of the Angle at the Moon fubtended by the mean Excentricity, or to the mean Equation of the Centre, the Equation 80'Sin (2 Dif. (O-m. An.) expreffing the Variation produc'd by the Change of Excentricity, and Libration of the Apogee.

XVIII.

is her Va

The Place of the Moon corrected for the fifth Time, is obtained by The fifth Equation applying to the Place of the Moon corrected for the fourth Time, the othe Equation called the Variation which was already found, to be always in Moon's the direct Ratio of the Sine of double the Angle expreffing the Diffance Motion, of the Moon from the Sun, and in the inverse Ratio of the Cube of the riation. Distance of the Earth from the Sun; this Equation, which is to be added in the first and third Quadrant (in counting from the Sun) and fubtracted in the second and fourth is 35' 10" when the Moon is in Оctant with the Sun, and the Earth in its mean Distance.

XIX.

tion.

The fixth Equation of the Motion of the Moon is proportional to Sixth Equathe Sine of the Angle which is obtained by adding the Distance of the Moon from the Sun, to the Distance of the Apogee of the Moon from that of the Sun. Its Maximum is 2' 20", and it is pofitive when this Sum is lefs than 180 Degrees, and negative if this Sum be greater.

XX.

The seventh and last Equation, which gives the true Place of the Seventh Moon in its Orbit, is proportional to the Distance of the Moon from Equation. the Sun; it is 2' 20" in its Maximum.

XXI.

Newton

has not as

It is fcarce poffible to trace the Road which could have conducted TheMethod Newton to all thofe Equations, except fome Corollaries of Prop. 66. made ufe where he fhews how to estimate the perturbating Forces of the Sun. It of in invefis eafy to perceive, that of those two Forces, the one which acts in the tigating the Direction of the Ray of the Orbit of the Moon, being joined to the Corrections foregoing. Force of the Earth, alters the inverfe Proportion of the Square of the Distances, and confequently should change not only the Curvature of the Orbit, but also the Time which the Moon employs in defcribing it :But how did Newton employ thofe Alterations of the central Force, and what Principles did he make use of to avoid or furmount the extreme Complication and the Difficulties of Computation which occur in this Inquiry is what has not as yet been discovered, at least after a fatisfactory Manner.

We find, it is true, in the first Book of the Principia, a Propofition concerning the Motion of the Apfides in general, by which we learn, that if to a Force which acts inverfely as the Square of the Distance, another Force which is inversely as the Cube of the Distance be joined, the Body will defcribe an Ellipfe whofe Plane revolves about the Centre

yet been

difcovered.

Saturn,

of the Forces. In the Corollaries of this Propofition, Newton extends his Conclufion to the Cafe in which the Force, added to the Force which follows the Law of the Square of the Distance, does not vary in the Triplicate, but in the Ratio of any Power of the Distance.

If therefore the perturbating Force of the Sun depended on the Diflance of the Moon from the Earth alone, by the Help of this Propofition, the Motion of the Apfides of the Moon could be determined; but as the Distance of the Moon from the Sun enters into the Expreffion of this Force, it is only by new Artifices, and perhaps as difficult to be found as the Determination of the entire Orbit of the Moon: the Propofition of Newton concerning the Motion of the Apfides in general, can be applied to the Moon. Senfible of which, the firft Mathematicians of the prefent Age, have abandoned in this, as in every other Point that regards the Theory of the Moon, the Road pursued by the Commentators of Newton, and have refumed the whole Theory from its very Beginning; they have investigated in a direct Manner, the Paths and Velocities of any three Bodies which attract each other mutually. The Success which has attended their united Efforts fhall be explained

hereafter.

XXII.

Theory of It is manifeft, that the Satellites of Jupiter, confidered feparately, the Satellites of Jupiter fhould be affected by the three Forces which actuate them, in the fame and thofe of Manner as the Moon; but their Number introduces a new Source of Inequalities, not only each of them is attracted by Jupiter and the Sun, but they attract each other mutually, and this mutual Attraction should produce very confiderable Variations in their Motions; Variations fo much the more difficult to be fubjected to exact Computations, as they depend on their different Pofitions with refpect to each other, which their different Distances and Velocities continually alter. However, the Laws of their Motions difcovered by Bradley, Wargentin and Maraldi, have enabled the eminent Mathematicians of this Age, to furmount those Difficulties, and to apply the Solution of the Problem of the three Bodies to the Investigation of the Inequalities of the Motions of thofe Satellites, with almoft the fame Success as they had already done to those of the Moon.

