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Which of

the two perturbating Forces of

the Sun he employs.

Law of the

Motion of

the Nodes.

Regreffion

and Progref

this circular Orbit into an Ellipfe, whofe Centre is in the Centre of the Earth, or rather into the Curve whose Construction we have already given by the Means of an Ellipfe. Of the two perturbating Forces of the Sun, that which urges the Moon towards the Earth, acting in the Plane of the Orbit, cannot produce any Motion in this Plane; he therefore only confiders that Force which acts parallel to the Line drawn from the Earth to the Sun, which he had fhewn to be proportional to the Cofine of the Angle formed by the Lines drawn from the Moon and the Earth to the Sun, and we fhall now explain how he employs this Force.

At the Extremity of the little Arc which the Moon describes in any fmall Portion of Time, he takes one equal to it, which would be that which the Moon would defcribe if the perturbating Force of the Moon ceased to act on her; and at the Extremity of this new Arc, he draws a Line parallel to that which joins the Centre of the Earth and the Sun, and he determines the Length of this ftraight Line, by the Measure already determined of the Force which acts in the fame Direction as it; which being done, the Diagonal of the Parallelogram, one of whofe Sides is the little Arc which the Moon would describe if the perturbating Force ceafed to act, and the other, the Arc the Moon would defcribe if this Force acted alone, is the real Arc the Moon would defcribe. There remains therefore no more to be done than to determine, how much the Plane which would pafs through this small Arc and the Earth, differs from the Plane which paffes through the first Side and the Earth.

The two Sides already mentioned, being produced until they meet the Plane of the Orbit of the Earth, and having drawn from their Points of Concourse with this Plane, two ftraight Lines to the Centre of the Earth, the Angle which those two ftraight Lines form, is the Motion of the Node during the fmall Portion of Time which the Moon employs in defcribing this fmall Arc, which we have been confidering. And Newton finds that the Meafure of this Angle, and confequently the Velocity or the inftantaneous Motion of the Node, is proportional to the Product of the Sines of three Angles, which exprefs the Distance of the Moon from the Quadrature, of the Moon from the Node, and of the Node from the Sun.

VIII.

It follows from hence, that when one of thofe three Sines becomes fion of the negative, the Motion of the Nodes which before was retrograde, beNodes comes direct. Wherefore when the Moon is between the Quadrature Revolution, and the nearest Node, the Motion of the Node is direct; in all other Cafes, its Motion is retrograde, but the retrograde Motion exceeding

in each.

the dire& Motion, it happens that in each Revolution of the Moon, At the End of every the Nodes are made to recede.

Revolution

When the Moon is in the Syfigies, and the Nodes in the Quadratures, the Nodes that is, 90 Degrees from the Sun, their Motion is 33" 10" 37v 12, recede. wherefore the horary Motion of the Nodes in every other Situation, is Formula to 33" 10" 27v 12, as the Product of the three Sines already mention- which gives ed to the Cube of Radius.

IX.

the horary Motion of the Nodes in any

of the

Nodes.

Suppofing the Sun and the Node to be in the fame Situation with Situation. refpect to the fixed Stars, whilft the Moon paffes fucceffively through Determinaits feveral Distances with refpect to the Sun. Newton investigates (Prop. tion of the 32. B. III.) the horary Motion of the Node, which is a Mean between mean Motiall the different Motions refulting from the foregoing Formula, and this mean Motion of the Node is 16" 33" 16v 36, when the Orbit is fuppofed circular, and the Nodes are in Quadrature with the Sun; in every other Situation of the Nodes, this Motion is to 16" 33" 16 36, as the Square of the Sine of the Distance of the Sun from the Node, is to the Square of the Radius. If the Orbit of the Moon be fuppofed to be an Ellipfe, having its Centre in the Centre of the Earth, the mean Motion of the Nodes in the Quadratures is only 16" 16" 37v 42, and in any other Situation of the Nodes, it depends likewife on the Square of the Sine of the Distance from the Sun.

In order to determine for any given Time, the mean Place of the Nodes, Newton takes a Medium between all the mean Motions already mentioned. He employs in this Inquiry, the Quadrature of Curves, and the Method of Series. By this Means he finds that the Motion of the Nodes in a fydereal Year, fhould be 19° 18' 1" 23", which only differs. 3' from that which results from astronomical Obfervations.

X.

