ANUA MATHEMATICA. VEL, Trigonometria Practica. SECTION VII. OF Theorical ASTRONOMY. I N the Fourth Section of this Thrid PART, you have the Doctrine of Triangles applyed to Practice in that part of Aftroпоту nomy which concerns the Diurnal Motion of the Sun and Fixed Stars: Now, in this Section, I fhall discourse something of the other Stars or Planets, which have a Secondary Motion of their own; fhewing alfo, how fubfervient Trigonometry is for the Calculating of their true Places at any time, as alfo in compute of their Magnitudes, Diftances, &c. And in order thereunto I fhall firft give you a brief defcription of the Planetary Syftem. TH CHA P. I. Of the Planetary System. HIS Syftem may be fufficiently represented as in the Figure thercof: Wherein, Firft, The Sun being in the Centre, hath only a Rotation from Weft to Eaft, upon its own Axis, in the space of 26 days, or Hii 2 much much thereabouts: His Centre being an immoveable Point; to which the Revolutions of the other Planets are referred.. Secondly, The Planets are moved about the Sim, from Weft to Eaft, infeveral Orbs, which return into themselves, every Planet in Time proportionable to the Magnitude of his Orb and Distance from the Sun, the Motions of the Primary Planets, h, ¥, d, e, 9, and, being uniform, perpetually conftant and regular. The Figure of the System. Fig. The Circle BCD, denotes the Way and Revolution of the LXX. Planet Mercury, about the Sun, in 88 Days-The Circle. 8 E FG, the Revolution of Venus in 225 Days--The Circle H IK, the Revolution of the Earth, with the Moon in one Year---The Circle L M N &, theRevolution of Mars-The Circle OPQ, the Revolution of Jupiter, with his four Companions (or Satellities) in twelve Years The Circle ↳ RST h, the Revolu tion of Saturn, with his Ring and Moon, in Thirty Years.The Moon alfo moves round the Earth, every Month- -Fupi-· ters four Companions, move about him according to their Diflances from him.- The Firft (and nearest to him) in One Day and 18 Hours- The Second, in Three Days and 13 Hours. The Third, in Seven Days and 4 Hours. 26 Days and 5 Hours The Fourth, in Saturn's Moon moveth about him in Sixteen Days: And all of them from Weft to Eaft, according to their Planets Revolution about the Sun. Thefe Planets, whofe Revolutions refpect the Sun only, as Saturn Jupiter, Mars, the Earth, Venus and Mercury, are called Primary Planets: The other that move about Saturn, Jupiter and the Earth, Secondary Planets. The Secondary Planets, are all of them much Lefs in Magnitude than their Primary; and all the Planets together, much Lej's than the Sim, from whom they receive their Light, Motion, &E СНАР. CHA P. II. Of the Theory of the Sun and the other Primary Planets. HAT the Motion of the Primary Planets about the Sun were T Elliptical, was firft difcovered by the learned Kepler, which he deduced from the acurate Obfervations of the Noble Dane Tychobra. And therefore, in fuch an Elliptical Figure, may be defcribed all fuck Points, Lines, Arches, &c. as are requifite to be known, and to inform the Fancy in the Trigonometrical Calculation of the Planets Places. But before I proceed fo far, I must fhew, How to defcribe an Ellipfis. Concerning the Definition of this Figure, only that it is one of the Sections of a Cone, and the many ways that it may be Artificially defcribed, I fhall fay nothing, referring the Reader to Midorges, and others that have largely written of the Sections of a Cone. But to Defcribe fuch a Figure Mechanically. Thus: About any Two Right-Lines Given, for the Two Diameters of the Ellipfis, to defcribe fuch an Ellipfis, Fig. LXX. Let the Line P Y be the Longer, and RM the Shorter Diameter Fig. given: Let them crofs one another at Right-angles in B. This LXXI. · done, Take half of the Longest Diameter BP or BY, in your Compaffes, and fetting