Algoritmo propuesto

Following the publication of the first edition of the Principia in 1687, Newton began to make corrections in his working copy of the text and to propose revisions and additions for a possible second edition. When, twenty-six years later, in 1713, the second edition was published, many of these hand-written revisions were incorporated. Several revisions, however, never appeared in printed form. Of particular interest are the unpublished revisions of the fundamental dynamics of Sections 2 and 3 of Book One. These revisions, if published, would have provided a dramatically different format for these fundamental sections. They have been masterfully reconstructed by D. T. Whiteside, the editor of Newton's mathematical papers. Whiteside sets out the vision that Newton had of a revision for the 1687 Principia : Newton came in the early 1690s to conceive a grand scheme of revision of the published Principia in which not only its particular verbal and mathematical errors were to be corrected but, much more radically, the redundant in its logical and expository framework was to be cut out and the flimsier portions of the remaining structure were to be strengthened and supported and (where necessary) completely rebuilt.^[1]

Newton was not the only person to suggest corrections and revisions to the Principia following its publication. A select group of scholars, both in Britain and on the continent, struggled with the work and were eager to note its failures as well as its successes. The

Scottish mathematician David Gregory had aspirations (unrequited) of having his notes on the work published in a revised edition or as a separate companion volume. During Gregory's visit to Cambridge in May 1694, Newton showed him the manuscript papers that contained the drafts of the proposed revisions. Whiteside notes

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that Newton "proved unprecedentedly expansive regarding his intentions, elaborating for him [Gregory] a detailed overview of his plans for revision."^[2] In a memorandum written in July 1694, Gregory summarized the revisions that Newton had discussed during his visit. Of particular interest to Newton's dynamics are the opening lines of Gregory's summary:

Many corrections are made near the beginning: some corollaries are added; the order of the propositions is changed and some of them are omitted and deleted. He [Newton] deduces the computation of the centripetal force of a body tending to the focus of a conic section from that of a centripetal force tending to the center, and this again from that of a constant centripetal force tending to the center of a circle; moreover the proofs given in propositions 7 to 13 inclusive now follow from it just like corollaries.^[3]

Gregory went on to list other revisions proposed by Newton for Books Two and Three. It is the fundamental revisions to the opening sections of Book One, however, that command our interest.

Whiteside has called these proposed revisions of the 1690s "radical restructurings." In the 1687 edition of the Principia , Newton employed only the linear dynamics ratio as a measure of the force in producing solutions to the direct problems set in Sections 2 and 3 of Book One.

In the proposed radical revision, however, he introduced two other related but distinct

methods for generating such solutions: the circular dynamics ratio and the comparison theorem.

In the published revised editions of the Principia of 1713 and 1726, Newton retains the wording of the statements of the propositions of the 1687 edition with only minimal changes.

In dramatic contrast, however, Newton considered major changes in the statements of the propositions in the proposed radical restructurings of the 1690s. He did not attempt in the unpublished radical revisions to conform to the general outline of the 1687 edition as he did in the revised editions that eventually were published in 1713 and 1726.

The statement of the proposed Proposition 1, Kepler's area law, remained in the unpublished revisions as it was in the 1687 edition, although six new corollaries were added. The

statements of the next four propositions also remained unchanged. The statement of the proposed Proposition 6, however, underwent a dramatic revision. In the 1687 edition,

Proposition 6 introduced the linear dynamics ratio and then was used to generate solutions for a series of direct problems in the following several propositions. The proposed revised

Proposition 6 was the first of three new propositions that introduced a new technique for solving direct problems, the comparison theorem, which bore no resemblance to that used in the original Proposition 6. The linear dynamics ratio of the original Proposition 6 was transferred to the proposed Proposition 9, to which

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was added yet another measure of force, the circular dynamics ratio. The proposed

Proposition 10 provided a measure of force for motion in a conic directed to any point, and the proposed Proposition 11 produced yet another method of attack. In what follows, we look in some detail at the new method of the proposed Propositions 6, 7, and 8, and of the method outlined in the proposed Proposition 11. Only the revisions of the proposed Proposition 9 appear in the published revised editions; they will be discussed in detail in the chapter to follow.

