eclipses of Jupiter's satellites a simple and rigorous solution of the
famous problem of the longitude, and active negotiations were
immediately commenced with the view of introducing the new method on
board the numerous vessels of Spain and Holland. These negotiations
failed. From the discussion it plainly appeared that the accurate
observation of the eclipses of the satellites would require powerful
telescopes; but such telescopes could not be employed on board a ship
tossed about by the waves.
The method of Galileo seemed, at any rate, to retain all its advantages
when applied on land, and to promise immense improvements to geography.
These expectations were found to be premature. The movements of the
satellites of Jupiter are not by any means so simple as the immortal
inventor of the method of longitudes supposed them to be. It was
necessary that three generations of astronomers and mathematicians
should labour with perseverance in unfolding their most considerable
perturbations. It was necessary, in fine, that the tables of those
bodies should acquire all desirable and necessary precision, that
Laplace should introduce into the midst of them the torch of
mathematical analysis.
In the present day, the nautical ephemerides contain, several years in
advance, the indication of the times of the eclipses and reappearances
of Jupiter's satellites. Calculation does not yield in precision to
direct observation. In this group of satellites, considered as an
independent system of bodies, Laplace found a series of perturbations
analogous to those which the planets experience. The rapidity of the
revolutions unfolds, in a sufficiently short space of time, changes in
this system which require centuries for their complete development in
the solar system.
Although the satellites exhibit hardly an appreciable diameter even when
viewed in the best telescopes, our illustrious countryman was enabled to
determine their masses. Finally, he discovered certain simple relations
of an extremely remarkable character between the movements of those
bodies, which have been called _the laws of Laplace_. Posterity will not
obliterate this designation; it will acknowledge the propriety of
inscribing in the heavens the name of so great an astronomer beside that
of Kepler.
Let us cite two or three of the laws of Laplace:--
If we add to the mean longitude of the first satellite twice that of the
third, and subtract from the sum three times the mean longitude of the
second, the result will be exactly equal to 180°.
Would it not be very extraordinary if the three satellites had been
placed originally at the distances from Jupiter, and in the positions,
with respect to each other, adapted for constantly and rigorously
maintaining the foregoing relation? Laplace has replied to this question
by showing that it is not necessary that this relation should have been
rigorously true at the origin. The mutual action of the satellites would
necessarily have reduced it to its present mathematical condition, if
once the distances and the positions satisfied the law approximately.
This first law is equally true when we employ the synodical elements. It
hence plainly results, that the first three satellites of Jupiter can
never be all eclipsed at the same time. Bearing this in mind, we shall
have no difficulty in apprehending the import of a celebrated
observation of recent times, during which certain astronomers perceived
the planet for a short time without any of his four satellites. This
would not by any means authorize us in supposing the satellites to be
eclipsed. A satellite disappears when it is projected upon the central
part of the luminous disk of Jupiter, and also when it passes behind the
opaque body of the planet.
The following is another very simple law to which the mean motions of
the same satellites of Jupiter are subject:
If we add to the mean motion of the first satellite twice the mean
motion of the third, the sum is exactly equal to three times the mean
motion of the second.[38]
This numerical coincidence, which is perfectly accurate, would be one of
the most mysterious phenomena in the system of the universe if Laplace
had not proved that the law need only have been approximate at the
origin, and that the mutual action of the satellites has sufficed to
render it rigorous.
The illustrious geometer, who always pursued his researches to their
most remote ramifications, arrived at the following result: The action
of Jupiter regulates the movements of rotation of the satellites so
that, without taking into account the secular perturbations, the time of
rotation of the first satellite plus twice the time of rotation of the
third, forms a sum which is constantly equal to three times the time of
rotation of the second.
Influenced by a deference, a modesty, a timidity, without any plausible
motive, our artists in the last century surrendered to the English the
exclusive privilege of constructing instruments of astronomy. Thus, let
us frankly acknowledge the fact, at the time when Herschel was
prosecuting his beautiful observations on the other side of the Channel,
there existed in France no instruments adapted for developing them; we
had not even the means of verifying them. Fortunately for the scientific
honour of our country, mathematical analysis is also a powerful
instrument. Laplace gave ample proof of this on a memorable occasion
when from the retirement of his chamber he predicted, he minutely
announced, what the excellent astronomer of Windsor would see with the
largest telescopes which were ever constructed by the hand of man.
When Galileo, in the beginning of the year 1610, directed towards Saturn
a telescope of very low power which he had just executed with his own
hands, he perceived that the planet was not an ordinary globe, without
however being able to ascertain its real form. The expression
_tri-corporate_, by which the illustrious Florentine designated the
appearance of the planet, implied even a totally erroneous idea of its
structure. Our countryman Roberval entertained much sounder views on the
subject, but from not having instituted a detailed comparison between
his hypothesis and the results of observation, he abandoned to Huyghens
the honour of being regarded as the author of the true theory of the
phenomena presented by the wonderful planet.
