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\DOI{10.5802/crphys.288}
\datereceived{2026-05-12}
\daterevised{2026-08-19}
\dateaccepted{2026-08-21}
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\section*{Declaration of interests}
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\COI{The authors do not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
have declared no affiliation other than their research organizations.}

\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{history-of-sciences}

\title{On the anticipation of lunar travel in the early 20th century: a
pedagogical exercise}

\dedicatory{\raggedleft Ch\'erif Matta dedicates this article to the
memory of his parents, Farid A. Matta and Nabila N. Abdel-Nour, whose
well-stocked family home library in Alexandria (Egypt) introduced him,
in his childhood, to the beautiful book of Alphonse Berget, \textit{Le
Ciel}. This book has profoundly influenced his lifelong trajectory in
science and interest in astronomy.\break\vspace*{1pc}}

%\dedicatory{Ch\'erif Matta d\'edie cet article \`a la m\'emoire de ses parents,
%Farid A. Matta et Nabila N. Abdel-Nour, dont la riche biblioth\`eque
%familiale, dans leur maison d'Alexandrie, lui fit d\'ecouvrir, durant
%son enfance, le magnifique ouvrage d'Alphonse Berget, \textit{Le Ciel}.
%Ce livre a profond\'ement influenc\'e le parcours scientifique de CFM
%tout au long de sa vie ainsi que son int\'er\^et pour lastronomie.}

\alttitle{Sur l'anticipation du voyage lunaire au d\'{e}but du XX\textsuperscript{e}
si\`{e}cle : un exercice p\'{e}dagogique}

\author{\firstname{Tina Ann} \lastname{Harriott}\CDRorcid{0000-0003-3469-8980}\IsCorresp}
\address{Department of Mathematics and Statistics, Mount Saint Vincent
University, Halifax, Nova Scotia, B3M 2J6, Canada}
\email[T. A. Harriott]{Tina.Harriott@msvu.ca}

\author{\firstname{Ch\'erif Farid} \lastname{Matta}\CDRorcid{0000-0001-8397-5353}\IsCorresp}
\address{Department of Chemistry and Physics,  Mount Saint Vincent
University, Halifax, Nova Scotia, B3M 2J6, Canada}
\email[C. F. Matta]{cherif.matta@msvu.ca}

\keywords{\kwd{Robert Esnault-Pelterie}\kwd{Alphonse
Berget}\kwd{\textit{Le Ciel} (book)}\kwd{History of
astronautics}\kwd{Lunar travel}\kwd{Interplanetary travel}\kwd{Pedagogy
of astrophysics}}

\altkeywords{\kwd{Robert Esnault-Pelterie}\kwd{Alphonse
Berget}\kwd{\textit{Le Ciel} (ouvrage)}\kwd{Histoire de
l'astronautique}\kwd{Voyage vers la Lune}\kwd{Voyages
interplan\'{e}taires}\kwd{P\'{e}dagogie de l'astrophysique}}

\begin{abstract}
This article examines, from historical and pedagogical perspectives,
Alphonse Berget's anticipation of Earth--Moon travel in \textit{Le Ciel}
(Larousse, 1923), decades before the beginning of the space age. The
discussion is triggered by \textit{Le Ciel}, a richly illustrated
French popular science work, which has a devoted chapter examining
lunar and interplanetary travel within Newtonian physics. Although
Berget's treatment was not developed in isolation and reflects a
broader early 20th century context that included pioneers such as
French aero-engineer Robert Esnault-Pelterie, the book provides a
striking pedagogical synthesis of elementary celestial mechanics and
scientific popularization.

Unlike earlier fictional treatments such as Jules Verne's \textit{De la
Terre \`{a} la Lune}, Berget approached space travel using physical
reasoning grounded in Newtonian gravitation. Using qualitative and
semi-quantitative arguments based on the inverse-square law, he
identified the principal phases of an Earth--Moon trajectory: escape
from Earth, inertial translunar motion, transition through competing
Earth--Moon gravitational fields, and final lunar capture and
deceleration. His estimated Earth--Moon travel time of approximately 49
h is of the same order of magnitude as Apollo missions transit times
(\textasciitilde{72 h}).

We compare these early ideas with modern elementary concepts of
astrodynamics and with the trajectories of the Apollo and Artemis
programs. We also examine Berget's discussion of interplanetary travel,
lunar landscapes, and human factors associated with prolonged voyages,
including confinement, food supply, and travel duration. The analysis
highlights the pedagogical value of historically grounded scientific
reasoning underpinning spaceflight mechanics.
\end{abstract}

\begin{altabstract}
Cet article examine, dans une perspective historique et
p\'{e}dagogique, l'anticipation par Alphonse Berget du voyage
Terre--Lune dans \textit{Le Ciel} (Larousse, 1923), plusieurs
d\'{e}cennies avant le d\'{e}but de l'\`{e}re spatiale. La
r\'{e}flexion prend pour point de d\'{e}part \textit{Le Ciel}, ouvrage
fran\c{c}ais de vulgarisation scientifique richement illustr\'{e}, qui
consacre un chapitre \`{a} l'\'{e}tude des voyages lunaires et
interplan\'{e}taires dans un cadre newtonien. Bien que l'approche de
Berget ne se soit pas d\'{e}velopp\'{e}e de mani\`{e}re isol\'{e}e et
s'inscrive dans un contexte plus large du d\'{e}but du XX\textsuperscript{e} si\`{e}cle,
auquel appartenaient des pionniers tels que l'ing\'{e}nieur
a\'{e}ronautique fran\c{c}ais Robert Esnault-Pelterie, l'ouvrage offre
une remarquable synth\`{e}se p\'{e}dagogique de m\'{e}canique
c\'{e}leste \'{e}l\'{e}mentaire et de vulgarisation scientifique.