Proposed Propositions 6, 7, and 8

An overview of the revisions that Newton considered making in the presentation of his basic dynamics follows. The center column is the location of the proposed propositions in the unpublished revisions that Newton generated after the publication of the 1687 edition (i.e., what Whiteside has called the "radical revisions"). The first and third columns give the location of the material in the 1687 edition and in the revised editions of 1713 and 1726, respectively.

1687 edition Proposed radical revision

1713 and 1726 editions

Does not appear New Proposition 6: Similarity

Theorem.^[4]Where bodies describe all similar parts of similar figures in proportional times, their centripetal forces are . . .

Does not appear

Does not appear New Proposition 7: Proportionality Does not appear

Theorem.^[5]If in two orbits

proportional ordinates stand at any given angles on proportional abscissas . . . the centripetal forces will be as . . .

Corollary: If one of the orbits be a circle and the other orbit any ellipse, and the point S the center of both . . . [the force to the center of the ellipseµ PC].

Does not appear New Proposition 8: Comparison Theorem.^[6] Ratio of forces at point P directed to two different centers of force, S and R , for a given common orbit.

(where PT passes through S parallel to the tangent at point P and cuts PR at

1687 edition Proposed radical revision 1713 and 1726 editions

Does not appear Corollary 1. If the orbit is an ellipse and force center S is at the center, then FSµ PS and

Proposition 7, Corollary 4

Does not appear Corollary 2. If the orbit is an ellipse and the force center R is at a focus, then the force is proportional to the inverse square of the distance.

Proposition 11, Alternate Method

Does not appear Corollary 3. If the orbit is a parabola and the force center R is at a focus, then the force is proportional to the inverse square of the distance.

Does not appear

Does not appear Corollary 4. If the orbit is a hyperbola and the force center R is at a focus, then the force is proportional to the inverse square of the distance.

Does not appear New Lemma 12. An expression is Proposition 10,

derived for the chords of the circle of curvature to a conic through the center and through a focus.^[8]

Chord through the center

Does not appear New Proposition 10. The expression for the force at any point in a conic, FRµPT³ / PR ² , which was obtained in the new Proposition 8, is now derived from the new Proposition 9 and the new Lemma 12.^[9]

Proposition 7, Corollary 4

Proposition 10 Corollary 1. If the center of force is at the center of the conic, then the force is directly proportional to the distance.

Proposition 10

Proposition 11, Proposition 12, Proposition 13

Corollary 2. If the center of force is at a focus of the conic, then the force is proportional to the inverse square of

Proposition 8 Corollary 3. If the center of force recedes to an infinite distance.

Proposition 8

Proposition 7, Body of the text

Corollary 4. If the conic passes into a circle and the center of force is on the circumference, then the force is proportional to the inverse fifth power of the distance.

Proposition 7, Corollary 1

The three specific basic revisions noted by Gregory in his summary statement are readily discernible in the comparison of the initial and proposed propositions:

1. "The computation of the centripetal force of a body tending to the focus of a conic section [is deduced] from that of a centripetal force tending to the center" [Proposed Proposition 8, Corollaries 2, 3, and 4].

2. "This [force toward the center of an ellipse] again from that of a constant centripetal force tending to the center of a circle" [Proposed Proposition 7, Corollary].

3. "The proofs given in Propositions 7 to 13 inclusive now follow from it just like corollaries"

[Proposed Proposition 10, Corollaries 1, 2, 3, and 4].

The Three Techniques

As Gregory also noted, Newton has dramatically changed the order of the propositions. The difficulties involved in producing a revised edition so reordered may well be the reason that the "grand scheme of revision" was never carried to completion. In the preface of the 1687

edition, Newton notes, "Some things found out after the rest, I chose to insert in places less suitable, rather than to change the number of the propositions as well as the citations." In the revisions just given, Newton has drastically revised the number, order, and nature of the opening propositions. The extensive revision of citations in the remaining propositions, even those that were not to be changed, would have presented an enormous editorial challenge. In the 1713 edition, Newton chose to continue with his earlier practice of inserting the new material "in places less suitable." In 1694, however, he was still toying with the idea of a dramatic revision. Newton's proposed revision did more than shift his dynamical foundations from one proposition to another; it increased the number of basic methods for solving the direct problems. In place of the single method of the 1687 edition,

New-― 171 New-―

ton used in the radical revision three related but distinct methods of generating such solutions.

