side V E,

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Of the Proportions of the Semidiameters of the Sun, Earth,
Moon, and the other Planets.

I. Of the Sun, Earth, &c. Moon;

Y the beft Telescope-Obfervations.

B To the mean diflance of the Earth from the Sun. 10.00000

The Semidameter of the Sun is of thofe Parts
The Semidiameter of the Earth

To the mean diftance of the Moon from the Earth
The Semidiameter of the Earth is

The Semidiameter of the Moon

46300

727

1.00000

1650 446

And from hence, at all times, the Diftance of the Lumina

ries being firft found, their apparent Semidiameters may be obtained: And for the Semidiameter of the Earths Shadow, in Lunar Eclipfes; I have here inferted the Diagram of Hyparcus.

In which Diagram.

Fig.
LXXIV. A D his Semidiameter.

A denotes the Centre of the Sun.

B the Centre of the Earth.

BD

BE her Semidiameter.

AED, or A BD, the apparent Semidiameter of the Sun.

A EH, or BD E, the Horizontal Parallax.

CGF equal to HED, the Semi-Angle of the Cone of the Earths
Shadow.

BC and BF, being equal to the Diftane of the Moon from
the Earth.

BTE, her Horizontal Parallax.

CBF, the apparent Semidiameter of the Earths Shadow.

From hence,

1. The Semidiameter of the Sun, the Horizontal Parallax being fubftracted, is equal to the Semi-angle of the Cone of the Earths Shadow.

SO A E DAEHHED

2. The Horizontal Parallax of the Moon, the Semi-angle of the Cone of the Earths Shadow being fubftracted, is equal to the apparent Semidiameter of the Shadow.

So BFE CGEG BF.

3. The Sum of the Horizontal Parallaxes of the Sun and Moon,
is equal to the Sum of the apparent Semidiameter of the Sun
and Shadow of the Earth.

So, BDFBFDABDCBF.
Therefore, From the Sum of the Horizontal Parallax of the
Sun and Moon, fubftract the apparent Semidiameter of the Sim;
and there will remain the apparent Semidiameter of the Earths
Shadow.

So, BFDBDF-ABD

CBF.

JI. For the Proportional Magnitudes of thefe Three Bodies.

Fig.

LXXV

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The Logar. of 258309

5.412141

So that the Body of the Sun exceeds the Body of the Earth 258309

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