The fame Curve the Quadrature of whofe Area determines the mean DeterminaVelocity of the Nodes, ferves alfo for finding the true Place of the tion of the Nodes for any given Time, (Prop. 33. B. III.)

true Place of the

Time.

The Refult of his Computation is as follows: Having made an Angle Nodes for equal to the Double of that which expreffes the Distance of the Sun any given from the mean Place of the Nodes, let the Sides of this Angle be to each other, as the mean annual Motion of the Nodes, which is 19° 49′ 3" 55", to the Half of their true mean Motion, when they are in the Quadratures, which is 0° 31' 2" 3", that is, as 38,3 to 1, which being done, and having completed the Triangle which will be given, fince this Angle and its two Sides are given, the Angle of this Triangle opposite to the least of those Sides, will exprefs with fufficient Accuracy, the Equation or Correction in the mean Motion of the Nodes for determining the true Motion required.

Variation

clination of

Orbit.

XI.

From the Investigation of the Motion of the Nodes, Newton paffes of the In- (Prop. 34. B. III.) to the Determination of the Variation in the Incli-. the Moon's nation of the Orbit of the Moon. By employing that one of the two perturbating Forces of the Sun which does not act in the Plane of the Orbit of the Moon, he obtains the Measure of the horary Variation in the Inclination of the Orbit of the Moon; this Variation, when the Horary VaOrbit is fuppofed circular, being to the horary Motion of the Nodes, riation of the Inclina- 33" 10" 3v 12, (the Nodes being in the Quadratures, and the Moon in the Syfigies) diminished in the Ratio of the Sine of the Inclination of the Orbit of the Moon to the Radius: as the Product of the Sine of the Distance of the Moon from the nearest Quadrature, the Sine of the Distance of the Sun from the Nodes, and the Sine of the Diftance of the Moon from the Nodes to the Cube of Radius. And this Quantity diminished by is the Variation correfponding to the Orbit rendered elliptic by the perturbating Force of the Sun.

tion.

Method for finding the Inclination of the Moon's

Orbit for any given Time.

Determina

tion of the Latitude of

the Moon.

XII.

The horary Variation of the Inclination of the Orbit of the Moon being thus determined, Newton employing the fame Method, and the fame Suppofitions by which he found the true Place of the Nodes for any given Time, determines (Prop. 35. B. III.) the Inclination of the Orbit for any given Inftant of Time; the Refult of his Computation is as follows.

Let there be taken from the fame Point of a straight Line, affumed as a Base, three Parts in geometrical Proportion, the first expreffing the least Inclination, the third the greateft; let there be afterwards drawn through the Extremity of the Second, a Line making with this Bafe an Angle equal to double the Distance of the Sun from the Node for the propofed Motion let this Line be produced until it meets the Semicircle defcribed on the Difference of the firft and third Lines in geometrical Proportion; which being done, the Interval comprised between the first Extremity of the Base, and the Perpendicular let fall from the common Section of the Circle and the Side of the Angle just mentioned, will exprefs the Inclination for the propofed Time.

From hence is deduced the Moon's Latitude corrected; for in a Right-angled fpherical Triangle is given, befides the Right-angle, the Hypothenufe, viz. the Moon's Distance from the Node, the Angle at the Node, viz. the Inclination of the Plane of the Moon's Orbit to the Plane of the Ecliptic, confequently the Side oppofite to this Angle, which expreffes the Latitude corrected, will be be alfo given.

But there is a more fimple Method for finding the Latitude of the Moon corrected. For the mean Latitude being computed, the Inclination of the Moon's Orbit to the Ecliptic being fuppofed conftant and equal to 5°. 9. 8". the Equation or Correction of the Latitude will be

8' 50" multiplied by the Sine of double the Distance of the Moon frem the Sun lefs the Distance from the Node.

XIII.

tics of the

Newton, after having expofed the Method by which he calculated that W at NewInequality in the Moon's Motion, called her Variation, and the Method to fays with regard he had followed in determining the Motion of her Nodes, and the Va- to the other riation of the Inclination of her Orbit to the Ecliptic, gives an Account Irregulariof what he fays he deduced from his Theory of Gravitation, with re- Moon's spect to the Motion of the Apogee, the Variation of the Excentricity, Motion. and all the other Irregularities in the Moon's Motion. It is in the Scholium of Prop. 35. B. III. he delivers thofe Theorems, which ferve as a Foundation to the Conftruction of the Tables of the Moon's Motion. The Substance of which is as follows.

