The Mathematical Principles of Natural Philosophy (1729)/The Laws of the Moon's Motion according to Gravity

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The Laws of the Moon's Motion according to Gravity (appended to The Mathematical Principles of Natural Philosophy (Isaac Newton, translated by Andrew Motte, 1729)
by John Machin

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593892The Laws of the Moon's Motion according to Gravity (appended to The Mathematical Principles of Natural Philosophy (Isaac Newton, translated by Andrew Motte, 1729)John Machin

Banner from Andrew Motte's 1729 translation of Newton's The Mathematical Principles of Natural Philosophy

THE

LAWS

OF THE

MOON's

MOTION

According to

GRAVITY.

Banner from Andrew Motte's 1729 translation of Newton's The Mathematical Principles of Natural Philosophy

The Laws of the

MOON's

MOTION.

n justice to the editor of this translation of Sir Isaac Newton's Principia, it is proper to acquaint the reader, that it was with my consent, he published an advertisement, at the end of a volume of miscellanies, concerning a small tract which I intended to add to his book by way of appendix; my design in which was to deliver some general elementary propositions, serving, as thought, to explain and demonstrate the truth of the rules in Sir Isaac Newton's Theory of the Moon.

The occasion of the undertaking was merely accidental; for he shewing me a paper which I communicated to the author, in the year 1717, relating to the motion of the nodes of the Moon's orbit; I recollected, that the method made use of in settling the Equation for that motion, was equally applicable to any other motion of revolution. And therefore I thought that it would not be unacceptable to a reader of the Principia, to see the uses of the said method explained in the other Equations of the Moon's motion: Especially since the greatest part of the Theory of the Moon is laid down without any proof; and since those propositions relating to the Moon's motion, which are demonstrated in the Principia, do generally depend upon calculations very intricate and abstruse, the truth of which is not easily examined, even by those that are most skilful; and which however might be easily deduced from other principles.

But in my progress in this design, happening to find several general propositions relating to the Moon's motions, which serve to determine many things, which have hitherto been taken from the observations of Astronomers: And having reason to think, that the Theory of the Moon might by these means, be made more perfect and compleat than it is at present; I retarded the publication of the book, 'till I could procure due satisfaction by examining observations on places of the Moon. But finding this to be a work requiring a considerable time, not only in procuring such places as are proper, but also in performing calculations, upon a new method, not yet accommodated to practise by convenient rules, or assisted by tables; I thought it therefore more convenient for the Bookseller, not to stop the publication of his impression any longer upon this account. But that I may in some measure, satisfy those who are well conversant in Sir Isaac Newton's Principia, (and I could wish that none but such would look over these papers,) that the said advertisement was not without some foundation; and that I may remove any suspicion that the design is entirely laid aside, I have put together, altho' in no order, as being done upon a sudden resolution, some of the Propositions, among many others, that I have by me, which seem chiefly to be wanting in a Theory of the Moon, as it is a speculation founded on a physical cause; and those are what relate to the stating of the mean motions. For altho' it be of little or no use in Astronomy to know the rules for ascertaining the mean motions of the Node or Apogee, since the fact is all that is wanting, and that is otherwise known by comparing the observations of former ages with those of the present; yet in matter of speculation, this is the chief and most necessary thing required: since there is no other way to know that the cause is rightly assigned, but by shewing that the motions are so much and no more than what they ought to be.

But that it may not be altogether without its use, I have added all the rules for the equation of the Moon's motion, except two; one of which is a monthly equation of the variation depending on the Moon's anomaly; and the other an equation arising from the Earth's being not in the focus of the Moon's orbit, as it has been supposed to be, in all the modern theories since Horrox.

For not having had time to examine over the observations which are necessary, but being oblig'd instead thereof, to take Sir Isaac Newton's theory for my chief guide and direction, I cannot venture to depart from it too far, in establishing equations entirely new; since I am well assured, upon the best authority, that it is never found to err more than seven or eight minutes.

And therefore, hoping that the reader, who considers the sudden occasion and necessity of my publishing these Propositions at this time, will make due allowance for the want of order and method, and look upon them only as so many distinct Rules and Propositions not connected: I shall begin, without any other preface, with shewing the origine of that inequality, which is called the Variation or Reflection of the Moon.

The variation or reflection is that monthly inequality in the Moon's motion, The Variation of the Moon.wherein it more manifestly differs from the laws of the motion of a planet in an elliptic orbit. Tycho Brahe makes this inequality to arise from a kind of libratory motion backwards and forwards, whereby the Moon is accelerated and retarded by turns, moving swifter in the first and third quarter, and slower in the second and fourth, which inequality is principally observed in the octants.

Sir Isaac Newton accounts for the variation from the different force of gravity of the Moon and Earth to the Sun, arising from the different distances of the Moon in its several aspects.

The mean gravity of the Moon to the Sun, he supposes, is satisfied by the annual motion of the Moon round the Sun; the gravity of the Moon to the Earth, he supposes, is satisfied by a revolution of the Moon about the Earth. But the difference of the Moon's gravity to the Sun more or less than the Earth's gravity, he supposes, produces two effects; for as this difference of force may be resolved into two forces, one acting in the way, or contrary to the way, of the Moon about the Earth, and the other acting in the line to or from the Earth: the first causes the Moon to describe a larger or smaller area in the same time about the Earth, according as it tends to accelerate or retard it; the other changes the form of the lunar orbit from what it ought to be merely from the Moon's gravity to the Earth, and both together make up that inequality which is called the variation.

But since the real motion of the Moon, tho' a simple motion, caused by a continual deflection from a streight line, by the joint force of its gravity to the Sun and Earth, thereby describing an orbit, which incloses not the Earth but the Sun, is yet considered as a compound motion, made from two motions, one about the Sun, and the other about the Earth; because two such motions are requisite to answer the two forces of its gravity, if separately considered: For the very same reason, the Moon's motion ought to be resolved into a third motion of revolution, since there remains a third force to be satisfied, and that is the force arising from the alteration of the Moon's gravity to the Sun. And this when considered, will require a motion in a small ellipsis, in the manner here described.

Fig 1.The circle $ADFH$ represents the orbit of the Moon about the Earth in the center $T$ , as it would be at a mean distance, supposing the Moon had no gravity to any other body but the Earth. The diameter $ATF$ divides that part of the orbit which is towards the Sun, suppose $ADF$ , from the part opposite to the Sun, suppose $AHF$ . The diameter at right angles $HTD$ , is the line of the Moon's conjunction with or opposition to the Sun. The figure $PQLK$ is an Ellipsis, whose center is carried round the Earth in the orbit $ABDEFH$ , having its longer axis $PL$ in length double of the shorter axis $QK$ , and lying always parallel to $TD$ , the line joining the centers of the Earth and Sun. Whilst the said figure is carried from $A$ to $B$ , the Moon revolves the contrary way from $Q$ to $N$ , so as to describe equal areas in equal times about the centre of it; and to perform its revolution in the same time as the center of the said Elliptic epicycle (if it may be so called,) performs its revolution; the Moon being always in the remoter extremity of its shorter axis in $Q$ and $K$ when it is in the quarters, and in the nearest extremity of its longer axis at the time of the new and full Moon.

The shorter semiaxis of this Ellipsis $AQ$ , is to the distance of its center from the Earth $AT$ , in the duplicate proportion of the Moon's periodical time about the Earth to the Sun's periodical time: Which proportion, if there be 2139 revolutions of the Moon to the Stars in 160 sydereal years, is that of 47 to 8400.

The figure which is described by this compound motion of the Moon in the Elliptic epicycle, whilst the center of it is carried round the Earth, very nearly represents the form of the Lunar orbit; supposing it without eccentricity, and that the plane was coincident with the plane of the ecliptic, and that the Sun continu'd in the same place during the whole revolution of the Moon about the Earth.

