It is by no means easy for the applied mathematician to decide how much importance he should attach to the more abstract and aesthetic side of his work and how much to the detailed applications to physics, astronomy, engineering or the design of instruments. Great mathematical ideas do not blossom in workshops, as a rule, but on the other hand the theorist should not divorce himself from a healthy and intimate connection with practical questions.

Sir William Rowan Hamilton (1805-1865) created a method in Geometrical Optics, which, after lying long in disuse, is at last finding its proper place in the science. To all appearances, Hamilton attached little importance to the practical applications of his method, and it was only with the publication of his Mathematical Papers, Vol. I (Cambridge, 1931), that it was possible to form a more correct and balanced judgment of Hamilton as an applied mathematician. Great indeed was the labour which he employed with a view to applying his method to the design of optical instruments, but for him the abstract and aesthetic side of his work was of so much greater public importance than its practical use that the details of application remained unpublished till long after his death and long after other workers had discovered equivalent processes.

Since it was left largely to those primarily interested in optical design to develop the subject of Geometrical Optics, it is only natural that the student of the subject soon finds himself immersed in details which tend to cloud his understanding of the underlying general principles. Now, just as it is widely recognized that in the teaching of mechanics a middle course must be steered between a completely abstract presentation and a technical approach, so it seems to me that the student of Geometrical Optics is most likely to understand the principles of Hamilton's method if he does not think too much at first of technical applications. But, at the same time, he should not be kept entirely remote from them.

Since editing, in collaboration with Professor Arthur Conway, F.R.S., Hamilton's papers on Geometrical Optics, I have had the opportunity of lecturing on the subject to graduate students and undergraduates in the University of Toronto. This book represents a course of twenty-five lectures to the latter. Although the reader may fail to find in it some things which he would naturally expect in a book on Geometrical Optics, no apology is offered on that account. If Hamilton's method is understood, the book serves its purpose. For that reason it is not necessary to defend the application of the method to problems which would admit shorter special solutions.

Hamilton was a master of mathematical notation, and he might in this respect be profitably studied by some modern writers in our subject. I have employed his notation in the main, changing the signs of the $W$ and $T$ functions to make their physical interpretation more obvious, and making some changes in nomenclature. It does not seem necessary or desirable to use the word "eikonal", which Bruns invented in 1895 in ignorance of Hamilton's work. Since one letter is just about as good as another, would it not be a harmless compliment to the genius of Hamilton for writers on Geometrical Optics to employ for the various characteristic functions the letters which he employed?

Although Hamilton himself started by considering the simpler case of isotropic media, it was not long before he saw that his method was also applicable to anisotropic media, and when he came to give his theory final form in his Third Supplement, he did so in all generality. This has done much to discourage those interested in the more practical aspects of his method, because in order to apply it they have been compelled to think in terms of (to them) unnecessary generality. To avoid a repetition of this error of policy, the theory of anisotropic media has been entirely omitted from this book. To compensate for this omission and for the fact that, although an attempt has been made to amplify Hamilton's work in the directions since found of most interest, these amplifications have not been sufficient to create an adequate text-book, a brief bibliography is given below. In some of these works Hamilton's characteristic functions are referred to as Bruns' eikonals, but there is no significant difference.

I have to thank three of my students, Messrs H R Roberts, P R Wallace and A White, for assistance in the preparation of the manuscript, and my colleagues, Professor A F Stevenson and Dr B A Griffith, for reading the proofs and making valuable suggestions. It is also a pleasure to pay tribute to the skill and accuracy of the Cambridge University Press.

J. L. S.
TORONTO
October 1937

A "perfect" scientific theory may be described as one which proceeds logically from a few simple hypotheses to conclusions which are in complete agreement with observation, to within the limits of accuracy of observation. But the theory is "useful" only in so far as it is possible to obtain conclusions from the hypotheses. As accuracy of observation increases, a theory ceases to be "perfect": modifications are introduced, making the theory more complicated and less "useful". Since we do not willingly surrender the wealth of approximate results furnished by the earlier form of the theory, we find ourselves in the unsatisfactory position of using one theory for one problem and another for another, although the two problems really belong to the same part of science. To rescue ourselves from intellectual confusion, we may admit theories called "ideal", in the sense that they deal with an ideal universe, resembling the actual universe to a fair degree of accuracy and usually corresponding to a limiting case of physical reality.

A critical examination of the history of mathematical physics shows that in truth man has always created "ideal" theories. Nature is much too complicated to be considered otherwise than in a simplified or idealized form, and it is inevitable that this idealization should lead to discrepancies between theoretical prediction and observation. As examples we may mention the mechanical theories of rigid bodies and perfect fluids; neither rigid bodies nor perfect fluids exist in nature. Or we may think of the Newtonian theory of gravitation, long regarded as "perfect", but now "ideal", physically replaced by the "perfect (but not so "useful") general theory of relativity.

Geometrical optics is an ideal theory and a useful one. The discovery that the propagation of light is an electromagnetic phenomenon made the subject of optics coextensive with electromagnetism. We may, however, study certain parts of the subject of optics without reference to electromagnetism, always understanding that there is a limit to the physical accuracy of the results so obtained. It is customary to use the name "physical optics" for the more complex and physically accurate theory, and "geometrical optics " for the simpler ideal theory with which we shall be concerned. It is possible to justify geometrical optics as a limiting case of physical optics, the wave-length of the light in question tending to zero; [M Born, Optik (Berlin, 1933), 45] but we shall be content with the development of geometrical optics on the basis of its own hypotheses, just as it is customary to develop the dynamics of rigid bodies as a separate theory, and not as a limiting case of the dynamics of elastic bodies whose elastic moduli tend to infinity.

E T Whittaker, The Theory of Optical Instruments (Cambridge Tracts, No. 7, 1907).

J G Leathem, The Elementary Theory of the Symmetrical Optical Instrument (Cambridge Tracts, No. 8, 1908).

J P C Southall, The Principles and Methods of Geometrical Optics (New York, 1913).

S Czapski and O Eppenstein, Grundzüge der Theorie der optischen Instrumente (Leipzig, 1924).

O C Steward, The Symmetrical Optical System (Cambridge Tracts, No. 25, 1928).

M Herzberger, Strahlenoptik (Berlin, 1931).

The Mathematical Papers of Sir W R Hamilton, Vol. I (edited by A W Conway and J L Synge, Cambridge, 1931).

W R Hamiltons Abhandlungen zur Strahlenoptik (translated and edited with notes by G Prange, Leipzig, 1933).

M Born, Optik (Berlin, 1933).

J L Synge, Hamilton's Method in Geometrical Optics, Journal of the Optical Society of America 27 (1937), 75-82.