\8
\8
\8
\8<[4 4] 0 setdash >
\8
\8<>
\8
\figfin
\eqfig{152}{100}{
\ins{117}{90}{$\st M $}
\ins{16}{3}{$\st S $}
\ins{49}{3}{$\st C $}
\ins{36}{23}{$\st \th $}
\ins{97}{23}{$\st \ell $}
\ins{78}{3}{$\st S_{equant} $}
\ins{60}{19}{$\st e a$}
\ins{0}{61}{$\st CM=a,\ SC= e a$}
%\ins{106}{62}{$\st Equant\ point\ \to\ S_{equant} $}
}
{e}{e}
%\eq(18)
\*
\0{\it Fig. e}: The equant construction of Ptolemy adapted to a
heliocentric theory of Mars; $S$ is the Sun, $M$ is Mars, $C$ the
center of the orbit and the equant point is $S_{equant}$.
\*
\0this means that the angle $\ell$ in the drawing rotates uniformly
and
%
\begin{equation}
\th=\ell-2e\sin \ell+ e^2 \sin 2\ell+O(e^3)\label{(19)}\end{equation}
%
Truncating the series in (\ref{(17)}) and (\ref{(19)}) to first order
in the eccentricity we obtain (\ref{(16)}) and hence a description in
terms of one deferent, two epicycles and an equant: it is a
description quite accurate of the motion of Mars with respect to the
Fixed Stars Sky and it is the theory that one finds in the {\it
Almagest}, after converting it to the inertial frame of reference
fixed with the Sun.
The motion of the Earth around the Sun (or viceversa if one prefers)
is similar except that the center of the deferent circle is directly
the equant point, see\cite{[Ne69]} p.192, see also\cite{[Ke09]}
Ch.2-4: this is usually quoted by saying the ``for the Earth Ptolemy
(Copernicus and Tycho) did not {\it bisect} the eccentricity'',
meaning that the center and the equant were identical and both $2\,e\,a$
away from the Sun: from\cite{nota-4} we deduce that this did not
matter for the Earth which has a much smaller eccentricity (than
Mars). Before discovering the ellipse Kepler had to redress this
``anomaly'' and he indeed bisected also the Earth eccentricity,
see\cite{nota-4}, making the Copernican Earth lose one more
distinguishing feature with respect to the other planets.\cite{[nota-Co]}
The above, however, {\it is not} the path followed by Kepler,
see\cite{nota-4} where the latter is discussed in some detail.
Thus bringing the development in $e$ to first order one reaches a
level of approximation quite satisfactory for the observations to which
Kepler had access, not only for the Sun but also for the more anomalous
planets like Mercury, Moon and Mars: to second order however the equant
becomes insufficient and Kepler realized that the ellipse had to be
described at constant area velocity with respect to the focus.
We can say that {\it the experimental data agree within a third order
error in the eccentricity with the hypothesis of an elliptical motion
and with a time law based on the area law: this, within a second order
error in the eccentricity, coincides with the Ptolemaic law of the
equant}.
\end{section}
\begin{section}{Modern times}
To realize better the originality of the Newtonian theory we must
observe that in the approximations in which Kepler worked it was
evident that the laws of Kepler were not absolutely valid: the
precession of the lunar node, of the lunar perigee and of the Earth
itself did require, to be explained, new epicycles: in a certain sense
the Keplerian ellipses became ``deferent'' motions that, if run with
the law of the areas, did permit us to avoid the use of equants and of
other Ptolemaic ``tricks''. The theory of Newtonian gravitation
follows after the abstraction made by Newton according to which the
laws of Kepler, manifestly in contrast with certain elementary
astronomical observations unless combined with suitable constructions
of epicycles as Kepler himself realized and applied to the theory of
the Moon,\cite{[St94]} were {\it rigorously exact} in the situation in
which we {\it could neglect the perturbations due to the other
planets}, \ie if we consider the ``two body problem'' originally
reinterpreting the Keplerian conception that the motion of a planet
was due mostly to a force due to the Sun and partly to a force due to
itself.
The theory of gravitation not only predicts that the motions of the
heavenly bodies are quasi periodic, apparently even in the
approximation in which one does not neglect the reciprocal
interactions between the planets, but it {\it gives us the algorithms}
for computing the functions $f(\f_1,\ldots,\f_n)$.
The {\it summa} of Laplace (1749-1827) on the {\it M\'ecanique
cel\`este} of 1799, \cite{[La66]}, makes us see how the description of
the solar system motions, also taking account of the interactions
between the planets, could be made in terms quasi periodic functions.
