The Famous Mr. Isaac Newton's THEORY OF THE MOON.

THIS Theory hath been long expected by the Lovers of Art, and is now pub⯑lish'd in Dr. Gregory's A⯑stronomy, in Mr. Newton's own Words.

[10]By this Theory, what by all Astro⯑nomers was thought most difficult and almost impossible to be done, the Ex⯑cellent Mr. Newton hath now effect⯑ed, viz. to determine the Moon's Place even in her Quadratures, and all other Parts of her Orbit, besides the Syzygys, so accurately by Calcu⯑lation, that the Difference between that and her true Place in the Hea⯑vens shall scarce be two Minutes, and is usually so small, that it may well enough be reckon'd only as a Defect in the Observation. And this Mr. Newton experienced by comparing it with very many Pla⯑ces of the Moon observ'd by Mr. Flamstead, and communicated to him.

The Royal Observatory at Green⯑wich is to the West of the Meridian of Paris 2°. 19′. Of Vraniburgh 12°. 51′. 30″. And of Gedanum 18°. 48′.

[11]The mean Motions of the Sun and Moon, accounted from the Vernal E⯑quinox at the Meridian of Greenwich, I make to be as followeth. The last Day of December 1680. at Noon (Old Stile) the mean Motion of the Sun was 9 ^Sign. 20°. 34′. 46″. Of the Sun's Apogaeum was 3 ^Sign. 7°. 23′. 30″.

The mean Motion of the Moon at that time was 6 ^Sign. 1°. 35′. 45″. And of her Apogee 8 ^Sign. 4°. 28′. 5″. Of the Ascending Node of the Moon's Orbit 5 ^Sign. 24°. 14′. 35″.

And on the last Day of December 1700. at Noon, the mean Motion of the Sun was 9 ^Sign. 20°. 43′. 50″. Of the Sun's Apogee 3 ^Sign. 7°. 44′. 30″. The mean Motion of the Moon was 10 ^Sign. 15°. 19′. 50″. Of the Moon's Apogee 11 ^Sign. 8°. 18′. 20″. And of her ascend⯑ing Node 4 ^Sign. 27°. 24′. 20″. For in 20 Julian Years or 7305 Days, [12] the Sun's Motion is 20 ^Revolut. 0 ^Sign. 0°. 9′. 4″. And the Motion of the Sun's Apogee 21′. 0″.

The Motion of the Moon in the same Time is 247 ^Rev. 4 ^Sign. 13°. 34′. 5″. And the Motion of the Lunar Apogee is 2 ^Revol. 3 ^Sign. 3°. 50′. 15″. And the Motion of her Node 1 ^Revol. 0 ^Sign. 26°. 50′. 15″.

All which Motions are accounted from the Vernal Equinox: Wherefore if from them there be subtracted the Recession or Motion of the Equinoctial Point in Antecedentia during that space, which is 16′. 0″. there will remain the Motions in reference to the Fixt Stars in 20 Julian Years; viz. the Sun's 19 ^Revol. 11 ^Sign. 29°. 52′. 24″. Of his Apogee 4′. 20″. And the Moon's 247 ^Revol. 4 ^Sign. 13°. 17′. 25″. Of her Apogee 2 ^Revol. 3 ^Sign. 3°. 33′. 35″. And of the Node of the Moon 1 ^Revol. 0 ^Sign. 27°. 6′. 55″.

[13]According to this Computation the Tropical Year is 365 ^Days. 5 ^Hours. 48′. 57″. And the Sydereal Year is 365 ^Days. 6 ^Hours. 9′. 14″.

These mean Motions of the Lu⯑minaries are affected with various Inequalities: Of which,

1. There are the Annual Equations of the aforesaid mean Motions of the Sun and Moon, and of the Apogee and Node of the Moon.

The Annual Equation of the mean Motion of the Sun depends on the Eccentricity of the Earth's Orbit round the Sun, which is 16 11/12 of such Parts, as that the Earth's mean Distance from the Sun shall be 1000: Whence 'tis called the Equation of the Centre; and is when greatest 1°. 56′. 20″.

