CONFIRMATION AND EXTENSION OF THE UNDULATORY THEORY.
AFTER the undulatory theory had been developed in all its main
features, by its great authors, Young and Fresnel, although it bore
marks of truth that could hardly be fallacious, there was still
here, as in the case of other great theories, a period in which
difficulties were to be removed, objections answered, men's minds
familiarized to the new conceptions thus presented to them; and in
which, also, it might reasonably be expected that the theory would
be extended to facts not at first included in its domain. This
period is, indeed, that in which we are living; and we might,
perhaps with propriety, avoid the task of speaking of our living
contemporaries. But it would be unjust to the theory not to notice
some of the remarkable events, characteristic of such a period,
which have already occurred; and this may be done very simply. {119}
In the case of this great theory, as in that of gravitation, by far
the most remarkable of these confirmatory researches were conducted
by the authors of the discovery, especially Fresnel. And in looking
at what he conceived and executed for this purpose, we are, it
appears to me, strongly reminded of Newton, by the wonderful
inventiveness and sagacity with which he devised experiments, and
applied to them mathematical reasonings.
1. _Double Refraction of Compressed Glass._--One of these
confirmatory experiments was the production of double refraction by
the compression of glass. Fresnel observes,[107\9] that though Sir
D. Brewster had shown that glass under compression produced colors
resembling those which are given by doubly-refracting crystals,
"very skilful physicists had not considered those experiments as a
sufficient proof of the bifurcation of the light." In the hypothesis
of moveable polarization, it is added, there is no apparent
connexion between these phenomena of coloration and double
refraction; but on Young's theory, that the colors arise from two
rays which have traversed the crystal with different velocities, it
appears almost unavoidable to admit also a difference of path in the
two rays.
[Note 107\9: _Ann. de Chim._ 1822, tom. xx. p. 377.]
"Though," he says, "I had long since adopted this opinion, it did
not appear to me so completely demonstrated, that it was right to
neglect an experimental verification of it;" and therefore, in 1819,
he proceeded to satisfy himself of the fact, by the phenomena of
diffraction. The trial left no doubt on the subject; but he still
thought it would be interesting actually to produce two images in
glass by compression; and by a highly-ingenious combination,
calculated to exaggerate the effect of the double refraction, which
is very feeble, even when the compression is most intense, he
obtained two distinct images. This evidence of the dependence of
dipolarizing structure upon a doubly-refracting state of particles,
thus excogitated out of the general theory, and verified by trial,
may well be considered, as he says, "as a new occasion of proving
the infallibility of the principle of interferences."
2. _Circular Polarization._--Fresnel then turned his attention to
another set of experiments, related to this indeed, but by a tie so
recondite, that nothing less than his clearness and acuteness of
view could have detected any connexion. The optical properties of
quartz had been perceived to be peculiar, from the period of the
discovery {120} of dipolarized colors by MM. Arago and Biot. At the
end of the Notice just quoted, Fresnel says,[108\9] "As soon as my
occupations permit me, I propose to employ a pile of prisms similar
to that which I have described, in order to study the double
refraction of the rays which traverse crystals of quartz in the
direction of the axis." He then ventures, without hesitation, to
describe beforehand what the phenomena will be. In the _Bulletin des
Sciences_[109\9] for December, 1822, it is stated that experiment
had confirmed what he had thus announced.
[Note 108\9: _Ann. de Chim._ 1822, tom. xx. p. 382.]
[Note 109\9: Ib. _Ann. de Chim._ 1822, tom. xx. p. 191.]
The phenomena are those which have since been spoken of as _circular
polarization_; and the term first occurs in this notice.[110\9] They
are very remarkable, both by their resemblances to, and their
differences from, the phenomena of _plane-polarized_ light. And the
manner in which Fresnel was led to this anticipation of the facts is
still more remarkable than the facts themselves. Having ascertained
by observation that two differently-polarized rays, totally
reflected at the internal surface of glass, suffer different
_retardations_ of their undulations, he applied the formulæ which he
had obtained for the polarizing effect of reflection to this case.
But in this case the formulæ expressed an impossibility; yet as
algebraical formulæ, even in such cases, have often some meaning, "I
interpreted," he says,[111\9] "in the manner which appeared to me
most natural and most probable, what the analysis indicated by this
imaginary form;" and by such an interpretation he collected the law
of the difference of undulation of the two rays. He was thus able to
predict that by two internal reflections in a _rhomb_, or
parallelopiped of glass, of a certain form and position, a polarized
ray would acquire a circular undulation of its particles; and this
constitution of the ray, it appeared, by reasoning further, would
show itself by its possessing peculiar properties, partly the same
as those of polarized light, and partly different. This
extraordinary anticipation was exactly confirmed; and thus the
apparently bold and strange guess of the author was fully justified,
or at least assented to, even by the most cautious philosophers. "As
I cannot appreciate the mathematical evidence for the nature of
circular polarization," says Prof. Airy,[112\9] "I shall mention the
experimental evidence on which I receive it." The conception has
since been universally adopted.
[Note 110\9: Ib. p. 194.]
[Note 111\9: _Bullet. des Sc._ 1823, p. 33.]
[Note 112\9: _Camb. Trans._ vol. iv. p. 81, 1831.]
But Fresnel, having thus obtained circularly-polarized rays, saw
{121} that he could account for the phenomena of quartz, already
observed by M. Arago, as we have noticed in Chap. ix., by supposing
two circularly-polarized rays to pass, with different velocities,
along the axis. The curious succession of colors, following each
other in right-handed or left-handed circular order, of which we
have already spoken, might thus be hypothetically explained.
But was this hypothesis of two circularly-polarized rays, travelling
along the axis of such crystals, to be received, merely because it
accounted for the phenomena? Fresnel's ingenuity again enabled him
to avoid such a defect in theorizing. If there were two such rays,
they might be visibly separated[113\9] by the same artifice, of a
pile of prisms properly achromatized, which he had used for
compressed glass. The result was, that he did obtain a visible
separation of the rays; and this result has since been confirmed by
others, for instance. Professor Airy.[114\9] The rays were found to
be in all respects identical with the circularly-polarized rays
produced by the internal reflections in Fresnel's rhomb. This kind
of double refraction gave a hypothetical explanation of the laws
which M. Biot had obtained for the phenomena of this class; for
example,[115\9] the rule, that the deviation of the plane of
polarization of the emergent ray is inversely as the square of the
length of an undulation for each kind of rays. And thus the
phenomena produced by light passing along the axis of quartz were
reduced into complete conformity with the theory.
[Note 113\9: _Bull. des Sc._ 1822, p. 193.]
[Note 114\9: _Cambridge Trans._ iv. p. 80.]
[Note 115\9: _Bull. des Sc._ 1822, p. 197.]
[2nd Ed.] [I believe, however, Fresnel did not deduce the phenomenon
from the mathematical formula, without the previous suggestion of
experiment. He _observed_ appearances which implied a difference of
retardation in the two differently-polarized rays at total
reflection; as Sir D. Brewster observed in reflection of metals
phenomena having a like character. The general fact being observed,
Fresnel used the theory to discover the law of this retardation, and
to determine a construction in which, one ray being a quarter of an
undulation retarded more than the other, circular polarization would
be produced. And this anticipation was verified by the construction
of his _rhomb_.
As a still more curious verification of this law, another of
Fresnel's experiments may be mentioned. He found the proper angles
for a circularly-polarizing glass rhomb on the supposition that
there were {122} _four_ internal reflections instead of two; two of
the four taking place when the surface of the glass was dry, and two
when it was wet. The rhomb was made; and when all the points of
reflection were dry, the light was not circularly polarized; when
two points were wet, the light was circularly polarized; and when
all four were wet, it was not circularly polarized.]
3. _Elliptical Polarization in Quartz._--We now come to one of the
few additions to Fresnel's theory which have been shown to be
necessary. He had accounted fully for the colors produced by the
rays which travel _along the axis_ of quartz crystals; and thus, for
the colors and changes of the central spot which is produced when
polarized light passes through a transverse plate of such crystals.
But this central spot is surrounded by rings of colors. How is the
theory to be extended to these?
This extension has been successfully made by Professor Airy.[116\9]
His hypothesis is, that as rays passing along the axis of a quartz
crystal are circularly polarized, rays which are oblique to the axis
are elliptically polarized, the amount of ellipticity depending, in
some unknown manner, upon the obliquity; and that each ray is
separated by double refraction into two rays polarized elliptically;
the one right-handed, the other left-handed. By means of these
suppositions, he not only was enabled to account for the simple
phenomena of single plates of quartz; but for many most complex and
intricate appearances which arise from the superposition of two
plates, and which at first sight might appear to defy all attempts
to reduce them to law and symmetry; such as spirals, curves
approaching to a square form, curves broken in four places. "I can
hardly imagine," he says,[117\9] very naturally, "that any other
supposition would represent the phenomena to such extreme accuracy.
I am not so much struck with the accounting for the continued
dilatation of circles, and the general representation of the forms
of spirals, as with the explanations of the minute deviations from
symmetry; as when circles become almost square, and crosses are
inclined to the plane of polarization. And I believe that any one
who shall follow my investigation, and imitate my experiments, will
be surprised at their perfect agreement."
[Note 116\9: _Camb. Trans._, iv. p. 83, &c.]
[Note 117\9: _Camb. Trans._, iv. p. 122.]
4. _Differential Equations of Elliptical Polarization._--Although
circular and elliptical polarization can be clearly conceived, and
their existence, it would seem, irresistibly established by the
phenomena, it {123} is extremely difficult to conceive any
arrangement of the particles of bodies by which such motions can
mechanically be produced; and this difficulty is the greater,
because some fluids and some gases impress a circular polarization
upon light; in which cases we cannot imagine any definite
arrangement of the particles, such as might form the mechanism
requisite for the purpose. Accordingly, it does not appear that any
one has been able to suggest even a plausible hypothesis on that
subject. Yet, even here, something has been done. Professor Mac
Cullagh, of Dublin, has discovered that by slightly modifying the
_analytical expressions_ resulting from the common case of the
propagation of light, we may obtain other expressions which would
give rise to such motions as produce circular and elliptical
polarization. And though we cannot as yet assign the mechanical
interpretation of the language of analysis thus generalized, this
generalization brings together and explains by one common numerical
supposition, two distinct classes of facts;--a circumstance which,
in all cases, entitles an hypothesis to a very favorable
consideration.
Mr. Mac Cullagh's assumption consists in adding to the two equations
of motion which are expressed by means of second differentials, two
other terms involving third differentials in a simple and
symmetrical manner. In doing this, he introduces a coefficient, of
which the magnitude determines both the amount of rotation of the
polarization of a ray passing along the axis, as observed and
measured by Biot, and the ellipticity of the polarization of a ray
which is oblique to the axis, according to Mr. Airy's theory, of
which ellipticity that philosopher also had obtained certain
measures. The agreement between the two sets of measures[118\9] thus
brought into connexion is such as very strikingly to confirm Mr. Mac
Cullagh's hypothesis. It appears probable, too, that the
confirmation of this hypothesis involves, although in an obscure and
oracular form, a confirmation of the undulatory theory, which is the
starting-point of this curious speculation.
[Note 118\9: _Royal I. A. Trans._ 1836.]
5. _Elliptical Polarization of Metals._--The effect of metals upon
the light which they reflect, was known from the first to be
different from that which transparent bodies produce. Sir David
Brewster, who has recently examined this subject very fully,[119\9]
has described the modification thus produced, as _elliptic
polarization_. In employing this term, "he seems to have been led,"
it has been observed,[120\9] "by a {124} desire to avoid as much as
possible all reference to theory. The laws which he has obtained,
however, belong to elliptically-polarized light in the sense in
which the term was introduced by Fresnel." And the identity of the
light produced by metallic reflection with the
elliptically-polarized light of the wave-theory, is placed beyond
all doubt, by an observation of Professor Airy, that the rings of
uniaxal crystals, produced by Fresnel's elliptically-polarized
light, are exactly the same as those produced by Brewster's metallic
light.