Contrairement \`{a} des traitements fictionnels ant\'{e}rieurs, tels
que \textit{De la Terre \`{a} la Lune} de Jules Verne, Berget aborde le
voyage spatial au moyen d'un raisonnement physique fond\'{e} sur la
gravitation newtonienne. \`{A} partir d'arguments qualitatifs et
semi-quantitatifs reposant sur la loi de l'inverse du carr\'{e} de la
distance, il distingue les principales phases d'une trajectoire
Terre--Lune : \'{e}chappement \`{a} l'attraction terrestre, mouvement
translunaire inertiel, transition \`{a} travers des champs
gravitationnels terrestre et lunaire concurrents, puis capture lunaire
et d\'{e}c\'{e}l\'{e}ration finale. Son estimation d'une dur\'{e}e
d'environ 49 h pour le voyage Terre--Lune est du m\^{e}me ordre de
grandeur que les temps de transit des missions Apollo (\textasciitilde{72 h}).

Nous comparons ces premi\`{e}res conceptions aux notions
\'{e}l\'{e}mentaires de l'astrodynamique moderne ainsi qu'aux
trajectoires des programmes Apollo et Artemis. Nous examinons
\'{e}galement les r\'{e}flexions de Berget sur les voyages
interplan\'{e}taires, les paysages lunaires et les facteurs humains
associ\'{e}s aux voyages de longue dur\'{e}e, notamment le confinement,
l'approvisionnement alimentaire et la dur\'{e}e du voyage. Cette
analyse met en \'{e}vidence la valeur p\'{e}dagogique d'un raisonnement
scientifique historiquement ancr\'{e} pour l'enseignement des principes
physiques qui sous-tendent la m\'{e}canique du vol spatial.
\end{altabstract}

\thanks{Natural Sciences and Engineering Council of Canada (NSERC),
Canadian Foundation for Innovation (CFI), Digital Research Alliance of
Canada, Mount Saint Vincent University}

%\input{CR-pagedemetas}

\maketitle

\end{noXML}

\dedication{Ch\'erif Matta dedicates this article to the
memory of his parents, Farid A. Matta and Nabila N. Abdel-Nour, whose
well-stocked family home library in Alexandria (Egypt) introduced him,
in his childhood, to the beautiful book of Alphonse Berget, \textit{Le
Ciel}. This book has profoundly influenced his lifelong trajectory in
science and interest in astronomy.}

\section{When physics precedes technology}\label{sec1}

There are rare moments in the history of science when the conceptual
structure of a technological achievement is laid out long before its
realization. 

Konstantin Tsiolkovsky (1857--1935) is often called the grandfather of
rocketry. He was a Russian schoolteacher who in a 1903 work titled
\textit{Exploration of Outer Space by Means of Rocket Devices} included
the now so called ``rocket equation'', proposed liquid propellants, and
suggested multi-stage rockets~\cite{1}. Indeed, 
56 years later and 24 years after Tsiolkovsky's death, 
the Soviet Union (USSR) was the first
country to successfully launch a spacecraft (Luna~1) that escaped
Earth's gravity, on January 2, 1959~\cite{2}.

The case of lunar travel is often associated with 20th century pioneers
such as Robert Goddard, who is credited with creating and building the
world's first liquid fueled rocket, successfully launched in
1926~\cite{3,4} and Wernher von Braun, who headed the team that
developed the Saturn V rocket in the 1960's that ultimately carried the
Apollo astronauts on their lunar missions~\cite{5}. These advances in
rocketry along with the development of digital computers, and use of
orbital mechanics led to the realization of human travel to the Moon,
yet its underpinnings are entirely contained within classical Newtonian
physics.

A notable popularizing review of the state of knowledge of the time by
Alphonse Berget in his 1923 beautifully illustrated monograph
``\textit{Le Ciel} (The Sky)''~\cite{6}, published by Larousse in Paris
(Figure~\ref{fig1}), shows that the idea of travel to the Moon is
already discussed in a qualitative but physically sound manner almost
half a century before humans set foot on the Moon in 1969.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Portrait of Alphonse Berget (left), together with
the cover and title page of his book \textit{Le Ciel} (centre and
right)~\cite{6}. Berget (1860--1933) was a French science writer and
educator associated with the Larousse publishing house, known for
synthesizing contemporary astronomical (and broader scientific)
knowledge for a broad audience. Trained in the sciences and active in
early 20th-century scientific communication, he combined rigorous
physical reasoning with pedagogical beautifully illustrated exposition.
In \textit{Le Ciel} (1923), he uses Newtonian dynamics to describe the
conditions and phases of a lunar travel decades before their
technological realization. (Public domain).}
\end{figure}

Berget credits his compatriot Robert Esnault-Pelterie (1881--1957),
Figure~\ref{fig2}, and uses his predictions, not in the speculative manner
of fiction but by using solid mathematical and physical arguments. His
discussion begins with the most elementary principle, and that is, the
inverse square law of gravity. From this, he constructs, not formally,
but unmistakably, the phases and milestones of a lunar voyage from
Earth with humans inside what he calls a ``projectile''. To estimate
the total time of the Earth--Moon trip, Berget suggests~\cite{6}:{\pagebreak}

\begin{quote}
La dur\'{e}e totale du voyage peut se faire par addition des
dur\'{e}es des trois phases {\ldots}

\smallskip\noindent
\textit{The total duration of the journey can be obtained by adding
the durations of the three phases {\ldots}} 
\end{quote}
This is not merely descriptive, but rather, it is the \textit{first step of a
dynamical decomposition} implying that motion must pass through
qualitatively distinct regimes. That insight alone places Berget beyond
the speculative imagination and within the domain of sound physical
reasoning. For clarity, three distinct levels of discussion are used
throughout this article. The first is \textit{historical}: what Berget
explicitly wrote in 1923. The second is \textit{interpretative}: how
the physical reasoning contained in that text can be expressed using
modern elementary mechanics. The third is \textit{pedagogical}: how the
comparison can be used today to introduce concepts of celestial
mechanics. Modern terminology and equations used below are therefore
analytical tools applied to Berget's text and should not be understood
as concepts or formalisms explicitly formulated by Berget himself.
\looseness=-1

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Robert Esnault-Pelterie (1881--1957) (left) and an
early aircraft design associated with his work (right).
Esnault-Pelterie was a pioneering French aviator and engineer who made
significant contributions to early aviation and was among the first to
develop a theory for space travel, including
independent derivations of the rocket equation in the 1910s. His work
predates and informs later discussions of astronautics in the early
20th Century. Image source: Wikipedia (public domain/Wikimedia
Commons).}
\end{figure}

\section{From the fiction of Jules Verne to the dynamics of Alphonse
Berget}\label{sec2}

A little over a century before the Apollo lunar missions, Jules Verne,
in 1865, published the fiction \textit{De la Terre \`{a} la Lune} (From
the Earth to the Moon)~\cite{7}---Figure~\ref{fig3}. Berget summarizes
the thesis of Jules Verne in a section that he starts (on p. 180) by
stating~\cite{6}:

\begin{quote}
Une premi\`{e}re m\'{e}thode a \'{e}t\'{e} rendue populaire par le
prestigieux roman de Jules Verne, \textit{De la Terre \`{a} la Lune}.

