Speaker 1: Chapter eighteen, 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: reading by Paul King p J K dot scrips dot
Speaker 1: m I T dot E d U forward slash p
Speaker 1: K J. A Short Account of the History of mathematics
Speaker 1: by W. W. Rouse Baul. Chapter eighteen Leibniz and the
Speaker 1: Mathematicians of the first half of the eighteenth century, Part two.
Speaker 1: Lambert Johann Heinrich Lambert was born at Mulhausen on August
Speaker 1: twenty eighth, seventeen twenty eight and died at Berlin on
Speaker 1: September twenty fifth, seventeen seventy seven. He was the son
Speaker 1: of a small tailor and had to rely on his
Speaker 1: own efforts for his education. From a clerk in some
Speaker 1: iron works, he got a place in a newspaper office,
Speaker 1: and subsequently, on the recommendation of the editor, he was
Speaker 1: appointed tutor in a private family, which secured him the
Speaker 1: use of a good library and sufficient leisure to use it.
Speaker 1: In seventeen fifty nine he settled in Augsburg, and in
Speaker 1: seventeen thirty three removed to Berlin, where he was given
Speaker 1: a small pension and finally made editor of the Prussian
Speaker 1: Astronomical Almanac. Lambel's most important work were one on optics,
Speaker 1: issued in seventeen fifty nine, which suggested to arago the
Speaker 1: lines of investigation. He subsequently pursued a treatise on perspective,
Speaker 1: published in seventeen fifty nine, to which in seventeen sixty
Speaker 1: eight an appendix given practical applications was added, and a
Speaker 1: treatise on comets printed in seventeen sixty one, containing the
Speaker 1: well known expression for the area of a focal sector
Speaker 1: of a conic in terms of the cord and the
Speaker 1: bounding race. Besides these, he communicated numerous papers to the
Speaker 1: Berlin Academy. Of the most important are his memoir in
Speaker 1: seventeen sixty eight on transcendental magnitudes, in which he proved
Speaker 1: that pi is incommensurable. The proof is given in Legendre's geometry,
Speaker 1: and is there extended to pi squared. His paper on trigonometry,
Speaker 1: read in seventeen sixty eight, in which he developed Demoffra's
Speaker 1: theorems on the trigonometry of complex variables, and introduced the
Speaker 1: hyperbolic sign and co sign, denoted by the symbols sign
Speaker 1: h x cohx. His essay entitled the Analytical Observations, published
Speaker 1: in seventeen seventy one, which is the earliest attempt to
Speaker 1: form the functional equations by expressing the given properties in
Speaker 1: the language of the differential calculus and then integrating. Lastly,
Speaker 1: his paper on viz Viva published in seventeen eighty three,
Speaker 1: in which for the first time he expressed Newton's second
Speaker 1: law of motion in the notation of the differential calculus.
Speaker 1: Of the other mathematicians above mentioned, I hear add a
Speaker 1: few words. Etienne Bezous born in Nemours on March thirty first,
Speaker 1: seventeen thirty and died on September twenty seventh, seventeen eighty three.
Speaker 1: Besides numerous minor works, wrote to Therrie Generreal de sequeigion Algebraque,
Speaker 1: published at Paris in seventeen seventy nine, which in particular
Speaker 1: contained much new and valuable matter on the theory of
Speaker 1: elimination and symmetrical functions of the roots of an equation.
Speaker 1: He used his determinants in a paper in the Histoire
Speaker 1: de leca Demi Royale seventeen sixty four, but did not
Speaker 1: treat of the general theory. Jean Trembrey born at Geneva
Speaker 1: in seventeen forty nine and died at September eighteenth, eighteen eleven,
Speaker 1: contributed to the development of differential equations, finite differences, and
Speaker 1: the calculus of probabilities. Louis Francois d'artoine Abregaste born a
Speaker 1: dal Sas on October fourth, seventeen fifty nine and died
Speaker 1: at Strasbourg, where he was professor, on April eighth, eighteen
Speaker 1: o three, wrote on series and the derivatives. Known by
Speaker 1: his name, he was the first writer to separate the
Speaker 1: symbols of operation from those of quantity. I do not
Speaker 1: wish to crowd my pages with an account of those
Speaker 1: who have not distinctly advanced the subject, but I have
Speaker 1: mentioned the above writers because their names are still well known.