[Note 119\9: _Phil. Trans._ 1830.]
[Note 120\9: Lloyd, _Report on Optics_, p. 372. (Brit. Assoc.)]
6. _Newton's Rings by Polarized Light._--Other modifications of the
phenomena of thin plates by the use of polarized light, supplied
other striking confirmations of the theory. These were in one case
the more remarkable, since the result was foreseen by means of a
rigorous application of the conception of the vibratory motion of
light, and confirmed by experiment. Professor Airy, of Cambridge,
was led by his reasonings to see, that if Newton's rings are
produced between a lens and a plate of metal, by polarized light,
then, up to the polarizing angle, the central spot will be black,
and instantly beyond this, it will be white. In a note,[121\9] in
which he announced this, he says, "This I anticipated from Fresnel's
expressions; it is confirmatory of them, and defies emission." He
also predicted that when the rings were produced between two
substances of very different refractive powers, the centre would
twice pass from black to white and from white to black, by
increasing the angle; which anticipation was fulfilled by using a
diamond for the higher refraction.[122\9]
[Note 121\9: Addressed to myself, dated May 28, 1831. I ought,
however, to notice, that this experiment had been made by M. Arago,
fifteen years earlier, and published: though not then recollected by
Mr. Airy.]
[Note 122\9: _Camb. Trans._ vol. ii. p. 409.]
7. _Conical Refraction._--In the same manner. Professor Hamilton of
Dublin pointed out that according to the Fresnelian doctrine of double
refraction, there is a certain direction of a crystal in which a
single ray of light will be refracted so as to form a _conical
pencil_. For the direction of the refracted ray is determined by a
plane which touches the wave surface, the rule being that the ray must
pass from the centre of the surface to the point of contact; and
though in general this contact gives a single point only, it so
happens, from the peculiar inflected form of the wave surface, which
has what is called _a cusp_, that in one particular position, the
plane can touch the surface in an entire circle. Thus the general rule
which assigns the path of {125} the refracted ray, would, in this
case, guide it from the centre of the surface to every point in the
circumference of the circle, and thus make it a cone. This very
curious and unexpected result, which Professor Hamilton thus obtained
from the theory, his friend Professor Lloyd verified as an
experimental fact. We may notice, also, that Professor Lloyd found the
light of the conical pencil to be polarized according to a law of an
unusual kind; but one which was easily seen to be in complete
accordance with the theory.
8. _Fringes of Shadows._--The phenomena of the _fringes of shadows_
of small holes and groups of holes, which had been the subject of
experiment by Fraunhofer, were at a later period carefully observed
in a vast variety of cases by M. Schwerd of Spires, and published in
a separate work,[123\9] _Beugungs-erscheinungen_ (Phenomena of
Inflection), 1836. In this Treatise, the author has with great
industry and skill calculated the integrals which, as we have seen,
are requisite in order to trace the consequences of the theory; and
the accordance which he finds between these and the varied and
brilliant results of observation is throughout exact. "I shall,"
says he, in the preface,[124\9] "prove by the present Treatise, that
all inflection-phenomena, through openings of any form, size, and
arrangement, are not only explained by the undulation-theory, but
that they can be represented by analytical expressions, determining
the intensity of the light in any point whatever." And he justly
adds, that the undulation-theory accounts for the phenomena of
light, as completely as the theory of gravitation does for the facts
of the solar system.
[Note 123\9: _Die Beugungs-erscheinungen, aus dem Fundamental-gesetz
der Undulations-Theorie analytisch entwickelt und in Bildern
dargestellt_, von F. M. Schwerd. Mannheim, 1835.]
[Note 124\9: Dated Speyer, Aug. 1835.]
9. _Objections to the Theory._--We have hitherto mentioned only
cases in which the undulatory theory was either entirely successful
in explaining the facts, or at least hypothetically consistent with
them and with itself. But other objections were started, and some
difficulties were long considered as very embarrassing. Objections
were made to the theory by some English experimenters, as Mr.
Potter, Mr. Barton, and others. These appeared in scientific
journals, and were afterwards answered in similar publications. The
objections depended partly on the measure of the _intensity_ of
light in the different points of the phenomena (a datum which it is
very difficult to obtain with accuracy {126} by experiment), and
partly on misconceptions of the theory; and I believe there are none
of them which would now be insisted on.
We may mention, also, another difficulty, which it was the habit of
the opponents of the theory to urge as a reproach against it, long
after it had been satisfactorily explained: I mean the
_half-undulation_ which Young and Fresnel had found it necessary, in
some cases, to assume as gained or lost by one of the rays. Though
they and their followers could not analyse the mechanism of
reflection with sufficient exactness to trace out all the
circumstances, it was not difficult to see, upon Fresnel's
principles, that reflection from the interior and exterior surface
of glass must be of opposite kinds, which might be expressed by
supposing one of these rays to lose half an undulation. And thus
there came into view a justification of the step which had
originally been taken upon empirical grounds alone.
10. _Dispersion, on the Undulatory Theory._--A difficulty of another
kind occasioned a more serious and protracted embarrassment to the
cultivators of this theory. This was the apparent impossibility of
accounting, on the theory, for the prismatic dispersion of color.
For it had been shown by Newton that the amount of refraction is
different for every color; and the amount of refraction depends on
the velocity with which light is propagated. Yet the theory
suggested no reason why the velocity should be different for
different colors: for, by mathematical calculation, vibrations of
all degrees of rapidity (in which alone colors differ) are
propagated with the same speed. Nor does analogy lead us to expect
this variety. There is no such difference between quick and slow
waves of air. The sounds of the deepest and the highest bells of a
peal are heard at any distance in the same order. Here, therefore,
the theory was at fault.
But this defect was far from being a fatal one. For though the
theory did not explain, it did not contradict, dispersion. The
suppositions on which the calculations had been conducted, and the
analogy of sound, were obviously in no small degree precarious. The
velocity of propagation might differ for different rates of
undulation, in virtue of many causes which would not affect the
general theoretical results.
Many such hypothetical causes were suggested by various eminent
mathematicians, as solutions of this conspicuous difficulty. But
without dwelling upon these conjectures, it may suffice to notice
that hypothesis upon which the attention of mathematicians was soon
concentrated. This was the _hypothesis of finite intervals_ between
the {127} particles of the ether. The length of one of those
undulations which produce light, is a very small quantity, its mean
value being 1⁄50,000th of an inch; but in the previous
investigations of the consequences of the theory, it had been
assumed that the distance from each other, of the particles of the
ether, which, by their attractions or repulsions, caused the
undulations to be propagated, is indefinitely less than this small
quantity;--so that its amount might be neglected in the cases in
which the length of the undulation was one of the quantities which
determined the result. But this assumption was made arbitrarily, as
a step of simplification, and because it was imagined that, in this
way, a nearer approach was made to the case of a continuous fluid
ether, which the supposition of distinct particles imperfectly
represented. It was still free for mathematicians to proceed upon
the opposite assumption, of particles of which the distances were
finite, either as a mathematical basis of calculation, or as a
physical hypothesis; and it remained to be seen if, when this was
done, the velocity of light would still be the same for different
lengths of undulation, that is, for different colors. M. Cauchy,
calculating, upon the most general principles, the motion of such a
collection of particles as would form an elastic medium, obtained
results which included the new extension of the previous hypothesis.
Professor Powell, of Oxford, applied himself to reduce to
calculation, and to compare with experiment, the result of these
researches. And it appeared that, on M. Cauchy's principles, a
variation in the velocity of light is produced by a variation in the
length of the wave, provided that the interval between the molecules
of the ether bears a sensible ratio to the length of an
undulation.[125\9] Professor Powell obtained also, from the general
expressions, a formula expressing the relation between the
refractive index of a ray, and the length of a wave, or the color of
light.[126\9] It then became his task to ascertain whether this
relation obtained experimentally; and he found a very close
agreement between the numbers which resulted from the formula and
those observed by Fraunhofer, for ten different kinds of media,
namely, certain glasses and fluids.[127\9] To these he afterwards
added ten other cases of crystals observed by M. Rudberg.[128\9] Mr.
Kelland, of Cambridge, also calculated, in a manner somewhat
different, the results of the same hypothesis of finite
intervals;[129\9] and, obtaining {128} formulæ not exactly the same
as Professor Powell, found also an agreement between these and
Fraunhofer's observations.
[Note 125\9: _Phil. Mag._ vol. vi. p. 266.]
[Note 126\9: Ib. vol. vii. 1835, p. 266.]
[Note 127\9: _Phil. Trans._ 1835, p. 249.]
[Note 128\9: Ib. 1836, p. 17.]
[Note 129\9: _Camb. Trans._ vol. vi. p. 153.]
It may be observed, that the refractive indices observed and
employed in these comparisons, were not those determined by the
color of the ray, which is not capable of exact identification, but
those more accurate measures which Fraunhofer was enabled to make,
in consequence of having detected in the spectrum the black lines
which he called B, C, D, E, F, G, H. The agreement between the
theoretical formulæ and the observed numbers is remarkable,
throughout all the series of comparisons of which we have spoken.
Yet we must at present hesitate to pronounce upon the hypothesis of
finite intervals, as proved by these calculations; for though this
hypothesis has given results agreeing so closely with experiment, it
is not yet clear that other hypotheses may not produce an equal
agreement. By the nature of the case, there must be a certain
gradation and continuity in the succession of colors in the
spectrum, and hence, any supposition which will account for the
general fact of the whole dispersion, may possibly account for the
amount of the intermediate dispersions, because these must be
interpolations between the extremes. The result of this hypothetical
calculation, however, shows very satisfactorily that there is not,
in the fact of dispersion, anything which is at all formidable to
the undulatory theory.
11. _Conclusion._--There are several other of the more recondite
points of the theory which may be considered as, at present, too
undecided to allow us to speak historically of the discussions which
they have occasioned.[130\9] For example, it was conceived, for some
time, that the vibrations of polarized light are perpendicular to
the plane of polarization. But this assumption was not an essential
part of the theory; and all the phenomena would equally allow us to
suppose the vibrations to be in the polarization plane; the main
requisite being, that light polarized in planes at right angles to
each other, should also have the vibrations at right angles.
Accordingly, for some time, this point was left undecided by Young
and Fresnel, and, more recently, some mathematicians have come to
the opinion that ether vibrates in the plane of polarization. The
theory of transverse vibrations is equally stable, whichever
supposition may be finally confirmed.
[Note 130\9: For on account of these, see Professor Lloyd's _Report
on Physical Optics_. (Brit. Assoc. Report, 1834.)]
We may speak, in the same manner, of the suppositions which, from
{129} the time of Young and Fresnel, the cultivators of this theory
have been led to make respecting the mechanical constitution of the
ether, and the forces by which transverse vibrations are produced.
It was natural that various difficulties should arise upon such
points, for transverse vibrations had not previously been made the
subject of mechanical calculation, and the forces which occasion
them must act in a different manner from those which were previously
contemplated. Still, we may venture to say, without entering into
these discussions, that it has appeared, from all the mathematical
reasonings which have been pursued, that there is not, in the
conception of transverse vibrations, anything inconsistent either
with the principles of mechanics, or with the best general views
which we can form, of the forces by which the universe is held
together.
