MOO
(S78)
MOO
lias a World, in great meafure, of his own difcovering, or rather fubduing.
From the Theory of Gravity l.e fhews, that the larger Planers revolving round the Sun, may carry along with 'em fmaller Planers revolving round themfelvea » and ftiews, ii priori^ that thefe fmaller mull: move in EUipfes having their Umbilici in the Centres of the larger; and have their Morion in their Orbit variously diilurbed by the Mo- tion of the Sun 5 and, in a word, mult be affected with thufe Inequalities which we actually obferve in the Maori. And from this Theory, argues analogous Irregularities in the Satellites of Saturn.
From this fame Theory he examines the force which the Sun has to difturb the Moo;:'.s Motion, determines the horary Increafe of the Area which the Moon would de- scribe m a circular Orbit by Radii drawn to the Earth her Diftance from the Earth the Horary Mo- tion in a circular and elliptic Orbit the mean Motion of
the Nodes the true Motion of rhe Nodes the horary
"Variation of the Inclination of the Moons Orbit to the Plane of the Ecliptic.
LalUy, From the fame Theory he has found the annual Equation of th&.Moon's mean Motion to arife from the various dilatation of her Orbit j and that Variation to arife from the Sun's iorcc, which being greater in the Perigee, diirends the Orbit; and being leis in the Apogee, fufrers it to" be "again contracted. In the dilated Orbit, the moves more ilowly ■> in the contracted; more fwiftly : and the annual Equation, whereby this lr equality is compenlated, in the Apogee, and Perigee is nothing at all; at a mode- rate diiiance from the Sun, amounts to ir, 50" J and in other places is proportional to the Equation of the Sun's Centre, and is added to the mean Motion of the Moon, when the Earth proceeds fiom its Aphelion to its Periheli- on; and lubilracled when in its Oppofite part. Supposing the Radius of the Orbis .magnps iccc, and the Earth's Ec- centricity 16 f; this Equation, when greateft, accurding to the Theory of Gravity, comes out 11', 49", S'". He adds, thar in ti:e Earth's Perihelion the Nodes move Iwifter than in the Aphelion", and that in a triplicate Ratio of the Earth's diilar.ee from the Sun, inverfely. Whence arife annual Equations of their Motions, proportionable to that of the Centre of the Sun. Now the Sun's Motion is in a duplicate Ratioof the Earth's Diftance from the Sun inverlefy, and the greateft Equation of rhe Centre wBich this Inequality occaflons, is i Q 56 , ±6', agreeable to the Sun's Eccen- tricity 16 yj If the Sun's Morion were in a triplicate Ratio of it£ Diftance iBverfely, this Inequality could generate the greateft: Equation 2 , 66', 9"; and therefore the greateft Equarinii,-- which the Inequalities of the Motions of the Moon's Apou.' e and Nodes occahon, are to 2 , 56', 9, as the mean diurnal Motion of the Moon's Apogee, and the mean diurnal Motion of her Nedes are to the mean diurna! Motion of the Sun. Whence the greateft Equation ot the mean Motion of the Apogee comes our 19", 52 j and the greater Equa- tion of the mean Motion of the Nodes 9' 27". The former Equation is added, and the latter fubftracted, when the Earth proceeds from its Perihelion to its Aphelion; and the contrary in the oppofite part of its Orbir.
From the fame Theory of Gravity it alfo appears, that the Sun's Action on the iJ/o<?« mult be fomewhat greater when the tranfverfe Diameter of the Lunar Orbit paffes through the Sun, than when it is at right Angles with the Line that joins the Earth and Sun : And, therefore, that the Lunar Orbit is fomewhat greater in the firfi cafe, than in the fecond. Hence arifes another Equation of rhe mean Lunar Motion, depending on rhe Situation of the Moon's Apogee with regard 10 the Sun, which is greateft when the moob's Apogee is in an Octant with rhe Sun; and none, when that arrives at the Quadrature, or Syzygies 5 and is added to the mean Motion, in the Paffageof the Moon's Apogee from the Quadrature to the Syzygies, and fubftracted in the Paffage of the Apogee from the Syzygies to the Quadra- ture. This Equation, which he calls Seme/iris, when greateft, viz. in the Octants of the Apogee, arifes to 3', 34", at a mean diftance of the Earth from the Sun , but it increafes and diniiruihes in a triplicate Ratio of the Sun's diiiance inverfely 5 and therefore in the Sun's greateft diiiance, is;', 34"; in the fmalleft, 3', 56", nearly. But when the Apogee of the Moon is without the Octants, it becomes lefs, and is to the greateft Equation, as the Sine of double the diiiance of the Moon's Apogee, from the next Syzygy or Quadrature, to the Radius.
