Back in the days before blogs and web forums when we used to discuss physics on usenet I was for a short while the editor of the Physics FAQ. Although the FAQ is still cared for by Don Koks it’s role has been mostly superseded by Wikipedia for better or for worse. In my time I wrote a number of articles for the FAQ with a special interest in the relativity questions. A lot of new discoveries have been made since then and some of them impact the cosmology sections. I’d like to update a few of my earlier articles and rewrite some others in my own words. I’ll post them here over time and I may even write some new ones. Feel free to add comments or ask further questions.
Why is “c” used for the speed of light?
The short answer
According to the science fiction writer Isaac Asimov, the reason why is used as the symbol for the speed of light is that stands for celeritas, the Latin word for speed , but is this really true?
Tracing the origin of the use of a given symbol in science is not always straight forward. Scientists rarely trouble to note the origin of the notations they use. When Einstein wrote his first papers on the theory of relativity he chose as his symbol for the speed of light . In 1907 he suddenly changed it to  without explanation but it is likely that he was being influenced by earlier usage rather than choosing his own notation. From that point on the notation was used by everyone.
When you try to track where this came from and why, three possible explanations come up. There is indeed some evidence to show that was sometimes used as a generic symbol for speed because it stands for celeritas, but there are two more specific influences that seem more concrete. One comes from the use of the letter as the speed variable in the wave equation. This can be traced back to the work of Euler who developed it in two and three dimensions during the eighteenth century. When he choose the symbol he had already used and as other variables so the most obvious reason for his choice was that was simply the next letter in the alphabet.
Another equally plausible explanation has its origins in a paper from 1856 by Weber and Kohlrausch on electricity and magnetism . They introduced a constant with dimensions of speed that was important to the interrelationships between the two forces. It is clear from the text and subsequent usage that simply stood for constant. Later this quantity was shown to be related to electromagnetic wave propagation and was then popularised as a symbol for the speed of light.
It is surprising that the notation seems to have more than one origin and it is not easy to determine which of these was the most influential, but probably all three played a part over time in establishing the latter as the universally accepted symbol for the speed of light.
Does c stand for celeritas?
In 1959 Isaac Asimov penned an article for a sci-fi magazine entitled “C for Celeritas” in which he claimed he knew why is used 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.” he wrote. The article was reprinted in some of his books  which sold many copies to budding scientists, so it is not surprising that since then, Asimov’s answer has become a factoid repeated in many other articles and books.
However, if you go back and read his essay you discover that Asimov merely stated his case in that one sentence. He made no further attempt to justify his theory for the origin of the notation. So is his claim really born out by history, or was 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 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?
It is certainly true that the letter has been used as a generic symbol for speed since at least the 18th century. For example it is also usually used as the standard symbol for the speed of sound. 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. However, their writing style was very geometric and descriptive. They did not tend to write down formulae where speed is denoted by a symbol. Possibly the earliest example of the letter being used for speed can be found in a work written in the eighteenth century. In 1716 Jacob Hermann published a Latin text called “Phoronomia”, meaning the science of motion . 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 stands for “celeritas” meaning speed, and stands for “potentia”, meaning force.
Physicists of the nineteenth century would have read the classic Latin texts on physics, and would have been aware that could stand for “celeritas”. As an example, Lorentz used in 1899 for the speed of the Earth through the ether . Typically Einstein had little interest in Latin at school considering it to be a useless subject. Yet we know that even Einstein used for speed outside relativity. In a letter to a friend about a patent for a flying machine, he used for the speed of air flowing at a mere 4.9 m/s .
All this lends plausibility to the argument that stands for celeritas but there is also persuasive evidence for other origins of the notation, so the best we can really say is that the generic use of as a symbol for speed helped the notation to prevail over other alternatives in use around the turn of the twentieth century.
Does c stand for constant?
The idea that might stand for constant rather than celeritas may seem a little too obvious to be true, but there is good reason to propose that this is really the correct origin of the notation.
Although is now the universal symbol for the speed of light, the most common symbol in the nineteenth century was an upper-case which Maxwell had started using in 1865 . Maxwell may have borrowed this from Foucault who had used the same symbol earlier in his thesis of 1853 . That was then the notation adopted by Einstein for his seminal papers on relativity from 1905.
The most convincing origin of the letter being used for the speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch . They defined and measured a quantity denoted by 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 became the symbol for the speed of electrodynamic waves . In optics Drude continued to follow Maxwell in using an upper-case for the speed of light. Progressively the 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 to in his papers, it had become the standard symbol for the speed of light in vacuum for electrodynamics, optics, thermodynamics and relativity.
In France and England the electromagnetic constant was often symbolised by a lower case rather than Drude’s . 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 for the former and for the latter. Maxwell described a whole suite of possible experiments in electromagnetism to determine . Those that had not already been done were performed one after the other in England and France over the three decades that followed . In this context, lower case was always used for the quantity measured. But using was doomed to pass away once authors had to write relativistic equations involving moving bodies, because 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 in this way, he also had a minor use for the symbol 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 . 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 , and the complete form was published by Weber in 1846 . 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 . To give a theoretical backing they rewrote Weber’s force law in terms of the measured constant and used the symbol . This 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  and another by Riemann  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 . The odd thing is that when Maxwell wrote down the force law, he changed the variable 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 for a value equal to the speed of light, although he defined it as the speed of electricity through wires instead.
