Speaker 1: Chapter seventeen of A Short Account of the History of Mathematics.
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Speaker 1: edu forward slash p k J. A Short Account of
Speaker 1: the History of Mathematics by W. W. Rowse Ball. Chapter seventeen.
Speaker 1: Leibniz and the Mathematicians of the first half of the
Speaker 1: eighteenth century, Part one. I have briefly traced in the
Speaker 1: last chapter the nature and extent of Newton's contributions to science.
Speaker 1: Modern analysis is, however, derived directly from the works of
Speaker 1: Leibniz and the elder Bernoulli's, and it is immaterial to
Speaker 1: us whether the fundamental ideas of it were by them
Speaker 1: from Newton or discovered independently. The English mathematicians of the
Speaker 1: years considered in this chapter continued to use the language
Speaker 1: and notation of Newton. They are thus somewhat distinct from
Speaker 1: their Continental contemporaries, and I have therefore grouped them together
Speaker 1: in a section by themselves. Leibnitz and the Bernoullis Leibnitz.
Speaker 1: Gottfried Wilhelm Leibnitz was born at Leipzig on June the
Speaker 1: twenty first, sixteen forty six and died at Hanover on
Speaker 1: November fourteenth, seventeen sixteen. His father died before he was six,
Speaker 1: and the teaching at the school to which he was
Speaker 1: then sent was inefficient, but his industry triumphed over all difficulties.
Speaker 1: By the time he was twelve, he had taught himself
Speaker 1: to read Latin easily and had begun Greek, and before
Speaker 1: he was twenty he had mastered the ordinary textbooks on mathematics, philosophy,
Speaker 1: theology and law. Refused the degree of Doctor of Laws
Speaker 1: at Leipzig by those who were jealous of his youth
Speaker 1: and learning, he moved to Nuremberg. An essay which he
Speaker 1: there wrote on the study of law was dedicated to
Speaker 1: the Elector of Mainz and led to his appointment by
Speaker 1: the Elector on a commission for the revision of some statutes,
Speaker 1: from which he was subsequently promoted to the diplomatic service.
Speaker 1: In the latter capacity, he supported unsuccessfully the claims of
Speaker 1: the German candidate for the Crown of Poland. The violent
Speaker 1: seizure of various small places in Alsas in sixteen seventy
Speaker 1: excited universal alarm in Germany as to the designs of
Speaker 1: Louis the fourteenth, and Leibnitz drew up a scheme by
Speaker 1: which it was proposed to offer German cooperation if France
Speaker 1: liked to take Egypt and use the possession of that
Speaker 1: country as a basis for attack against Holland and Asia,
Speaker 1: on the condition that Germany was to be left undisturbed
Speaker 1: by France. This bears a curious resemblance to the similar
Speaker 1: plan by which Napoleon the First proposed to attack England
Speaker 1: in sixteen seventy two. Leibnitz went to Paris on the
Speaker 1: invitation of the French government to explain the details of
Speaker 1: the scheme, but nothing came of it. At Paris he
Speaker 1: met Huygens, who was then residing there, and their conversation
Speaker 1: led him to study geometry, which he described as opening
Speaker 1: a new world to him, though he had, as a
Speaker 1: matter of fact, previously written some tracts on various minor
Speaker 1: points in mathematics, the most important of them being a
Speaker 1: paper on combinations written in sixteen sixty eight and a
Speaker 1: description of a new calculating machine. In January sixteen seventy
Speaker 1: three he was sent on a political mission to London,
Speaker 1: where he stopped some months and made the acquaintance of Oldenburg,
Speaker 1: Collins and others. It was at this time that he
Speaker 1: communicated the memoir to the Royal Society, in which he
Speaker 1: was found to have been forestalled by Mouton. In sixteen
Speaker 1: seventy three, the Elector of Mainz died, and in the
Speaker 1: following year Leibnitz entered the service of the Brunswick family.
Speaker 1: In sixteen seventy six he again visited London and then
Speaker 1: moved to Hanover, where till his death he occupied the
Speaker 1: well paid post of librarian in the Ducal Library. His
Speaker 1: pen was thenceforth employed in all the political matters which
Speaker 1: affected the Hanoverian family, and his services were recognized by
Speaker 1: honors and distinctions of various kinds. His memoranda on the
Speaker 1: various political, historical and theological questions which concerned the dynasty
Speaker 1: during the forty years from sixteen seventy three to seventeen
Speaker 1: thirteen form a valuable contribution to the history of that time.
