[1] The Med. Gr. [Greek: lazourion], Med. Lat. _lazurius_ or
_lazulus_, as the names of this mineral substance, were adaptations
of the Arab. _al-lazward_, Pers. _lajward_, blue colour, lapis
lazuli. The same word appears in Med. Lat. as _azura_, whence O.F.
azur, Eng. "azure," blue, particularly used of that colour in
heraldry (q.v.) and represented conventionally in black and white by
horizontal lines.
LAPITHAE, a mythical race, whose home was in Thessaly in the valley of
the Peneus. The genealogies make them a kindred race with the Centaurs,
their king Peirithoüs being the son, and the Centaurs the grandchildren
(or sons) of Ixion. The best-known legends with which they are connected
are those of Ixion (q.v.) and the battle with the Centaurs (q.v.). A
well-known Lapith was Caeneus, said to have been originally a girl named
Caenis, the favourite of Poseidon, who changed her into a man and made
her invulnerable (Ovid, _Metam._ xii. 146 ff). In the Centaur battle,
having been crushed by rocks and trunks of trees, he was changed into a
bird; or he disappeared into the depths of the earth unharmed. According
to some, the Lapithae are representatives of the giants of fable, or
spirits of the storm; according to others, they are a semi-legendary;
semi-historical race, like the Myrmidons and other Thessalian tribes.
The Greek sculptors of the school of Pheidias conceived of the battle of
the Lapithae and Centaurs as a struggle between mankind and mischievous
monsters, and symbolical of the great conflict between the Greeks and
Persians. Sidney Colvin (_Journ. Hellen. Stud._ i. 64) explains it as a
contest of the physical powers of nature, and the mythical expression of
the terrible effects of swollen waters.
LA PLACE (Lat. _Placaeus_), JOSUÉ DE (1606?-1665), French Protestant
divine, was born in Brittany. He studied and afterwards taught
philosophy at Saumur. In 1625 he became pastor of the Reformed Church at
Nantes, and in 1632 was appointed professor of theology at Saumur, where
he had as his colleagues, appointed at the same time, Moses Amyraut and
Louis Cappell. In 1640 he published a work, _Theses theologicae de statu
hominis lapsi ante gratiam_, which was looked upon with some suspicion
as containing liberal ideas about the doctrine of original sin. The view
that the original sin of Adam was not imputed to his descendants was
condemned at the synod of Charenton (1645), without special reference
being made to La Place, whose position perhaps was not quite clear. As a
matter of fact La Place distinguished between a direct and indirect
imputation, and after his death his views, as well as those of Amyraut,
were rejected in the _Formula consensus of_ 1675. He died on the 17th of
August 1665.
La Place's defence was published with the title _Disputationes
academicae_ (3 vols., 1649-1651; and again in 1665); his work _De
imputatione primi peccati Adami_ in 1655. A collected edition of his
works appeared at Franeker in 1699, and at Aubencit in 1702.
LAPLACE, PIERRE SIMON, MARQUIS DE (1749-1827), French mathematician and
astronomer, was born at Beaumont-en-Auge in Normandy, on the 28th of
March 1749. His father was a small farmer, and he owed his education to
the interest excited by his lively parts in some persons of position.
His first distinctions are said to have been gained in theological
controversy, but at an early age he became mathematical teacher in the
military school of Beaumont, the classes of which he had attended as an
extern. He was not more than eighteen when, armed with letters of
recommendation, he approached J. B. d'Alembert, then at the height of
his fame, in the hope of finding a career in Paris. The letters remained
unnoticed, but Laplace was not crushed by the rebuff. He wrote to the
great geometer a letter on the principles of mechanics, which evoked an
immediate and enthusiastic response. "You," said d'Alembert to him,
"needed no introduction; you have recommended yourself; my support is
your due." He accordingly obtained for him an appointment as professor
of mathematics in the École Militaire of Paris, and continued zealously
to forward his interests.
