# De motu corporum in gyrum

De motu corporum in gyrum (Latin: "On the motion of bodies in an orbit") is a manuscript by Isaac Newton sent to Edmund Halley in November 1684. It derived the three laws of Kepler assuming an inverse square law of force, and generalized the answer to conic sections. It tried to set out the foundations of modern dynamics and extended its methodology by adding to the derivation of Kepler's laws the solution of a problem on the motion of a body through a resisting medium. Halley reported these results to the Royal Society on 1684-12-10 (Julian calendar). Three versions of the manuscript exist: they differ from each other in some crucial respects. The book Philosophiae Naturalis Principia Mathematica, commonly known as Principia Mathematica, is a correction and an expansion of this note.

Two important definitions in the first version of this work are worth drawing attention to in the light that they throw on the development of Newton's thoughts on dynamics -

1. that of centripetal force, accepting totally the tutorial on the role of inertia in circular motion given by Robert Hooke in his letter of 1674,

2. that of force inherent in a body which forces it to move in a straight line, showing that the nature of inertia was still not clear.

This error was further compounded by adding this inherent force to an external force by the parallelogram law.

The first version was reported to the Royal Society, but was not published since Newton wanted to revise it. The second version (possibly dating from December or January) contained minor corrections. The last version of this note attempts a reconstruction of dynamics by stating five laws of dynamics -

Law 1

stated that a body moved uniformly by inherent force alone.

Law 2

asserted that the "change in the state of moving or resting is proportional to" the impressed force and is made in the direction of the line in which the force acts.

Law 3

held that the motion in a given space did not depend on any rectilinear uniform motion of that space.

Law 4

stated that mutual interactions of bodies do not change the motion of their center of mass.

Law 5

contained an empirical statement about the resistance of media.

In addition he added to this last version an explicit hypothesis about an absolute frame of reference, with respect to which the motion of bodies could be determined: this in spite of already undermining this notion by the above Law 3.

Two separate papers of revision followed the last version of De motu. In these he reduced the number of laws of motion to 3. He sharpened the law of inertia without giving up an inherent force, but introduced a distinction between inherent and impressed force, clarifying that the latter only changes the motion of the body, but is not added to the inherent force. The inherent force disappeared in the Principia Mathematica.

Halley's question

When Edmund Halley visited Newton in 1684, he asked Newton, "…what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it." In the original tract of De motu corporum in gyrum, Newton wrote, as Problem 3, "A body orbits in an ellipse: there is required the law of centripetal force tending to a focus of the ellipse." Halley had asked what the orbital curve would be if the attractive force is inverse square. Newton's tract, however, considered what the attractive force would be if the orbital curve is an ellipse.

In De motu, Problem 3, Newton wrote:

Therefore the centripetal force is reciprocally as L X SP², that is, (reciprocally) in the doubled ratio of the distance … .

In the following Scholium, he asserted:

The major planets orbit, therefore, in ellipses having a focus at the centre of the Sun, and with their radii (vectores) drawn to the Sun describe areas proportional to the times, exactly as Kepler supposed."

In Problem 4, Newton then claimed that the elliptical orbits of the planets were merely the results of the planets' speed. If the body were a comet, moving faster than a planet, then the orbit would describe a different conic section.

This argument holds when the figure is an ellipse. It can, of course, happen that the body moves in a parabola or hyperbola. Specifically, if the speed of the body is so great that the latus rectum L is equal to 2SP + 2KP, the figure will be a parabola having its focus at point S and all its diameters parallel to the line PH. But if the body is released at a still greater speed, it will move in a hyperbola having one focus at the point S, the second at the point H taken on the opposite side of the point P, and its transverse axis equal to the difference of the lines PS and PH.

Thus, the different orbital curves depend on the different speeds of the body. For all three conic sections, though, the attractive force is inverse square.

* Isaac Newton, Galileo, Descartes, Robert Hooke and Christian Huygens

* Philosophiae Naturalis Principia Mathematica and classical mechanics

References

1. ^ Quoted in Richard S. Westfall's Never at Rest, Chapter 10, Page 403.

2. ^ Whiteside's Mathematical Papers of Isaac Newton, Vol. 6

Bibliography

* Never at rest: a biography of Isaac Newton, by R. S. Westfall, Cambridge University Press, 1980 [ISBN 0-521-23143-4]

* The Mathematical Papers of Isaac Newton, Vol. 6, pp. 30-91, ed. by D. T. Whiteside, Cambridge University Press, 1974 [ISBN 0-521-08719-8]