I willingly speak as briefly as the nature of my undertaking allows,
of those points of the undulatory theory which are still under
deliberation among mathematicians. With respect to these, an
intimate acquaintance with mathematics and physics is necessary to
enable any one to understand the steps which are made from day to
day; and still higher philosophical qualifications would be
requisite in order to pronounce a judgment upon them. I shall,
therefore, conclude this survey by remarking the highly promising
condition of this great department of science, in respect to the
character of its cultivators. Nothing less than profound thought and
great mathematical skill can enable any one to deal with this
theory, in any way likely to promote the interests of science. But
there appears, in the horizon of the scientific world, a
considerable class of young mathematicians, who are already bringing
to these investigations the requisite talents and zeal; and who,
having acquired their knowledge of the theory since the time when
its acceptation was doubtful, possess, without effort, that
singleness and decision of view as to its fundamental doctrines,
which it is difficult for those to attain whose minds have had to go
through the hesitation, struggle, and balance of the epoch of the
establishment of the theory. In the hands of this new generation, it
is reasonable to suppose the Analytical Mechanics of light will be
improved as much as the Analytical Mechanics of the solar system was
by the successors of Newton. We have already had to notice many of
this younger race of undulationists. For besides MM. Cauchy,
Poisson, and Ampère, M. Lamé has been more recently following these
researches in France.[131\9] In {130} Belgium, M. Quetelet has given
great attention to them; and, in our own country, Sir William
Hamilton, and Professor Lloyd, of Dublin, have been followed by Mr.
Mac Cullagh. Professor Powell, of Oxford, has continued his
researches with unremitting industry; and, at Cambridge, Professor
Airy, who did much for the establishment and diffusion of the theory
before he was removed to the post of Astronomer Royal, at Greenwich,
has had the satisfaction to see his labors continued by others, even
to the most recent time; for Mr. Kelland,[132\9] whom we have
already mentioned, and Mr. Archibald Smith,[133\9] the two persons
who, in 1834 and 1836, received the highest mathematical honors
which that university can bestow, have both of them published
investigations respecting the undulatory theory.
[Note 131\9: Prof. Lloyd's _Report_, p. 392.]
[Note 132\9: _On the Dispersion of Light, as explained by the
Hypothesis of Finite Intervals._ Camb. Trans. vol. vi. p. 153.]
[Note 133\9: _Investigation of the Equation to Fresnel's Wave
Surface_, ib. p. 85. See also, in the same volume, _Mathematical
Considerations on the Problem of the Rainbow_, showing it to belong
to Physical Optics, by R. Potter, Esq., of Queen's College.]
We may be permitted to add, as a reflection obviously suggested by
these facts, that the cause of the progress of science is
incalculably benefited by the existence of a body of men, trained
and stimulated to the study of the higher mathematics, such as exist
in the British universities, who are thus prepared, when an abstruse
and sublime theory comes before the world with all the characters of
truth, to appreciate its evidence, to take steady hold of its
principles, to pursue its calculations, and thus to convert into a
portion of the permanent treasure and inheritance of the civilized
world, discoveries which might otherwise expire with the great
geniuses who produced them, and be lost for ages, as, in former
times, great scientific discoveries have sometimes been.
The reader who is acquainted with the history of recent optical
discovery, will see that we have omitted much which has justly
excited admiration; as, for example, the phenomena produced by glass
under heat or pressure, noticed by MM. Lobeck, and Biot, and
Brewster, and many most curious properties of particular minerals.
We have omitted, too, all notice of the phenomena and laws of the
absorption of light, which hitherto stand unconnected with the
theory. But in this we have not materially deviated from our main
design; for our end, in what we have done, has been to trace the
advances of Optics {131} towards perfection as a theory; and this
task we have now nearly executed as far as our abilities allow.
We have been desirous of showing that the _type_ of this progress,
in the histories of the two great sciences, Physical Astronomy and
Physical Optics, is the same. In both we have many _Laws of
Phenomena_ detected and accumulated by acute and inventive men; we
have _Preludial_ guesses which touch the true theory, but which
remain for a time imperfect, undeveloped, unconfirmed: finally we
have the _Epoch_ when this true theory, clearly apprehended by great
philosophical geniuses, is recommended by its fully explaining what
it was first meant to explain, and confirmed by its explaining what
it was not meant to explain. We have then its _Progress_ struggling
for a little while with adverse prepossessions and difficulties;
finally overcoming all these, and moving onwards, while its
triumphal procession is joined by all the younger and more vigorous
men of science.
It would, perhaps, be too fanciful to attempt to establish a
parallelism between the prominent persons who figure in these two
histories. If we were to do this, we must consider Huyghens and
Hooke as standing in the place of Copernicus, since, like him, they
announced the true theory, but left it to a future age to give it
development and mechanical confirmation; Malus and Brewster,
grouping them together, correspond to Tycho Brahe and Kepler,
laborious in accumulating observations, inventive and happy in
discovering laws of phenomena; and Young and Fresnel combined, make
up the Newton of optical science.
[2nd Ed.] [In the _Report on Physical Optics_, (_Brit. Ass.
Reports_, 1834,) by Prof. Lloyd, the progress of the mathematical
theory after Fresnel's labors is stated more distinctly than I have
stated it, to the following effect. Ampère, in 1828, proved
Fresnel's mathematical results directly, which Fresnel had only
proved indirectly, and derived from his proof Fresnel's beautiful
geometrical construction. Prof. Mac Cullagh not long after gave a
concise demonstration of the same theorem, and of the other
principal points of Fresnel's theory. He represents the elastic
force by means of an ellipsoid whose axes are inversely proportional
to those of Fresnel's generating ellipsoid, and deduces Fresnel's
construction geometrically. In the third Supplement to his _Essay on
the Theory of Systems of Rays_ (_Trans. R. I. Acad._ vol. xvii.),
Sir W. Hamilton has presented that portion of Fresnel's theory which
relates to the fundamental problem of the determination of the
velocity and polarization of a plane wave, in a very elegant and
analytical form. This he does by means of what he calls the {132}
_characteristic function_ of the optical system to which the problem
belongs. From this function is deduced the _surface of
wave-slowness_ of the medium; and by means of this surface, the
direction of the rays refracted into the medium. From this
construction also Sir W. Hamilton was led to the anticipation of
_conical refraction_, mentioned above.
The investigations of MM. Cauchy and Lamé refer to the laws by which
the particles of the ether act upon each other and upon the
particles of other bodies;--a field of speculation which appears to
me not yet ripe for the final operations of the analyst.
Among the mathematicians who have supplied defects in Fresnel's
reasoning on this subject, I may mention Mr. Tovey, who treated it
in several papers in the _Philosophical Magazine_ (1837-40). Mr.
Tovey's early death must be deemed a loss to mathematical science.
Besides investigating the motion of symmetrical systems of particles
which may be supposed to correspond to biaxal crystals, Mr. Tovey
considered the case of unsymmetrical systems, and found that the
undulations propagated would, in the general case, be elliptical;
and that in a particular case, circular undulations would take
place, such as are propagated along the axis of quartz. It appears
to me, however, that he has not given a definite meaning to those
limitations of his general hypothesis which conduct him to this
result. Perhaps if the hypothetical conditions of this result were
traced into detail, they would be found to reside in a _screw-like_
arrangement of the elementary particles, in some degree such as
crystals of quartz themselves exhibit in their forms, when they have
plagihedral faces at both ends.
Such crystals of quartz are, some like a right-handed and some like
a left-handed screw; and, as Sir John Herschel discovered, the
circular polarization is right-handed or left-handed according as
the plagihedral form is so. In Mr. Tovey's hypothetical
investigation it does not appear upon what part of the hypothesis
this difference of right and left-handed depends. The definition of
this part of the hypothesis is a very desirable step.
When crystals of Quartz are right-handed at one end, they are
right-handed at the other end: but there is a different kind of
plagihedral form, which occurs in some other crystals, for instance,
in Apatite: in these the plagihedral faces are right-handed at the
one extremity and left-handed at the other. For the sake of
distinction, we may call the former _homologous_ plagihedral faces,
since, at both ends, they have the same name; and the latter
_heterologous_ plagihedral faces. {133}
The homologous plagihedral faces of Quartz crystals are accompanied
by homologous circular polarization of the same name. I do not know
that heterologous circular polarization has been observed in any
crystal, but it has been discovered by Dr. Faraday to occur in
glass, &c., when subjected to powerful magnetic action.
Perhaps it was presumptuous in me to attempt to draw such
comparisons, especially with regard to living persons, as I have
done in the preceding pages of this Book. Having published this
passage, however, I shall not now suppress it. But I may observe
that the immense number and variety of the beautiful optical
discoveries which we owe to Sir David Brewster makes the comparison
in his case a very imperfect representation of his triumphs over
nature; and that, besides his place in the history of the Theory of
Optics, he must hold a most eminent position in the history of
Optical Crystallography, whenever the discovery of a True Optical
Theory of Crystals supplies us with the _Epoch_ to which his labors
in this field form so rich a _Prelude_. I cordially assent to the
expression employed by Mr. Airy in the _Phil. Trans._ for 1840, in
which he speaks of Sir David Brewster as "the Father of Modern
Experimental Optics."]
{{135}}
BOOK X.
_SECONDARY MECHANICAL SCIENCES._
(CONTINUED.)
HISTORY
OF
THERMOTICS AND ATMOLOGY.
Et primum faciunt ignem se vortere in auras
Aëris; hinc imbrem gigni terramque creari
Ex imbri; retroque a terrâ cuncta revorti,
Humorem primum, post aëra deinde calorem;
Nec cessare hæc inter se mutare, meare,
De cœlo ad terram de terrâ ad sidera mundi.
LUCRETIUS, i. 783.
Water, and Air, and Fire, alternate run
Their endless circle, multiform, yet one.
For, moulded by the fervor's latent beams,
Solids flow loose, and fluids flash to steams,
And elemental flame, with secret force,
Pursues through earth, air, sky, its stated course.
{{137}}
INTRODUCTION.
_Of Thermotics and Atmology._
I EMPLOY the term _Thermotics_ to include all the doctrines
respecting Heat, which have hitherto been established on proper
scientific grounds. Our survey of the history of this branch of
science must be more rapid and less detailed than it has been in
those subjects of which we have hitherto treated: for our knowledge
is, in this case, more vague and uncertain than in the others, and
has made less progress towards a general and certain theory. Still,
the narrative is too important and too instructive to be passed over.
The distinction of Formal Thermotics and Physical Thermotics,--of
the discovery of the mere Laws of Phenomena, and the discovery of
their causes,--is applicable here, as in other departments of our
knowledge. But we cannot exhibit, in any prominent manner, the
latter division of the science now before us; since no general
theory of heat has yet been propounded, which affords the means of
calculating the circumstances of the phenomena of conduction,
radiation, expansion, and change of solid, liquid, and gaseous form.
Still, on each of these subjects there have been proposed, and
extensively assented to, certain general views, each of which
explains its appropriate class of phenomena; and, in some cases,
these principles have been clothed in precise and mathematical
conditions, and thus made bases of calculation.
These principles, thus possessing a generality of a limited kind,
connecting several observed laws of phenomena, but yet not
connecting all the observed classes of facts which relate to heat,
will require our separate attention. They may be described as the
Doctrine of Conduction, the Doctrine of Radiation, the Doctrine of
Specific Heat, and the Doctrine of Latent Heat; and these, and
similar doctrines respecting heat, make up the science which we may
call _Thermotics proper_.
But besides these collections of principles which regard heat by
itself, the relations of heat and moisture give rise to another and
important collection of laws and principles, which I shall treat of
in connexion with Thermotics, and shall term _Atmology_, borrowing
{138} the term from the Greek word (ἄτμος,) which signifies _vapor_.