Laplace had not yet completed his twenty-fourth year when he entered
upon the course of discovery which earned him the title of "the Newton
of France." Having in his first published paper[1] shown his mastery of
analysis, he proceeded to apply its resources to the great outstanding
problems in celestial mechanics. Of these the most conspicuous was
offered by the opposite inequalities of Jupiter and Saturn, which the
emulous efforts of L. Euler and J. L. Lagrange had failed to bring
within the bounds of theory. The discordance of their results incited
Laplace to a searching examination of the whole subject of planetary
perturbations, and his maiden effort was rewarded with a discovery which
constituted, when developed and completely demonstrated by his own
further labours and those of his illustrious rival Lagrange, the most
important advance made in physical astronomy since the time of Newton.
In a paper read before the Academy of Sciences, on the 10th of February
1773 (_Mém. présentés par divers savans_, tom, vii., 1776), Laplace
announced his celebrated conclusion of the invariability of planetary
mean motions, carrying the proof as far as the cubes of the
eccentricities and inclinations. This was the first and most important
step in the establishment of the stability of the solar system. It was
followed by a series of profound investigations, in which Lagrange and
Laplace alternately surpassed and supplemented each other in assigning
limits of variation to the several elements of the planetary orbits. The
analytical tournament closed with the communication to the Academy by
Laplace, in 1787, of an entire group of remarkable discoveries. It
would be difficult, in the whole range of scientific literature, to
point to a memoir of equal brilliancy with that published (divided into
three parts) in the volumes of the Academy for 1784, 1785 and 1786. The
long-sought cause of the "great inequality" of Jupiter and Saturn was
found in the near approach to commensurability of their mean motions; it
was demonstrated in two elegant theorems, independently of any except
the most general considerations as to mass, that the mutual action of
the planets could never largely affect the eccentricities and
inclinations of their orbits; and the singular peculiarities detected by
him in the Jovian system were expressed in the so-called "laws of
Laplace." He completed the theory of these bodies in a treatise
published among the Paris _Memoirs_ for 1788 and 1789; and the striking
superiority of the tables computed by J. B. J. Delambre from the data
there supplied marked the profit derived from the investigation by
practical astronomy. The year 1787 was rendered further memorable by
Laplace's announcement on the 19th of November (_Memoirs_, 1786), of the
dependence of lunar acceleration upon the secular changes in the
eccentricity of the earth's orbit. The last apparent anomaly, and the
last threat of instability, thus disappeared from the solar system.
With these brilliant performances the first period of Laplace's
scientific career may be said to have closed. If he ceased to make
striking discoveries in celestial mechanics, it was rather their
subject-matter than his powers that failed. The general working of the
great machine was now laid bare, and it needed a further advance of
knowledge to bring a fresh set of problems within reach of
investigation. The time had come when the results obtained in the
development and application of the law of gravitation by three
generations of illustrious mathematicians might be presented from a
single point of view. To this task the second period of Laplace's
activity was devoted. As a monument of mathematical genius applied to
the celestial revolutions, the _Mécanique céleste_ ranks second only to
the _Principia_ of Newton.
The declared aim of the author[2] was to offer a complete solution of
the great mechanical problem presented by the solar system, and to
bring theory to coincide so closely with observation that empirical
equations should no longer find a place in astronomical tables. His
success in both respects fell little short of his lofty ideal. The
first part of the work (2 vols. 4to, Paris, 1799) contains methods for
calculating the movements of translation and rotation of the heavenly
bodies, for determining their figures, and resolving tidal problems;
the second, especially dedicated to the improvement of tables,
exhibits in the third and fourth volumes (1802 and 1805) the
application of these formulae; while a fifth volume, published in
three instalments, 1823-1825, comprises the results of Laplace's
latest researches, together with a valuable history of progress in
each separate branch of his subject. In the delicate task of
apportioning his own large share of merit, he certainly does not err
on the side of modesty; but it would perhaps be as difficult to
produce an instance of injustice, as of generosity in his estimate of
others. Far more serious blame attaches to his all but total
suppression in the body of the work--and the fault pervades the whole
of his writings--of the names of his predecessors and contemporaries.