From the above construction it appears, that the proportion between the mean distance of the Moon and its greatest or least distances, is easily assigned; being something larger than that which is assigned by Sir Isaac Newton in the 28th^{[errata 1]} proposition of his third book. But as the computation there given, depends upon the solution of a biquadratio equation, affected with numeral coefficients; which renders it impossible to compare the proportions with each other, so as to see their agreement or disagreement, except in a particular application to numbers; I shall therefore set down a rule, in general terms, derived from his method, which will be exact enough, unless the periods of the Sun and Moon should be much nearer equal than they are. Let $L$ be the periodical time of the Moon, $S$ the period of the Sun, $M$ the synodical period of the Moon to the Sun, and $D$ be the difference of the periods of the Sun and Moon; then, according to Sir Isaac Newton's method, the difference of the two axes of the Moon's elliptic orbit, as it is contracted by the action of the Sun, is to the sum of the said axes as $3L\times {\frac {M+L}{2}}$ to $4DD-SS$ . But according to the construction before laid down, the said proportion is as $3LL$ to $2SS-LL$ .

By Sir Isaac Newton's rule, the difference will be to the sum, nearly as 5 to 694; and consequently the diameters will be nearly as 689 to 699, or 69 to 70: But by the latter rule, the difference will be to the sum, nearly as 1 to 119; and the diameters or distances of the Moon, in its conjunction and quadrature with the Sun, will be as 59 to 60. Dr. Halley, (who in his remarks upon the Lunar theory, at the end of his catalogue of the Southern stars, first took notice of this contraction of the Lunar orbit in the Syzygies from the phenomena of the Moon's motion) makes the difference of the diameters to the sum, as 1 to 90; and consequently the greater axis to the lesser, as 45⁠1/2⁠ to 44⁠1/2⁠.

But the difference, in these proportions of the extream distances, tho' it may appear considerable, is not, however, to be distinguish'd by the observations on the diameters of the Moon, whilst the variations of the diameters, from this cause, are intermixt with the other much greater variations, arising from the eccentricity of the orbit.

Fig. 1The angle of the Moon's elongation from the center, designed by $BTN$ , is properly the variation or reflection of the Moon. The properties of which are evident from the description.

First, It is as the sine of the double distance of the Moon from the quadrature or conjunction with the Sun: For it is the difference of the two angles $BTA$ and $NTA$ , whose tangents, by the construction, are in a given proportion.

Secondly, The variation is, cæteris paribus, in the duplicate proportion of the synodical time of the Moon's revolution to the Sun. For the variation is in proportion to the mean diameter of the epicycle, and that is in the duplicate proportion of the synodical time of revolution.

The greatest variation is an angle, whose sine is to the radius, as the difference of the greatest and least distances $TQ$ and $TL$ , that is $3AQ$ , to their sum. According to the proportion of the lines before described, this rule makes the elongation near 29 minutes; which would be the variation, supposing the Moon perform'd its revolution to the Sun in the time of its revolution round the Earth. But if that elongation of 29 Minutes be increased in the duplicate proportion of the synodical time to the periodical time of revolution, it will produce near 34 minutes for the variation.

It is to be noted, that what is said of the epicycle, is upon supposition, that the Earths orbit round the Sun is a circle; if the eccentricity of the annual orbit be considered, the mean diameter of the epicycle must increase or diminish reciprocally in the triplicate proportion of the Sun's distance.

The method of finding the inequalities in any Revolution.The construction which I communicated to Sir Isaac Newton, for the annual motion of the nodes of the Moon's orbit, (which is printed in the scholium to the 33d proposition of his 3d book) is a case of a general method, for shewing the inequality of any motion round a center, when the hourly motion or velocity of the object varies, according to any rule, depending on its aspect to some other object. For in any revolution, the mean motion and inequality are to be assigned by means of a curvilinear figure, wherein equal areas are described about the center in equal times; the property of which figure is, that the rays from the center, are always reciprocally in the subduplicate proportion of the hourly motion or velocity about the center.

Fig. 2.Thus in the figure described in my construction, where $TN$ is the line of the nodes, $TA$ the line drawn to the Sun, is supposed to revolve round the center $T$ , with the velocity of the Sun's motion from the node; and the ray $TB$ , which is taken always in the subduplicate proportion of that velocity, will describe equal areas in equal times; so that the sector $NTB$ will be the mean motion of the Sun; the sector $NTA$ the motion of the Sun from the node; and consequently the area $NAB$ the motion of the node; which will be a retrograde motion if the area be within the circle, and direct if it falls without. From whence it follows,

1. That the periodical time of the Sun's revolution to the node, will be to the periodical time of the Sun's revolution, as the area of the curvilinear figure, to the area of the circle.

2. That if a circle be described, whose area is equal to the area of the curvilinear figure, it will cut that figure in the place where the Sun has the mean motion from the node.

3. If an angle $NTF$ be made, which shall comprehend an area in the said circle, equal to the sector $NTB$ in the figure, that angle will be the mean motion of the Sun from the node. And consequently,

4. The angle $FTB$ , which is the difference between the Sun's true motion from the node, designed by $ATN$ , and the Sun's mean motion from the node, designed by $FTN$ , will be the equation for the Sun's motion from the node, when the Sun's position to the node is designed by the angle $ATN$ .

From all which it appears, that what is said of the Sun's motion from the node, will hold as to any other motion round a center; as of the Sun from the Moon, or the Moon from the node or apogee. In any such revolution, a curvilinear figure may be described about the center, by the areas of which, the relation between the mean and true motion may be shewn; and consequently the inequality or equation of the motion.

And as in every revolution there is a certain figure which is proper to shew this relation, such a figure may be call'd an Equant for that motion or revolution.

And in every revolution where the Equant is a figure of the same property, the inequalities or equations will alter according to the same rule.

Thus, if the Equant be an ellipsis about the center, as in that for the motion of the Sun from the node,

First, The mean motion in the whole revolution, will be a geometrical mean proportional, between the greatest motion in the extremity of the lesser axis, and the least motion in the extremity of the longer axis: For the radius of the circle, which is equal to an ellipsis, is a mean proportional between the two semiaxes.

Secondly, The tangents of the angles of the mean and true motion, are in the given proportion of the two axes of the ellipsis. Thus the tangents of the angles of the true and mean motion of the Sun from the node, viz. Fig. 2.the tangents of the angles $ATN$ and $FTN$ , are in proportion as the ordinates $BG$ and $FG$ , that is, as the semiaxes $TH$ and $TN$ .

Thirdly, The sine of the angle of the greatest inequality in the octants is to the radius, as half the sum of the axes to half their difference.

It is to be noted, that the equant is an ellipsis about the center, in every motion, where the excess of the velocity about the center above the least velocity, is always in the duplicate proportion of the sine of the angle of the true motion, from the place where the velocity about the center is least. From which remark, upon examination it will appear, that the following motions are to be reduced to an Elliptic equant described about the center.

The monthly motion of the Moon from the node.

The annual motion of the Sun from the node.

The motion of the Moon from the Sun, as it is accelerated or retarded, by the alteration of the area describ'd about the Earth, according to Sir Isaac Newton's 26th prop. 3d book.

And the annual Motion of the Sun from the apogee. How these several equants are determin'd will appear by what follows.

The motion of the Nodes.The node is in its swiftest retrograde motion, when the Sun and Moon are in conjunction or opposition, and in a quadrature with the line of the nodes. According to Sir Isaac Newton's method, (explain'd at the end of the thirtieth proposition of the third book) the force of the Sun to produce a motion in the node, at this time, is equal to three times the mean Solar force; that is, by the construction of the elliptic epicycle, equal to a force, which is to the force of gravity, as $3AQ$ to $AT$ , or three times the lesser Fig. 1. semiaxis of the ellipsis to the distance of its center from the center of the Earth. But if the Moon revolve in the elliptic epicycle as before described, the force to make a motion in the node at the time mention'd, will be to the force of gravity, as $3DL$ to $DT$ , or three times the longer semiaxis to the distance of the center; which is the double of the former force. But then, according to Sir Isaac's method, the motion of the node at this time, is to the Moon's motion, as the solar force to create a motion in the node is to the force of gravity. But if the Moon be conceived as revolving in a circle, with the velocity of its motion from the node at this time, when the node moves swiftest, and the plane of the said circle be supposed to have a rotation upon an axis perpendicular to the plane of the ecliptic, and the contrary way to the motion of the Moon, so as to produce the motion of the node, and leave the Moon to move with its own motion about the Earth; the force to make a motion in the node seems to be the difference of the forces to retain it with the velocity of its motion in the moveable and immoveable planes: But the velocities of bodies revolving in circles are in the subduplicate proportion of the central forces. From whence it follows, that

The motion of the Moon from the node at this time, when the node moves swiftest, is to the motion of the Moon, in the subduplicate proportion of the sum of the forces to the force of gravity, or as the sum of ${\mbox{TD}}$ and ${\mbox{3DL}}$ to ${\mbox{TD}}$ .

