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By Philip Gibbs, 1997, 2004.

Why is c the symbol for the speed of light?

"As for c, that is the speed of light in vacuum, and if you ask why c,
the answer is that it is the initial letter of celeritas, the Latin word meaning
speed." Isaac Asimov in "C for Celeritas (1959)" [1]

A Short Answer
==============

Although c is now the universal symbol for the speed of light, the most common
symbol in the nineteenth century was an upper-case V which Maxwell had started
using in 1865. That was the notation adopted by Einstein for his first few
papers on relativity from 1905. The origins of the letter c being used for the
speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch
[2]. They defined and measured a quantity denoted by c that they used in an
electrodynamics force law equation. It became known as Weber's constant and was
later shown to have a theoretical value equal to the speed of light times the
square root of two. In 1894 Paul Drude modified the usage of Weber's constant so
that the letter c became the symbol for the speed of electrodynamic waves
[3]. In optics Drude continued to follow Maxwell in using an upper-case V for
the speed of light. Progressively the c notation was used for the speed of light
in all contexts as it was picked up by Max Planck, Hendrik Lorentz and other
influential physicists. By 1907 when Einstein switched from V to c in his
papers, it had become the standard symbol for the speed of light in vacuum for
electrodynamics, optics, thermodynamics and relativity.

Weber apparently meant c to stand for "constant" in his force law, but there is
evidence that physicists such as Lorentz and Einstein were accustomed to a
common convention that c could be used as a variable for velocity. This usage
can be traced back to the classic Latin texts in which c stood for "celeritas"
meaning "speed". The uncommon English word "celerity" is still used when
referring to the speed of wave propagation in fluids. The same Latin root is
found in more familiar words such as acceleration and even celebrity, a word
used when fame comes quickly.

Although the c symbol was adapted from Weber's constant, it was probably thought
appropriate for it to represent the velocity of light later on because of this
Latin interpretation. So history provides an ambiguous answer to the question
"Why is c the symbol for the speed of light?", and it is reasonable to think of
c as standing for either "constant" or "celeritas".

The Long Answer
===============

In 1992 Scott Chase wrote on sci.physics that "anyone who read hundreds of books
by Isaac Asimov knows that the Latin word for `speed' is `celeritas', hence the
symbol `c' for the speed of light". Asimov had written an article entitled "C
for Celeritas" in a sci-fi magazine in 1959 and had reprinted it in some of his
later books [1]. Scott was the first editor of the Physics FAQ on Usenet and
Asimov's explanation was later included in the relativity section as the
"probable" answer to the question "Why is c the symbol for the speed of light?".
Since then, Asimov's answer has become a factoid repeated in many articles and
books. But if you go back and read his essay you discover that Asimov merely
stated his case in one sentence, and made no further attempt to justify his
theory for the origin of the "c" notation. So is his claim really born out by
history, or was c originally introduced as a variable standing for something
else? The special theory of relativity is based on the principle that the speed
of light is constant; so did c stand for "constant", or did it simply appear by
accident in some text where all the other likely variables for speed had already
been used up? These questions have been asked repeatedly on usenet, and now
after much searching through old papers and books the answers can be revealed.

A lower-case c has been consistently used to denote the speed of light in
textbooks on relativity almost without exception since such books started to be
written. For example, the notation was used in the earliest books on relativity
by Lorentz (1909) [4], Carmichael (1913) [5], Silberstein (1914) [6], Cunningham
(1915) [7], and Tolman (1917) [8]. That was not the case just a few years
before. In his earliest papers on relativity from 1905 to 1907, Einstein began
by using an upper-case V for the speed of light [9]. At that time he was also
writing papers about the thermodynamics of radiation, and in those he used up
upper-case L [10]. All of these papers appeared in volumes of the German
periodical Annalen Der Physik. Einstein's notation changed suddenly in 1907 in
a paper for the Journal Jahrbuch der Radioaktivität und Elektronik [11]. There
he used the lower case c, and his most famous equation E=mc² came
into being.

