Homersham Cox (1857–1918) was an English mathematician.
Life
He was the son of
Homersham Cox (1821–1897) and brother of
Harold Cox
Harold Cox (1859 – 1 May 1936) was a Liberal MP for Preston from 1906 to 1910.
Early life
The son of Homersham Cox, a County Court judge, Cox was educated at Tonbridge School in Kent and was scholar and later fellow at Jesus College, Cam ...
and was educated at
Tonbridge School
(God Giveth the Increase)
, established =
, closed =
, type = Public schoolIndependent day and boarding
, religion =
, president =
, head_label ...
(1870–75). At
Trinity College, Cambridge
Trinity College is a 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 college at either Cambridge or Oxford. ...
, he graduated B.A. as 4th
wrangler in 1880, and
MA in 1883. He became a
fellow
A fellow is a concept whose exact meaning depends on context.
In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements.
Within the context of higher education ...
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 research into the transfer of mathematical knowledge.
Although re ...
with a book on
arithmetic in 1885. His ''Principles of Arithmetic'' included
binary numbers,
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
s, and
permutations.
Contracted to teach mathematics at
Muir Central College
Muir Central College in 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 Allahabad University.
...
, Cox became a resident of
Allahabad, 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
1881–1883 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 g ...
.
For instance, in his 1881 paper (which was published in two parts in 1881 and 1882)
he described homogeneous coordinates for hyperbolic geometry, 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 hyperbolo ...
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 Mü ...
(1879) and
Henri Poincaré (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 velo ...
s leaving invariant the quadratic form
, and in addition also for
. 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 i ...
which he described as a transfer of the origin in the hyperbolic plane, on page 194:
:
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,
quaternions and
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, he provided the following formula describing a transfer of point P to point Q in the hyperbolic plane, on page 86
:
together with
with
for elliptic space, and
with
for parabolic space. On page 88, he identified all these cases as
quaternion multiplications. The variant
is now called a
hyperbolic number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
, the whole expression on the left can be used as a hyperbolic
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
. Subsequently, that paper was described by
Alfred North Whitehead (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 2
n 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 2
n 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. Schoolbooks are textbook ...
''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 publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, ...
Works
References
Bibliography
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{{DEFAULTSORT:Cox, Homersham
1857 births
1918 deaths
19th-century English mathematicians
20th-century English mathematicians
Geometers
Fellows of Trinity College, Cambridge
English expatriates in India
People educated at Tonbridge School
Alumni of Trinity College, Cambridge