Homersham Cox (mathematician)

Homersham Cox (1857–1918) was an English mathematician.^[1]^[2]

Life

He was the son of Homersham Cox (1821–1897) and brother of Harold Cox and was educated at Tonbridge School (1870-75). At Trinity College, Cambridge, he graduated B.A. as 4th wrangler in 1880, and M.A. in 1883. He became a fellow in 1881. He obtained a professorship of mathematics at Muir Central College from 1891 to 1918.

Work on non-Euclidean geometry

1881–1883 he published papers on non-Euclidean geometry,^{[c 1]}^{[c 2]}^{[c 3]}^{[c 4]} and in 1885 the textbook "Principles of Arithmetic".^{[c 5]}

For instance, in his 1881 paper (which was published in two parts in 1881 and 1882)^{[c 1]}^{[c 2]} he described homogeneous coordinates for hyperbolic geometry, now called Weierstrass coordinates of the hyperboloid model introduced by Wilhelm Killing (1879) and Henri Poincaré (1881)). Similar formulas have been used by Gustav von Escherich in 1874, whom Cox mentions on page 186. Like Poincaré in 1881, Cox wrote the general Lorentz transformations leaving invariant the quadratic form $z^{2}-x^{2}-y^{2}=1$ , and in addition also for $w^{2}-x^{2}-y^{2}-z^{2}=1$ (see History of Lorentz transformations#Cox). He also formulated the Lorentz boost which he described as a transfer of the origin in the hyperbolic plane, on page 194:

{\begin{aligned}X&=x\cosh p-z\sinh p\\Z&=-x\sinh p+z\cosh p\end{aligned}}

and

{\begin{aligned}x&=X\cosh p+Z\sinh p\\z&=X\sinh p+Z\cosh p\end{aligned}}

In his 1882/1883 paper^{[c 3]}^{[c 4]}, which deals with Non-Euclidean geometry, quaternions and exterior algebra, he provided the following formula describing a transfer of point P to point Q in the hyperbolic plane, on page 86

{\begin{matrix}QP^{-1}=\cosh \theta +\iota \sinh \theta \\QP^{-1}=e^{\iota \theta }\end{matrix}}\left(\iota ^{2}=1\right)

together with $\cos \theta +\iota \sin \theta$ with $\iota ^{2}=-1$ for elliptic space, and $1-\iota \theta$ with $\iota ^{2}=0$ for parabolic space. On page 88, he identified all these cases as quaternion multiplications. The variant $\iota ^{2}=1$ is now called a hyperbolic number, the whole expression on the left can be used as a hyperbolic versor. Subsequently, that paper was described by Alfred North Whitehead (1898) as follows:^[3]

Homersham Cox constructs a linear algebra [cf. 22] analogous to Clifford's Biquaternions which applies to Hyperbolic Geometry of two and three and higher dimensions. He also points out the applicability of Grassmann's Inner Multiplication for the expression of the distance formulae both in Elliptic and Hyperbolic Space; and applies it to the metrical theory of systems of forces. His whole paper is most suggestive.

References

↑ Steed, H. E., ed. (1911). The register of Tonbridge School from 1826 to 1910. Rivingtons. p. 150.
↑ "Cox, Homersham (CS875H)". A Cambridge Alumni Database. University of Cambridge.
↑ Whitehead, A. (1898). A Treatise on Universal Algebra. Cambridge University Press. p. 370.

Works

1 2 Cox, H. (1881). "Homogeneous coordinates in imaginary geometry and their application to systems of forces". The quarterly journal of pure and applied mathematics. 18 (70): 178–192.
1 2 Cox, H. (1882) [1881]. "Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued)". The quarterly journal of pure and applied mathematics. 18 (71): 193–215.
1 2 Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Trans. Camb. Phil. Soc. 13: 69–143.
1 2 Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Proc. Camb. Phil. Soc. 4: 194–196.
↑ Cox, H. (1885). Principles of Arithmetic. Deighton.

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[1] Steed, H. E., ed. (1911). The register of Tonbridge School from 1826 to 1910. Rivingtons. p. 150.

[2] "Cox, Homersham (CS875H)". A Cambridge Alumni Database. University of Cambridge.

[8] Whitehead, A. (1898). A Treatise on Universal Algebra. Cambridge University Press. p. 370.

[cox1-3] 1 2 Cox, H. (1881). "Homogeneous coordinates in imaginary geometry and their application to systems of forces". The quarterly journal of pure and applied mathematics. 18 (70): 178–192.

[cox2-4] 1 2 Cox, H. (1882) [1881]. "Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued)". The quarterly journal of pure and applied mathematics. 18 (71): 193–215.

[cox3-5] 1 2 Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Trans. Camb. Phil. Soc. 13: 69–143.

[cox4-6] 1 2 Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Proc. Camb. Phil. Soc. 4: 194–196.

[7] Cox, H. (1885). Principles of Arithmetic. Deighton.