Speaker 1: Chapter seventeen, Part two of A short account of the
Speaker 1: History of mathematics. This is a LibriVox recording. All LibriVox
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Speaker 1: ed U forward slash p k J. A Short account
Speaker 1: of the history of mathematics by W. W. Rowse Ball,
Speaker 1: Chapter seventeen, Libinitz and the Mathematicians of the first half
Speaker 1: of the eighteenth century. Part two, the development of analysis
Speaker 1: on the continent. Leaving for a moment the English mathematicians
Speaker 1: of the first half of the eighteenth century, we come
Speaker 1: to a number of continental writers who barely escape mediocrity,
Speaker 1: and to whom it will be necessary to devote but
Speaker 1: few words. Their writings mark the steps by which analytics, geometry,
Speaker 1: and the differential and integral calculus were perfected and made
Speaker 1: familiar to mathematicians. Nearly all of them were pupils of
Speaker 1: one or other of the two elder Bernoullis, and they
Speaker 1: were so nearly contemporaries that it is difficult to arrange
Speaker 1: them chronologically. The most eminent of them are Cramer, Degois
Speaker 1: de Montemret, Fagagnon Lo Pital, NiCoT, parent Riccouti, Sarin and
Speaker 1: Verignon lo Pital, Guillon, Francois Antoine lo Pital. Marquis de
Speaker 1: Saint Mes born at Paris in sixteen sixty one and
Speaker 1: died there on February second, seventeen o four, was among
Speaker 1: the earliest pupils of John Bernoulli, who in sixteen ninety
Speaker 1: one spent some months at Lopital's house in Paris for
Speaker 1: the purpose of teaching them the new calculus. It seems strange,
Speaker 1: but it is substantial true that a knowledge of the
Speaker 1: infinitesimal calculus and the power of using it was then
Speaker 1: confined to Newton, Leibnitz and the two elder Bernoulli's, and
Speaker 1: it will be noticed that they were the only mathematicians
Speaker 1: who solve the more difficult problems then proposed as challenges.
Speaker 1: There was at the time no textbook on the subject,
Speaker 1: and the credit of putting together the first treatise, which
Speaker 1: explained the principles and use of the method, is due
Speaker 1: to Lopitale. It was published in sixteen ninety six under
Speaker 1: the title Annelis des infinitemon petit. This contains a partial
Speaker 1: investigation of the limiting value of the ratio of functions which,
Speaker 1: for a certain value of the variable take the indeterminate
Speaker 1: form zero over zero, a problem solved by John Bernoulli
Speaker 1: in seventeen o four. This work had a wide circulation.
Speaker 1: It brought the differential notation into universally use in France
Speaker 1: and helped to make it generally known in Europe. A
Speaker 1: supplement containing a similar treatment of the integral calculus, together
Speaker 1: with the editions of the differential calculus which had been
Speaker 1: made in the following half century, was published at Paris
Speaker 1: seventeen fifty four to seventeen fifty six by L. A.
Speaker 1: De Bourganville. Lopitale took part in most of the challenges
Speaker 1: by Leibniz, the Bernoullis, and other continental mathematicians of the time.
Speaker 1: In particular, he gave a solution of the burchistochrone and
Speaker 1: investigated the form of the solid of least resistance of
Speaker 1: which Newton and the Principia had stated the result. He
Speaker 1: also wrote a treatise on analytical conics, which was published
Speaker 1: in seventeen o seven and for nearly a century was
Speaker 1: deemed a standard work. On the subject. Verignon Pierre Verignon
Speaker 1: born at Caine in sixteen fifty four and died in
Speaker 1: Paris on dad December twenty second, seventeen twenty two was
Speaker 1: an intimate friend of Newton, Leibniz and the Bernoullis, and
Speaker 1: after Lo Pitale, was the earliest and most powerful advocate
Speaker 1: in France of the use of the differential calculus. He
Speaker 1: realized the necessity of obtaining a test for examining the
Speaker 1: convergency of series, but the analytical difficulties were beyond his powers.