The Newtonian mechanics allows us to compute approximately the $3N$
coordinates $\V A=(A_1,\ldots,A_{3N})$ and the $3N$ angles
$\f_1,\ldots,\f_{3N}$ and the $3N$ angular velocities $\o_1(\V
A),\ldots,\o_{3N}(\V A)$ in terms of which the motion simply consists
of $3N$ uniform rotations of the $3N$ angles while the $\V A$ remain
constant.
Laplace makes us see that there is an algorithm that allows us to
compute the $A_i,\o_i,\f_i$ by successive approximations in a series
of powers in several parameters (ratios of masses of heavenly bodies,
eccentricities, ratios of the planets radii to their orbits radii \etc),
that will be denoted here with the only symbol $\e$, for simplicity.
After Laplace approximately 80 years elapse during which the technique
and the algorithms for the construction of the {\it heavenly} series
are developed and refined leading to the construction of the formal
structure of analytic mechanics. And Poincar\'e hws able to see
clearly the new phenomenon that marks the first true and definitive
blow to the Greek conception of motion: with a simple proof,
celebrated but somehow little known, he showed that the algorithms
that had obtained so many successes in the astronomy of the 1800's
were in general {\it nonconvergent} algorithms, \cite{[Po87]}.
Few did realize the depth and the revolutionary character of
Poincar\'e's discovery: among them Fermi that tried of deduce from the
method of Poincar\'e the proof of the ergodicity of the motions of
Hamiltonian dynamical systems that were not too special. The proof of
Fermi, very instructive and witty although strictly speaking not
conclusive from a physical viewpoint, remained one of the few attempts
made in the first sixty years of the 1900's by theoretical physicists,
to understand the importance of of Poincar\'e's theorem.
Fermi himself, at the end of his history, came back on the subject to
reach conclusions very different from the ones of his youth (with the
celebrated numerical experiment of Fermi-Pasta-Ulam).\cite{[FPU55]}
And the Greek conception of motion finds one of its last (quite
improbable, {\it a priori}) ``advocates'' in Landau that, still in the
1950's, proposes it as a base of his theory of turbulence in
fluids.\cite{[LL71]} His conception that has been criticized by Ruelle
and Takens (and apparently by others)\cite{[RT71a],[RT71b]} on the
base of the ideas that, at the root, went back to Poincar\'e.
The alternative proposed by them began the modern research on the
theory of the development of the turbulence and the renewed attempts
at the theory of developed turbulence.
The attitude was quite different among the mathematicians who, with
Birkhoff, Hopf, Siegel in particular, started from Poincar\'e to
begin the construction of the corpus that is today called the {\it
theory of chaos}.
But only around the middle of the 1950's it has been possible to solve
the paradox consisting in the dichotomy generated by Poincar\'e:
\*
{(1) } on the one hand the successes of classical astronomy based on
Newtonian mechanics and the perturbation theory of Laplace, Lagrange,
\etc\ seemed to confirm the validity of the quasi periodic conception
of motions (recall for instance Laplace's theory of the World, or
Gauss' ``rediscovery'' of Ceres,\cite{[Ga71]} and the discovery of
Neptune, \cite{[Gr79]}).
{(2) } on the other hand the theorem of Poincar\'e excluded the
convergence of the series used in (1)).
\*
The fundamental new contribution came from
Kol\-mo\-go\-rov,\cite{[Ko54],[Ga84]}: he stressed the existence of
two ways of performing perturbation theory, \cite{[Ga71]}. In the
first way, the classical one, one fixes the initial data and lets them
evolve with the equations of motion. Such equations, in all
applications, depend by several small parameters (ratios of masses,
\etc) denoted above generically by $\e$. And for $\e=0$ the equations
can be solved exactly and explicitly, because they reduce to a
Newtonian problem of two bodies or, in not {\it heavenly} problems, to
other {\it integrable} systems. One then tries to show that the
perturbed motion, with $\e\ne0$, is still quasi periodic, simply by
trying to compute the periodic functions $f$ that should represent the
motion with the given initial data (and the corresponding phases
$\f_i$, angular velocities $\o_i$, and the constants of motion $A_i$)
by means of power series in $\e$. Such series, however, do not
converge or sometimes even contain divergent terms, deprived of
meaning, see\cite{[Ga84]} Sec. 5.10.
