LEIPZIG veut dire en slave la :ville des tilleuls

Elle s’appellait Lipsk à l’origine, à l’origine de lipa =tilleul

C’est une racine qui se trouve dans un grand nombre de
familles d’origine polonaise :**lipski**

Elle se trouve aussi dans le nom du célèbre philosophe **Leibniz**,qui
est la forme germanique du mot slave Lipnica

Nom d’une bourgade de l’actuelle Slovénie , ex Yougoslavie.

Tout comme Legnica peut tout avoir a faire avec Lydien.

http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html

*From
`A Short Account of the History of Mathematics' (4th edition, 1908) by W. W.
Rouse Ball.*

*Gottfried Wilhelm
Leibnitz* (or *Leibniz*)
was born at Leipzig on June 21 (O.S.), 1646, and died in Hanover on November 14, 1716. His father died before he was six, and the
teaching at the school to which he was then sent was inefficient, but his
industry triumphed over all difficulties; by the time he was twelve he had
taught himself to read Latin easily, and had begun Greek; and before he was
twenty he had mastered the ordinary text-books on mathematics, philosophy,
theology and law. Refused the degree of doctor of laws at Leipzig by those who
were jealous of his youth and learning, he moved to Nuremberg. An essay which
there wrote on the study of law was dedicated to the Elector of Mainz, and led
to his appointment by the elector on a commission for the revision of some
statutes, from which he was subsequently promoted to the diplomatic service. In
the latter capacity he supported (unsuccessfully) the claims of the German
candidate for the crown of Poland. The violent seizure of various small places
in Alsace in 1670 excited universal alarm in Germany as to the designs of Louis
XIV.; and Leibnitz drew up a scheme by which it was proposed to offer German
co-operation, if France liked to take Egypt, and use the possessions of that
country as a basis for attack against Holland in Asia, provided France would
agree to leave Germany undisturbed. This bears a curious resemblance to the
similar plan by which Napoleon I. proposed to attack England. In 1672 Leibnitz
went to Paris on the invitation of the French government to explain the details
of the scheme, but nothing came of it.

At Paris he met Huygens who
was then residing there, and their conversation led Leibnitz to study geometry,
which he described as opening a new world to him; though as a matter of fact he
had previously written some tracts on various minor points in mathematics, the
most important being a paper on combinations written in 1668, and a description
of a new calculating machine. In January, 1673, he was sent on a political
mission to

In 1673 the Elector of
Mainz died, and in the following year Leibnitz entered the service of the
Brunswick family; in 1676 he again visited London, and then moved to Hanover,
where, till his death, he occupied the well-paid post of librarian in the ducal
library. His pen was thenceforth employed in all the political matters which
affected the Hanoverian family, and his services were recognized by honours and
distinctions of various kinds, his memoranda on the various political, historical,
and theological questions which concerned the dynasty during the forty years
from 1673 to 1713 form a valuable contribution to the history of that time.

Leibnitz's appointment in
the Hanoverian service gave him more time for his favourite pursuits. He used
to assert that as the first-fruit of his increased leisure, he invented the
differential and integral calculus in 1674, but the earliest traces of the use
of it in his extant note-books do not occur till 1675, and it was not till 1677
that we find it developed into a consistent system; it was not published till
1684. Most of his mathematical papers were produced within the ten years from
1682 to 1692, and many of them in a journal, called the Acta Eruditorum,
founded by himself and Otto Mencke in 1682, which had a wide circulation on the
continent.

Leibnitz occupies at least
as large a place in the history of philosophy as he does in the history of
mathematics. Most of his philosophical writings were composed in the last
twenty or twenty-five years of his life; and the points as to whether his views
were original or whether they were appropriated from Spinoza, whom he visited
in 1676, is still in question among philosophers, though the evidence seems to
point to the originality of Leibnitz. As to Leibnitz's system on philosophy it
will be enough to say that he regarded the ultimate elements of the universe as
individual percipient beings whom he called monads.
According to him the monads are centres of force, and substance is force, while
space, matter, and motion are merely phenomenal; finally, the existence of God
is inferred from the existing harmony among the monads. His services to
literature were almost as considerable as those to philosophy; in particular, I
may single out his overthrow of the then prevalent belief that Hebrew was the
primeval language of the human race.

