A new and most accurate theory of the moon's motion: whereby all her irregularities may be solved, ... Written by ... Mr. Isaac Newton, and published in Latin by Mr. David Gregory in his excellent Astronomy. — Theory of the Moon's motion. Latin.

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- [titlepage]

- TO THE READER·

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

THIS Theory hath been long expected by the Lovers of Art, and is now publish'd in Dr. Gregory's Astronomy, in Mr. Newton's own Words.

By this Theory, what by all Astronomers was thought most difficult and almost impossible to be done, the Excellent Mr. Newton hath now effected, viz. to determine the Moon's Place even in her Quadratures, and all other Parts of her Orbit, besides the Syzygys, so accurately by Calculation, that the Difference between that and her true Place in the Heavens 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 Places of the Moon observ'd by Mr. Flamstead, and communicated to him.

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

The mean Motions of the Sun and Moon, accounted from the Vernal Equinox 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 ascending Node 4 Sign. 27°. 24′. 20″. For in 20 Julian Years or 7305 Days, 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″.

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 Luminaries 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″.

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 Apogee of the Moon will be E, and that of the Node F.

Only observe here, that if the Equation of the Sun's Centre be required to be added; then the Equation of the Moon's mean Motion must be subtracted, that of her Apogee must be added, and that of the Node subducted. And on the contrary, if the Equation of the Sun's Centre were to be subducted, the Moon's Equation must be added, the Equation of her Apogee subducted, and that of her Node added.

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 the Moon's Apogee is any where but in the Octants, this Equation grows less, and is mostly at the same distance 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 Quadrature.

There is moreover another Equation 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 vanishes quite, when they come to their Quadratures or Syzygys. This Equation is proportional to the Sine of the double Distance of the 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 Syzygys with the Sun to their Quadratures 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 Lunar 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 Apogee may be compar'd after the manner following (which takes place also in the Computation of any other intermediate Equations.)

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 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 Apogee required.

In order to whose Determination 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 according 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″.

Having from these Principles made a Table of the Equation of the Moon's Apogee, and of the Eccentricitys 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 Apogee) may easily be had for any Time required; let the Equation thus found be added, to the first Equated 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 corresponding to any given Time. Moreover, from this mean Anomaly of the Moon, and the before-found Eccentricity of her Orbit, may be found (by means of a Table of Equations of the Moon's Centre made to every degree of the mean Anomaly, and some Eccentricitys, viz. 45000, 50000, 55000, 60000, and 65000) the Prostaphaeresis or Equation 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, reciprocally 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 Differences of this Variation in the Octants be made reciprocally as the Cubes of the Distances of the Sun from the Earth; and so let a Table 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 Anomaly: 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 (accounting 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 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 encreasing, add, but when decreasing, subtract; 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 produced by the mean Quantity 2′. 20″. is not always of the same Magnitude, but is encreased and diminished according to the Position of the Lunar Apogee. For if the Moon's Apogee be in Conjunction with the Sun's, the aforesaid Equation is about 54″. greater: but when the Apogees are in opposition, 'tis about as much less; and it librates between its greatest Quantity 3′. 14″. and its least 1′. 26″. And this is when the Lunar Apogee is in Conjunction or Opposition with the Sun's: But in the Quadratures the aforesaid Equation is to be lessen'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 mentioned, I dare determine nothing certain, for the same Reason, viz. the want of Observation accurately made.

If the sixth and seventh Equations are augmented or diminished in a reciprocal Ratio of the Distance of the Moon from the Earth, i. e. in a direct Ratio of the Moon's Horizontal Parallax; they will become 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 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 ascending Node, equated as above; the Remainder shall be the Annual Argument of the Node, whence its second Equation may be computed after the following manner in the preceding Figure:

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 Argument of the Node before found: So shall the Angle BTF be the second Equation of the Ascending Node: which must be added when the Node is passing from a Quadrature to a Syzygy with the Sun, and subducted when the Node moves from a Syzygy towards a Quadrature. 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 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 Ecliptick 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 Latitude thus found, and the given Obliquity 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′. 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 Geographical 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 Semidiameter of the Earth's Shadow projected 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 Incomparable Mathematician. And if we had many Places of the Moon accurately observed, especially about her Quadratures, and these well compared with her Places at the same time calculated according to this Theory; it would then appear whether there yet remain any other sensible Equations, which when accounted for, might serve to improve and enlarge this Theory.