Every person knows, in the present day, that Saturn consists of a globe
about 900 times greater than the earth, and a ring. This ring does not
touch the ball of the planet, being everywhere removed from it at a
distance of 20,000 (English) miles. Observation indicates the breadth of
the ring to be 54,000 miles. The thickness certainly does not exceed 250
miles. With the exception of a black streak which divides the ring
throughout its whole contour into two parts of unequal breadth and of
different brightness, this strange colossal bridge without piles had
never offered to the most experienced or skilful observers either spot
or protuberance adapted for deciding whether it was immovable or endued
with a movement of rotation.
Laplace considered it to be very improbable, if the ring was immovable,
that its constituent parts should be capable of resisting by their mere
cohesion the continual attraction of the planet. A movement of rotation
occurred to his mind as constituting the principle of stability, and he
hence deduced the necessary velocity. The velocity thus found was
exactly equal to that which Herschel subsequently deduced from a course
of extremely delicate observations.
The two parts of the ring being placed at different distances from the
planet, could not fail to experience from the action of the sun,
different movements of rotation. It would hence seem that the planes of
both rings ought to be generally inclined towards each other, whereas
they appear from observation always to coincide. It was necessary then
that some physical cause should exist which would be capable of
neutralizing the action of the sun. In a memoir published in February,
1789, Laplace found that this cause must reside in the ellipticity of
Saturn produced by a rapid movement of rotation of the planet, a
movement the existence of which Herschel announced in November, 1789.
The reader cannot fail to remark how, on certain occasions, the eyes of
the mind can supply the want of the most powerful telescopes, and lead
to astronomical discoveries of the highest importance.
Let us descend from the heavens upon the earth. The discoveries of
Laplace will appear not less important, not less worthy of his genius.
The phenomena of the tides, which an ancient philosopher designated in
despair as _the tomb of human curiosity_, were connected by Laplace with
an analytical theory in which the physical conditions of the question
figure for the first time. Accordingly calculators, to the immense
advantage of the navigation of our maritime coasts, venture in the
present day to predict several years in advance the details of the time
and height of the full tides without more anxiety respecting the result
than if the question related to the phases of an eclipse.
There exists between the different phenomena of the ebb and flow of the
tides and the attractive forces which the sun and moon exercise upon the
fluid sheet which covers three fourths of the globe, an intimate and
necessary connection from which Laplace, by the aid of a series of
twenty years of observations executed at Brest, deduced the value of the
mass of our satellite. Science knows in the present day that
seventy-five moons would be necessary to form a weight equivalent to
that of the terrestrial globe, and it is indebted for this result to an
attentive and minute study of the oscillations of the ocean. We know
only one means of enhancing the admiration which every thoughtful mind
will entertain for theories capable of leading to such conclusions. An
historical statement will supply it. In the year 1631, the illustrious
Galileo, as appears from his _Dialogues_, was so far from perceiving the
mathematical relations from which Laplace deduced results so beautiful,
so unequivocal, and so useful, that he taxed with frivolousness the
vague idea which Kepler entertained of attributing to the moon's
attraction a certain share in the production of the diurnal and
periodical movements of the waters of the ocean.
Laplace did not confine himself to extending so considerably, and
improving so essentially, the mathematical theory of the tides; he
considered the phenomenon from an entirely new point of view; it was he
who first treated of the stability of the ocean. Systems of bodies,
whether solid or fluid, are subject to two kinds of equilibrium, which
we must carefully distinguish from each other. In the case of stable
equilibrium the system, when slightly disturbed, tends always to return
to its original condition. On the other hand, when the system is in
unstable equilibrium, a very insignificant derangement might occasion an
enormous dislocation in the relative positions of its constituent parts.
If the equilibrium of waves is of the latter kind, the waves engendered
by the action of winds, by earthquakes, and by sudden movements from the
bottom of the ocean, have perhaps risen in past times and may rise in
the future to the height of the highest mountains. The geologist will
have the satisfaction of deducing from these prodigious oscillations a
rational explanation of a great multitude of phenomena, but the public
will thereby be exposed to new and terrible catastrophes.
Mankind may rest assured: Laplace has proved that the equilibrium of the
ocean is stable, but upon the express condition (which, however, has
been amply verified by established facts), that the mean density of the
fluid mass is less than the mean density of the earth. Every thing else
remaining the same, let us substitute an ocean of mercury for the actual
ocean, and the stability will disappear, and the fluid will frequently
surpass its boundaries, to ravage continents even to the height of the
snowy regions which lose themselves in the clouds.