The greatest Annual Equation of the Moon's mean Motion is 11′. 49″. of her Apogee 20′. and of her Node 9′. 30″.

[14]And these four Annual Equations are always mutually proportional one to another: Wherefore when any of them is at the greatest, the other three will also be greatest; and when any one lessens, the other three will also be diminished in the same Ratio.

The Annual Equation of the Sun's Centre being given, the three other corresponding Annual Equations will be also given; and therefore a Table of that will serve for all. For if the Annual Equation of the Sun's Centre be taken from thence, for any Time, and be called P, and let 1/10 P = Q, Q + 1/60 Q = R, ⅙ P= D, D + 1/30 D = E, and D− 1/60 D = 2 F; then shall the Annual Equation of the Moon's mean Motion for that time be R, that of the Apo⯑gee of the Moon will be E, and that of the Node F.

Only observe here, that if the E⯑quation of the Sun's Centre be re⯑quired [15] to be added; then the Equa⯑tion of the Moon's mean Motion must be subtracted, that of her A⯑pogee must be added, and that of the Node subducted. And on the contra⯑ry, if the Equation of the Sun's Centre were to be subducted, the Moon's Equation must be added, the Equation of her Apogee sub⯑ducted, and that of her Node ad⯑ded.

There is also an Equation of the Moon's mean Motion depending on the Situation of her Apogee in respect of the Sun; which is greatest when the Moon's Apogee is in an Octant with the Sun, and is nothing at all when it is in the Quadratures or Syzygys. This Equation, when greatest, and the Sun in Perigaeo, is 3′. 56″. But if the Sun be in Apogaeo, it will never be above 3′. 34″. At other Distances of the Sun from the Earth, this Equation, when greatest, is reciprocally as the Cube of such Distance. But when [16] the Moon's Apogee is any where but in the Octants, this Equation grows less, and is mostly at the same di⯑stance between the Earth and Sun, as the Sine of the double Distance of the Moon's Apogee from the next Quadrature or Syzygy, to the Radius.

This is to be added to the Moon's Motion, while her Apogee passes from a Quadrature with the Sun to a Syzygy; but is to be subtracted from it, while the Apogee moves from the Syzygy to the Quadra⯑ture.

There is moreover another Equa⯑tion of the Moon's Motion, which depends on the Aspect of the Nodes of the Moon's Orbit with the Sun: and this is greatest when her Nodes are in Octants to the Sun, and va⯑nishes quite, when they come to their Quadratures or Syzygys. This Equation is proportional to the Sine of the double Distance of the [17] Node from the next Syzygy or Quadrature; and at greatest is but 47″. This must be added to the Moon's mean Motion, while the Nodes are passing from their Sy⯑zygys with the Sun to their Quadra⯑tures with him; but subtracted while they pass from the Quadratures to the Syzygys.

From the Sun's true Place take the equated mean Motion of the Lu⯑nar Apogee, as was above shewed, the Remainder will be the Annual Argument of the said Apogee. From whence the Eccentricity of the Moon. and the second Equation of her Apo⯑gee may be compar'd after the man⯑ner following (which takes place also in the Computation of any other inter⯑mediate Equations.)

[18]

Let T represent the Earth, TS a Right Line joining the Earth and Sun, TACB a Right Line drawn from the Earth to the middle or mean Place of the Moon's Apogee, equated, as above: Let the Angle STA be the Annual Argument of the aforesaid Apogee, TA the least Eccentricity of the Moon's Orbit, TB the greatest. Bissect AB in C; and on the Centre C with the Distance AC describe a Circle AFB, and make the Angle BCF = to the [19] double of the Annual Argument. Draw the Right Line TF, that shall be the Eccentricity of the Moon's Orbit; and the Angle BTF is the second Equation of the Moon's Apo⯑gee required.

In order to whose Determina⯑tion let the mean Distance of the Earth from the Moon, or the Semidiameter of the Moon's Orbit, be 1000000; then shall its greatest Eccentricity TB be 66782 such Parts; and the least TA, 43319. So that the greatest Equation of the Orbit, viz. when the Apogee is in the Syzygys, will be 7°. 39′. 30″. or perhaps 7°. 40′. (for I suspect there will be some Alteration ac⯑cording to the Position of the Apogee in ♋ or in ♑.) But when it is in Quadrature to the Sun, the greatest Equation aforesaid will be 4°. 57′. 56″. and the greatest Equation of the Apogee 12°. 15′. 4″.

