George Osborn (Mathematician)

George Osborn (1864-1932) was an English mathematician, known for Osborn’s rule that deals with Hyperbolic functions identities.
== Life ==
George Osborn was born in 1864 in Manchester, England. He attended Emmanuel College, Cambridge, University of Cambridge in 1884. Where in 1887 he received the 17th Wrangler (University of Cambridge) award for achieving a 1st in his mathematics degree. He then attended The Leys School, Cambridge in 1888. In 1891 George Osborn became assistant headmaster and senior science master at The Leys School. He continued to work at the school until his retirement in 1926. Apart from mathematics, George Osborn took his time to study the New Testament owing to his grand farther George Osborn the president of the Methodist Conference in 1863 and 1881. In addition to this, George Osborn enjoyed reading Spanish literature and was an avid chess player up until his death on October 14th, 1932.
== Work ==
From 1902-1925 George Osborn would write numerous articles for The Mathematical Gazette. In his submissions he covered a range of topics from sums of cubes to series expansions. However, his most notable entry was in July of 1902 titled: Mnemonic for Hyperbolic functions. In this publication George Osborn outlined a rule, that he found useful for teaching, when converting between Trigonometric functions and Hyperbolic functions identities. In conjunction with this George Osborn published various books with his colleague Charles Henry French, who was the head of mathematics at The Leys School, Cambridge. The titles of their joint work include: Elementary Algebra, First Year’s Algebra and The Graphical Representation of Algebraic function.
== Osborn's Rule ==
Osborn’s Rule which was outlined by George Osborn in his 1902 The Mathematical Gazette publication: Mnemonic for Hyperbolic functions. Aids in the conversion between Trigonometric functions and hyperbolic trigonometric identities. To convert a Trigonometric functions identity to the equivalent hyperbolic Trigonometric functions, Osborn’s rule states to first write out all the cosine and sine compound angles terms to their expanded constituent parts. Then exchange all the cosine and sine terms to cosh and sinh terms. However, for all products or implied products of two sine terms replace it with the negative product of two sinh terms. This is because i\sin(ix) is equivalent to \sinh(x), so when multiplied to together the sign switched when compared to the regular Trigonometric functions. Due to this fact however, for terms which have a product of a multiple of four sinh terms the sign does not change when compared to the regular Trigonometric functions.
Trigonometric Identity:
\cos^2(x)+\sin^2(x)=1
1+\tan^2(x)=\sec^2(x)
\cot^2(x)+1=\csc^2(x)
\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)
\cos(2x)=1-2\sin(x)
Hyperbolic Trigonometric Identity:
\cosh^2(x)-\sinh^2(x)=1
1-\tanh^2(x)=\operatorname{sech^2}(x)
-\coth^2(x)+1=-\operatorname{csch^2}(x)
\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)
\cosh(2x)=1+2\sinh(x)
Category:1864 births
Category:1932 deaths
Category:Mathematician
Category:People from Manchester
Category:Alumni of Emmanuel College, Cambridge
Category:Cambridge University alumni