He married Adelaide Catherine Wilkin of Walton, Suffolk, on August 22, 1855.

The Rev. Robert Hartley provides us with the most complete portrait of the mathematician James Cockle with an obituary. He communicated Cockle's work in differential equations and saw the passion that he expressed for mathematics. However, since James served Australia as the first Chief Justice of Queensland, his fame as a judge rises to the fore.

This biography high-lights Cockle's contributions to the birth of linear algebra. In particular, his invention of tessarines and coquaternions came before Cayley's matrix algebra. The Presidential Address he gave when serving the London Mathematical Society displays his philosophy of mathematics.

James Cockle and Arthur Cayley each contributed early developments in linear algebra. Another great impetus came from Hamilton's quaternions, which came in 1843. The matrix product
( a_{i j} ) ( b_{i j} ) = ( c _{k m} ) where c_{k m} = ∑_i a_{k i} b_{i m}
was put down by Cayley's hand in 1855 , but by 1848 and 1849 James Cockle had exhibited two linear algebras beyond Hamilton's quaternions. The gestation of general linear algebra was a long process anchored in known structure of complex numbers, quaternions, and a few other particular linear algebras.

James Cockle was the first Chief Justice of Queensland. Serving from 1863 to 1879 at this post in the newly formed state on the continent of Australia merited him knighthood and a pension. His wife Adelaide bore him eight children; the family returned to England where James frequented the London Mathematical Society with Arthur Cayley. Though Cayley grew up in Russia, they had attended Trinity College, Cambridge University, where they became acquainted. They passed away January 25 and 26, 1895.

James Cockle

Arthur Cayley

One must appreciate the context of this linear algebra in the nineteenth century when the categories of mathematics were not yet well-demarcated. In particular, it wasn't until 1868 that Beltrami showed Lobachevski-Bolyai geometry consistent by exhibiting a mathematical model. Even Hamilton's quaternions of 1843 were incompletely understood because the mapping ideas now prevalent in complex function theory (e.g. conformal mapping) were only slowly becoming widespread. In some ways the ocean of linear possibilities amounts to a cultural breakdown when the Euclidean presumptions are stripped away. The matrix is an extraordinary number, stimulating imaginative mathematics for its comprehension. It has gained substance by proving its utility with now over a century of increasing usage. Coincidentally, Charles Darwin promulgated the theory of natural selection while linear agebra was getting started; both lines of thought undercut traditional notions. Getting to know the motor plane enables one to see that the realm of linear algebra goes beyond Euclidean distance concept. For instance, the 1882 upper half-plane HP model of ?hyperbolic geometry? is a commonly used facility for the basic exercises needed to transcend Euclid.

James Cockle invented or discovered two four-dimensional linear algebras. In each case assume the first row of a 2 x 2 complex matrix is ( w , z )
The tessarines (1848) are matricies with second row ( z , w ) .
The coquaternions (1849 , 1852 ) are matricies with second row ( z* , w * ) ; ( z* conjugate to z ) .
The matrix notation for a tessarine or coquaternion follows the Cayley method ordinarily seen. Of course, writing in the London-Edinburgh-Dublin Philosophical Magazine he could not make this modern presentation since the notation was not yet known. Instead,for coquaternions he used Hamilton's method with the quaternionic group underlying the linear space: He exhibited a group of eight basis elements which pair-wise span a linear 4-space and give it a product. (When a group has even order, it has Z₂ for a subgroup, so pairs can be taken as positive and negative units from an origin along a line.)
The first abstract linear algebra appeared in 1855 with the first hint from Arthur Cayley of his matrix product: It appeared in Crelle's Journal fur die reine und ungewandte Mathematik (Band 50, p. 282). The note was titled "Remarques sur la notation des fonctions algebraiques", and was the third of seven "differents memoires d'analyse" that appeared consecutively written by A. Cayley.
The main evidence that Cockle's algebras presaged Cayley's is that they inoculated the mathematical reader to the existence of new types of algebraic entities idempotents and nilpotents.
The tessarines introduce idempotent elements e that satisfy ee = e . Anybody would notice that the numbers zero(0) and one (1) have this property, but the introduction of more idempotents into the abstract algebra began with the tessarines.
Furthermore, when coquaternions were found as an associative 4-algebra and an alternative to Hamilton's quaternions, one had to contend with nilpotent elements n which satisfy nn = 0 . These are non-zero algebraic entities which abound in Cayley's system but were first found by Cockle. A complex plane generated by the real line and a nilpotent adjunct is now called a dual number plane; the concept was advanced at the dawn of the 20th Century by Eduard Study and Josef Grunwald as an aid to kinematics. In planar linear algebra then, the dual numbers demand consideration alongside the ordinary complex plane and the motor plane.