And this would be the greatest motion of the node, upon supposition that the plane of the Moon's orbit was almost co-incident with the plane of the ecliptic; but if the inclination be considered, the motive force for the node must be diminished, in the proportion of the sine-complement of the inclination to the radius. How much this motion is, will appear by the following short calculation.

The distance $TD$ being as before equal to 8400, and $3DL$ being 282, the inclination of the plane in this position is 4°. 59′. 35″; the sine-complement of which is to the radius, as 525 to 527 nearly; therefore the force of gravity is to the motive force for the node thus diminished, in the compound proportion of 8400 to 282, and of 527 to 525, that is, in the proportion of 4216 to 141. So that the greatest motion of the Moon from the node is to the motion of the Moon, in the subduplicate proportion of 4357 to 4216, that is, in the proportion nearly of 613 to 603. According to which calculation, the greatest hourly motion of the node ought to be $32''.47'''$ . By Sir Isaac Newton's method, it amounts to $33''.10'''$ ⁠1/2⁠.

This is the swiftest retrograde motion of the node, when the line of the nodes is in a quadrature with the Sun, and the Moon is in its greatest latitude in conjunction or opposition to the Sun. But the equant for the motion of the Moon from the node in this month, when the line of the nodes is in quadrature with the Sun, is an ellipsis about the center; and therefore the mean motion in this month will be known by the following rule:

The mean motion of the Moon from the node, in that month when the line of the nodes is in a quadrature with the Sun, is a geometrical mean proportional, between the greatest motion of the Moon from the node and the motion of the Moon.

And therefore this mean motion, will be to the motion of the Moon, in the subduplicate proportion of 613 to 603, that is, nearly in the proportion of 1221 to 1211. So that the mean motion of the node in this month, will be to the motion of the Moon, as 10 to 1211, which makes the mean hourly motion $16''.\,19'''.$ ⁠1/40⁠. According to Sir Isaac Newton it amounts to $16''.\,35'''$ ; but, by the corrections which he afterwards uses, it is reduced to $16''.\,16'''$ ⁠2/3⁠.

But the equant for the annual motion of the Sun from the node being also an ellipsis, it follows, that

The mean motion of the Sun from the node, is a geometrical mean proportional, between the motion of the Sun and the mean motion of the Sun from the node, in the month when the line of the nodes is in quadrature with the Sun.

How near this rule agrees with the observations, will appear by this calculation.

Since the mean motion of the node in that month, when the line of nodes is in quadrature to the Sun, was before shewn to be to the Moon's mean motion, as 10 to 1211; and the motion of the Sun is to the motion of the Moon, as 160 to 2139: it follows, that the motion of the node and the motion of the Sun will be in the proportion of 154 and 1395; and therefore, by the rule, the Sun's mean motion from the node, is to the Sun's mean motion, in the subduplicate proportion of 1549 to 1395, that is, nearly as 98 to 93. Which corresponds with the observations; there being 98 revolutions of the Sun to the node in 93 revolutions of the Sun. The subduplicate proportion taken more nearly, is as 941 to 893, which will produce 19°. 21′. 3″, for the motion of the node from the fix'd Stars, in a sydereal year. The motion (as observ'd) is 19°. 21′. 22″.

Had the calculation from the rule, been more exactly made in large numbers, the annual motion produced would be 19°. 21′. 07″⁠1/2⁠, which is 14″ less than the motion, as observed by the Astronomers.

Which difference may very probably arise from the Sun's parallax; and if so, it may perhaps furnish the best and most certain method of adjusting and fixing the true distance of the Sun. For the Sun's force being something more on that half of the orb which is towards the Sun, than what it is on the other half, the elliptic epicycle is accordingly larger in the first case, than in the latter. And by calculation, I find that the mean motion of the node, arising after consideration is had of this difference, is more than the mean motion from the mean magnitude of the epicycle, by near 2″ in the year, for every minute in the parallactic angle of the orbit of the Moon, or for every second of the Sun's parallax. And by the best computation I have yet made, this difference of 14″, in the annual motion of the node, will arise from about 8″ of parallax; which will make the Sun's distance above 25000 semi-diameters of the Earth.

In like manner as the equant for the motion of the node, in that month when the line of the nodes is in quadrature with the Sun, is an ellipsis; so in any other month it is also an ellipsis: the motion of the node being direct and retrograde by turns, in the Moon's passing from the quadrature to the Sun to the place of its node, and from the place of its node to the quadrature.

But these elliptic equants do not only serve to shew the inequality of the motion of the node, The Inclination of the Plane of the Moon's orbit to the Plane of the Ecliptic.but also the inclination of the plane of the Moon's orbit to the plane of the ecliptic. Thus the rays in the elliptic equants, for the motion of the Moon from the node in each month, design the inclinations of the plane of its orbit to the plane of the ecliptic, in the several respective positions of the Moon to the line of the nodes. And the rays of the elliptic equant for the annual motion of the Sun from the node, in my Construction, (in the schol. to prop. 33. book 3. of Sir Isaac Newton's Principia) design the different mean inclinations of the said plane, to the plane of the ecliptic in each month, when the Sun is in each respective afpect to the line of the nodes.

Fig. 2.Thus if $NT$ (the semi-transverse axis of the elliptic equant for the motion of the Sun from the node,) design the mean inclination of the plane, or, which is the same thing, if it represent the mean distance between the pole of the ecliptic and the pole of the Moon's orbit, in that month when the Sun is in the line of the nodes; $TH$ , the semiconjugate axis of the said ellipsis, will design the mean inclination or mean distance of the poles in that month when the line of nodes is in quadrature to the Sun; and $TB$ , any other semidiameter of the said ellipsis, will represent the mean distance between the said poles, when the Sun is in that aspect to the line of the nodes, which is designed by the angle $NTA$ . For example, if the least inclination, designed by the shorter semiaxis $TH$ be 5°. 00′. 00″; since $TH$ is to $TK$ as the motion of the Sun to the mean motion of the Sun from the node, by the property of this equant; and since there are 98 revolutions of the Sun to the node in 93 revolutions of the Sun; it follows, that $HK$ , the difference between the greatest and least of the mean inclinations in the several months of the year, is to $TH$ the least, as 5 to 93; by which proportion, the said difference will amount to 16′. 10″. According to Sir Isaac Newton's computation in the 35th prop. of the third book, it is 16′. 23″⁠1/2⁠. But if the said number be lessen'd in the proportion of 69 to 70, according to the author's note at the end of the 34th prop. the said difference will become 16′. 9″.

And in like manner, the inclinations of the plane of the Moon's orbit, in that month when the motion of the node is swiftest, (being situated in the line of quadratures with the Sun,) are determined by the equant for the motion of the Moon from the node, in that month.

Fig.2.Thus, let $TH$ be to $TN$ in the subduplicate proportion of the Moon's motion, to its greatest motion from the node, when the Moon is in the conjunction in $TH$ ; that is, (as was before determined) let $TH$ be to $TN$ in the proportion of 1211 to 1221; and the ellipsis described on the semiaxes $TH$ and $TN$ , will be the equant for the motion of the Moon from the node in that month. And the rays of the said equant will design the inclinations of the plane in the several aspects of the Moon to the line of the nodes. That is, if $TN$ be the inclination of the plane, or the distance of the pole of the ecliptic from the pole of the Moon's orbit, when the Moon is in $TN$ the line of the nodes, the ray $TB$ will represent the distance of the said poles, or the inclination of the plane, in that aspect which is designed by the angle $NTB$ .

Which being laid down, it follows that the whole variation of the inclination, in the time the Moon moves from the line of the nodes to its quadrature in $THK$ , is to the least inclination, as $KH$ to $TH$ , that is, as 10 to 1211. Wherefore if the least inclination be 4°. 59′. 35″, the whole variation will be 2′. 29″. This is upon supposition that the Sun continued in the same position to the line of the nodes, during the time that the Moon moves from the node to its quadrature. But the Sun's motion protracting the time of the Moon's period to the Sun, in the proportion of 13 to 12; the variation must be increased in the same proportion, and will therefore be 2′. 41″. According to Sir Isaac Newton's computation, as delivered in the corollaries to the 34th prop. of the 3d book, for stating this greatest variation, (the intermediate variations in this or any other month not being computed or shewn by any method) it amounts to 2′. 43″. But if the said quantity be diminish'd in the proportion of 70 to 69, according to his note at the end of the said proposition, it will become the same precisely as it is here deriv'd from the equant.