The _Atmosphere_ was so named by the Greeks, as being a sphere of
vapor; and, undoubtedly, the most general and important of the
phenomena which take place in the air, by which the earth is
surrounded, are those in which water, of one _consistence_ or other
(ice, water, or steam,) is concerned. The knowledge which relates to
what takes place in the atmosphere has been called _Meteorology_, in
its collective form: but such knowledge is, in fact, composed of
parts of many different sciences. And it is useful for our purpose
to consider separately those portions of Meteorology which have
reference to the laws of aqueous vapor, and these we may include
under the term Atmology.
The instruments which have been invented for the purpose of
measuring the moisture of the air, that is, the quantity of vapor
which exists in it, have been termed _Hygrometers_; and the
doctrines on which these instruments depend, and to which they lead,
have been called _Hygrometry_; but this term has not been used in
quite so extensive a sense as that which we intend to affix to
_Atmology_.
In treating of Thermotics, we shall first describe the earlier
progress of men's views concerning Conduction, Radiation, and the
like, and shall then speak of the more recent corrections and
extensions, by which they have been brought nearer to theoretical
generality.
{{139}}
THERMOTICS PROPER.
CHAPTER I.
THE DOCTRINES OF CONDUCTION AND RADIATION.
_Section_ 1.--_Introduction of the Doctrine of Conduction._
BY _conduction_ is meant the propagation of heat from one part to
another of a continuous body; or from one body to another in contact
with it; as when one end of a poker stuck in the fire heats the
other end, or when this end heats the hand which takes hold of it.
By _radiation_ is meant the diffusion of heat from the surface of a
body to points not in contact. It is clear in both these cases,
that, in proportion as the hot portion is hotter, it produces a
greater effect in warming the cooler portion; that is, it
_communicates more Heat_ to it, if _Heat_ be the abstract conception
of which this effect is the measure. The simplest rule which can be
proposed is, that the heat thus communicated in a given instant is
proportional to the excess of the heat of the hot body over that of
the contiguous bodies; there are no obvious phenomena which
contradict the supposition that this is the true law; and it was
thence assumed by Newton as the true law for radiation and by other
writers for conduction. This assumption was confirmed approximately,
and afterwards corrected, for the case of Radiation; in its
application to Conduction, it has been made the basis of calculation
up to the present time. We may observe that this statement takes for
granted that we have attained to a measure of heat (or of
_temperature_, as heat thus measured is termed), corresponding to
the law thus assumed; and, in fact, as we shall have occasion to
explain in speaking of the _measures_ of sensible qualities, {140}
the thermometrical scale of heat according to the expansion of
liquids (which is the measure of temperature here adopted), was
constructed with a reference to Newton's law of radiation of heat;
and thus the law is necessarily consistent with the scale.
In any case in which the parts of a body are unequally hot, the
temperature will vary _continuously_ in passing from one part of the
body to another; thus, a long bar of iron, of which one end is kept
red hot, will exhibit a gradual diminution of temperature at
successive points, proceeding to the other end. The law of
temperature of the parts of such a bar might be expressed by the
ordinates of a _curve_ which should run alongside the bar. And, in
order to trace mathematically the consequences of the assumed law,
some of those processes would be necessary, by which mathematicians
are enabled to deal with the properties of curves; as the method of
infinitesimals, or the differential calculus; and the truth or
falsehood of the law would be determined, according to the usual
rules of inductive science, by a comparison of results so deduced
from the principle, with the observed phenomena.
It was easily perceived that this comparison was the task which
physical inquirers had to perform; but the execution of it was
delayed for some time; partly, perhaps, because the mathematical
process presented some difficulties. Even in a case so simple as
that above mentioned, of a linear bar with a stationary temperature
at one end, _partial differentials_ entered; for there were three
variable quantities, the time, as well as the place of each point
and its temperature. And at first, another scruple occurred to M.
Biot when, about 1804, he undertook this problem.[1\10] "A
difficulty," says Laplace,[2\10] in 1809, "presents itself, which
has not yet been solved. The quantities of heat received and
communicated in an instant (by any point of the bar) must be
infinitely small quantities of the same order as the excess of the
heat of a slice of the body over that of the contiguous slice;
therefore the _excess_ of the heat received by any slice over the
heat communicated, is an infinitely small quantity of the second
order; and the accumulation in a finite time (which depends on this
excess) cannot be finite." I conceive that this difficulty arises
entirely from an arbitrary and unnecessary assumption concerning the
relation of the infinitesimal parts of the body. Laplace resolved
the difficulty by further reasoning founded upon the same assumption
which occasioned {141} it; but Fourier, who was the most
distinguished of the cultivators of this mathematical doctrine of
conduction, follows a course of reasoning in which the difficulty
does not present itself. Indeed it is stated by Laplace, in the
Memoir above quoted,[3\10] that Fourier had already obtained the
true fundamental equations by views of his own.
[Note 1\10: Biot, _Traité de Phys._ iv. p. 669.]
[Note 2\10: Laplace, _Mém. Inst._ for 1809, p. 332.]
[Note 3\10: Laplace, _Mém. Inst._ for 1809, p. 538.]
The remaining part of the history of the doctrine of conduction is
principally the history of Fourier's labors. Attention having been
drawn to the subject, as we have mentioned, the French Institute, in
January, 1810, proposed, as their prize question, "To give the
mathematical theory of the laws of the propagation of heat, and to
compare this theory with exact observations." Fourier's Memoir (the
sequel of one delivered in 1807,) was sent in September, 1811; and
the prize (3000 francs) adjudged to it in 1812. In consequence of
the political confusion which prevailed in France, or of other
causes, these important Memoirs were not published by the Academy
till 1824; but extracts had been printed in the _Bulletin des
Sciences_ in 1808, and in the _Annales de Chimie_ in 1816; and
Poisson and M. Cauchy had consulted the manuscript itself.
It is not my purpose to give, in this place,[4\10] an account of the
analytical processes by which Fourier obtained his results. The
skill displayed in these Memoirs is such as to make them an object
of just admiration to mathematicians; but they consist entirely of
deductions from the fundamental principle which I have
noticed,--that the quantity of heat conducted from a hotter to a
colder point is proportional to the excess of heat, modified by the
_conductivity_, or conducting power of each substance. The equations
which flow from this principle assume nearly the same forms as those
which occur in the most general problems of hydrodynamics. Besides
Fourier's solution, Laplace, Poisson, and M. Cauchy have also
exercised their great analytical skill in the management of these
formulæ. We shall briefly speak of the comparison of the results of
these reasonings with experiment, and notice some other consequences
to which they lead. But before we can do this, we must pay some
attention to the subject of radiation. {142}
[Note 4\10: I have given an account of Fourier's mathematical
results in the _Reports of the British Association_ for 1835.]
_Sect._ 2.--_Introduction of the Doctrine of Radiation._
A HOT body, as a mass of incandescent iron, emits heat, as we
perceive by our senses when we approach it; and by this emission of
heat the hot body cools down. The first step in our systematic
knowledge of the subject was made in the _Principia_. "It was in the
destiny of that great work," says Fourier, "to exhibit, or at least
to indicate, the causes of the principal phenomena of the universe."
Newton assumed, as we have already said, that the rate at which a
body cools, that is, parts with its heat to surrounding bodies, is
proportional to its heat; and on this assumption he rested the
verification of his scale of temperatures. It is an easy deduction
from this law, that if times of cooling be taken in arithmetical
progression, the heat will decrease in geometrical progression.
Kraft, and after him Richman, tried to verify this law by direct
experiments on the cooling of vessels of warm water; and from these
experiments, which have since been repeated by others, it appears
that for differences of temperature which do not exceed 50 degrees
(boiling water being 100), this geometrical progression represents,
with tolerable (but not with complete) accuracy, the process of
cooling.
This principle of radiation, like that of conduction, required to be
followed out by mathematical reasoning. But it required also to be
corrected in the first place, for it was easily seen that the rate
of cooling depended, not on the absolute temperature of the body,
but on the excess of its temperature above the surrounding objects
to which it communicated its heat in cooling. And philosophers were
naturally led to endeavor to explain or illustrate this process by
some physical notions. Lambert in 1765 published[5\10] an _Essay on
the Force of Heat_, in which he assimilates the communication of
heat to the flow of a fluid out of one vessel into another by an
excess of pressure; and mathematically deduces the laws of the
process on this ground. But some additional facts suggested a
different view of the subject. It was found that heat is propagated
by radiation according to straight lines, like light; and that it
is, as light is, capable of being reflected by mirrors, and thus
brought to a focus of intenser action. In this manner the radiative
effect of a body could be more precisely traced. A fact, however,
came under notice, which, at first sight, appeared to {143} offer
some difficulty. It appeared that cold was reflected no less than
heat. A mass of ice, when its effect was concentrated on a
thermometer by a system of mirrors, made the thermometer fall, just
as a vessel of hot water placed in a similar situation made it rise.
Was cold, then, to be supposed a real substance, no less than heat?
[Note 5\10: _Act. Helvet._ tom. ii. p. 172.]
The solution of this and similar difficulties was given by Pierre
Prevost, professor at Geneva, whose theory of radiant heat was
proposed about 1790. According to this theory, heat, or _caloric_,
is constantly radiating from every point of the surface of all
bodies in straight lines; and it radiates the more copiously, the
greater is the quantity of heat which the body contains. Hence a
constant exchange of heat is going on among neighboring bodies; and
a body grows hotter or colder, according as it receives more caloric
than it emits, or the contrary. And thus a body is cooled by
rectilinear rays from a cold body, because along these paths it
sends rays of heat in greater abundance than those which return the
same way. This _theory of exchanges_ is simple and satisfactory, and
was soon generally adopted; but we must consider it rather as the
simplest mode of expressing the dependence of the communication of
heat on the excess of temperature, than as a proposition of which
the physical truth is clearly established.
A number of curious researches on the effect of the different kinds
of surface of the heating and of the heated body, were made by
Leslie and others. On these I shall not dwell; only observing that
the relative amount of this radiative and receptive energy may be
expressed by numbers, for each kind of surface; and that we shall
have occasion to speak of it under the term _exterior conductivity_;
it is thus distinguished from _interior conductivity_, which is the
relative rate at which heat is conducted in the interior of
bodies.[6\10]
[Note 6\10: The term employed by Fourier, _conductibility_ or
_conducibility_, suggests expressions altogether absurd, as if the
bodies could be called _conductible_, or _conducible_, with respect
to heat: I have therefore ventured upon a slight alteration of the
word, and have used the abstract term which analogy would suggest,
if we suppose bodies to be _conductive_ in this respect.]
_Sect._ 3.--_Verifications of the Doctrines of Conduction and
Radiation._
THE interior and exterior conductivity of bodies are numbers, which
enter as elements, or _coefficients_, into the mathematical
calculations founded on the doctrines of conduction and radiation.
These {144} coefficients are to be determined for each case by
appropriate experiments: when the experimenters had obtained these
data, as well as the mathematical solutions of the problems, they
could test the truth of their fundamental principles by a comparison
of the theoretical and actual results in properly-selected cases.
This was done for the law of conduction in the simple cases of
metallic bars heated at one end, by M. Biot,[7\10] and the
accordance with experiment was sufficiently close. In the more
complex cases of conduction which Fourier considered, it was less
easy to devise a satisfactory mode of comparison. But some rather
curious relations which he demonstrated to exist among the
temperatures at different points of an _armille_, or ring, afforded
a good criterion of the value of the calculations, and confirmed
their correctness.[8\10]
[Note 7\10: _Tr. de Phys._ iv. 671.]
[Note 8\10: _Mém. Inst._ 1819, p. 192, published 1824.]
We may therefore presume these doctrines of radiation and conduction
to be sufficiently established; and we may consider their
application to any remarkable case to be a portion of the history of
science. We proceed to some such applications.