\smallskip\noindent
\textit{A first method was popularized by Jules Verne's celebrated
novel} From the Earth to the Moon.
\end{quote}

Verne's novel is celebrated for its pre-science as it is anchored in a
pre-dynamical concept whereby the journey is reduced to a single act,
that of the launch. In this context, in Chapter VII of the novel, Verne
writes:

\begin{quote}
{\ldots} il r\'{e}sulte de mes calculs indiscutables que tout
projectile dou\'{e} d'une vitesse initiale de douze mille yards par
seconde, et dirig\'{e} vers la Lune, arrivera n\'{e}cessairement
jusqu'\`{a} elle.

\smallskip\noindent
\textit{{\ldots} it results from my indisputable calculations that
any projectile endowed with an initial velocity of twelve thousand
yards per second, and directed towards the Moon, will necessarily reach
it.}
\end{quote}
Berget summarizes Jules Verne's method of lunar travel as the launch of
a projectile from Earth with sufficiently high initial velocity to
carry passengers toward the Moon. The journey is conceived essentially
as a ballistic motion initiated by a single powerful impulse, after
which the projectile continues through space largely under its acquired
momentum. Berget recognizes the imaginative power of this idea but
implicitly contrasts it with a more physically grounded treatment based
on the continuous action of gravitational forces throughout the voyage.
While Verne emphasizes the launch itself, Berget shifts the focus
toward the successive dynamical regimes governing the entire
Earth--Moon trajectory.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Illustrations from 1865 novel \textit{De la Terre
\`{a} la Lune} (From the Earth to the Moon) and its sequel by Jules
Verne~\cite{7}. Left: preparation and emplacement of the projectile at
Stone's Hill, launched by cannon toward the Moon. Centre: the
projectile in flight toward the lunar surface. Right: imagined arrival
and exploration on the Moon. Verne's depiction assumes an impulsive
launch with large initial velocity and treats the trajectory as
essentially ballistic, without a continuous dynamical analysis of
motion under gravitational forces. These images illustrate an early
conceptual stage of lunar travel, preceding the physically grounded
treatment developed and summarized nicely by Berget. (Public domain).}
\end{figure}

Jules Verne's approach is one totally determined by the initial
velocity ($v_{0}$) condition:
{\begin{equation}\label{eq1}
v(t = 0) = v_{0},
\end{equation}}\unskip
after which the motion is largely ``taken for granted''. There is no
analysis of the governing equation:
{\begin{equation}\label{eq2}
\frac{\mathrm{d}^{2}r}{\mathrm{d}t^{2}} = -\dfrac{GM_{\oplus}}{r^{2}}
\end{equation}}\unskip
where $M_{\oplus}$ is the mass of the Earth, $r$ is the (changing)
separation of the rocket and the center of the Earth and $G$ the
universal constant of gravitation ($G=6.67 \times 10^{-11}$
m\tsup{3}${\cdot}$kg$^{-1}{\cdot}$s$^{-2}$).
Neither is there any discussion of the growing influence of the Moon's
gravity as it is approached. In Verne, gravity is present but not
structurally organizing. In Berget--Esnault-Pelterie, it becomes the
central driving force.

In \textit{Le Ciel} the idea of a single decisive impulse is replaced
with a \textit{continuous evolution under force}, reflecting the
prevalent view in the 1920's. This is a conceptual shift of great
importance. It transforms lunar travel from a single event into an
evolving phased process.

\section{Escape from Earth: the energy barrier}
\label{sec3}

The first physical obstacle for travel from the Earth to the Moon is
the gravitational attraction of Earth on the object undergoing the
travel. The total energy of a body of mass $m$ in Earth's
gravitational field, in standard notation, is:
{\begin{equation}\label{eq3}
E = \dfrac{1}{2} mv^{2} - \dfrac{GM_{\oplus}m}{r},
\end{equation}}\unskip
where $m$ is the mass of the object undergoing the travel and
$M_{\oplus}$ is the mass of the Earth. That is, the total energy of
the object is given by the sum of its kinetic energy and its
gravitational potential energy (due to the Earth's gravitational
field).

Escape requires that this energy be non-negative. Setting $E = 0$  at
Earth's surface yields
{\begin{equation}\label{eq4}
v_{\mathrm{esc}} = \sqrt{\frac{2GM_{\oplus}}{R_{\oplus}}},
\end{equation}}\unskip
where the separation at the Earth's surface is $r = R_{\oplus}$, where
$R_{\oplus}$ is the radius of the Earth. Numerically, this translates
into $v_{\mathrm{esc}} \approx 11.2$ km/s.

While Berget does not present this equation explicitly, he explains
that a sufficiently large initial velocity is required to overcome
Earth's attraction. The presence of the concept is what is significant,
and not the absence of the formula. In modern terminology, Berget's
qualitative argument can be interpreted as an energy problem: the
spacecraft must climb out of Earth's gravitational potential well.
Berget does not himself formulate the problem in these terms or write
the corresponding energy equation.

\section{The decay of gravitational influence and the take-over by
inertia}\label{sec4}

Once the spacecraft has departed Earth, the magnitude of the
gravitational force, $F(r)$, between the Earth and the object,
governing its motion decreases rapidly:
{\begin{equation}\label{eq5}
F(r) = \dfrac{GM_{\oplus}m}{r^{2}}.
\end{equation}}\unskip
The ratio
{\begin{equation}\label{eq6}
\dfrac{F(r)}{F(R_{\oplus})} = \left(\dfrac{R_{\oplus}}{r}\right)^{2}
\end{equation}}\unskip
reveals the scale of this decay. At ten Earth radii, the influence of
Earth's gravity is already reduced by two orders of magnitude.

