\makeatletter
\@ifundefined{HCode}
{\documentclass[screen,CRMECA,Unicode,biblatex,published]{cedram}
\addbibresource{crmeca20260053.bib}
\newenvironment{noXML}{}{}%
\def\tsup#1{\textsuperscript{#1}}
\def\tsub#1{\textsubscript{#1}}
\def\stbfrac#1#2{{(#1)}/{(#2)}}
\def\sbfrac#1#2{{#1}/{(#2)}}
\def\sfrac#1#2{{#1}/{#2}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\xsection{}
\def\mathbi#1{\text{\textbf{\textit{#1}}}}
\def\xtbody{\noalign{\relax}}
\def\tabnote#1{\vskip4pt\parbox{.95\linewidth}{#1}}
\def\xxtabnote#1{\vskip4pt\parbox{.96\linewidth}{#1}}
\usepackage{upgreek}
\usepackage[T1]{fontenc}
\usepackage{multirow} 
\newcounter{runlevel}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20260053}
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\def\textemail#1{\href{mailto:#1}{#1}}
\csdef{Seqnsplit}{\\}
\def\botline{\\\hline}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}} 
\def\0{\phantom{0}}
\def\mn{\phantom{$-$}}
\def\refinput#1{}
\def\back#1{}
\let\Lbreak\break
\allowdisplaybreaks
\def\xmorerows#1#2{{#2}}
\DOI{10.5802/crmeca.374}
\datereceived{2026-01-22}
\daterevised{2026-05-28}
\dateaccepted{2026-06-04}
\ItHasTeXPublished
\renewenvironment{quote}{\unskip
\topsep=0pt
\parskip=0pt
\leftskip=0pt
\rightskip=0pt
\parfillskip=0pt
\small\list{}{\unskip
\leftmargin 3.5em
\rightmargin 2.5em
}\unskip\item
\relax
}
{\endlist}
}
{\documentclass[crmeca]{article}
\usepackage{upgreek}
\usepackage[T1]{fontenc} 
\def\CDRdoi{10.5802/crmeca.374}
\def\centering{}
\def\newline{\unskip\break}
\def\hyperlink#1#2{#2}
\def\hypertarget#1#2{#2}
\def\sfrac#1#2{{#1}/{#2}}
\def\sbfrac#1#2{{#1}/{(#2)}}
\def\stbfrac#1#2{{(#1)}/{(#2)}}
\def\xxtabnote#1{\tabnote{#1}}
\def\href#1#2{\url[#1]{#2}}
\newcommand\@coi{}
\newcommand\COI[1]{\gdef\@coi{#1}}
\newcommand\printCOI{\ifx\@coi\@empty\else%
\section*{Declaration of interests}
\@coi\fi
}
}
\makeatother

\COI{The author does not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
has declared no affiliations other than their research organization.}

\dateposted{2026-07-21}
\begin{document}

%\dateposted{2026-02-16}

\begin{noXML}

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

\editornote{This article follows a presentation by the author at the conference “250 Years of Ampère: An Electric World Tomorrow?”, held on December 11–12, 2025, in Lyon.}
\alteditornote{ Cet article fait suite à une intervention de l'auteur au colloque \og 250 ans d'Ampère : Demain, un monde électrique ?\fg, qui s'est tenu les 11-12 décembre 2025 à Lyon.}

\title{The providential timing of scientific discovery: Andr\'{e}-Marie
Amp\`{e}re in 1820}                                             

\alttitle{Le moment providentiel de la d\'{e}couverte scientifique :
Andr\'{e}-Marie Amp\`{e}re en 1820}

\author{\firstname{James Robert} \lastname{Hofmann}\CDRorcid{0000-0001-6890-2148}}
\address{California State University, Fullerton, Fullerton, CA 92835, USA}
\email{jhofmann@fullerton.edu}

\keywords{\kwd{Equilibrium experiments}
\kwd{Electrodynamics}
\kwd{Serendipity}
\kwd{Romanticism}}

