**Pierre-Simon,
Marquis de Laplace was a French mathematician and astronomer who
put the final capstone on mathematical astronomy by summarizing
and extending the work of his predecessors in his five volume Mécanique
Céleste (Celestial Mechanics) (1799-1825). This masterpiece
translated the geometrical study of mechanics used by Newton to
one based on calculus, known as physical mechanics. **
**He
is also the discoverer of Laplace's equation. Although the Laplace
transform is named in honor of Pierre-Simon Laplace, who used
the transform in his work on probability theory, the transform
was discovered originally by Leonhard Euler, the prolific eighteenth-century
Swiss mathematician. The Laplace transform appears in all branches
of mathematical physics - a field he took a leading role in forming.
He became count of the Empire in 1806 and was named a marquis
in 1817 after the restoration of the Bourbons. Pierre-Simon Laplace
was among the most influential scientists in history.**
**The
Laplacian differential operator, much relied-upon in applied mathematics,
is named after him. While still a teenager, having studied mathematics
only briefly, he quickly impressed d'Alembert with his mathematical
ability, who made effort to procure him a professorship - the
undertaking was found with ease owing to his newfound pupil's
genius. He is remembered as one of the greatest scientists of
all time (sometimes referred to as a French Newton) with a natural
phenomenal mathematical faculty possessed by none of his contemporaries.
**
**It
does appear that Laplace was not modest about his abilities and
achievements, and he probably failed to recognise the effect of
his attitude on his colleagues. Lexell visited the Académie
des Sciences in Paris in 1780-81 and reported that Laplace let
it be known widely that he considered himself the best mathematician
in France. The effect on his colleagues would have been only mildly
eased by the fact that Laplace was right!**
**Laplace
had a wide knowledge of all sciences and dominated all discussions
in the Académie. Quite uniquely for a mathematical prodigy
of his skill, Laplace viewed mathematics as nothing in itself
but a tool to be called upon in the investigation of a scientific
or practical inquiry. Further, Laplace would often omit details
of proof in many of his works, stating that one can easily show
their validity, although, in fact, the proof would require the
keenest analytical mind to comprehend - one such as his. This
habit of his would often necessitate him to rework many of his
results later for reference, sometimes taking days to complete.
**
**Laplace
spent much of his life working on mathematical astronomy that
culminated in his masterpiece on the proof of the dynamic stability
of the solar system with the assumption that it consists of a
collection of rigid bodies moving in a vacuum. He independently
formulated the nebular hypothesis and was one of the first scientists
to postulate the existence of black holes and the notion of gravitational
collapse. While he conducted much research in physics, another
major theme of his life's endeavors was probability theory. In
his Essai philosophique sur les probabilités, Laplace set
out a mathematical system of inductive reasoning based on probability,
which we would today recognise as Bayesian. **
**One
well-known formula arising from his system is the rule of succession.
Suppose that some trial has only two possible outcomes, labeled
"success" and "failure". Under the assumption
that little or nothing is known a priori about the relative plausibilities
of the outcomes, Laplace derived a formula for the probability
that the next trial will be a success.It is still used as an estimator
for the probability of an event if we know the event space, but
only have a small number of samples.**
**The
rule of succession has been subject to much criticism, partly
due to the example which Laplace chose to illustrate it. He calculated
that the probability that the sun will rise tomorrow, given that
it has never failed to in the past. This result has been derided
as absurd, and some authors have concluded that all applications
of the Rule of Succession are absurd by extension. However, Laplace
was fully aware of the absurdity of the result; immediately following
the example, he wrote, "But this number [i.e., the probability
that the sun will rise tomorrow] is far greater for him who, seeing
in the totality of phenomena the principle regulating the days
and seasons, realizes that nothing at the present moment can arrest
the course of it."**
**Laplace
strongly believed in causal determinism, which is expressed in
the following quote from the introduction to the Essai:**
*"We
may regard the present state of the universe as the effect of