It is not difficult to find where the upper case V had come from. Maxwell used
it extensively in his publications on electrodynamics from as early as 1865
[12]. It was the principal symbol for the speed of light in his 1873 treatise
on electrodynamics [13]. By the 1890s Maxwell's book was in wide circulation
around the world and there were translations available in French and German. It
is no surprise then that the upper-case V is found in use in such papers as the
1887 report of Michelson and Morley on their attempt to find seasonal variations
in the speed of light [14]. That was written in the United States, but the same
notation was also found across Europe, from papers by Oliver Lodge [15] and
Joseph Lamor [16] in England, to the lecture notes of Poincaré in France [17],
and the textbooks of Paul Drude in Germany [18] and Lorentz in the Netherlands
[19]. Einstein's education at the Polytechnik in Zurich had not covered
Maxwell's theory of Electrodynamics in the detail he would have liked. But he
had read a number of extra textbooks on the new Electrodynamics as self study,
so he would have been familiar with the standard notations. From 1905 he wrote
his first papers on relativity, and there is nothing extraordinary in his choice
of the symbol V for the speed of light [9].

Why then, did he change it to c in 1907? At that time he still worked as a
clerk in the Bern patent office, but for the previous two years he had been in
regular correspondence with eminent physicists such as Max Laue, Max Planck,
Wilhelm Wien and Johannes Stark. Stark was the editor of the Jahrbuch, and had
asked Einstein to write the article in which he was to first use the letter c.
Einstein mentioned to Stark that it was hard for him to find the time to read
published scientific articles to acquaint himself with all the work others have
done in the field, but he had seen papers by Lorentz, Kohn, Monsegeil and Planck
[20]. Lorentz and Planck in particular had been using c for the speed of light
in their work. Lorentz had won the 1902 Nobel prize for physics, and it is not
surprising that physicists in Germany had now taken up the same notation. It is
also not surprising that Einstein, who was looking for an academic position,
aligned himself to the same conventions at that time. Another reason for him to
make the switch was that the letter c is simply more practical. The upper-case
V would have been easily confused with the lower case v appearing in the
equations of relativity for the velocity of moving bodies or frames of
reference. Einstein must have found this confusion inconvenient, especially in
his hand written notes.

Looking back at papers of the late 1890s, we find that Max Planck and Paul Drude
in particular were using the symbol c at that time. The name of Drude is less
well known to us today. He worked on relations between the physical constants
and high precision measurements of their value. These were considered to be
highly worthy pursuits of the time. Drude had been a student of Voigt, who
himself had used a Greek ω for the speed of light when he wrote down an
almost complete form of the Lorentz transformations in 1887 [43]. Voigt's
ω was later used by a few other physicists [44, 45], but Drude did not use
his teacher's notation. Drude first used the symbol c in 1894, and in doing so
he referenced a paper by Kirchhoff [3]. As already mentioned, Paul Drude also
used V. In fact he made a distinction of using V in the theory of optics for
the directly-measured speed of light in vacuum, whereas he used c for the
electromagnetic constant that was the theoretical speed of electromagnetic
waves. This is seen especially clearly in his book "Theory of Optics" of 1900
[21], which is divided into two parts with V used in the first and c in the
second part. Although Maxwell's theory of light predicted that they had the
same value, it was only with the theory of relativity that these two things were
established as fundamentally the same constant. Other notations vied against
Drude's and Maxwell's for acceptance. Herglotz [46] opted for an elaborate
script B, while Himstedt [47], Helmholtz [48] and Hertz [49] wrote the equations
of electrodynamics with the letter A for the reciprocal of the speed of light.
In 1899 Planck backed Drude by using c, when he wrote a paper introducing what
we now call the Planck scale of units based on the constants of electrodynamics,
quantum theory and gravity [22]. Drude and Planck were both editors of the
prestigious journal Annalen Der Physik, so they would have had regular contact
with most of the physicists of central Europe.