\altkeywords{\kwd{Exp\'{e}riences d'\'{e}quilibre}
\kwd{\'{E}lectrodynamique}
\kwd{S\'{e}rendipit\'{e}}
\kwd{Romantisme}}

\begin{abstract} 
The 2025 commemoration of the 250th anniversary of Andr\'{e}-Marie
Amp\`{e}re's birth in 1775 was inspired in large part by the legacy he
left through his creation of the new branch of physics he called
electrodynamics. Primarily accomplished between 1820 and 1826,
Amp\`{e}re's achievement was precipitated by Hans Christian Oersted's
1820 discovery of an unexpected interaction between a magnetic needle
and an electric current. While many factors contributed to Amp\`{e}re's
successful response to Oersted's discovery, there was some serendipity
to its timing that is best appreciated by locating it within the
biographical context of Amp\`{e}re's academic career, his tumultuous
personal life, his extensive philosophical investigations, and his
tenuous religious faith. It is noteworthy that Oersted's discovery came
at a point in time when Amp\`{e}re's status on all these fronts was
relatively stable. The fortuitous state of equipoise he enjoyed in 1820
facilitated his engagement with electrodynamics to an extent that would
have been extremely difficult to achieve during the previous decade.
From Amp\`{e}re's religious perspective, the perspicacious timing of
Oersted's discovery could readily be attributed to divine providence.
The timely balance between the romantic and analytic components of
Amp\`{e}re's mentality in 1820 is aptly symbolized by the first of the
four equilibrium apparatuses he invented to ground his derivation of
the force law at the heart of his electrodynamics. 
\end{abstract} 

\begin{altabstract} 
La comm\'{e}moration en 2025 du 250\textsuperscript{e} anniversaire de la naissance
d'Andr\'{e}-Marie Amp\`{e}re en 1775 s'inspire en grande partie de
l'h\'{e}ritage qu'il a laiss\'{e} gr\^{a}ce \`{a} la cr\'{e}ation de la
nouvelle branche de la physique qu'il a nomm\'{e}e
\'{e}lectrodynamique. Principalement r\'{e}alis\'{e}e entre 1820 et
1826, cette avanc\'{e}e majeure d'Amp\`{e}re a \'{e}t\'{e}
pr\'{e}cipit\'{e}e par la d\'{e}couverte, en 1820, par Hans Christian
Oersted, d'une interaction inattendue entre une aiguille aimant\'{e}e
et un courant \'{e}lectrique. Si de nombreux facteurs ont contribu\'{e}
\`{a} la r\'{e}ussite de la r\'{e}ponse d'Amp\`{e}re \`{a} la
d\'{e}couverte d'Oersted, le caract\`{e}re fortuit de cette
co\"{i}ncidence s'explique au mieux par le contexte biographique de sa
carri\`{e}re universitaire, sa vie personnelle tumultueuse, ses vastes
recherches philosophiques et sa foi religieuse fragile. Il est \`{a}
noter que la d\'{e}couverte d'Oersted est survenue \`{a} un moment
o\`{u} la situation d'Amp\`{e}re \'{e}tait relativement stable dans
tous ces domaines. L'\'{e}quilibre providentiel dont il jouissait en
1820 facilita son exploration de l'\'{e}lectrodynamique, chose qui
aurait \'{e}t\'{e} extr\^{e}mement difficile \`{a} r\'{e}aliser durant
la d\'{e}cennie pr\'{e}c\'{e}dente. Du point de vue religieux
d'Amp\`{e}re, la perspicacit\'{e} avec laquelle la d\'{e}couverte
d'Oersted survint pouvait ais\'{e}ment \^{e}tre attribu\'{e}e \`{a} la
providence divine. L'\'{e}quilibre opportun entre les composantes
romantique et analytique de la pens\'{e}e d'Amp\`{e}re en 1820 est
parfaitement symbolis\'{e} par le premier des quatre appareils
d'\'{e}quilibre qu'il inventa pour \'{e}tayer sa d\'{e}monstration de
la loi de force, au c\oe{}ur de son \'{e}lectrodynamique.
\end{altabstract} 