Speaker 1: We may, however, say that the discoveries of Euler and
Speaker 1: Lagrange in the subjects which they treated, were so complete
Speaker 1: and far reaching that what their less gifted contemporary he's
Speaker 1: added is not of sufficient importance to require mention in
Speaker 1: a book of this nature. Lagrange. Joseph Louis Lagrange, the
Speaker 1: greatest mathematician of the eighteenth century, was born at Turin
Speaker 1: on January twenty fifth, seventeen thirty six, and died at
Speaker 1: Paris on April tenth, eighteen thirteen. His father, who had
Speaker 1: the charge of the Sardinian Military Chest, was of good
Speaker 1: social position and wealthy, but before his son grew up,
Speaker 1: he had lost most of his property in speculations, and
Speaker 1: young Lagrange had to rely for his position on his
Speaker 1: own abilities. He was educated at the College of Tyrann,
Speaker 1: but was not until he was seventeen that he chewed
Speaker 1: any taste for mathematics, his interest in the subject being
Speaker 1: first excited by a memoir by Halley across which he
Speaker 1: came by accident, alone and unaided. He threw himself into
Speaker 1: mathematical studies, and at the end of a year's incessant toil,
Speaker 1: he was already an accomplished mathematician, and was made a
Speaker 1: lecturer in the artillery school. The first of these labors
Speaker 1: was his letter, written when he was still only nineteen
Speaker 1: to Euler, in which he solved the isoparametrical problem, which
Speaker 1: for more than half a century had been the subject
Speaker 1: of discussion to effect a solution in which he sought
Speaker 1: to determine the form of a function so that a
Speaker 1: formula in which it entered should satisfy a certain condition.
Speaker 1: He enunciated the principles of the calculus of variations. Euler
Speaker 1: recognized the generality of the method adopted and its superiority
Speaker 1: to that used by himself, and with rare courtesy he
Speaker 1: withheld a paper that he had previously written, which covered
Speaker 1: some of the same ground, in order that the young
Speaker 1: Italian might have time to complete his work and claim
Speaker 1: the undisputed invention of the new calculus. The name of
Speaker 1: this branch of analysis was suggested by Euler. This memoir
Speaker 1: at once placed Lagrange in the front rank of mathematicians.
Speaker 1: Then living. In seventeen fifty eight, Lagrange established, with the
Speaker 1: aid of his pupils, a society which was subsequently incorporated
Speaker 1: as the Turan Academy, and in the five volumes of
Speaker 1: its transactions, usually known as the Miscellanea Tourernensia, most of
Speaker 1: his early writings are to be found. Many of these
Speaker 1: are elaborate works. The first volume contains a memoir of
Speaker 1: the theory of the propagation of sound and in this
Speaker 1: he indicates a mistake made by Newton, obtains a general
Speaker 1: differential equation for the motion, and integrates it for motion
Speaker 1: in a straight line. This volume also contains the complete
Speaker 1: solution of the problem of a string vibrating transversely. In
Speaker 1: this paper, he points out a lack of generality in
Speaker 1: the solutions previously given by Taylor, de Lambert and Euler,
Speaker 1: and arrives at the con inclusion that form of a
Speaker 1: curve at any time T is given by the equation
Speaker 1: y equals a sign mx sign nt. The article concludes
Speaker 1: with a masterly discussion of echoes, beats, and compound sounds.