In 1700 the academy of
Berlin was created on his advice, and he drew up the first body of statutes for
it. On the accession in 1714 of his master, George I., to the throne of
England, Leibnitz was thrown aside as a useless tool; he was forbidden to come
to England; and the last two years of his life were spent in neglect and
dishonour. He died at Hanover in 1716. He was overfond of money and personal
distinctions; was unscrupulous, as perhaps might be expected of a professional
diplomatist of that time; but possessed singularly attractive manners, and all
who once came under the charm of his personal presence remained sincerely
attached to him. His mathematical reputation was largely augmented by the
eminent position that he occupied in diplomacy, philosophy and literature; and
the power thence derived was considerably increased by his influence in the
management of the Acta Eruditorum.

The
last years of his life - from 1709 to 1716 - were embittered by the long
controversy with John Keill, Newton, and others, as to whether he had
discovered the differential calculus independently of Newton's previous
investigations, or whether he had derived the fundamental idea from Newton, and
merely invented another notation for it. The controversy occupies a place in the
scientific history of the early years of the eighteenth century quite
disproportionate to its true importance, but it so materially affected the
history of mathematics in western Europe, that I feel
obliged to give the leading facts, though I am reluctant to take up so much
space with questions of a personal character.

The ideas of the
infinitesimal calculus can be expressed either in the notation of fluxions or in
that of differentials. The former was used by Newton in 1666, but no distinct
account of it was printed till 1693. The earliest use of the latter in the
note-books of Leibnitz may probably be referred to 1675, it was employed in the
letter sent to Newton in 1677, and an account of it was printed in the memoir
of 1684 described below. There is no question that the differential notation is
due to Leibnitz, and the sole question is as to whether the general idea of the
calculus was taken from Newton or discovered independently.

The case in favour of the
independent invention by Leibnitz rests on the ground that he published a
description of his method some years before Newton printed anything on
fluxions, that he always alluded to the discovery as being his own invention,
and that for some years this statement was unchallenged; while of course there
must be a strong presumption that he acted in good faith. To rebut this case it
is necessary to shew (i) that he saw some of Newton's papers on the subject in or
before 1675, or at least 1677, and (ii) that he thence derived the fundamental
ideas of the calculus. The fact that his claim was unchallenged for some years
is, in the particular circumstances of the case, immaterial.

That Leibnitz saw some of
Newton's manuscripts was always intrinsically probable; but when, in 1849, C.
J. Gerhardt examined Leibnitz's papers he found among them a manuscript copy,
the existence of which had been previously unsuspected, in Leibnitz's
handwriting, of extracts from Newton's De Analysi per Equationes Numero
Terminorum Infinitas (which was printed in the De Quadratura
Curvarum in 1704), together with the notes on their expression in the
differential notation. The question of the date at which these extracts were
made is therefore all important. It is known that a copy of Newton's manuscript
had been sent to Tschirnhausen in May, 1675, and as in that year he and
Leibnitz were engaged together on a piece of work, it is not impossible that
these extracts were made then. It is also possible that they may have been made
in 1676, for Leibnitz discussed the question of analysis by infinite series
with Collins and Oldenburg in that year, and it is *a priori* probable
that they would have then shewn him the manuscript of Newton on that subject, a
copy of which was possessed by one or both of them. On the other hand it may be
supposed that Leibnitz made the extracts from the printed copy in or after
1704. Leibnitz shortly before his death admitted in a letter to Conti that in
1676 Collins had shewn him some Newtonian papers, but implied that they were of
little or no value, - presumably he referred to Newton's letters of June 13 and
Oct. 24, 1676, and to the letter of Dec. 10, 1672, on the method of tangents,
extracts from which accompanied the letter of June 13, - but it is remarkable
that, on the receipt of these letters, Leibnitz should have made no further
inquiries, unless he was already aware from other sources of the method
followed by Newton.