So 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 for the speed of light itself. Firstly, when Maxwell wrote Weber’s force law in his treatise in 1873, he modified the scale of 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 , the paper by Kirchhoff that he cited  was using 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 or invented it indepednently, but he did go one step further in explicitly naming his 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 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  Weber’s constant was introduced with these words “.. and the constant represents that relative speed, that the electrical masses e and e must have and keep, if they are not to affect each other.” This was written in German and the modern German word for constant is “konstant” but at that time before spelling was completely fixed it was often written in German with an initial “c”. This was the case here. So it appears that originated as a letter standing for “constant” rather than “celeritas”. However, it had nothing to do with the constancy of the speed of light until much later.
Why is c used in the wave equation?
The speed of light is not the only place in physics where is commonly used to represent speed. It is also the standard notation for the speed of sound and is used in the linear wave equation. In its simplest scalar form it can be written like this
You might think that this form of the wave equation is used because physicists most commonly think of the wave equation in the context of relativistic physics where represents the speed of light. It would be a natural progression to then use the same symbol for other wave equations even if then stands for a different speed such as the speed of sound or water waves. The literature shows that this is not the correct explanation because this usage in the wave equation goes back before light was even known as a wave phenomena.
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 . The general solution is y = f(x – at) + f(x + at), representing a supposition of two waves of fixed shape travelling in opposite directions with velocity .
Euler was one of the most prolific mathematicians of all time. He wrote hundreds of manuscripts and most of them were in Latin. Was it Euler then who established a convention for using for “celeritas”? 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 now the velocity constant .
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 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 . This predates relativity and can be found, for example, in Lord Rayleigh’s classic text “Theory of Sound” published in 1877 .
So can we tell why Euler used the letter ? Most of his work on the wave equation was written in French rather than Latin and there is no reason to suppose he was thinking of the Latin word celeritas. It is true that the French sometimes use the word célérité to mean the phase velocity in the context of the wave equation. They even sometimes talk of “célérité du son” and “ccélérité de la lumière” for the speed of sound and light, but there is no evidence that this goes back to the time of Euler. Euler just used “vitesse” the more common French word for speed. It is more likely that the French usage was adopted later as a consequence of Euler’s use of in the wave equation.
Euler’s style of notation for algebra and calculus was very similar to basic usage in mathematics and physics today. He commonly picked sequences of letters such as a, b, c or x, y, z for variables, sometimes changing to upper case or Greek letters with little indication that the letters actually stood for anything in particular. In fact Euler set many of the notations we use today including the use of the Greek letter for 3.14159… One possibility for his choice of is that when he moved from the one dimensional wave equation to the 2-dimensional case, he may have used and for the components of the wave velocity, then writing for the magnitude. Unfortunately there is no written evidence to support this theory either.
There is however a case for an even more mundane answer. When he introduced the variable for possibly the first time he defined it with an equation that he wrote
and were simply two physical variables he had already introduced, so seems to be chosen as merely the next letter in the alphabet.
Why did Einstein switch notation for the speed of light?
A lower-case 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) , Carmichael (1913) , Silberstein (1914) , Cunningham (1915) , and Tolman (1917) . That was not the case just a few years before. In his earliest papers on relativity from 1905—1907 Einstein began by using an upper-case for the speed of light . At that time he was also writing papers about the thermodynamics of radiation, and in those he used up upper-case . 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 . There he used the lower case , and his most famous equation finally took it’s familiar form.
It is not difficult to find where the upper case had come from. Maxwell used it extensively in his publications on electrodynamics from as early as 1865 . It was the principal symbol for the speed of light in his 1873 treatise on electrodynamics . 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 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 . That was written in the United States, but the same notation was also found across Europe, from papers by Oliver Lodge  and Joseph Lamor  in England, to the lecture notes of Poincaré in France , and the textbooks of Paul Drude in Germany  and Lorentz in the Netherlands . 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 for the speed of light .
Why then, did he change it to 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 . Einstein mentioned to Stark that it was hard for him to find the time to read published scientific articles in order to acquaint himself with all the work others have done in the field, but he had seen papers by Lorentz, Kohn, Monsegeil and Planck . Lorentz and Planck in particular had been using 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 is simply more practical. The upper-case 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 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 for the speed of light when he wrote down an almost complete form of the Lorentz transformations in 1887 . Voigt’s was later used by a few other physicists [37, 38,50], but Drude did not use his teacher’s notation. Drude first used the symbol in 1894, and in doing so he referenced a paper by Kirchhoff . As already mentioned, Paul Drude also used . In fact he made a distinction of using in the theory of optics for the directly-measured speed of light in vacuum, whereas he used 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 , which is divided into two parts with used in the first and 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  opted for an elaborate script B, while Himstedt , Helmholtz  and Hertz  wrote the equations of electrodynamics with the letter for the reciprocal of the speed of light. In 1899 Planck backed Drude by using , 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 . 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 , but then switched to Maxwell’s upper case . He wrote a book in 1895  that contained the equations for length contraction, and was cited by Einstein in his 1907 paper. While Drude had started to use , Lorentz was still using in this book. He continued to use until 1899 , but by 1903 when he wrote an encyclopedia article on electrodynamics  he too used . Max Abraham was another early user of the symbol in 1902, in a paper that was seen by Einstein . From Drude’s original influence, followed by Planck and Lorentz, by 1907 the symbol had become the prevailing notation in Germanic science and it made perfect sense for Einstein to adopt it too.
What was the earliest symbol used for the speed of light?
It is hard to be certain when a symbol was first used for the speed of light, but Doppler used the Greek letter in his paper on the frequency shift of light with velocity in 1842 . There were other papers written about the speed of light before then but none of them seems to have used any symbol to represent it.
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