Speaker 1: Other articles in this volume are on recurring series probabilities
Speaker 1: and the calculus of variations. The second volume containing a
Speaker 1: long paper embodying the results of several memoirs in the
Speaker 1: first volume on the theory and notation of the calculus
Speaker 1: of variations, and he illustrates its use by deducing the
Speaker 1: principle of least action and also by solutions of various
Speaker 1: problems in dynamics. The third volume includes a solution of
Speaker 1: several dynamical problems by means of the calculus of variations.
Speaker 1: Some papers on the integral calculus, a solution of Fermat's
Speaker 1: problem mentioned above, and on the general differential equations of
Speaker 1: motion for three bodies moved under their mutual attractions. In
Speaker 1: seventeen sixty one, Lagrange stood without arrival as the foremost
Speaker 1: mathematician living, But the unceasing labour of the preceding nine
Speaker 1: years had seriously affected his health, and the doctors refused
Speaker 1: to be responsible for his reason or life unless he
Speaker 1: would take rest and exercise. Although his health was temporarily restored,
Speaker 1: his nervous system never quite recovered its tone, and henceforth
Speaker 1: he constantly suffered from attacks of profound melancholy. The next
Speaker 1: work he produced was in seventeen sixty four on the
Speaker 1: libration of the Moon and an explanation as to why
Speaker 1: the same face was always turned to the Earth, a
Speaker 1: problem which he treated by aid of virtual work. His
Speaker 1: solution is especially interesting as containing the germ of the
Speaker 1: idea of generalized equations of motion equations, which he first
Speaker 1: formally proved in seventeen eighty. He now started to go
Speaker 1: on a visit to London, but on the way fell
Speaker 1: ill at Paris. There he was received with the most
Speaker 1: marked honour, and it was with regret that he left
Speaker 1: the brilliant society of that city to return to his
Speaker 1: provincial life at Tehran. His further stay in Piedmont was,
Speaker 1: however short. In seventeen sixty six, Euler left Berlin, and
Speaker 1: Frederick the Great immediately wrote expressing the wish of the
Speaker 1: greatest king in Europe to have the greatest mathematician in
Speaker 1: Europe resident in his court. Lagrange accepted the offer and
Speaker 1: spent the next twenty years at Prussia, where he produced
Speaker 1: not only the long series of memoirs published at the
Speaker 1: Berlin and Tyran Transactions, but his monumental work Mechanique Anarritique.
Speaker 1: His residence at Berlin commenced with an unfortunate mistake, finding
Speaker 1: most of his colleagues married and assured by their wives
Speaker 1: that it was the only way to be happy. He married,
Speaker 1: his wife soon died, but the union was not a
Speaker 1: happy one. Lagrange was a favorite of the King, who
Speaker 1: used frequently to discourse to him on the advantage of
Speaker 1: perfect regularity of life. The lesson went home, and thenceforth
Speaker 1: Lagrange studied his mind and body as though they were machines,
Speaker 1: and found by experiment the exact amount of work which
Speaker 1: he was able to do without breaking down. Every night
Speaker 1: he set himself a definite task for the next day,
Speaker 1: and on completing any branch of a subject, he wrote
Speaker 1: a short analysis to see what points in the demonstrations
Speaker 1: or in the subject matter were capable of improvement. He
Speaker 1: always thought out the subject of his papers before he
Speaker 1: began to compose them, and usually wrote them straight off
Speaker 1: without a single erasure or correction. His mental activity during
Speaker 1: these twenty years was amazing. Not only did he produce
Speaker 1: his splendid mechanique and rietique, but he contributed between one
Speaker 1: and two hundred papers to the academies of Berlin, Tyrann
Speaker 1: and Paris. Some of these are complete treatises, and all,
Speaker 1: without exception, are of Hahaigh order of excellence, except for
Speaker 1: a short time when he was ill, and produced on
Speaker 1: an average about one memoir a month. Of these, I
Speaker 1: note the following as among the most important. First, his
Speaker 1: contributions to the fourth and fifth volumes seventeen sixty six
Speaker 1: to seventeen seventy three of the Miscellanea Taurernensia, of which
Speaker 1: the most important was the one in seventeen seventy one,
Speaker 1: in which he discussed how numerous astronomical observations should be
Speaker 1: combined so as to give the most probable result, and
Speaker 1: later his contributions to the first two volumes seventeen eighty
Speaker 1: four to seventeen eighty five of the Transactions of the
Speaker 1: Turin Academy, to the first of which he contributed a
Speaker 1: paper on the pressure exerted by fluids in motion, and
Speaker 1: to the second an article on integration by infinite series
Speaker 1: and the kind of problems for which it is suitable.
