of Clairaut, D'Alembert, and Lagrange. We shall be somewhat less concise
in noticing the labours of Laplace.
After having enumerated the various forces which must result from the
mutual action of the planets and satellites of our system, even the
great Newton did not venture to investigate the general nature of the
effects produced by them. In the midst of the labyrinth formed by
increases and diminutions of velocity, variations in the forms of the
orbits, changes of distances and inclinations, which these forces must
evidently produce, the most learned geometer would fail to discover a
trustworthy guide. This extreme complication gave birth to a
discouraging reflection. Forces so numerous, so variable in position, so
different in intensity, seemed to be incapable of maintaining a
condition of equilibrium except by a sort of miracle. Newton even went
so far as to suppose that the planetary system did not contain within
itself the elements of indefinite stability; he was of opinion that a
powerful hand must intervene from time to time, to repair the
derangements occasioned by the mutual action of the various bodies.
Euler, although farther advanced than Newton in a knowledge of the
planetary pertubations, refused also to admit that the solar system was
constituted so as to endure for ever.
Never did a greater philosophical question offer itself to the inquiries
of mankind. Laplace attacked it with boldness, perseverance, and
success. The profound and long-continued researches of the illustrious
geometer established with complete evidence that the planetary ellipses
are perpetually variable; that the extremities of their major axes make
the tour of the heavens; that, independently of an oscillatory motion,
the planes of their orbits experienced a displacement in virtue of which
their intersections with the plane of the terrestrial orbit are each
year directed towards different stars. In the midst of this apparent
chaos there is one element which remains constant or is merely subject
to small periodic changes; namely, the major axis of each orbit, and
consequently the time of revolution of each planet. This is the element
which ought to have chiefly varied, according to the learned
speculations of Newton and Euler.
The principle of universal gravitation suffices for preserving the
stability of the solar system. It maintains the forms and inclinations
of the orbits in a mean condition which is subject to slight
oscillations; variety does not entail disorder; the universe offers the
example of harmonious relations, of a state of perfection which Newton
himself doubted. This depends on circumstances which calculation
disclosed to Laplace, and which, upon a superficial view of the subject,
would not seem to be capable of exercising so great an influence.
Instead of planets revolving all in the same direction in slightly
eccentric orbits, and in planes inclined at small angles towards each
other, substitute different conditions and the stability of the universe
will again be put in jeopardy, and according to all probability there
will result a frightful chaos.[31]
Although the invariability of the mean distances of the planetary
orbits has been more completely demonstrated since the appearance of the
memoir above referred to, that is to say by pushing the analytical
approximations to a greater extent, it will, notwithstanding, always
constitute one of the admirable discoveries of the author of the
_Mécanique Céleste_. Dates, in the case of such subjects, are no luxury
of erudition. The memoir in which Laplace communicated his results on
the invariability of the mean motions or mean distances, is dated
1773.[32] It was in 1784 only, that he established the stability of the
other elements of the system from the smallness of the planetary masses,
the inconsiderable eccentricity of the orbits, and the revolution of the
planets in one common direction around the sun.
The discovery of which I have just given an account to the reader
excluded at least from the solar system the idea of the Newtonian
attraction being a cause of disorder. But might not other forces, by
combining with attraction, produce gradually increasing perturbations as
Newton and Euler dreaded? Facts of a positive nature seemed to justify
these fears.
A comparison of ancient with modern observations revealed the existence
of a continual acceleration of the mean motions of the moon and the
planet Jupiter, and an equally striking diminution of the mean motion
of Saturn. These variations led to conclusions of the most singular
nature.
In accordance with the presumed cause of these perturbations, to say
that the velocity of a body increased from century to century was
equivalent to asserting that the body continually approached the centre
of motion; on the other hand, when the velocity diminished, the body
must be receding from the centre.
Thus, by a strange arrangement of nature, our planetary system seemed
destined to lose Saturn, its most mysterious ornament,--to see the
planet accompanied by its ring and seven satellites, plunge gradually
into unknown regions, whither the eye armed with the most powerful
telescopes has never penetrated. Jupiter, on the other hand, the planet
compared with which the earth is so insignificant, appeared to be moving
in the opposite direction, so as to be ultimately absorbed in the
incandescent matter of the sun. Finally, the moon seemed as if it would
one day precipitate itself upon the earth.
There was nothing doubtful or speculative in these sinister forebodings.
The precise dates of the approaching catastrophes were alone uncertain.
It was known, however, that they were very distant. Accordingly, neither
the learned dissertations of men of science nor the animated
descriptions of certain poets produced any impression upon the public
mind.