Speaker 1: He simplified the proofs of many of the leading propositions
Speaker 1: in mechanics, and in sixteen eighty seven recast the treatment
Speaker 1: of the subject, basing it on the composition of forces.
Speaker 1: His works were published at Paris in seventeen twenty five.
Speaker 1: For further details see the erouge by b. De Fontinelles, Paris,
Speaker 1: seventeen sixty six. Des Montemur Pierre Raymond de Montemur born
Speaker 1: at Paris on October twenty seventh, sixteen seventy eight and
Speaker 1: died there on October seventh, seventeen nineteen was interested in
Speaker 1: the subject of finite differences. He determined in seventeen thirteen,
Speaker 1: the sum of n terms of a finite series of
Speaker 1: the form en choose one times a plus, en choose
Speaker 1: two times delta a plus, en choose three times delta
Speaker 1: squared day, and so on, a theorem which seems to
Speaker 1: have been independently rediscovered by Christian Goldbach in seventeen eighteen.
Speaker 1: Nicole Francois Nicole, who was born at Paris on December
Speaker 1: twenty third, sixteen eighty three and died there on January eighteenth,
Speaker 1: seventeen fifty eight, was the first to publish a systematic
Speaker 1: treatise on the finite differences. Taylor had regarded the differential
Speaker 1: coefficient as the ratio of two infinitesimal differences as the
Speaker 1: limiting value of the ratio of two finite differences, a
Speaker 1: method which is still used by many English writers, though
Speaker 1: it has been generally abandoned on the DVD continent, and
Speaker 1: thus had been led to give a sketch of the
Speaker 1: subject in his Methodists, published in seventeen fifteen. Nicoles Treete
Speaker 1: de Quercuur de di France Finice was published in seventeen seventeen.
Speaker 1: It is a well arranged book and contains rules both
Speaker 1: for forming differences and for affecting the summation of given series.
Speaker 1: Besides this, in seventeen o six he wrote a book
Speaker 1: on Roulette's especially spherical epicycloids, and in seventeen twenty nine
Speaker 1: and seventeen thirty one he published memoirs on Newton's essay
Speaker 1: on curves of the third degree. Parents Antoine Parents born
Speaker 1: at Paris on September sixteenth, sixteen sixty six, and died
Speaker 1: there on September twenty six, seventeen sixteen. Wrote in seventeen
Speaker 1: hundred on analytical geometry of three dimensions. His works were
Speaker 1: collected and published in three volumes at Paris in Sea
Speaker 1: seventeen thirteen. Sarrans Joseph Sarran, born at Corteizan in sixteen
Speaker 1: fifty nine and died at Paris on December twenty ninth,
Speaker 1: seventeen thirty seven, was the first to show how the
Speaker 1: tangents at the multiple points of curves could be determined
Speaker 1: by analysis. De Nois Jean Paul de GUIs d'ar Mauves
Speaker 1: was born at Carcasson in seventeen thirteen and died at
Speaker 1: Paris on June second, seventeen eighty five. He published in
Speaker 1: seventeen forty a work on analytical geometry, in which he
Speaker 1: applied it without the aid of the differential calculus, to
Speaker 1: find the tangents, asymptotes, and various singular points of an
Speaker 1: algebraical curve, and he further shoed how singular points and
Speaker 1: isolated loops were affected by conical projection. He gave the
Speaker 1: proof of Descartes's rule of signs, which is to be
Speaker 1: found in most modern works. It is not clear whether
Speaker 1: Descartes ever proved its own strictly, and Newton seems to
Speaker 1: have regarded it as obvious. Kramer Gabrielle Kramer born at
Speaker 1: Geneva in seventeen o four and died at Bagnol in
Speaker 1: seventeen fifty two, was professor at Geneva. The work by
Speaker 1: which he is best known is his Treatise on Algebraical Curves,
Speaker 1: published in seventeen fifty, which, as far as it goes,
Speaker 1: is fairly complete. It contains the earliest demonstration that a
Speaker 1: curve of the nth degree is in general determined if
Speaker 1: one half end times en plus three points on it
Speaker 1: to be given. This work is still sometimes read. Besides this,
Speaker 1: he edited that the works of the two elder Bernoullis,
Speaker 1: and wrote on the physical cause of the spheroidal shapes
Speaker 1: of the planets and the motion of their apses seventeen
Speaker 1: thirty and on Newton's treatment of cubic curves seventeen forty six. Riccati,
Speaker 1: Jacopo Francesco Count Riccati, born at Venice on May twenty eighth,
Speaker 1: sixteen seventy six and died at Treves on April fifteenth,
Speaker 1: seventeen fifty four, did a great deal to disseminate a
Speaker 1: knowledge of the Newtonian philosophy in Italy. Besides the equation
Speaker 1: known by his name, certain cases of which he succeeded
Speaker 1: in integrating, he discussed the question of the possibility of
Speaker 1: lowering the order of a given differential equation. His works
Speaker 1: were published at Treves in four volumes in seventeen fifty eight.
