*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.

Links:

D.R. Wilkins(

School of Mathematics

Trinity College, Dublin