1. _Newton's Corpuscular Theory._--Until the beginning of the 19th
century physicists were divided between two different views concerning
the nature of optical phenomena. According to the one, luminous bodies
emit extremely small corpuscles which can freely pass through
transparent substances and produce the sensation of light by their
impact against the retina. This _emission_ or _corpuscular theory_ of
light was supported by the authority of Isaac Newton,[8] and, though it
has been entirely superseded by its rival, the _wave-theory_, it remains
of considerable historical interest.
2. _Explanation of Reflection and Refraction._--Newton supposed the
light-corpuscles to be subjected to attractive and repulsive forces
exerted at very small distances by the particles of matter. In the
interior of a homogeneous body a corpuscle moves in a straight line as
it is equally acted on from all sides, but it changes its course at the
boundary of two bodies, because, in a thin layer near the surface there
is a resultant force in the direction of the normal. In modern language
we may say that a corpuscle has at every point a definite potential
energy, the value of which is constant throughout the interior of a
homogeneous body, and is even equal in all bodies of the same kind, but
changes from one substance to another. If, originally, while moving in
air, the corpuscles had a definite velocity v0, their velocity v in the
interior of any other substance is quite determinate. It is given by the
equation ½mv² - ½mv0² = A, in which m denotes the mass of a corpuscle,
and A the excess of its potential energy in air over that in the
substance considered.
A ray of light falling on the surface of separation of two bodies is
reflected according to the well-known simple law, if the corpuscles
are acted on by a sufficiently large force directed towards the first
medium. On the contrary, whenever the field of force near the surface
is such that the corpuscles can penetrate into the interior of the
second body, the ray is refracted. In this case the law of Snellius
can be deduced from the consideration that the projection w of the
velocity on the surface of separation is not altered, either in
direction or in magnitude. This obviously requires that the plane
passing through the incident and the refracted rays be normal to the
surface, and that, if [alpha]1 and [alpha]2 are the angles of
incidence and of refraction, v1 and v2 the velocities of light in the
two media,
sin [alpha]1/sin [alpha]2 = w/v1 : w/v2 = v2/v1. (1)
The ratio is constant, because, as has already been observed, v1 and
v2 have definite values.
As to the unequal refrangibility of differently coloured light, Newton
accounted for it by imagining different kinds of corpuscles. He
further carefully examined the phenomenon of total reflection, and
described an interesting experiment connected with it. If one of the
faces of a glass prism receives on the inside a beam of light of such
obliquity that it is totally reflected under ordinary circumstances,
a marked change is observed when a second piece of glass is made to
approach the reflecting face, so as to be separated from it only by a
very thin layer of air. The reflection is then found no longer to be
total, part of the light finding its way into the second piece of
glass. Newton concluded from this that the corpuscles are attracted by
the glass even at a certain small measurable distance.
3. _New Hypotheses in the Corpuscular Theory._--The preceding
explanation of reflection and refraction is open to a very serious
objection. If the particles in a beam of light all moved with the same
velocity and were acted on by the same forces, they all ought to follow
exactly the same path. In order to understand that part of the incident
light is reflected and part of it transmitted, Newton imagined that each
corpuscle undergoes certain alternating changes; he assumed that in some
of its different "phases" it is more apt to be reflected, and in others
more apt to be transmitted. The same idea was applied by him to the
phenomena presented by very thin layers. He had observed that a gradual
increase of the thickness of a layer produces periodic changes in the
intensity of the reflected light, and he very ingeniously explained
these by his theory. It is clear that the intensity of the transmitted
light will be a minimum if the corpuscles that have traversed the front
surface of the layer, having reached that surface while in their phase
of easy transmission, have passed to the opposite phase the moment they
arrive at the back surface. As to the nature of the alternating phases,
Newton (_Opticks_, 3rd ed., 1721, p. 347) expresses himself as
follows:--"Nothing more is requisite for putting the Rays of Light into
Fits of easy Reflexion and easy Transmission than that they be small
Bodies which by their attractive Powers, or some other Force, stir up
Vibrations in what they act upon, which Vibrations being swifter than
the Rays, overtake them successively, and agitate them so as by turns to
increase and decrease their Velocities, and thereby put them into those
Fits."
4. _The Corpuscular Theory and the Wave-Theory compared._--Though Newton
introduced the notion of periodic changes, which was to play so
prominent a part in the later development of the wave-theory, he
rejected this theory in the form in which it had been set forth shortly
before by Christiaan Huygens in his _Traité de la lumière_ (1690), his
chief objections being: (1) that the rectilinear propagation had not
been satisfactorily accounted for; (2) that the motions of heavenly
bodies show no sign of a resistance due to a medium filling all space;
and (3) that Huygens had not sufficiently explained the peculiar
properties of the rays produced by the double refraction in Iceland
spar. In Newton's days these objections were of much weight.
