# Interpretation of Principia

Principia are a so indigestible work which it is advisable to harness to make criticism of it: that at all does not call in question the imagination and the capacity for work of Newton. Recently a second reading of the new methods of the celestial mechanics of Henri Poincaré was carried out, without calling into Poincaré question. In 2005, a second reading of the literature of the restricted Relativité around 1905 was not useless, when it was released from any passion.

It is simply a question of demystifying work: it is not good in science to rent a work and not to help to make it lireCet article could be entitled: to read Principia.

The notation will use that usual: translation of Cohen: " variorum" or more simply " var" ; dimensions-Cajori translation: " principles" or more simply " princ" ; translation of the marchioness of Châtelet: " cat".

## Why read the Principia ?

1. In France, the reading of the basic principle of dynamics is slightly different from that of England, for historical reasons, which undoubtedly go back to Descartes versus Locke. The Of gravitatione of Newton in is a partial explanation by Newton, itself.

2. One knows that Newton maintained the reports/ratios tended with Hooke. On the quarrel Hooke-Newton, all or almost was known as on the famous letter of November 1679, gràce with Koyré. But one knows the intellectual atmosphere little 1684 -1687. And did one examine closely the claim of Hooke on the law of the universal gravitation? and on the statement of the “remarkable theorems” (theorems of Gauss - Newton)?
3. One knows that Newton maintained the reports/ratios tended with Huygens. The principle of the “boat of Huygens” defended later by McLaurin integrated forever in the Principia , creating an epistemological obstacle to study the principle of Galiléenne Relativity: what perhaps explains the very late character of the experiment of Foucault (1851).
4. At the time of the successive editions (1687, 1713,1726), of the subtle modifications occurred: which slips really represented in the thought of Newton and that of the mechanics of the time (Huygens, Leibniz, Hermann, Jean Bernouilli, Varignon)?
5. In France, we remain largely marked by the influence of Alexandre Koyré; there it is advisable still to be dissociated from this giant, without disavowing it: many English studies took place; it is advisable to integrate them in our culture, particularly the masterly studies of Whiteside and Bernard Cohen.
6. the thesis of François de Gandt brought an innovative breath still little exploited: Archimedes, said the Trio of Torricelli, showed the formula of the surface of the surface of the sphere by the calculus (prohibited in Greece) and thus had to reformulate its demonstration so that it is accepted. In the same way, of Gandt supports that, the time of reception of the calculus not being run out, Newton reformulated its De Motu in geometrical language (more limit of the ultimate reason (0/0, to say in short)), then asked time to write the Principia properly. Very the beautiful book of Chandrasekhar explains a little that, but remains little diffused in France (but it is not at all a book of history of sciences). Just as Dana Densmore (Newton' S principia: the central argument, ISBN 1-888-009-00-4).

## How to proceed?

An interpretation wiki is rich only in its plurality .

The pages of discussion could be invaluable.

The masterly Bernard Cohen (& Anne Whitman) (1999, ISBN 0-520-08816-6) many useful councils give (chap.10, p293) and a bibliography with accompanying notes impressive.

## An example: Reading of Proposition.VI

That is to say a trajectory (T) of a material point P, mass m, under the action of a central field of center O.

Problem: to find the force me a' , acting as this point P.

Answer: To take the point Q close to P in later time. To trace the demitangente out of P; then the parallel with COp, carried out Q, which comes to cut the tangent as a R.

Graphically , to trace for example segment COp " vertical" , P: (x=0; z=10), the portion of tangent PR towards the line, let us say R: (x= 2; z= 8). to trace the vertical segment " descendant" RQ (Q: (x=2; z=7)). To close trapezoid OPRQ by plotting straight line QO. To finish the figure by tracing the arc (~ parabolic) curve (T), is arc PQ.

the figure resembles then surprisingly that of Torricelli, in 1641 (?) , in its book deMotu, presented to Castelli, then sent to Galileo.

The reasoning will be practically the same one, with this close Newton has a clock WHICH IS NOT the X-COORDINATE of R (or of Q), because the field is CENTRAL (gold Newton already located this trap in 1679 (cf Déviation towards the East): this clock is the LAW of the SURFACES: surface OPQ = C .dt.

The reasoning then is identical: PR = V (P) .dt and the fall is RQ : = H = 1/2 has (P) . ²: = 1/2. G. ²

Then, WHATEVER THE PR, has (P) = 2 RQ/(aire/C) ²; has (P) does not depend on speed V (P)! By taking notations in Torricelli, to then obtain the formula very simple to retain:

H: = RQ = 1/2. G (P). ² = 1/2. G (P). ²

According to the cases, this surface will be expressed by 1/2 X (Q) .OP, or via the podaire, like 1/2. p.PQ.

This REMARKABLE theorem is relatively little known in France, in spite of its relative simplicity!