Whether Leibnitz made no
use of the manuscript from which he had copied extracts, or whether he had
previously invented the calculus, are questions on which at this distance of
time no direct evidence is available. It is, however, worth noting that the
unpublished Portsmouth Papers shew that when, in 1711, Newton went carefully
into the whole dispute, he picked out this manuscript as the one which had
probably somehow fallen into the hands of Leibnitz. At that time there was no
direct evidence that Leibnitz had seen this manuscript before it was printed in
1704, and accordingly Newton's conjecture was not published; but Gerhardt's
discovery of the copy made by Leibnitz tends to confirm the accuracy of
Newton's judgement in the matter. It is said by those who question Leibnitz's
good faith that to a man of his ability the manuscript, especially if
supplemented by the letter of Dec. 10, 1672, would supply sufficient hints to
give him a clue as to the methods of the calculus, though as the fluxional
notation is not employed in it anyone who used it would have to invent a
notation; but this is denied by others.

There was at first no
reason to suspect the good faith of Leibnitz; and it was not until the
appearance in 1704 of an anonymous review of Newton's tract on quadrature, in
which it was implied that Newton had borrowed the idea of the fluxional
calculus from Leibnitz, that any responsible mathematician questioned the
statement that Leibnitz had invented the calculus independently of Newton. (In
1699 Duillier had accused Leibnitz of plagiarism from Newton, but Dullier was
not a person of much importance) It is universally admitted that there was no
justification or authority for the statements made in this review, which was
rightly attributed to Leibnitz. But the subsequent discussion led to a critical
examination of the whole question, and doubt was expressed as to whether
Leibnitz had not derived the fundamental idea from Newton. The case against
Leibnitz as it appeared to Newton's friends was summed up in the Commercium
Epistolicum issued in 1712, and detailed references are given for all
the facts mentioned.

No such summary (with
facts, dates, and references) of the case for Leibnitz was issued by his
friends; but John Bernoulli attempted to indirectly weaken the evidence by
attacking the personal character of Newton; this was in a letter dated June 7,
1713. The charges were false, and when pressed for an explanation of them,
Bernoulli most solemnly denied having written the letter. In accepting the
denial Newton added in a private letter to him the following remarks, which are
interesting as giving Newton's account of why he was at last induced to take
any part in the controversy. ``I have never,'' said he, ``grasped at fame among
foreign nations, but I am very desirous to preserve my character for honesty,
which the author of that epistle, as if by the authority of a great judge, had
endeavoured to wrest from me. Now that I am old, I have
little pleasure in mathematical studies, and I have never tried to propagate my
opinions over the world, but I have rather taken care not to involve myself in
disputes on account of them.''

Leibnitz's defence or explanation of his silence is given in the following letter, dated April 9, 1716, from him to Conti. ``Pour répondre de point en point à l'ouvrage publié contre moi, il falloit entrer dans un grand détail de quantité de minutiés passées il y a trente à quarante ans, dont je ne me souvenois guère: il me falloit chercher mes vieilles lettres, dont plusiers se sont perdus, outre que le plus souvent je n'ai point gardé les minutes des miennes: et les autres sont ensevelies dans un grand tas de papiers, que je ne pouvois débrouiller qu'avec du temps et de la patience; mais je n'en avois guère le loisir, étant chargé présentement d'occupations d'une toute autre nature.''