A second approach consists in fixing, instead of given initial data
(note that it is in any case illusory to imagine knowing them
exactly), the angular velocities (or {\it frequencies})
$\o_1,\ldots,\o_n$ of the quasi periodic motions that one wants to
find. Then it is often possible to construct by means of power series
in $\e$ the functions $f$ and the variables $\V A,\V \f$, in terms of
which one can represent quasi periodic motions, with the prefixed
frequencies.
In other words, and making an example, we ask the possible question:
given the system Sun, Earth, Jupiter and imagining for simplicity the
Sun fixed and Jupiter on a Keplerian orbit around it, is it or not
possible that {\it in eternity} (or also only up to just a few
billion years) the Earth evolves with a period of rotation around to
Sun of about $1$ year, of revolution around its axis of about $1$ day,
of precession around the heavenly poles of about $25.500$ years,
\etc?
One shall remark that this second type of question is much more similar to the
ones that the Greek astronomers asked themselves when trying to deduce
from the periods of the several motions that animated a heavenly body the
equations of the corresponding quasi periodic motion.
The answer of Kolmogorov is that if $\o_1,\ldots,\o_n$ are the $n$
angular velocities of the motion of which we investigate the existence
it will happen that for the {\it most part} of the choices of the $\o_i$
there actually exists a quasi periodic motion with such frequencies and
its equations can be constructed by means of a power series in $\e$,
{\it convergent} for $\e$ small.\cite{[Ko54]}
The set of the initial data that generate quasi periodic motions has a
complement of measure that tends to zero as $\e\to0$, in every bounded
part of the phase space contained in a region in which the
unperturbed motions are already quasi periodic.
One cannot say, therefore, whether a preassigned initial datum actually
undergoes a quasi periodic motion, but one can say that near it there
are initial data that generate quasi periodic motions. And the closer
the smaller $\e$ is.
By the theorem of continuity of solutions of equations of motion
with respect to variations of initial data it follows that every
motion can be simulated, for a long time as $\e\to0$, by a quasi
periodic motion.
But obviously there remains the problem:
\*
{(1) } are there, really, initial data which follow motions that, in
the long run, reveal themselves to be not quasi periodic?
{(2) } if yes, is it possible that in the long run the motion of a system
differs substantially from that of the (abundant) quasi periodic motions
that develop starting with initial data near it?
\*
The answer to these questions is affirmative: in many systems motions
that are not quasi periodic do exist and become easily visible as $\e$
increases. Since electronic computers became easily accessible it is
easy for everybody to observe personally on computer screens the very
complex drawings generated by such motions (as {\it seen} by Poincar\'e,
Birkhoff, Hopf \etc, {\it without} using a computer).
Furthermore quasi periodic motions although being, at least for $\e$
small, very common and almost dense in phase space probably do not
constitute an obstacle to the fact that the not quasi periodic motions
evolve very far, in the long run, from the points visited by the quasi
periodic motions to which the initial data were close. This is the
phenomenon of {\it Arnold's diffusion} of which there exist quite a
few examples: it is a phenomenon of wide interest. For example, if in
the theory of the solar system diffusion was possible it would be
conceivable the occurrence of important variations of quantities such as
the radii of the orbits of the planets, with obvious (dramatic)
consequences on the stability of the solar system.
In this last question the true problem is the evaluation of the time
scale on which the diffusion in phase space could be observable. In
systems simpler than the solar system (to which, strictly speaking,
Kolmogorov's theorem does not directly apply, for some reasons that we
shall not attempt to analyze here,\cite{[Ga84]} Sec. 5.10) one thinks
that a sudden transition, as the intensity $\e$ of the perturbation
increases, is possible from a regime in which the diffusion times are
super astronomical in correspondence of the interesting values of the
parameters (\ie times of several orders of magnitude larger than the
age of the Universe) to a regime in which such times become so short
to be observable on human scales. This is one of the central themes
of the present day research on the subject, \cite{[MM87]}.
\end{section}
\begin{references}
\bibitem{nota1} This is, in part, a translation of the text of a
conference at the University of Roma given around 1989, circulated in
the form of a preprint since. The Italian text, not intended for
publication, has circulated widely and I still receive requests of
copies. I decided to translate it into English and make it available
more widely also because I finally went into more detail in the part
about Kepler, that I considered quite superficial as presented in the
original text. Therefore this preprint differs from the previous,
see\cite{[Ga89]}, mainly (but not only) for the long new part (in
footnote\cite{nota-4}) about {\it Astronomia nova}, which might be of
independent interest.