The Variation of the Area described by the Moon about the Earth.The motion of the Moon from the Sun, as it is accelerated or retarded by the increment of the area described about the Earth, (according to the 26th prop. of the 3d book) is also to be reduced to an elliptic equant; by taking the shorter axis to the longer axis, in the subquadruplicate proportion of the force of the Moon's gravity to the Earth, to the said force added to three times the mean Solar force, Fig. 1.that is, as $TA$ to the first of three mean proportionals between $TA$ and $TA+3AQ$ . And in the same proportion is the area described by the Moon about the Earth, when in quadrature with the Sun, to the mean area, or as the mean area to the area described in the syzygies: So that the greatest area in the syzygies is to the least in the quadratures, in the subduplicate proportion of $TA+3AQ$ to $TA$ , or as ${\sqrt {8541}}$ to ${\sqrt {8400}}$ . This is upon supposition, that the Moon revolves to the Sun in the same time as it revolves about the Earth; which will be found to agree very nearly with Sir Isaac Newton's computation, in the before-cited proposition.

The Motion of the Apogee.And after the same manner an elliptic equant might be constructed, which would very nearly shew the mean motion of the apogee, according to the rules deliver'd by Sir Isaac Newton (in the corollaries of the 45th prop. of the first book) for stating the motion of the apogee, namely, by taking the greatest retrograde motion of the apogee, from the force of the Sun upon the Moon in the quarters; and the greatest direct motion, from the force of the Sun upon the Moon when in the conjunction or opposition; each according to his rule, deliver'd in the second corollary to the said proposition. And if an ellipsis be made whose axes are in the subduplicate proportion of the Moon's motion from the apogee, when in the said swiftest direct and retrograde motions, the said ellipsis will be nearly the equant for the motion of the Moon from the apogee, and will be found to be nearly of the form of that above for the increment of the area.

But the motion of the apogee, according to this method, will be found to be no more than 1°. 37′. 22″, in the revolution of the Moon from apogee to apogee, which (according to the observations) ought to be 3°. 4′. 7″⁠1/2⁠.

So that it seems there is more force necessary to account for the motion of the Moon's apogee, than what arises from the variation of the Moon's gravity to the Sun, in its revolution about the Earth.

But if the cause of this motion be supposed to arise from the variation of the Moon's gravity to the Earth, as it revolves round in the elliptic epicycle, this difference of force, which is near double the former, will be found to be sufficient to account for the motion; but not with that exactness as ought to be expected. Neither is there any method that I have ever yet met with upon the commonly received principles, which is perfectly sufficient to explain the motion of the Moon's apogee.

The rules which follow concerning the motion of the apogee, and the alteration of the eccentricity, are founded upon other principles, which I may have occasion hereafter to explain, it being, as I apprehend, impossible to derive these, and many other such propositions from the laws of centripetal forces.

Fig. 1.Let $TC$ (in the above construction of the Lunar orbit) be the mean distance of the Moon, or half the sum of its greatest and least distances, viz. $TQ$ and $TL$ ; and let $CL$ be the mean semidiameter of the elliptic epicycle, or half the sum of the semiaxes; and take a distance $LM$ , on the other side towards the centre, equal to $CL$ ; then,

The mean motion of the Moon from its apogee, is to the mean motion of the Moon, in the subduplicate proportion of $TM$ to $TC$ .

For example, Half the shorter axis or $DC$ is 23⁠1/2⁠; therefore $TC$ the mean distance is 8376⁠1/2⁠; $CM$ or $2CL$ , the sum of the semiaxes, is 141; so that $TM$ is 8235⁠1/2⁠. Wherefore the motion of the Moon from the apogee is to the motion of the Moon, in the subduplicate proportion of 8235⁠1/2⁠ to 8376⁠1/2⁠, or of 16471 to 16753, that is, nearly as 117 to 118, or more nearly, as 352 to 355; or yet more nearly, as 1877 to 1893; so that there ought to be about 16 revolutions of the apogee in 1893 revolutions of the Moon; which agrees to great preciseness with the most modern numbers of Astronomy; according to which proportion, the mean motion of the apogee, in a sydereal year, ought to be 40°. 40′. 40⁠1/2⁠″. But by the numbers in Sir Isaac Newton's theory of the Moon, the said motion is 40° 40′. 43″. According to the numbers of Tycho Brahe, it ought to be 40° 40′. 47″.

The mean motion of the apogee being stated, The Variation of the Eccentricity.I find the following rule for the alteration of the eccentricity.

The least eccentricity is to the mean eccentricity, in the duplicate proportion of the Sun's mean motion from the apogee of the Moon's orbit, to the Sun's mean motion. Or in the duplicate proportion of the periodical time of the Sun's revolution, to the mean periodical time of its revolution to the Moon's apogee.

By the foregoing rule for the mean motion of the apogee, there are 16 revolutions of the apogee in 1893 revolutions of the Moon; but there being 254 revolutions of the Moon in 19 revolutions of the Sun; there must be about 7 revolutions of the apogee in about 62 revolutions of the Sun, or rather about 20 in 177. So that the periods of the Sun to the Stars, and of the Sun to the Moon's apogee, are in proportion nearly as the numbers 157 and 177. The duplicate of which proportion is that of 107 to 136; which, according to the rule, ought to be the proportion of the least eccentricity to the mean eccentricity.

So that by this rule, the mean eccentricity, (or half the sum of the greatest and least,) ought to be to the difference of the mean from the least, (or half the difference of the greatest and the least,) as 136 to 29.

How near this agrees with the Observations, will appear from the numbers of Mr. Horrox or Mr. Flamsted, and of Sir Isaac Newton.

The mean eccentricity according to Mr. Flamsted or Mr. Horrox is 0.055236, half the difference between the greatest and least is 0.011617; which numbers are in the proportion of 135⁠1/2⁠ to 28, nearly.

According to Sir Isaac Newton the mean eccentricity is 0.05505, half the difference of the greatest and least is 0.01173; which numbers are in proportion nearly as 135⁠47/48⁠ to 28⁠47/48⁠, each of which proportions is very near that above assigned.

But it is to be noted, that the rule, which is here laid down, is true only upon supposition that the eccentricity is exceeding small. There is another rule derived from a different method, which presupposes the knowledge of the quantity of the mean eccentricity; and which will not only determine the variation of the eccentricity according to the laws of gravity, with greater exactness, but serve also to correct an hypothesis in the modern theories of the Moon, in which their greatest error seems to consist; and that is, in placing the earth in the focus of that ellipsis, which is described on the extreme diameters of the lunar orbit; whereas it ought to be in a certain point nearer the perigee, as I may have occasion to explain more fully hereafter.

The greatest and least eccentricity being determined; The Equation of the Apogee.the equant for the motion of the Sun from the apogee is an ellipsis, whose greater and lesser axes are the greatest and least eccentricities: and therefore, by the property of such an equant as before laid down,

The sine of the greatest equation of the apogee will be to the radius, as the difference of the axes of the equant is to their sum; that is, as the difference of the greatest and least eccentricities to their sum.

For example, since the difference is to the sum as 29 to 136, by what was determined in the foregoing article, the greatest equation of the apogee will be about 12°. 18′. 40″. Sir Isaac Newton has determined it from the observations to be 12º. 18′.

The greatest and least eccentricities being determined; the eccentricity and equation of the apogee, in any given aspect of the Sun, are determined by the equant, in the following manner.

Let $TN$ be the greatest eccentricity, Fig. 2. $TH$ the least, the ellipsis on the semi-axes $TN$ and $TH$ , the equant for the motion of the apogee.

Then if the angle $NTF$ , be made equal to the mean distance or mean motion of the Sun from the apogee, the angle $NTB$ will be the true distance or motion of the Sun from the apogee; the difference $BTF$ , the equation of the apogee; and the ray $TB$ , the eccentricity of the orbit, in that aspect of the Sun to the apogee designed by the angle $NTB$ . Hence arises this rule.

The tangent of the mean distance, viz. NTF, is to the tangent of the true distance NTB, in the given proportion of the greatest eccentricity TN to the least TH, that is, as 165 to 107.