its past and the cause of its future. An intellect which at a
certain moment would know all forces that set nature in motion,
and all positions of all items of which nature is composed, if
this intellect were also vast enough to submit these data to analysis,
it would embrace in a single formula the movements of the greatest
bodies of the universe and those of the tiniest atom; for such
an intellect nothing would be uncertain and the future just like
the past would be present before its eyes."*
**This
intellect is often referred to as Laplace's demon (in the same
vein as Maxwell's demon). Note that the description of the hypothetical
intellect described above by Laplace as a demon does not come
from Laplace, but from later biographers: Laplace saw himself
as a scientist that hoped that humanity would progress in a better
scientific understanding of the world, which, if and when eventually
completed, would still need a tremendous calculating power to
compute it all in a single instant. While Laplace saw foremost
practical problems for mankind to reach this ultimate stage of
knowledge and computation, later interpretations of quantum mechanics,
which were adopted by philosophers defending the existence of
free will, also leave the theoretical possibility of such an "intellect"
contested: for a further discussion of this issue, see also: determinism.**
**There
has recently been proposed a limit on the computational power
of the universe, ie the ability of Laplace's Demon to process
an infinite amount of information. The limit is based on the maximum
entropy of the universe, the speed of light, and the minimum amount
of time taken to move information across the Planck length, and
the figure turns out to be 2130 bits. Accordingly, anything that
requires more than this amount of data cannot be computed in the
amount of time that has lapsed so far in the universe. (An actual
theory of everything might find an exception to this limit, of
course.)**
**Biography**
Simon Laplace was born at Beaumont-en-Auge in Normandy on March
23, 1749, and died at Paris on March 5, 1827. He was the son of
a small cottager or perhaps a farm-labourer, and owed his education
to the interest excited in some wealthy neighbours by his abilities
and engaging presence. Very little is known of his early years,
for when he became distinguished he had the pettiness to hold
himself aloof both from his relatives and from those who had assisted
him.
**It
would seem from a pupil he became an usher in the school at Beaumont;
but, having procured a letter of introduction to D'Alembert, he
went to Paris to push his fortune. A paper on the principles of
mechanics excited D'Alembert's interest, and on his recommendation
a place in the military school was offered to Laplace.**
**Secure
of a competency, Laplace now threw himself into original research,
and in the next seventeen years, 1771-1787, he produced much of
his original work in astronomy. This commenced with a memoir,
read before the French Academy in 1773, in which he shewed that
the planetary motions were stable, and carried the proof as far
as the cubes of the eccentricities and inclinations. This was
followed by several papers on points in the integral calculus,
finite differences, differential equations, and astronomy. **
During the years 1784-1787 he produced some memoirs of exceptional
power. Prominent among these is one read in 1784, and reprinted
in the third volume of the Méchanique céleste, in
which he completely determined the attraction of a spheroid on
a particle outside it. This is memorable for the introduction
into analysis of spherical harmonics or Laplace's coefficients,
as also for the development of the use of the potential - a name
first given by Green in 1828.
**If
the co-ordinates of two points be (r,µ,?) and (r',µ',?'),
and if r' = r, then the reciprocal of the distance between them
can be expanded in powers of r/r', and the respective coefficients
are Laplace's coefficients. Their utility arises from the fact
that every function of the co-ordinates of a point on the sphere
can be expanded in a series of them. It should be stated that
the similar coefficients for space of two dimensions, together
with some of their properties, had been previously given by Legendre
in a paper sent to the French Academy in 1783. Legendre had good
reason to complain of the way in which he was treated in this
matter.**
**This
paper is also remarkable for the development of the idea of the
potential, which was appropriated from Lagrange, who had used
it in his memoirs of 1773, 1777 and 1780. Laplace shewed that
the potential always satisfies a certain differential equation
and on this result his subsequent work on attractions was based.