XIV.

of the Mo

The mean Motion of the Moon fhould be corrected by an Equation Annual depending on the Distance of the Sun from the Earth. This Equation or Equations Correction, called the annual one, is greatest when the Sun is in his Peri- tion of the gee, and least when in his Apogee. Its Maximum is 11'51", and in the other Moon, of Cafes, it is proportional to the Equation of the Centre of the Sun. It is to the Apogee be added to the mean Motion of the Moon in the six first Signs, counted Nodes. from the Apogee of the Sun, and to be fubtracted in the fix other Signs.

The mean Places of the Apogee and of the Nodes fhould be also each corrected by an Equation of the fame Kind, depending on the Distance of the Sun from the Earth, and proportional to the Equation of the Centre of the Sun. The Equation of the Apogee in its Maximum is 19′ 43′′, and is to be added from the Perihelion to the Aphelion of the Earth; the Equation for the Node is to be fubtracted from the Aphelion to the Perihelion of the Earth, and in its Maximum amounts to 9' 24".

XV.

tion of the

Moon.

The mean Motion of the Moon requires a second Correction, depend- First femef ing at once on the Distance of the Sun from the Earth, and on the Situ- trial Equaation of the Apogee of the Moon with refpect to the Sun; this Equa- mean Motion, which is in the direct Ratio of the Sine of double the Angle ex- tion of the preffing the Distance of the Sun from the Apogee of the Moon, and in the inverse Ratio of the Cube of the Distance of the Sun from the Earth, is called the Semeftrial Equation; it is 3' 45" when the Apogee of the Moon is in Octants with the Sun, and the Earth is in its mean Distance. It is to be added, when the Apogee of the Moon advances from its Quadrature with the Sun to its Syfigie: and is to be fubftracted when the Apogee paffes from the Syfigie to the Quadrature.

Second

XVI.

The mean Motion of the Moon requires a third Correction, dependfemeftrial ing on the Situation of the Sun with refpect to the Nodes, as also on Equation. the Distance of the Sun from the Earth; this Correction or Equation,

Determination of the

Place of the

which Newton calls the fecond Semestrial Equation, is in the direct Ratio of the Sine of double the Distance of the Node from the Sun, and in the inverfe Ratio of the Cube of the Distance of the Earth from the Sun it amounts to 47" when the Node is in Octant with the Sun and the Earth in its mean Distance; it is to be added when the Sun recedes in Antecedentia from the nearest Node, and is to be fubtracted when the Sun advances in Confequentia.

XVII.

After those three firft Equations of the Moon's Motion, follows that which is called her Equation of the Centre; but this Equation cannot be obtained as that of the other Planets, by the Help of one Table, becaufe her Excentricity varies every Inftant, and the Motion of her Apogee is very irregular. In order therefore to obtain the Equation of the Centre of the Moon, the Excentricity and the true Place of the Apogee of the Moon is firft to be determined, which is effected by the Help of Tables founded on the following Propofition.

A ftraight Line being taken to exprefs the mean Excentricity of the Orbit of the Moon, which is 5505 Parts of the 100000 into which the Apogee, and mean Distance of the Moon from the Earth is fuppofed to be divided; at of the Ex- the Extremity of this ftraight Line affumed as a Bafe, an Angle is centricity. made equal to double of the annual Argument, or of double the Distance of the Sun from the mean Place of the Moon once corrected, as has been already directed. The Length of the Side of this Angle is afterwards determined by making it equal to 11724, half the Difference between the least and greatest Excentricity. The Triangle being then completed, the other Angle at the Bafe, expreffes the Equation or Correction to be made to the Place of the Apogee already once corrected; and the other Side of the Triangle which is oppofite to the Angle made equal to double of the annual Argument, will exprefs the Excentricity correfponding to the propofed Time. The Equation of the Apogee being added to its Place already corrected, if the annual Argument be lefs than 90, or between 180 and 270, or being fubtracted in every other Equation of Cafe, the true Place of the Apogee will be obtained, which is to be the Centre, fubducted from the Place of the Moon corrected by the three Equations already mentioned, in order to have the mean Anomaly of the Moon. of the Place With this Anomaly, and the Excentricity, the Equation of the Centre by the ufual Methods will be obtained, and confequently the Place of the Moon corrected for the fourth Time.

or fourth Correction

of the

Moon.

The Equation of the Centre may be obtained without fuppofing the Excentricity variable, or a Motion in the Apogee, by applying to double

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