De motu corporum in gyrum

De motu corporum in gyrum (from Latin: "On the motion of bodies in an orbit"; abbreviated De Motu) was a manuscript by Isaac Newton that he sent to Edmond Halley in November 1684. This work began after Halley visited Newton earlier that year and asked him about some tricky problems that Halley and his friends in London, including Sir Christopher Wren and Robert Hooke, were discussing.

In this manuscript, Newton shared important math ideas related to what we now call "Kepler's laws of planetary motion". Before Newton's work, these ideas weren't widely seen as solid scientific rules. Halley told the Royal Society about Newton's ideas on December 10, 1684. Later, with more encouragement from Halley, Newton turned these ideas into his famous book, Philosophiæ Naturalis Principia Mathematica.

One of the surviving copies of De Motu was made by being entered in the Royal Society's register book, and its (Latin) text is available online.

For ease of cross-reference to the contents of De Motu that appeared again in the Principia, there are online sources for the Principia in English translation, as well as in Latin.

De motu corporum in gyrum is short enough to set out here the contents of its different sections. It contains 11 propositions, labelled as 'theorems' and 'problems', some with corollaries. Before reaching this core subject-matter, Newton begins with some preliminaries:

3 Definitions:

1: 'Centripetal force' (Newton originated this term, and its first occurrence is in this document) impels or attracts a body to some point regarded as a center.

2: 'Inherent force' of a body is defined in a way that prepares for the idea of inertia.

3: 'Resistance': the property of a medium that regularly impedes motion.

4 Hypotheses:

1: Newton indicates that in the first 9 propositions below, resistance is assumed nil.

2: By its intrinsic force (alone) every body would progress uniformly in a straight line to infinity unless something external hinders that.

3: Forces combine by a parallelogram rule.

4: In the initial moments of effect of a centripetal force, the distance is proportional to the square of the time.

Then follow two more preliminary points:

2 Lemmas:

1: Newton briefly sets out continued products of proportions involving differences.

2: All parallelograms touching a given ellipse are equal in area.

Then follows Newton's main subject-matter, labelled as theorems, problems, corollaries and scholia:

Theorem 1

Theorem 1 demonstrates that where an orbiting body is subject only to a centripetal force, it follows that a radius vector, drawn from the body to the attracting center, sweeps out equal areas in equal times.

Theorem 2

Theorem 2 considers a body moving uniformly in a circular orbit, and shows that for any given time-segment, the centripetal force is proportional to the square of the arc-length traversed, and inversely proportional to the radius.

Corollary 1 then points out that the centripetal force is proportional to V²/R, where V is the orbital speed and R the circular radius.

Corollary 2 shows that, putting this in another way, the centripetal force is proportional to (1/P²) * R where P is the orbital period.

Corollary 3 shows that if P² is proportional to R, then the centripetal force would be independent of R.

Corollary 4 shows that if P² is proportional to R², then the centripetal force would be proportional to 1/R.

Corollary 5 shows that if P² is proportional to R³, then the centripetal force would be proportional to 1/(R²).

A scholium then points out that the Corollary 5 relation is observed to apply to the planets in their orbits around the Sun, and to the Galilean satellites orbiting Jupiter.

Theorem 3

Theorem 3 now evaluates the centripetal force in a non-circular orbit, using another geometrical limit argument.

A corollary then points out how it is possible in this way to determine the centripetal force for any given shape of orbit and center.

Problem 1 then explores the case of a circular orbit, assuming the center of attraction is on the circumference of the circle.

Problem 2 explores the case of an ellipse, where the center of attraction is at its center, and finds that the centripetal force to produce motion in that configuration would be directly proportional to the radius vector.

Problem 3 again explores the ellipse, but now treats the further case where the center of attraction is at one of its foci.

A scholium then points out that this Problem 3 proves that the planetary orbits are ellipses with the Sun at one focus.

Theorem 4

Theorem 4 shows that with a centripetal force inversely proportional to the square of the radius vector, the time of revolution of a body in an elliptical orbit with a given major axis is the same as it would be for the body in a circular orbit with the same diameter as that major axis.

A scholium points out how this enables determining the planetary ellipses and the locations of their foci by indirect measurements.

Problem 4 then explores, for the case of an inverse-square law of centripetal force, how to determine the orbital ellipse for a given starting position, speed, and direction of the orbiting body. Newton points out here, that if the speed is high enough, the orbit is no longer an ellipse, but is instead a parabola or hyperbola.

A scholium then remarks that a bonus of this demonstration is that it allows definition of the orbits of comets and enables an estimation of their periods and returns where the orbits are elliptical.

Lastly, Newton attempts to extend the results to the case where there is atmospheric resistance, considering first the effects of resistance on inertial motion in a straight line, and then the combined effects of resistance and a uniform centripetal force on motion towards/away from the center in a homogeneous medium.

Commentaries on the contents

In "De Motu," Newton sometimes used proofs in a way that assumed the opposite statements were also true without fully explaining why. This was especially noticeable in "Problem 3." Newton’s explanations were often short, assuming that some ideas would seem obvious to readers.

Scholars have long debated whether Newton’s methods for showing these opposite statements were clear enough. While no one doubts the truth of these statements, there has been discussion about whether Newton provided enough detail in his proofs.

Halley's question

Edmund Halley visited Isaac Newton in 1684 and asked him an important question about how planets move. Halley wanted to know what path planets would follow if the force pulling them toward the Sun got weaker the farther away they were.

Newton later remembered Halley asking him if he knew the shape of the planets’ paths around the Sun and wanted to see Newton’s proof. Because these stories were told many years later, we are not completely sure exactly what words Halley used at that meeting.

Role of Robert Hooke

In 1679, the scientist Robert Hooke wrote to Isaac Newton, asking about his ideas on how planets move. This sparked Newton’s interest in studying the motions of heavenly bodies. Later, Hooke claimed that Newton used his ideas, especially about a special law of attraction, but Newton disagreed. Newton said he only got a nudge to think more about these problems from Hooke’s letters, and that many ideas came from others or his own work. Scholars still discuss today how much Newton learned from Hooke.

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