Speaker 1: Most of the memoirs sent to Paris were on astronomical questions,
Speaker 1: and among these I ought particularly to mention his memoir
Speaker 1: on the Jovian System in seventeen sixty six, his essay
Speaker 1: on the problem of the three bodies in seventeen seventy two,
Speaker 1: his work on the secular equation of the Moon in
Speaker 1: seventeen seventy three, and his treatise on cometary perturbations in
Speaker 1: seventeen seventy eight. These were all written on subjects proposed
Speaker 1: by the French Academy, and in each case the prize
Speaker 1: was awarded to him. The greater number of his papers
Speaker 1: during the same time were, however, contributed to the Berlin Academy.
Speaker 1: Several of them deal with questions on algebra. In particular.
Speaker 1: I may mention one his discussion of the solution of
Speaker 1: indeterminate equations on integers in seventeen seventy with special notice
Speaker 1: of indeterminate quadratics in seventeen sixty nine, two, His tract
Speaker 1: on the Theory of Elimination seventeen seventy three, His Memoirs
Speaker 1: on the General Process for solving in algebraical equations of
Speaker 1: any degree seventeen seventy and seventeen seventy one. This method
Speaker 1: fails for equations of an order above the fourth, because
Speaker 1: then it involves a solution of an equation of higher
Speaker 1: dimensions than the one proposed, but it gives all the
Speaker 1: solutions of his predecessors as modifications of a single principle.
Speaker 1: He found, however, the complete solution of the binomial equations
Speaker 1: of any degree and fourth. Lastly, in seventeen seventy three
Speaker 1: he treated of determinants of the second and third order.
Speaker 1: Several of his early papers also deal with questions connected
Speaker 1: with the neglected but singularly fascinating suchbject of the theory
Speaker 1: of numbers. Among these are one his proof of the
Speaker 1: theorem that every integer which is not a square can
Speaker 1: be expressed as a sum of either two, three, or
Speaker 1: four integral squares seventeen seventy His proof of Wilson's theorem
Speaker 1: the off n b prime, then n minus one factorial
Speaker 1: plus one is always a multiple of nds seventeen seventy one. Three,
Speaker 1: His memoirs of seventeen seventy three, seventeen seventy five, and
Speaker 1: seventeen seventy seven, which give the demonstrations of several results
Speaker 1: enunciated by format and not previously proved. Four And lastly,
Speaker 1: his method for determining the factors of numbers of the
Speaker 1: form x squared plus a y squared. There are also
Speaker 1: numerous articles on various points of analytical geometry. In two
Speaker 1: of them, in seventeen ninety two and seventeen ninety three,
Speaker 1: he reduced the equations of the quadrix or conochoids to
Speaker 1: their canonical forms. During the years from seventeen seventy two
Speaker 1: to seventeen eighty five he contributed a long series of
Speaker 1: memoirs which created the science of deferential equations at any rate,
Speaker 1: as far as partial differential equations are concerned, I do
Speaker 1: not think that any previous writer had done anything beyond
Speaker 1: considering equations of some particular form. A large part of
Speaker 1: these results were collected in the second edition of Euler's
Speaker 1: Integral Calculus, which was published in seventeen ninety four. His
Speaker 1: papers on mechanics require no separate mention here, as the
Speaker 1: results arrived at are embodied in the mechanique analytique, which
Speaker 1: is described below. Lastly, there are numerous memoirs on problems
Speaker 1: in astronomy. Of these, the most important are the following one,
Speaker 1: The Attraction of ellipsoids seventeen seventy three. This is founded
Speaker 1: on mclaurin's work two on the secular equation of the
Speaker 1: Moon seventeen seventy three. Also noticeable for the earliest introduction
Speaker 1: of the idea of the potential. The potential of a
Speaker 1: body at any point is the sum of the mass
Speaker 1: of every element of the body when divided by its
Speaker 1: distance from the point. Lagrant should that if the potential
Speaker 1: of a body at an external point were known, the
Speaker 1: attraction in any direction could be at once found. The
Speaker 1: theory of the potential was elaborated in a paper sent
Speaker 1: to Berlin in seventeen seventy seven three on the motion
Speaker 1: of the nodes of a planet's orbit seventeen seventy four.