Lorentz was next to change notation. When he started writing about light speed in
1887 he used an upper case A [23], but then switched to Maxwell's upper case
V [24]. He wrote a book in 1895 [25] that contained the equations for length
contraction, and was cited by Einstein in his 1907 paper. While Drude had started to
use c, Lorentz was still using V in this book. He continued to use
V until 1899 [26], but by 1903 when he wrote an encyclopedia article on
electrodynamics [27] he too used c. Max Abraham was another early user of the
symbol c in 1902, in a paper that was seen by Einstein [28]. From Drude's
original influence, followed by Planck and Lorentz, by 1907 the c symbol had become
the prevailing notation in Germanic science and it made perfect sense for Einstein to
adopt it too.

In France and England the electromagnetic constant was symbolised by a lower case
v rather than Drude's c. This was directly due to Maxwell, who wrote
up a table of experimental results for direct measurements of the speed of light on the
one hand and electromagnetic experiments on the other. He used V for the
former and v for the latter. Maxwell described a whole suite of possible
experiments in electromagnetism to determine v. Those that had not already
been done were performed one after the other in England and France over the three decades
that followed [29]. In this context, lower case v was always used for the
quantity measured. But using v was doomed to pass away once authors had to
write relativistic equations involving moving bodies, because v was just too common
a symbol for velocity. The equations were much clearer when something more distinct
was used for the velocity of light to differentiate it from the velocity of moving
bodies.

While Maxwell always used v in this way, he also had a minor use for the symbol
c in his widely read treatise of 1873. Near the end he included a section
about the German electromagnetic theory that had been an incomplete precursor to his own
formulation [30]. This theory, expounded by Gauss, Neumann, Weber, and Kirchhoff,
attempted to combine the laws of Coulomb and Ampère into a single action-at-a-distance
force law. The first versions appeared in Gauss's notes in 1835 [31], and the
complete form was published by Weber in 1846 [32]. Many physicists of the time were
heavily involved in the process of defining the units of electricity. Coulomb's law
of electrostatic force could be used to give one definition of the unit of charge while
Ampère's force law for currents in wires gave another. The ratio between these units
had the dimension of a velocity, so it became of great practical importance to measure its
value. In 1856 Weber and Kohlrausch published the first accurate measurement
[2]. To give a theoretical backing they rewrote Weber's force law in terms of the
measured constant and used the symbol c. This c appeared in numerous
subsequent papers by German physicists such as Kirchhoff, Clausius, Himstedt, and
Helmholtz, who referred to it as "Weber's constant". That continued until the
1870s, when Helmholtz discredited Weber's force law on the grounds of energy
conservation, and Maxwell's more complete theory of propagating waves prevailed.

Two papers using Weber's force law are of particular note. One by Kirchhoff [33]
and another by Riemann [34] related Weber's constant to the velocity at which electricity
propagated. They found this speed to be Weber's constant divided by the square root
of two and it was very close to the measured speed of light. It was already known
from experiments by Faraday that light was affected by magnetic fields, so there was
already much speculation that light could be an electrodynamic phenomenon. This was
the inspiration for Maxwell's work on electrodynamics, so it is natural that he finally
included a discussion of the force law in his treatise [30]. The odd thing is that
when Maxwell wrote down the force law, he changed the variable c so that it was
smaller than Weber's constant by a factor of the square root of two. So Maxwell was
probably the first to use c for a value equal to the speed of light, although he
defined it as the speed of electricity through wires instead.

So c was used as Weber's constant having a value of the speed of light times the
square root of two, and this can be related to the later use of c for the speed of
light itself. Firstly, when Maxwell wrote Weber's force law in his treatise in 1873,
he modified the scale of c in the equation so that it reduced by a factor of the
square root of two. Secondly, when Drude first used c in 1894 for the speed
of light [3], the paper by Kirchhoff that he cited [35] was using c for Weber's
constant, so Drude had made the same adjustment as Maxwell. It is impossible to say
if Drude copied the notation from Maxwell, but he did go one step further in explicitly
naming his c as the velocity of electrodynamic waves which by Maxwell's theory was
also the speed of light. He seems to have been the first to do so, with Lorentz,
Planck, and others following suit a few years later.

So to understand why c became the symbol for the speed of light we now have to
find out why Weber used it in his force law. In the paper of 1856 [2] Weber's
constant was introduced with these words "and the constant c represents that
relative speed, that the electrical masses e and e must have and keep, if they are not to
affect each other." So it appears that c originated as a letter standing for
"constant" rather than "celeritas". Nevertheless, it had nothing to do with the constancy
of the speed of light until much later.