Speaker 1: He had two sons who wrote on several minor points
Speaker 1: connected with the integral calculus and differential equations, and applied
Speaker 1: the calculus to several mechanical questions. These were Vincenzo, who
Speaker 1: was born in seventeen o seven and died in seventeen
Speaker 1: seventy five, and Giordano who was born in seventeen o
Speaker 1: nine and died in seventeen ninety. Fagnano Giulio Carlo, Count
Speaker 1: Fagnano and Marquise de Toci, born at Senegaglia on December sixth,
Speaker 1: sixteen eighty two and died on September twenty sixth, seventeen
Speaker 1: sixty six, may be said to have been the first
Speaker 1: writer who directed attention to the theory of elliptic functions.
Speaker 1: Failing to rectify the ellipse or hyperbolea, Faniano attempted to
Speaker 1: determine arcs whose difference should be rectifiable. He also pointed
Speaker 1: out the remarkable analogy existing between the integrals which represent
Speaker 1: the arc of a circle and the arc of a
Speaker 1: Lemnas gate. Finally, he proved the formula pie equals two
Speaker 1: times i times the log of one minus i divided
Speaker 1: by one plus i, where I stands for the square
Speaker 1: root of minus one. His works were collected and published
Speaker 1: in two volumes in Persero in seventeen fifty. It was
Speaker 1: inevitable that some mathematicians should object to methods of analysis
Speaker 1: founded on the infinitesimal calcis. The most prominent of these
Speaker 1: were Viviani de la Here and roll chronologically they come here,
Speaker 1: but they flourished half a century after the date to
Speaker 1: which their writings properly belong. Viviani Vincenzo Viviani, a pupil
Speaker 1: of Galileo and Torricelli, born at Florence on April fifth,
Speaker 1: sixteen twenty two, and died there on September twenty second.
Speaker 1: Seventeen o three, brought out in sixteen fifty nine a
Speaker 1: restoration of the lost Book of Apollonius on conic sections,
Speaker 1: and in seventeen oh one a restoration of the work
Speaker 1: of Erestius. He explained in sixteen seventy seven how an
Speaker 1: angle could be trisected by the aid of an echilateral
Speaker 1: hyperbola or the conchoid. In sixteen ninety two he proposed
Speaker 1: the problem to construct four windows in a hemispherical vault
Speaker 1: so that the remainder of the surface can be accurate determined,
Speaker 1: a celebrated problem of which analytical solutions were given by
Speaker 1: Wallace Leibnitz, David Gregory, and James Bernoulli. De la Here
Speaker 1: Philippe de la Here or Lahire born in Paris on
Speaker 1: March eighteen, sixteen forty and died there on April twenty first,
Speaker 1: seventeen nineteen, wrote on graphical methods sixteen seventy three, on
Speaker 1: the conic sections sixteen eighty five, a treatise on epicycloids
Speaker 1: sixteen ninety four, one on roulettes seventeen o two, and
Speaker 1: lastly another on conchoids seventeen oh eight. His works on
Speaker 1: conic sections and epicycloids were founded on the teaching of
Speaker 1: de Rogues, whose favorite pupil he was. He also translated
Speaker 1: the essay of Muscopolis on magic squares and collected many
Speaker 1: of the theorems on them which were previously known. This was
Speaker 1: published in seventeen oh five. Roule Michel Rol born at
Speaker 1: Ambert on April twenty first, sixteen fifty two and died
Speaker 1: in Paris on November eighth, seventeen nineteen. Wrote an algebra
Speaker 1: in sixteen eighty nine which contains the theorem on the
Speaker 1: position of roots of an equation, which is known by
Speaker 1: his name. He published in sixteen ninety six a treatise
Speaker 1: on the solution of equations whether determinate or indeterminate, and
Speaker 1: he produced several other minor works. He taught that differential
Speaker 1: calculus was nothing but a collection of ingenious fallacies so far,