Theorems and formulae are appropriated wholesale without
acknowledgment, and a production which may be described as the
organized result of a century of patient toil presents itself to the
world as the offspring of a single brain. The _Mécanique céleste_ is,
even to those most conversant with analytical methods, by no means
easy reading. J. B. Biot, who assisted in the correction of its proof
sheets, remarked that it would have extended, had the demonstrations
been fully developed, to eight or ten instead of five volumes; and he
saw at times the author himself obliged to devote an hour's labour to
recovering the dropped links in the chain of reasoning covered by the
recurring formula. "Il est aisé à voir."[3]
The _Exposition du système du monde_ (Paris, 1796) has been styled by
Arago "the _Mécanique céleste_ disembarrassed of its analytical
paraphernalia." Conclusions are not merely stated in it, but the methods
pursued for their attainment are indicated. It has the strength of an
analytical treatise, the charm of a popular dissertation. The style is
lucid and masterly, and the summary of astronomical history with which
it terminates has been reckoned one of the masterpieces of the language.
To this linguistic excellence the writer owed the place accorded to him
in 1816 in the Academy, of which institution he became president in the
following year. The famous "nebular hypothesis" of Laplace made its
appearance in the _Système du monde_. Although relegated to a note
(vii.), and propounded "Avec la défiance que doit inspirer tout ce qui
n'est point un résultat de l'observation ou du calcul," it is plain,
from the complacency with which he recurred to it[4] at a later date,
that he regarded the speculation with considerable interest. That it
formed the starting-point, and largely prescribed the course of thought
on the subject of planetary origin is due to the simplicity of its
assumptions, and the clearness of the mechanical principles involved,
rather than to any cogent evidence of its truth. It is curious that
Laplace, while bestowing more attention than they deserved on the crude
conjectures of Buffon, seems to have been unaware that he had been, to
some extent, anticipated by Kant, who had put forward in 1755, in his
_Allgemeine Naturgeschichte_, a true though defective nebular cosmogony.
The career of Laplace was one of scarcely interrupted prosperity.
Admitted to the Academy of Sciences as an associate in 1773, he became a
member in 1785, having, about a year previously, succeeded E. Bezout as
examiner to the royal artillery. During an access of revolutionary
suspicion, he was removed from the commission of weights and measures;
but the slight was quickly effaced by new honours. He was one of the
first members, and became president of the Bureau of Longitudes, took a
prominent place at the Institute (founded in 1796), professed analysis
at the École Normale, and aided in the organization of the decimal
system. The publication of the _Mécanique céleste_ gained him world-wide
celebrity, and his name appeared on the lists of the principal
scientific associations of Europe, including the Royal Society. But
scientific distinctions by no means satisfied his ambition. He aspired
to the rôle of a politician, and has left a memorable example of genius
degraded to servility for the sake of a riband and a title. The ardour
of his republican principles gave place, after the 18th Brumaire, to
devotion towards the first consul, a sentiment promptly rewarded with
the post of minister of the interior. His incapacity for affairs was,
however, so flagrant that it became necessary to supersede him at the
end of six weeks, when Lucien Bonaparte became his successor. "He
brought into the administration," said Napoleon, "the spirit of the
infinitesimals." His failure was consoled by elevation to the senate, of
which body he became chancellor in September 1803. He was at the same
time named grand officer of the Legion of Honour, and obtained in 1813
the same rank in the new order of Reunion. The title of count he had
acquired on the creation of the empire. Nevertheless he cheerfully gave
his voice in 1814 for the dethronement of his patron, and his
"suppleness" merited a seat in the chamber of peers, and, in 1817, the
dignity of a marquisate. The memory of these tergiversations is
perpetuated in his writings. The first edition of the _Système du monde_
was inscribed to the Council of Five Hundred; to the third volume of the
_Mécanique céleste_ (1802) was prefixed the declaration that, of all the
truths contained in the work, that most precious to the author was the
expression of his gratitude and devotion towards the "pacificator of
Europe"; upon which noteworthy protestation the suppression in the
editions of the _Théorie des probabilités_ subsequent to the
restoration, of the original dedication to the emperor formed a fitting
commentary.