It was not so with our scientific societies, the members of which
regarded with regret the approaching destruction of our planetary
system. The Academy of Sciences called the attention of geometers of all
countries to these menacing perturbations. Euler and Lagrange descended
into the arena. Never did their mathematical genius shine with a
brighter lustre. Still, the question remained undecided. The inutility
of such efforts seemed to suggest only a feeling of resignation on the
subject, when from two disdained corners of the theories of analysis,
the author of the _Mécanique Céleste_ caused the laws of these great
phenomena clearly to emerge. The variations of velocity of Jupiter,
Saturn, and the Moon flowed then from evident physical causes, and
entered into the category of ordinary periodic perturbations depending
upon the principle of attraction. The variations in the dimensions of
the orbits which were so much dreaded resolved themselves into simple
oscillations included within narrow limits. Finally, by the powerful
instrumentality of mathematical analysis, the physical universe was
again established on a firm foundation.
I cannot quit this subject without at least alluding to the
circumstances in the solar system upon which depend the so long
unexplained variations of velocity of the Moon, Jupiter, and Saturn.
The motion of the earth around the sun is mainly effected in an ellipse,
the form of which is liable to vary from the effects of planetary
perturbation. These alterations of form are periodic; sometimes the
curve, without ceasing to be elliptic, approaches the form of a circle,
while at other times it deviates more and more from that form. From the
epoch of the earliest recorded observations, the eccentricity of the
terrestrial orbit has been diminishing from year to year; at some future
epoch the orbit, on the contrary, will begin to deviate from the form of
a circle, and the eccentricity will increase to the same extent as it
previously diminished, and according to the same laws.
Now, Laplace has shown that the mean motion of the moon around the
earth is connected with the form of the ellipse which the earth
describes around the sun; that a diminution of the eccentricity of the
ellipse inevitably induces an increase in the velocity of our satellite,
and _vice versâ_; finally, that this cause suffices to explain the
numerical value of the acceleration which the mean motion of the moon
has experienced from the earliest ages down to the present time.[33]
The origin of the inequalities in the mean motions of Jupiter and Saturn
will be, I hope, as easy to conceive.
Mathematical analysis has not served to represent in finite terms the
values of the derangements which each planet experiences in its movement
from the action of all the other planets. In the present state of
science, this value is exhibited in the form of an indefinite series of
terms diminishing rapidly in magnitude. In calculation, it is usual to
neglect such of those terms as correspond in the order of magnitude to
quantities beneath the errors of observation. But there are cases in
which the order of the term in the series does not decide whether it be
small or great. Certain numerical relations between the primitive
elements of the disturbing and disturbed planets may impart sensible
values to terms which usually admit of being neglected. This case occurs
in the perturbations of Saturn produced by Jupiter, and in those of
Jupiter produced by Saturn. There exists between the mean motions of
these two great planets a simple relation of commensurability, five
times the mean motion of Saturn, being, in fact, very nearly equal to
twice the mean motion of Jupiter. It happens, in consequence, that
certain terms, which would otherwise be very small, acquire from this
circumstance considerable values. Hence arise in the movements of these
two planets, inequalities of long duration which require more than 900
years for their complete development, and which represent with
marvellous accuracy all the irregularities disclosed by observation.
Is it not astonishing to find in the commensurability of the mean
motions of two planets, a cause of perturbation of so influential a
nature; to discover that the definitive solution of an immense
difficulty--which baffled the genius of Euler, and which even led
persons to doubt whether the theory of gravitation was capable of
accounting for all the phenomena of the heavens--should depend upon the
fortuitous circumstance of five times the mean motion of Saturn being
equal to twice the mean motion of Jupiter? The beauty of the conception
and the ultimate result are here equally worthy of admiration.[34]
We have just explained how Laplace demonstrated that the solar system
can experience only small periodic oscillations around a certain mean
state. Let us now see in what way he succeeded in determining the
absolute dimensions of the orbits.
What is the distance of the sun from the earth? No scientific question
has occupied in a greater degree the attention of mankind;
mathematically speaking, nothing is more simple. It suffices, as in
common operations of surveying, to draw visual lines from the two
extremities of a known base to an inaccessible object. The remainder is
a process of elementary calculation. Unfortunately, in the case of the
sun, the distance is great and the bases which can be measured upon the
earth are comparatively very small. In such a case the slightest errors
in the direction of the visual lines exercise an enormous influence upon
the results.