The quantity has been termed the concentration of and its value
at any point indicates the excess of the value of there over its
mean value in the neighbourhood of the point. **
**Laplace's
equation, or the more general form , appears in all branches of
mathematical physics. According to some writers this follows at
once from the fact that is a scalar operator; or the equation
may represent analytically some general law of nature which has
not been yet reduced to words; or possibly it might be regarded
by a Kantian as the outward sign of one of the necessary forms
through which all phenomena are perceived.**
**This
memoir was followed by another on planetary inequalities, which
was presented in three sections in 1784, 1785, and 1786. This
deals mainly with the explanation of the "great inequality"
of Jupiter and Saturn. Laplace shewed by general considerations
that the mutual action of two planets could never largely affect
the eccentricities and inclinations of their orbits; and that
the peculiarities of the Jovian system were due to the near approach
to commensurability of the mean motions of Jupiter and Saturn:
further developments of these theorems on planetary motion were
given in his two memoirs of 1788 and 1789. It was on these data
that Delambre computed his astronomical tables.**
**The
year 1787 was rendered memorable by Laplace's explanation and
analysis of the relation between the lunar acceleration and the
secular changes in the eccentricity of the earth's orbit: this
investigation completed the proof of the stability of the whole
solar system on the assumption that it consists of a collection
of rigid bodies moving in a vacuum. All the memoirs above alluded
to were presented to the French Academy, and they are printed
in the Mémoires présentés par divers savans.**
**Laplace
now set himself the task to write a work which should "offer
a complete solution of the great mechanical problem presented
by the solar system, and bring theory to coincide so closely with
observation that empirical equations should no longer find a place
in astronomical tables." The result is embodied in the Exposition
du système du monde and the Méchanique céleste.**
**The
former was published in 1796, and gives a general explanation
of the phenomena, but omits all details. It contains a summary
of the history of astronomy: this summary procured for its author
the honour of admission to the forty of the French Academy; it
is commonly esteemed one of the masterpieces of French literature,
though it is not altogether reliable for the later periods of
which it treats.**
**The
nebular hypothesis was here enunciated. According to this hypothesis
the solar system has been evolved from a globular mass of incandescent
gas rotating around an axis through its centre of mass. As it
cooled this mass contracted and successive rings broke off from
its outer edge. These rings in their turn cooled, and finally
condensed into the planets, while the sun represents the central
core which is still left. On this view we should expect that the
more distant planets would be older than those nearer the sun.
The subject is one of great difficulty, and though it seems certain
that the solar system has a common origin, there are various features
which appear almost inexplicable on the nebular hypothesis as
enunciated by Laplace.**
**Another
theory which avoids many of the difficulties raised by Laplace's
hypothesis has recently found favour. According to this, the origin
of the solar system is to be found in the gradual aggregation
of meteorites which swarm through our system, and perhaps through
space. These meteorites which are normally cold may, by repeated
collisions, be heated, melted, or even vaporized, and the resulting
mass would, by the effect of gravity, be condensed into planet-like
bodies - the larger aggregations so formed becoming the chief
bodies of the solar system. **
**To
account for these collisions and condensations it is supposed
that a vast number of meteorites were at some distant epoch situated
in a spiral nebula, and that condensations and collisions took
place at certain knots or intersections of orbits. As the resulting
planetary masses cooled, moons or rings would be formed either
by collisions of outlying parts or in the manner suggested in
Laplace's hypothesis. This theory seems to be primarily due to
Sir Norman Lockyer. It does not conflict with any of the known
facts of cosmical science, but as yet our knowledge of the facts
is so limited that it would be madness to dogmatize on the subject.
Recent investigations have shown that our moon broke off from
the earth while the latter was in a plastic condition owing to
tidal friction. Hence its origin is neither nebular nor meteoric.**
**Probably
the best modern opinion inclines to the view that nebular condensation,
meteoric condensation, tidal friction, and possibly other causes
as yet unsuggested, have all played their part in the evolution
of the system. The idea of the nebular hypothesis had been outlined
by Kant in 1755, and he had also suggested meteoric aggregations
and tidal friction as causes affecting the formation of the solar
system: it is probable that Laplace was not aware of this.**
**According
to the rule published by Titius of Wittemberg in 1766-but generally
known as Bode's Law, from the fact that attention was called to
it by Johann Elert Bode in 1778 - the distances of the planets
from the sun are nearly in the ratio of the numbers 0 + 4, 3 +
4, 6 + 4, 12+4, etc., the (n+2)th term being ( 3) + 4. It would
be an interesting fact if this could be deduced from the nebular,
meteoric, or any other hypotheses, but so far as I am aware only
one writer has made any serious attempt to do so, and his conclusion
seems to be that the law is not sufficiently exact to be more
than a convenient means of remembering the general result.**
**Laplace's
analytical discussion of the solar system is given in his Méchanique
céleste published in five volumes. An analysis of the contents
is given in the English Cyclopaedia. The first two volumes, published
in 1799, contain methods for calculating the motions of the planets,
determining their figures, and resolving tidal problems. The third
and fourth volumes, published in 1802 and 1805, contain applications
of these methods, and several astronomical tables. **
**The
fifth volume, published in 1825, is mainly historical, but it
gives as appendices the results of Laplace's latest researches.