Does not the reader remark how each of the analytical investigations of
Laplace serves to disclose the harmony and duration of the universe and
of our globe!
It was impossible that the great geometer, who had succeeded so well in
the study of the tides of the ocean, should not have occupied his
attention with the tides of the atmosphere; that he should not have
submitted to the delicate and definitive tests of a rigorous calculus,
the generally diffused opinions respecting the influence of the moon
upon the height of the barometer and other meteorological phenomena.
Laplace, in effect, has devoted a chapter of his splendid work to an
examination of the oscillations which the attractive force of the moon
is capable of producing in our atmosphere. It results from these
researches, that, at Paris, the lunar tide produces no sensible effect
upon the barometer. The height of the tide, obtained by the discussion
of a long series of observations, has not exceeded two-hundredths of a
millimètre, a quantity which, in the present state of meteorological
science, is less than the probable error of observation.
The calculation to which I have just alluded, may be cited in support
of considerations to which I had recourse when I wished to establish,
that if the moon alters more or less the height of the barometer,
according to its different phases, the effect is not attributable to
attraction.
No person was more sagacious than Laplace in discovering intimate
relations between phenomena apparently very dissimilar; no person showed
himself more skilful in deducing important conclusions from those
unexpected affinities.
Towards the close of his days, for example, he overthrew with a stroke
of the pen, by the aid of certain observations of the moon, the
cosmogonic theories of Buffon and Bailly, which were so long in favour.
According to these theories, the earth was inevitably advancing to a
state of congelation which was close at hand. Laplace, who never
contented himself with a vague statement, sought to determine in numbers
the rapid cooling of our globe which Buffon had so eloquently but so
gratuitously announced. Nothing could be more simple, better connected,
or more demonstrative, than the chain of deductions of the celebrated
geometer.
A body diminishes in volume when it cools. According to the most
elementary principles of mechanics, a rotating body which contracts in
dimensions ought inevitably to turn upon its axis with greater and
greater rapidity. The length of the day has been determined in all ages
by the time of the earth's rotation; if the earth is cooling, the length
of the day must be continually shortening. Now there exists a means of
ascertaining whether the length of the day has undergone any variation;
this consists in examining, for each century, the arc of the celestial
sphere described by the moon during the interval of time which the
astronomers of the existing epoch called a day,--in other words, the
time required by the earth to effect a complete rotation on its axis,
the velocity of the moon being in fact independent of the time of the
earth's rotation.
Let us now, after the example of Laplace, take from the standard tables
the least considerable values, if you choose, of the expansions or
contractions which solid bodies experience from changes of temperature;
search then the annals of Grecian, Arabian, and modern astronomy for the
purpose of finding in them the angular velocity of the moon, and the
great geometer will prove, by incontrovertible evidence founded upon
these data, that during a period of two thousand years the mean
temperature of the earth has not varied to the extent of the hundredth
part of a degree of the centigrade thermometer. No eloquent declamation
is capable of resisting such a process of reasoning, or withstanding the
force of such numbers. The mathematics have been in all ages the
implacable adversaries of scientific romances.
The fall of bodies, if it was not a phenomenon of perpetual occurrence,
would justly excite in the highest degree the astonishment of mankind.
What, in effect, is more extraordinary than to see an inert mass, that
is to say, a mass deprived of will, a mass which ought not to have any
propensity to advance in one direction more than in another, precipitate
itself towards the earth as soon as it ceased to be supported!
Nature engenders the gravity of bodies by a process so recondite, so
completely beyond the reach of our senses and the ordinary resources of
human intelligence, that the philosophers of antiquity, who supposed
that they could explain every thing mechanically according to the
simple evolutions of atoms, excepted gravity from their speculations.
Descartes attempted what Leucippus, Democritus, Epicurus, and their
followers thought to be impossible.
He made the fall of terrestrial bodies depend upon the action of a
vortex of very subtle matter circulating around the earth. The real
improvements which the illustrious Huyghens applied to the ingenious
conception of our countryman were far, however, from imparting to it
clearness and precision, those characteristic attributes of truth.
Those persons form a very imperfect estimate of the meaning of one of
the greatest questions which has occupied the attention of modern
inquirers, who regard Newton as having issued victorious from a struggle
in which his two immortal predecessors had failed. Newton did not
discover the cause of gravity any more than Galileo did. Two bodies
placed in juxtaposition approach each other. Newton does not inquire
into the nature of the force which produces this effect. The force
exists, he designates it by the term attraction; but, at the same time,
he warns the reader that the term as thus used by him does not imply any
definite idea of the physical process by which gravity is brought into
existence and operates.