During the later years of his life, Laplace lived much at Arcueil, where
he had a country-place adjoining that of his friend C. L. Berthollet.
With his co-operation the Société d'Arcueil was formed, and he
occasionally contributed to its _Memoirs_. In this peaceful retirement
he pursued his studies with unabated ardour, and received with uniform
courtesy distinguished visitors from all parts of the world. Here, too,
he died, attended by his physician, Dr Majendie, and his mathematical
coadjutor, Alexis Bouvard, on the 5th of March 1827. His last words
were: "Ce que nous connaissons est peu de chose, ce que nous ignorons
est immense."
Expressions occur in Laplace's private letters inconsistent with the
atheistical opinions he is commonly believed to have held. His
character, notwithstanding the egotism by which it was disfigured, had
an amiable and engaging side. Young men of science found in him an
active benefactor. His relations with these "adopted children of his
thought" possessed a singular charm of affectionate simplicity; their
intellectual progress and material interests were objects of equal
solicitude to him, and he demanded in return only diligence in the
pursuit of knowledge. Biot relates that, when he himself was beginning
his career, Laplace introduced him at the Institute for the purpose of
explaining his supposed discovery of equations of mixed differences, and
afterwards showed him, under a strict pledge of secrecy, the papers,
then yellow with age, in which he had long before obtained the same
results. This instance of abnegation is the more worthy of record that
it formed a marked exception to Laplace's usual course. Between him and
A. M. Legendre there was a feeling of "more than coldness," owing to his
appropriation, with scant acknowledgment, of the fruits of the other's
labours; and Dr Thomas Young counted himself, rightly or wrongly,
amongst the number of those similarly aggrieved by him. With Lagrange,
on the other hand, he always remained on the best of terms. Laplace left
a son, Charles Emile Pierre Joseph Laplace (1789-1874), who succeeded to
his title, and rose to the rank of general in the artillery.
It might be said that Laplace was a great mathematician by the original
structure of his mind, and became a great discoverer through the
sentiment which animated it. The regulated enthusiasm with which he
regarded the system of nature was with him from first to last. It can be
traced in his earliest essay, and it dictated the ravings of his final
illness. By it his extraordinary analytical powers became strictly
subordinated to physical investigations. To this lofty quality of
intellect he added a rare sagacity in perceiving analogies, and in
detecting the new truths that lay concealed in his formulae, and a
tenacity of mental grip, by which problems, once seized, were held fast,
year after year, until they yielded up their solutions. In every branch
of physical astronomy, accordingly, deep traces of his work are visible.
"He would have completed the science of the skies," Baron Fourier
remarked, "had the science been capable of completion."
It may be added that he first examined the conditions of stability of
the system formed by Saturn's rings, pointed out the necessity for
their rotation, and fixed for it a period (10^h 33^m) virtually
identical with that established by the observations of Herschel; that
he detected the existence in the solar system of an invariable plane
such that the sum of the products of the planetary masses by the
projections upon it of the areas described by their radii vectores in
a given time is a maximum; and made notable advances in the theory of
astronomical refraction (_Méc. cél._ tom. iv. p. 258), besides
constructing satisfactory formulae for the barometrical determination
of heights (_Méc. cél._ tom. iv. p. 324). His removal of the
considerable discrepancy between the actual and Newtonian velocities
of sound,[5] by taking into account the increase of elasticity due to
the heat of compression, would alone have sufficed to illustrate a
lesser name. Molecular physics also attracted his notice, and he
announced in 1824 his purpose of treating the subject in a separate
work. With A. Lavoisier he made an important series of experiments on
specific heat (1782-1784), in the course of which the "ice
calorimeter" was invented; and they contributed jointly to the
_Memoirs_ of the Academy (1781) a paper on the development of
electricity by evaporation. Laplace was, moreover, the first to offer
a complete analysis of capillary action based upon a definite
hypothesis--that of forces "sensible only at insensible distances";
and he made strenuous but unsuccessful efforts to explain the
phenomena of light on an identical principle. It was a favourite idea
of his that chemical affinity and capillary attraction would
eventually be included under the same law, and it was perhaps because
of its recalcitrance to this cherished generalization that the
undulatory theory of light was distasteful to him.