In the beginning of the last century Halley remarked that certain
interpositions of Venus between the earth and the sun, or, to use an
expression applied to such conjunctions, that the _transits_ of the
planet across the sun's disk, would furnish at each observatory an
indirect means of fixing the position of the visual ray very superior in
accuracy to the most perfect direct methods.[35]
Such was the object of the scientific expeditions undertaken in 1761 and
1769, on which occasions France, not to speak of stations in Europe, was
represented at the Isle of Rodrigo by Pingré, at the Isle of St. Domingo
by Fleurin, at California by the Abbé Chappe, at Pondicherry by
Legentil. At the same epochs England sent Maskelyne to St. Helena, Wales
to Hudson's Bay, Mason to the Cape of Good Hope, Captain Cooke to
Otaheite, &c. The observations of the southern hemisphere compared with
those of Europe, and especially with the observations made by an
Austrian astronomer Father Hell at Wardhus in Lapland, gave for the
distance of the sun the result which has since figured in all treatises
on astronomy and navigation.
No government hesitated in furnishing Academies with the means, however
expensive they might be, of conveniently establishing their observers in
the most distant regions. We have already remarked that the
determination of the contemplated distance appeared to demand
imperiously an extensive base, for small bases would have been totally
inadequate to the purpose. Well, Laplace has solved the problem
numerically without a base of any kind whatever; he has deduced the
distance of the sun from observations of the moon made in one and the
same place!
The sun is, with respect to our satellite, the cause of perturbations
which evidently depend on the distance of the immense luminous globe
from the earth. Who does not see that these perturbations would diminish
if the distance increased; that they would increase on the contrary, if
the distance diminished; that the distance finally determines the
magnitude of the perturbations?
Observation assigns the numerical value of these perturbations; theory,
on the other hand, unfolds the general mathematical relation which
connects them with the solar parallax, and with other known elements.
The determination of the mean radius of the terrestrial orbit then
becomes one of the most simple operations of algebra. Such is the happy
combination by the aid of which Laplace has solved the great, the
celebrated problem of parallax. It is thus that the illustrious geometer
found for the mean distance of the sun from the earth, expressed in
radii of the terrestrial orbit, a value differing only in a slight
degree from that which was the fruit of so many troublesome and
expensive voyages. According to the opinion of very competent judges the
result of the indirect method might not impossibly merit the
preference.[36]
The movements of the moon proved a fertile mine of research to our
great geometer. His penetrating intellect discovered in them unknown
treasures. He disentangled them from every thing which concealed them
from vulgar eyes with an ability and a perseverance equally worthy of
admiration. The reader will excuse me for citing another of such
examples.
The earth governs the movements of the moon. The earth is flattened, in
other words its figure is spheroidal. A spheroidal body does not attract
like a sphere. There ought then to exist in the movement, I had almost
said in the countenance of the moon, a sort of impression of the
spheroidal figure of the earth. Such was the idea as it originally
occurred to Laplace.
It still remained to ascertain (and here consisted the chief
difficulty), whether the effects attributable to the spheroidal figure
of the earth were sufficiently sensible not to be confounded with the
errors of observation. It was accordingly necessary to find the general
formula of perturbations of this nature, in order to be able, as in the
case of the solar parallax, to eliminate the unknown quantity.
The ardour of Laplace, combined with his power of analytical research,
surmounted all obstacles. By means of an investigation which demanded
the most minute attention, the great geometer discovered in the theory
of the moon's movements, two well-defined perturbations depending on the
spheroidal figure of the earth. The first affected the resolved element
of the motion of our satellite which is chiefly measured with the
instrument known in observatories by the name of the transit instrument;
the second, which operated in the direction north and south, could only
be effected by observations with a second instrument termed the mural
circle. These two inequalities of very different magnitudes connected
with the cause which produces them by analytical combinations of totally
different kinds have, however, both conducted to the same value of the
ellipticity. It must be borne in mind, however, that the ellipticity
thus deduced from the movements of the moon, is not the ellipticity
corresponding to such or such a country, the ellipticity observed in
France, in England, in Italy, in Lapland, in North America, in India, or
in the region of the Cape of Good Hope, for the earth's materials having
undergone considerable upheavings at different times and in different
places, the primitive regularity of its curvature has been sensibly
disturbed by this cause. The moon, and it is this circumstance which
renders the result of such inestimable value, ought to assign, and has
in reality assigned the general ellipticity of the earth; in other
words, it has indicated a sort of mean value of the various
determinations obtained at enormous expense, and with infinite labour,
as the result of long voyages undertaken by astronomers of all the
countries of Europe.
I shall add a few brief remarks, for which I am mainly indebted to the
author of the _Mécanique Céleste_. They seem to be eminently adapted for
illustrating the profound, the unexpected, and almost paradoxical
character of the methods which I have just attempted to sketch.
What are the elements which it has been found necessary to confront with
each other in order to arrive at results expressed even to the precision
of the smallest decimals?
On the one hand, mathematical formulæ, deduced from the principle of
universal attraction; on the other hand, certain irregularities observed
in the returns of the moon to the meridian.