Speaker 1: no one of the school of Leibnitz and the two
Speaker 1: elder Bernoullis had shewn any exceptional ability. But by the
Speaker 1: action of a number of second rate writers, the methods
Speaker 1: and language of analytical geometry and the differential calculus had
Speaker 1: become well known. By about seventeen forty. The close of
Speaker 1: this school is marked by the appearance of Cleirot de
Speaker 1: l'embert and Danielle Bernoulli. Their lives overlap the period considered
Speaker 1: in the next chapter. But though it is difficult to
Speaker 1: draw a sharp dividing line which shall separate by a
Speaker 1: definite date, the mathematicians there considered from those whose writings
Speaker 1: are discussed in this chapter, I think that on the whole,
Speaker 1: the works of these three writers are best treated here.
Speaker 1: Claiaut Alexis Claude Clat was born at Paris on May thirteenth,
Speaker 1: seventeen thirteen and died there on May seventeenth, seventeen sixty five.
Speaker 1: He belongs to the small group of children who, though
Speaker 1: of exceptional precaucity, survive and maintain their powers when grown up.
Speaker 1: As early as the age of twelve, he wrote a
Speaker 1: memoir on four geometrical curves, but his first important work
Speaker 1: was a treatise on tortuous curves, published when he was eighteen,
Speaker 1: a work which first procured for him immediate admission to
Speaker 1: the French Academy. In seventeen thirty one, he gave a
Speaker 1: demonstration of the fact noted by Newton that all curves
Speaker 1: of the third order were projections of one of five parabolas.
Speaker 1: In seventeen forty one Clairot went on a scientific expedition
Speaker 1: to measure the length of a meridian degree on the
Speaker 1: Earth's surface, and on his return in seventeen forty three
Speaker 1: he publishes THII de la figure de lataire. This is
Speaker 1: founded on a paper by Maclaurin, where it has been
Speaker 1: shown that a mass of homogeneous fluids set in rotation
Speaker 1: about a line through its center of mass, would, under
Speaker 1: the mutual attraction of its particles, take the form of
Speaker 1: a spheroid. This work of cliat treated of heterogeneous spheroids
Speaker 1: and contains the proof of his formula for the accelerating
Speaker 1: effect of gravity in a place of latitude L namely,
Speaker 1: G equals big g multiplied by one plus five halves
Speaker 1: m minus epsilon multiplied by sience squaredel where g is
Speaker 1: the value of equatorial gravity, m the ratio of the
Speaker 1: centrifugal force to gravity at the equator, and epsilon the
Speaker 1: epelicity of a meridian section of the Earth. In eighteen
Speaker 1: forty nine, Professor Stokes shewed that the same result was
Speaker 1: true whatever was the interior construction or density of the Earth,
Speaker 1: provided the surface was a spheroid of equilibrium of small
Speaker 1: ellipticity impressed by it the power of geometry, as shewn
Speaker 1: in the writing of Newton and McLaurin. Clarat abandoned analysis,
Speaker 1: and his next work, The Theore de l Lune, published
Speaker 1: in seventeen fifty two, is strictly Newtonian in character. This
Speaker 1: contains the explanation of the motion of the apse, which
Speaker 1: had previously puzzled astronomers, and which Clarat had at first
Speaker 1: deemed so inexplicable that he was on the point of
Speaker 1: publishing a new hype apothesis as to the law of
Speaker 1: attraction when it occurred to him to carry the approximation
Speaker 1: to the third order, and he thereupon found that the
Speaker 1: result was in accordance with observations. This was followed in
Speaker 1: seventeen fifty four by some lunar tables. Claiat subsequently wrote
Speaker 1: various papers on the orbit of the Moon and on
Speaker 1: the motion of the comets as affected by the perturbation
Speaker 1: of the planets, particularly on the path of Halley's comet.