The investigation of the figure of equilibrium of a rotating fluid
mass engaged the persistent attention of Laplace. His first memoir was
communicated to the Academy in 1773, when he was only twenty-four, his
last in 1817, when he was sixty-eight. The results of his many papers
on this subject--characterized by him as "un des points les plus
intéressans du système du monde"--are embodied in the _Mécanique
céleste_, and furnish one of the most remarkable proofs of his
analytical genius. C. Maclaurin, Legendre and d'Alembert had furnished
partial solutions of the problem, confining their attention to the
possible figures which would satisfy the conditions of equilibrium.
Laplace treated the subject from the point of view of the gradual
aggregation and cooling of a mass of matter, and demonstrated that the
form which such a mass would ultimately assume must be an ellipsoid of
revolution whose equator was determined by the primitive plane of
maximum areas.
The related subject of the attraction of spheroids was also signally
promoted by him. Legendre, in 1783, extended Maclaurin's theorem
concerning ellipsoids of revolution to the case of any spheroid of
revolution where the attracted point, instead of being limited to the
axis or equator, occupied any position in space; and Laplace, in his
treatise _Théorie du mouvement et de la figure elliptique des
planètes_ (published in 1784), effected a still further generalization
by proving, what had been suspected by Legendre, that the theorem was
equally true for any confocal ellipsoids. Finally, in a celebrated
memoir, _Théorie des attractions des sphéroides et de la figure des
planètes_, published in 1785 among the Paris _Memoirs_ for the year
1782, although written after the treatise of 1784, Laplace treated
exhaustively the general problem of the attraction of any spheroid
upon a particle situated outside or upon its surface.
These researches derive additional importance from having introduced
two powerful engines of analysis for the treatment of physical
problems, Laplace's coefficients and the potential function. By his
discovery that the attracting force in any direction of a mass upon a
particle could be obtained by the direct process of differentiating a
single function, Laplace laid the foundations of the mathematical
sciences of heat, electricity and magnetism. The expressions
designated by Dr Whewell, Laplace's coefficients (see SPHERICAL
HARMONICS) were definitely introduced in the memoir of 1785 on
attractions above referred to. In the figure of the earth, the theory
of attractions, and the sciences of electricity and magnetism this
powerful calculus occupies a prominent place. C. F. Gauss in
particular employed it in the calculation of the magnetic potential of
the earth, and it received new light from Clerk Maxwell's
interpretation of harmonics with reference to poles on the sphere.
Laplace nowhere displayed the massiveness of his genius more
conspicuously than in the theory of probabilities. The science which
B. Pascal and P. de Fermat had initiated he brought very nearly to
perfection; but the demonstrations are so involved, and the omissions
in the chain of reasoning so frequent, that the _Théorie analytique_
(1812) is to the best mathematicians a work requiring most arduous
study. The theory of probabilities, which Laplace described as common
sense expressed in mathematical language, engaged his attention from
its importance in physics and astronomy; and he applied his theory,
not only to the ordinary problems of chances, but also to the inquiry
into the causes of phenomena, vital statistics and future events.
The device known as the method of least squares, for reducing numerous
equations of condition to the number of unknown quantities to be
determined, had been adopted as a practically convenient rule by Gauss
and Legendre; but Laplace first treated it as a problem in
probabilities, and proved by an intricate and difficult course of
reasoning that it was also the most advantageous, the mean of the
probabilities of error in the determination of the elements being
thereby reduced to a minimum.
Laplace published in 1779 the method of generating functions, the
foundation of his theory of probabilities, and the first part of his
_Théorie analytique_ is devoted to the exposition of its principles,
which in their simplest form consist 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. A _direct_ and
an _inverse_ calculus is thus created, the object of the former being
to determine the coefficients from the generating function, of the
latter to discover the generating function from the coefficients. The
one is a problem of interpolation, the other a step towards the
solution of an equation in finite differences. The method, however, is
now obsolete owing to the more extended facilities afforded by the
calculus of operations.