What follows from this is crucial. As the force weakens, acceleration
tends to zero:
{\begin{equation}\label{eq7}
\frac{\mathrm{d}^{2}r}{\mathrm{d}t^{2}}\rightarrow 0,
\end{equation}}\unskip
and thus its velocity $v \rightarrow$ constant. 

Berget does not state this explicitly, but his narrative implies it by
emphasizing that the spacecraft's motion becomes essentially inertial.
This corresponds directly to the translunar coast phase in Apollo
missions. At this phase, the spacecraft is no longer ``driven''
outward; it drifts. This is a profound shift in physical regime, and
Berget captures it without formalism.

\section{The gravitational competition: Earth vs. Moon}
\label{sec5}

The analysis reviewed qualitatively in \textit{Le Ciel} recognizes that
there exists a region where Earth and Moon exert comparable influences.
This is not obvious as it requires understanding that gravitational
forces from different bodies can compete in space.

The condition for equality in the magnitude of these forces (which are
opposite in direction) is that the size of the Earth's gravitational
attraction equals the Moon gravitational attraction, {i.e.},
{\begin{equation}\label{eq8}
\dfrac{GM_{\oplus}}{r_b^{2}} = \dfrac{GM_{\mathrm{Moon}}}{(d-r_b)^{2}},
\end{equation}}\unskip
where $r_b$ is the distance from the center of the Earth to the point
where the magnitudes of the gravitational attractions of Earth and Moon
become equal (the ``balance point''), $d=$\ Earth--Moon distance 
($d\approx 3.84\times 10^{8}$ m), and, clearly, ($d-r_b$) is simply the
distance from this balance point to the center of the Moon. Solving for
$r_b$ we get:
{\begin{equation}\label{eq9}
r_b = \dfrac{d}{1 + \sqrt{\frac{M_{\mathrm{Moon}}}{M_{\oplus}}}}.
\end{equation}}\unskip
Using $M_{\mathrm{Moon}} = 7.3477\times 10^{22}$ kg, and  $M_{\oplus} =
5.9722\times 10^{24}$ kg, one obtains
{\begin{equation}\label{eq10}
\sqrt{\frac{M_{\mathrm{Moon}}}{M_{\oplus}}} \approx 0.11,
\end{equation}}\unskip
and hence $r_b\approx 0.90d$.

Berget argues that there exists a region where the two attractions
nearly compensate one another. In modern language, this provides a
natural pedagogical entry point for discussing competing gravitational
fields. It should not, however, be interpreted as an attribution to
Berget of the later formalism of the restricted three-body problem,
Lagrange points, or Hill spheres.

\vspace*{-2pt}

\section{Capture by the Moon: a second dynamical regime}
\label{sec6}

\vspace*{-2pt}

Beyond the transition region, the Moon becomes dominant. The equation
of motion changes character, and the spacecraft acceleration is now
directed primarily toward the Moon:
{\begin{equation}\label{eq11}
\dfrac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}} = -
\dfrac{GM_{\mathrm{Moon}}}{|\mathbf{r}|^{2}}\widehat{\mathbf{r}},
\end{equation}}\unskip
where $\mathbf{r}$ is the spacecraft position vector relative to the
Moon, and $\hat{\mathbf{r}}$ is the radial unit vector directed away
from the Moon.

The spacecraft is now in a new gravitational potential well. To remain
bound to the Moon, its mechanical energy relative to the Moon must
satisfy
{\begin{equation}\label{eq12}
E_{\mathrm{Spaceccraft}} = \dfrac{1}{2}mv^{2} - 
\dfrac{GM_{\mathrm{Moon}}m}{r} < 0,
\end{equation}}\unskip
where $r$ is the spacecraft distance from the Moon and $v$ its total
speed relative to the Moon (the magnitude of the spacecraft velocity
vector $\mathbf{v}$ in the Moon-centered frame). This condition implies
that the spacecraft velocity must be sufficiently reduced upon
approaching the Moon in order for the total mechanical energy to become
negative. In practice, this requires a deceleration maneuver to prevent
the spacecraft from simply passing by the Moon on an unbound
trajectory.

Berget notes qualitatively that the velocity would need to be adjusted
near the Moon. In modern astronautics, this corresponds to the lunar
orbit insertion maneuver.

\vspace*{-2pt}

\section{A qualitative comparison with the Apollo missions: 
similarities and limits}\label{sec7}

\vspace*{-2pt}

Apollo missions provide a qualitative benchmark for assessing the
staged structure of the lunar voyage described in \textit{Le Ciel},
while also revealing the many elements absent from Berget's
popularizing review. According to the author, the voyage begins with
the \textit{initial powered phase}, during which the projectile is
accelerated to a very high velocity of approximately 11,800~m/s, a
value close to the modern escape velocity of ${\approx}$11,200~m/s
given by Equation~(\ref{eq4}).  This phase is short in duration,
lasting, according to the author, only {ca}.\ 24~min, corresponding to
overcoming Earth's gravitational attraction.

The motion then enters a \textit{long inertial (coasting) phase},
during which the projectile travels under the combined influence of
Earth and Moon gravity. Berget notes that the velocity gradually
decreases to a minimum near the region where the attractions of the two
bodies balance, after which it increases again as the projectile falls
toward the Moon. This phase dominates the total travel time and is
estimated to last about 48.5~h.

Finally, Berget introduces a \textit{third phase of deceleration
(``freinage'' (braking))}, required to ensure that the projectile
reaches the lunar surface with a manageable velocity. Without this
braking, the impact velocity would remain high. He estimates a short
braking interval of a few minutes, during which the speed is reduced
sufficiently for landing.