_Sect._ 4.--_The Geological and Cosmological Application of
Thermotics._
BY far the most important case to which conclusions from these
doctrines have been applied, is that of the globe of the earth, and
of those laws of climate to which the modifications of temperature
give rise; and in this way we are led to inferences concerning other
parts of the universe. If we had any means of observing these
terrestrial and cosmical phenomena to a sufficient extent, they
would be valuable facts on which we might erect our theories; and
they would thus form part, not of the corollaries, but of the
foundations of our doctrine of heat. In such a case, the laws of the
propagation of heat, as discovered from experiments on smaller
bodies, would serve to explain these phenomena of the universe, just
as the laws of motion explain the celestial movements. But since we
are almost entirely without any definite indications of the
condition of the other bodies in the solar system as to heat; and
since, even with regard to the earth, we know only the temperature
of the parts at or very near the surface, our knowledge of the part
which heat plays in the earth and the heavens must be in a great
measure, not a generalization of observed facts, but a deduction
from theoretical principles. Still, such knowledge, whether obtained
{145} from observation or from theory, must possess great interest
and importance. The doctrines of this kind which we have to notice
refer principally to the effect of the sun's heat on the earth, the
laws of climate,--the thermotical condition of the interior of the
earth,--and that of the planetary spaces.
1. _Effect of Solar Heat on the Earth._--That the sun's heat passes
into the interior of the earth in a variable manner, depending upon
the succession of days and nights, summers and winters, is an
obvious consequence of our first notions on this subject. The mode
in which it proceeds into the interior, after descending below the
surface, remained to be gathered, either from the phenomena, or from
reasoning. Both methods were employed.[9\10] Saussure endeavored to
trace its course by digging, in 1785, and thus found that at the
depth of about thirty-one feet, the annual variation of temperature
is about 1⁄12th what it is at the surface. Leslie adopted a better
method, sinking the bulbs of thermometers deep in the earth, while
their stems appeared above the surface. In 1813, '16, and '17, he
observed thus the temperatures at the depths of one, two, four, and
eight feet, at Abbotshall, in Fifeshire. The results showed that the
extreme annual oscillations of the temperature diminish as we
descend. At the depth of one foot, the yearly range of oscillation
was twenty-five degrees (Fahrenheit); at two feet it was twenty
degrees; at four feet it was fifteen degrees; at eight feet it was
only nine degrees and a half. And the time at which the heat was
greatest was later and later in proceeding to the lower points. At
one foot, the maximum and minimum were three weeks after the
solstice of summer and of winter; at two feet, they were four or
five weeks; at four feet, they were two months; and at eight feet,
three months. The mean temperature of all the thermometers was
nearly the same. Similar results were obtained by Ott at Zurich in
1762, and by Herrenschneider at Strasburg in 1821, '2, '3.[10\10]
[Note 9\10: Leslie, art. _Climate_, Supp. _Enc. Brit._ 179.]
[Note 10\10: Pouillet, _Météorol._ t. ii. p. 643.]
These results had already been explained by Fourier's theory of
conduction. He had shown[11\10] that when the surface of a sphere is
affected by a periodical heat, certain alternations of heat travel
uniformly into the interior, but that the extent of the alternation
diminishes in geometrical progression in this descent. This
conclusion applies to the effect of days and years on the
temperature of the earth, and shows that such facts as those
observed by Leslie are both exemplifications of {146} the general
circumstances of the earth, and are perfectly in accordance with the
principles on which Fourier's theory rests.
[Note 11\10: _Mém. Inst._ for 1821 (published 1826), p. 162.]
2. _Climate._--The term _climate_, which means _inclination_, was
applied by the ancients to denote that inclination of the axis of
the terrestrial sphere from which result the inequalities of days in
different latitudes. This inequality is obviously connected also
with a difference of thermotical condition. Places near the poles
are colder, on the whole, than places near the equator. It was a
natural object of curiosity to determine the law of this variation.
Such a determination, however, involves many difficulties, and the
settlement of several preliminary points. How is the temperature of
any place to be estimated? and if we reply, by its _mean_
temperature, how are we to learn this mean? The answers to such
questions require very multiplied observations, exact instruments,
and judicious generalizations; and cannot be given here. But certain
first approximations may be obtained without much difficulty; for
instance, the mean temperature of any place may be taken to be the
temperature of deep springs, which is probably identical with the
temperature of the soil below the reach of the annual oscillations.
Proceeding on such facts, Mayer found that the mean temperature of
any place was nearly proportional to the square of the cosine of the
latitude. This, as a law of phenomena, has since been found to
require considerable correction; and it appears that the mean
temperature does not depend on the latitude alone, but on the
distribution of land and water, and on other causes. M. de Humboldt
has expressed these deviations[12\10] by his map of _isothermal
lines_, and Sir D. Brewster has endeavored to reduce them to a law
by assuming two _poles of maximum cold_.
[Note 12\10: British Assoc. 1833. Prof. Forbes's _Report on
Meteorology_, p. 215.]
The expression which Fourier finds[13\10] for the distribution of
heat in a homogeneous sphere, is not immediately comparable with
Mayer's empirical formula, being obtained on a certain hypothesis,
namely, that the equator is kept constantly at a fixed temperature.
But there is still a general agreement; for, according to the
theory, there is a diminution of heat in proceeding from the equator
to the poles in such a case; the heat is propagated from the equator
and the neighboring parts, and radiates out from the poles into the
surrounding space. And thus, in the case of the earth, the solar
heat enters in the tropical {147} parts, and constantly flows
towards the polar regions, by which it is emitted into the planetary
spaces.
[Note 13\10: Fourier. _Mém. Inst._ tom. v. p. 173.]
Climate is affected by many thermotic influences, besides the
conduction and radiation of the solid mass of the earth. The
atmosphere, for example, produces upon terrestrial temperatures
effects which it is easy to see are very great; but these it is not
yet in the power of calculation to appreciate;[14\10] and it is
clear that they depend upon other properties of air besides its
power to transmit heat. We must therefore dismiss them, at least for
the present.
[Note 14\10: _Mém. Inst._ tom. vii. p. 584]
3. _Temperature of the Interior of the Earth._--The question of the
temperature of the interior of the earth has excited great interest,
in consequence of its bearing on other branches of knowledge. The
various facts which have been supposed to indicate the fluidity of
the central parts of the terrestrial globe, belong, in general, to
geological science; but so far as they require the light of
thermotical calculations in order to be rightly reasoned upon, they
properly come under our notice here.
The principal problem of this kind which has been treated of is
this:--If in the globe of the earth there be a certain original
heat, resulting from its earlier condition, and independent of the
action of the sun, to what results will this give rise? and how far
do the observed temperatures of points below the surface lead us to
such a supposition? It has, for instance, been asserted, that in
many parts of the world the temperature, as observed in mines and
other excavations, increases in descending, at the rate of one degree
(centesimal) in about forty yards. What inference does this justify?
The answer to this question was given by Fourier and by Laplace. The
former mathematician had already considered the problem of the
cooling of a large sphere, in his Memoirs of 1807, 1809, and 1811.
These, however, lay unpublished in the archives of the Institute for
many years. But in 1820, when the accumulation of observations which
indicated an increase of the temperature of the earth as we descend,
had drawn observation to the subject, Fourier gave, in the Bulletin
of the Philomathic Society,[15\10] a summary of his results, as far
as they bore on this point. His conclusion was, that such an
increase of temperature in proceeding towards the centre of the
earth, can arise from nothing but the remains of a primitive
heat;--that the heat which the sun's action would communicate,
would, in its final and {148} permanent state, be uniform in the
same vertical line, as soon as we get beyond the influence of the
superficial oscillations of which we have spoken;--and that, before
the distribution of temperature reaches this limit, it will
decrease, not increase, in descending. It appeared also, by the
calculation, that this remaining existence of the primitive heat in
the interior of the earth's mass, was quite consistent with the
absence of all perceptible traces of it at the surface; and that the
same state of things which produces an increase of one degree of
heat in descending forty yards, does not make the surface a quarter
of a degree hotter than it would otherwise be. Fourier was led also
to some conclusions, though necessarily very vague ones, respecting
the time which the earth must have taken to cool from a supposed
original state of incandescence to its present condition, which time
it appeared must have been very great; and respecting the extent of
the future cooling of the surface, which it was shown must be
insensible. Everything tended to prove that, within the period which
the history of the human race embraces, no discoverable change of
temperature had taken place from the progress of this central
cooling. Laplace further calculated the effect[16\10] which any
contraction of the globe of the earth by cooling would produce on
the length of the day. He had already shown, by astronomical
reasoning, that the day had not become shorter by 1⁄200th of a
second, since the time of Hipparchus; and thus his inferences agreed
with those of Fourier. As far as regards the smallness of the
perceptible effect due to the past changes of the earth's
temperature, there can be no doubt that all the curious conclusions
just stated are deduced in a manner quite satisfactory, from the
fact of a general increase of heat in descending below the surface
of the earth; and thus our principles of speculative science have a
bearing upon the history of the past changes of the universe, and
give us information concerning the state of things in portions of
time otherwise quite out of our reach.
[Note 15\10: _Bullet. des Sc._ 1820, p. 58.]
[Note 16\10: _Conn. des Tems_, 1823.]
4. _Heat of the Planetary Spaces._--In the same manner, this portion
of science is appealed to for information concerning parts of space
which are utterly inaccessible to observation. The doctrine of heat
leads to conclusions concerning the temperatures of the spaces which
surround the earth, and in which the planets of the solar system
revolve. In his Memoir, published in 1827,[17\10] Fourier states
that he conceives it to follow from his principles, that these
planetary spaces {149} are not absolutely cold, but have a "proper
heat" independent of the sun and of the planets. If there were not
such a heat, the cold of the polar regions would be much more
intense than it is, and the alternations of cold and warmth, arising
from the influence of the sun, would be far more extreme and sudden
than we find them. As the cause of this heat in the planetary
spaces, he assigns the radiation of the innumerable stars which are
scattered through the universe.
[Note 17\10: _Mém. Inst._ tom. vii. p. 580.]
Fourier says,[18\10] "We conclude from these various remarks, and
principally from the mathematical examination of the question," that
this is so. I am not aware that the mathematical calculation which
bears peculiarly upon this point has anywhere been published. But it
is worth notice, that Svanberg has been led[19\10] to the opinion of
the same temperature in these spaces which Fourier had adopted (50
centigrade below zero), by an entirely different course of
reasoning, founded on the relation of the atmosphere to heat.
[Note 18\10: _Mém. Inst._ tom. vii. p. 581.]
[Note 19\10: Berzel. _Jahres Bericht_, xi. p. 50.]
In speaking of this subject, I have been led to notice incomplete
and perhaps doubtful applications of the mathematical doctrine of
conduction and radiation. But this may at least serve to show that
Thermotics is a science, which, like Mechanics, is to be established
by experiments on masses capable of manipulation, but which, like
that, has for its most important office the solution of geological
and cosmological problems. I now return to the further progress of
our thermotical knowledge.
_Sect._ 5.--_Correction of Newton's Law of Cooling._
IN speaking of the establishment of Newton's assumption, that the
temperature communicated is proportional to the excess of
temperature, we stated that it was approximately verified, and
afterwards corrected (chap. i., sect. 1.)**. This correction was the
result of the researches of MM. Dulong and Petit in 1817, and the
researches by which they were led to the true law, are an admirable
example both of laborious experiment and sagacious induction. They
experimented through a very great range of temperature (as high as
two hundred and forty degrees centigrade), which was necessary
because the inaccuracy of Newton's law becomes considerable only at
high temperatures. They removed the effect of the surrounding
medium, by making their experiments in a vacuum. They selected with
great {150} judgment the conditions of their experiments and
comparisons, making one quantity vary while the others remained
constant. In this manner they found, that _the quickness of cooling
for a constant excess of temperature, increases in geometrical
progression, when the temperature of the surrounding space increases
in arithmetical progression_; whereas, according to the Newtonian
law, this quickness would not have varied at all. Again, this
variation being left out of the account, it appeared that _the
quickness of cooling, so far as it depends on the excess of
temperature of the hot body, increases as the terms of a geometrical
progression diminished by a constant number, when the temperature of
the hot body increases in arithmetical progression_. These two laws,
with the coefficients requisite for their application to particular
substances, fully determine the conditions of cooling in a vacuum.
