In February this year there appeared in Physics World an article entitled Constant Failure by Robert P Crease of Stony Brook University, in which he showed in how many formulae of physics and mathematics 2π turns up, rather than π. This article struck a chord with me, since even after many years I remember the feeling of "cognitive dissonance" when being taught that the formula was 2πR rather than πD.
I felt it a bit much, though, suggesting that Archimedes might have been mistaken in choosing to calculate the ratio of circumference to diameter rather than to radius. In those days, the fundamental dichotomy seems to have been between the geometers who thought of circumference, diameter and their ratio, and the astronomers who used the radius in their calculation of chord tables.
Hipparchus used a radius of 3438 which is the nearest integer to the number of minutes in 1 radian, but Ptolemy preferred 3600 as this is easier to calculate within the sexagesimal system. The work of these astronomers, further developed by Hindu and Arabic mathematicians, gives us our trigonometry of today.
In particular, Aryabhata published in 499 A.D. the Aryabhatiya in which he invented the sine function (radius!) as more convenient than the chord, but nevertheless computed the most accurate value of π (diameter!) known in ancient times. However al-Kashi, who was very much an astronomer and trigonometer, set a new record in precision in his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places.
The Greek geometers did not think of their ratio as a number. To them, number, magnitude and ratio were three distinct concepts. Then who first did? As it might say at the beginning of a tale from The Thousand and One Nights, "there were three brothers from Baghdad", namely the Banu Musa in the 9th Century, who are first recorded as having described this ratio as a number.
The first person to use π to represent the ratio of the circumference to the diameter (3.14159...) was the Welshman William Jones in 1706. But the radius fought back, with the word 'radian' first appeared in print in 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast.
He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University in Scotland, hesitated between 'rad', 'radial' and 'radian', adopting 'radian' after consultation with James Thomson. (A Welshman, an Irishman, and a Scotsman – is it a Celtic conspiracy?)
Even the difference between the two versions of Planck's constant ℎ and ℏ (aka the Dirac constant) depends on whether one is thinking physically in terms of frequency ν or mathematically in terms of angular velocity ω. Physics is not Applied Mathematics!