From what has been laid down concerning the general property of an equant, that it is a curve line described about the center, whose rays are reciprocally in the subduplicate proportion of the velocity at the center, or the velocity of revolution, it will not be difficult to describe the proper curve for any motion that is proposed; and where the inequality of the motion throughout the revolution is but small, there is no need of any nice or scrupulous exactness in the quadrature of the curve for shewing what the equation is. Thus all the small annual equations of the Moon's motion arising from the different distances of the Sun, at different times of the year, may be reduced to one rule exact enough for the purpose.

For since the Sun's force to create these annual alterations, is reciprocally in the triplicate proportion of the distance; the rays of the equant for such a motion, will be in the sesquiplicate proportion of the distance. From whence it will not be difficult to prove, that if the revolution of the motion to be equated, were performed in the time of the Sun's revolution, the equation would be to the equation of the Sun's center, nearly as 3 to 2: and so if the force decreased as any other power of the Sun's distance, suppose that whose index is $m$ , the equation would be to that of the Sun's center as $m$ to 2. But if the motion be performed in any other period, the equation will be more or less, in the proportion of the period of the revolution to the Sun, to the period of the revolution of the motion to be equated. Thus if it were the node or apogee of the Moon's orbit, the equation is to the former as the period of the Sun to the node or apogee, to the period of the node or apogee. Which rule makes the greatest equation for the node about 8′. 56″, being a small matter less than that in Sir Isaac Newton's theory; and the greatest equation for the apogee about 21′. 57″, being something larger than that in the same theory.

The like rule will serve for the annual equation of the Moon's mean motion. If instead of the equation for the Sun's center, another small equation be taken in proportion to it as the force, by Sir Isaac Newton called the mean solar force, to the force of the Moon's gravity, or as 47 to 8400; the said equation increased in the proportion of the Sun's period to the mean synodical period of the Moon to the Sun, or of 99 to 8, will be the annual equation of the Moon's mean motion. According to this, the equation, when greatest, will be 12′. 5″.

What is said may be sufficient for the present purpose, which is only to lay down the principal laws and rules of the several motions of the Moon, according to gravity. Some other propositions, which seem no less necessary than the former, for compleating the theory of the Moon's motion, as to its astronomical use, I reserve to another time.

But to make some amends for the shortness and confusedness of the preceeding propositions, I shall add one example to shew the use of the equant more at large, in what is commonly called the solution of the Keplerian problem; that being one of the things which I proposed to explain, when the elements for the theory of the Moon were advertised.

An example of the use of the equant in finding the equation of the center.

Fig. 3.Let the figure $ADP$ be the orbit in which a body revolves, describing equal areas in equal times by lines drawn from a given point $S$ ; and let it be propos'd to find the equant for the apparent motion of the said body, about any other place within the orbit, suppose $F$ .

Let there be a line $FR$ indefinitely produc'd, which revolves with the body as it moves through the arch $AR$ ; and in the said line take a distance $Fp$ , which shall be to $FR$ , the distance of the body from the given point $F$ , in the subduplicate proportion of the perpendicular let fall upon the tangent of the orbit at $R$ from the point $S$ , to the perpendicular on the said tangent let fall from the given point $F$ ; and the curvilinear figure, describ'd by the point $p$ , so taken every where, will be the equant for the motion of the body about the point $F$ .

For since the areas described at the distances $Fp$ and $FR$ are in the duplicate proportion of those lines, that is, by the construction, in the proportion of the perpendiculars on the tangents let fall from $S$ and $F$ ; the areas which the body describes, in moving through the arch $AR$ about the points $S$ and $F$ , are in the proportion of the same perpendiculars. And therefore the area described by the revolution of the line $Fp$ in the figure, will be equal to that which is described by the revolution of the line $SR$ in the orbit. So that the areas described in the figure will be equal in equal times, as they are in the orbit. And consequently the rays $Fp$ of the figure will constantly be in the subduplicate proportion of the velocity of the motion, as it appears at the center $F$ , which is the property of the equant.

From which construction, it will be easy to shew, that in the cafe where a body describes equal areas in equal times about a fixed point, there may be a place found out within the orbit, about which the body will appear to revolve with a motion more uniform than about any other place.

Thus suppose the orbit $ADP$ was a figure, wherein the remotest and nearest apsis $A$ and $P$ were diametrically opposite, in a line passing through the point $S$ , viz. the point about which the equal areas are described; then if the point $F$ be taken at the same distance from the remotest apsis $A$ , as the point $S$ is from the nearest apsis $P$ , the said center $F$ will be the place, about which the body will appear to have the most uniform motion. For in this case the point $F$ will be in the middle of the figure $LpDl$ , which is the equant for the motion about that point. So that the body will appear to move about the center $F$ , as swift when it is in its slowest motion in the remoter apsis $A$ , as it does when it is in its swiftest motion in the nearest apsis $P$ .

For by the construction, when the body is at $A$ , the ray of the equant $FL$ is a mean proportional between $AF$ and $AS$ ; and when the body is at $P$ , the ray of the equant $Fl$ is a mean proportional between the two distances $PS$ and $PF$ , which are respectively equal to the former.

And in like manner in an orbit of any other given form, a place may be found about which the motion is most regular.

If what has been said be applied to the case of a body revolving in an elliptic orbit, and describing equal areas in equal times about one of the foci, as is the case of a planet about the Sun, and a secondary planet about the primary one; it will serve to shew the foundation of the several hypotheses and rules which have been invented by the modern Astronomers, for the equating of such motions; and likewise shew how far each of them are deficient or imperfect.

For if the ellipsis $ADP$ be the orbit of a planet describing equal areas about the Sun in the focus $S$ , the other focus, suppose $F$ , will be the place about which the motion is most regular, from what has been already said; that focus being at the same distance from the aphelion $A$ , as the Sun at $S$ is from the perihelion $P$ . And by the construction, each ray ( $Fp$ ) of the equant will always be a mean proportional between $FR$ and $RS$ , the two distances of the planet from the two foci, in that place where the ray $Fp$ is taken. For the rays $SR$ and $RF$ , making equal angles with the tangent at $R$ , by the property of the ellipsis, are in the proportion of the perpendiculars from $S$ and $F$ , let fall on those tangents. And therefore $Fp$ being to $FR$ in the subduplicate proportion of $SR$ to $FR$ , it will be a mean proportional between those distances.

1. Hence when the planet is in the aphelion $A$ , or perihelion $P$ , the rays of the equant $FL$ and $Fl$ are the shortest, each being equal to $CD$ , the lesser semi-axis of the orbit: For by the property of the ellipsis, the rectangle of the extream distances from the focus is equal to the square of the lesser semi-axis.

2. When the planet is at its mean distance from the Sun in $D$ or $d$ , the extremities of the lesser axis, the equant cuts the orbit in the same place; the rays of the equant being then the longest, being each equal to the greater semi-axis $CA$ . For in those points of the orbit, the distances from the foci and the mean proportional are the same.

From which form of the equant it appears,

1. That the velocity of the revolution about the focus $F$ diminishes, in the motion of the planet from the aphelion or perihelion to the mean distance; and increases in passing from the mean distance to the perihelion or aphelion. For the rays of the equant increase in the first case, and diminish in the latter; and the velocity of revolution increases in the duplicate proportion, as the rays diminish.

2. In any place of the orbit, suppose $R$ , the velocity of the revolution about the focus $F$ , is in proportion to the mean velocity, as the rectangle of the semi-axes of the orbit $CD$ and $CA$ , to the rectangle of the focal distances $RF$ and $RS$ . For the equant and the orbit, being figures of the same area, are each equal to a circle, whose radius is a mean proportional between the two semi-axes $CD$ and $CA$ . But the mean motion about the focus $F$ , is in those places, where the said circle cuts the equant; and in other places, the velocity of the revolution is reciprocally as the square of the distance, that is, reciprocally as the rectangle of the focal distances $RF$ and $RS$ .

3. So that the planet is in its mean velocity of revolution about the focus $F$ , in four places of the orbit, that is, where the rectangle of the focal distances is equal to the rectangle of the semi-axes; which places in orbits nearly circular, such as those of the planets, are about 45 degrees from the aphelion or perihelion; but may be assigned in general, if need be, by taking a point in the orbit, suppose $R$ , whose nearest distance from the lesser axis of the orbit $CD$ is to the longer semi-axis $CA$ , in the subduplicate proportion of the longer axis to the sum of the two axes; as may be easily proved.