Speaker 1: Four on the Stability of Planetary orbits seventeen seventy six, five.
Speaker 1: Two memoirs in which the method of determining the orbit
Speaker 1: of a comet from three observations is completely worked out
Speaker 1: seventeen seventy eight and seventeen eighty three. This is not
Speaker 1: indeed proved practically available, but his system of calculating the
Speaker 1: perturbations by means of mechanical quadratures has formed the basis
Speaker 1: of most subsequent researches on the subject. Six His Determination
Speaker 1: of the secular and periodic variations of the elements of
Speaker 1: the planets seventeen eighty one to seventeen eighty four. The
Speaker 1: upper limits assigned for these agree closely with those obtained
Speaker 1: later by Leverrier, and he proceeded as far as the
Speaker 1: knowledge then possessed of the masses of the planets permitted seven.
Speaker 1: Three Memoirs on the method of interpolation seventeen eighty three,
Speaker 1: seventeen ninety two, and seventeen ninety three. The part of
Speaker 1: finite differences dealing Therewith is now in the same stage
Speaker 1: as that in which Lagrange left it. Over and above
Speaker 1: these various papers, he composed his great treatise Mechanique Analytique.
Speaker 1: In this he lays down the law of virtual work,
Speaker 1: and from that one fundamental principle, by the age of
Speaker 1: the calculus of variations, he deduces the whole of mechanics,
Speaker 1: both of solids and fluids. The object of the book
Speaker 1: is to shew that the subject is implicitly included in
Speaker 1: a single principle, and to give the general formulae from
Speaker 1: which any particular result can be obtained. The method of
Speaker 1: generalized coordinates by which he obtained this result is perhaps
Speaker 1: the most brilliant result of his analysis. Instead of following
Speaker 1: the motion of each individual part of a material system,
Speaker 1: as de Lambert and Euler had done, he shewed that
Speaker 1: if we determine its configuration by a sufficient number of
Speaker 1: variables whose number is the same as that of the
Speaker 1: degrees of freedom possessed by the system, then the connecticut
Speaker 1: potential energies of the system can be expressed in terms
Speaker 1: of these, and the differential equations of motions then deduced
Speaker 1: by simple differentiation, for example, in dynamics of a rigid system.
Speaker 1: He replaces the consideration of the particular problem by the
Speaker 1: general equation, which is no now usually written in the
Speaker 1: form of D by dt times the partial derivative of
Speaker 1: capital T with respect to theta minus, the partial derivative
Speaker 1: of capital T with respect to theta plus the partial
Speaker 1: derivative of V with respect to theta equals zero, amongst
Speaker 1: other minor theorems. Here given, I may mention the proposition
Speaker 1: that the kinetic energy imparted by given impulses to a
Speaker 1: material system under given constraints is a maximum, and the
Speaker 1: principle of least action. All the analysis is so elegant
Speaker 1: that Sir William Rowan Hamilton said the work could be
Speaker 1: only described as a scientific poem. It may be interesting
Speaker 1: to note that Legrange remarked that the mechanics was really
Speaker 1: a branch of pure mathematics, analogous to a geometry of
Speaker 1: four dimensions, namely the time and the three coordinates of
Speaker 1: the point in space. And it is said that he
Speaker 1: he prided himself that from the beginning to the end
Speaker 1: of the work there was not a single diagram. At first,
Speaker 1: no printer could be found who had published the book.