The death of Leibnitz in
1716 only put a temporary stop to the controversy which was bitterly debated
for many years later. The question is one of difficulty; the evidence is
conflicting and circumstantial; and every one must judge for himself which opinion
seems most reasonable. Essentially it is a case of Leibnitz's word against a
number of suspicious details pointing against him. His unacknowledged
possession of a copy of part of one of Newton's manuscripts may be explicable;
but the fact that on more than one occasion he deliberately altered or added to
important documents (*ex. gr.* the letter of June 7, 1713, in the Charta
Volans, and that of April 8, 1716, in the Acta Eruditorum),
before publishing them, and, what is worse, that a material date in one of his
manuscripts has been falsified (1675 being altered to 1673), makes his own
testimony on the subject of little value. It must be recollected that what he
is alleged to have received was rather a number of suggestions than an account
of the calculus; and it is possible that as he did not publish his results of
1677 until 1684, and that as the notation and subsequent development of it were
all of his own invention, he may have been led, thirty years later, to minimize
any assistance which he had obtained originally, and finally to consider that
it was immaterial. During the eighteenth century the prevalent opinion was
against Leibnitz, but to-day the majority of writers incline to think it more
likely that the inventions were independent.

If we must confine
ourselves to one system of notation then there can be little doubt that that
which was invented by Leibnitz is better fitted for most of the purposed to
which the infinitesimal calculus is applied than that of fluxions, and for some
(such as the calculus of variations) it is indeed almost essential. It should
be remembered, however, that at the beginning of the eighteenth century the
methods of the infinitesimal calculus had not been systematized, and either
notation was equally good. The development of that calculus was the main work
of the mathematicians of the first half of the eighteenth century. The
differential form was adopted by continental mathematicians. The application of
it by Euler, Lagrange, and Laplace to the principles of mechanics laid down in
the Principia was the great achievement of the last half of that
century, and finally demonstrated the superiority of the differential to the
fluxional calculus. The translation of the Principia into the
language of modern analysis, and the filling in of the details of the Newtonian
theory by the aid of that analysis, were effected by Laplace.

The controversy with
Leibnitz was regarded in England as an attempt by foreigners to defraud Newton
of the credit of his invention, and the question was complicated on both sides
by national jealousies. It was therefore natural, though it was unfortunate,
that in England the geometrical and fluxional methods as used by Newton were
alone studied and employed. For more than a century the English school was thus
out of touch with continental mathematicians. The consequence was that, in
spite of the brilliant band of scholars formed by

Leaving now this long
controversy I come to the discussion of the mathematical papers produced by
Leibnitz, all the more important of which were published in the Acta
Eruditorum. They are mainly concerned with various questions on
mechanics.

The only papers of
first-rate importance which he produced are those on the differential calculus.
The earliest of these was one published in the Acta Eruditorum for
October, 1684, in which he enunciated a general method for finding maxima and
minima, and for drawing tangents to curves. One inverse problem, namely, to
find the curve whose subtangent is constant, was also discussed. The notation
is the same as that with which we are familiar, and the differential
coefficients of
and of products and quotients are determined. In 1686 he wrote a
paper on the principles of the new calculus. In both of these papers the
principle of continuity is explicitly assumed, while his treatment of the
subject is based on the use of infinitesimals and not on that of the limiting
value of ratios. In answer to some objections which were raised in 1694 by
Bernard Nieuwentyt, who asserted that *dy*/*dx* stood
for an unmeaning quantity like 0/0, Leibnitz explained, in the same way that
Barrow had previously done, that the value of *dy*/*dx* in
geometry could be expressed as the ratio of two finite quantities. I think that
Leibnitz's statement of the objects and methods of the infinitesimal calculus
as contained in these papers, which are the three most important memoirs on it
that he produced, is somewhat obscure, and his attempt to place the subject on
a metaphysical basis did not tend to clearness; but the fact that all the
results of modern mathematics are expressed in the language invented by
Leibnitz has proved the best monument of his work. Like Newton, he treated
integration not only as a summation, but as the inverse of differentiation.

In 1686 and 1692 he wrote
papers on osculating curves. These, however, contain some bad blunders, as, for
example, the assertion that an osculating circle will necessarily cut a curve
in four consecutive points: this error was pointed out by John Bernoulli, but
in his article of 1692 Leibnitz defended his original assertion, and insisted
that a circle could never cross a curve where it touched it.

In 1692 Leibnitz wrote a
memoir in which he laid the foundation of the theory of envelopes. This was
further developed in another paper in 1694, in which he introduced for the
first time the terms ``co-ordinates'' and ``axes of co-ordinates.''