One of the consequences of having tessarines, is that there are algebraic entities called motors. Today the structural repercussions of these motors has led to a special mathematical literature.

The Presidential Address

As President of the London Mathematical Society in 1888, James Cockle had the duty and privilege of delivering a Presidential Address.It can be compared to Arthur Cayley's Southport Address delivered in 1882 when he was president of the British Association. Both Cockle and Cayley develop the convergence in philosophy between the concepts of space and time. This subject, opened by other writers of the time, including Bernard Riemann, W.K. Clifford, H.J.S. Smith (as interpreter of Clifford), and Hermann Helmholtz, can be considered a proto-relativity -- setting the stage for the Twentieth Century's theory of spacetime according to H.A. Lorentz, A. Einstein, and H. Minkowski.

Both Cayley and Cockle root their inquiries in classical literature: Proclus, Euclid, Aristotle. James Cockle leans heavily on Francis Bacon's Advancement of Learning and cites all the writers mentioned with more than fifty footnotes for its nine pages. His title is "Confluences and Bifurcations in Certain Theories", while the text makes clear the theories he has in mind are arithmetic and geometry, space and time, as well as cause and effect.His experience showed a convergence of geometry with algebra, and a bifurcation of planar geometric algebra according to the square (+1 or -1) of the unit element perpendicular to the real axis. He lets his years in jurisprudence show by allusion to the phrase "there and then" which appears in indictments. While Cayley, in 1882, went into mathematical details, Cockle does not include any particular mathematics, alluding only to "modern methods" at one point. According to Alexander MacFarlane, Cayley was blind to the alternative complex planes. Cockle, on the other hand, had played with tessarines and coquaternions while young; he had a healthy respect for their potency. By writing the "impossible equation" (1+j)(1-j) = 0 in 1852 he seems to have spooked himself and others since the topic was dropped by James and, except for WK Clifford, zero-divisors had to wait for the 20th Century for respect. They seemed then to be horrible lacuna in the terra firma of reason, suggesting possible sinkholes in reality.

Cockle assembles the clues available and presses the point, via Bacon, that appropriate axioms are in order. By considering first the convergences in arithmetic and geometry, he is inviting those looking into this scholarly document to investigate Clifford's motor plane, a concommitant of tessarines and coquaternions. But Cockle stops short of spelling out the consequent deformations: length contraction and time dilation, perhaps because he saw the ambiguities of a four-dimensional system. So in this philosophical work, Cockle puts in the hands of the London Mathematical Society the task of creating an adequate philosophy of space and time for the electric age. The following year Heinrich Hertz, collaborator of Hermann Helmholtz, successfully demonstrated a wireless transmitter-receiver system, citing the science of J.C. Maxwell, one writer that Cockle does not cite. While Maxwell has had his time in the sun, James Cockle's science of geometry, algebra, and differential equations is once more under consideration for mathematical models in physics.

Hartley says Cockle was "never really strong after his return home". Nevertheless, he was socially active, especially in the London Mathematical Society over which he presided as President 1886-8. He also served on Council of the Royal Astronomical Society 1888-92 where he had been a member since 1854. At the 1886 Colonial and Indian Exposition in London he was Commissioner of the Queensland section, so at age 67 he reflected the life of a traveller he had begun when he was 17.

