Read an Excerpt
Force and Geometry in Newton's Principia
By François De Gandt, Curtis Wilson PRINCETON UNIVERSITY PRESS
Copyright © 1995 Princeton University Press
All rights reserved.
ISBN: 978-0-691-03367-9
CHAPTER 1
THE DE MOTU OF 1684
* * *
THE ELEMENTS OF NEWTON'S SOLUTION
I turn now to the demonstration actually given by Newton toward the end of 1684 in the little treatise that so delighted Dr Halley
How to pass beyond the vague idea of an attractive force to its geometrical expression—to its evaluation along a trajectory"? Newton saw in the mathematical elaboration of this problem the difference that separated him from Hooke It is one thing to propose a conjecture concerning the variation of force, and quite another to enter into the detail of the geometrical determination, the observations, and the calculations Hooke had spoken and written as if all of those—the working out of the mathematical details—were only a subsidiary chore, a piece of drudgery that he did not himself have the time to carry out To carry out the task, it would in fact have been necessary to construct the entire edifice of a theory of central forces, starting from materials scattered throughout the scientific writings of the seventeenth century
Inertia and Deviation
In a schematic view of the text of the De motu, the elements of Newton's solution are the following
Left to themselves, the planets would move uniformly in straight lines (this is called the principle of inertia, which will be Law I of the Principia)
The incurvation of their trajectories is due to an exterior force directed toward the Sun (Newton speaks of a deviation or deflection)
To evaluate the force, it is necessary to measure the incurvation, that is, the difference between the virtual rectilinear trajectory and the real, incurved trajectory
The supposition that the deviation enables evaluation of the force, because force and deviation are proportional to one another, remains implicit in the first manuscripts of 1684 but will become Law II of the Principia:
The change of motion is proportional to the motive force impressed.
The segment QR is therefore the index and geometrical measure of the force that draws P toward S.
The Generalization of Galileo's Law of Fall
How then to measure the deviation QR? Besides the intensity of the force, it is necessary to take account of other factors to determine how much the moving body departs from a rectilinear trajectory. The separation between the tangent PR and the curve PQ is greater if, for example, the moving body is farther from P along the arc. Newton chose time as the basic variable: the deviation depends on the time elapsed.
The De motu poses the hypothesis that the deviation QR is proportional to the square of the elapsed time. Whence comes this relation? It is a generalization of Galileo's law of fall: the space traversed by a body in free fall starting from rest is proportional to the square of the time. To be able to apply this law in the present case, it is necessary to accept several assumptions:
The force that attracts the planets toward the sun is analogous to terrestrial weight.
The length QR represents a kind of path of descent.
In other words, the curved trajectory PQ must be considered as a combination of two motions: one that is uniform and rectilinear from P to R, and another that is accelerated from P towards S. The length QR represents this second component, viewed by itself and in abstraction from the first component.
It was Galileo, again, who had made such a decomposition possible and legitimate. In the Fourth Day of the Discorsi, he had shown how the trajectories of projectiles could be analyzed into a uniform, rectilinear motion (either horizontal or oblique) and a vertical, accelerated motion. This operation of abstraction in the decomposition of motions had become—notably in the work of Huygens—a very precious tool.
Between the case of the planets and that of terrestrial weight, however, there are several important differences. The fall or quasi-fall of the planet is not vertical with constant direction but is instead directed toward a point, the fixed center S. Moreover, the intensity of the force varies from point to point in space. With the planet's change in position, its "weight" is no longer the same; the moving body is subject to a force that varies, though perhaps only slightly, along its path. In the planetary case, then, Galileo's law is applicable only in the infinitely small, very close to point P. As Newton put it, the law of fall is here true only "at the commencement of the motion." His reasoning is thus valid only if arc PQ is very small or "nascent."
In sum, in figure 1.3—a figure inspired by those in Newton's text—the length QR, which is the index and measure of the force directed toward S, is proportional to this force and proportional also to the square of the time, provided that Q is very close to P.
The Measurement of Time by Area
But what to say of the time itself? How to introduce it into the diagrams and the calculations? The geometrical diagram shows the path traversed but not the time elapsed. If length of path were exactly proportional to time, the latter could be replaced by the length of path traversed; the planet, however, does not move constantly at the same speed. To evaluate its velocity it would be necessary to know its initial velocity and the force at the different points of its path.