Starting from this determination, MM. Dulong and Petit proceeded to
ascertain the effect of the medium, in which the hot body is placed,
upon its rate of cooling; for this effect became a _residual
phenomenon_,[20\10] when the cooling in the vacuum was taken away.
We shall not here follow this train of research; but we may briefly
state, that they were led to such laws as this;--that the rapidity
of cooling due to any gaseous medium in which the body is placed, is
the same, so long as the excess of the body's temperature is the
same, although the temperature itself vary;--that the cooling power
of a gas varies with the elasticity, according to a determined law;
and other similar rules.
[Note 20\10: See _Phil. Ind. Sciences_, B. xiii. c. 7, Sect. iv.]
In reference to the process of their induction, it is worthy of
notice, that they founded their reasonings upon Prevost's law of
exchanges; and that, in this way, the second of their laws above
stated, respecting the quickness of cooling, was a mathematical
consequence of the first. It may be observed also, that their
temperatures are measured by means of the air-thermometer, and that
if they were estimated on another scale, the remarkable simplicity
and symmetry of their results would disappear. This is a strong
argument for believing such a measure of temperature to have a
natural prerogative of simplicity. This belief is confirmed by other
considerations; but these, depending on the laws of _expansion_ by
heat, cannot be here referred to; and we must proceed to finish our
survey of the mathematical theory of heat, as founded on the
phenomena of radiation and conduction, which alone have as yet been
traced up to general principles.
We may observe, before we quit this subject, that this correction of
{151} Newton's law will materially affect the mathematical
calculations on the subject, which were made to depend on that law
both by Fourier, Laplace, and Poisson. Probably, however, the
general features of the results will be the same as on the old
supposition. M. Libri, an Italian mathematician, has undertaken one
of the problems of this kind, that of the armil, with Dulong and
Petit's law for his basis, in a Memoir read to the Institute of
France in 1825, and since published at Florence.[21\10]
[Note 21\10: _Mém. de Math. et de Phys._ 1829.]
_Sect._ 6.--_Other Laws of Phenomena with respect to Radiation._
THE laws of radiation as depending upon the surface of radiating
bodies, and as affecting screens of various kinds interposed between
the hot body and the thermometer, were examined by several
inquirers. I shall not attempt to give an account of the latter
course of research, and of the different laws which luminous and
non-luminous heat have been found to follow in reference to bodies,
whether transparent or opaque, which intercept them. But there are
two or three laws of the phenomena, depending upon the effects of
the surfaces of bodies, which are important.
1. In the first place, the powers of bodies to _emit_ and to
_absorb_ heat, as far as depends upon their surface, appear to be in
the same proportion. If we blacken the surface of a canister of hot
water, it radiates heat more copiously; and in the same measure, it
is more readily heated by radiation.
2. In the next place, as the radiative power increases, the power of
reflection diminishes, and the contrary. A bright metal vessel
reflects much heat; on this very account it does not emit much; and
hence a hot fluid which such a vessel contains, remains hot longer
than it does in an unpolished case.
3. The heat is emitted from every point of the surface of a hot body
in all directions; but by no means in all directions with equal
intensity. The intensity of the heating ray is as the sine of the
angle which it makes with the surface.
The last law is entirely, the two former in a great measure, due to
the researches of Leslie, whose _Experimental Inquiry into the
Nature and Propagation of Heat_, published in 1804, contains a great
number of curious and striking results and speculations. The laws
now just {152} stated bear, in a very important manner, upon the
formation of the theory; and we must now proceed to consider what
appears to have been done in this respect; taking into account, it
must still be borne in mind, only the phenomena of conduction and
radiation.
_Sect._ 7.--_Fourier's Theory of Radiant Heat._
THE above laws of phenomena being established, it was natural that
philosophers should seek to acquire some conception of the physical
action by which they might account, both for these laws, and for the
general fundamental facts of Thermotics; as, for instance, the fact
that all bodies placed in an inclosed space assume, in time, the
temperature of the inclosure. Fourier's explanation of this class of
phenomena must be considered as happy and successful; for he has
shown that the supposition to which we are led by the most simple
and general of the facts, will explain, moreover, the less obvious
laws. It is an obvious and general fact, that bodies which are
included in the space tend to acquire the same temperature. And this
identity of temperature of neighboring bodies requires an
hypothesis, which, it is found, also accounts for Leslie's law of
the sine, in radiation.
This hypothesis is, that the radiation takes place, not from the
surface alone of the hot body, but from all particles situated
within a certain small depth of the surface. It is easy to
see[22\10] that, on this supposition, a ray emitted obliquely from
an internal particle, will be less intense than one sent forth
perpendicular to the surface, because the former will be intercepted
in a greater degree, having a greater length of path within the
body; and Fourier shows, that whatever be the law of this
intercepting power, the result will be, that the radiative intensity
is as the sine of the angle made by the ray with the surface.
[Note 22\10: _Mém. Inst._ t. v. 1821, p. 204.]
But this law is, as I have said, likewise necessary, in order that
neighboring bodies may tend to assume the same temperature: for
instance, in order that a small particle placed within a spherical
shell, should finally assume the temperature of the shell. If the
law of the sines did not obtain, the final temperature of such a
particle would depend upon its place in the inclosure;[23\10] and
within a shell of ice we should have, at certain points, the
temperature of boiling water and of melting iron.
[Note 23\10: _An. Chim._ iv. 1817, p. 129.]
This proposition may at first appear strange and unlikely; but it
may {153} be shown to be a necessary consequence of the assumed
principle, by very simple reasoning, which I shall give in a general
form in a Note.[24\10]
[Note 24\10: The following reasoning may show the connexion of the
law of the sines in radiant heat with the general principle of
ultimate identity of neighboring temperatures. The equilibrium and
identity of temperature between an including shell and an included
body, cannot obtain upon the whole, except it obtain between each
pair of parts of the two surfaces of the body and of the shell; that
is, any part of the one surface, in its exchanges with any part of
the other surface, must give and receive the same quantity of heat.
Now the quantity exchanged, so far as it depends on the receiving
surface, will, by geometry, be proportional to the sine of the
obliquity of that surface: and as, in the exchanges, each may be
considered as receiving, the quantity transferred must be
proportional to the sines of the two obliquities; that is, to that
of the giving as well as of the receiving surface.
Nor is this conclusion disturbed by the consideration, that all the
rays of heat which fall upon a surface are not absorbed, some being
reflected according to the nature of the surface. For, by the other
above-mentioned laws of phenomena, we know that, in the same measure
in which the surface loses the power of admitting, it loses the
power of emitting, heat; and the superficial parts gain, by
absorbing their own radiation, as much as they lose by not absorbing
the incident heat; so that the result of the preceding reasoning
remains unaltered.]
This reasoning is capable of being presented in a manner quite
satisfactory, by the use of mathematical symbols, and proves that
Leslie's law of the sines is rigorously and mathematically true on
Fourier's hypothesis. And thus Fourier's theory of _molecular
extra-radiation_ acquires great consistency.
_Sect._ 8.--_Discovery of the Polarization of Heat._
THE laws of which the discovery is stated in the preceding Sections
of this Chapter, and the explanations given of them by the theories
of conduction and radiation, all tended to make the conception of a
material heat, or _caloric_, communicated by an actual flow and
emission, familiar to men's minds; and, till lately, had led the
greater part of thermotical philosophers to entertain such a view,
as the most probable opinion concerning the nature of heat. But some
steps have recently been made in thermotics, which appear to be
likely to overturn this belief, and to make the doctrine of emission
as untenable with regard to heat, as it had been found to be with
regard to light. I speak of the discovery of the polarization of
heat. It being ascertained that rays of heat are polarized in the
same manner as rays of {154} light, we cannot retain the doctrine
that heat radiates by the emanation of material particles, without
supposing those particles of caloric to have poles; an hypothesis
which probably no one would embrace; for, besides that the ill
fortune which attended that hypothesis in the case of light must
deter speculators from it, the intimate connexion of heat and light
would hardly allow us to suppose polarization in the two cases to be
produced by two different kinds of machinery.
But, without here tracing further the influence which the
polarization of heat must exercise upon the formation of our
theories of heat, we must briefly notice this important discovery,
as a law of phenomena.
The analogies and connexions between light and heat are so strong,
that when the polarization of light had been discovered, men were
naturally led to endeavor to ascertain whether heat possessed any
corresponding property. But partly from the difficulty of obtaining
any considerable effect of heat separated from light, and partly
from the want of a thermometrical apparatus sufficiently delicate,
these attempts led, for some time, to no decisive result. M. Berard
took up the subject in 1813. He used Malus's apparatus, and
conceived that he found heat to be polarized by reflection at the
surface of glass, in the same manner as light, and with the same
circumstances.[25\10] But when Professor Powell, of Oxford, a few
years later (1830), repeated these experiments with a similar
apparatus, he found[26\10] that though the heat which is conveyed
along with light is, of course, polarizable, "simple radiant heat,"
as he terms it, did not offer the smallest difference in the two
rectangular azimuths of the second glass, and thus showed no trace
of polarization.
[Note 25\10: _Ann. Chim._ March, 1813.]
[Note 26\10: _Edin. Journ. of Science_, 1830, vol. ii. p. 303.]
Thus, with the old thermometers, the point remained doubtful. But
soon after this time, MM. Melloni and Nobili invented an apparatus,
depending on certain galvanic laws, of which we shall have to speak
hereafter, which they called a _thermomultiplier_; and which was
much more sensitive to changes of temperature than any
previously-known instrument. Yet even with this instrument, M.
Melloni failed; and did not, at first, detect any perceptible
polarization of heat by the tourmaline;[27\10] nor did M.
Nobili,[28\10] in repeating M. Berard's experiment. But in this
experiment the attempt was made to polarize heat by reflection from
glass, as light is polarized: and the quantity {155} reflected is so
small that the inevitable errors might completely disguise the whole
difference in the two opposite positions. When Prof. Forbes, of
Edinburgh, (in 1834,) employed mica in the like experiments, he
found a very decided polarizing effect; first, when the heat was
transmitted through several films of mica at a certain angle, and
afterwards, when it was reflected from them. In this case, he found
that with non-luminous heat, and even with the heat of water below
the boiling point, the difference of the heating power in the two
positions of opposite polarity (parallel and _crossed_) was
manifest. He also detected by careful experiments,[29\10] the
polarizing effect of tourmaline. This important discovery was soon
confirmed by M. Melloni. Doubts were suggested whether the different
effect in the opposite positions might not be due to other
circumstances; but Professor Forbes easily showed that these
suppositions were inadmissible; and the property of a difference of
_sides_, which at first seemed so strange when ascribed to the rays
of light, also belongs, it seems to be proved, to the rays of heat.
Professor Forbes also found, by interposing a plate of mica to
intercept the ray of heat in an intermediate point, an effect was
produced in certain positions of the mica analogous to what was
called _depolarization_ in the case of light; namely, a partial
destruction of the differences which polarization establishes.
[Note 27\10: _Ann. de Chimie_, vol. lv.]
[Note 28\10: _Bibliothèque Universelle_.]
[Note 29\10: _Ed. R. S. Transactions_, vol. xiv.; and _Phil. Mag._
1835, vol. v. p. 209. Ib. vol. vii. p. 349.]