The first method, the linear dynamics ratio, is the initial measure of force that was introduced in Theorem 3 of On Motion and continued into Proposition 6 of the 1687 edition (i.e., QR / QT² × SP² ). This measure of the force appears as Corollary 1 of the proposed Proposition 9, and it could have been applied to the same direct problems (circle/circumference, spiral/pole, and conics) as in the 1687 edition. The draft of Proposition 9 does close with the promise to provide examples of the procedure in the problems to follow, but they are not given.

Whiteside speculates that if it had been Newton's intention to reproduce them, then he would have done so in a more abbreviated form than he used for the solutions in the 1687 edition.^[10]

The second method, the circular dynamics ratio, was the technique alluded to in his early writings on curvature in the Waste Book in late 1664 or early 1665 when he wrote the following:

If the body b moved in an Ellipsis then its force in each point (if its motion in that point bee given ) may bee found by a tangent circle of Equall crookednesse with that point of the Ellipsis .^[11]

No examples of an application of curvature by Newton to the solution of elliptical motion have been found before the revisions of 1690. It is clear, however, that curvature played an important role in Newton's thoughts on dynamics, even here in 1664. It is tempting to

speculate what use Newton did make of this suggestion before he developed the area law and the linear dynamics ratio in 1679. In 1690, the second measure of force, 1/SY² × PV , appears in Corollary 3 of the proposed Proposition 9 and Newton employed it to develop an entire set of alternate solutions for the exemplary problems that followed the initial Proposition 6 in the Principia . In contrast to the parabolic approximation of the first measure, in which a

vanishingly small arc of an arbitrary curve is replaced by a vanishingly small parabolic arc, the second measure arose from a circular approximation, in which a vanishingly small arc of an arbitrary curve is replaced by a vanishingly small arc of the circle of curvature at that point. In figure 8.1, the line segment SY is the normal from the tangent to the center of force S and the line segment PV is the chord of the circle of curvature at the point P through the center of force S . The proposed Lemma 12, which followed this proposed Proposition 9, is concerned with chords of circles of curvature in conics, and stands in contrast to the initial Lemma 12, which is concerned with circumscribed areas about conics. In the proposed Proposition 10, the results of the previous proposition and lemma are applied to the problems that were set in the initial Propositions 7, 8, 10, 11, 12, and 13, and the solutions appear as Corollaries 1, 2, 3, and 4

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Figure 8.1

The circle DPV is the circle of curvature at point P of the general curve AP . It defines the chord PV and the diameter PD .

of the proposed Proposition 10, as Gregory noted when he wrote, "The proofs given in propositions 7 to 13 [of the 1687 Principia ] inclusive now follow from it [the proposed Proposition 10] just like corollaries."

The third method, the comparison theorem, is a measure of force that gives the ratio of forces to two different force centers for any given orbit. The proposed Propositions 6 and 7 consider two different orbits with a common center of force and provide the basis for the

demonstration in the corollary to the proposed Proposition 8. In that corollary Newton obtained the force directed to the center of an ellipse from that of the force directed to the center of a circle, which Gregory had reported as "this [force to the center of an ellipse is found] again from that of a constant centripetal force tending to the center of a circle." The proposed Proposition 8 extends the comparison from two given orbits with a single force center to a single given orbit with two different force centers. Thus, as Gregory reported of Newton's proposed revisions, one could obtain "the centripetal force of a body tending to the focus of a conic section from that of a centripetal force tending to the center." In Corollaries 2, 3, and 4 of the proposed Proposition 8, Newton gave solutions to the problems set for conics in the initial Propositions 11, 12, and 13.

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The first technique or method has been discussed in the analysis of Theorem 3 of On Motion in chapter 4 of this book and in the analysis of Proposition 6 in the 1687 edition in chapter 8.