A particularly instructive passage in Chapter XIII of \textit{Le Ciel}
follows the decomposition of the Earth--Moon journey into the
above-mentioned phases the prelude of which reads (p.~183) \cite{6}:

\begin{quote}
La dur\'{e}e totale du voyage peut se faire par addition des
dur\'{e}es des trois phases {\ldots} 

\smallskip\noindent
\textit{The total duration of the journey can be obtained by adding
the durations of the three phases {\ldots}}
\end{quote}
followed immediately by the following estimates~\cite{6}:
{\begin{eqnarray*}
\mbox{Premi\`{e}re phase}/\mathit{First~phase:}  && \mbox{0 h 24 m 9 s} \\
\mbox{Deuxi\`{e}me phase}/\mathit{Second~phase:} && \mbox{48 h 30 m} \\
\mbox{Troisi\`{e}me phase}/\mathit{Third~phase:} && \mbox{0  h 3  m 46 s} \\
\mbox{Dur\'{e}e totale}/\mathit{Total~duration:} && \mbox{48 h 58 m}~(\mbox{environ}/\mathit{approximately})
\end{eqnarray*}}\unskip

The above tabulation explicitly identifies three regimes (acceleration,
inertial motion under gravity, and terminal deceleration) that
correspond precisely to the modern phases of translunar flight.
Furthermore, the total duration is of the correct order of magnitude,
despite the absence of any formal orbital mechanics. Accordingly, the
total duration of the journey, obtained by summing these three phases,
is of roughly 48 h 58 min ($t_{\mathrm{Berget}}\approx 49$ h),
remarkably close in order of magnitude to modern translunar flight
times ($t_{\mathrm{Apollo}}\approx 72$ h) with an error of only ${-}$32\%.

The Apollo missions provided the first empirical realization of lunar
travel. Their trajectories followed a canonical sequence consisting of
launch into Earth orbit, translunar injection, translunar coast, lunar
orbit insertion, descent, and landing~\cite{8,9,10}.

This sequence corresponds approximately to the phases described in
\textit{Le Ciel}. The primary difference lies in the mathematical
description. Apollo trajectories are governed by orbital mechanics, in
which the motion is described by conic sections and characterized by
quantities such as the semi-major axis $a$, of an ellipse, with the
total mechanical energy, $E$, of the spacecraft of mass, $m$, given
by~\cite{11,12}:
{\begin{equation}\label{eq13}
E = -\dfrac{GMm}{2a},
\end{equation}}\unskip
where $M$ is the mass of the central body (Earth or Moon). The equation
is that of the well-known bound Keplerian orbit and obtained from the
sum of kinetic and gravitational potential energies averaged over the
orbit and satisfying the virial theorem (which states that potential
energy is equal to twice the negative of the kinetic energy averaged
over the elliptical orbit's period). 

For typical spacecraft of mass $m$, $m\ll M$, so the motion, to a very
good approximation may be treated as that of a test particle orbiting a
fixed central mass. Thus, Berget's success arises from the correct use
of inverse-square law, scaling arguments, and, importantly, the
decomposition of motion into phases. It is limited, however, by the
absence of detailed orbital mechanics formalism, {e.g.},
Equation~(\ref{eq13}), the absence of propulsion constraints, and the
lack of an explicit discussion of angular momentum:
{\begin{equation}\label{eq14}
\mathbf{L} = \mathbf{r} \times \mathbf{p},
\end{equation}}\unskip
and, therefore, cannot describe orbital motion. These limitations are
central and prevent Berget's account from being treated as a modern
astro-dynamical analysis; its value is instead pedagogical and
historical.

In 1923, aviation was still in its formative stage. Aircraft were
fragile biplanes (Figures~\ref{fig2} and~\ref{fig4}) with limited
range, low speed, and minimal reliability. Commercial air travel,
though initiated in a few isolated and short routes, had not yet
developed into a stable or widespread system. Typical flight speeds of
order 10\tsup{2} km/h were  \textit{two orders of magnitude below the
velocities required for escape from Earth's gravitational field},
$v_{\mathrm{esc}}\approx 40{,}000$ km/h. Under these conditions, the
very idea of systematic travel beyond the atmosphere remained
technologically remote.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Contrast between the state of aviation in 1923 and
the launch capability achieved during the Apollo program. Left:
aviation context in the early 1920s. Top: passengers and crew boarding
a propeller-driven aircraft ({ca.}\ 1920s). Bottom: interior of
an early commercial aircraft cabin, illustrating limited capacity and
rudimentary comfort. Right: launch of the Saturn V rocket (Apollo era,
late 1960s), capable of delivering payloads to translunar trajectories
with velocities approaching escape velocity. The juxtaposition
highlights the gap between early aviation and spaceflight. (Left
images: public domain (early aviation photography, {ca.}\ 1920s).
Right image: NASA, Saturn V launch (public domain)).}
\end{figure}

It is in this context the early 20th Century leaps of imagination
captured in Berget's book must be appreciated. At a time when sustained
intercontinental flight was not yet operational and intercontinental
travel was by sea, he approached lunar travel as a concrete real future
possibility governed by Newtonian mechanics and correctly identified in
qualitative form several physically meaningful stages, their duration,
and structure. 

The contrast with the Saturn V launch vehicle of the Apollo program,
shown in Figure~\ref{fig4}, underscores the magnitude of this leap. By
1968, propulsion technology had advanced to the point where velocities
of several kilometers per second could be achieved, enabling translunar
injection and controlled spaceflight.

In \textit{Le Ciel} we find an imagined view of the Earth as seen from
the lunar surface, rendered with a realism that is remarkable for its
time. The terrestrial globe is depicted with a large apparent angular
diameter, suspended in a black sky devoid of atmospheric scattering,
and set above a stark, high-contrast landscape. A comparison of the
diameters of the Earth seen from the Moon and the Moon seen from the
Earth, displayed side by side on p.~184 of the book, reveals that the
angular diameter of the Earth as seen from the Moon is approximately
$\theta\approx 2$\textdegree, about four times that of the Moon seen
from Earth. The absence of atmosphere ensures sharp contrasts between
illuminated and shadowed regions as opposed to the Earthly landscape
juxtaposed to it (Figure~\ref{fig5}).