What has been said, may be enough to shew the form of the equant, and the manner of the motion about the upper focus in general. But the precise determination of the inequality of the motion, requires the knowledge of the quadrature of the several sectors of the equant, or at least, if any other method be taken, of that which is equivalent to such a quadrature.

There are divers methods for shewing the relation between the mean and true motion of a planet round the Sun, or round the other focus, some more exact than others. But the following seems the most proper for exhibiting in one view, all the several hypotheses, and rules, which are in common use in the modern Astronomy, whereby it may easily appear, how far they agree or differ from each other, and how much each of them errs from the precise determination of the motion, according to the true law of an equal description of areas about the Sun.

Upon the center $F$ describe the ellipsis $LNl$ , equal and similar to the elliptic orbit $ADP$ ; but having its axes $FN$ and $FL$ contrarily posited, that is, the shorter axis $LF$ lying in the longer axis of the orbit $AP$ , and the longer axis $FN$ parallel to the shorter $CD$ . Let the focus of the said ellipsis be in $f$ . And suppose two other ellipsis $LBl$ and $Lfl$ , to be drawn upon the common axis $Ll$ , one passing through the point $B$ , where the perpendicular $FN$ intersects the orbit, and the other through the focus $f$ . Let the line $FR$ , revolving with the planet in the orbit, be indefinitely produced, till it intersect the first ellipsis $LNl$ (which was similar to the orbit) in $Q$ , the equant in $p$ , and the ellipsis $LBl$ (drawn through the intersection $B$ ,) in $K$ . From the point $K$ let fall $KH$ perpendicular to the line of apsides $AP$ , and let it be produced till it intersect the first ellipsis $LNl$ in $O$ , and the ellipsis $Lfl$ (passing through the focus $f$ ) in $E$ . And lastly, in the ellipsis $LNl$ , let $GM$ be an ordinate equal and parallel to $EH$ . In which construction it is to be noted, that the ellipsis $Lfl$ and $LBl$ are supposed as drawn only to divide the line $OKH$ in given proportions, that $KH$ may be to $OH$ , as the latus rectum of the orbit to the transverse axis; and that $EH$ or $GM$ , the base of the elliptic segment $GLM$ , may be to $OH$ , as the distance of the foci to the transverse axis.

Which being premised, it will be easy to prove, that the sector $pFL$ in the equant, or, which is the same thing, the sector $RSA$ in the orbit, is equal to the curvilinear area $OKFMG$ , that is, equal to the elliptic sector $QFL$ , deducting the segment $LMG$ , and adding or subducting the trilinear space $QKO$ , according as the angle $RFA$ is less or greater than a right angle. Wherein it is to be noted, that these signs of addition and subduction are to be used in general, if the angle $AFR$ is taken from the aphelion in the first semi-circle, but towards the aphelion in the latter semi-circle. But if the angle $AFR$ be taken the same way throughout the whole revolution, as is the method in Astronomical calculations, then the segment and the trilinear space in the latter semi-circle must be taken with the contrary signs to what are laid down.

Hence it appears, that the inequality in the motion of a planet about the upper focus $F$ , consists of three parts.

I. The first and principal of which is the inequality in the alteration of the angle $QFL$ , in making equal areas in the ellipsis $LNl$ . Fig. 3.For if a circle equal to the ellipsis be described upon the center $F$ , since the radius (being a mean proportional between the two semi-axes) will fall without the ellipsis about the line of apsides, and within it about the middle distances, the angle $QFL$ , which is proportional to the area described in the circle, will therefore increase faster about the line of apsides, and slower about the middle distances, in describing equal areas in the ellipsis, than it ought to do in the hypothesis of Bishop Ward, who makes the planet revolve uniformly about the focus. The equation to rectify this inequality is determined by the following rule.

The tangent of the angle $QFL$ , is to the tangent of the angle in the circle including the same area, as the longer axis of the ellipsis to the shorter axis; and the difference of the angles, whose tangents are in this proportion, is the equation; as is manifest from what was before said on the properties of an elliptic equant. From the same it also follows, that

1. The greatest equation is an angle, whose sine is to the radius as the difference of the axes to their sum, or, which is the same thing, as the square of the distance of the foci, to the square of half the sum of the axes. So that in ellipsis nearly circular, of different eccentricities, this greatest equation will vary nearly in the duplicate proportion of the eccentricity.

2. In ellipsis nearly circular, the equation at any given angle $QFL$ , is to the greatest equation, nearly as the sine of the double of the given angle to the radius; which follows from hence, that the equation is the difference of two angles, whose tangents are in a given proportion, and nearly equal.

3. This equation adds to the mean motion in the first and third quadrant of mean anomaly, and subducts in the second and fourth; as will easily appear from that the line $QF$ , in describing equal areas in the ellipsis, makes the angle to the line of the apsides, less acute than it would be in an uniform revolution.

This is the equation which is accounted for in the hypothesis of Bullialdus. For he supposes the motion of the planet in its orbit to be so regulated about the upper focus, that the tangents of the angles, from the lines of apsides, shall always be to the tangents of the angles answering to the mean anomaly, in the proportion of the ordinates in the ellipsis to the ordinates in the circle circumscribed; which in effect is the same, as if he had made the true equant for its motion about the focus $F$ ; to be the ellipsis as above described.

The same equation is also used by Sir Isaac Newton, in his solution of the Keplerian problem, in the scholium to the 31st prop. of the 1st book, and is there designed by the letter $V$ .

But since the true equant $LDl$ coincides with the elliptic equant in the extremities of the shorter axis at $L$ and $l$ , and falls within the same at its intersection with the longer axis $FN$ , it follows, that the motion of the planet in the semi-circle about the aphelion, is swifter than according to the hypothesis of an equal description of areas in the ellipsis $LNl$ , and for the same reason slower in the other semi-circle about the perihelion; the velocity about the center $F$ being always reciprocally in the duplicate proportion of the distance.

Which leads to the second part of the inequality of the motion about the focus.

II. The equation to rectify this inequality, is an angle answering to the segment $GLM$ ; which angle is to be added to the mean anomaly, to make the area of the elliptic sector $QFL$ .

This angle or equation is determined by the following rule. Let $R$ be an angle subtended by an arch equal in length to the radius of the circle, viz. 57,29578 degrees; and let $A$ be an angle, whose sine is to the radius as $GM$ , the base of the segment, to $FN$ the semi-transverse axis; also let $B$ be an arch in proportion to $R$ , as the sine of the double of the angle $A$ to the radius: Then the equation for the segment will be equal to $A-{\tfrac {1}{2}}B$ .

This equation is at its maximum, when the angle $LFQ$ is a right angle; the base of the segment becoming equal to $Ff$ , half the distance of the foci, and the angle $A$ , being in this case half the angle $FDS$ formed at the extremity of the lesser axis, and subtended by $FS$ , the distance of the foci; which is commonly called the greatest equation of the center. And consequently the arch $B$ , in this case, is to $R$ , as the sine of the said greatest equation of the center, is to the radius. So that according to this rule, for the measure of the segment, it will follow, That

1. This greatest equation is in proportion to the greatest equation of Bullialdus, as found in the preceding article for the elliptic equant, nearly as three times the transverse axis, to eight times the distance of the foci. Or, otherwise, the greatest equation is to the angle designed by $R$ , as twice the cube of the distance between the foci, to three times the cube of the transverse axis. Either of which rules may be derived from the true angle, as before determined; or by taking ${\tfrac {2}{3}}$ of the rectangle of $GM$ and $LM$ , the base and height of the segment, for the measure of that segment.

So that in elliptic orbits nearly circular, this greatest equation for the segment is in the triplicate proportion of the eccentricity.

2. This equation at any given angle $QFL$ , is to the greatest equation, in the triplicate proportion of the ordinate $OH$ to the semi-transverse; that is, nearly as the cube of the sine of the mean anomaly joined to the double of Bullialdus's equation to the cube of the radius. For the segment $GML$ , which is proportional to the equation, is in the triplicate proportion of its base nearly; and the base is proportional to the ordinate $OH$ , by the construction.