The second technique will be the subject of the analysis of the revised Proposition 6 in the 1713 and 1726 editions in chapter 9 to follow. The third technique, the new proposed Theorem 7, the comparison theorem, is the core of the proposed reconstruction, and it is carried forward to the revised published editions only in a disjointed form. To help the reader

to identify it in its ultimate published but distorted form, it is presented below in the form Newton gave it in the proposed radical revision of the 1690s.

The Comparison Theorem

Figure 8.2A is based on the diagram for the new proposed Proposition 8, Theorem 7, in which Newton considers the ratio of the forces necessary to maintain motion in a single given orbit ANPM about two different centers of force, points S and R .^[12] As with the linear dynamics ratio, the force is given as proportional to the ratio of the linear deviation and the square of the time. The two extracts from Newton's basic diagram in figure 8.2B show the deviations from the tangential motion as Pf = yN and Pe = xN (where the deviations yN and xN are parallel to the lines of force PS and PR ) and the times are proportional to the shaded areas SNM and RNM . Thus, the ratio of the forces is given as follows:

From similar triangles, Newton notes that the displacement Pf is proportional to the line PS and the displacement Pe is proportional to the line PT . Moreover, the area SNM is found to be proportional to PT² and the area RNM to PR² :

Thus, as the theorem states, the ratio of the forces is "as the product of the height of the first body SP and the square of the height of the second body PR . . . to the cube of the straight line PT " (i.e., FS / F R = SP × PR ² / PT³ ).

The solution to the Kepler problem then follows in two short corollaries. In Corollary 1 the orbits are restricted to either circles or ellipses. Thus, if the force FS is directed to the center of the ellipse, then it is known from the corollary to the new proposed Theorem 6 that it is proportional to SP . Thus, the force FR directed to any other point is given by the following:

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Figure 8.2A

Based on Newton's diagram for the proposed Proposition 8.

Figure 8.2B

The deviations Pf and Pe extracted from the diagram for the proposed Proposition 8.

or

Thus, Newton obtained a general expression for the force FR required to maintain elliptical motion directed to any general point R .

Now in Corollary 2 of the new proposed Proposition 8, Theorem 7, Newton obtains the solution to the distinguished Kepler problem simply by noting that if the point R is a focus of the ellipse, then the distance PT is a constant of the ellipse.^[13] That result is all that is required to demonstrate that the force FR is proportional to the inverse square of the distance from the body to the focal force center PR (i.e., FµPT³ / PR ²µ 1 / PR² , since PT is a constant). The solution is simplicity itself, particularly when contrasted to the solution in the 1687 edition that employs the linear

dy-― 175 dy-―

namics ratio. Corollaries 3 and 4 of the new proposed Proposition 8 extend the result to parabolic and hyperbolic paths.

Thus, the new proposed Theorems 5 and 6 demonstrate that the force to the center of an ellipse is directly proportional to the distance, and the new Theorem 7 extends the result to a focal point and to all the other conic sections. The basics of the theorems flow smoothly out of the parabolic approximation and the area law. The proposed revision is a paradigm of directness and compactness.

The Proposed Propositions 9 and 10

Newton was not satisfied in the projected revisions with simply giving the solution for the Kepler problem as obtained from the comparison ratio. In the first corollary of the new proposed Proposition 9 of the radical revision of 1694, he presents the original linear

dynamics ratio of the initial Proposition 6 from the 1687 edition. Thus, he would have offered the solutions to the central and focal conic section direct problems that appeared in the 1687 edition as a second and complementary set of solutions to those provided by the new

comparison theorem. More significant, however, is the appearance of a third set of solutions for the dynamic problems, the circular dynamics ratio, in the third and fifth corollary of the new proposed Proposition 9.

Following the new circular dynamics theorem of the proposed Proposition 9, Newton proposes a new Lemma 12. In all of the published editions Lemma 12 is concerned with the circumscribed area around an ellipse. The proposed version of Lemma 12, however,

developed relationships between an ellipse and chords of its circle of curvature through the

developed relationships between an ellipse and chords of its circle of curvature through the