%\input{CR-pagedemetas}

%\def\thanksname{Note}
%\thanks{This article is based upon the author's presentation at the
%``Colloque historique et scientifique en homage \`{a}  Andr\'{e}-Marie
%Amp\`{e}re'', Lyon, 11 December 2025.}

\maketitle

\end{noXML}

\section{Introduction: Amp\`{e}re's complex mentality}\label{sec1}
While historians of science understandably emphasize Andr\'{e}-Marie
Amp\`{e}re's scientific accomplishments, his complex personality has
always been an additional source of interest. For example, although his
devout Catholicism sometimes wavered, Amp\`{e}re always remained
convinced that divine providence played a crucial role in his life. He
was often dismayed when his ambitions and desires were thwarted by what
he considered to be a supernatural intervention, and he was plagued by
anxiety over how his conduct might have merited these setbacks.
Amp\`{e}re's response to providential events was conditioned by his
mercurial personality, torn between romantic inclinations to emotional
exuberance on the one hand and the competing demands of intellectual
rigor on the other. This essential tension in his character is a large
part of what makes Amp\`{e}re a fascinating historical 
figure~\cite{1,2,3}. Another way to express this duality is to say that
he was a unique product of both early nineteenth-century French
romanticism and the scientific milieu of mathematical physics we
associate with more single-minded personalities such as Cauchy,
Laplace, Poisson, or Biot.

With this context in mind, it is noteworthy that the impetus for
Amp\`{e}re's most creative scientific work, the experimental
exploration and mathematical formalization of a new branch of physics
he called electrodynamics, came at a most propitious time, a point in
his life when his romantic and analytic inclinations were in a rare
state of harmony. This turning point came in 1820 when Amp\`{e}re was
informed of Hans Christian Oersted's discovery that a suspended magnet
changes its orientation in proximity to an electric current. He
immediately began his own research and soon demonstrated that electric
current bearing wires either attract or repel each other depending upon
the direction of the currents. Had Oersted's revolutionary discovery
come at a different time than it did, Amp\`{e}re might very well not
have been in a position to take up this research and carry it out so
brilliantly.

Let us first consider in more detail the two facets of Amp\`{e}re's
mentality. On the one hand, in addition to his highly volatile
personality, Amp\`{e}re had many other characteristics usually
associated with a romantic temperament. In keeping with Rousseau's
skepticism about materialism and technological progress,
nineteenth-century romantics reacted against the artistic conventions
of neo-classical art and musical classicism, the imperialism of
Napoleon, and the dehumanizing aspects of the industrial revolution.
Perhaps most importantly, Romanticism represented a reaction against
Enlightenment ideals of objectivity, predictive power, quantitative
analysis, and mechanical cause and effect determinism. With an acute
awareness of the limits of reason, Romantics called attention to
emotion, freedom, subjectivity, spirituality, unpredictable
individualism, and nostalgia for an idealized past. Amp\`{e}re was
drawn to many of these attitudes. He maintained a preference for
bucolic rural environments rather than urban modernity. He shared with
his philosopher friend Maine de Biran an introspective disposition for
metaphysics and moody psychological analysis. Although periodically
tempted by the deism typical of Enlightenment  figures such as
Voltaire, Amp\`{e}re was resolute in his rejection of materialism in
favor of a tenuous but devout commitment to Catholicism and some
receptivity to the intervention of supernatural providence. While
heaven is sometimes thought of as a peaceful state of resolution, for
Romantic Catholics such as Amp\`{e}re, striving for an unattainable
sanctity typically resulted in a life of anxiety plagued by doubt and
guilt. The image he often presented to public scrutiny was that of a
bemused eccentric, largely indifferent to social norms and the
practical concerns of daily life. The traumatic loss of his father to
the guillotine in 1793 and the early death of his beloved wife Julie
Carron in 1803 are well-known episodes in the early life of Amp\`{e}re,
and they left permanent emotional scars. His innate sensitivity later
made him ill at ease in the competitive social and scientific
environment of Paris. All these characteristics make Amp\`{e}re an
obvious candidate for the ``romantic'' label. 

