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The Hamiltonian Revival

Published online by Cambridge University Press:  03 November 2016

E. T Whittaker
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One of the most agreeable fruits of the renaissance which has followed the achievement of national independence in Ireland has been the decision to make accessible, in a form worthy of their importance, the writings of the greatest Irish mathematician and natural philosopher. The responsibility for the work was undertaken by the Royal Irish Academy, with the cooperation of the University of Dublin and the National University of Ireland, and with assistance from the Royal Society and the editorship was entrusted to Professor A. W Conway, F.R.S., of University College, Dublin, and Professor J L. Synge of Trinity College, Dublin. The first volume, devoted to Hamilton’s writings on Geometrical Optics, was published in 1931, and we have now to congratulate the editors (Professor Synge having meanwhile been succeeded by Professor A. J McConnell) on the appearance of the second volume which contains the papers and manuscripts on Dynamics.

Type
Research Article
Information
The Mathematical Gazette , Volume 24 , Issue 260 , July 1940 , pp. 153 - 158
Copyright
Copyright © Mathematical Association 1940

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References

Page 153 of note * The Mathematical Papers of Sir William Rowan Hamilton, Volume II. Dynamics. Pp. xv, 656. 70s. net. 1940. (Camb. Univ. Press.)

Page 155 of note * Other types of algebra quickly followed: Boole’s Algebra of Logic (1847) retains the commutative law, but has x 2=x and has complete duality between addition and multiplication, which are distributive with respect to each other. The Theory of Matrices, which is non-commutative, was discovered by Cayley (1855), but is the same in essence as Hamilton’s theory of Linear Vector Operators, which he gave in his Lectures on Quaternions in 1852. Non-associative algebras, in which a bc is not necessarily equal to ab c, were suggested by De Morgan. Grassmann’s Ausdehnungslehre, the first sketch of which appeared the year after the publication of Hamilton’s theory, should also be mentioned.

Page 155 of note † For the only linear associative algebras over the field of real numbers in which division is uniquely possible are the field of real numbers, the field of ordinary complex numbers, and real quaternions.

Page 155 of note ‡ I include geometry in theoretical physics, as is right pace Whitehead and Russell.

Page 156 of note * Though it must be acknowledged that now and then, e.g. in Tait’s proof of Ampère’s laws for the ponderomotive force between elements of electric currents (Proc. R.S.E., 1873-4) and in Heaviside’s beautifully simple expression for the Maxwell stresses in an electric field (Phil. Trans., 1892), quaternions made a very good showing.

Page 157 of note * Afterwards published in Proc. L.M.S. (2) 7 (1909), 338.

Mention may here be made of an incident in the history of geometrical optics. Bruns—the author of the famous theorem on the non-existence of further algebraic integrals in the problem of three bodies—published in 1895 a method of describing and calculating the aberrations of optical instruments, depending on a function which he called the eikonal. This, as Klein pointed out, was essentially nothing more or less than the “characteristic function” of Hamilton’s optical papers: and since then the Hamiltonian method has been used extensively in practical calculations relating to the aberrations of the symmetrical optical instrument. From a very different point of view, the optical papers came to be recognised by geometers as the origin of the theory of the general rectilinear congruence.

Page 158 of note * This discovery led to the publication of an enormous amount of nonsense about determinism and free-will. Actually the uncertainty-principle has nothing whatever to do with determinism or free-will: it simply expresses the fact that the classical concept of a particle (i.e. a very small body having a definite position and velocity) does not represent anything in nature: what is realised in nature is the quantum concept of a particle, which is something quite different.

Page 158 of note † Proc. R.S., 162 (1937), 145.