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

FINIS.

A New and most Accurate THEORY OF THE Moon's Motion; Whereby all her Irregularities may be solved, and her Place truly calculated to Two Minutes.

Written by That Incomparable Mathematician Mr. ISAAC NEWTON, And Published in Latin by Mr. DAVID GREGORY in his Excellent Astronomy.

LONDON, Printed, and sold by A. Baldwin in Warwick-lane. 1702.

TO THE READER·

THE Irregularity of the Moon's Motion hath been all along the just Complaint of Astronomers; and indeed I have always look'd upon it as a great Misfortune that a Planet so near us as the Moon is, and which might be so wonderfully useful to us by her Motion, as well as her Light and Attraction (by which our Tides are chiefly occasioned) should have her Orbit so unaccountably various, that it is in a manner vain to depend on any Calculation of an Eclipse, a Transit, or an Appulse of her, tho never so accurately made. Whereas could her Place be but truly calculated, the Longitudes of Places would be found every where at Land with great Facility, and might be nearly guess'd at Sea without the help of a Telescope, which cannot there be used.

This Irregularity of the Moon's Motion depends (as is now well known, since Mr. Newton hath demonstrated the Law of Universal Gravitation) on the Attraction of the Sun, which perturbs the Motion of the Moon (and of all other Satellites or secondary Planets) and makes her move sometimes faster and sometimes slower in her Orbit; and makes consequently an Alteration in the Figure of that Orbit, as well as of its Inclination to the Plain of the Ecliptick. But this being now to be accounted for, and reduced to a Rule; by this Theory such Allowances are made for it, as that the Place of the Planet shall be truly Equated.

This therefore being perfectly New, and what the Lovers of Astronomy have a long while been put in hopes to receive from the Great Hand that hath now finished it; I thought it would be of good service to our Nation to give it an English Dress, and publish it by it self: For as Dr. Gregory's Astronomy is a large and scarce Book, it is neither every ones Money that can purchase it, nor Acquaintance that can procure it; and besides I hope we have a great many Persons in England that have Skill and Patience enough to calculate a Planet's Place, who yet it may be don't well enough understand the Latin Tongue to make themselves Masters of this Theory in the Author's own Words. At least I perswade my self, that a Theory so easy and plain as this, which carries along with it such a Pretence to Exactness, will encourage many Persons to imploy themselves in Astronomical Calculation, which before possibly they neglected, because they judged there was but little Exactness to be attained in it. And this would be a very useful way of spending their leisure Hours; and if they would oblige us with the Publication of good Ephemerides, Tables, &c. they would soon enflame others with a Desire of pursuing these kind of Studies.

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A new and most accurate theory of the moon's motion: whereby all her irregularities may be solved, ... Written by ... Mr. Isaac Newton, and published in Latin by Mr. David Gregory in his excellent Astronomy.

Theory of the Moon's motion. Latin.

Newton, Isaac, Sir, 1642-1727.

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http://ota.ox.ac.uk/id/3806

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1106008057

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9781106008053

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K029412.000

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CW108853224

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0888000300

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— A new and most accurate theory of the moon's motion: whereby all her irregularities may be solved, ... Written by ... Mr. Isaac Newton, and published in Latin by Mr. David Gregory in his excellent Astronomy., Theory of the Moon's motion. Latin., Newton, Isaac, Sir, 1642-1727.. 29,[1]p. : ill. ; 8⁰. London :: printed, and sold by A. Baldwin,. 1702.. [Extracted from David Gregory's 'Astronomiae physicae & geometricae elementa', Oxford, 1702.] [Reproduction of original from the British Library.] [English Short Title Catalog, ESTCT18650.] [Electronic data. Farmington Hills, Mich. : Thomson Gale, 2003. Page image (PNG). Digitized image of the microfilm version produced in Woodbridge, CT by Research Publications, 1982-2002 (later known as Primary Source Microfilm, an imprint of the Gale Group).]

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