Radiant Radian concept
Radiant Radian concept
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the center of a plane circle by an arc that is equal in length to the radius. The unit is defined in the SI as the coherent unit for plane angle, as well as for phase angle. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.
https://en.m.wikipedia.org/wiki/Radian
In 1714, British mathematician Roger Cotes (1682–1716) published a paper titled "Logometria." In it, he developed the idea of an angular unit derived from the ratio of an arclength along the circumference of a circle to the radius of the circle. He defined this unit, now known as the radian, as the angle corresponding to an arclength that is equal to the radius of the circle. This measurement of angular separation is approximately equal to 57.3 degrees. The unit of radians uses no arbitrary definitions as to what it means. It can therefore easily be used in mathematical applications ranging from trigonometry to calculus. In trigonometry, the basic functions of the sine, cosine, and tangent of an angle are defined as the ratios between the lengths of two sides of a right triangle. Since the radian is a ratio of two lengths, it has no units and can be used in trigonometry. In calculus, the need for a unit-less quantity was clear when trying to calculate simple harmonic motions of molecules and other simple oscillators.
https://www.ebsco.com/research-starters/mathematics/radians-and-degrees
Roger Cotes FRS (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also devised the quadrature formulas known as Newton–Cotes formulas, which originated from Newton's research,[4] and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula.[5] He was the first Plumian Professor at Cambridge University from 1707 until his death.
https://en.m.wikipedia.org/wiki/Roger_Cotes
RADIAN. According to Cajori1 (1919, page 484):
An isolated matter of interest is the origin of the term "radians", used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was the father of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between "rads," "radials" and "radians." In 1874, T. Muir adopted "radians" after a consultation with James Thomson.
In a footnote, Cajori gives a reference to Nature, Vol. 83, pp. 156, 217, 459, 460 [Julio González Cabillón].
In fact Cajori made a mistake. The James Thomson above was the elder brother and not the father of Lord Kelvin. This James Thomson eventually became professor of engineering in Glasgow, and, as mentioned below, it was his son (also called James Thomson) who debated with Muir in the Letters pages of Nature in 1910.
In a letter appearing in the April 7, 1910, Nature, Thomas Muir wrote: I wrote to him [i.e., to Alexander J. Ellis, in 1874], and he agreed at once for the form "radians," on the ground that it could be viewed as a contraction for "radial angles."
In a letter appearing in the June 16, 1910, Nature, James Thomson (the son of James Thomson) wrote: "I shall be very pleased to send Dr. Muir a copy of my father's examination questions of June, 1873, containing the word "radians." ... It thus appears that "radians" was thought of independently by Dr. Muir and my father, and, what is really more important than the exact form of the name, they both independently thought of the necessity of giving a name to the unit-angle" [Dave Cohen].
According to W. N. Roseveare, "The radian is an uninteresting angle—Lord Kelvin introduced the word merely as a convenience in lecturing, to avoid the long phrase 'angle whose circular measure is.'" This quotation appears on page 133 of W. N. Roseveare, "On 'circular measure' and the product forms of the sine and cosine," Mathematical Gazette 3 #49 (January 1905), 129-137. [Dave L. Renfro]
In 1867, William Thomson and Peter Guthrie Tait Treatise on Natural Philosophy (page 31) mention the unit radian equal to 180 deg/π:
For brevity we shall call this angle a radian.
A post on the Internet indicated that Thomas Muir (1844-1934) claimed to have coined the term in 1869, and that Muir and Ellis proposed the term as a contraction of "radial angle" in 1874. A reference given was: Michael Cooper, "Who named the radian?", Mathematical gazette 76, no. 475 (1992) 100-101. I have not seen this article.
A 1991 Prentice-Hall high school textbook, Algebra 2, by Bettye C. Hall and Mona Fabricant has: "James Muir, a mathematician, and James T. Thomson, a physicist, were working independently during the late nineteenth century to develop a new unit of angle measurement. They met and agreed on the name radian, a shortened form of the phrase radial angle. Different names were used for the new unit until about 1900. Today the term radian is in common usage."
In 1876-79, the Globe encyclopaedia of universal information has, in the Circle article: "The unit, called a radian by Professor James Thomson, is that angle whose subtending arc is equal in length to the radius" [University of Michigan Digital Library].
1 Florian Cajori 1859 - 1930 Florian Cajori was a Swiss mathematician best known for his influential History of Mathematics.