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Fig. Now, forafmuch as the Planets are moved in Ellipfes, and not LXXII. in perfect Circles, the next thing, therefore, to be enquired is, the Planets Place in the Ellipfis: And to that purpose, about Z, let there be defcribed a little Circle K ON, upon which, from K to N, I number the Anomaly of the Epicicle 59.034 deg. before found, and then will the Angle PBZ, be equal to the corrected Anomaly; and fo is N the place of the Planet in the Ellipfis ; and the Angle Z B N, the Equotion of the Epicicle; or (which is the fame) the Difference between the Place of a Planet in the Circk, and in the Ellipfis: And may be found Trigonometrically, thus. For,

Thirdly, In the Triangle B Z N, you have given, (1) The Side B Z, the common Radius 100000 P. (2.) The Side Z N, 00008 P, the Semidiameter of the Epicicle. (3) The Angle BZN 120.98 Deg. by which you may find, the Angles BNZ and ZB N, (by CASE II. of O. A. R. T.) Thus,

The Angle N ZB being 120.98 deg. the Complement to 180 deg. is 59.02 deg. the half whereof is 29.51 deg. The Side B Z is 100000 P. the Side Z N 00008 P, their Sum 100008 P, their Difference 99992. Then,

(3) As the Sum of the Sides B Z and Z N, 100008


Is to their Difference 99992


So is the Tang. of half Z BN and ZNB 29.505 d.




To the Tang. of the half Difference 29.502 d. This balf Difference, added to the balf Sum, gives 59.007 deg. for the Angla Z NB: And fubftracted from it, leaves 0.003 deg. for the Angle NB Z.

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To the Side B N 100004 P.

From the Angle BPZ 29.502
Subftract NBZ







There remains PBN 29.497 Whofe Complement to 180

deg. is 150.503 deg. and is the Angle SBN.


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


As the Sum of BN and B S, 101689.

Is to their Difference 98319

So is the Tangent of half B S N and BNS 14.77d.

To the Tangent of 14.30 d. the half difference.




Which added and fubftracted from the half Sun, gives
29.07 deg. for BSN the Angle at the Sim.
And 0.47 deg. for the Opt. Equation B NS.
Then from the Mean Anomaly P XH 30.00 deg.
Subftract the Angle at the Sun BSN,

There remains the Proftapherifis.



: 0.93

As the Sine of BSN, 29.04 de.


Is to B N, 100004.


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


To 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 thefe Points; it hath Variation more or lefs, and is at its greatest when the Elongation is 45 deg. from any of thofe Points.- And note alfo, 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 fuppofed in the First Quarter of the Ellipfis, the fame is to be understood and performed, if the Planet were in the Second, Third or Fouth Quadrant thereof, as is plain by the Figure.

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

HE Sim feems to be moved under the Plain of the EclipTrick, But it is the Earth that performeth this Motion about the Sim, who is feated in the Focie of the Earths Ellipfis: So that, as the Earth 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 fubject to no other Inequality, than what is produced by their own Simple Motion in the Ellipfis: And this is true alfo, in all the Primary Planets as in refpect to the Sun; who is feen, 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 pofitions of their Orbs) they feem to us to be fubject to a fecond Inequality; which is not effential in their Motions, but accidental only; being caufed through the great diftance of the Point to which their Places are referred: So that the Planets feem to us to have a different fpecies of Motion from the Natural Motion in their refpective 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 caufe of all which various Paf fions will manifeftly appear from the Calculation of their Places, which fhall be the Work of the following Chapter.


To Calculate the true Place of a Planet Trigonometrically.

Retaining the fame Method deliver'd in the laft Chapter, I fhall 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 Mars from his Aphelion point 55.47 deg.


The reft, as in the following Table. And from thefe things gi- Fig. LXXI. 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

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the Figure the Semicircle P HY, is half the Orbit of Mars, P Fig. his Aphelion point, Hhis 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 H X B. the Comple ment of the mean Anomaly PH: 55.47 to 180 deg. uiz, 124.53 deg.

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

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


Is to their Difference 13793.


So is the Tang. of half XH B and X BH 27-73 d.



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To the Tangent of 23.50d.*
Which fubftracted 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-

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LXXII. As Radius

Which found; fay,


To the Sine of the greateft Variation of Mars,0.25 d. 7.639816 So is the Sine of Variat. laft 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 X B Z

The double whereof is of the Epicicle a Z N.

51.32 deg.


x 7.629195

0.244 the Variation.

51.564 deg.

103.128 And is the Motion

Defcribe the Epicicle, and upon it fet the Motion of the Epicicle 103.128 deg. from a to N, fo will the Angle a ON be equal to the corrected Anomaly PBZ; fo that N is the Place of the Planet Mars in the Ellipfis: And the Angle ZB N the Equation of the Epicicle; which is the difference between the place of a Planet in the Circle, and in his Ellipfis; 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


Is to their Difference 99992


So is the Tang. of half ZN B and Z B N, 51.32 de.



To the Tangent of 51.31 d.


Which added to 51.32, gives 102.63 de. for the Angle
Z NB; and fubftracted therefrom, leaves o.or deg. for

the Angle N B Z.

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