By Philip Gibbs, 1997, 2004.

"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]

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".

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* = *mc*^{2}
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 d*c* = *p* d*t*, 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.

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- [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) - [3] P. Drude, "Zum Studium des elektrischen Resonators", Göttingen Nachrichten (1894), pgs 189–223
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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) - [10] A. Einstein, e.g. "On the Theory of Light Production and Light Absorption",
Annalen Der Physik,
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Jahrbuch der Radioaktivität und Elektronik
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Philos. Trans. Roy. Soc.
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Lumineux", Arch. Neerl.
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einer Kupferkugel", Nachrichten von der Gesellschaft
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