But the ordinate $OH$ (in a circle described upon the radius $FN$ ,) becomes the sine of an angle, whose tangent is to the tangent of the angle $QFL$ , in the proportion of the transverse axis to the conjugate; but the tangent of the same angle $QFL$ , is to the tangent of the mean motion, answering to the area of the elliptic equant $QFL$ in the same proportion. So that the ordinate $OH$ is to the sine of that angle of mean motion, in the duplicate of the said proportion; and consequently the ordinate $OH$ , in the circle on the radius $FN$ , is the sine of an angle, nearly equal to the mean anomaly joined to the double of Bullialdus's equation.

3. This equation adds to the mean motion in passing from the aphelion to the perihelion, and subducts in passing from the perihelion to the aphelion; as is evident from the transit of the point of intersection $E$ round the periphery of the ellipsis $Lfl$ .

In Sir Isaac Newton's rule (in the before-cited scholium to the 31st prop. 1st book,) the angle $X$ answers to this equation for the segment; excepting that it is there taken in the triplicate proportion of the sine of the mean anomaly, instead of the triplicate proportion of the ordinate $OH$ . The error of this rule makes

III. The third part of the inequality, answering to the trilinear space $OKQ$ , being the difference of the elliptic sector $OFQ$ and the triangle $OFK$ .

The sector $OQF$ is proportional to an angle, which is the difference of two angles, whose tangents are in the given proportion of the semi-latus rectum $FB$ and the semi-transverse $FN$ , or in the duplicate proportion of the lesser axis to the axis of the orbit. So that this sector, when at a maximum, is as an angle, whose sine is to the radius, as the difference of the latus rectum and transverse to their sum; or as the difference of the squares of the semi-axes to their sum.

The triangle $OFK$ is proportional to the rectangle of the co-ordinates $OH$ and $HF$ ; that is, as the rectangle of the sine $OH$ and its cosine, in the circle on the radius $FN$ ; or as the sine of the double of that angle, whose sine is $OH$ ; that is, the double of the angle, whose tangent is to the tangent of the angle $QFL$ , in the given ratio of the greater to the lesser axis; or whose tangent is the tangent of the angle of mean motion answering to the elliptic sector QFL, in the duplicate of the said ratio. But this triangle $OFK$ , when at a maximum, makes an angle of mean motion, which is to the angle called $R$ , as $BN$ , half the difference between the latus rectum and transverse axis, is to the double of the transverse axis.

So that the sector or triangle in orbits nearly circular, is always nearly equal to the double of Bullialdus's equation.

The triangle and sector being thus determined, the equation for the trilinear space is accordingly determined. From what has been said, it appears, that

1. This equation for the trilinear space $OKQ$ , is to that for the triangle $OKF$ , in a ratio compounded of $BN$ , the difference between the semi-transverse and semi-latus rectum to the semi-latus rectum, and of the duplicate proportion of the sine $OH$ to the radius; or $OKQ$ is to $OKF$ , in a proportion compounded of the duplicate proportion of the distance of the foci to the square of the lesser axis, and the duplicate proportion of the line $OH$ to the radius. For the trilinear figure $OKQ$ and the triangle $OKF$ , are nearly as $OK$ and $KH$ , which are in that proportion, and consequently it holds in this proportion to the double of Bullialdus's equation. 2. This equation, in different angles, is as the content under the sine complement and the cube of the sine. For the triangle $OKF$ , is as the rectangle of the sine and the sine complement.

3. It is at a maximum, at an angle whole sine complement is to the radius, as the square of the greater axis is to the sum of the squares of the two axes; which in orbits nearly circular, is about 60 degrees of mean anomaly.

4. In orbits of different eccentricities, it increases in the quadruplicate proportion of the eccentricity.

5. It observes the contrary signs to that for the elliptic equant, called Bullialdus's equation; subducting from the mean motion in the first and third quadrants, and adding in the second and fourth, if the motion is reckoned from the aphelion.

The use of these equations, in finding the place of a planet from the upper focus, will appear from the following rules, which are easily proved from what has been said.

Let $t$ be equal to $CA$ the semi-transverse, $c$ equal to $FC$ the distance of the center from the focus, $b$ equal to $CD$ the semi-conjugate, and $R$ an angle subtended by an arch equal to the radius, viz, 57°. 17'. 44". 48"', or 57, 2957795 degrees. Take an angle $T={\frac {cc}{2tt}}R;E={\frac {b}{2t}}T;S={\frac {4c}{3b}}T.$

The angle $T$ be will the greatest equation for the triangle $OFK$ ; the angle $S$ will be the greatest equation for the segment $LMG$ ; and the angle $E$ will be the greatest equation for the area $OKFL$ . Which greatest equations being found, the equations at any angle of mean anomaly, will be determined by the following rules.

Let $M$ be the mean anomaly; and let $\tau$ be to $T$ as the sine of the angle $2M$ to the radius: In which proportion, as also in the following, there is no need of any great exactness, it being sufficient to take the proportions in round numbers.

Take $e$ to $E$ as the sine of $2M\pm 2\tau$ to the radius; and $s$ to $S$ as the cube of the sine of $M\pm \tau$ to the cube of the radius.

Then the angle $QFL$ is equal to $M+e+s$ , in the first quadrant $LN$ , or $M-e+s$ , in the second quadrant $Nl$ , or $M+e-s$ in the third quadrant, or $M-e-s$ in the fourth quadrant.

Note, That the small equation $\tau$ is always of the same sign with the equation $e$ ; and in the case of the planets, always near the double of that equation.

The angle RFA at the upper focus F being known, the angle RSA at the Sun in the other focus, is found by the common rule of Bishop Ward; viz. the tangent of half the angle RSA, is to be to the tangent of half the angle RFA, always in the given proportion of the perihelion distance SP to the aphelion distance SA. How these equations are in the several eccentricities of the Moon's orbit, will appear by the following Table.

Eccentr.	E.	S.
	′″	′
0.040	1.23	09
0.045	1.45	13
0.050	2.09	17
0.055	2.36	23
0.060	3.06	30
0.065	3.38	38
0.070	4.14	47

To add one example; suppose the eccentricity 0.060, the mean anomaly 30°. The sine of the double of the mean anomaly, that is, the sine of 60 is to the radius, nearly as 87 to 100; whence, if the equation E = 3′.06″, be divided in that proportion, it will produce 2′.40″ nearly, for the equation $e$ : the sine of $M$ is, in this case, equal to ${\tfrac {1}{2}}$ the radius, the cube is ${\tfrac {1}{8}}$ of the cube of the radius; whence if the equation $S=30''$ be divided in the same proportion, it will produce near 4″ for the equation $s$ . Therefore the angle $RFA$ , which is $M+e+s$ , will be 30°.2′.44″; and the half is 15°.1′.22″; wherefore if the tangent of this angle be diminished, in the proportion of 1.06, the aphelion distance, to 94 the perihelion distance, it will produce the tangent of 13°.23′.13″; the double of which 26°.46′.26″, is the true anomaly or angle at the Sun $RSA$ . And consequently, the equation of the center is 3°.13′.34″ to be subducted, at 30 degrees mean anomaly.

When the place of a planet is found by this, or any other method; the place may be corrected to any degree of exactness by the common property of the equant, viz. that the rays are reciprocally in the duplicate proportion of the velocity about the center. For in this case, if there be a difference between the mean motion belonging to the angle assumed at the upper focus, and the given mean motion, the error of the angle assumed is to the difference, as the rectangle of the semi-axes to the rectangle of the distances from the foci. But in orbits like those of the planets, the rules as they are delivered above are sufficient of themselves without further correction.

Banner from Andrew Motte's 1729 translation of Newton's The Mathematical Principles of Natural Philosophy

POSTSCRIPT.

Upon reviewing these few sheets after they were printed off, which happened a little sooner than I expected, I fear the apology I have offered for delivering the propositions relating to the Moon's motion, in this rude manner, without giving any proof of them, or so much as mentioning the fundamental principles of their demonstration, will scarcely pass as a satisfactory one; especially since there are among these propositions, some which, I am apt to think, cannot easily be proved to be either true or false, by any methods which are now in common use.

Wherefore to render some satisfaction in this article, I shall add a few words concerning the principles from whence these propositions, and others of the like nature are derived: and also take the opportunity to subjoin a few remarks, which ought to have been made in their proper places.

First, There is a law of motion, which holds in the case where a body is deflected by two forces, tending constantly to two fixed points.

