Of the Laws of Motion

If the Body C was alone undisturbed in the Place ’tis represented in, it would by the first self-centering Principle always remain there: But if its Situation be within the Sphere of Attraction, of a much greater or more powerful Body A, it will be by the same Law that keeps its own Parts in firm Union, endeavour to meet the Body A, and the Point of Contact will be G.

If the Body C had no natural Tendency to the Body A, or more properly, if the Body A were removed to an infinite Distance from that of C, so as to render its Virtue of no Effect at C; and if C by any external Impulse become a moving Body along the Line CH; then C now no way adhering to A or to any other Body, in the same Space of Time (the Motion being continued) it would reach to H; by the same Law, i.e. of its uniform Progression continued on, it will move thro’ an equal Space of the same right-line Direction HI, and so on ad Infinitum.

But if the Body A assume its former State, and be allowed to exert its Power upon C, and if at the same Time C begins to exert its new-born Force of Levity or Freedom from A, in the Direction of CI, it will still be compell’d to adhere to the more powerful Principles of A, whilst it continues within the Sphere of A’s Activity, and so perpetually deflect from the Direction of its rectelineal Motion to that of a Curve.

Let the Power of Gravity in C towards A for any given Space of Time, be as the Part of the Line, which joints their Centers CL; and let a projectile Force be given to C at the same Instant at right Angles to it, which in the same Time alone would remove the Body to H: Now both these Forces acting together with different Directions; in the Time the Body C under the Conduct of either, and independent of the other, would be carried respectively from C to H or L, by the joint Influence of both it would be conveyed to K; for K is equally distant from H and from L, as H and L is from C, and joining two Forces together by the line CK, form what the Mathematicians call an Astronomical Area, viz. the Triangle CKA, which is always equal to that of its simple Generation CHA, and ever proportional to the periodic Times.

To illustrate this great Problem first proposed, and limited by Kepler; reduced to Theorem in Lemmata, by Gallileus;² and solv’d in all its various consequent Effects, by Sir Isaac Newton.

Let us suppose the Ball or Body C, to be thrown or projected with any determine’d Force towards H; Gravity immediately acting upon it, it will deflect from Line CH, and describe a curvelineal Arch, but meeting with Resistance superior to its cetrifugal Power, it may fall at D; if a greater original Force be given it toward I, before it is overcome by Gravity, ’tis possible it may reach to E: If a still greater rectelineal Force, it may arrive at F; and if greater still, ’tis probable it may come to G. But if at Length a Force of Levity be given it, equal to that of Gravity, it will at last return to C; and may possibly still retain so powerful a Propensity for continuing its now uniform Motion, as to be able to perform a Repetition of this Orbit, and after a third Revolution, and so on, with its Velocity undiminish’d to the End of Time.

Thus the Moon, as at BBB, perpetually renews her monthly Course about the Earth, and all the primary Planets, &c. their unvaried Orbits round the Sun, and where either of the above Powers are least predominant, that Part of the Orbit most deflects from a right Line, or if otherwise comes nearest to a perfect Circle; but the Excentricity of any Orbit is ever more or less, according to the original Direction of the Body and Quantity of the Force imprest, and very often a circular Orbit may change its Form and become an Ellipse from its Approach or too near Proximity to any external and powerful Body; thus the Moon’s Orbit is continually changing its Species from the Opposition to the Quadratures of the Sun, &c.

But in all Orbits generated by the mutual Gravity of any two Bodies, if undisturbed by a third, the Areas described are proportional to the Times, that is to say,

In the concentric Circle POHCK the Motion being always uniform about the Center ☉, the Areas, such as PO☉ described in equal Times, in all Parts of the Cicrcle, must of consequence be equal; and also in the excentric Circle RQPONMD, where it is supposed not equal, but accelerated toward ☉ and retarted from it, in the same Time the moving Body is passing from M to S, it would also pass from Q to R, and the Spaces or Areas QR☉ and N☉M described by the Radii drawn to ☉ in those equal Times, would likewise be equal. The same is to be understood of the two Ellipses, the Sun being supposed the common Focus of both; in one of which, the Spaces LK☉ and I☉H are equal and generated in the same Time; and in the other Areas E☉F, C☉D and G☉B are all equal and also produced in equal Time. Thus the great Comet of 1680, which in its Aphelion was so inactive an indolent, as scarce to move at all, when in its Perihelion (by the undeniable Principle of Geometry) is demonstrated to have mov’d with a Velocity, much swifter than that of a Cannon Ball when first exploded, and almost equal to that of Light.