Speaker 1: His appointment in the Hanoverian service gave him an increased
Speaker 1: leisure for his favorite pursuits. Leibnets used to assert that
Speaker 1: as the first fruit of his increased leisure, he invented
Speaker 1: the differential and integral calculus in sixteen seventy four, but
Speaker 1: the earliest traces of the use of it in his
Speaker 1: extant notebooks do not occur till sixteen seventy five, and
Speaker 1: it was not till sixteen seventy seven that we find
Speaker 1: it developed into a consistent system. He was not published
Speaker 1: until sixteen eighty four. Nearly all his mathematical papers were
Speaker 1: produced within the ten years from sixteen eighty two to
Speaker 1: sixteen ninety two, most of them in a journal called
Speaker 1: the Acta Eruditorum, which he and Ottomenkad founded in sixteen
Speaker 1: eighty two and which had a wide circulation on the continent.
Speaker 1: Leibnitz occupies at least as large a place in the
Speaker 1: history of philosophy as he does in the history of mathematics.
Speaker 1: Most of his philosophical writings were composed in the last
Speaker 1: twenty or twenty five years of his life, and the
Speaker 1: point as to whether his views were original or whether
Speaker 1: they were appropriated from Spinoza, whom he visited in sixteen
Speaker 1: seventy six, is still in question among philosophers, though the
Speaker 1: evidence seems to point to the originality of Leibnitz. As
Speaker 1: to Leibnitz's system of philosophy, it will be enough to
Speaker 1: say that he regarded the ultimate elements of the universe
Speaker 1: as individual precipient beings, whom he called monads. According to him,
Speaker 1: the monads are centers of force and substances force, while space, matter,
Speaker 1: and motion are merely phenomenal. Finally, the existence of God
Speaker 1: is inferred from the existing harmony among the monads. His
Speaker 1: services to literature were almost as considerable as those to philosophy.
Speaker 1: In particular, I may single out his overthrow of the
Speaker 1: then prevalent belief that Hebrew was the primeval language of
Speaker 1: the entire human race. In seventeen hundred, the Academy of
Speaker 1: Berlin was created on his advice, and he drew up
Speaker 1: the first body of statutes for it. On the accession
Speaker 1: in seventeen fourteen of his master, George First to the
Speaker 1: throne of England, Leibnitz was practically thrown aside as a
Speaker 1: useless tool. He was forbidden to come to England, and
Speaker 1: the last two years of his life were spent in
Speaker 1: neglect and dishonor. He died at Hanover in seventeen sixteen.
Speaker 1: He was over fond of money and personal distinctions, was unscrupulous,
Speaker 1: as might be expected of a professional diplomat of that time,
Speaker 1: but possessed singularly attractive manners, and all who once came
Speaker 1: under the charm of his personal presence remained sincerely attached
Speaker 1: to him. His mathematical reputation was largely augmented by the
Speaker 1: eminent position that he occupied in diplomacy, philosophy, and literature,
Speaker 1: and the power thence derived was considerably increased by his
Speaker 1: influence in the management of the Acta Rodatorum, which I
Speaker 1: believe was the only private scientific journal of the time.
Speaker 1: The last years of his life, from seventeen o nine
Speaker 1: to seventeen sixteen were embittered by the long controversy with
Speaker 1: John Kyle Newton and others as to whether he had
Speaker 1: discovered the differential calculus independently of Newton's previous investigations, or
Speaker 1: whether he had derived the fundamental idea from Newton and
Speaker 1: merely invented another notation for it. The controversy occupies a
Speaker 1: place in the scientific history of the early years of
Speaker 1: the eighteenth century quite disproportionate to its true importance, but
Speaker 1: it so materially affected the history of mathematics in Western
Speaker 1: Europe that I feel obliged to give the leading facts,
Speaker 1: though I am reluctant to take up so much space
Speaker 1: with questions of a personal character. The case in favor
Speaker 1: of the independent invention by Leibnitz is stated in Gerhaltz
Speaker 1: Leibenismus mathematische Schriften in Biatt and Leefort's edition of the
Speaker 1: Commercium Epistocolum Paris eighteen fifty six. The arguments on the
Speaker 1: other side are given in H. Sloman's Lebanistan's unspruckhofti Erfindong
Speaker 1: der di frienzia Rechnung Leipezig, eighteen fifty seven, of which
Speaker 1: an English translation with the editions by doctor Sloman was
Speaker 1: published at Cambridge in eighteen sixty. The history of the
Speaker 1: invention of the calculus is given in an article of
Speaker 1: it in the ninth edition of the Encyclopedia Britannica, and
Speaker 1: in P. Manson's is Quiste lestraal t Calcula Infinities mal
Speaker 1: Gand eighteen eighty seven, end of footnote. The ideas of
Speaker 1: the infinitesimal calculus can be expressed either in the notation
Speaker 1: of fluxions or in that of differentials. The former was
Speaker 1: used by Newton in sixteen sixty six and communicated in
Speaker 1: manuscript to his friends and pupils from sixteen sixty nine onwards,
Speaker 1: but no distinct account of it was printed until sixteen
Speaker 1: ninety three. The earliest use of the latter in the
Speaker 1: notebooks of Leibnitz is dated sixteen seventy five, and was
Speaker 1: employed in the letters sent to Newton in sixteen seventy seven,
Speaker 1: and an account of it was printed in the Memoir
Speaker 1: of sixteen eighty four described below. There is no question
Speaker 1: that the differential notation is due to Libanates, and that
Speaker 1: the sole question is as to whether the general idea
Speaker 1: of the calculus was taken from Newton or discovered independently.