Speaker 1: His growing popularity in society hindered his scientific work on Gaget,
Speaker 1: says beaussu a de supei a de viller entarnaier parent
Speaker 1: gudvive poulefame Voulain alais brasier, a stravau denaire a padire
Speaker 1: pou la sante enfan villellages de Saint Conden de Lambert.
Speaker 1: Jean de ren d' lambert was born at Paris on
Speaker 1: November sixteenth, seventeen seventeen, and died there on October twenty ninth,
Speaker 1: seventeen thirteen. He was the illegitimate child of the Chevalier Destouche,
Speaker 1: being abandoned by his mother on the steps of the
Speaker 1: little church of Saint Jean Leran, which then nestled under
Speaker 1: the great porch of nottal Dame, he was taken to
Speaker 1: the parish commissary, who, following the usual practice in such cases,
Speaker 1: gave him the Christian name of Jean laran I. Do
Speaker 1: not know by what title. He subsequently assumed the right
Speaker 1: to prefix DA to his name. He was boarded out
Speaker 1: by the parish with the wife of a glazier in
Speaker 1: a small way of business who lived near the cathedral,
Speaker 1: and here he seems to have found a real home,
Speaker 1: though a very humble one. His father appears to have
Speaker 1: looked after him and paid for his going to a
Speaker 1: school where he obtained a fair mathematical education. An essay
Speaker 1: written in seventeen thirty eight on the integral calculus, and
Speaker 1: another in seventeen forty on ducks and drakes or ricochets,
Speaker 1: attracted some attention, and in the same year he was
Speaker 1: elected a member of the French Academy. This was probably
Speaker 1: due to the influence of his father. It is to
Speaker 1: his credit that he absolutely refused to leave his adopted mother,
Speaker 1: with whom he continued to live until her death in
Speaker 1: seventeen fifty seven. It cannot be said that she sympathized
Speaker 1: with his success, for at the height of his fame
Speaker 1: she remonstrated with him for wasting his talents on such work.
Speaker 1: Vuniserreemeh kun Philissov said she e keskun Philissov Situnfu quissitour
Speaker 1: Monte padas Seville pour compard de luis LUs kilne serre plus.
Speaker 1: Nearly all his mathematical works were produced within the years
Speaker 1: seventeen forty three to seventeen fifty four. The first of
Speaker 1: these was his Treete de di neemik public in seventeen
Speaker 1: forty three, in which he enunciates the principle known by
Speaker 1: his name, namely that the internal forces of inertia i e.
Speaker 1: The forces which resist acceleration must be equal and opposite
Speaker 1: to the forces which produce the acceleration. This is a
Speaker 1: particular case of Newton's second reading of his third Law
Speaker 1: of motion, but the full consequence of it had not
Speaker 1: been realized previously. The application of this principle enables us
Speaker 1: to obtain the differential equations of motion of any rigid system.