Despite this, there could still be some substance to Asimov's claim that c is
the initial letter of "celeritas". It is true, after all, that c is also
often used for the speed of sound, and it is commonly used as the velocity constant in the
wave equation. Furthermore, this usage was around before relativity.

Starting with the Latin manuscripts of the 17th century, such as Galileo's "De Motu
Antiquiora" or Newton's "Principia", we find that they often use the word "celeritas" for
speed. But their writing style was very geometric and descriptive, and so they
did not tend to write down formulae where speed is given a symbol. But an example of
the letter c being used for speed can be found from the eighteenth century.
In 1716 Jacob Hermann published a Latin text called Phoronomia, meaning the science of
motion [36]. In it he developed Newton's mechanics in a form more familiar to us
now, except for the Latin symbols. His version of the basic newtonian equation
F=ma was dc = p dt, where c stands
for "celeritas" meaning speed, and p stands for "potentia", meaning force.

Apart from in relativity, the most pervasive use of c to represent a speed today
is in the wave equation. In 1747 Jean d'Alembert made a mathematical study of the
vibrating string and discovered the one dimensional wave equation, but he wrote it without
the velocity constant. Euler generalised d'Alembert's equation to include the
velocity, denoting it by the letter a [38]. The general solution is y
= f(x - at) + f(x + at), representing two waves of fixed shape
travelling in opposite directions with velocity a.

Euler was one of the most prolific mathematicians of all time. He wrote hundreds
of manuscripts and most of them were in Latin. If anyone established a convention
for using c for "celeritas", it has to have been Euler. In 1759 he studied
the vibrations of a drum, and moved on to the 2-dimensional wave equation. This he
wrote in the form we are looking for with c now the velocity constant [39].

The wave equation became a subject of much discussion, being investigated by all the
great mathematicians of the époque including Lagrange, Fourier, Laplace, and
Bernoulli. Through their works, Euler's form of the wave equation with c for
the speed of wave propagation was carved in stone for good. To a first
approximation, sound waves are also governed by the same wave equation in three
dimensions, so it is not surprising that the speed of sound also came to be denoted by the
symbol c. This predates relativity and can be found, for example, in Lord
Rayleigh's classic text "Theory of Sound" [40]. Physicists of the nineteenth century
would have read the classic Latin texts on physics, and would have been aware that
c could stand for "celeritas". As an example, Lorentz used c in 1899
for Earth's speed through the ether [41]. We even know that Einstein used
it for speed outside relativity, because in a letter to a friend about a patent for a
flying machine, he used c for the speed of air flowing at a mere 4.9 m/s [42].

In conclusion, although we can trace c back to Weber's force law where it most
likely stood for "constant", it is possible that its use persisted because c could
stand for "celeritas" and had therefore become a conventional symbol for speed. We
cannot tell for sure how Drude, Lorentz, Planck or Einstein thought about their notation,
so there can be no definitive answer for what it stood for then. The only logical
answer is that when you use the symbol c, it stands for whatever possibility you
prefer.

<h2>References</h2>

<dl>
<dd>[1] Isaac Asimov "C for Celeritas" in "The Magazine of Fantasy and Science Fiction",
Nov-59 (1959), reprinted in "Of Time, Space, and Other Things", Discus (1975), and
"Asimov On Physics", Doubleday (1976)</dd> 

<dd>[2] R. Kohlrausch and W.E. Weber, "Ueber die Elektricitätsmenge, welche bei
galvanischen Strömen durch den Querschnitt der Kette fliesst", Annalen der Physik,
99, pg 10 (1856)</dd> 

<dd>[3] P. Drude, "Zum Studium des elektrischen Resonators", Göttingen Nachrichten (1894),
pgs 189–223 </dd> 

<dd>[4] H.A. Lorentz, "The theory of Electrons and its applications to the phenomena of
light and radiant heat". A course of lectures delivered in Columbia
University, New York, in March and April 1906, Leiden (1909)</dd> 