Speaker 1: The case in favor of the independent invention by Libnates
Speaker 1: rests on the ground that he published a description of
Speaker 1: his methods some years before Newton printed anything on fluctions,
Speaker 1: that he always alluded to the discovery as being his
Speaker 1: own invention, and that for many years this statement was unchallenged,
Speaker 1: While of course there must be a strong presumption that
Speaker 1: he acted in good faith to Rebut this case, it
Speaker 1: is necessary to shew that one he saw some of
Speaker 1: Newton's papers on the subject in or before sixteen seventy five,
Speaker 1: or at least sixteen seventy seven, and two that E
Speaker 1: thence derived the fundamental ideas of the calculus. The fact
Speaker 1: that his claim was unchallenged for some years, in my opinion,
Speaker 1: in the particular circumstances of the case, is immaterial. That
Speaker 1: Leibnitz saw some of Newton's manuscripts was always intrinsically probable.
Speaker 1: But when in eighteen forty nine C. J. Gerhardt examined
Speaker 1: Leibnitz's papers, he found among them a manuscript copy, the
Speaker 1: existence of which had been previously unsuspected, in Leibanez's handwriting
Speaker 1: of extracts from Newton's Di Analisi periquezion is numero timenurum infinitas,
Speaker 1: which was printed in the Di Quatrata Corverum in seventeen
Speaker 1: oh four, together with notes on their expression in the
Speaker 1: differential notation. The question of the date at which these
Speaker 1: extracts were made is therefore all important. It is known
Speaker 1: that a copy of Newton's manuscript had been sent to
Speaker 1: Chernhausen in May sixteen seventy five, and that in that
Speaker 1: year he and Leibnitz were engaged together in a piece
Speaker 1: of work. It is not impossible that these extracts were
Speaker 1: made then. It is also possible that they may have
Speaker 1: been made in sixteen seventy six, for Leibnitz discuss the
Speaker 1: question of analysis by infinite series with Collins and Oldenburg
Speaker 1: in that year, and it is a priori probable that
Speaker 1: they would have then shewn him the manuscript of Newton
Speaker 1: on that subject, a copy of which was possessed by
Speaker 1: one or both of them. On the other hand, it
Speaker 1: may have been supposed that Leibanetz made the extracts from
Speaker 1: the printed copy in or after seventeen o four. Libanutz,
Speaker 1: shortly before his death, admitted in a letter to Conti
Speaker 1: that that in sixteen seventy six Collins had shown him
Speaker 1: some Newtonian papers, but implied that they were of little
Speaker 1: or no value. Presumably he referred to Newton's letters of
Speaker 1: June thirteenth and October twenty fourth, sixteen seventy six, and
Speaker 1: to the letter of December tenth, sixteen seventy two, on
Speaker 1: the method of tangents extracts from which accompanied the letter
Speaker 1: of June thirteenth. But it was curious that on the
Speaker 1: receipt of these letters Lybanuts should have made no further
Speaker 1: inquiries unless he was already aware from other sources of
Speaker 1: the method followed by Newton. Whether Leibanetz made no use
Speaker 1: of the manuscript from which he had copied extracts, or
Speaker 1: whether he had previously invented the calculus, are questions on which,
Speaker 1: at this distance of time, no direct evidence is available.
Speaker 1: It is, however, worth noting that the unpublished Portsmouth papers
Speaker 1: schue that when in seventeen eleven Newton went carefully into
Speaker 1: the whole dispute, he picked out this manuscript as the
Speaker 1: one which had probably somehow fallen into the hands of Leibniz.