But this designation must be tempered by recognition of another facet
of Amp\`{e}re's personality, his atypical aptitude and passion for
mathematics and mathematical physics and his ability to
compartmentalize these interests and pursue them with minimal input
from his romantic inclinations. As a childhood prodigy, especially in
mathematics, he was immediately recognized as a remarkably brilliant
intellect. For those who have experienced a formal institutional
education, it is difficult to imagine the intellectual freedom of
Amp\`{e}re's childhood. His literally ``encyclopedic'' reading in his
father's library may have been partly responsible for his lifelong
fascination with classification schemes and his inclination to abruptly
shift the focus of his investigations in response to his eclectic
interests rather than more pragmatic obligations.

\section{Amp\`{e}re in 1814}\label{sec2} 
In addition to his own state of mind, Amp\`{e}re's reaction to
scientific opportunity was of course also affected by circumstances
beyond his control. The pivotal moment of his scientific career came at
a particularly opportune time in 1820. It was at that point, in
response to Oersted's discovery of the mutual interaction between an
electric current and a magnet, that Amp\`{e}re began investigations
into what he would call the science of electrodynamics. It has been a
persistent puzzle among historians to account for his amazing
creativity in so rapidly laying the foundations for this new branch of
physics. His metaphysical presuppositions, his outsider status with
respect to the predominant Laplacian paradigm for French physics, and
his enthusiastic investigation of what historian Friedrich Steinle has
called ``exploratory experiments'' have all been appropriately
documented as contributing  factors~\cite{4,5,6,7}. However, there was
a providential timing to Oersted's discovery that was also highly
significant. As an historical exercise, consider how difficult
Amp\`{e}re's response would have been if this discovery had come just
six years earlier in 1814. The detrimental factors he would have had to
overcome fall into several categories.

1814 was of course the year prior to Napoleon's final defeat.
Amp\`{e}re followed the events leading to the fall of the Empire very
closely and his dread of the future of France and the human condition
in general kept him in an even more heightened state of anxiety than
usual. On a more personal level, his domestic life had degenerated due
to his rash 1806 marriage to Jeanne Potot, a woman Joseph Janin once
described as a ``pimb\^{e}che  pr\'{e}tentieuse''~\cite[p.~15]{8}. The
marriage was an unmitigated disaster and promptly dissolved into
prolonged legal proceedings over their daughter Albine, born on 6 July
1807 and initially treated with shameless indifference by her mother.
But 1814 also included the painful ending of another romantic
relationship with a woman Amp\`{e}re referred to in correspondence only
as \textit{la constante amiti\'{e}}. Amp\`{e}re seems to have adopted
this term due to his fascination with an engraving by that name based
upon a painting by Jean-Baptiste  Mallet~\cite[vol.~1, p.~373]{9}.
Amp\`{e}re's angst concerning the complicated nature of his breakup
with this woman may have driven him to thoughts of suicide. In a letter
to his Lyon friend, Claude-Julien Bredin, on 14 March 1814 he referred
to ``ce jour affreux'' and reassured Bredin that ``Je te reverrai,
j'en suis s\^{u}r \`{a} pr\'{e}sent. Ne crains rien pour ma 
vie~!''~\cite[vol.~2, p.~465]{9}. Although Amp\`{e}re's penchant for
dramatic language may have given the impression that he was closer to
suicide than he actually was, he was clearly burdened by a severe and
debilitating sense of guilt over how this affair ended. He found some
consolation in believing that ``aucune volont\'{e} humaine n'a
pr\'{e}vu ni ordonn\'{e} de cet \'{e}v\'{e}nement;
mais il est une volont\'{e} toute puissante qui a appesanti sur moi les
coups de sa vengeance. Ah~! sans doute je m\'{e}ritais ce qui m'est
arriv\'{e}, puisqu'elle l'a  permis''~\cite[vol.~2, p.~465]{9}.