The Proof of these Mathematical Conclusions may be comprehended in Fig. III.

The Body B in an uninterrupted rectelineal Motion, would move from O to C in the same Time it would move from B to O; but being drawn off from its rectelineal Direction, by Virtue of the Body A immediately acting upon it, and at right Angles to CD, i will describe the uniting Line or Diagonal of the Forces PB; but the Triangles BOA and PBA are upon the same common Base BA, and betwixt the same Parallels OQ consequently (by Euclid, B. I. Prop. 38) the Areas will be absolutely equal.

In like Manner may be prov’d, that the double Quantity of simple Motion CAB, is also equal to the double Quantity, or Area, of compoud Motion RBA, adn that likewise the second Space PS, described in a like Interval of Time, from an equal Area both with RAP and PAQ.

In a third Particle of Time, the same Body would move from S to K; then on to M, and so to N, ever describing Areas proportional to the generating Times. But here, the Force of Gravity decreasing, as the Square of the Distance increases, in thsi Part of the Orbit the Motion is retarded, and the revolving Body instantaneously moves slower.

As also the Body at F in the Ellipsis FEGH, tho’ there the projectile Force ce, &c. seems to conspire with the curvelineal Direction: But towards G, the moving Force is every Moment accelerated, the projectile Power gw, and that of Gravity wy, both conspiring with the acquir’d Velocity and the natural Direction of the Curve, as is evident from the great Obliquity of the rhomboidical Spaces iyxo, oxYZ, &c.

At the Absis E, which is called the perihelion Point of all the solar Planets, the Body moves fastest of all; and at the other Absis I, the aphelion Point, the slowest, Gravity, bh, there being not only the weakest, but also the centrifugal Force dh the most avers’d.

Besides the real progressive Motion of the several Planets round the Sun from West to East, in respect of one another they sometimes appear to stand still, and at other Times seem to move contrary, as from East to West, and in effect contradict their own proper Motions.

Thus the Inhabitants upon the Earth, like the Crew of a Ship at Sea, may think all the Objects they see are flyin gfrom them, while their own real Motion in a contrary Direction, is all the while effecting the Deceptio Visus, and is the only true Cause of it.

To solve this Phænomena, with regard to the Planets, (See Fig. [IV].)

Let A represent an inferior Planet, and B a superior One, or suppose A the Earth and B Jupiter or Saturn, &c. Now the Motion of A being much swifter than that of B, while B is moving to H, I, or K, A will have got respectively to E, F, or G: Consequently B now supposed to be removed, which from the Point A appeared amongst the fix’d Stars at x, when the eye is in E, F or G will be seen successively at w v and t. Here B will apparently move, as from Aries to Taurus, and in this Case be direct, or as the Astronomers term it, moving in Consequentia; as will also A at the same Time appear to do from Eye at B.

But if B were in any other Part of its Orbit, in respect of A, or which is all one, if A were removed to M; whilst B is in passing from I to S or K, A would advance from N to O or M, and here the two Bodies would seem amongst the Staras to stand still; in this Case, a Planet not appearing to move at all, is said to be stationary.

Lastly, if B was posited at Q while A was at N, by that Time B was got to P, A would have come to M; in this Position, B from A, and consequently A from B, would appear to go backwards, i.e. from ♉︎ unto ♈︎, having first from N appeared at r, and then soon after to O and M in s and t.

This is called Retrogradation, or the Motion of the Planets in Antecedentria, they thus appearing to move from East to West.

Thus Jupiter to an Eye at Mars in O or M, at R and P will be found to be retrograde, as from o to p, but direct at z: Venus to Jupiter or Saturn at L, will appear likewise to be retrograde; and most sensibly so in her Transit from k to a and b, but to Mars in the same Part of her Orbit from V, W and T, she will appear first direct near h and i, then stationary about k, and afterwards retrograde towards a and b, as appears by the Position of the Lines of Direction to l, m, n, &c.³

The Poles of these alternate Motions are such as D and C, change their Situation every Revolution.