Speaker 1: At that time, there was no direct evidence that Leibnitz
Speaker 1: had seen this manuscript before it was printed in seventeen
Speaker 1: o four, and accordingly Newton's conjecture was not published. But
Speaker 1: Gerhardt's discovery of the copy made by Leibanez tends to
Speaker 1: confirm the accuracy of Newton's judgment in the matter. It
Speaker 1: is said by some that to a man of Leibnitz's ability,
Speaker 1: the manuscript, especially if supplemented by the letter of December tenth,
Speaker 1: sixteen seventy two, would supply sufficient hints to give him
Speaker 1: a clue to the methods of the calculus, though as
Speaker 1: fluctional notation is not employed in it, anyone who used
Speaker 1: it would have to invent a notation, but this is
Speaker 1: denied by others. There was at first no reason to
Speaker 1: suspect the good faith of Leibnitz, and it was not
Speaker 1: until the appearance in seventeen o four of an anonymous
Speaker 1: review of Newton's Tract on Quadrature, in which it was
Speaker 1: implied that Newton had borrowed the idea of the flectional
Speaker 1: calculus from Leibnitz, that any responsible mathematician questioned the statement
Speaker 1: that Leibnitz had invented the calculus independently of Newton. It
Speaker 1: is universally admitted that there was no justification or authority
Speaker 1: for the statements made in this review, which was rightly
Speaker 1: attributed to Leibnitz, But the subsequent discussion led to a
Speaker 1: critical examination of the whole question, and doubt was expressed
Speaker 1: as to whether Leibanates had not derived the fundamental idea
Speaker 1: from Newton. The case against Libyanates, as it appeared to
Speaker 1: Newton's friends, was summed up in the Commercium Epistilicolum, issued
Speaker 1: in seventeen twelve. The evidence there collected may be inconclusive,
Speaker 1: but at any rate, detailed references are given for all
Speaker 1: facts mentioned. No such summary with fact states and references
Speaker 1: of the case for Leibniz was issued by his friends,
Speaker 1: But John Berneulli attempted to indirectly weaken the evidence by
Speaker 1: attacking the personal character of Newton. This was in a
Speaker 1: letter dated June seventh, seventeen thirteen. The charges were false,
Speaker 1: and when pressed for an explanation of them, Bernoulli most
Speaker 1: solemnly denied having written the letter. In accepting the denial,
Speaker 1: Newton added in a private letter to him the following remarks,
Speaker 1: which are interesting as giving Newton's account of why he
Speaker 1: was last induced to take any part in the controversy.
Speaker 1: I have never said he grasped at the fame among
Speaker 1: foreign nations. But I am very desirous to preserve my
Speaker 1: character for honesty for which the author of that epistle,
Speaker 1: as if by the authority of a great judge, has
Speaker 1: endeavored to wrest from me. Now that I am old
Speaker 1: I have little pleasure in mathematical studies, and I have
Speaker 1: never tried to propagate my opinions over the world. But
Speaker 1: I have rather taken care not to involve myself in
Speaker 1: disputes on account of them. Leibnitz's defense or explanation of
Speaker 1: his silence is given in the following letter, dated April nineteenth,
Speaker 1: seventeen sixteen, from him to Conti poor ris pondre de
Speaker 1: point a point aluvreche pour bles, contremis il fulois, un
Speaker 1: ottre jos souis grand pro le, moile sluillo, il ferois,
Speaker 1: entre danze grand details de conte te de minis, passe ille,
Speaker 1: trene carn don jeon, meer sauvignois, guerre, il me felix
Speaker 1: cherche mer vielle lettre d'ant plusier ser sant perdu utre cole, pleu,
Speaker 1: su vange, pointe garde, le mignet, demens, eluzotto, sona salvier,
Speaker 1: don zun grand has pepier qujeune, pavois de bourier, cuevic
Speaker 1: dutamp e de la pacience, mei gen en avois, guerre,
Speaker 1: loisier etints charges presentmento capassion juneutres otto natur. The death
Speaker 1: of Leibyanitz in seventeen sixteen only put a temporary stop
Speaker 1: to the controversy, which was bitterly debated for many years later.
Speaker 1: The question is one of great difficulty. The evidence is
Speaker 1: conflicting and circumstantial, and everyone must form for themselves the
Speaker 1: opinion which seems most probable. I think the majority of
Speaker 1: modern writers would accept the view that probably Leibane's invention
Speaker 1: of the calculus was independent of that of Newton, and
Speaker 1: everyone will hope that they are right. For myself, I cannot, however,
Speaker 1: but think it probable that Leibyanez read Newton's manuscript d
Speaker 1: Analsi before sixteen seventy seven, and was materially assisted by it.
Speaker 1: His unacknowledged possession of a copy of part of one
Speaker 1: of Newton's manuscripts may be explicable. But the admitted fact
Speaker 1: that on more than one occasion he deliberately altered or
Speaker 1: added to important documents before publishing them seems to me
Speaker 1: to make his own testimony of little value. In mitigation
Speaker 1: of his conduct, I can only say that it must
Speaker 1: be recollected that what he is alleged to have received
Speaker 1: was rather a series of hints than an account of
Speaker 1: the calculus. And it seems to me that the facts
Speaker 1: that he did not publish his results of sixteen seventy
Speaker 1: seven until sixteen eighty four, and that the notation and
Speaker 1: subsequent development of it were all of his own invention,
Speaker 1: may have led him thirty years later to minimize any
Speaker 1: assistance which he obtained originally, and finally, consider that it
Speaker 1: was immaterial. If we must confine in ourselves to one
Speaker 1: system of notation, then there can be no doubt that
Speaker 1: that which was invented by Leibniz is better fitted for
Speaker 1: most of the purposes to which the infinitesimal calculus is
Speaker 1: applied than that of fluxions, and for some as such
Speaker 1: as the calculus of variations, it is indeed almost essential.