As to the Satellites of Saturn, Aftronomers have not been able to determine the Phenomena of their Motions with any Degree of Accuracy on Account of their great Distance; hence the Theory of those Planets is reduced to fhew, that the Forces with which they act on each other, or that with which the Sun acts on them, and disturbs their Motions, are very inconfiderable when compared with the Force with which they tend towards their principal Planet; and that this Attraction is inversely proportional to the Squares of the Distances.

THO

THEORY of the COM ET 3.

I.

HOUGH the Comets have in all Ages, drawn the Attention of The PeripaPhilofophers, yet it is only fince the laft Century and even teticks refince garded the Newton, they can be faid to be known. Seneca feemed to have forefeen the Comets as Discoveries which one Day would be made concerning those Bodies, but Meteors. the Germ of the true Principles which he had fown, were ftifled by the Doctrine of the Peripateticks, who, tranfmitting from Age to Age, the Errors of their Mafter, maintained that the Comets were Meteors or tranfient Fires.

II.

above the

Several Aftronomers, but particularly Ticho, proved this Opinion to Ticho provbe erroneous, by fhewing by their Obfervations, that thofe Bodies were ed that they fituated far above the Moon, they deftroyed at the fame Time, the folid were fituated Heavens, invented by the fcholaftic Philofophers, and propofed Views Moon. concerning the Syftem of the World, which were much more conformable to Reason and Obfervation. But their Conjectures were yet very far from that Point, to which the Geometry of Newton alone could attain.

III.

them as Pla

Defcartes, to whom the Sciences are so much indebted, did not fucceed Descartes better than his Predeceffors in his Enquiries concerning the Comets; regarded he neither thought of employing the Obfervations which were fo eafy nets wanderfor him to collect, nor Geometry to which it was fo natural to have Re- ing from course, and which he had carried to fo great a Point of Perfection; he Vortex to confidered them as Planets wandering through the different Vortices, which, compofed according to him, the Universe; and did not imagine that their Motions were regulated by any Law.

IV.

Vortex.

the Comets

Newton, aided by his Theory of the Planets, and by the Obfer- Newton difvations which taught him that the Comets defcended into our planetary covered that System, foon perceived that thofe Bodies were of the fame Nature revolve with the Planets, and fubject to the fame Laws.

about the

Every Body placed in our planetary System, should, according to the Sun, and are fubjected to Theory of Newton, be attracted by the Sun, with a Force reciprocally the fame proportional to the Squares of the Distances, which combined with a Laws as the Force of Projection, would make it defcribe a Conic Section about the Planete. Sun placed in the Focus. According therefore to this Theory, the Comets fhould revolve in a Conic Section about the Sun, and defcribe Areas proportional to the Times.

V.

Calculation and Obfervation, the faithful Guides of this great Man, enabled him to verify his Conjecture. He folved this fine Aftronomicogeometrical Problem. Three Places of a Comet which is fuppofed to

He determines the

Orbit of a

move in a parabolic Orbit, defcribing round the Sun Areas proportional to the Times, being given, with the Places of the Earth in the Ecliptic Comet from correfponding to thofe Times, to find the Vertex and Parameter of this thr e Obier- Parabola, its Nodes, the Inclination of its Plane to that of the Ecliptic, and the Paffage of the Comet at the Perihelion, which are the Elements neceffary for determining the Pofition and Dimensions of the Parabolar

vations.

Rules for

This Problem, already of very great Difficulty in a parabolic Orbit, was fo extremely complicated in the Ellipfe and Hyperbola, that it was neceffary to reduce it to this Degree of Simplicity. Befides the Hypopothefis of a parabolic Orbit, answered in Practice, the fame End as that of the Ellipfe, because the Comets during the Time they are visible, defcribing but a very fmall Portion of their Orbit, move in very excentric Ellipfes, and it is demonftrated that the Portions of fuch Curves. which are near their Foci, may be confidered without any sensible Error as parabolic Arcs.

VI.

The Refult of his Solution of this important Problem is as follows. determining From the obferved Distances of the Comet from the fixed Stars, whose the Elements right Afcenfions and Declinations are known, deduce the right Afcenfiof a Comet. on and Declination, and from thence the Longitude of the Comet reduced to the Ecliptic, and its Latitude, correfponding to each ObferPreliminary vation: Compute the Longitude of the Sun at the Time of each ObferComputati vation, take the Difference (A, A', A") between the Longitude of the Comet and that of the Sun, correfponding to each Obfervation, which is the Elongation of the Comet reduced to the Ecliptic. Compute alfo the Distance (B, B', B") of the Earth from the Sun at the Time of cach Obfervation.

ons.