Kepler's law of areas provides an escape from this circular trap. It asserts that the time required for a planet to traverse an arc can be evaluated by measuring the area of the sector swept out by a radius connecting the planet to the sun. Whatever the intensity of the force acting at P, the surface area swept out by the radius SP (traditionally called the radius vector) is always proportional to the time that elapses. The infinitely small triangle SPQ is therefore a measure of the time required to traverse the arc PQ. In place of the time it is thus possible to substitute the area of the triangle SPQ, and if this area is expressed in terms of the base SP of the triangle and its height QT, the length QR is proportional to the square of the product of SP and QT.
The General Formula
Thus a relation is obtained that connects the force, the deflection QR, and the time as represented by the area of the triangle:
QR is as the force and as (SP × QT),
or equivalently,
the force is as the deflection QR and inversely as the square of the area SP × QT.
Here is a completely geometrical expression of the force—at least if, into geometry, are admitted "nascent" arcs and traversed paths that are "very small." In order to evaluate the force to which a body is subject at a point P of its trajectory, it suffices to determine the value of QR/(SP2 × QT2). And to prove that the force varies from point to point according to a certain law, for example, as a function of distance from the center of force, it suffices to calculate how QR/(SP2 × QT2) depends on the distance SP when P varies.
It is this that Newton did—in contrast to the three other scientists—in the writing that he sent to Halley. The variation of QR/(SP2 × QT2) when P traverses an ellipse and the force is directed toward a focus S is inversely proportional to the square of the distance SP:
QR/[SP2 × QT2] [varies] 1/SP2.
Thus he demonstrated that if the planets move in ellipses, with the sun at a focus of the ellipse in each case, then the force attracting them to the sun diminishes as the square of the distance.
Why Read the De Motu?
It is this fundamental schema of reasoning in the De motu that Newton follows in the Principia, but in the latter work it is less prominent because of all the complements, amplifications, refinements in demonstration, and philosophical scholia that Newton added to his pristine text. Between 1684 and its publication in 1687 he expanded the text inordinately—from eleven propositions to more than two hundred.
Newton was already ahead of his time in 1684 when he arrived at a mathematical formulation of the idea of attraction—the idea that his three colleagues had been discussing. He was yet more advanced in 1687 when he at last submitted Book III of the Principia to the printer—so far had his speculations led him beyond his contemporaries. In three years he had produced an immense work, which the eighteenth century would have the task of understanding, developing, and verifying. What Proust has said of literary works is true here also: a veritably original work must create its own public. Newton's true contemporaries, Euler and Laplace, belonged to the next century.
Few seventeenth-century readers found the Principia accessible. The main results became known indirectly through reviews, popularizations, and simplified expositions. Even the most determined and enthusiastic readers found their patience tried by Newton's book. John Locke, an admirer, wanted to comprehend it, and at length Newton composed for him in four pages a demonstration that the planets by their gravity move in ellipses. For the benefit of discouraged readers, Newton declared in the preamble to Book III:
However, since the propositions there [in Books I and II] are many, and could cause too much delay even to readers well trained in mathematics, I do not suggest that anyone read them all. It is enough if one carefully reads the definitions, the laws of motion, and the first three sections of Book I, then passes to this book concerning the system of the world, and consults at pleasure such of the remaining propositions of the first two books as are cited here. (Princ., 386)
The De motu is much more modest in its proportions and much more accessible as well. There is great advantage in reading one of its versions before plunging into the meanders and refinements of the Principia. An initial orientation is indispensable—a pinpointing of principal results, connections, and methods—to follow the Principia.
The first pages of the Principia have been much read and discussed, especially by philosophers: the preamble of the book, setting forth the laws of motion and giving the scholia on absolute time and space, has nourished debates on the foundations of mechanics and on the nature of space and time. From the point of view of this discussion, these texts no longer occupy so essential a place. The De motu, in all its early versions, contains neither the three famous laws of motion nor any mention of absolute time and space but presents, above all, the first evidence of a geometrical translation of the concept of central force. This is its primary interest: how did force come to be expressed geometrically, and how did mathematics become capable of translating dynamics?