Before this discovery, M. Melloni had already proved by experiment
that heat is _refracted_ by transparent substances as light is. In
the case of light, the _depolarizing_ effect was afterwards found to
be really, as we have seen, a _dipolarizing_ effect, the ray being
divided into two rays by _double refraction_. We are naturally much
tempted to put the same interpretation upon the dipolarizing effect
in the case of heat; but perhaps the assertion of the analogy
between light and heat to this extent is as yet insecure.
It is the more necessary to be cautious in our attempt to identify
the laws of light and heat, inasmuch as along with all the
resemblances of the two agents, there are very important
differences. The power of transmitting light, _the diaphaneity_ of
bodies, is very distinct from their power of transmitting heat,
which has been called _diathermancy_ by M. Melloni. Thus both a
plate of alum and a plate of rock-salt transmit nearly the whole
light; but while the first stops nearly the whole heat, the second
stops very little of it; and a plate of opake {156} quartz, nearly
impenetrable by light, allows a large portion of the heat to pass.
By passing the rays through various media, the heat may be, as it
were, _sifted_ from the light which accompanies it.
[2nd Ed.] [The diathermancy of bodies is distinct from their
diaphaneity, in so far that the same bodies do not exercise the same
powers of selection and suppression of certain rays on heat and on
light; but it appears to be proved by the investigations of modern
thermotical philosophers (MM. De la Roche, Powell, Melloni, and
Forbes), that there is a close analogy between the absorption of
certain colors by transparent bodies, and the absorption of certain
kinds of heat by diathermanous bodies. Dark sources of heat emit
rays which are analogous to blue and violet rays of light; and
highly luminous sources emit rays which are analogous to red rays.
And by measuring the angle of total reflection for heat of different
kinds, it has been shown that the former kind of calorific rays are
really less refrangible than the latter.[30\10]
[Note 30\10: See Prof. Forbes's _Third Series of Researches on
Heat_, _Edinb. R.S. Trans._ vol. xiv.]
M. Melloni has assumed this analogy as so completely established,
that he has proposed for this part of thermotics the name
_Thermochroology_ (Qu. _Chromothermotics_?); and along with this
term, many others derived from the Greek, and founded on the same
analogy. If it should appear, in the work which he proposes to
publish on this subject, that the doctrines which he has to state
cannot easily be made intelligible without the use of the terms he
suggests, his nomenclature will obtain currency; but so large a mass
of etymological innovations is in general to be avoided in
scientific works.
M. Melloni's discovery of the extraordinary power of _rock-salt_ to
transmit heat, and Professor Forbes's discovery of the extraordinary
power of _mica_ to polarize and depolarize heat, have supplied
thermotical inquirers with two new and most valuable
instruments.[31\10]]
[Note 31\10: For an account of many thermotical researches, which I
have been obliged to pass unnoticed here, see two Reports by Prof.
Powell on the present state of our knowledge respecting Radiant
Heat, in the _Reports of the British Association_ for 1832 and 1840.]
Moreover, besides the laws of conduction and radiation, many other
laws of the phenomena of heat have been discovered by philosophers;
and these must be taken into account in judging any theory of heat.
To these other laws we must now turn our attention. {157}
CHAPTER II.
THE LAWS OF CHANGES OCCASIONED BY HEAT.
_Sect._ 1.--_Expansion by Heat.--The Law of Dalton and Gay-Lussac
for Gases._
ALMOST all bodies expand by heat; solids, as metals, in a small
degree; fluids, as water, oil, alcohol, mercury, in a greater
degree. This was one of the facts first examined by those who
studied the nature of heat, because this property was used for the
measure of heat. In the _Philosophy of the Inductive Sciences_, Book
iv., Chap. iv., I have stated that secondary qualities, such as
Heat, must be measured by their effects: and in Sect. 4 of that
Chapter I have given an account of the successive attempts which
have been made to obtain measures of heat. I have there also spoken
of the results which were obtained by comparing the rate at which
the expansion of different substances went on, under the same
degrees of heat; or as it was called, the different _thermometrical
march_ of each substance. Mercury appears to be the liquid which is
most uniform in its thermometrical march; and it has been taken as
the most common material of our thermometers; but the expansion of
mercury is not proportional to the heat. De Luc was led, by his
experiments, to conclude "that the dilatations of mercury follow an
accelerated march for equal augmentations of heat." Dalton
conjectured that water and mercury both expand as the square of the
_real temperature_ from the point of greatest contraction: the real
temperature being measured so as to lead to such a result. But none
of the rules thus laid down for the expansion of solids and fluids
appear to have led, as yet, to any certain general laws.
With regard to gases, thermotical inquirers have been more
successful. Gases expand by heat; and their expansion is governed by
a law which applies alike to all degrees of heat, and to all gaseous
fluids. The law is this: that _for equal increments of temperature
they expand by the same fraction of their own bulk_; which fraction
is _three-eights_ {158} in proceeding from freezing to boiling
water. This law was discovered by Dalton and M. Gay-Lussac
independently of each other;[32\10] and is usually called by both
their names, _the law of Dalton and Gay-Lussac_. The latter
says,[33\10] "The experiments which I have described, and which have
been made with great care, prove incontestably that oxygen,
hydrogen, azotic acid, nitrous acid, ammoniacal acid, muriatic acid,
sulphurous acid, carbonic acid, gases, expand equally by equal
increments of heat." "Therefore," he adds with a proper inductive
generalization, "the result does not depend upon physical
properties, and I collect that _all gases expand equally by heat_."
He then extends this to vapors, as ether. This must be one of the
most important foundation-stones of any sound theory of heat.
[Note 32\10: _Manch. Mem._ vol. v. 1802; and _Ann. Chim._ xliii.
p. 137.]
[Note 33\10: Ib. p. 272.]
[2nd Ed.] Yet MM. Magnus and Regnault conceive that they have
overthrown this law of Dalton and Gay-Lussac, and shown that the
different gases do not expand alike for the same increment of heat.
Magnus found the ratio to be for atmospheric air, 1∙366; for
hydrogen, 1∙365; for carbonic acid, 1∙369; for sulphurous-acid gas,
1∙385. But these differences are not greater than the differences
obtained for the same substances by different observers; and as this
law is referred to in Laplace's hypothesis, hereafter to be
discussed, I do not treat the law as disproved.
Yet that the rate of expansion of gas in certain circumstances is
different for different substances, must be deemed very probable,
after Dr. Faraday's recent investigations _On the Liquefaction and
Solidification of Bodies generally existing as Gases_,[34\10] by
which it appears that the elasticity of vapors _in contact with
their fluids_ increases at different rates in different substances.
"That the force," he says, "of vapor increases in a geometrical
ratio for equal increments of heat is true for all bodies, but the
ratio is not the same for all. . . . For an increase of pressure
from two to six atmospheres, the following number of degrees require
to be added to the bodies named:--water 69°, sulphureous acid 63°,
cyanogen 64°∙5, ammonia 60°, arseniuretted hydrogen 54°,
sulphuretted hydrogen 56°∙5, muriatic acid 43°, carbonic acid 32°∙5,
nitrous oxide 30°."]
[Note 34\10: _Phil. Trans._ 1845, Pt. 1.]
We have already seen that the opinion that the air-thermometer is a
true measure of heat, is strongly countenanced by the symmetry
which, by using it, we introduce into the laws of radiation. If we
{159} accept the law of Dalton and Gay-Lussac, it follows that this
result is independent of any peculiar properties in the air
employed; and thus this measure has an additional character of
generality and simplicity which make it still more probable that it
is the true standard. This opinion is further supported by the
attempts to include such facts in a theory; but before we can treat
of such theories, we must speak of some other doctrines which have
been introduced.
_Sect._ 2.--_Specific Heat.--Change of Consistence._
IN the attempts to obtain measures of heat, it was found that bodies
had different capacities for heat; for the same quantity of heat,
however measured, would raise, in different degrees, the temperature
of different substances. The notion of different capacities for heat
was thus introduced, and each body was thus assumed to have a
specific _capacity for heat_, according to the quantity of heat
which it required to raise it through a given scale of heat.[35\10]
The term "capacity for heat" was introduced by Dr. Irvine, a pupil
of Dr. Black. For this term, Wilcke, the Swedish physicist,
substituted "specific heat;" in analogy with "specific gravity."
[Note 35\10: See Crawford, _On Heat_, for the History of Specific
Heat.]
It was found, also, that the capacity of the same substance was
different in the same substance at different temperatures. It
appears from experiments of MM. Dulong and Petit, that, in general,
the capacity of liquids and solids increases as we ascend in the
scale of temperature.
But one of the most important thermotic facts is, that by the sudden
contraction of any mass, its temperature is increased. This is
peculiarly observable in gases, as, for example, common air. The
amount of the increase of temperature by sudden condensation, or of
the cold produced by sudden rarefaction, is an important datum,
determining the velocity of sound, as we have already seen, and
affecting many points of meteorology. The coefficient which enters
the calculation in the former case depends on the ratio of two
specific heats of air under different conditions; one belonging to
it when, varying in density, the pressure is constant by which the
air is contained; the other, when, varying in density, it is
contained in a constant space.
A leading fact, also, with regard to the operation of heat on bodies
{160} is, that it changes their _form_, as it is often called, that
is, their condition as solid, liquid, or air. Since the term "form"
is employed in too many and various senses to be immediately
understood when it is intended to convey this peculiar meaning, I
shall use, instead of it, the term _consistence_, and shall hope to
be excused, even when I apply this word to gases, though I must
acknowledge such phraseology to be unusual. Thus there is a change
of consistence when solids become liquid, or liquids gaseous; and
the laws of such changes must be fundamental facts of our
thermotical theories. We are still in the dark as to many of the
laws which belong to this change; but one of them, of great
importance, has been discovered, and to that we must now proceed.
_Sect._ 3.--_The Doctrine of Latent Heat._
The Doctrine of Latent Heat refers to such changes of consistence as
we have just spoken of. It is to this effect; that during the
conversion of solids into liquids, or of liquids into vapors, there
is communicated to the body heat which is not indicated by the
thermometer. The heat is absorbed, or becomes _latent_; and, on the
other hand, on the condensation of the vapor to a liquid, or the
liquid to a solid consistency, this heat is again given out and
becomes sensible. Thus a pound of ice requires twenty times as long
a time, in a warm room, to raise its temperature seven degrees, as a
pound of ice-cold water does. A kettle placed on a fire, in four
minutes had its temperature raised to the boiling point, 212°: and
this temperature continued stationary for twenty minutes, when the
whole was boiled away. Dr. Black inferred from these facts that a
large quantity of heat is absorbed by the ice in becoming water, and
by the water in becoming steam. He reckoned from the above
experiments, that ice, in melting, absorbs as much heat as would
raise ice-cold water through 140° of temperature: and that water, in
evaporating, absorbs as much heat as would raise it through 940°.
That snow requires a great quantity of heat to melt it; that water
requires a great quantity of heat to convert it into steam; and that
this heat is not indicated by a rise in the thermometer, are facts
which it is not difficult to observe; but to separate these from all
extraneous conditions, to group the cases together, and to seize
upon the general law by which they are connected, was an effort of
inductive insight, which has been considered, and deservedly, as one
of the most striking {161} events in the modern history of physics.
Of this step the principal merit appears to belong to Black.
[2nd Ed.] [In the first edition I had mentioned the names of De Luc
and of Wilcke, in connexion with the discovery of Latent Heat, along
with the name of Black. De Luc had observed, in 1755, that ice, in
melting, did not rise above the freezing-point of temperature till
the whole was melted. De Luc has been charged with plagiarizing
Black's discovery, but, I think, without any just ground. In his
_Idées sur la Météorologique_ (1787), he spoke of Dr. Black as "the
first who had attempted the determinations of the quantities of
latent heat." And when Mr. Watt pointed out to him that from this
expression it might be supposed that Black had not discovered the
fact itself, he acquiesced, and redressed the equivocal expression
in an Appendix to the volume.[36\10]
[Note 36\10: See his _Letter_ to the Editors of the _Edinburgh
Review_, No. xii. p. 502, of the _Review_.]