The force of attraction being once admitted as a fact, Newton studies it
in all terrestrial phenomena, in the revolutions of the moon, the
planets, satellites, and comets; and, as we have already stated, he
deduced from this incomparable study the simple, universal, mathematical
characteristics of the forces which preside over the movements of all
the bodies of which our solar system is composed.
The applause of the scientific world did not prevent the immortal
author of the _Principia_ from hearing some persons refer the principle
of gravitation to the class of occult qualities. This circumstance
induced Newton and his most devoted followers to abandon the reserve
which they had hitherto considered it their duty to maintain. Those
persons were then charged with ignorance who regarded attraction as an
essential property of matter, as the mysterious indication of a sort of
charm; who supposed that two bodies may act upon each other without the
intervention of a third body. This force was then either the result of
the tendency of an ethereal fluid to move from the free regions of
space, where its density is a maximum, towards the planetary bodies
around which there exists a greater degree of rarefaction, or the
consequence of the impulsive force of some fluid medium.
Newton never expressed a definitive opinion respecting the origin of the
impulse which occasioned the attractive force of matter, at least in our
solar system. But we have strong reasons for supposing, in the present
day, that in using the word _impulse_, the great geometer was thinking
of the systematic ideas of Varignon and Fatio de Duillier, subsequently
reinvented and perfected by Lesage: these ideas, in effect, had been
communicated to him before they were published to the world.
According to Lesage, there are, in the regions of space, bodies moving
in every possible direction, and with excessive rapidity. The author
applied to these the name of ultra-mundane corpuscles. Their totality
constituted the gravitative fluid, if indeed, the designation of a fluid
be applicable to an assemblage of particles having no mutual connexion.
A single body placed in the midst of such an ocean of movable
particles, would remain at rest although it were impelled equally in
every direction. On the other hand, two bodies ought to advance towards
each other, since they would serve the purpose of mutual screens, since
the surfaces facing each other would no longer be hit in the direction
of their line of junction by the ultra-mundane particles, since there
would then exist currents, the effect of which would no longer be
neutralized by opposite currents. It will be easily seen, besides, that
two bodies plunged into the gravitative fluid, would tend to approach
each other with an intensity which would vary in the inverse proportion
of the square of the distance.
If attraction is the result of the impulse of a fluid, its action ought
to employ a finite time in traversing the immense spaces which separate
the celestial bodies. If the sun, then, were suddenly extinguished, the
earth after the catastrophe would, mathematically speaking, still
continue for some time to experience its attractive influence. The
contrary would happen on the occasion of the sudden birth of a planet; a
certain time would elapse before the attractive force of the new body
would make itself felt on the earth.
Several geometers of the last century were of opinion that the force of
attraction is not transmitted instantaneously from one body to another;
they even assigned to it a comparatively inconsiderable velocity of
propagation. Daniel Bernoulli, for example, in attempting to explain how
the spring tide arrives upon our coasts a day and a half after the
sizygees, that is to say, a day and a half after the epochs when the sun
and moon are most favourably situated for the production of this
magnificent phenomenon, assumed that the disturbing force required all
this time (a day and a half) for its propagation from the moon to the
ocean. So feeble a velocity was inconsistent with the mechanical
explanation of attraction of which we have just spoken. The explanation,
in effect, necessarily supposes that the proper motions of the celestial
bodies are insensible compared with the motion of the gravitative fluid.
After having discovered that the diminution of the eccentricity of the
terrestrial orbit is the real cause of the observed acceleration of the
motion of the moon, Laplace, on his part, endeavoured to ascertain
whether this mysterious acceleration did not depend on the gradual
propagation of attraction.
The result of calculation was at first favourable to the plausibility of
the hypothesis. It showed that the gradual propagation of the attractive
force would introduce into the movement of our satellite a perturbation
proportional to the square of the time which elapsed from the
commencement of any epoch; that in order to represent numerically the
results of astronomical observations it would not be necessary to assign
a feeble velocity to attraction; that a propagation eight millions of
times more rapid than that of light would satisfy all the phenomena.
Although the true cause of the acceleration of the moon is now well
known, the ingenious calculation of which I have just spoken does not
the less on that account maintain its place in science. In a
mathematical point of view, the perturbation depending on the gradual
propagation of the attractive force which this calculation indicates has
a certain existence. The connexion between the velocity of perturbation
and the resulting inequality is such that one of the two quantities
leads to a knowledge of the numerical value of the other. Now, upon
assigning to the inequality the greatest value which is consistent with
the observations after they have been corrected for the effect due to
the variation of the eccentricity of the terrestrial orbit, we find the
velocity of the attractive force to be fifty millions of times the
velocity of light!