In addition to his personal turmoil, Amp\`{e}re's efforts to make a
significant contribution to philosophy had reached a turning point by
1814. Throughout the previous decade he had participated in vigorous
discussions of Kant's \textit{Critique of Pure Reason}, the new German
epistemology that was under scrutiny in France. Although Amp\`{e}re
agreed with the consensus among French philosophers that innate ideas
should not be posited and that knowledge originates through sensation,
he was driven by an obsessive need to justify knowledge that is not
limited to subjective phenomena. In his lengthy correspondence with
Maine de Biran, he persistently argued that the discovery of relations
(\textit{rapports}) among phenomena, especially when mathematically
expressed, can serve as a bridge from the subjectivity of sensations to
an objective reality he referred to as the world of \textit{noumena},
adopting a term from Kant. He found it very difficult to find a
convincing justification for this ``structural realism'', as it has
been  termed~\cite{10}. His efforts included analysis of faculty
psychology together with dubious speculations that the spatial aspect
of sensation suggests a physiologically based alternative to Kant's
analysis of space and  time~\cite[pp.~247--248]{10}. In response to
criticisms from Maine de Biran and an enthusiastic disciple of Kant,
S\'{e}bastien Falquet de  Planta~\cite{11}, Amp\`{e}re was forced to
retreat to a contingent hypothetical assertion of the desired analogy
between phenomenal and noumenal relations. By 1814 his argument
amounted to the claim that only some form of realism could adequately
explain the empirical success of science. This meant that the
justification of one of his strongest philosophical convictions was
contingent upon confirmation by the development of successful
scientific theories. Consumed~as he was by these metaphysical issues,
Amp\`{e}re was severely distracted from concentrating on the scientific
research he needed to justify his own epistemology. 

During the first half of 1814 it appeared that chemistry might be the
domain in which this support would come, and Amp\`{e}re threw himself
into the debates of the day with his customary passion. Unfortunately,
his outsider status with respect to the highly centralized and
competitive community of chemists in Paris ensured that his insights
into the discovery of new elements were marginalized. Furthermore,
during the waning days of the Empire, his interactions with the British
chemist Humphrey Davy were frowned upon and further isolated him. Once
again, Amp\`{e}re saw the workings of providential retribution as he
explained in another letter to Bredin in February  1814~\cite[vol.~2,
pp.~458--459]{9}.

\begin{quote}
La mal\'{e}diction du ciel semble attach\'{e}e sur moi~; mais c'est
toujours par ma faute, que je me suis pr\'{e}cipit\'{e} dans le
malheur. Toujours je ne sais quelle providence me mettait dans une
situation qui m'\'{e}clairait, qui me donnait le temps de
r\'{e}fl\'{e}chir, et ensuite un \'{e}v\'{e}nement subit me venait
entrainer dans le moment o\`{u} je semblais n'avoir plus rien \`{a}
craindre.
\end{quote}

He did manage to publish one 1814 paper on molecular structure and
atomism in the \textit{Annales de Chimie}. Because of the geometric
foundation of his approach, he had high hopes that it was destined to
advance his metaphysical program. Amp\`{e}re posited a ``representative
form'' for each particle of a chemical element or compound, a
polyhedron with atoms (``molecules'' in Amp\`{e}re's 1814 terminology)
located at each vertex. For the simplest of these forms, Amp\`{e}re
drew upon the ``primitive forms'' Ha\"{u}y had discovered in his study
of crystallography. As a purely mathematical set of relations, this
model was ideally suited to illustrate Amp\`{e}re's belief in a
scientific bridge between the phenomenal and noumenal worlds. By
studying how his geometrical particles could be superimposed and
combined to agree with empirical data such as Gay-Lusaac's law of
combining volumes for gases, Amp\`{e}re essentially tried to reduce
chemistry to a branch of geometry. Unfortunately, although he did
manage to construct appropriate geometrical models for some relatively
simple chemical reactions, his approach was of little value for more
complicated cases, and it generated little interest. 