Speaker 1: It should be remembered, however, that at the beginning of
Speaker 1: the eighteenth century, the methods of the infinitesimal calculus had
Speaker 1: not been systematized, and either notation was equally good. The
Speaker 1: development of that calculus was the main work of the
Speaker 1: mathematician of the first half of the eighteenth century. The
Speaker 1: differential form was adopted by continental mathematicians, the application of
Speaker 1: it by Euler, Lagrange and Laplace to the principles of
Speaker 1: mechanics laid down in the Principia was the great achievement
Speaker 1: of the last half of that century, and finally demonstrated
Speaker 1: the superiority of the differential to the Fluck calculus. The
Speaker 1: translation of the Principia into the language of modern analysis,
Speaker 1: and the filling in of the details of the Newtonian
Speaker 1: theory by the aid of that analysis, were affected by Laplace.
Speaker 1: The controversy with Leibniz was regarded in England as an
Speaker 1: attempt by foreigners to defraud Newton of the credit of
Speaker 1: his invention, and the question was complicated on both sides
Speaker 1: by national jealousies. It was therefore natural, though it was
Speaker 1: unfortunate that in England the geometrical and fluctional methods as
Speaker 1: used by Newton were alone studied and employed for more
Speaker 1: than a century. The English school was thus out of
Speaker 1: touch with continental mathematicians. The consequence was that, in spite
Speaker 1: of the brilliant band of scholars formed by Newton, the
Speaker 1: improvements in the methods of analysis gradually effected on the
Speaker 1: continent were almost unknown in Britain. It was not until
Speaker 1: eighteen twenty that the value of an Olye methods was
Speaker 1: fully recognized in England, and that Newton's countrymen again took
Speaker 1: any large share in the development of mathematics. Leaving now
Speaker 1: this long controversy, I come to the discussion of the
Speaker 1: mathematical papers produced by Libanitz, all the more important of
Speaker 1: which were published in the Acta Eruditorum. They are mainly
Speaker 1: concerned with the applications of the infinitesimal calculus and with
Speaker 1: various questions on mechanics. The only papers of first rate
Speaker 1: importance which he produced are those on the differential calculus.
Speaker 1: The earliest of these was one published in the Acta
Speaker 1: Eruditorum for October sixteen eighty four, in which he enunciated
Speaker 1: a general method for finding maxima and minima and for
Speaker 1: drawing tangents to curves. One inverse problem, namely to find
Speaker 1: the curve whose subtangent is constant, was also discussed. The
Speaker 1: notation is the same as that with which we are familiar,
Speaker 1: and the differential coefficients of x to the power N
Speaker 1: and of products and quotients are determined. In sixteen eighty
Speaker 1: six he wrote a paper on the principles of the
Speaker 1: new calculus. In both of these papers, the principle of
Speaker 1: continuity is explicitly assumed, while his treatment of the subject
Speaker 1: is based on the use of infinitesimals and not on
Speaker 1: that of the limiting value of ratios. In answer to
Speaker 1: some objections which are raised in sixteen ninety four by
Speaker 1: Bernard Neuantite, who asserted that d y by dx stood
Speaker 1: for an unmeaning quantity like zero over zero, Leibnitz explained,
Speaker 1: in the same way as Barrow had previously done, that
Speaker 1: the value of d y by dx in geometry could
Speaker 1: be expressed as the ratio of two finite quantities. I
Speaker 1: think leibnitz a statement of the objects and methods of
Speaker 1: the infinitesimal calculus, as can contained in these papers, which
Speaker 1: are the three most important memoirs on it, that he
Speaker 1: had produced, is somewhat obscure, and his attempt to place
Speaker 1: the subject on a metaphysical basis did not tend to clearness.