Yet his own theory had many weaknesses. It explained the propagation in
straight lines, but it could assign no cause for the equality of the
speed of propagation of all rays. It adapted itself to a large variety
of phenomena, even to that of double refraction (Newton says
[ibid.]:--"... the unusual Refraction of Iceland Crystal looks very much
as if it were perform'd by some kind of attractive virtue lodged in
certain Sides both of the Rays, and of the Particles of the Crystal."),
but it could do so only at the price of losing much of its original
simplicity.
In the earlier part of the 19th century, the corpuscular theory broke
down under the weight of experimental evidence, and it received the
final blow when J. B. L. Foucault proved by direct experiment that the
velocity of light in water is not greater than that in air, as it should
be according to the formula (1), but less than it, as is required by the
wave-theory.
5. _General Theorems on Rays of Light._--With the aid of suitable
assumptions the Newtonian theory can accurately trace the course of a
ray of light in any system of isotropic bodies, whether homogeneous or
otherwise; the problem being equivalent to that of determining the
motion of a material point in a space in which its potential energy is
given as a function of the coordinates. The application of the dynamical
principles of "least and of varying action" to this latter problem leads
to the following important theorems which William Rowan Hamilton made
the basis of his exhaustive treatment of systems of rays.[9] The total
energy of a corpuscle is supposed to have a given value, so that, since
the potential energy is considered as known at every point, the velocity
v is so likewise.
(a) The path along which light travels from a point A to a point B is
determined by the condition that for this line the integral [int]v ds,
in which ds is an element of the line, be a minimum (provided A and B
be not too near each other). Therefore, since v = µv0, if v0 is the
velocity of light _in vacuo_ and µ the index of refraction, we have
for every variation of the path the points A and B remaining fixed,
[delta][int]µ ds = 0. (2)
(b) Let the point A be kept fixed, but let B undergo an infinitely
small displacement BB´ (=q) in a direction making an angle [theta]
with the last element of the ray AB. Then, comparing the new ray AB´
with the original one, it follows that
[delta][int]µ ds = µ_B q cos [theta], (3)
where µ_B is the value of µ at the point B.
6. _General Considerations on the Propagation of Waves._--"Waves," i.e.
local disturbances of equilibrium travelling onward with a certain
speed, can exist in a large variety of systems. In a theory of these
phenomena, the state of things at a definite point may in general be
defined by a certain directed or vector quantity P,[10] which is zero in
the state of equilibrium, and may be called the disturbance (for
example, the velocity of the air in the case of sound vibrations, or the
displacement of the particles of an elastic body from their positions of
equilibrium). The components P_x, P_y, P_z of the disturbance in the
directions of the axes of coordinates are to be considered as functions
of the coordinates x, y, z and the time t, determined by a set of
partial differential equations, whose form depends on the nature of the
problem considered. If the equations are homogeneous and linear, as they
always are for sufficiently small disturbances, the following theorems
hold.
(a) Values of P_x, P_y, P_z (expressed in terms of x, y, z, t) which
satisfy the equations will do so still after multiplication by a
common arbitrary constant.
(b) Two or more solutions of the equations may be combined into a new
solution by addition of the values of P_x, those of P_y, &c., i.e. by
compounding the vectors P, such as they are in each of the particular
solutions.
In the application to light, the first proposition means that the
phenomena of propagation, reflection, refraction, &c., can be produced
in the same way with strong as with weak light. The second proposition
contains the principle of the "superposition" of different states, on
which the explanation of all phenomena of interference is made to
depend.
In the simplest cases (monochromatic or homogeneous light) the
disturbance is a simple harmonic function of the time ("simple
harmonic vibrations"), so that its components can be represented by
P_x = a1 cos (nt + f1),
P_y = a2 cos (nt + f2),
P_z = a3 cos (nt + f3).
The "phases" of these vibrations are determined by the angles nt + f1,
&c., or by the times t + f1/n, &c. The "frequency" n is constant
throughout the system, while the quantities f1, f2, f3, and perhaps
the "amplitudes" a1, a2, a3 change from point to point. It may be
shown that the end of a straight line representing the vector P, and
drawn from the point considered, in general describes a certain
ellipse, which becomes a straight line, if f1 = f2 = f3. In this
latter case, to which the larger part of this article will be
confined, we can write in vector notation