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Comparative views of Earth and Moon across
imagination and observation. Left: Earth above the lunar horizon during
the Apollo program (top: Apollo 8 ``Earthrise,'' 1968; bottom: Apollo
17, 1972)~\cite{13}. Credit: NASA (public domain). Centre and right:
Earth as seen from the Moon and the Moon as seen from Earth,
respectively, from \textit{Le Ciel} (1923)~\cite{6}, p. 184. (Berget
figures: public domain; NASA images: public domain).} 
\end{figure}

The famous Apollo 8 ``Earthrise'' image and the Apollo 17 views of
Earth above the lunar horizon confirm the same geometric and
photometric features anticipated in \textit{Le Ciel} half a century
earlier.

\section{A modern reconstruction of the transit time reviewed in 
\textit{Le Ciel}}\label{sec8}

The following section is explicitly a modern pedagogical
reconstruction. It asks how the transit time discussed by Berget can be
examined using elementary modern orbital mechanics. The equations and
concepts introduced below are not attributed to Berget
and do not imply that he used Keplerian transfer theory or modern
astrodynamical formalism~\cite{11,14}.

To reconstruct Berget's second phase, consider a Hohmann-like transfer.
A Hohmann transfer is an orbital maneuver used to transfer a spacecraft
between two orbits of different altitudes around a central body---in
this case the Earth. This maneuver moves the spacecraft from an
altitude that is approximately equal to the Earth's radius
$r_{1}\approx R_{\oplus}$ to one at the lunar distance, $d$, so that
$r_{2}\approx d$. The semi-major axis is:
{\begin{equation}\label{eq15}
a = \dfrac{r_{1} + r_{2}}{2}.
\end{equation}}\unskip
For a minimum-energy Earth--Moon transfer orbit, the time of flight for
half the ellipse is:
{\begin{equation}\label{eq16}
t = \uppi \sqrt{\frac{a^{3}}{\mu _{\oplus}}},
\end{equation}}\unskip
where $a\approx (6.37\times 10^{6} + 3.84\times 10^{8})/{2} \approx
1.95\times 10^{8}$ m is the semi-major axis of the transfer ellipse and
$\mu_{\oplus} = GM_{\oplus} = 3.986 \times 10^{14}$
m\tsup{3}${\cdot}$s$^{-2}$ is Earth's gravitational parameter. Using
these accepted values yields Apollo's one-way travel time
$t_{\mathrm{Apollo}}\approx 4.3\times 10^{5}~\mathrm{s}\approx
119~\mathrm{h}\approx 5$ days. This idealized two-body estimate is not
the actual Apollo trajectory. Apollo used a faster trajectory with
higher injection energy and mission-specific constraints designed to
reduce travel time, rather than the slower minimum-energy Hohmann-like
transfer considered here, with typical Earth-to-Moon transit times of
approximately 72 h (3 days).

The shorter estimate in \textit{Le Ciel} implies a trajectory with
higher specific energy, or larger $v_{\infty}$, which reduces the
transit time. If the spacecraft departs faster than required for a
minimum-energy transfer, the time of flight decreases. A simple radial
approximation captures this point. Applying conservation of specific
mechanical energy (energy per unit mass), gives~\cite{12,15}:
{\begin{equation}\label{eq17}
\dfrac{1}{2} \left(\dfrac{\mathrm{d}r}{\mathrm{d}t}\right)^{2} = 
\dfrac{GM_{\oplus}}{r} + \varepsilon ,
\end{equation}}\unskip
where $\varepsilon$ is the specific orbital energy. For escape
trajectories, $\varepsilon>0$. Rearranging and integrating the
differential equation given in (\ref{eq17}), the travel time from
$r_{1}$ to $r_{2}$ is given by
{\begin{equation}\label{eq18}
t = \bigints _{r_{1}}^{r_{2}}
\dfrac{1}{\sqrt{2\left(\dfrac{GM_{\oplus}}{r} + \varepsilon 
\right)}}\,\mathrm{d}r.
\end{equation}}\unskip
For sufficiently large $\varepsilon$, over much of the trajectory,
$\varepsilon \gg (GM_{\oplus})/{r}$, so that:
{\begin{equation}\label{eq19}
\dfrac{GM_{\oplus}}{r} + \varepsilon \approx \varepsilon 
\end{equation}}\unskip
and then the integral (\ref{eq18}) simplifies to:
{\begin{equation}\label{eq20}
t \approx \displaystyle \int _{r_{1}}^{r_{2}}
\dfrac{1}{\sqrt{2\varepsilon}}\,\mathrm{d}r = 
\dfrac{r_{2}-r_{1}}{\sqrt{2\varepsilon }}.
\end{equation}}\unskip
Defining:
{\begin{equation}\label{eq21}
v_{\infty} = \sqrt{2\varepsilon},
\end{equation}}\unskip
we obtain
{\begin{equation}\label{eq22}
t\approx \dfrac{r_{2}-r_{1}}{v_{\infty}}.
\end{equation}}\unskip

This is motion at approximately constant speed. Assuming a value of
$v_{\infty} \sim 2$--3~km/s, one obtains $t \sim (3.8 \times
10^{8}~\mathrm{m}/ 2.5 \times 10^{3}~\mathrm{m}{\cdot}\mathrm{s}^{-1})
\approx 1.5\times 10^{5}~\mathrm{s}\approx 42~\mathrm{h}$ which is
of the same order of magnitude as Berget's estimate once allowance
is made for gravitational
deceleration and acceleration along the path. Thus, Berget's value of
${\approx}$48.5~h \cite{6} for the coasting phase is consistent with a
high-energy, non-minimum transfer trajectory. Berget's value is a
physically plausible value, and it is acceptable that it does not match
the Apollo value exactly---because real missions include parking
orbits, injection geometry, midcourse corrections, lunar targeting, and
mission-specific safety constraints.

Qualitatively, what happens is that if the spacecraft leaves Earth with
only slightly positive energy, gravity continues to strongly modify the
velocity during transit. The spacecraft slows substantially while
climbing out of Earth's potential well. Therefore, motion is highly
non-uniform. In contrast, if the spacecraft has large excess energy,
then the gravitational potential term $GM_{\oplus}/r$ quickly becomes
small compared with the excess specific energy $\varepsilon$. Gravity
then acts only as a perturbation. The speed varies little during the
trip, so the motion becomes approximately inertial or
``quasi-uniform''.