Speaker 1: But the notation he introduced is superior to that of Newton,
Speaker 1: and the fact that all the results of modern mathematics
Speaker 1: are expressed in the language invented by Leibnitz has proved
Speaker 1: to be the best monument of his work. In sixteen
Speaker 1: eighty six, and sixteen ninety two he wrote papers on
Speaker 1: osculating curves. These, however, contained some bad blunders, as for example,
Speaker 1: the assertion that an osculating circle will necessarily cut a
Speaker 1: curve in four consecutive points. This error was pointed out
Speaker 1: by John Bernoulli, but in his article of sixteen ninety
Speaker 1: two Leibnitz defended his original assertion and insisted that a
Speaker 1: circle could never cross a curve it touched it. In
Speaker 1: sixteen ninety two, Leibnitz wrote a memoir and which he
Speaker 1: laid the foundation of the theory of envelopes. This was
Speaker 1: further developed in another paper in sixteen ninety four, in
Speaker 1: which he introduced for the first time the terms coordinates
Speaker 1: and axes of coordinates. Leibnitz also published a good many
Speaker 1: papers on mechanical subjects, but some of them contain mistakes
Speaker 1: which shew that he did not understand the principles of
Speaker 1: the subject. Thus, in sixteen eighty five he wrote a
Speaker 1: memoir to find the pressure exerted by a sphere of
Speaker 1: wight w placed between two inclined planes of complimentary inclinations,
Speaker 1: placed so that the lines of the greatest slope are
Speaker 1: perpendicular to the line of the intersection of the plains.
Speaker 1: He asserted that the pressure on each plane must consist
Speaker 1: of two components unum coot di crevetere di santre tensit
Speaker 1: al teremko planum di crive premith. He further said that
Speaker 1: for metaphysical reasons, that the sum of the two pressures
Speaker 1: must be equal to W. Hence, if R and r
Speaker 1: prime be the required pressures, and alpha and one half
Speaker 1: prime minus alpha the inclination of the planes, he finds
Speaker 1: that R minus one half W multiplied by the quantity
Speaker 1: one minus sign alpha plus cos alpha, and our prime
Speaker 1: equals one half W multiplied by one minus coas alpha
Speaker 1: plus sign alpha. The true values are R equals w
Speaker 1: cos alpha and r prime equals w sign alpha. Nevertheless,
Speaker 1: some of his papers on mechanics are valuable. Of these,
Speaker 1: the most important were two in sixteen eighty nine and
Speaker 1: sixteen ninety four, in which he solved the problem of
Speaker 1: finding an isochronous curve one in sixteen ninety seven on
Speaker 1: the curve of quickest descent. This was the problem sent
Speaker 1: as a challenge to Newton, and two in sixteen ninety
Speaker 1: one and sixteen ninety two, in which he stated the
Speaker 1: intrinsic equation of the curve assumed by a flexible rope
Speaker 1: suspended from two points i e. The caternary, but gave
Speaker 1: no proof. This last problem had been originally proposed by Galileo.
Speaker 1: In sixteen eighty nine, that as two years after the
Speaker 1: Principia has been published, he wrote on the movements of
Speaker 1: the planets, which he stated were produced by a motion
Speaker 1: of the ether. Not only were the equations of motion
Speaker 1: which he obtained wrong, but his deductions from them were
Speaker 1: not even in accordance with his own axioms. In another
Speaker 1: memoir of in seventeen o six, that is, nearly twenty
Speaker 1: years after the Principia had been written, he admitted that
Speaker 1: he had made some mistakes in his former paper, but
Speaker 1: adhered to his previous conclusions and some of the matter
Speaker 1: up by saying, it is certain that gravitation generates a
Speaker 1: new force at each instant to the center, but the
Speaker 1: centrifugal force also generates another away from the center. The
Speaker 1: centrifugal force may be considered in two aspects, according as
Speaker 1: the movement is treated as along the tangent to the curve,
Speaker 1: or as along the arc of the circle. Itself. It
Speaker 1: seems clear from this paper that he did not really
Speaker 1: understand the manner in which Newton had reduced the dynamics
Speaker 1: to an exact science. It is hardly necessary to consider
Speaker 1: his work on dynamics in further detail. Much of it
Speaker 1: is vitiated by a constant confusion between momentum and kinetic energy.
Speaker 1: When the force is passive, he uses the first, which
Speaker 1: he calls his vis mortua, as the measure of force.
Speaker 1: When the force is active, he uses the latter, the
Speaker 1: double of which he calls the vizviva. The series quoted
Speaker 1: by Leibnitz comprise those of E to the X and
Speaker 1: log of one plus x, sign of x, verse sign
Speaker 1: of x, and the arc tangent of x. All of
Speaker 1: these have been previously published, and he rarely, if ever
Speaker 1: added any demonstrations. Leibneates, like Newton, recognize the importance of
Speaker 1: James Gregory's remarks on the necessity of examining whether infinite
Speaker 1: series are convergent or divergent, and proposed a test to
Speaker 1: distinguish the series whose terms are alternatively positive and negative.
Speaker 1: In sixteen ninety three, he explained the method of expansion
Speaker 1: by indetermined coefficients though his applications were not free from error.
Speaker 1: To sum the matter up briefly, it seems to me
Speaker 1: that Leibnitz's work exhibits great skill in analysis, but much
Speaker 1: of it is unfinished, and when he leaves his symbols
Speaker 1: and attempts to interpret his results, he frequently commits blunders.