[20]Having from these Principles made a Table of the Equation of the Moon's Apogee, and of the Eccen⯑tricitys of her Orbit to each degree of the Annual Argument, from whence the Eccentricity TF, and the Angle BTF (viz. the second and principal Equation of the Apo⯑gee) may easily be had for any Time required; let the Equation thus found be added, to the first E⯑quated Place of the Moon's Apogee, if the Annual Argument be less than 90°, or greatear than 180°, and less than 270; otherwise it must be subducted from it: and the Sum or Difference shall be the Place of the Lunar Apogee secondarily equated; which being taken from the Moon's Place equated a third time, shall leave the mean Anomaly of the Moon corre⯑sponding to any given Time. More⯑over, from this mean Anomaly of the Moon, and the before-found Eccentricity of her Orbit, may be found (by means of a Table of E⯑quations [21] of the Moon's Centre made to every degree of the mean Ano⯑maly, and some Eccentricitys, viz. 45000, 50000, 55000, 60000, and 65000) the Prostaphaeresis or Equa⯑tion of the Moon's Centre, as in the common way: and this being taken from the former Semicircle of the middle Anomaly, and added in the latter to the Moon's Place thus thrice equated, will produce the Place of the Moon a fourth time equated.

The greatest Variation of the Moon (viz. that which happens when the Moon is in an Octant with the Sun) is, nearly, reciprocal⯑ly as the Cube of the Distance of the Sun from the Earth. Let that be taken 37′. 25″. when the Sun is in Perigaeo, and 33′. 40″. when he is in Apogaeo: And let the Dif⯑ferences of this Variation in the Oc⯑tants be made reciprocally as the Cubes of the Distances of the Sun from the Earth; and so let a Table [22] be made of the aforesaid Variation of the Moon in her Octants (or its Logarithms) to every Tenth, Sixth, or Fifth Degree of the mean Anoma⯑ly: And for the Variation out of the Octants, make, as Radius to the Sine of the double Distance of the Moon from the next Syzygy or Quadrature ∷ so let the aforefound Variation in the Octant be to the Variation congruous to any other Aspect; and this added to the Moon's Place before found in the first and third Quadrant (account⯑ing from the Sun) or subducted from it in the second and fourth, will give the Moon's Place equated a fifth time.

Again, as Radius to the Sine of the Sum of the Distances of the Moon from the Sun, and of her Apogee from the Sun's Apogee (or the Sine of the Excess of that Sum above 360°.) ∷ so is 2′. 10″. to a sixth Equation of the Moon's Place, which must be subtracted, if [23] the aforesaid Sum or Excess be less than a Semicircle, but added, if it be greater. Let it be made also, as Radius to the Sine of the Moon's Distance from the Sun ∷ so 2′. 20″. to a seventh Equation: which, when the Moon's Light is encreas⯑ing, add, but when decreasing, sub⯑tract; and the Moon's Place will be equated a seventh time, and this is her Place in her proper Orbit.

Note here, the Equation thus pro⯑duced by the mean Quantity 2′. 20″. is not always of the same Magnitude, but is encreased and di⯑minished according to the Position of the Lunar Apogee. For if the Moon's Apogee be in Conjunction with the Sun's, the aforesaid Equa⯑tion is about 54″. greater: but when the Apogees are in oppositi⯑on, 'tis about as much less; and it librates between its greatest Quan⯑tity 3′. 14″. and its least 1′. 26″. And this is when the Lunar Apogee is in Conjunction or Opposition with [24] the Sun's: But in the Quadratures the aforesaid Equation is to be les⯑sen'd about 50″. or one Minute, when the Apogees of the Sun and Moon are in Conjunction; but if they are in Opposition, for want of a sufficient number of Observations, I cannot determine whether it is to be lessen'd or increas'd. And even as to the Argument or Decrement of the Equation 2′. 20″. above men⯑tioned, I dare determine nothing certain, for the same Reason, viz. the want of Observation accurately made.