For any of the Apollo lunar missions, the two-body approximation of
spacecraft in the field of the Earth
(Equations~(\ref{eq15})--(\ref{eq17})) works remarkably well because
its trajectory meant that the motion was relatively fast and direct.
Most of the motion can therefore be treated as motion in Earth's
gravitational field slightly perturbed by the Moon. The Artemis
program, in contrast, operates within the restricted\footnote{The
term ``restricted'' in the \textit{restricted three-body problem}
refers to a simplification of the \textit{full three-body problem}
whereby one of the three bodies is assumed to have a negligible mass
compared to the two others. Therefore, the body with negligible mass
does not affect the motion of the two massive bodies, although it
remains fully subject to their gravitational field.} three-body
problem~\cite{14,16,17}. However, the transit time to lunar distance
remains comparable to Apollo values (about five days), confirming that
Berget's \mbox{estimate} pertains to the fundamental Earth--Moon transfer
timescale rather than mission-specific architecture. Specifically,
Artemis used a low-energy trajectory and not a direct Hohmann-like
transfer unlike the fast trajectory/high energy Apollo voyage.

An important development beyond the elementary two-body description is
the use of restricted three-body dynamics in modern lunar mission
design~\cite{14,16,17}. This improved approximation introduces dynamical structures
absent from Berget's account, including Lagrange points, halo orbits,
and distant retrograde orbits. Because these developments postdate
Berget and are not required for the central historical comparison,
their detailed discussion is available in Appendix~\ref{appA}. They are
considered there only as an example of an important ingredient that has
entered lunar trajectory design since 1923.

\section{Interplanetary travel beyond the Moon}\label{sec9}

The same chapter of \textit{Le Ciel}, extends space travel to Mars and
other planets, thereby projecting Newtonian mechanics far beyond any
technological capability of his era. Using scaling arguments based on
planetary distances and sustained motion through interplanetary space,
Berget concludes that voyages to Mars would require durations measured
in several (three) months while that to Venus would take around 47
days, implicitly recognizing the immense scale of the Solar System and
the impossibility of rapid transit under classical propulsion. This
estimate is remarkably consistent with modern Mars missions.
Contemporary spacecraft typically require approximately 6--9 months to
reach Mars using Hohmann-type transfer trajectories, depending on
launch geometry and mission architecture~\cite{11,14,15,18}.

\section{Pedagogical use of the historical example}\label{sec10}

The principal pedagogical value of Berget's account is that a
qualitative historical text can be translated progressively into the
language of elementary mechanics without requiring students first to
master modern astrodynamics. The three phases provide natural starting
points for three familiar physical ideas: escape from Earth introduces
conservation of mechanical energy; the weakening of terrestrial
attraction introduces the distinction between the $1/r$ gravitational
potential and the $1/r^{2}$ force; and the transition from
Earth-dominated to Moon-dominated motion introduces competing
gravitational fields and different dynamical regimes.

A simple classroom or public-outreach exercise can therefore begin
with Berget's original three travel times and ask what elementary
physics is required to rationalize each phase. The resulting estimates
can then be compared with Apollo data. The purpose is not to ask
whether Berget ``predicted'' Apollo correctly, but to demonstrate how
much can be inferred from elementary Newtonian reasoning, where that
reasoning reaches its limits, and what additional concepts were
subsequently required by modern astronautics.

\section{Conclusion: physics before engineering}\label{sec11}

Read through modern elementary mechanics, the account summarized by
Berget presents the journey to the Moon as a
succession of physically distinct stages associated with departure from
Earth, translunar motion, changing Earth--Moon gravitational influence,
and final approach to the Moon. What these pioneers lacked were
modern tools of calculation such as angular momentum formalism,
propulsion theory, numerical integration, digital computation, and
contemporary astrodynamics. However, their insight by taking
space-travel seriously enough to estimate the necessary time and phases
of the voyage was remarkable. The comparison with Jules
Verne highlights the transition from literary imagination to physically
grounded reasoning.

The interest of the triphasic decomposition summarized in \textit{Le
Ciel} lies not in intellectual priority nor in quantitative precision,
but in its pedagogical clarity and beauty of its exposition. It shows
how early twentieth century writers could use elementary Newtonian
reasoning to anticipate realistic lunar travel phased into
acceleration, coast, transition, and braking stages. Modern mission
design is, clearly, \textit{vastly} more complex, requiring orbital
mechanics, propulsion theory, guidance, numerical integration, and
three-body dynamics. The comparison with Apollo and Artemis is
therefore not a claim of direct anticipation, but an instructive
historical exercise showing how a simple physical model can foreshadow
some qualitative features of later spaceflight. 

Berget's discussion is also remarkable for extending beyond celestial
mechanics to the human realities of prolonged space travel. In the same
chapter, he considers food supply, confinement, duration, and the
physiological consequences of long voyages. Berget also recognized that
interplanetary space constituted a physically hostile environment and
discussed the effects of radiation, heat, and the exposure to radiation
on travelers, although within early twentieth-century physics rather
than the modern understanding of cosmic rays and high-energy particle
radiation. From a modern perspective, several of the practical
questions discussed by Berget such as travel duration, consumable
resources, confinement, and the physical conditions of prolonged
travel, can be recognized as questions that later became central to
astronautics and human spaceflight. In 1923, when aviation itself was
still in an early stage, Berget was already contemplating issues that
now dominate planning for lunar and Martian missions, including
life-support systems, consumable resources, and the physical and
psychological endurance required for interplanetary travel. 