An observing geometer who, from his infancy, had never quitted his
chamber of study, and who had never viewed the heavens except through a
narrow aperture directed north and south, in the vertical plane in which
the principal astronomical instruments are made to move,--to whom
nothing had ever been revealed respecting the bodies revolving above his
head, except that they attract each other according to the Newtonian law
of gravitation,--would, however, be enabled to ascertain that his narrow
abode was situated upon the surface of a spheroidal body, the equatorial
axis of which surpassed the polar axis by a _three hundred and sixth
part_; he would have also found, in his isolated immovable position, his
true distance from the sun.
I have stated at the commencement of this Notice, that it is to
D'Alembert we owe the first satisfactory mathematical explanation of the
phenomenon of the precession of the equinoxes. But our illustrious
countryman, as well as Euler, whose solution appeared subsequently to
that of D'Alembert, omitted all consideration of certain physical
circumstances, which, however, did not seem to be of a nature to be
neglected without examination. Laplace has supplied this deficiency. He
has shown that the sea, notwithstanding its fluidity, and that the
atmosphere, notwithstanding its currents, exercise the same influence on
the movements of the terrestrial axis as if they formed solid masses
adhering to the terrestrial spheroid.
Do the extremities of the axis around which the earth performs an entire
revolution once in every twenty-four hours, correspond always to the
same material points of the terrestrial spheroid? In other words, do the
poles of rotation, which from year to year correspond to different
stars, undergo also a displacement at the surface of the earth?
In the case of the affirmative, the equator is movable as well as the
poles; the terrestrial latitudes are variable; no country during the
lapse of ages will enjoy, even on an average, a constant climate;
regions the most different will, in their turn, become circumpolar.
Adopt the contrary supposition, and every thing assumes the character of
an admirable permanence.
The question which I have just suggested, one of the most important in
Astronomy, cannot be solved by the aid of mere observation on account of
the uncertainty of the early determinations of terrestrial latitude.
Laplace has supplied this defect by analysis. The great geometer has
demonstrated that no circumstance depending on universal gravitation can
sensibly displace the poles of the earth's axis relatively to the
surface of the terrestrial spheroid. The sea, far from being an obstacle
to the invariable rotation of the earth upon its axis, would, on the
contrary, reduce the axis to a permanent condition in consequence of the
mobility of the waters and the resistance which their oscillations
experience.
The remarks which I have just made with respect to the position of the
terrestrial axis are equally applicable to the time of the earth's
rotation which is the unit, the true standard of time. The importance of
this element induced Laplace to examine whether its numerical value
might not be liable to vary from internal causes such as earthquakes and
volcanoes. It is hardly necessary for me to state that the result
obtained was negative.
The admirable memoir of Lagrange upon the libration of the moon seemed
to have exhausted the subject. This, however, was not the case.
The motion of revolution of our satellite around the earth is subject to
perturbations, technically termed _secular_, which were either unknown
to Lagrange or which he neglected. These inequalities eventually place
the body, not to speak of entire circumferences, at angular distances of
a semi-circle, a circle and a half, &c., from the position which it
would otherwise occupy. If the movement of rotation did not participate
in such perturbations, the moon in the lapse of ages would present in
succession all the parts of its surface to the earth.
This event will not occur. The hemisphere of the moon which is actually
invisible, will remain invisible for ever. Laplace, in fact, has shown
that the attraction of the earth introduces into the rotatory motion of
the lunar spheroid the secular inequalities which exist in the movement
of revolution.
Researches of this nature exhibit in full relief the power of
mathematical analysis. It would have been very difficult to have
discovered by synthesis truths so profoundly enveloped in the complex
action of a multitude of forces.
We should be inexcusable if we omitted to notice the high importance of
the labours of Laplace on the improvement of the lunar tables. The
immediate object of this improvement was, in effect, the promotion of
maritime intercourse between distant countries, and, what was indeed far
superior to all considerations of mercantile interest, the preservation
of the lives of mariners.
Thanks to a sagacity without parallel, to a perseverance which knew no
limits, to an ardour always youthful and which communicated itself to
able coadjutors, Laplace solved the celebrated problem of the longitude
more completely than could have been hoped for in a scientific point of
view, with greater precision than the art of navigation in its utmost
refinement demanded. The ship, the sport of the winds and tempests, has
no occasion, in the present day, to be afraid of losing itself in the
immensity of the ocean. An intelligent glance at the starry vault
indicates to the pilot, in every place and at every time, his distance
from the meridian of Paris. The extreme perfection of the existing
tables of the moon entitles Laplace to be ranked among the benefactors
of humanity.[37]