\bibitem{[Ga89]} The original Italian text can be found and freely
downloaded from {\tt http://ipparco.roma1.infn.it}, at the page
``$\le1994$''.
\bibitem{[Dr53]} A general history of astronomy is in:
J. Dreyer,: {\it A history of astronomy}, Dover, 1953.
\bibitem{[Ne69]} For a simple introduction to the Ptolemaic system
see: Neugebauer, O.: {\it The exact sciences in antiquity}, Dover,
1969. The figures of the text are taken from this volume: see
p. 193--197.
\bibitem{[Ne75]} A critical and commented version of Ptolemy's theory,
both of the {\it Almagest} and of the {\it Planetary Hypothesis}, is
in: Neugebauer, O.: {\it A history of ancient mathematical astronomy},
part 2, Springer--Verlag, 1975.
\bibitem{[Pt84]} A recent edition of the {\it Almagest}, with comment, is:
{\it Ptolemy's Almagest}, edited by G. Toomer, Springer Verlag, 1984.
\bibitem{[Ne79]} Some critiques to Ptolemy are in: R. Newton: {\it The
crime of Claudius Ptolemy}, John's Hopkins Univ. Press, 1979.
\bibitem{nota1-1} \ie no linear combination of them with rational
coefficients can vanish unless all coefficients vanish.
\bibitem{[NS84]} Recent is the volume on Copernicus: O. Neugebauer,
N. Swerdlow: {\it Mathematical astronomy in Copernicus' de
revolutionibus}, vol. 1,2, Springer Verlag, 1984.
\bibitem{[Sc26]} The interpretation that the Fourier transform (and
(\ref{(13)})) has in terms of deferent and epicyclical motions has
been noted by many; the more ancient that I could retrieve is in a
memory of 1874 of G. Schiaparelli, reprinted in G. Schiaparelli, {\it
Scritti sulla storia dell' astronomia antica}, part I, tomo II, {\it
Le sfere omocentriche di Eudosso, di Callippo e di Aristotele},
p. 11, Zanichelli, Bologna, 1926. It should be stressed that the above
``reduction'' of a quasi periodic motion to an epicyclic series is {\it
not unique} and other paths can be followed: this will be very clear
by the examples below\cite{nota-4}.
\bibitem{[Co30]} The work of Copernicus and Newton can be easily found
in English as there are plenty of reprints; in Italian I quote the
collection printed by UTET directed by L. Geymonat: N. Copernicus,
{\it Opere}, ed. F. Barone, UTET, Torino, 1979. We find here in
particular the so called {\it Commentariolus} that presents the plan
of the Copernican work, as optimistically viewed by the young (and
perhaps still naive) Copernicus himself before he really confronted
himself with a work of the dimensions of the {\it Almagest}. Great
were the difficulties that he then met, while dedicating the rest of
his life to a complete realization of the program sketched in the {\it
Commentariolus} ($\sim$1530), so that the {\it De revolutionibus}
presents solutions quite more elaborate than those programmed in the
quoted work. Nevertheless the copernican revolution appears already
clearly from this brief and illuminating work: here one finds the
passage quoted in the text (p.108 of the Italian edition).
\bibitem{[St69]} And for a detailed treatment, also of the notion of
average motion, see in particular (D. Boccaletti pointed out to me
this bibliographic note, together with the preceding
one\cite{[Sc26]}): S. Sternberg, {\it Celestial mechanics}, vol. 1,
Benjamin, 1969.
\bibitem{[AA68] } Arnold, V.I., Avez, A.: {\sl Ergodic problems
of classical mechanics}, Benjamin, New York, 1968.
\bibitem{[He81]} It is interesting, not only for the personality of
Aristarchus of Samos, to read the volume: T. Heath: {\it Aristarchus
of Samos. The ancient Copernicus}, Dover, 1981.
\bibitem{[He91]} Heath, T.L.: {\it Greek Astronomy}, Dover, 1991. This
contains a very illuminating collection of translations of fragments
from greek originals.
\bibitem{nota3a} Neugebauer assesses very lucidly Copernicus'
contribution: see\cite{[Ne69]} p. 205.