Let us notice the scaling immediately formula: if the curve is the spiral équiangulaire of Torricelli-Bernouilli, then h/aire ² varies like 1/r ³ ~ G (R)! it is proposal 9 (cf also Spirale logarithmic curve of Newton)!

## Some Corollaries

Corollary : the first which comes to mind is to check at least the case of the circle, radius R, center O, i.e. the formula of Huygens (Horologium 1657).

One finds easily has (P) = 2 C ² RQ/(surface) ² = $2 \ pi^ 2 r ^4/T^2 \ cdot \ frac \left\{4 RQ\right\} \left\{RP^ 2 r ^2\right\} = \ omega^ 2 r$.

It is initially this formula which gives the third law of Kepler for a force in 1/r ² and generally $\ omega^2 \ cdot r^ \left\{\left(n+1\right)\right\} = cste$ for a law in 1/r^n.

As well, Newton never asserted it anteriority on the law in 1/r ², but on the other hand, the " theorems remarquables" law of gravitation are well of him and not of Hooke!

Corollary, proposal 7 : the trajectory is a circle of radius has, the center of force on the circumference. To make the figure and to deduce immediately that H (surface) ² varies like 1/r^5. (cf central Force in 1/r ⁵)

Corollary, proposal 10 , variorum115: the law of Hooke in - k/m' r' gives an ellipse of Hooke well:

demonstration: H/(surface) ² = R (2p ² .OD ²), with OD length of the combined semi-diameter, and p the length podaire. However theorem VII.31 of Apollonius (geometry closely connected of the ellipse) known as that the surface of the parallelogram built on 2 combined semi-diameters is constant: p.OD = cste = a.b; thus G (R) = K R; CQFD.

Note: the Symétrie of Corinne immediately gives the solution in the repulsive case: a hyperbole of center O.

Notice 2: Did Hooke undoubtedly have anteriority on this problem (< 1684?), via a discrete métode (cf Mechanical Newtonian discrete)

Corollary, the proposal 11 , variorum125: the law in 1/r ² gives again the laws of Kepler well:

It is what made the glory of Newton in the deMotu of 1684. More exactly, Newton shows that if (T) is an ellipse of Kepler, then G (R) = k/r ². It will show the reciprocal one (edition 1713, proposal 17) by using the theorem of Cauchy intuitively: two C.I. (initial conditions) position, speed determine a single ellipse. Caution: trap in the case of the spiral, and Newton avoids it!

It thus should be shown that the report/ratio falls (surface) ² ~1/r ²: cf Keplerian Movement

## Corollary on the Transmutation of the force

Celebrate theorem where Newton shows that any trajectory of central field S can be a trajectory of central field of center Q, with a law of different force of course. In particular, one can transmute the law of Hooke with center in the center of the ellipse, in a law into 1 FP ², with for center of force the hearth F of the ellipse: this theorem was a wild snook with Hooke, who could never go back some: forever, it was that which had known to pass only from the ellipses of Hooke to the ellipsoids; it did not have the mathematical power of Newton, and if it were very gifted in drawing, that did not enable him obviously to pass to the " limit of the ultimate quotient 0/0" !

Let us point out this splendid statement :

• Stated: That is to say a central field of center S of force F (R) producing a movement of trajectory (T), described according to the law of the surfaces (second law of Kepler).

Then, this even trajectory (T) exists as solution of a problem of central field of unspecified center (but in the concavity of (T), certainly), of force F' (r') different obviously:

F' (r') = F (R). (factor of transmutation)

This factor of transmutation is worth: SG^3/(R. r' ²), where SG is the segment parallel with the vector M, located between S and tangent-in-MR. at the trajectory (T).

Let us point out the historical consequence of 1684 : Hooke => Kepler.

Indeed = O is S centers ellipse and F (R) = - K R (law known as of Hooke);

And is = F hearth of the same ellipse, then SG = cste = has,

the factor of transmutation becomes has ³/(R. r' ²) and thus the central force of Hearth F is - K has ³ /r' ² in 1/r' ². Knocked-out of Hooke , who never would not have had the culture necessary for this geometrical turn-of-force!

## See too

Summary bibliography:
• Densmore, Newton' S Principia, 1995, ED Green Lion Close, ISBN 1-888009-01-2
• Brackenridge, Key to Newton' S dynamics, 1995, U California p, ISBN 0-520-20065-9
• Guicciardini, reading the Principia, 1999, CUP, ISBN 0-521-64066-0
• CORDANI, the Kepler problem, 2003, ED Birkhauser, ISBN 3-7643-6902-7
• Cohen
• Whiteside
• Biarnais
• of Gandt

the Galileo project undertaken on a worldwide scale is an good example of what one can try, if there are the means. Here the company is to show the STRENGTH of the concept wiki.

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