Laplace's own investigations embodied in it are so numerous and
valuable that it is regrettable to have to add that many results
are appropriated from writers with scanty or no acknowledgement,
and the conclusions - which have been described as the organized
result of a century of patient toil - are frequently mentioned
as if they were due to Laplace.**
**The
matter of the Méchanique céleste is excellent, but
it is by no means easy reading. Biot, who assisted Laplace in
revising it for the press, says that Laplace himself was frequently
unable to recover the details in the chain of reasoning, and,
if satisfied that the conclusions were correct, he was content
to insert the constantly recurring formula, "Il est aisé
à voir." The Méchanique céleste is not
only the translation of the Principia into the language of the
differential calculus, but it completes parts of which Newton
had been unable to fill in the details. F. F. Tisserand's recent
work may be taken as the modern presentation of dynamical astronomy
on classical lines, but Laplace's treatise will always remain
a standard authority.**
**Laplace
went in state to beg Napoleon to accept a copy of his work, and
the following account of the interview is well authenticated,
and so characteristic of all the parties concerned that I quote
it in full. Someone had told Napoleon that the book contained
no mention of the name of God; Napoleon, who was fond of putting
embarrassing questions, received it with the remark, "M.
Laplace, they tell me you have written this large book on the
system of the universe, and have never even mentioned its Creator."
**
**Laplace,
who, though the most supple of politicians, was as stiff as a
martyr on every point of his philosophy, drew himself up and answered
bluntly, "Je n'avais pas besoin de cette hypothèse-là
(I did not need to make such an assumption)." Napoleon, greatly
amused, told this reply to Lagrange, who exclaimed, "Ah!
c'est une belle hypothèse; ça explique beaucoup
de choses (Ah! that is a beautiful assumption; it explains many
things)."**
**Laplace
also came close to propounding the concept of the black hole.
He pointed out that there could be massive stars whose gravity
is so great that not even light could escape from their surface.
Laplace also speculated that some of the nebulae revealed by telescopes
may not be part of the Milky Way and might actually be galaxies
themselves. Thus, he anticipated the major discovery of Edwin
Hubble, some 100 years before it happened.**
**In
1812 Laplace issued his Théorie analytique des probabilités.
The theory is stated to be only common sense expressed in mathematical
language. The method of estimating the ratio of the number of
favourable cases to the whole number of possible cases had been
indicated by Laplace in a paper written in 1779. It consists in
treating the successive values of any function as the coefficients
in the expansion of another function with reference to a different
variable. **
**The
latter is therefore called the generating function of the former.