\bibitem{nota3} See Ptolemy, C.: {\it The Almagest}, ed. G.J. Toomer,
Springer Verlag, New York, 1984. See also Theon, {\it Commentaires de
Pappus et de Th\'eon d'Alexandrie sur l'Almag\`este}, T. II,
ed. annot\'ee par A. Rome, Biblioteca Apostolica Vaticana, Citt\'a del
Vaticano, 1936. I often wonder whether it is possible that this
passage has been contaminated by later commentators. Although
certainly not much later, because it is already commented by Theon of
Alexandria in the second half of the fourth century: however two
centuries is a {\it very} long time for Science (if one thinks to what
happened since Laplace). In a way this and the argument that follows
it is much too rough compared to the level of the rest of the
Almagest. Nevertheless if one attributes, as it seems right to
do,\cite{[Ne75]} p. 900, the {\it Planetary Hypotheses} books to
Ptolemy, then one is led to think that the passage is indeed
original. This is perhaps also proved by the fact that Ptolemy does
not seem to realize that the heliocentric hypothesis would have
allowed a clear determination of the average radii of the orbits,
missing in his work. In turn this makes us wonder which exactly was
the famous heliocentric hypothesis of Aristarchus and if it went
beyond a mere qualitative change of coordinates. Had it been the same
as Copernicus' he could have determined the sizes of the
orbits:\cite{nota3a} a problem in which he had a strong interest as he
dedicated a book\cite{[He81],[He91]} to his determination of the
distance of the Moon to the Earth, and this should have been reported
by Ptolemy. From the extant information about Aristarchus' theory
there is, strangely, no trace of an application of the heliocentric
system to planets other than the Earth and the Sun, see\cite{[He81]}
p. 299 and following, although it would be surprising that there was
none. For a critical account of the {\it Planetary Hypotheses}
see\cite{[Ne75]}.
\bibitem{nota2} Note that, from Newtonian mechanics and as discussed
below, the motions of the $8$ classical planets (the Fixed Stars Sky,
Sun, Mercury, Venus, Moon, Mars, Jupiter, Saturn not counting the
Earth (whose rigid motions are described by those of the Fixed Stars),
or alternatively not counting the Sun and the Fixed Stars but
regarding the Earth as having $6$ degrees of freedom, requires a
maximum of $24=3\times 8$ independent ``fundamental'' frequencies
namely three for each planet: so that both in Ptolemy and in
Copernicus there must be epicycles rotating at speeds multiple of the
fundamental frequencies or at least at speeds which are linear
combinations with integer coefficients of the fundamental frequencies:
hence the $43$ frequencies cannot be rationally independent of each
other; see for instance the figure on Copernicus' Moon and note that
the ``number of epicycles'' in Ptolemy's theory could be counted
differently.
\bibitem{[AA89]} {\it The astronomical almanac for the year 1989},
issued by the National Almanac Office, US government printing office,
1989.
\bibitem{[Ke09]} Kepler, J.: {\it Astronomia nova}, reprinted french
translation by J. Peyroux, Blanchard, Paris, 1979.
\bibitem{[St94]} Stephenson, B.: {\sl Kepler's physical astronomy},
Princeton University Press, 1994.
\bibitem{[nota-Co]} It is interesting to compare in detail the theory
of Mars of Copernicus and that of Ptolemy (reduced to a heliocentric
one). The first has a deferent of radius $a$ on which a first epicycle
of radius $\frac32 e a$ counter--rotates at equal speed and, {\it on
it}, a second epicycle rotates at twice the speed; the starting
configuration being the first epicycle at aphelion and the second
opposite to the aphelion of the first. In other words the position $z_C$
from the aphelion is, at average anomaly $\ell$ given by $z_C$
$$z_C=a e^{i\ell} +\frac32 a e-\frac12 e a e^{2i\ell}=$$
$$=e a + a e^{i\ell}\,(1-i e\sin\ell)$$
%
Ptolemy has the planet on an eccentric circle, centered $e a$ away
from the Sun, whose center rotates at constant speed around the equant
point which, in turn, is $ea$ further away from the center of the
orbit. Hence if $\x$ is the eccentric anomaly (\ie the longitudinal
position of the planet on the orbit as seen from the center) it is
$$z_T=e a + a e^{i\x}$$
%
and from Fig. e we see that the relation between the average anomaly
$\ell$ (\ie the longitudinal position of the planet as seen from the
equant point) is related to $\x$ by
$$\sin(\ell-\x)=e\,\sin\ell\ \to\ \x=\ell-e\sin\ell+O(e^3)$$
%
so that
$$z_T=ea+ a e^{i\ell}(1-i e \sin\ell-\frac12 e^2\sin^2\ell)+O(e^3)$$
%
and we see that the longitudinal difference (\ie the difference of the
true anomalies or longitudes from the Sun po\-si\-tion $S$) is ${\rm
arg}\,\frac{z_T}{z_C}$ or
$${\rm arg}\, \Big(1-\frac12\frac{e^2\sin^2\ell}{e\, e^{-i\ell}+1-i e
\sin\ell}\Big)=O(e^3\sin^3\ell)$$
%
However the difference in distance is $|z_C|-|z_T|$ of the order
$O(\frac12 e^2\sin^2\ell)$ so that Copernicus' epicycles are
equivalent to Ptolemy's equant within $O(e^3)$, or about $4'$, in
longitude measurements and within $O(e^2)$, or about $40'$ in distance
measurements (\ie to match distances at quadrature, say, one should
alter by about $40'$ the average anomaly, which means to delay the
observations by about one day since Mars period is about $2$ years
(provided the distances could be measured accurately enough, which was
not the case at Ptolemy's time).