one foot in R or M, the other will reach upon the Longest Diameter to the Points Z and S, which will be the Centres upon which the Ellipfis PRY M must be defcribed. Wherefore, In the two Points Z and S, Fix two Pins, Nails, or the like; and about them put a String fo long, that being doubled it may reach from the Pin at S, to the end of the Diameter at P; or from the Pin at Z to the end. Y; fo then the whole length of the String, will be twice as long as the Distance SP or Z Y; which String at that length, joyn at both ends: Then, putting this. String over the two Pins, at Z and S, with a Third Pin (which may be a Black-Lead Pencil, or fuch like) move the String about upon the two Fixed Pins Z and S; and it will by its Motion defcribe.an Ellipfis, fuch as-is the Figure PCREYFMDP. CHAP. Fig. LXXI. Ν CHAP. III. Of the Motion of the Planets in the Ellip IN this Elliptical Figure, the Line P Y is called the (or Longer,) And R M, the Conjugate (or Shorter of the Ellipfis. S the Lower, and Z the Upper Focus of fis; upon which two Points the Elliptical Figure was d And the Elliptical Figure it felf PCREYFMDP b of the Earth, or any other of the Primary Planets :---of the Sun; (the common Node, and Centre of the Pla to which the true Motions of the Planets are referred,) ir Focus of the Ellipfis. Now, a Planet moving in t when it shall be in P, it is in Aphelion, or at its Great from the Sun-- When in Y Perihelion, or at its Near to the Sun About the upper Foci of the Ellipfis Z Motion of the Planet is regulated: B is the common Ce Figure BZ or BS, the Excentricity: And SZ the centricity. The Figure thus defcribed, the next thing to be know what manner the true Place of a Planet may be therein. ་ First, The mean Anomaly of a Planet, is its equal 1 P, the Aphelion Point of that Planet: And this Anom accounted from P, to P again. And here it is to b that a Planet moving in the Elliptical Arch PRYM Focus Z, is as long time tracing the leffer Arch CP3 the greater CYM; the Reafon is, for that (the being made upon Z) the Planet defcribeth equal An Intervals of Time: And it is alfo evident, That if t! equal upon one Focus Z, of the Ellipfis, it must be t the other S, in which the Sun is feated. For inftance: Suppofe the Planet to be in C, whi from the Aphelion Point P, then will the Angle of it maly be PZC, but the Co-equated Anomaly will be BC: And the Angle at the Sim PSC, which is th found. CHA P. IV. How to find the Angle that a Planet (in any part of the Fig. Ellipfis makes with the Sun; and alfo, the Planets true LXXII. Place in the Ellipfis. L Et S be the Sun, X the Centre of the Planets Mean Motion; the Points H, A, T, feveral Places of the Planet in its Orb, from the Aphelion point P, then will the feveral Angles P XH, PXA, PXT, reprefent the Anomaly of the Planet: And the Angles PSN, and PS M; the Angles at the Sun. Now these An-gles may be found as followeth. Suppofe the Mean Anomaly of the Earth being at H,to be 30 deg. thisis reprefented by the Angle PXH, whofe Complement BXH, is 150 deg. By help whereof, with the Common Radius, and Ex-centricity, the true Place of the Planet may be found by Trigonometrical Calculation. For, Firft, In the Triangle H X B, there is Given, (1) The Angle HX B, 150.00 d. the Complement of the Planets Mean Anomaly to 180 deg. (2) The Side B H, the Common Radius 100000. And (3) The Side B X, 1685, equal to half X S, the Excentricity. By which you may find the Angle B H X, By CASE I. of O. A. P. T. Then, (1) As the Common Radius, B H, 100000 Parts To XB (half the Excentricity) 1685 P. Anomaly HZ P, 30 d. to 180 deg. 5.000900 3.226599€ So is the Sine of H XB, the Complement of the 9.698970 To the Sine of BH X 0.483 deg.. 12.925569 7.925569 And therefore the Angle XBH is 29.518 deg, the double whereof, 59.036 deg. is the Anomaly of Variation. |