<dd>[5] R.D. Carmichael, "The Theory of Relativity", John Wiley & Sons (1913)</dd> 

<dd>[6] L. Silberstein, "The Theory of Relativity", Macmillan (1914)</dd> 

<dd>[7] E. Cunningham, "The Principle of Relativity", Cambridge University Press
(1914)</dd> 

<dd>[8] R.C. Tolman, "The Theory of the Relativity of Motion", University of California
Press (1917)</dd> 

<dd>[9] A. Einstein, From "The Collected Papers, Vol 2, The Swiss Years: Writings,
1900–1909", English Translation, he wrote five papers using V, e.g. "On
the Electrodynamics of Moving Bodies", Annalen Der Physik 17, pgs 891–921
(1905), "On the Inertia of Energy Required by the Relativity Principle", Annalen
Der Physik 23, pgs 371–384 (1907)</dd> 

<dd>[10] A. Einstein, e.g. "On the Theory of Light Production and Light Absorption",
Annalen Der Physik, 20, pgs 199–206 (1906)</dd> 

<dd>[11] A. Einstein, "On the Relativity Principle and the Conclusions Drawn From It",
Jahrbuch der Radioaktivität und Elektronik 4, pgs 411–462 (1907)</dd> 

<dd>[12] J. Clerk Maxwell, "A dynamical theory of the electromagnetic field",
Philos. Trans. Roy. Soc. 155, pgs 459–512 (1865). Abstract:
Proceedings of the Royal Society of London 13, pgs 531–536
(1864)</dd> 

<dd>[13] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon
Press (1873)</dd> 

<dd>[14] A.A. Michelson and E.W. Morley, "On the Relative Motion of the Earth and the
Luminiferous Ether", Amer. J. Sci. 34, pgs 333–345 (1887),
Philos. Mag. 24, pgs 449–463 (1887)</dd> 

<dd>[15] O. Lodge, "Aberration Problems", Phil. Trans. Roy. Soc. 184, pgs 729–804
(1893)</dd> 

<dd>[16] J. Larmor, "A Dynamical Theory of the Electric and Luminiferous Medium I",
Phil. Trans. Roy. Soc. 185, pgs 719–822 (1894)</dd> 

<dd>[17] H. Poincaré, "Cours de physique mathématique. Electricité et optique.
La lumière et les théories électrodynamiques" (1900)</dd> 

<dd>[18] P. Drude, "Physik des Äthers auf elektromagnetischer Grundlage", Verlag F. Enke,
Stuttgart (1894)</dd> 

<dd>[19] H. Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen
in bewegten Körpern", Leiden (1895)</dd> 

<dd>[20] A. Einstein, from "The Collected Papers, Vol 5, The Swiss Years: Correspondence,
1902–1914", English Translation, Doc 58. </dd> 

<dd>[21] P. Drude, "The theory of optics", translated from German by C.R. Mann and
R.A. Millikan, New York, Longmans, Green, and Co. (1902) </dd> 

<dd>[22] M. Planck, "Uber irreversible Strahlungsvorgange", Verl. d. Kgl. Akad. d. Wiss.
(1899) </dd> 

<dd>[23] H.A. Lorentz, "De l'Influence du Mouvement de la Terre sur les Phenomenes
Lumineux", Arch. Neerl. 21, pg 103 (1887)</dd> 

<dd>[24] H.A. Lorentz, "On the Reflection of Light by Moving Bodies",
Versl. Kon. Akad. Wetensch Amsterdam I, 74 (1892)</dd> 

<dd>[25] H.A. Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen
in bewegten Körpern", Leiden (1895)</dd> 

<dd>[26] H. A. Lorentz, "Théorie simplifiée des phenomènes electriques et optiques dans
des corps en mouvement", Proc. Roy. Acad. Amsterdam I 427 (1899)</dd> 

<dd>[27] H.A. Lorentz, "Maxwells elektromagnetische Theorie" Encyclopädie der
Mathematischen Wissenschaften. Leipzig, Teubner (1903)</dd> 

<dd>[28] M. Abraham, "Prinzipien der Dynamik des Elektrons", Annalen der Physik 10,
pgs 105–179 (1903)</dd> 