Legacy

The service in the British Empire that quaternionist James Cockle evidenced on the bench in Queensland may have served to inspire other quaternionists. In the following generation there were three other quaternionists that traveled the globe in connection with their scientific work.
The most prominent was C.G. Knott who went to the Imperial University in Tokyo. Later, when returned to Edinburgh, he was the editor of a quaternion text by Tait and Kellend when a new edition was needed. A second was Alexander MacAulay, who went to Australia. While teaching at Ormond College, Melbourne (1892) he wrote "On the Mathematical Theory of Electromagnetism". Later, at the University of Tasmania, he wrote "Octonions" (1895).

The third was Alexander Macfarlane who had a sensational carreer in the American states and in retirement facilitated quaternion research with a textbook, journal articles, a bibliography, attending congresses, and leading an international organization in the project of fathoming 4-algebras like Hamilton's and its cousins.

Two appreciations of Sir James Cockle's contribution to Australian jurisprudence are found in books:
Founders of the Law in Australia by L.A. Whitfeld (Butterworths, 1971).
J.M. Bennett has penned a book Sir James Cockle, First Chief Justice of Queensland. This is the first (2003) in a series of books on Australian Chief Justices to be published by Federation Press. Within the legal profession the Cockle family name was perpetuated during the twentieth century by the following reference work: Ernest Cockle and W. Nembhard Hibbert (1929) Leading Cases on Common Law, Sweet & Maxwell, London.

Before he left Brisbane, Cockle endowed the local school with funds to perpetually award a prize to a top mathematics student. These Australian efforts immortalized him; in contrast the aura of Arthur Cayley in London and Cambridge eclipsed his later mathematical efforts, until now.

Ibagimov citations: Elementary Lie Group Analysis and Ordinary Differential Equations by Nail H. Ibragimov. On page 257 of his text, Ibragimov recalls James Cockle's contributions to invariants of differential equations. The three papers listed below are from p.339 of the text. Furthermore, the hyperbolic rotations of the plane of real tessarines form a one-parameter group of "Lorentz boosts". See page 149 of this book for a general table consisting of generators, invariants, and canonical variables.

One can find an obituary in the Proceedings of the Royal Society at London, v. LIX, pp. xxx - xxix, but tessarines and coquaternions do not figure there.

The 1911 Encyclopedia Britannica on James Cockle.

The Six Cockle References in MacFarlane's 1904 Bibliography 1848 - On Certain Functions Resembling Quaternions, and on a New Imaginary Algebra, Phil. Mag. (3) 33: 435-39. 1849 - On a New Imaginary in Algebra, Phil. Mag. (3) 34: 37-47. 1849 - On the Symbols of Algebra and on the Theory of Tessarines, Phil. Mag. (3) 34: 406-410. 1849 - On Systems of Algebra Involving more than one Imaginary and on Equations of the Fifth Degree, Phil. Mag. (3) 35: 434-437. 1850 - On the True Amplitude of a Tessarine, Phil. Mag. (3) 38: 290-292. 1850 - On Impossible Equations, on Impossible Quantities and on Tessarines, Phil. Mag (3) 37: 281-283.

1. J.C. (1862)"Correlations of Analysis", Philosophical Magazine 24(4):532.
2. J.C. (1876) "On Linear Differential Equations of the Third Order", Quarterly Journal of Mathematics 15:340-53.
3. R. Hartley (1884) "Professor Malet's Classes of Invariants identified with Sir James Cockle's Criticoids", Proceedings of the Royal Society of London 38:45-57.

XIV 18-22 Explicit Integration & Differential Resolvents
XVIII 180-202 Equation of Riccati
XIX 257-78 On the General Linear Differential Equation of the Second Order
XI 123-31 On a Binomial Biordinal and the constants of its complete solution
XII Supplement on Binomial Biordinals
XX 4-14 On the Confluences and Bifurcations of Certain Theories

See also Tony Crilly (2006) Arthur Cayley - Mathematician Laureate of the Victorian Age Johns Hopkins University Press.
Furthermore, JJ O'Connor and EF Robertson have now included Cockle in their list of biographies: see MacTutor on James Cockle.
Read Andrew Warwick (2003) Masters in Theory for the story of the Cambridge wranglers, their tutors, and programs of physical fitness.

Forward your comments to cocklebio@yahoo.ca

First posted: 2003 July 15
Last modified: 2009 October 17