# Account of a Theory of Systems of Rays

By William Rowan Hamilton

[From the Life of Sir William Rowan Hamilton (Volume I, Chapter VI, pp. 228-231) by Robert Perceval Graves (Dublin University Press, 1882).]

It appears proper to give some accurate notions of what is meant by a System of Rays, and of what (mainly) has been done by me towards forming a Theory of such Systems.

A Ray, in Optics, is to be considered here as a straight or bent or curved line, along which light is propagated; and a System of Rays as a collection or aggregate of such lines, connected by some common bond, some similarity of origin or production, in short some optical unity. Thus the rays which diverge from a luminous point compose one optical system, and, after they have been reflected at a mirror, they compose another. To investigate the geometrical relations of the rays of a system of which we know (as in these simple cases) the origin and history, to inquire how they are disposed among themselves, how they diverge or converge, or are parallel, what surfaces or curves they touch or cut, and what angles of section, how they can be combined in partial pencils, and how each ray in particular can be determined and distinguished from every other, is to study that System of Rays. And to generalise this study of one system so as to become able to pass, without change of plan, to the study of other systems, to assign general rules and a general method whereby these separate optical arrangements may be connected and harmonised together, is to form a Theory of Systems of Rays. Finally, to do this in such a manner as to make available the powers of the modern mathesis, replacing figures by functions and diagrams by formulae, is to construct an Algebraic Theory of such Systems, or an Application of Algebra to Optics.

Towards constructing such an application it is natural, or rather necessary, to employ the method introduced by Descartes for the application of Algebra to Geometry. That great and philosophical mathematician conceived the possibility, and employed the plan, of representing or expressing algebraically the position of any point in space by three co-ordinate numbers which answer respectively the questions how far the point is in three rectangular directions (such as north, east and west), from some fixed point or origin selected or assumed for the purpose; the three dimensions of space receiving thus their three algebraical equivalents, their appropriate conceptions and symbols in the general science of progression. A plane or curve surface became thus algebraically defined by the assigning as its equation the relation connecting the three co-ordinates of any point upon it, and common to all those points: and a line, straight or curved, was expressed according to the same method, by the assigning two such relations, correspondent to two surfaces of which the line might be regarded as the intersection. In this manner it became possible to conduct general investigations respecting surfaces and curves, and to discover properties common to all, through the medium of general investigations respecting equations between three variable numbers: every geometrical problem could be at least algebraically expressed, if not at once resolved, and every improvement or discovery in Algebra became susceptible of application or interpretation in Geometry. The sciences of Space and Time (to adopt here a view of Algebra which I have elsewhere ventured to propose) became intimately intertwined and indissolubly connected with each other. Henceforth it was almost impossible to improve either science without improving the other also. The problem of drawing tangent to curves led to the discovery of Fluxions or Differentials: those of rectification and quadrature to the invention of Fluents or Integrals: the investigation of curvatures of surfaces required the Calculus of Partial Differentials: the isoperimetrical problems resulted in the formation of the Calculus of Variations. And reciprocally, all these great steps in Algebraic Science had immediately their applications to Geometry, and led to the discovery of new relations between points or lines or surfaces. But even if the applications of the method had not been so manifold and important, there would still have been derivable a high intellectual pleasure from the contemplation of it as a method.

The first important application of this algebraical method of co-ordinates to the study of optical systems was made by Malus, a French officer of engineers who served in Napoleon's army in Egypt, and who has acquired celebrity in the history of Physical Optics as the discoverer of the polarisation of light by reflexion. Malus presented to the Institute of France, in 1807, a profound mathematical work which is of the kind above alluded to, and is entitled Traité d'Optique. The method employed in that treatise may be thus described:---The direction of a straight ray of any final optical system being considered as dependent on the position of some assigned point upon that ray, according to some law which characterises the particular system and distinguishes it from others; this law may be algebraically expressed by assigning three expressions for the three co-ordinates of some other point of the ray, as functions of the three co-ordinates of the point proposed. Malus accordingly introduces general symbols denoting three such functions (or at least three functions equivalent to these), and proceeds to draw several important general conclusions, by very complicated but yet symmetric calculations; many of which conclusions, along with many others, were also obtained afterwards by myself, when, by a method nearly similar, without knowing what Malus had done, I began my own attempts to apply Algebra to Optics. But my researches soon conducted me to substitute, for this method of Malus, a very different, and (as I conceive that I have proved) a much more appropriate one, for the study of optical systems, by which, instead of employing the three functions above mentioned, or at least their two ratios, it becomes sufficient to employ one function, which I call characteristic or principal. And thus, whereas he made his deductions by setting out with the two equations of a ray, I on the other hand establish and employ the one equation of a system.

The function which I have introduced for this purpose, and made the basis of my method of deduction in mathematical Optics, had, in another connexion, presented itself to former writers as expressing the result of a very high and extensive induction in that science. This known result is usually called the law of least action, but sometimes also the principle of least time, and includes all that has hitherto been discovered respecting the rules which determine the forms and positions of the line along which light is propagated, and the changes of direction of those lines produced by reflexion, or refraction, ordinary or extraordinary. A certain quantity which in one physical theory is the action, and in another the time, expended by light in going from any first to any second point, is found to be less than if the light had gone in any other than its actual path, or at least to have what is technically called its variation null, the extremities of the path being unvaried. The mathematical novelty of my method consists in considering this quantity as a function of the co-ordinates of these extremities, which varies when they vary, according to a law which I have called the law of varying action; and in reducing all researches respecting optical systems of rays to the study of this single function: a reduction which presents mathematical Optics under an entirely novel view, and one analogous (as it appears to me) to the aspect under which Descartes presented the application of Algebra to Geometry.