Speaker 1: In seventeen forty four, Dalembert publishest t rete de EquiLibre
Speaker 1: in de movement de fluids, in which he applies his
Speaker 1: principle to fluids. This led to partial differential equations, which
Speaker 1: he was then unable to solve. In seventeen forty five
Speaker 1: he developed that part of the Subjects, which dealt with
Speaker 1: the motion of air in his Thera General de Van,
Speaker 1: and this again led him to partial difference equations. A
Speaker 1: second edition of this in seventeen forty six was dedicated
Speaker 1: to Frederick the Great of Prussia, and procured an invitation
Speaker 1: to Berlin and the offer of a pension. He declined
Speaker 1: the former, but subsequently, after some pressing, pocketed his pride
Speaker 1: and the latter. In seventeen forty seven he applied the
Speaker 1: differential calculus to the problem of a vibrating string, and
Speaker 1: again arrived at a partial differential equation. His analysis had
Speaker 1: three times brought him to an equation of the form
Speaker 1: the second derivative of you with respect to T equals
Speaker 1: the second derivative of U with respect to X, and
Speaker 1: he now succeeded in shewing that it was satisfied by
Speaker 1: U equals Fi of x plus T plus Sci of
Speaker 1: x minus t, Where Pi and Si are arbitrary functions.
Speaker 1: It may be interesting to give his solution, which was
Speaker 1: published in the Transactions of the Berlin Academy for seventeen
Speaker 1: forty seven. He begins by saying that if the partial
Speaker 1: derivative of U with respect to X be denoted by P,
Speaker 1: and the partial derivative of u with respect to t
Speaker 1: by q, then du equals pdx plus qdt. But by
Speaker 1: the given equation, the partial derivative of q with respect
Speaker 1: to T equals the partial derivative of p with respect
Speaker 1: to x, and therefore pdt plus qdx is also an
Speaker 1: exact differential. Denote it by DV. Therefore, dv equals pdt
Speaker 1: plus qdx. Hence du plus dv equals pdx plus qdt
Speaker 1: plus pdt plus qdx equals p plus q multiplied by
Speaker 1: dx plus dt, and du minus dv equals pdx plus
Speaker 1: qdt minus. The quantity pdt plus qdx equals the quantity
Speaker 1: p minus q multiplied the quantity d x minus d T.
Speaker 1: Thus U plus v must be a function of x
Speaker 1: plus t, and U minus v must be a function
Speaker 1: of x minus t. We may therefore put U plus
Speaker 1: v equals two times fi of x plus t, and
Speaker 1: U minus v equals to ci of x minus t,
Speaker 1: Hence U equals fi of x plus t plus ci
Speaker 1: of x minus t. De Lambert added that the conditions
Speaker 1: of the physical problem of a vibrating string demand that
Speaker 1: when x equals zero, U should vanish for all values
Speaker 1: of T, Hence, identically fi of T plus Si of
Speaker 1: negative T equals zero. Assuming that both functions can be
Speaker 1: expanded in integral powers of t, this requires that they
Speaker 1: should contain only odd powers. Hence si of negative T
Speaker 1: equals negative, Fi of T equals fi of negative T.
Speaker 1: Therefore U equals pi of x plus T plus fi
Speaker 1: of x minus t. Euler now took the matter up
Speaker 1: and shewed that the equation of the form of the
Speaker 1: string was the second derivative of U with respect to
Speaker 1: T equals a squared multiplied by the second derivative of
Speaker 1: U with respect to x, and that the general integral
Speaker 1: was U equals pi of x minus at plus pi
Speaker 1: of x plus at, where pi and pi are arbitrary functions.
Speaker 1: The chief remaining contributions of de Lambert to mathematics were
Speaker 1: on physical astronomy, especially on the precession of the equinoxes,
Speaker 1: and on variations of the obliquity of the ecliptic. These
Speaker 1: were collected on his Sistem des Mondes, published in three
Speaker 1: volumes in seventeen fifty four. During the latter part of
Speaker 1: his life he was mainly occupied with the d S
Speaker 1: Great French Encyclopedia. For this he wrote the introduction, and
Speaker 1: numerous philosophical and mathematical articles. The best are those on
Speaker 1: geometry and on probabilities. His style is brilliant but not polished,
Speaker 1: and faithfully reflects his character, which was bold, honest and frank.
Speaker 1: He defended a severe criticism which he had offered on
Speaker 1: some mediocre work by the remark Jeans musettin Seville Connuier,
Speaker 1: and with his dislike of sickophants and boors, it is
Speaker 1: not surprising that during his life he had more enemies
Speaker 1: than friends. End of section twenty eight. Recording by Paul King, Oakville, Ontario,
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