Black never published his own account of the doctrine of Latent
Heat: but he delivered it every year after 1760 in his Lectures. In
1770, a surreptitious publication of his Lectures was made by a
London bookseller, and this gave a view of the leading points of Dr.
Black's doctrine. In 1772, Wilcke, of Stockholm, read a paper to the
Royal Society of that city, in which the absorption of heat by
melting ice is described; and in the same year, De Luc of Geneva
published his _Recherches sur les Modifications de l'Atmosphère_,
which has been alleged to contain the doctrine of latent heat, and
which the author asserts to have been written in ignorance of what
Black had done. At a later period, De Luc, adopting, in part.
Black's expression, gave the name of _latent fire_ to the heat
absorbed.[37\10]
[Note 37\10: See _Ed. Rev._ No. vi. p. 20.]
It appears that Cavendish determined the amount of heat produced by
condensing steam, and by thawing snow, as early as 1765. He had
perhaps already heard something of Black's investigations, but did
not accept his term "latent heat".**[38\10]]
[Note 38\10: See Mr. V. Harcourt's _Address_ to the Brit. Assoc. in
1839, and the _Appendix_.]
The consequences of Black's principle are very important, for upon
it is founded the whole doctrine of evaporation; besides which, the
principle of latent heat has other applications. But the relations
of aqueous vapor to air are so important, and have been so long a
{162} subject of speculation, that we may with advantage dwell a
little upon them. The part of science in which this is done may be
called, as we have said, Atmology; and to that division of Thermotics
the following chapters belong.
{{163}}
ATMOLOGY.
CHAPTER III.
THE RELATION OF VAPOR AND AIR.
_Sect._ 1.--_The Boylean Law of the Air's Elasticity._
IN the Sixth Book (Chap. iv. Sect. 1.) we have already seen how the
conception on the laws of fluid equilibrium was, by Pascal and others,
extended to air, as well as water. But though air presses and is
pressed as water presses and is pressed, pressure produces upon air an
effect which it does not, in any obvious degree, produce upon water.
Air which is pressed is also _compressed_, or made to occupy a smaller
space; and is consequently also made more dense, or _condensed_; and
on the other hand, when the pressure upon a portion of air is
diminished, the air expands or is rarefied. These broad facts are
evident. They are expressed in a general way by saying that air is an
_elastic_ fluid, yielding in a certain degree to pressure, and
recovering its previous dimensions when the pressure is removed.
But when men had reached this point, the questions obviously offered
themselves, in what degree and according to what law air yields to
pressure; when it is compressed, what relation does the density bear
to the pressure? The use which had been made of tubes containing
columns of mercury, by which the pressure of portions of air was
varied and measured, suggested obvious modes of devising experiments
by which this question might be answered. Such experiments
accordingly were made by Boyle about 1650; and the result at which
he arrived was, that when air is thus compressed, the density is as
the pressure. Thus if the pressure of the atmosphere in its common
state be equivalent to 30 inches of mercury, as shown by the
barometer; if air included in a tube be pressed by 30 additional
inches of {164} mercury, its density will be doubled, the air being
compressed into one half the space. If the pressure be increased
threefold, the density is also trebled; and so on. The same law was
soon afterwards (in 1676) proved experimentally by Mariotte. And
this law of the air's elasticity, that the density is as the
pressure, is sometimes called the _Boylean Law_, and sometimes the
_Law of Boyle and Mariotte_.
Air retains its aerial character permanently; but there are other
aerial substances which appear as such, and then disappear or change
into some other condition. Such are termed _vapors_. And the
discovery of their true relation to air was the result of a long
course of researches and speculations.
[2nd Ed.] [It was found by M. Cagniard de la Tour (in 1823), that at
a certain temperature, a liquid, under sufficient pressure, becomes
clear transparent vapor or gas, having the same bulk as the liquid.
This condition Dr. Faraday calls the _Cagniard de la Tour_ state,
(the _Tourian_ state?) It was also discovered by Dr. Faraday that
carbonic-acid gas, and many other gases, which were long conceived
to be permanently elastic, are really reducible to a liquid state by
pressure.[39\10] And in 1835, M. Thilorier found the means of
reducing liquid carbonic acid to a solid form, by means of the cold
produced in evaporation. More recently Dr. Faraday has added several
substances usually gaseous to the list of those which could
previously be shown in the liquid state, and has reduced others,
including ammonia, nitrous oxide, and sulphuretted hydrogen, to a
solid consistency.[40\10] After these discoveries, we may, I think,
reasonably doubt whether all bodies are not capable of existing in
the three _consistencies_ of solid, liquid, and air.
[Note 39\10: _Phil. Trans._ 1823.]
[Note 40\10: Ib. Pt. 1. 1845.]
We may note that the law of Boyle and Mariotte is not exactly true
near the limit at which the air passes to the liquid state in such
cases as that just spoken of. The diminution of bulk is then more
rapid than the increase of pressure.
The transition of fluids from a liquid to an airy consistence
appears to be accompanied by other curious phenomena. See Prof.
Forbes's papers on the _Color of Steam under certain circumstances_,
and on the _Colors of the Atmosphere_, in the _Edin. Trans._ vol.
xiv.] {165}
_Sect._ 2.--_Prelude to Dalton's Doctrine of Evaporation._
VISIBLE clouds, smoke, distillation, gave the notion of Vapor; vapor
was at first conceived to be identical with air, as by Bacon.[41\10]
It was easily collected, that by heat, water might be converted into
vapor. It was thought that air was thus produced, in the instrument
called the _æolipile_, in which a powerful blast is caused by a
boiling fluid; but Wolfe showed that the fluid was not converted
into air, by using camphorated spirit of wine, and condensing the
vapor after it had been formed. We need not enumerate the doctrines
(if very vague hypotheses may be so termed) of Descartes, Dechales,
Borelli.[42\10] The latter accounted for the rising of vapor by
supposing it a mixture of fire and water; and thus, fire being much
lighter than air, the mixture also was light. Boyle endeavored to
show that vapors do not permanently float _in vacuo_. He compared
the mixture of vapor with air to that of salt with water. He found
that the pressure of the atmosphere affected the heat of boiling
water; a very important fact. Boyle proved this by means of the
air-pump; and he and his friends were much surprised to find that
when air was removed, water only just warm boiled violently. Huyghens
mentions an experiment of the same kind made by Papin about 1673.
[Note 41\10: Bacon's _Hist. Nat._ Cent. i. p. 27.]
[Note 42\10: They may be seen in Fischer, _Geschichte der Physik_,
vol. ii. p. 175.]
The ascent of vapor was explained in various ways in succession,
according to the changes which physical science underwent. It was a
problem distinctly treated of, at a period when hydrostatics had
accounted for many phenomena; and attempts were naturally made to
reduce this fact to hydrostatical principles. An obvious hypothesis,
which brought it under the dominion of these principles, was, to
suppose that the water, when converted into vapor, was divided into
small hollow globules;--thin pellicles including air or heat. Halley
gave such an explanation of evaporation; Leibnitz calculated the
dimensions of these little bubbles; Derham managed (as he supposed)
to examine them with a magnifying glass: Wolfe also examined and
calculated on the same subject. It is curious to see so much
confidence in so lame a theory; for if water became hollow globules
in order to rise as vapor, we require, in order to explain the
formation of these globules, new laws of nature, which are not even
hinted at by {166} the supporters of the doctrine, though they must
be far more complex than the hydrostatical law by which a hollow
sphere floats.
Newton's opinion was hardly more satisfactory; he[43\10] explained
evaporation by the repulsive power of heat; the parts of vapors,
according to him, being small, are easily affected by this force,
and thus become lighter than the atmosphere.
[Note 43\10: _Opticks_, Qu. 31.]
Muschenbroek still adhered to the theory of globules, as the
explanation of evaporation; but he was manifestly discontented with
it; and reasonably apprehended that the pressure of the air would
destroy the frail texture of these bubbles. He called to his aid a
rotation of the globules (which Descartes also had assumed); and,
not satisfied with this, threw himself on electrical action as a
reserve. Electricity, indeed, was now in favor, as hydrostatics had
been before; and was naturally called in, in all cases of
difficulty. Desaguliers, also, uses this agent to account for the
ascent of vapor, introducing it into a kind of sexual system of
clouds; according to him, the male fire (heat) does a part, and the
female fire (electricity) performs the rest. These are speculations
of small merit and no value.
In the mean time, Chemistry made great progress in the estimation of
philosophers, and had its turn in the explanation of the important
facts of evaporation. Bouillet, who, in 1742, placed the particles
of water in the interstices of those of air, may be considered as
approaching to the chemical theory. In 1748, the Academy of Sciences
of Bourdeaux proposed the ascent of vapors as the subject of a
prize; which was adjudged in a manner very impartial as to the
choice of a theory; for it was divided between Kratzenstein, who
advocated the bubbles, (the coat of which he determined to be
1⁄50,000th of an inch thick,) and Hamberger, who maintained the
truth to be the adhesion of particles of water to those of air and
fire. The latter doctrine had become much more distinct in the
author's mind when seven years afterwards (1750) he published his
_Elementa Physices_. He then gave the explanation of evaporation in
a phrase which has since been adopted,--the _solution of water in
air_; which he conceived to be of the same kind as other chemical
solutions.
This theory of solution was further advocated and developed by Le
Roi;[44\10] and in his hands assumed a form which has been
extensively adopted up to our times, and has, in many instances,
tinged the language commonly used. He conceived that air, like other
solvents, {167} might be _saturated_; and that when the water was
beyond the amount required for saturation, it appeared in a visible
form. The saturating quantity was held to depend mainly on warmth
and wind.
[Note 44\10: _Ac. R. Sc._ Paris, 1750.]
This theory was by no means devoid of merit; for it brought together
many of the phenomena, and explained a number of the experiments
which Le Roi made. It explained the facts of the transparency of
vapor, (for perfect solutions are transparent,) the precipitation of
water by cooling, the disappearance of the visible moisture by
warming it again, the increased evaporation by rain and wind; and
other observed phenomena. So far, therefore, the introduction of the
notion of the chemical solution of water in air was apparently very
successful. But its defects are of a very fatal kind; for it does
not at all apply to the facts which take place when air is excluded.
In Sweden, in the mean time,[45\10] the subject had been pursued in
a different, and in a more correct manner. Wallerius Ericsen had, by
various experiments, established the important fact, that water
evaporates in a _vacuum_. His experiments are clear and
satisfactory; and he inferred from them the falsity of the common
explanation of evaporation by the solution of water in _air_. His
conclusions are drawn in a very intelligent manner. He considers the
question whether water can be changed into air, and whether the
atmosphere is, in consequence, a mere collection of vapors; and on
good reasons, decides in the negative, and concludes the existence
of permanently-elastic air different from vapor. He judges, also,
that there are two causes concerned, one acting to produce the first
ascent of vapors, the other to support them afterwards. The first,
which acts in a vacuum, he conceives to be the mutual repulsion of
the particles; and since this force is independent of the presence
of other substances, this seems to be a sound induction. When the
vapors have once ascended into the air, it may readily be granted
that they are carried higher, and driven from side to side by the
currents of the atmosphere. Wallerius conceives that the vapor will
rise till it gets into air of the same density as itself, and being
then in equilibrium, will drift to and fro.
[Note 45\10: Fischer, _Gesch. Phys._ vol. v. p. 63.]