FIRST HY

Angle at the
Comet.

Thofe preleminary Calculations being performed, affuming by ConPOTHESIS. jecture, the Distances (Y and Z) of the Comet from the Sun, reduced to the Ecliptic at the Time of the first and fecond Obfervation, determine the true Distances by the Means of the two following Proportions, as the affumed Diftance (Y or Z) of the Comet from the Sun in the firft or Second Obfervation, is to the Sine of the obferved Elongation, (A or A') fo is the Distance (B or B') of the Earth from the Sun at the Time of the first or fecond Obfervation, to the Sine of the Angle (C or C') contained by the firaight Lines drawn from the Earth and the Sun to the Comet. Add this Angle (Cor C) to the Elongation (A or A') their Sum will be the Supplement of the Angle of Commutation (D or D). And then fay as the Sine of the Angle of Elongation (A or A') is to the Sine of the Angle of Commutation (D or D'), Jo is the Tangent of the obferved geocentric Latitude of the Comet correlHeliocentric ponding to the firft or fecond Obfervation, to the Tangent of the corref fonding heliocentric Latitude of the Comet (E or E').

Latitude.

Each of the curt Distances Y and Z divided by the Cofine of the Vector correfponding heliocentric Latitude E and E' gives the true Distances (V, Rays. V) of the Comet from the Sun.

Find the Angle contained by those Distances thus: Add to (a) or fub · fract from the Places of the Earth, the correfponding Angles of Commutation (D, D') which will give the two heliocentric Longitudes (L,L') of the Comet, whofe Difference (F) is the heliocentric Motion of the Comet in the Plane of the Ecliptic. Then fay, As Radius, is to the Cofine of the Motion (F) of the Comet in the Ecliptic, fo is the Cotangent of the greatest of the two beliocentric Latitudes, to the Tangent of an Arc X. Subftract this Arc X from the Complement of the leaft heliocentric Latitude, and Motion of call the Remainder X'. Then the Cofine of the firft Arc X, will be to the in its Orbit. Cofine of the fecond Arc X', as the Sine of the greatest of the two Latitudes, to the Cofine of the Angle contained by the two vector Rays of the Comet.

the Comet

Which being done, determine the Place of the Perihelion by the fo!lowing Rule: fubftract the Logarithm of the least vector Ray from that of the greatest, take half the Remainder, to whofe Characteristic, 10 being added, it will be the Tangent of an Angle, from which fubducting 45°, the Logarithm of the Tangent of the Remainder, added to the Log. of the Cotangent of of the Motion of the Comet in its Orbit, will be the Logarithm of the Tangent of an Angle, to which of the Motion of the 44 Comet in its Orbit being added, the Sum will be the Half of the greatest true Anomaly, and their Difference will be Half the least of the two true True AnoAnomalies. Double thofe Quantities to obtain the two true Anomalies, which will be both on the fame Side of the Perihelion, when their Difference is the whole Motion of the Comet, but on different Sides of it, when it is their Sum, which is equal to the whole Motion of the Comet.

malies.

Find the Perihelion Distance by adding twice the Logarithm of the Perihelion Cofine of the greatest of the Halfs of the two true Anomalies, to that Distance. of the greatest of the two vector Rays, which will be the Logarithm of the Perihelion Diftance required.

Interval of

`Determine the Time which the Comet fhould employ in defcribing the Angle contained by the two vector Rays, by the following Rule: To the conftant Logarithm 1,9149328, add the Logarithm of the Tangent Time emof half of each true Anomaly. Add the Triple of this fame Logarithm of ployed in the Tangent to the conflant Logarithm 1,4378116, the Sum of the two describing the Angle Numbers correfponding to thofe two Sums of Logarithms, will be the exact contained Number of Days correfponding to each true Anomaly in a Parabola whofe by the two peribelion Diflance is 1. Take the Logarithm of the Difference or Sum vector Rays. of thofe two Numbers, according as the two Anomalies are fituated on the fame Side, or on different Sides of the Peribelion. To this Logarithm add the of the Log. of the perihelion Diflance, the Sum will be Log. of the (a) According to the Pofition of the Comet with respect to the Signs of the Zodiac.