Fourthly, In the Triangle BSN, there is given, (1) BN Fig. 100004. (2) BS 1685. (3) The Angle contained by them, SBN LXXII. 150.50 deg.

Then,

As the Sum of BN and B S, 101689.

5.007278

Is to their Difference 98319

4.992637

So is the Tangent of half BSN and BNS 14.77d.

9.421029

14.413666

To the Tangent of 14.30 d. the half difference.

9.406388

Which added and substracted from the half Sun, gives

29.07 deg. for BSN the Angle at the Sun.

And 0.47 deg. for the Opt. Æquation BNS.

Then from the Mean Aromaly PXH 30.00 deg.

Substract the Angle at the Sun BSN,

29.07

5.000021

9.692338

14.692359

5.006232

There remains the Profstapherifis.

As the Sine of BSN, 29.04 de.

Is to BN, 100004.

So is the Sine of BSN, 150.50 deg. (or 29.50.)

Το SN, 101564.

Note that when a Planet is in any of the Points P, A, or Y, there is no Variation, but being out of any of these Points; it hath Variation more or less, and is at its greatest when the Elongation is 45 deg. from any of those Points. And note also, That this Variation augments the Places of all the Planets (Venus and the Moon excepted) in the First and Third Quadrants of the Orbit; But in the Second and Fourth; it diminishes the Planets place; according to the quantity of the Angle of Variation. And as this Example was wrought, the Earth or other Planet being supposed in the First Quarter of the Ellipsis; the fame is to be understood. and performed, if the Planet were in the Second, Third or Fourth Quadrant thereof, as is plain by the Figure.

Fig. LXXII.

CHAP. V.

Of the Two-fold Inequality of the other Primary Planets.

T

HE Sun feems to be moved under the Plain of the Ecliptick; ; But it is the Earth that performeth this Motion about the Sun, who is seated in the Focis of the Earths Ellipsis: So that, as the Eath is moved under the Ecliptick Orb, fo much the Sun appears to move on the contrary part thereof: And from hence may be concluded, that the Sun (or Earth) can be subject to no other Inequality, than what is produced by their own Simple Motion in the Ellipsis: And this is true also, in all the Primary Planets as in refpect to the Sun; who is seen, from them, to change his Place, according to the quantity of their Motions.

Now, For as much as the Earth is far diftant from the common Centre of their Orbs (which is the Sun) therefore it is, that (by their different Motions, and various positions of their Orbs) they feem to us to be subject to a fecond Inequality; which is not effential in their Motions, but accidental only; being caused through the great distance of the Point to which their Places are referred: So that the Planets seem to us to have a different species of Motion from the Natural Motion in their respective Orbs: they appearing at one time Direct, at other times Retrograde, and fometimes not to move at all, but appear as Stationary: To be nearer to the Earth at one time than at another; and appearing to the Eye of a Greater and Leffer Magnitude: The cause of all which various Paffions will manifestly appear from the Calculation of their Places, which shall be the Work of the following Chapter.

CHAP. VI.

To Calculate the true Place of a Planet Trigonometrically.

R

shall give an Example in the finding out of the place of the Planet Mars. The Sun being in 5.85 deg. of Scorpio, and the mean Anomaly of Murs from his Aphelion point 55.47 deg..

The

The rest, as in the following Table. And from these things gi- Fig. ven, the true place of the Planet Mars may be found.

A Table of the Semidiameters of the Orbits and Epicicles; the
Excentricities; the Greatest Variations and Inclinations of
the Planets, in order to the Trigonometrical Calculation of their
Places.

LXXI.

N the Figure the Semicircle P HY, is is half the Orbit of Mars, P Fig. his Aphelion point, H his place in his Orbit, B the Centre, BX LXXII.)

the Excentricity: H B the Semidiameter of the Orbit.

>

I. Then, In the Triangle XH B, there is given, (1) The Side H B, the Semidiameter of the Orbit, 15204. (2) The Side X B, the Excentricity 1411. (3) The Angle HX B. the Complement of the mean Anomaly PH: 55.47 to 180 deg. viz. 124.53 deg.

By which you may find the Angles BHX, and XBH.
(By CASE II. of O. A. P. T.

As the Sum of the Sides HB and X B 16615.

Is to their Difference 13793.

4.2205003

4.1396587

So is the Tang. of half X HB and X BH 27.73 d.

9.7207213

13.8603800

To the Tangent of 23.50d.

9.6398797

Which substracted from 27.73 (the half Sum) leaves 4.15
de. for the Angle at H: And added to 27.73 d. gives
51.31 deg. for the Angle XBH. And that doubled,
makes 102.52 d. Which is the Anomaly of Varia-

Fig.

LXXII. As Radius

Which found; say,

10.

To the Sine of the greatest Variation of Mars, 0.25 d. 7.639816

So is the Sine of Variat. last found 102.62 (77.38)

To the Sine of HBZ, 0.244 deg. the Variat.

To the Angle XBH

Add the Angle HBZ

The Sum is XBZ

The double whereof is

of the Epicicle a Z N.

9.989379

x 7.629195

51.32 deg.

0.244 the Variation.

Describe the Epicicle, and upon it fet the Motion of the Epicicle 103.128 deg. from a to N, so will the Angle a ON be equal to the corrected Anomaly PBZ; so that Nis the Place of the Planet Mars in the Ellipsis: And the Angle ZBN the Equation of the Epicicle; which is the difference between the place of a Planet in the Circle, and in his Ellipsis; which may be thus found. For,

In the Triangle B Z N there is given, (1) The Side Z B, the Semidiameter of the Circle 100000, (2) The Side Z N, the Semidiameter of the Epicicle 33. (3) The Angle BZN, the Complement of the Motion of the Epicicle N Za (103.64) viz. 77.36 de.

And by thefe may be found, the Angle N B Z.

Thus :

As the Sum of the Sides Z B and Z N, 100008

5.00008

Is to their Difference 99992

4.99996

So is the Tang. of half ZN Band Z BN, 51.32 de. 10.09659

15.09655

To the Tangent of 51.31 d.

10.09647

Which added to 51.32, gives 102.63 de. for the Angle

ZNB; and substracted therefrom, leaves 0.01 deg. for

the Angle N B Z.

9.989362

9.989345

5.000000

14.989345

And then,

In the Triangle BNS, there is Given, (1) The Side BN 99996. (2) The Side BS, 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

Is to the Difference of those Sides 98585.

4.993811

So Tang. of half the Angles BSN and BNS, 25.54 d. 9.679276

14.673087

To the Tangent of 24.92 deg.

9.667005

Which added to 25.54 de. gives, 51.08 deg for the Angle BSN ;

and substracted, leaves 0.62 de. for the Angle. BN S.

Fig. LXXII.