Which is, That the body, in such a case, will describe, by lines drawn from the two fixt points, egual solids in equal times, about the line joining the said fixt points.

The law of Kepler, that bodies describe equal areas in equal times, about the center of their revolution, is the only general principle, in the modern doctrine of centripetal forces.

But since this law, as Sir Isaac Newton has proved, cannot hold, whenever a body has a gravity or force to any other than one and the same point; there seems to be wanting some such law as I have here laid down, that may serve to explain the motions of the Moon and Satellites, which have a gravity towards two different centers.

It follows as a corollary to the law here laid down, that if a body, gravitating towards two fixt centers, be supposed, for given small intervals of time, as moving in a plane passing through one of the fixt centers, the inclination of the said plane, to the line joining the centers, will vary according to the area described; that is, if the area be greater, the inclination will be less; and if the area be less, the inclination will be greater, in order to make the solids equal.

This corollary, when rightly applied, will serve to explain the variation of the inclination of the plane of the Moon's orbit to the plane of the ecliptic.

And how extremely difficult it is to compute the variation of the inclination in any particular case, without the knowledge of some such principle as this is, will best appear, if any one consider the intricacy of the calculations, used in the corollaries to the 34 prop. of the third book of the Principia, in order to state the greatest quantity of variation, in that month, when the line of the nodes is in quadrature with the Sun, and that only in particular Numbers, whereby it is determined to be 2′.43″.

Whereas, there is a plain and general rule in this case, which follows from what is laid down, though not immediately; namely, that the greatest variation in the said position of the Moon's orbit, is to the mean inclination of the plane as the difference of the greatest and least areas described in the same time by the Moon about the earth, when in the conjunction and in the quarters to the mean area.

Wherefore, if $S$ be to $L$ , as the Sun's period to the Moon's period: The greatest area is to the least, as ${\sqrt {SS+3LL}}$ to $S$ , or as $S+{\frac {3LL}{2S}}$ to $S$ nearly, by what is said on this article in the 29th page. So that the difference of areas is to the mean area, as ${\tfrac {1}{2}}LL$ to $SS+{\tfrac {3}{4}}LL$ ; and in the same proportion is the greatest variation of the inclination of the plane in this month to the mean inclination, which agrees nearly with Sir Isaac's computation.

Secondly, There is a general method for assigning the laws of the motion of a body to and from the center, abstractly consider'd, from its motion about the center.

The motion to and from the center is called by Kepler a Libratory motion; the knowledge of which seems absolutely requisite, to define the laws of the revolution of a body, in respect of the apsides of its orbit.

For the revolution of a body, from apsis to apsis, is performed in the time of the whole libratory motion; the apsides of the orbit being the extreme points, wherein the libratory motion ceases.

So that, according to this method, the motion of a body round the center, is not consider'd as a continued deflection from a streight line; but as a motion compounded of a circulatory motion round the center, and a rectilinear motion to or from the center.

Each of which motions require a proper Equant. Of the equant for the motion round the center, I have already given several examples. And in the case of all motions, which are governed by a gravity or force tending to a fixt point; the real orbit in which the body moves, is the equant for this motion. In all other cases it is a different figure.

The Equant for the libratory motion, is a curve line figure, the areas of which serve to shew the time wherein the several spaces of the libration are performed.

Which figure is to be determined, by knowing the law of the gravity to the center: For the libratory force, to accelerate or retard the motion to or from the center, is the difference between the gravity of the body to the center, and the centrifugal force arising from the circulatory motion. But the latter is always under one rule: For in all revolutions round a center, in any curve line, whether described by a centripetal force or not, the centrifugal force is directly in the duplicate proportion of the area described in a given small time, and reciprocally in the triplicate proportion of the distance; which is an immediate consequence of a known proposition of Mr. Huygens. The like proportion also holds as to the centripetal force in all circular motions, from a known proposition of Sir Isaac Newton. But what is true of the centripetal force in circles, is universally true of the other force in orbits of any form.

So that by knowing the gravity of the body, since the other force is always known, the difference, which is the absolute force to move the body to or from the center, will be known; and from thence the velocity of the motion, and the space described in any given time, may be found, and the equant described. These hints may be sufficient to shew what the method is.

To add an example. If the gravity be reciprocally as the square of the distance; the equant for the libratory motion, will be found to be an ellipsis similar to the orbit, whose longer axis is the double of the eccentricity; the center of the libratory motion, that is the place where it is swiftest, will be in the focus; the time of the libration, through the several spaces, is to be measured by sectors of the said ellipsis, similar to those described by the body round the focus of the orbit; and the period of the libratory motion will be the same with the period of the revolution.

In any other law of gravity, the equant for the libratory motion, will either be of a form different from the orbit, or if it be of the same form, it must not be similarly divided.

I may just mention, that the equant for the libratory motion, in the case of the Moon, is a curve of the third kind, or whose equation is of four dimensions; but is to be described by an ellipsis, the center of the libration not being in the focus.

From this method of resolving the motion, it will not be difficult to shew the general causes of the alteration of the eccentricity and inequality in the motion of the apogee. For when the line of apsides is moving towards the Sun, it may be easily shewn, that since the external force in the apsides, is then centrifugal, it will contribute to lengthen the space and time of the libration; by lengthening the space, it increases the eccentricity; and by lengthening the time of the libration, it protracts the time of the revolution to the apsis, and causes what is improperly called a motion of the apsis forward. But when the line of apsides is moving to the quadratures, the external force in the apsides, is at that time centripetal; which will contribute to shorten the space and time of libration; and by shortening the space will thereby lessen the eccentricity, and by shortening the time of libration, will thereby contract the time of the revolution to the apsis; and cause what is improperly called a retrograde motion of the apsis.

I shall only add a few remarks, which ought to have been made in their proper places.

As to the motion of the Moon in the elliptic epicycle (page 9.) it should have been mentioned, that there is no need of any accurate and perfect description of the curve called an ellipsis, it being only to shew the elongation of the Moon, from the center of the epicycle; which doth not require any such accurate description.

Fig. 1.It should have been said, that when the Moon is in any place of its orbit, suppose somewhere at $N$ , in that half of the orbit which is next the Sun, it then being nearer the Sun than the Earth, has thereby a greater gravity to the Sun than the Earth: which excess of gravity, according to Sir Isaac Newton's method, consists of two parts; one acting in the line $NV$ parallel to that which joins the Earth and Sun; and the other acting in the line $VB$ directed to the Earth; and these two forces, being compounded into one, make a force directed in the line $NB$ ; which is in proportion to the force of gravity, as that line $NB$ is to $TB$ nearly. Wherefore, as there is a force constantly impelling the Moon somewhere towards the point $B$ , this force is supposed to inflect the motion of the Moon into a curve line about that point; for the same reason as the gravity of it to the Earth, is supposed to inflect its motion into a curve line about the Earth: not that the Moon can actually have so many distinct motions, but the one simple motion of the Moon round the Sun is supposed to arise from a composition of these several motions.

In the last article on the small annual equations, (page 38.) these rules ought to have been added.

Let Æ be the equation of the Sun's center; P the mean periodical time of the node or apogee; S the mean synodical time of the Sun's revolution to the node or apogee: Then will ${\frac {3S}{2P}}$ Æ be the annual equation of the node or apogee, according as S and P are expounded.

The like rule will serve for the annual equation of the Moon's mean motion. If S be put for the Sun's period; P for the mean synodical period of the Moon to the Sun; and L for the Moon's period to the Stars: The annual equation of the Moon's mean motion will be ${\frac {3LL}{2PS}}$ Æ.

According to these rules when expounded, the equation for the node will be found to be always in proportion to the equation of the Sun's center, nearly as 1 to 13.

The equation of the apogee to the equation of the Sun's center, as 10 to 53.

And the equation of the Moon's mean motion to the same, as 8 to 77.

THIS FIGURE IS INCOMPLETE - A COMPLETE VERSION NEEDS TO BE SOURCED

It may be throughout observed, that the propositions are in general terms, so as to serve, mutatis mutandis, for any other satellite, as well as the Moon.

There might have been several other observations and remarks made in many other places, had there been sufficient time for it. But perhaps what I have already said may be too much, considering the manner in which it is delivered.

FINIS

Erratum

↑ Original: 8th was amended to 28th: detail

[1] Original: 8th was amended to 28th: detail

[errata 1]