TRANSLATION AND COMMENTARY
The Definitions of Force in the De Motu
In the version discussed in this book (the De motu corporum in gyrum as given in MS B of Hall and text I, para. 1 of NMP , vol. 6), which is probably the earliest, the De motu is divided into four parts:
Three fundamental theorems (the law of areas, a formula for the force in uniform circular motion and then a formula for the force in any curvilinear motion whatever)
Three problems illustrating the preceding theorems
Various complements (Kepler's third law, the determination of orbits, the inverse problem, Kepler's problem, rectilinear fall)
Study of motion in a resisting medium
I leave aside the fourth part, because, being the germ of Book II of the Principia, it has no implications for Book I and because the analysis of the theorems on resisting media would stray too far afield.
The work is presented in a deductive form, starting from three definitions and three hypotheses. The definitions are as follows:
Definition 1. Centripetal force I call that by which a body is attracted or impelled towards some point viewed as a center.
Definition 2. And the force of a body or the force innate in a body, that by which the body endeavors to persevere in its motion along a straight line.
Definition 3. And resistance, a regular impeding due to a medium. (Add. 3965.7, fol. 55r)
The reader who seeks here a definition of force will be disappointed. Newton defines, not force, but the terms centripetal, innate, and resistance. The two first definitions concern the adjectives: "Centripetal force I call that by which ..." and "the force innate in a body, that by which ..." The third proceeds likewise on the basis of a tacit understanding of the concept of force. These definitions, therefore, bear not on force per se, taken generally, but only on the specifications that Newton has chosen to give to it.
The word force at the time Newton was writing was a vague term, unspecialized and poorly defined. John Wallis, one of the few authors at this time to specify a meaning for the word, defined force (vis) as a "power productive of motion" (potentia efficiendi motum). It will be assumed for the moment that this definition covers the usage that Newton made of the word.
Most people know more or less what a force is. Newton's definitions specify what constitutes a centripetal force, an innate force, and a resistance. Innate force (vis insita), or the force of a body, was not an absolute novelty in 1684; the concept had been accepted and the term received. Descartes, Galileo's disciples, Wallis, and Huygens spoke commonly of the force with which—or by virtue of which—a body continues its motion in a straight line with constant speed. The term force was thus not reserved for an external cause. Uniform rectilinear motion was associated with a force; like all motion it required a power that produced it and maintained it. Today it would be said: of itself the body perseveres in its uniform rectilinear motion. Newton and his contemporaries preferred to say: by the force that is inherent in it, which is properly its own, the body perseveres.
The adjective inherent (insita) had already been employed by Kepler and Gassendi to designate the force naturally implanted in, or belonging properly to, bodies: insita is the participle of the verb that means "to graft," "to implant." If this force is inherent, it is because the other force is not: centripetal force is not essential to bodies—it is not inherent or implanted. The opposition of the two forces underlies the use of the word insita. In later editions and in Opticks, Newton makes clear that weight does not belong essentially to matter and that the only truly inherent, or truly natural or essential force, is the "passive" force by which a body continues its uniform rectilinear motion and resists changes of motion.
The notion of centripetal force is the great innovation of this text, and as Definition 1, it is given pride of place. Newton invented the word in an act of conscious imitation: Huygens had conceived of "centrifugal force," and Newton honored him and corrected him in inverting the point of view. The force that accounts for curvilinear motions is directed toward the center, "tending toward the center" (centripeta).
The mode of action of this force is left indeterminate; it could be a push or a pull (i.e., an attraction): impellitur vel attrahitur. Various explanations remain possible. But a dynamical analysis of incurved motions does not require determination of the "physical" cause that curves the moving body in toward the central point.
Very little is known about the central point itself. It is not a center in a strict geometrical sense but is any point that can be considered as a center because the trajectory incurves around it; the path need not be perfectly circular. Is there a virtue that emanates from this privileged point, a flux of magnetism that gushes forth from it? This question cannot be answered. However, it is known that the centripetal force resembles weight. Hypothesis 4 of the De motu stipulates that the effects of centripetal force are similar to those of weight, at least locally. Centripetal force varies from point to point, but at each point its action, like that of weight, causes the body to traverse a distance proportional to the square of the time. The relationship between weight or gravity and centripetal force was so essential in Newton's eyes that he substituted the one for the other in different versions of the De motu.
(Continues...)
Excerpted from Force and Geometry in Newton's Principia by François De Gandt, Curtis Wilson. Copyright © 1995 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