Amp\`{e}re partially overcame this disappointment by turning to
mathematics during the second half of 1814. Despite his innate
mathematical ability, he rarely pursued pure mathematics except for
pragmatic purposes of career advancement. This was certainly the case
in 1814. In desperate need of publications to support his candidacy for
an opening in the mathematics section of the First Class of the
\textit{Institut National}, Amp\`{e}re wrote three significant
mathematics papers and submitted them on 11 July, 12 September, and 17
October 1814. After several months of trepidation, he was elected as a
member of the mathematics section on 28 November. His papers involved
highly technical analysis and classification of partial differential
equations, a branch of mathematics Amp\`{e}re had not published
previously. They required some intense concentration on his part, and
he complained that the time and energy they consumed would have been
more productively spent on philosophy and chemistry. 

One more important facet of Amp\`{e}re's state of mind in 1814
concerned his religious faith. Amp\`{e}re's son, Jean-Jacques, recalled
that his father often insisted that the celebration of his first
communion was one of the three most influential events of his 
childhood~\cite[p.~67]{12}. Nevertheless, his faith was intermittent
after that point. Raised in a devout home, he had a crisis of faith
just prior to his first marriage but was reconciled during his last
years in Lyon. This was the period in which he participated in the
\textit{Soci\'{e}t\'{e} chr\'{e}tienne} and wrote some of his most
quoted expressions of personal piety. In one famous example, probably
written in 1805 shortly after his move from Lyon to Paris, Amp\`{e}re
pleaded with himself to reach a state of harmony in which his spiritual
and intellectual impulses would be in  equilibrium~\cite[vol.~2,
p.~541]{9}.

\begin{quote}
\'{E}tudie les choses de ce monde, c'est le devoir de ton \'{e}tat~;
mais ne les regarde que d'un \oe{}il~; que ton autre \oe{}il soit
constamment fix\'{e} sur la lumi\`{e}re \'{e}ternelle~!
\end{quote} 

Amp\`{e}re did not always achieve this goal of balance and moderation,
but he did acknowledge it as an ideal to be sought. Although he would
return to Catholicism by 1817, in 1814 he was spiritually adrift. This
state did not sit well with him, and he agonized that it was in some
way responsible for his personal and professional setbacks.

So, let us return to our hypothetical question: would Amp\`{e}re have
been able to successfully devote himself to electrodynamics if
Oersted's discovery had come in 1814, just six years earlier than it
did? The evidence indicates that he would have been in no position to
do so. Untethered from the religious mooring of his youth, entangled in
romantic and domestic drama, embroiled in complex disputes with
chemists and philosophers who denied his relevance, and compelled to
publish detailed memoirs on partial differential equations in pursuit
of election to the \textit{Institut}, Amp\`{e}re was too overwhelmed to
do justice to an additional agenda in 1814, especially in a competitive
environment in which every such opportunity was seized upon by
experienced researchers intent upon their own career advancement. 

\section{Amp\`{e}re in 1820}\label{sec3} 
By 1820 Amp\`{e}re's life had taken on a new stability that contributed
to his more optimistic frame of mind. In 1818 he reluctantly sold his
property in Poleymieux and purchased a home at 19 rue Foss\'{e}s-Saint
Victor near the \'{E}cole Polytechnique. With his legal disputes with
Jenny Potot resolved and no longer interested in additional romantic
entanglements, Amp\`{e}re enjoyed a temporary period of domestic
tranquility. His unmarried sister Jos\'{e}phine took care of his
household and helped nurture his daughter Albine and his son
Jean-Jacques. Both children would cause him considerable consternation
in the future, but these developments were unforeseen in 1820. He also
periodically rented out a room to academic boarders who provided the
scientific and philosophical conversation he craved.