From the fame Theory of Gravity it follous, that the Sun's Aclion on the Moon is fomewhat greater when a Line right drawn through the Moon's Nodes paffes through the Sun, than when that Line is at right Angles with another joining rhe Sun and Earth : And hence arifes another JB'quaVton of the AToon's mean Motion, which he calls Se- curid.i Semejiris, and which is greateft when the Nodes are in the Sun's Oitar's, and vahjJtes when they are in the Sjzygies, or Quadratures; and in other Situations of the
Nodes is proportionable to the Sine of double the diftance of either Node from the next Syzygy, or Quadrature; it is added to the Moon's mean Motion while the Nodes ate in their Paffage from the Sun's Quadratures to the next Syzygy, and fubftracted in their Paffage from the Syzyg| es to the Quadratures in the Octants. When it is greateft, it amounts to 47", at a mean diitance of the Earth from the Sun 5 as appears from the Theory of Gravity : At other diftances of the Sun, this Equation in the Octants of the Nodes is reciprocally as the Cube of the Sun's diitance front the Earth 5 and therefore in the Sun's Perigee is 45 j in his Apogee nearly 49".
By the fame Theory of Gravity, the Moon's Apogee proceeds the fa ft eft when either in Conjunction with the bun, or in Opposition to it; and returns when it makes a Quadrature with the Sun. In the former Cafe, the Ex- cenrricity is greateft, and in the latter fmalleft. Thefe In- equalities are very considerable, and generate the principal Equation of the Apogee, which he calls Seme/iris, or Semi- mettjirutd. The greateft 'Semi-menftrual Equation is about
12°, l8'.
Ron-ox firft obferv'd the Moon to revolve in an Ellipsis round the Earth placed in the lower Umbilicus: And Halky placed the Centre of the EUipfts in an Epicycle whole Centre revolves uniformly about the Earth : And from the Motion in the Epicycle arife the Inequalities now obitrved in the Progrefs and Regrels of the Apogee, and the Quantity of the Eccentricity.
Suppose the mean diitance of rhe Moon from the Earth divided into jococc, and let T (Plate Astronomy, fig. 1 -.) reprefent the Earth, and T C the mean Eccentricity of the Moon 5 505 pans 5 produce T C to B, that C B may be the Sine of the greateft Semi-menftrual Equation iz Q , 18' to the Radius T C 5 the Circle B D A, defcribed on the Centre C, with the Interval C B, will be the Epicycle wherein the Centre of the Lunar Orb is placed, and wherein it revolves according to the Order ot the Letters B DA. Take the Angle BCD equal to double the annual Argument, or double the diitance of the true Place ot the Sun from the Moon's Apogee once equated, and C T D will be the Semi-menftrual Equation of the Moore's Apogee, andTD the Eccentricity of its Orbit tending to the Apogee equated a fecond time. Now the Moon'b mean Motion, Apogee, and Eccentricity, as alfo the greater Axis of its Orbit -coaco 5 the Moon's true place, as alfo her diftance from the Earth are found, and that by the commoneft Methods,
In thv Earth's Perihelion, by reafon of the gteater force of the Sun, the Centre of the Moon's Orbit will move more fwiftly about the Centre C, than in the Aphelion, and that in a triplicate Ratio of the Earth's diftance from the Sun inverfely. By reafon of the Equation of the Centre ot the Sun, comprehended in the annual Argument, the Centre of the Moon's Orbit will move more fwiftly in the Epicycle B D A 7 ma duplicate Ratio of the diftance of the Earth from the Sun inverfely. That the fame may ftill move more fwiftly in a fimple Ratio of the diftance inverfely from the Centre of the Orbit D, draw D E to- wards the Mom's Apogee, or parallel to T C; and take the Angle E D G equal to the Excefs of the annual Ar- gument, above the Diftance of the Moon's Apogee from the Sun's Perigee in Confequentia; or which is the fame, take the Angle CDF equal to the Complement of the true Anomaly of the Sun to 3<Jo Q 5 and let D F be to D C as double the Eccentricity of the Orbis magrnis to the mean diftance of the Sun from the Earth, and the mean diurnal Motion of the Sun from the Moon's Apogee, to the mean diurnal Motion of the Sun from its own Apogee, conjunctly, i.e. as 33! isto igco, and 52',* 7", 16" 1059' 8" ^"'con- junctly; or as 3 to 100. Conceive the Centre of the Moon's Orbit placed in the Point E, and to revolve in an Epicycle whofe Centre is D, and Radius D F, while D proceeds in the Circumference of the Circle D A B D : Thus the Velocity wherewith the Centre of the Moon's Orbit moves in a certain Curve, defcribed about the Centre C, will be reciprocally as the Cube of the Sun's diftance from the Earth. The Computation of this Motion is diffi- cult, but will be made eafy by the following Approxima- tion. If the Moon's mean diftance from the Earth be ioooco parts, and its Eccentricity TC 5505 of thofe parts, the right Line C B or C D will be found 1 1 7 x $, and the right Line D F 3 5 f. This right Line at the diftance T C, fub tends an Angle to the Earth, which the transferring of the Centre of the Orbir from the place D to F generates in the Motion of this Centre; and the fame right Line doubled, in a parallel Situation, at the diftance of the up- ' per Umbilicus of the Moon's Orbit from the Earth, fub- tends the fame Angle, generated by that tranflation in the Motion of the Umbilicus j and at the diftance of the Moon from the Earth fubtends an Angle which the fame tranflation generates in the Motion of the Moon j and which may therefore be call'd the Second Equation of the
Centre.