The first formal proof of Lagrange's theorem for the development in a
series of an implicit function was furnished by Laplace, who gave to
it an extended generality. He also showed that every equation of an
even degree must have at least one real quadratic factor, reduced the
solution of linear differential equations to definite integrals, and
furnished an elegant method by which the linear partial differential
equation of the second order might be solved. He was also the first to
consider the difficult problems involved in equations of mixed
differences, and to prove that an equation in finite differences of
the first degree and the second order might always be converted into a
continued fraction.
In 1842, the works of Laplace being nearly out of print, his widow was
about to sell a farm to procure funds for a new impression, when the
government of Louis Philippe took the matter in hand. A grant of
40,000 francs having been obtained from the chamber, a national
edition was issued in seven 4to vols., bearing the title _Oeuvres de
Laplace_ (1843-1847). The _Mécanique céleste_ with its four
supplements occupies the first 5 vols., the 6th contains the _Système
du monde_, and the 7th the _Th. des probabilités_, to which the more
popular _Essai philosophique_ forms an introduction. Of the four
supplements added by the author (1816-1825) he tells us that the
problems in the last were contributed by his son. An enumeration of
Laplace's memoirs and papers (about one hundred in number) is rendered
superfluous by their embodiment in his principal works. The _Th. des
prob._ was first published in 1812, the _Essai_ in 1814; and both
works as well as the _Système du monde_ went through repeated
editions. An English version of the _Essai_ appeared in New York in
1902. Laplace's first separate work, _Théorie du mouvement et de la
figure elliptique des planètes_ (1784), was published at the expense
of President Bochard de Saron. The _Précis de l'histoire de
l'astronomie_ (1821), formed the fifth book of the 5th edition of the
_Système du monde_. An English translation, with copious elucidatory
notes, of the first 4 vols. of the _Mécanique céleste_, by N.
Bowditch, was published at Boston, U.S. (1829-1839), in 4 vols. 4to.;
a compendium of certain portions of the same work by Mrs Somerville
appeared in 1831, and a German version of the first 2 vols. by
Burckhardt at Berlin in 1801. English translations of the _Système du
monde_ by J. Pond and H. H. Harte were published, the first in 1809,
the second in 1830. An edition entitled _Les Oeuvres complètes de
Laplace_ (1878), &c., which is to include all his memoirs as well as
his separate works, is in course of publication under the auspices of
the Academy of Sciences. The thirteenth 4to volume was issued in 1904.
Some of Laplace's results in the theory of probabilities are
simplified in S. F. Lacroix's _Traité élémentaire du calcul des
probabilités_ and De Morgan's _Essay_, published in Lardner's _Cabinet
Cyclopaedia_. For the history of the subject see _A History of the
Mathematical Theory of Probability_, by Isaac Todhunter (1865).
Laplace's treatise on specific heat was published in German in 1892 as
No. 40 of W. Ostwald's _Klassiker der exacten Wissenschaften_.
AUTHORITIES.--Baron Fourier's _Éloge, Mémoires de l'institut_, x.
lxxxi. (1831); _Revue encyclopédique_, xliii. (1829); S. D. Poisson's
Funeral Oration (_Conn. des Temps_, 1830, p. 19); F. X. von Zach,
_Allg. geographische Ephemeriden_, iv. 70 (1799); F. Arago, _Annuaire
du Bureau des Long_. 1844, p. 271, translated among Arago's
Biographies of Distinguished Men (1857); J. S. Bailly, _Hist. de
l'astr. moderne_, t. iii.; R. Grant, _Hist. of Phys. Astr._ p. 50,
&c.; A. Berry, _Short Hist. of Astr._ p. 306; Max Marie, _Hist. des
sciences_ t. x. pp. 69-98; R. Wolf, _Geschichte der Astronomie_; J.
Mädler, _Gesch. der Himmelskunde_, i. 17; W. Whewell, _Hist. of the
Inductive Sciences_, ii. _passim_; J. C. Poggendorff, _Biog-lit.
Handwörterbuch._ (A. M. C.)