With his election to the mathematics section of the Institut secured in
1814, Amp\`{e}re was no longer burdened by publishing demands in a
field he found uninspiring. He did write some short mathematical
analyses when a topic sparked his interest. In 1816 he also wrote an
idiosyncratic chemistry memoir in which he set out to provide a
``natural'' classification of the 48 recognized elements into fifteen
\textit{genres}. Due in part to his lack of attention to the issue of
atomic weights, this memoir generated even less interest than his 1814
paper. Amp\`{e}re's frustrated efforts to revolutionize chemistry came
to an end at this point and he admitted in a letter to Bredin ``dans
les sciences je ne puis prendre un vif int\'{e}r\^{e}t qu'aux travaux
qui se pr\'{e}sentent comme pouvant en changer la face \`{a} quelques 
\'{e}gards''~\cite[vol.~3, p.~887]{9}.

Amp\`{e}re's return to his Catholic faith was particularly fortifying
during this period. In one of several devotional passages written
during 1817 and 1818, he pledged to remain  faithful~\cite[vol.~2,
p.~533]{9}.

\begin{quote}
Dieu m'a r\'{e}v\'{e}l\'{e} aujourd'hui 5 octobre, 19\tsup{e}
dimanche apr\`{e}s la Pentec\^{o}te, \`{a} quoi
tenait mon salut \'{e}ternel [{\ldots}] Grand Saint Joseph, \`{a}
l'intercession duquel je dois surtout cette gr\^{a}ce, Sainte Marie,
m\`{e}re de Dieu dont j'ai re\c{c}u le nom \`{a} mon bapt\^{e}me et
\`{a} qui j'ai aussi ce don  inexprimable [sic], interc\'{e}dez
toujours aupr\`{e}s de Dieu pour qu'il me le conserve et pour que je
m'en rende digne~!
\end{quote}

As Charles Braverman has pointed out, Amp\`{e}re's renewed faith also
helped him to reconcile himself to the inconclusive status of his
philosophical  efforts~\cite[p.~195]{10}. His failure to provide a
convincing argument for his structural realism was set aside in favor
of a more pragmatic acceptance of it as a \textit{de facto} methodology
supported by both scientific progress and a belief in divine
providence. Rather than continue to try to establish his realism
metaphysically, Amp\`{e}re subsumed it into the four ``points of view''
through which he eventually structured his \mbox{classification} of the
sciences in his 1834 \textit{Essai sur la Philosophie des sciences}.
This project was just beginning in 1820, and it would benefit from
Amp\`{e}re's experience creating electrodynamics during the~1820s.
\looseness=1

Amp\`{e}re certainly never lost the emotional intensity that was one of
his innate characteristics. In 1826 when he was 51 years old,
Amp\`{e}re burst into tears when an experiment he had designed to
support his theory of magnetism did not give the result he had
predicted. Shortly thereafter, when Jean-Daniel Colladon paid a
midnight visit to inform him that a revised version of the experiment
had succeeded, Amp\`{e}re insisted upon immediately marching to the
Coll\`{e}ge de France demonstration room to witness the result in 
person~\cite[p.~119]{13}. But this enthusiasm was simply an expression
of the exuberance Amp\`{e}re brought to scientific investigation and
life in general; beneath this superficial turbulence he was closer to
an overall state of equipoise than he had been for many years. 

Immediately following Oersted's discovery and his own investigation of
forces between current carrying wires, Amp\`{e}re launched into an
ambitious effort to reduce all electrodynamic phenomena to their most
basic constituents. A crucial stage in that project was his
determination of an expression for the electrodynamic force between
infinitesimal electric circuit elements. Amp\`{e}re eventually
formulated the strength of this force as:
{$$
r^{-2 }(\sin \alpha \sin \beta \cos \gamma - 
\tfrac{1}{2} \cos \alpha \cos \beta)
\mathrm{i}\; \mathrm{d}s\; \mathrm{i}'\;\mathrm{d}s'
$$}\unskip

As shown in Figure~\ref{fig1},  $\mathrm{d}s$ and  $\mathrm{d}s'$ are
two tiny circuit elements of currents with strength  $\mathrm{i}$ and
$\mathrm{i}'$ separated by a distance  $r$; $\alpha$, 
$\beta$ and $\gamma$ are angles that depend upon the mutual
orientation of  $\mathrm{d}s$ and   $\mathrm{d}s'$~\cite[Figure~2.15a,
p.~45]{14}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}The geometry of the mutual orientation of two
current elements.}
\end{figure}