If the sixth and seventh Equati⯑ons are augmented or diminish⯑ed in a reciprocal Ratio of the Di⯑stance of the Moon from the Earth, i. e. in a direct Ratio of the Moon's Horizontal Parallax; they will be⯑come more accurate: And this may readily be done, if Tables are first made to each Minute of the said Parallax, and to every sixth or fifth Degree of the Argumennt of the [25] sixth Equation for the Sixth, as of the Distance of the Moon from the Sun, for the Seventh Equation.

From the Sun's Place take the mean Motion of the Moon's ascen⯑ding Node, equated as above; the Remainder shall be the Annual Ar⯑gument of the Node, whence its second Equation may be computed after the following manner in the pre⯑ceding Figure:

[26]Let T as before represent the Earth, TS a Right Line conjoining the Earth and Sun: Let also the Line TACB be drawn to the Place of the Ascending Node of the Moon, as above equated; and let STA be the Annual Argument of the Node. Take TA from a Scale, and let it be to AB ∷ as 56 to 3, or as 18 ⅔ to 1. Then bissect BA in C, and on C as a Centre, with the Distance CA, describe a Circle as AFB, and make the Angle BCF equal to double the Annual Argu⯑ment of the Node before found: So shall the Angle BTF be the se⯑cond Equation of the Ascending Node: which must be added when the Node is passing from a Quadra⯑ture to a Syzygy with the Sun, and subducted when the Node moves from a Syzygy towards a Quadra⯑ture. By which means the true Place of the Node of the Lunar Orbit will be gained: whence from Tables made after the common way, the [27] Moon's Latitude, and the Reduction of her Orbit to the Ecliptick, may be computed, supposing the Inclination of the Moon's Orbit to the Eclip⯑tick to be 4°. 59′. 35″. when the Nodes are in Quadrature with the Sun; and 5°. 17′. 20″. when they are in the Syzygys.

And from the Longitude and La⯑titude thus found, and the given Ob⯑liquity of the Ecliptick 23°. 29′. the Right Ascension and Declination of the Moon will be found.

The Horizontal Parallax of the Moon, when she is in the Syzygys at a mean Distance from the Earth, I make to be 57′. 30″. and her Horary Motion 33′. 32″. 32‴. and her apparent Diameter 31′. 30″. But in her Quadratures at a mean Distance from the Earth, I make the Horizontal Parallax of the Moon to be 56′. 40″. her Horary Motion 32′. 12″. 2‴. and her apparent Diameter 31′. [28] 3″. The Moon in an Octant to the Sun, and at a mean Distance, hath her Centre distant from the Centre of the Earth about 60 2/9. of the Earth's Semi-Diameters.

The Sun's Horizontal Parallax I make to be 10″. and its apparent Diameter at a mean Distance from the Earth, I make 32′. 15″.

The Atmosphere of the Earth, by dispersing and refracting the Sun's Light, casts a Shadow as if it were an Opake Body, at least to the height of 40 or 50 Geographical Miles (by a Geogra⯑phical Mile I mean the sixtieth Part of a Degree of a great Circle, on the Earth's Surface) This Shadow falling upon the Moon in a Lunar Eclipse, makes the Earth's Shadow be the larger or broader. And to each Mile of the Earth's Atmosphere is correspondent a Second in the Moon's Disk, so that the Semidiame⯑ter of the Earth's Shadow projected [29] upon the Disk of the Moon is to be encreased about 50 Seconds: or which is all one, in a Lunar Eclipse, the Horizontal Parallax of the Moon is to be encreased in the Ratio of about 70 to 69.

Thus far the Theory of this In⯑comparable Mathematician. And if we had many Places of the Moon accurately observed, especially about her Quadratures, and these well com⯑pared with her Places at the same time calculated according to this Theory; it would then appear whe⯑ther there yet remain any other sensible Equations, which when ac⯑counted for, might serve to improve and enlarge this Theory.

Dr. Greg. Astr. Elem. Phys. & Geom. p. 336.