Leibnitz also published a
good many papers on mechanical subjects; but some of them contain mistakes
which shew that he did not understand the principles of the subject. Thus, in
1685, he wrote a memoir to find the pressure exerted by a sphere of weight *W*
placed between two inclined planes of complementary inclinations, placed so
that the lines of greatest slope are perpendicular to the line of the
intersection of the planes. He asserted that the pressure on each plane must
consist of two components, ``unum quo decliviter descendere tendit, alterum quo
planum declive premit.'' He further said that for metaphysical reasons the sum
of the two pressures must be equal to *W*. Hence, if *R* and *R'*
be the required pressures, and
and
the inclinations of the planes, he finds that

and

.

The true
values are *R* = *W* cos
and *R'* = *W* sin .
Nevertheless some of his papers on mechanics are valuable. Of these the most
important were two, in 1689 and 1694, in which he solved the problem of finding
and isochronous curve; one, in 1697, on the curve of quickest descent (this was
the problem sent as a challenge to Newton); and two, in 1691 and 1692, in which
he stated the intrinsic equation of the curve assumed by a flexible rope
suspended from two points, that is, the catenary, but gave no proof. This last
problem had been originally proposed by Galileo.

In 1689, that is, two years
after the Principia had been published, he wrote on the movements
of the planets which he stated were produced by a motion of the ether. Not only
were the equations of motion which he obtained wrong, but his deductions from
them were not even in accordance with his own axioms. In another memoir in
1706, that is, nearly twenty years after the Principia had been
written, he admitted that he had made some mistakes in his former paper, but
adhered to his previous conclusions, and summed the matter up by saying ``it is
certain that gravitation generates a new force at each instant to the centre,
but the centrifugal force also generates another away from the centre.... The
centrifugal force may be considered in two aspects according as the movement is
treated as along the tangent to the curve or as along the arc of the circle
itself.'' It seems clear from this paper that he did not really understand the
principles of dynamics, and it is hardly necessary to consider his work on the
subject in further detail. Much of it is vitiated by a constant confusion
between momentum and kinetic energy: when the force is ``passive'' he uses the
first, which he calls the *vis** mortua*,
as the measure of a force; when the force is ``active'' he uses the latter, the
double of which he calls the *vis viva*.

The series quoted by
Leibnitz comprise those for ,
log (1 + *x*), sin *x*, vers *x* and ;
all these had been previously published, and he rarely, if ever, added any
demonstrations. Leibnitz (like Newton) recognised the importance of James
Gregory's remarks on the necessity of examining whether infinite series are
convergent or divergent, and proposed a test to distinguish series whose terms
are alternately positive and negative. In 1693 he explained the method of expansion
by indeterminate coefficients, though his applications were not free from
error.

To sum the matter up
briefly, it seems to me that Leibnitz's work exhibits great skill in analysis,
but much of it is unfinished, and when he leaves his symbols and attempts to
interpret his results he frequently commits blunders. No doubt the demands of
politics, philosophy, and literature on his time may have prevented him from
elaborating any problem completely or writing a systematic exposition of his
views, though they are no excuse for the mistakes of principle which occur in
his papers. Some of his memoirs contain suggestions of methods which have now
become valuable means of analysis, such as the use of determinants and of
indeterminate co-efficients; but when a writer of manifold interests like
Leibnitz throws out innumerable suggestions, some of them are likely to turn
out valuable, and to enumerate these (which he did not work out) without
reckoning the others (which are wrong) gives a false impression of the value of
his work. But in spite of this, his title to fame rests on a sure basis, for by
his advocacy of the differential calculus his name is inseparably connected
with one of the chief instruments of analysis, as that of Descartes - another
philosopher - is similarly connected with analytical geometry.

This page is included in a collection of mathematical
biographies taken
from A Short Account of the History of Mathematics by W. W. Rouse
Ball (4th Edition, 1908).

Transcribed by

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