DISCOVERY OF THE LAW OF REFRACTION.
WE have seen in the former part of this history that the Greeks had
formed a tolerably clear conception of the refraction as well as the
reflection of the rays of light; and that Ptolemy had measured the
amount of refraction of glass and water at various angles. If we
give the names of the _angle of incidence_ and the _angle of
refraction_ respectively to the angles which a ray of light makes
with the line perpendicular to surface of glass or water (or any
other medium) within and without the medium, Ptolemy had observed
that the angle of refraction is always less than the angle of
incidence. He had supposed it to be less in a given proportion, but
this opinion is false; and was afterwards rightly denied by the
Arabian mathematician Alhazen. The optical views which occur in the
work of Alhazen are far sounder than those of his predecessors; and
the book may be regarded as the most considerable monument which we
have of the scientific genius of the Arabians; for it appears, for
the most part, not to be borrowed from Greek authorities. The author
not only asserts (lib. vii.), that refraction takes place towards
the perpendicular, and refers to experiment for the truth of this:
and that the quantities of the refraction differ according to the
magnitudes of the angles which the directions of the incidental rays
(_primæ lineæ_) make with the perpendiculars to the surface; but he
also says distinctly and decidedly that the angles of refraction do
not follow the proportion of the angles of incidence.
[2nd Ed.] [There appears to be good ground to assent to the
assertion of Alhazen's originality, made by his editor Risner, who
says, "Euclideum hic vel Ptolemaicum nihil fere est." Besides the
doctrine of reflection and refraction of light, the Arabian author
gives a description of the eye. He distinguishes three fluids,
_humor aqueus_, _crystallinus_, _vitreus_, and four coats of the
eye, _tunica adherens_, _cornea_, _uvea_, _tunica reti similis_. He
distinguishes also three kinds of vision: "Visibile percipitur aut
solo visu, aut visu et syllogismo, aut visu et anticipatâ notione."
He has several propositions relating to what we sometimes call the
Philosophy of Vision: for instance this: "E visibili sæpius viso
remanet in anima generalis notio," &c.] {55}
The assertion, that the angles of refraction are not proportional to
the angles of incidence, was an important remark; and if it had been
steadily kept in mind, the next thing to be done with regard to
refraction was to go on experimenting and conjecturing till the true
law of refraction was discovered; and in the mean time to apply the
principle as far as it was known. Alhazen, though he gives
directions for making experimental measures of refraction, does not
give any Table of the results of such experiments, as Ptolemy had
done. Vitello, a Pole, who in the 13th century published an
extensive work upon Optics, does give such a table; and asserts it
to be deduced from experiment, as I have already said (vol. i.). But
this assertion is still liable to doubt in consequence of the table
containing impossible observations.
[2nd Ed.] [As I have already stated, Vitello asserts that his Tables
were derived from his own observations. Their near agreement with
those of Ptolemy does not make this improbable: for where the
observations were only made to half a degree, there was not much
room for observers to differ. It is not unlikely that the
observations of refraction out of air into water and glass, and out
of water into glass, were actually made; while the impossible values
which accompany them, of the refraction out of water and glass into
air, and out of glass into water, were calculated, and calculated
from an erroneous rule.]
The principle that a ray refracted in glass or water is turned
towards the perpendicular, without knowing the exact law of
refraction, enabled mathematicians to trace the effects of
transparent bodies in various cases. Thus in Roger Bacon's works we
find a tolerably distinct explanation of the effect of a convex
glass; and in the work of Vitello the effect of refraction at the
two surfaces of a glass globe is clearly traceable.
Notwithstanding Alhazen's assertion of the contrary, the opinion was
still current among mathematicians that the angle of refraction was
proportional to the angle of incidence. But when Kepler's attention
was drawn to the subject, he saw that this was plainly inconsistent
with the observations of Vitello for large angles; and he convinced
himself by his own experiments that the true law was something
different from the one commonly supposed. The discovery of this true
law excited in him an eager curiosity; and this point had the more
interest for him in consequence of the introduction of a correction
for atmospheric refraction into astronomical calculations, which had
been made by Tycho, and of the invention of the telescope. In {56}
his _Supplement to Vitello_, published in 1604, Kepler attempts to
reduce to a rule the measured quantities of refraction. The reader
who recollects what we have already narrated, the manner in which
Kepler attempted to reduce to law the astronomical observations of
Tycho,--devising an almost endless variety of possible formulæ,
tracing their consequences with undaunted industry, and relating,
with a vivacious garrulity, his disappointments and his hopes,--will
not be surprised to find that he proceeded in the same manner with
regard to the Tables of Observed Refractions. He tried a variety of
constructions by triangles, conic sections, &c., without being able
to satisfy himself; and he at last[3\9] is obliged to content
himself with an approximate rule, which makes the refraction partly
proportional to the angle of incidence, and partly, to the secant of
that angle. In this way he satisfies the observed refractions within
a difference of less than half a degree each way. When we consider
how simple the law of refraction is, (that the ratio of the sines of
the angles of incidence and refraction is constant for the same
medium,) it appears strange that a person attempting to discover it,
and drawing triangles for the purpose, should fail; but this lot of
missing what afterwards seems to have been obvious, is a common one
in the pursuit of truth.