Speaker 1: No doubt, the demands of politics, philosophy, and literature on
Speaker 1: his time may have prevented him from elaborating any scientific
Speaker 1: subject completely or writing any systematic exposition of his views,
Speaker 1: though they are no excuse for the mistakes of principle
Speaker 1: which occur so frequently in his papers. Some of his
Speaker 1: memoirs contain suggestions of methods which have now become valuable
Speaker 1: means of analysis, such as the use of determinants and
Speaker 1: of indeterminate coefficients. But when a writer of many fold
Speaker 1: interests like Leibanis throws out innumerable suggestions, some of them
Speaker 1: are likely to turn out valuable, and to enumerate these
Speaker 1: which he never worked out without reckoning the others which
Speaker 1: are wrong gives a false impression of the value of
Speaker 1: his work. But in spite of this, his title to
Speaker 1: fame rests on a sure basis, for it was he
Speaker 1: who brought the differential calculus into general use, and his
Speaker 1: name is inseparably connected with one of the chief instruments
Speaker 1: of analysis, just as that of Descartes, another philosopher, is
Speaker 1: with analytical geometry. Leibniz was only one amongst several continental
Speaker 1: writers whose papers in the Acta Eriditorum familiarize mathematicians with
Speaker 1: the use of differential calculus. The most important of these
Speaker 1: were James and John Bernoulli, both of whom were farm
Speaker 1: friends and admirers of Leibanitz, and to their devoted advocacy
Speaker 1: his reputation is largely due. Not only did they take
Speaker 1: a prominent part in nearly every mathematical question they discussed,
Speaker 1: but nearly all the leading mathematicians on the continent for
Speaker 1: the first half of the eighteenth century came directly or
Speaker 1: indirectly under the influence of one or both of them.
Speaker 1: The Bernoullis were a family of Dutch origin who were
Speaker 1: driven from Holland by the Spanish persecutions and finally settled
Speaker 1: at Bail in Switzerland. The first member of the family
Speaker 1: who attained any marked distinction in mathematics was James James Bernoulli,
Speaker 1: jacub or James Bernoulli was born at Bail in December
Speaker 1: twenty seventh, sixteen fifty four. In sixteen eighty seven he
Speaker 1: was appointed to a chair of mathematics in the university there,
Speaker 1: and occupied it until his death on August sixteenth, seventeen
Speaker 1: o five. He was one of the earliest to realize
Speaker 1: how powerful as an instrument of analysis was the infinitesimal calculus,
Speaker 1: and he applied it to several problems, but he did
Speaker 1: not himself invent any new processes. His great influence was
Speaker 1: uniformly and successfully exerted in favor of the use of
Speaker 1: differential calculus, and his lessons on it, which were written
Speaker 1: in the form of two essays in sixteen ninety one
Speaker 1: and are published in Volume two of his works, shew
Speaker 1: how completely he had even then grasped the principles of
Speaker 1: the new analysis. These lectures, which contain the earliest use
Speaker 1: of the term integral, were the first published attempt to
Speaker 1: construct an integral calculus, for Leibnitz had treated each problem
Speaker 1: by itself and had not laid down any general rules
Speaker 1: on the subject. The most important discoveries of James Bernoulli
Speaker 1: were his solution of the problem to find a n
Speaker 1: isochronous curve, proof that the construction for the catternary which
Speaker 1: had been given by Leibnitz was correct, and his extension
Speaker 1: of this to strings of variable density under a central force.
Speaker 1: His determination of the form had taken by an elastic
Speaker 1: rod fixed at one end and acted on by a
Speaker 1: given force in the other the elastica also of a
Speaker 1: flexible rectangular sheet with two sides fixed horizontally and filled
Speaker 1: with a heavy liquid the lintearia, and lastly of a
Speaker 1: sail filled with wind the valeria. In sixteen ninety six
Speaker 1: he offered a reward for the general solution of isoperimetrical figures,
Speaker 1: ie the determination of a figure of a given species
Speaker 1: which should include a maximum area its perimeter being given.
Speaker 1: His own solution, published in seventeen oh one, is correct
Speaker 1: as far as it goes. In sixteen ninety eight he
Speaker 1: published an essay on the differential calculus and his applications
Speaker 1: to geometry. He here investigated the chief properties of the
Speaker 1: U s equiangular spiral, and especially noticed the manner in
Speaker 1: which the various curves deduced from it reproduced the original curve.
Speaker 1: Struck by this fact, he begged that in imitation of Archimedes.