Long before the Artemis~\cite{18,19} or Apollo lunar missions, or
even the first V2 rockets of World War II, early twentieth century
scientists and engineers had already captured a number of essential
physical aspects of space travel. They provided reasonable estimates of
spaceflight duration, considered extended travel to Mars and other
planets, and anticipated the human challenges of prolonged voyages
through space. \textit{Le Ciel} stands as a beautiful historical and
pedagogical document, reflecting what was known at the time of its
publication, and a historically noteworthy vision of humanity's
future beyond Earth. It is fitting to close this article with the
somewhat poetic finale of Alphone Berget's \textit{Le Ciel}:

\begin{quote}
Oui, tournons-nous vers le Ciel : il symbolise l'\'{e}ternelle
V\'{e}rit\'{e}; sa contemplation nous \'{e}l\`{e}vera au-dessus des
petitesses de la Terre; son \'{e}tude nous consolera des turpitudes
humaines. Et nous en sortirons meilleurs, parce que nous y aurons
appris que l'homme, grain de sable dans le double infini de l'Espace et
du Temps, ne doit pas s'enorgueillir des d\'{e}couvertes acquises, mais
s'efforcer toujours \`{a} p\'{e}n\'{e}trer plus avant dans le domaine
myst\'{e}rieux de la Science, dans ce domaine toujours plus vaste \`{a}
mesure qu'on y progresse davantage, dans ce domaine o\`{u} l'esprit
trouve les seules satisfactions vraiment compl\`{e}tes qu'il nous soit
donn\'{e} de go\^{u}ter sur cette Terre.

\smallskip\noindent
\textit{Yes, let us turn toward the Sky: it symbolizes eternal Truth;
its contemplation will raise us above the pettiness of the Earth; its
study will console us for human turpitudes. And we shall emerge from it
improved, because we shall have learned there that man, a grain of sand
within the double infinity of Space and Time, must not take pride in
discoveries already attained, but must always strive to penetrate
further into the mysterious domain of Science---a domain that grows
ever larger as one advances within it, a domain in which the mind finds
the only truly complete satisfactions granted to us on this Earth.}
\end{quote}

\section*{Acknowledgments}

Ch\'erif Matta is grateful to Alphonse Berget for, \textit{Le Ciel}, and for the
enormous influence this beautiful legacy had, and still has, on him.
The financial support of the Natural Sciences and Engineering Council
of Canada (NSERC), the Canadian Foundation for Innovation (CFI), the
Digital Research Alliance of Canada, and Mount Saint Vincent University
is gratefully acknowledged.

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\appendix{}
\def\appendixlabel{Appendix \Alph{section}}

\def\theequation{\thesection.\arabic{equation}}
\setcounter{equation}{0}

\section{Beyond Berget: the restricted three-body problem in modern
lunar flight}\label{appA}

The following discussion is included for physical and pedagogical
completeness and is not part of the reconstruction of Berget's
reasoning. Berget did not formulate the restricted three-body problem
and did not discuss Lagrange points, halo orbits, or distant retrograde
orbits. These concepts are presented here solely to illustrate an
important development in lunar trajectory design that lies beyond the
approach discussed in \textit{Le Ciel}.

In the Artemis case, the gravitational attraction of both Earth and
Moon must be treated simultaneously throughout much of the trajectory
{\begin{equation}\label{eqA1}
\dfrac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}} = - 
\dfrac{GM_{\oplus}}{r^{3}}\mathbf{r} - 
\dfrac{GM_{\mathrm{Moon}}}
{\vert \mathbf{r} - \mathbf{r}_{\mathrm{M}}\vert^{3}}
(\mathbf{r} - \mathbf{r}_{\mathrm{M}}),
\end{equation}}\unskip
where $\mathbf{r}_{\mathrm{M}}$ is the position vector of the Moon and
$\vert\mathbf{r} - \mathbf{r}_{\mathrm{M}}\vert$ is the spacecraft
distance from the Moon. The first term represents the gravitational
attraction of the Earth while the second represents the attraction of
the Moon. The spacecraft therefore moves in a gravitational field that
continuously changes in both magnitude and direction. This introduces
new richer dynamical structures such as Lagrange points, halo orbits
and distant retrograde orbits which are non-existent in the simple
two-body problem.

\textit{Lagrange points} are equilibrium points for small mass objects
moving under the influence of two larger masses~\cite{14}. For example,
the combined gravitational effects of the Earth and Moon balance
the orbital motion of a spacecraft, allowing it to remain at a fixed
position with respect to the rotating Earth--Moon system. There are five
such equilibrium points, three are unstable and two are stable.

\textit{Halo orbits} are three-dimensional looping trajectories around
the three unstable Lagrange points which are not simple ellipses. These
complex orbits arise from the nonlinear coupling of the Earth and Moon
gravitational fields.

The Artemis program also employs \textit{distant retrograde orbits
(DROs)}. In such trajectories, the spacecraft follows a large orbit
around the Moon whose sense is retrograde relative to the Moon's
orbital motion around Earth. These orbits are dynamically unusual
because they derive their stability not from the Moon alone, but from
the combined Earth--Moon gravitational system. These trajectories are
clearly considerably more complex than the classical Keplerian ellipses
of Apollo. For their calculation, they require numerical integration,
nonlinear dynamical analysis, stability theory, and three-body orbital
mechanics~\cite{14,16,17}. Yet despite this increased complexity, the
fundamental physical sequence identified qualitatively by Berget
remains unchanged. Any Earth-to-Moon trajectory must still proceed
through the same broad regimes:
\begin{enumerate}[(1)]
\item Escape from Earth's strong gravitational field. 
\item Transit through interplanetary space under weakening Earth attraction. 
\item Entry into a region where Earth and Moon compete gravitationally. 
\item Final capture or controlled motion near the Moon. 
\end{enumerate}
Apollo missions realized this sequence through relatively direct
Keplerian transfers. The Artemis program realizes it through
trajectories arising from a more sophisticated analysis of the
three-body problem. The mathematical description changes, but the
underlying physical structure remains the same.

Berget did not possess the mathematical machinery of modern celestial
mechanics. He knew nothing of Lagrange points, halo orbits, or distant
retrograde orbits. Nevertheless, he identified, in qualitative
pedagogical form, several physically meaningful stages imposed by the
Earth--Moon gravitational system itself. Modern astrodynamics places
this qualitative picture within a far more rigorous terms, adding
orbital geometry, propulsion constraints, trajectory optimization, and
multi-body dynamics absent from Berget's account.

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