Applications of this electrodynamic force law to complicated arrays of
circuits and magnets was the crux of Amp\`{e}re's research program. In
his most comprehensive electrodynamics publication, his 1826
\textit{Th\'{e}orie des ph\'{e}nom\`{e}nes \'{e}lectro-dynamiques,
uniquement d\'{e}duite de l'exp\'{e}rience}, Amp\`{e}re would base his
derivation of the force law on four experimental demonstrations, each
of which displayed a static state of equilibrium in which all the
acting electrodynamic forces cancel each other out and leave a
potentially mobile part of the apparatus stationary. These equilibrium
demonstrations were unusual in that they did not involve the recording
of any quantitative data. Instead, a potential state of equilibrium is
designed, and when it is observed experimentally, mathematical
consequences are drawn from an analysis of the balanced forces\break in play.

Amp\`{e}re designed the first of these equilibrium demonstrations in
December 1820. In the final version of the equipment, as shown in 
Figure~\ref{fig2}, a potentially mobile rectangular circuitry
\textbf{GCDHG} is suspended and is free to rotate around a vertical
axis through \textbf{FI}, and the segment \textbf{GH} is exposed to two
electrodynamic forces, one due to a current in a straight vertical wire
\textbf{QP}, and the other due to another vertical current \textbf{SR}
in a wire twisted into a series of small ``sinuosities''. These two
vertical wires are located at equal distances from the vertical portion
\textbf{GH} of the suspended circuitry \textbf{GCDHG} and repel it in
opposite  directions~\cite[Planche~6, Figure~16]{15}.

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig02}
\vspace*{-4pt}
\caption{\label{fig2}The final design of Amp\`{e}re's first equilibrium
apparatus.}
\vspace*{-4pt}
\end{figure}

If the two oppositely directed forces acting on \textbf{GH} are of
equal magnitudes, the rectangular suspended circuitry \textbf{GCDHG}
will remain in equilibrium and will not rotate. Following his
successful demonstration of this phenomenon, Amp\`{e}re could
incorporate a mathematical expression of the observed state of
equilibrium into a derivation of his electrodynamic force law between
current  elements~\cite[pp.~77--82]{16}. Although Amp\`{e}re did not
immediately make equilibrium demonstrations his preferred mode of
argumentation, he continued to design and revise them until they
eventually became the foundation of his 1826 \textit{Th\'{e}orie des
ph\'{e}nom\`{e}nes \'{e}lectro-dynamiques}.

\section{Conclusion}\label{sec4} 
The conceptualization and successful operation of the apparatus shown
in  Figure~\ref{fig2} were important accomplishments  during an early
phase of Amp\`{e}re's most creative period of research. However, in
addition to the scientific import it had during the 1820s, the
equilibrium demonstration can be given symbolic significance as well.
From our perspective 250 years after Amp\`{e}re's birth, his first
equilibrium demonstration is symbolic of an elusive state of emotional
stability that Amp\`{e}re rarely achieved. Just as his electrodynamic
apparatus maintained equilibrium only if the forces in action were
balanced, so too Amp\`{e}re occasionally found himself in a healthy
state of equipoise when his romantic and rational impulses were
synchronized and his religious, philosophical and scientific impulses
were in harmony. Fortunately, this was the case in 1820, and Amp\`{e}re
was ideally positioned to focus his energy on electrodynamics, driven
by what his son Jean-Jacques called his characteristic ``soif de 
certitude''~\cite[p.~12]{12}. He may never have recovered the happiness
he experienced during brief periods of his youth in Poleymieux, but he
did accomplish a remarkably innovative contribution to scientific
inquiry, facilitated by the fact that Oersted's suggestive discovery
came at the providential time that it did.

\printCOI

\back{}

\printbibliography
\refinput{crmeca20260053-reference.tex}

\end{document}