CHA P. V.

The Ufe of the Theories, and by them to find the Places of the Earth and Planets, both in Longitude and Latitude, &c. for any time propofed. As for MARS. and the EARTH, on the 18th of October 1705.

HE Orbits of the EARTH and MARS being de

Tfcribed in the Theory, as is before thewed how to do; Firf, Collect the equal Anomalies for the Earth and Mars, for the Time propofed, and their Reductions to the Ecliptick, as is here done.

The Equal Anomalies of the Earth and Mars, for the 18th' Day of October 1705 at Noon.

Secondly, The Equal Anomaly of the Earth, reduced to the E-cliptick, being in 5. 12 de. of Taurus, lay a Ruler to the Centre O, and 5.12 deg., (as at c) and where it croffeth the Orbit of the Earth, inake, for that is the Place of the Earth in his Or- bit at that time.

Thirdly, The Equal Anomaly of M ARS, reduced to the Ecliptick Circle, being in 17.68 deg, of Libra: Lay a Ruler to the

Centre

Centre, and 17.68 deg (as at d,) and where it croffes the Orbit of MARS, inake; for that is the place of MARS in his Orbit at that time.

Fourthly, Draw the Right-Lines,, 0.3 and 30, conftituting the Right-lined Triangle

The Theory thus finished, I come now to fhew,

1. To find the Sun's true Place in the Ecliptick, for the time propofed.

Lay a Ruler upon the Line, and it will cut the Ecliptick Circle in bb, in 5. 51 deg. of Scorpio; and that is the Sun's Longitude (or Place in the Ecliptick) at the time proposed.

II. To find the Longitude (or true Place in the Ecliptick) of
MARS for the fame time.

Through the Point in the Centre, draw a Right-Line parallel to the Line, as the fhort Line a da, which will cross the Ecliptick Circle in 25.00 deg of Libra, where make the Character of MARS; for that is his true Place of Longitude in the Ecliptick, for the time propofed.

Mark now the Triangle, made upon the Theory, by the Centre, and the Point of and o in their Orbits: For in that Triangle,

The Side O >is the diftance of

Sun.

from the Sun.

Earth.

All which are to be Measured upon Proper Scales for the feve ral Planets.

And

30 is the An- SSun
Earth or of Commitation.
Elongation.
eSgle at the PlanetS

The Angle o

Paral. of the Orb

Thefe Argles may be Measured (very near the Truth by help of yoar Scale of Chords; but more accurately (when

the Sides

which hereafter.

Of

But

But for the measuring of the Sides, Proper Scales must be made Fig. for each Planet, and the Earth. And, LXXVI: The Proportions between the Length of the Aphelial Lines of the feveral Planets, (taken from the Center of the Theories, to the Aphelial Points) and the Scales by which to meafure their Difrances from the Sun or Earth; are thefe exhibited in this Table, viz.

Which Numbers will make Proper Scales for meafuring the Difances in each Theory. And the Scale for the Earth, will ferve, alfo, for the Theories of Venus and Mercury.

And note farther, That all Diftances measured upon thefe Proper Scales, are

And thefe Products fhall Reduce the Distances measured Proper Scales, into Semidiameters of the Earth.

upon the

III. To find the Distance of MARS from the Sun and from the Earth, for the Time propofed.

Take in your Compaffes the length of the Aphelial Line Ap. ,, and opening the Sector in the Points 56.73 and 56.73. The Sector, as fo opened, fhall be the Scale whereby to measure the Distances in the Theory of MARS. Wherefore, if you take in your Compaffes (out of the Theory) of Mars.

Parts

S55. 50 288. 20

IV. To find the Angle of Commination,, or Angle at the Sun) 300.

Lay a Ruler to the Centre, and the Point in its Orbit; and it will cut the Ecliptick Circle in the Point c: Alfo lay a Ruler upon the Centre,and the Point Mars in its Orbit, and it will cut the Ecliptick in the Point d; then will the Degrees of the Ecliptick, intercepted between c and d, be the quantity of the Angle ♂, the Angle at the Sun (or of Commination) and will be found to be 162.00 deg..

V. To find the Angles of Elongation and Parallas of the Orb
(or the Angles at the Earth and Planet) and
O ♂

Thefe (as I faid before) may be meafüred upon the Theory by the affiftance of the Scale of Chords: But the Angle of Commination and the distance of the Earth and Mars being known, they may more exactly be found by Trigonometrical Calculation: Thus ;

As the Sum of the Sides &

and O. 9025—2.954447

Is to their Difference 2075. 3.317018 So is the Tang. of half the Angles at 3 and 9.00 d. 9.199712

,

12.516730

8.5.62283

..

To the Tangent of 2,09 deg,
Which added to 9.00 deg. gives 11.09 deg. for the Angle at the
Earth (or of Elongation) And fubftracted from
3.
9.00 deg. leaves 6.91 deg. for the Angle at Mars (or of the
Parallax of the Orb..

1