<dd>[29] e.g. J.J. Thomson and G.F.C. Searle, "A Determination of `v', the Ratio of the
Electromagnetic Unit of Electricity to the Electrostatic Unit", Proc. Roy. Soc.
Lond. 181, pg 583 (1890), M. Hurmuzescu, "Nouvelle determination du
rapport v entre les unites electrostatiques et electromagnetiques", Ann. de
Chim. et de Phys., 7a serie T. X April 1897, pg 433. (1897)</dd> 

<dd>[30] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon
Press, Vol II; Chapter 23, section 849 (1873)</dd> 

<dd>[31] K.F. Gauss, "Zur mathematischen Theorie der elektrodynamischen Wirkung" (1835),
in "Werke", Göttingen 1867; Vol. V, pg 602</dd> 

<dd>[32] W. Weber, "Elektrodynamische Maassbestimmingen uber ein allgemeines Grundgesetz
der elektrischen Wirkung", Abh. Leibnizens Ges., Leipzig (1846)</dd> 

<dd>[33] G. Kirchhoff, "Ueber die Bewegung der Elektricität in Leitern"
Ann. Phys. Chem. 102, 529–544 (1857)</dd> 

<dd>[34] G.F.B. Riemann, "Ein Beitrag zur Elektrodynamik", Annalen der Physik und Chemie,
pg 131 (1867)</dd> 

<dd>[35] G. Kirchhoff, "Zur Theorie der Entladung einer Leydener Flasche",
Pogg. Ann. 121 (1864)</dd> 

<dd>[36] J. Hermann, "Phoronomia", Amsterdam, Wetsten, (1716)</dd> 

<dd>[37] J. d'Alembert, "Recherches sur les cordes vibrantes", LAcadémie Royal des
Sciences (1747)</dd> 

<dd>[38] L. Euler, "De La Propagation Du Son" Memoires de l'acadamie des sciences de
Berlin [15] (1759), 1766, pgs 185–209, in "Opera physica miscellanea
epistolae. Volumen primum", pg 432 </dd> 

<dd>[39] L. Euler, "Eclaircissemens Plus Detailles Sur La Generation et La Propagation Du
Son Et Sur La Formation De L'Echo", "Memoires de l'acadamie des sciences de
Berlin" [21] (1765), 1767, pgs 335–363 in "Opera physica miscellanea
epistolae. Volumen primum", pg 540 </dd> 

<dd>[40] J.W. Strutt, "Theory of Sound" Vol 1, pg 251, McMillan and Co. (1877)</dd> 

<dd>[41] H.A. Lorentz, "Stokes' Theory of Aberration in the Supposition of a Variable
Density of the Aether", Proc. Roy. Acad. Amsterdam I, pg 443 (1899)</dd> 

<dd>[42] A. Einstein, "The Collected Papers, Vol 5, The Swiss Years: Correspondence,
1902–1914", English Translation, Doc 86 (1907)</dd> 

<dd>[43] W. Voigt, "Ueber das Doppler'sche Princip", Goett. Nachr. 2, pg 41
(1887)</dd> 

<dd>[44] E. Cohn, "Zur Elektrodynamik bewegter Systeme. II", Sitzungsberichte der
Königlich Preussischen Akademie der Wissenschaften zu Berlin, der
physikalisch-mathematischen Classe (1904)</dd> 

<dd>[45] M. Brillouin, "Le mouvement de la Terre et la vitesse de la lumière", comptes
rendu 140, pg 1674 (1905)<dd> 

<dd>[46] G. Herglotz, "Zur Elektronentheorie", Nachrichten von der Gesellschaft 6,
pg 357 (1903)</dd> 

<dd>[47] F. Himstedt, "Ueber die Schwingungen eines Magneten unter dem dämpfenden Einfluß
einer Kupferkugel", Nachrichten von der Gesellschaft 11, pg 308
(1875)</dd> 

<dd>[48] H. Helmholtz, Berlin: Verl. d. Kgl. Akad. d. Wiss. (1892)</dd> 

<dd>[49] H. Hertz, "Electric Waves", Macmillan (1893)</dd> 

</dl>

 

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