Before Tycho (relative) differences of $O(e^2)$ could not be
appreciated experimentally: but their existence was derived from the
theories, and used, by Kepler who discussed them at the beginning of
his book in Ch.4, where the above calculation is performed for
$\ell=\frac\pi2$, where the discrepancy is maximal, see\cite{[Ke09]}
p. 23 (or p. 16 in the original edition).
Kepler's theory differs from both to order $O(e^2)$. He first derived
a better theory for the longitudinal observations (which turned out
eventually to agree with the complete theory already to $O(e^3)$) and
used it to find the ``correct'' theory agreeing within $O(e^3)$ with
the data for the distance measurements (that had become possible,
see\cite{nota3a}, after Copernicus).
\bibitem{nota-4} An account of Kepler's approach, in \cite{[Ke09]}, is
indeed as follows, see\cite{[St94]} for a very careful and detailed
analysis of Kepler's discoveries from where I derive most of what
follows. A key point to keep in mind in this footnote is that the
resolution of the observations available to Ptolemy (and Copernicus)
was of the order of $10'$ so that errors were in the order of tens of
primes: this meant that one could observe first order corrections in
the eccentricity of Mars but the second order corrections (of order
$e^2\simeq 10^{-2}$ or about $30'$) were barely non observable (the
ensuing difficulties in interpreting the data earned Mars the name of
{\it inobservabile sidus}, unobservable star, after Plinius). However
the observations of Tycho were of the order of a few primes so that
second order corrections were clearly observable, see\cite{[Dr53]}
p. 385, because the third order amounts to about $3'$.
Another major point to keep in mind is, as clearly stressed
in\cite{[St94]}, that Kepler was the first to have (perhaps since Greek
times) a {\it physical theory} to check: his language is not the one we
have become accustomed to after Newton, but he had very precise laws in
mind which he kept following very faithfully until the end of his
work. The main one, for our purposes, was the (vituperated) law that the
``speed of the motion due to the Sun is inversely proportional to the
distance to the Sun'', see below.
After ascertaining that the Earth and Mars orbits lie on planes
through the Sun (rather than through the mean Sun, as in Copernicus
and Tycho) he tried to ``imitate the ancients'' assigning to Mars an
orbit, on an eccentric circle that I will call the {\it deferent}, and
an equant: he noted that a very good approximation of the longitudes
followed if one {\it abandoned} Ptolemy's theory of the center $C$ of
the orbit being half way between the Sun $S$ and the equant $E$. His
equant was set, to save the phenomena, a little closer to the center
(with respect to the Ptolemaic equant point) at distance $e' a$ from
the center $C$ of the deferent rather than at distance $e a$. If $z$
denotes the position with respect to the center $C$ in the plane of
the orbit with $x$--axis along the apsidal line of Mars and $\z$
denotes the position with respect to the Sun $S$ (eccentric by $e a$
away from $C$) and if $\x$ denotes the position on the deferent of the
planet, called the {\it eccentric anomaly}, and if $\o$ is the angular
velocity with respect to the equant point $E$ this means (using the
complex numbers notation of Sec. 1, Eq. \ref{(5)} with $x$--axis along
the apsides line perihelion--aphelion)
$$z=a e^{i\x},\ \x=\ell-{\rm arcsin\,}
e'\sin\ell=\ell-e'\sin\ell+O(e^{\prime3}),$$
$$\ell=\o t$$
$$\z=e a +a e^{i\x},\ |\z|=a\, ((\e+\cos\x)^2+\sin^2\x)^{\frac12}=$$
$$=a\,(1+e^2+2 e\cos \x)^{\frac12}$$
which was called the {\it vicarious hypothesis}, illustrated in Fig. k1
\figini{k1}
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