Laplace then shews how, by means of interpolation, these coefficients
may be determined from the generating function. Next he attacks
the converse problem, and from the coefficients he finds the generating
function; this is effected by the solution of an equation in finite
differences. The method is cumbersome, and in consequence of the
increased power of analysis is now rarely used.**
**This
treatise includes an exposition of the method of least squares,
a remarkable testimony to Laplace's command over the processes
of analysis. The method of least squares for the combination of
numerous observations had been given empirically by Gauss and
Legendre, but the fourth chapter of this work contains a formal
proof of it, on which the whole of the theory of errors has been
since based. **
**This
was effected only by a most intricate analysis specially invented
for the purpose, but the form in which it is presented is so meagre
and unsatisfactory that in spite of the uniform accuracy of the
results it was at one time questioned whether Laplace had actually
gone through the difficult work he so briefly and often incorrectly
indicates.**
**In
1819 Laplace published a popular account of his work on probability.
This book bears the same relation to the Théorie des probabilités
that the Système du monde does to the Méchanique
céleste.**
**Amongst
the minor discoveries of Laplace in pure mathematics I may mention
his discussion (simultaneously with Vandermonde) of the general
theory of determinants in 1772; his proof that every equations
of an even degree must have at least one real quadratic factor;
his reduction of the solution of linear differential equations
to definite integrals; and his solution of the linear partial
differential equation of the second order. **
**He
was also the first to consider the difficult problems involved
in equations of mixed differences, and to prove that the solution
of an equation in finite differences of the first degree and the
second order might be always obtained in the form of a continued
fraction. Besides these original discoveries he determined, in
his theory of probabilities, the values of a number of the more
common definite integrals; and in the same book gave the general
proof of the theorem enunciated by Lagrange for the development
of any implicit function in a series by means of differential
coefficients.**
**In
theoretical physics the theory of capillary attraction is due
to Laplace, who accepted the idea propounded by Hauksbee in the
Philosophical Transactions for 1709, that the phenomenon was due
to a force of attraction which was insensible at sensible distances.
The part which deals with the action of a solid on a liquid and
the mutual action of two liquids was not worked out thoroughly,
but ultimately was completed by Gauss: Neumann later filled in
a few details. In 1862 Lord Kelvin (Sir William Thomson) shewed
that, if we assume the molecular constitution of matter, the laws
of capillary attraction can be deduced from the Newtonian law
of gravitation.**
**Laplace
in 1816 was the first to point out explicitly why Newton's theory
of vibratory motion gave an incorrect value for the velocity of
sound. The actual velocity is greater than that calculated by
Newton in consequence of the heat developed by the sudden compression
of the air which increases the elasticity and therefore the velocity
of the sound transmitted. Laplace's investigations in practical
physics were confined to those carried on by him jointly with
Lavoisier in the years 1782 to 1784 on the specific heat of various
bodies.**
**Laplace
seems to have regarded analysis merely as a means of attacking
physical problems, though the ability with which he invented the
necessary analysis is almost phenomenal As long as his results
were true he took but little trouble to explain the steps by which
he arrived at them; he never studied elegance or symmetry in his
processes, and it was sufficient for him if he could by any means
solve the particular question he was discussing.**
**It
would have been well for Laplace's reputation if he had been content
with his scientific work, but above all things he coveted social
fame. The skill and rapidity with which he managed to change his
politics as occasion required would be amusing had they not been
so servile. As Napoleon's power increased Laplace abandoned his
republican principles (which, since they had faithfully reflected
the opinions of the party in power, had themselves gone through
numerous changes) and begged the first consul to give him the
post of minister of the interior. Napoleon, who desired the support
of men of science, agreed to the proposal; but a little less than
six weeks saw the close of Laplace's political career. **
**Although
Laplace was removed from office it was desirable to retain his
allegiance. He was accordingly raised to the senate, and to the
third volume of the Méchanique céleste he prefixed
a note that of all the truths therein contained the most precious
to the author was the declaration he thus made of his devotion
towards the peacemaker of Europe. In copies sold after the restoration
this was struck out. **
**In
1814 it was evident that the empire was falling; Laplace hastened
to tender his services to the Bourbons, and on the restoration
was rewarded with the title of marquis: the contempt that his
more honest colleagues felt for his conduct in the matter may
be read in the pages of Paul Louis Courier. His knowledge was
useful on the numerous scientific commissions on which he served,
and probably accounts for the manner in which his political insincerity
was overlooked; but the pettiness of his character must not make
us forget how great were his services to science.**
**That
Laplace was vain and selfish is not denied by his warmest admirers;
his conduct to the benefactors of his youth and his political
friends was ungrateful and contemptible; while his appropriation
of the results of those who were comparatively unknown seems to
be well established and is absolutely indefensible - of those
whom he thus treated three subsequently rose to distinction (Legendre
and Fourier in France and Young in England) and never forgot the
injustice of which they had been the victims. **
**On
the other side it may be said that on some questions he shewed
independence of character, and he never concealed his views on
religion, philosophy, or science, however distasteful they might
be to the authorities in power; it should be also added that towards
the close of his life, and especially to the work of his pupils,
Laplace was both generous and appreciative, and in one case suppressed
a paper of his own in order that a pupil might have the sole credit
of the investigation.**
**Quotes**
"What we know is not much. What we do not know is immense.
"
**"I
have no need of that hypothesis." as a reply to Napoleon,
who had asked why he hadn't mentioned God in his book on astronomy)
**
**"The
weight of evidence for an extraordinary claim must be proportioned
to its strangeness." (known as the principle of Laplace)** |