Speaker 1: An equiangular spiral should be engraved on his tombstone with
Speaker 1: the inscription iyadem numero mutata resurgo. He also brought out
Speaker 1: in sixteen ninety five an edition of Descartes geometri. In
Speaker 1: his Arskojectandi, published in seventeen thirteen, he established the fundamental
Speaker 1: principles of the calculus of probabilities. In the course of
Speaker 1: the work, he defined the numbers known by his name
Speaker 1: and explained their use, and he also gave some theorems
Speaker 1: on finite differences. His higher lectures were mostly on the
Speaker 1: theory of series. These were published by Nicholas Bernoulli in
Speaker 1: seventeen thirteen. John Bernoulli Johann Bernoulli, the brother of James Bernoulli,
Speaker 1: was born at Bail on August seventh, sixteen sixty seven,
Speaker 1: and died there on January first, seventeen forty eight. He
Speaker 1: occupied the chair of mathematics at Groningen from sixteen ninety
Speaker 1: five to seventeen o five and at Bail, where he
Speaker 1: succeeded his brother from seventeen o five to seventeen forty eight.
Speaker 1: To Hall. He did not acknowledge his merits in a
Speaker 1: manner commensurate with his own view of their importance. He
Speaker 1: behaved most unjustly. As an illustration of his character, it
Speaker 1: may be mentioned that he attempted to substitute for an
Speaker 1: incorrect solution of his own on isoparametrical curves anothers stolen
Speaker 1: from his brother James, while he expelled his son Daniel
Speaker 1: from his house for obtaining a prize from the French
Speaker 1: Academy which he had expected to receive himself. After the
Speaker 1: deaths of Leibaniz and Lopital, he claimed the merit of
Speaker 1: some of their discoveries. These claims are now known to
Speaker 1: be false. He was, however, the most successful teacher of
Speaker 1: his aige, and had the faculty of inspiring his pupils
Speaker 1: with almost as passionate a zeal for mathematics as he
Speaker 1: felt for himself. The general adoption on the continent of
Speaker 1: the differential rather than the fluctional notation was largely due
Speaker 1: to his influence, Leaving out of account his innumerable controversies.
Speaker 1: The chief discoveries of John Bernoulli were the exponential calculus,
Speaker 1: the treatment of trigonometry as a branch of analysis, the
Speaker 1: conditions for a geodesic, the determination of orthogonal trajectories, the
Speaker 1: solution of the rachistochrone, and the statement that a ray
Speaker 1: of light traversed such a path that sigma MUDs was
Speaker 1: a minimum, and the enunciation of the principle of virtual work.
Speaker 1: I believe that he was the first to denote the
Speaker 1: accelerating effect of gravity by an algebraical sign G, and
Speaker 1: he thus arrived at the formula V squared equals to
Speaker 1: g h. The same result would have been previous expressed
Speaker 1: by the proportion V one squared is to V two squared,
Speaker 1: as H one is to H two. The notation phi
Speaker 1: x to indicate a function of x was introduced by
Speaker 1: him in seventeen eighteen and displace the notation chai or
Speaker 1: zi proposed by him in sixteen ninety eight. But the
Speaker 1: general adoption of symbols like little F, big F, phi,
Speaker 1: and psi to represent functions seem to be mainly due
Speaker 1: to Euler and Lagrange. Several members of the same family,
Speaker 1: but of a younger generation, enriched mathematics by their teachings
Speaker 1: and writings. The most important of these were the three
Speaker 1: sons of John, namely Nicholas, Daniel, and John the Younger,
Speaker 1: and the two sons of John the Younger, who bore
Speaker 1: the names of John and James. To make the account complete,
Speaker 1: I adhere their respective dates. Nicholas Bernoulli, the eldest of
Speaker 1: the three sons of John, was born on January twenty seventh,
Speaker 1: sixteen ninety five, and was drowned at Saint Petersburg, where
Speaker 1: he was professor, on July twenty six, seventeen six. Daniel Bernoulliy,
Speaker 1: the second son of John, was born on February ninth,
Speaker 1: seventeen hundred and died on March seventeenth, seventeen eighty two.
Speaker 1: He was professor first at Saint Petersburg and afterwards at Bail,
Speaker 1: and shares with Euler the unique distinction of having gained
Speaker 1: the prize proposed annually by the French Academy no less
Speaker 1: than ten times. I refer to him again a few
Speaker 1: pages later. John Berneully the Younger, a brother of Nicholas
Speaker 1: and Daniel, was born on May eighteenth, seventeen ten and
Speaker 1: died in seventeen ninety. He was also a professor at Bail.
Speaker 1: He left two sons, John and James. Of these, the former,
Speaker 1: who was born on December fourth, seventeen forty four, and
Speaker 1: died on July tenth, eighteen o seven was Astronomer Royal
Speaker 1: and Director of mathematical Studies at Berlin, while the latter,
Speaker 1: who was born on October seventeenth, seventeen fifty nine and
Speaker 1: died in July seventeen eighty nine, was successively professor at bail,
Speaker 1: Verona and Saint Petersburg. End of section twenty seven. Recording
Speaker 1: by Paul King, Oakville, Ontario, p j K dot scripts
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