[Note 3\9: L. U. K. _Life of Kepler_, p. 115.]
The person who did discover the Law of the Sines, was Willebrord
Snell, about 1621; but the law was first published by Descartes, who
had seen Snell's papers.[4\9] Descartes does not acknowledge this
law to have been first detected by another; and after his manner,
instead of establishing its reality by reference to experiment, he
pretends to prove _à priori_ that it must be true,[5\9] comparing,
for this purpose, the particles of light to balls striking a
substance which _accelerates_ them.
[Note 4\9: Huyghens, _Dioptrica_, p. 2.]
[Note 5\9: _Diopt._ p. 53.]
[2nd Ed.] [Huyghens says of Snell's papers, "Quæ et nos vidimus
aliquando, et Cartesium quoque vidisse accepimus, et hinc fortasse
mensuram illam quæ in sinibus consistit elicuerit." Isaac Vossius,
_De Lucis Naturâ et Proprietate_, says that he also had seen this
law in Snell's unpublished optical Treatise. The same writer says,
"Quod itaque (Cartesius) habet, refractionum momenta non exigenda
esse ad angulos sed ad lineas, id tuo Snellio, acceptum ferre
debuisset, cujus nomen _more solito_ dissimulavit." "Cartesius got
his law from Snell, and _in his usual way_, concealed it." {57}
Huyghens' assertion, that Snell did not _attend to_ the proportion
of the sines, is very captious; and becomes absurdly so, when it is
made to mean that Snell did not _know_ the law of the sines. It is
not denied that Snell knew the true law, or that the true law is the
law of the sines. Snell does not use the trigonometrical term
_sine_, but he expresses the law in a geometrical form more simply.
Even if he _had_ attended to the law of the sines, he might
reasonably have preferred his own way of stating it.
James Gregory also independently discovered the true law of
refraction; and, in publishing it, states that he had learnt that it
had already been published by Descartes.]
But though Descartes does not, in this instance, produce any good
claims to the character of an inductive philosopher, he showed
considerable skill in tracing the consequences of the principle when
once adopted. In particular we must consider him as the genuine
author of the explanation of the rainbow. It is true that
Fleischer[6\9] and Kepler had previously ascribed this phenomenon to
the rays of sunlight which, falling on drops of rain, are refracted
into each drop, reflected at its inner surface, and refracted out
again: Antonio de Dominis had found that a glass globe of water,
when placed in a particular position with respect to the eye,
exhibited bright colors; and had hence explained the circular form
of the bow, which, indeed, Aristotle had done before.[7\9] But none
of these writers had shown why there was a narrow bright circle of a
definite diameter; for the drops which send rays to the eye after
two refractions and a reflection, occupy a much wider space in the
heavens. Descartes assigned the reason for this in the most
satisfactory manner,[8\9] by showing that the rays which, after two
refractions and a reflection, come to the eye at an angle of about
forty-one degrees with their original direction, are far more dense
than those in any other position. He showed, in the same manner,
that the existence and position of the _secondary bow_ resulted from
the same laws. This is the complete and adequate account of the
state of things, so far as the brightness of the bows only is
concerned; the explanation of the colors belongs to the next article
of our survey.
[Note 6\9: Mont. i. 701.]
[Note 7\9: _Meteorol._ iii. 3.]
[Note 8\9: _Meteorum_, cap. viii. p. 196.]
The explanation of the rainbow and of its magnitude, afforded by
Snell's law of sines, was perhaps one of the leading points in the
verification of the law. The principle, being once established, was
applied, by the aid of mathematical reasoning, to atmospheric
refractions, {58} optical instruments, _diacaustic_ curves, (that
is, the curves of intense light produced by refraction,) and to
various other cases; and was, of course, tested and confirmed by
such applications. It was, however, impossible to pursue these
applications far, without a due knowledge of the laws by which, in
such cases, colors are produced. To these we now proceed.
[2nd Ed.] [I have omitted many interesting parts of the history of
Optics about this period, because I was concerned with the
_inductive_ discovery of laws, rather than with mathematical
_deductions_ from such laws when established, or _applications_ of
them in the form of instruments. I might otherwise have noticed the
discovery of Spectacle Glasses, of the Telescope, of the Microscope,
of the Camera Obscura, and the mathematical explanation of these and
other phenomena, as given by Kepler and others. I might also have
noticed the progress of knowledge respecting the Eye and Vision. We
have seen that Alhazen described the structure of the eye. The
operation of the parts was gradually made out. Baptista Porta
compares the eye to his _Camera Obscura_ (_Magia Naturalis_, 1579).
Scheiner, in his _Oculus_, published 1652, completed the Theory of
the Eye. And Kepler discussed some of the questions even now often
agitated; as the causes and conditions of our seeing objects single
with two eyes, and erect with inverted images.]