Homersham Cox (mathematician)
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Homersham Cox (1857–1918) was an English mathematician.


Life

He was the son of Homersham Cox (1821–1897) and brother of Harold Cox and was educated at
Tonbridge School Tonbridge School is a public school (English fee-charging boarding and day school for boys aged 13–18) in Tonbridge, Kent, England, founded in 1553 by Sir Andrew Judde (sometimes spelt Judd). It is a member of the Eton Group and has clo ...
(1870–75). At
Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...
, he graduated B.A. as 4th wrangler in 1880, and MA in 1883. He became a
fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...
in 1881. His younger sister Margaret, described him as a man often completely lost in his thoughts. He was married to Amy Cox. Later they separated and she started working as a governess in Russia in 1907. Cox wrote four papers applying algebra to physics, and then turned to
mathematics education In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...
with a book on
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
in 1885. His ''Principles of Arithmetic'' included
binary number A binary number is a number expressed in the Radix, base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may ...
s,
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, and
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s. Contracted to teach mathematics at
Muir Central College Muir Central College in Prayagraj (formerly Allahabad) in northern India was a college of higher education founded by William Muir in 1872. It had a separate existence to 1921, when as a result of the Allahabad University Act it was merged into U ...
, Cox became a resident of
Allahabad Prayagraj (, ; ISO 15919, ISO: ), formerly and colloquially known as Allahabad, is a metropolis in the Indian state of Uttar Pradesh.The other five cities were: Agra, Kanpur, Kanpur (Cawnpore), Lucknow, Meerut, and Varanasi, Varanasi (Benar ...
, Uttar Pradesh from 1891 till his death in 1918. He was married to Amy Cox, by whom he had a daughter, Ursula Cox.


Work on non-Euclidean geometry

From 1881 to 1883, he published papers on
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
. For instance, in his 1881 paper (which was published in two parts in 1881 and 1882) he described
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
for
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
, now called Weierstrass coordinates of the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
introduced by
Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of M ...
(1879) and
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
(1881)). Like Poincaré in 1881, Cox wrote the general
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s leaving invariant the quadratic form z^2-x^2-y^2=1, and also for w^2-x^2-y^2-z^2=1. He also formulated the
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
which he described as a transfer of the origin in the hyperbolic plane, on page 194: :\beginX & =x\cosh p-z\sinh p\\ Z & =-x\sinh p+z\cosh p \end \quad \text \quad \beginx & =X\cosh p+Z\sinh p\\ z & =X\sinh p+Z\cosh p \end Similar formulas have been used by Gustav von Escherich in 1874, whom Cox mentions on page 186. In his 1882/1883 paper, which deals with non-Euclidean geometry,
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s and
exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
, he provided the following formula describing a transfer of point P to point Q in the hyperbolic plane, on page 86 : \begin QP^ & =\cosh\theta+\iota\sinh\theta\\ QP^ & =e^ \end \quad (\iota^2=1) 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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
multiplications. The variant \iota^2=1 is now called a
hyperbolic number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+yj ...
, the whole expression on the left can be used as a hyperbolic
versor In mathematics, a versor is a quaternion of Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 conditi ...
. Subsequently, that paper was described by
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...
(1898) as follows:


Cox's chain

In 1891 Cox published a chain of theorems in Euclidean geometry of three dimensions: (i) In space of three dimensions take a point 0 through which pass sundry planes ''a, b, c, d, e'',.... (ii) Each two planes intersect in a line through 0. On each such line a point is taken at random. The point on the line of intersection of the planes ''a'' and ''b'' will be called the point ''ab''. (iii) Three planes ''a, b, c'', give three points ''bc, ac, ab''. These determine a plane. It will be called the plane ''abc''. Thus the planes ''a, b, c, abc'', form a tetrahedron with vertices ''bc, ac, ab'', 0. (iv) Four planes ''a, b, c, d'', give four planes ''abc, abd, acd, bcd''. It can be proved that these meet in a point. Call it the point ''abcd''. (v) Five planes ''a, b, c, d, e'', give five points such as ''abcd''. It can be proved that these lie in a plane. Call it the plane ''abcde''. (vi) Six planes ''a, b, c, d, e, f'', give six planes such as ''abcde''. It can be proved that these meet in a point. Call it the point . And so on indefinitely. The theorem has been compared to Clifford's circle theorems since they both are an infinite chain of theorems. In 1941 Richmond argued that Cox's chain was superior: :Cox's interest lay in the discovery of applications of Grassmann's Ausdehnungslehre and he uses the chain to that end. Any present-day geometer (to whom many of Cox's properties of circles in a plane must appear not a little artificial) would agree that his figure of points and planes in space is simpler and more fundamental than that of circles in a plane which he derives from it. Yet this figure of 2n circles shows beyond a doubt the superiority of Cox's chain over Clifford's; for the latter is included as a special case when half the circles in the former shrink into points. Cox's plane figure of 2n circles can be derived by elementary methods. H. S. M. Coxeter derived Clifford's theorem by exchanging the arbitrary point on a line ''ab'' with an arbitrary sphere about 0 which then intersects ''ab''. The planes ''a, b, c'', ... intersect this sphere in circles which can be projected stereographically into a plane. The planar language of Cox then translates to the circles of Clifford. In 1965 Cox's first three theorems were proven in Coxeter's
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...
''Introduction to Geometry''.H. S. M Coxeter (1965) ''Introduction to Geometry'', page 258,
John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...


Works


References


Bibliography

* * * {{DEFAULTSORT:Cox, Homersham 1857 births 1918 deaths 19th-century English mathematicians 20th-century English mathematicians British geometers Fellows of Trinity College, Cambridge English expatriates in India People educated at Tonbridge School Alumni of Trinity College, Cambridge