Speaker 1: Be Legendre at last persuaded a Paris firm to undertake it,
Speaker 1: and it was issued under his supervision in seventeen eighty eight.
Speaker 1: In seventeen eighty seven, Frederick died, and Lagrange, who had
Speaker 1: found the climate of Berlin trying, gladly accepted the offer
Speaker 1: of Louis the sixteenth to migrate to Paris. He received
Speaker 1: similar invitations from Spain and Naples. In France, he was
Speaker 1: received with every mark of distinction, and special apartments in
Speaker 1: the Louverre were prepared for his reception. For the first
Speaker 1: two years of his residence here he was seized with
Speaker 1: an attack of melancholy, and even the printed copy of
Speaker 1: his mechanique on which he had worked for a quarter
Speaker 1: of a century, lay for more than two years unopened
Speaker 1: on his desk. Curiosity as to the results so the
Speaker 1: French Revolution first stirred him out of his lethargy, a
Speaker 1: curiosity which soon turned to alarm as the revolution developed.
Speaker 1: It was about the same time, seventeen ninety two, that
Speaker 1: the unaccountable sadness of his life and timidity moved the
Speaker 1: compassion of a young girl, who insisted on marrying him
Speaker 1: and proved a devoted wife to whom he became warmly attached.
Speaker 1: Although the decree of October seventeen ninety three, which ordered
Speaker 1: all foreigners to leave France, specially exempted him by name.
Speaker 1: He was preparing to escape when he was offered the
Speaker 1: presidency of the Commission for the Reform of Weights and Measures.
Speaker 1: The choice of the units finally selected was largely due
Speaker 1: to him, and it was mainly owing to his influence
Speaker 1: that the decimal subdivision was accepted by the Commission of
Speaker 1: seventeen ninety nine. The general idea of the decimal system
Speaker 1: was taken from a work by Thomas Williams entitled Method
Speaker 1: for Fixing a Universal Standard for Weights and Measures, published
Speaker 1: in London in seventeen eighty eight. This almost unknown writer
Speaker 1: has hardly received the credit due to his suggestion. Though
Speaker 1: Lagrange had determined to escape from France while there was
Speaker 1: yet time, he was never in any danger, and the
Speaker 1: different revolutionary governments and at a later time Napoleon loaded
Speaker 1: him with honors and distinctions. A striking testimony to the
Speaker 1: respect in which he was held was shewn in seventeen
Speaker 1: ninety six when the French Commissary in Italy ordered to
Speaker 1: attend in full state on Lagrange's father and tender the
Speaker 1: congratulations of the Republic on the achievements of his son,
Speaker 1: who had done honor to all mankind by his genius,
Speaker 1: and whom it was a special glory of Piedmont to
Speaker 1: have produced. In seventeen ninety five, Lagrange was appointed to
Speaker 1: a mathematic chair at the newly established Echol Noomal which
Speaker 1: only enjoyed a brief existence of four months. His lectures
Speaker 1: here were quite elementary and contain nothing of any special importance.
Speaker 1: But they were published because the professors had to pledge
Speaker 1: themselves to the representatives of the people and to each other,
Speaker 1: neither to read nor to repeat from memory, and the
Speaker 1: discourses were ordered to be taken down in shorthand in
Speaker 1: order to enable the deputies to see how the professors
Speaker 1: acquitted themselves. On the establishment of the Ecole Polytechnique in
Speaker 1: seventeen ninety seven, Lagrange was made a professor, and his
Speaker 1: lectures there are described by mathematicians who had the good
Speaker 1: fortune to be able to attend them, as almost perfect,
Speaker 1: both in form and matter. Beginning with the merest elements,
Speaker 1: he led his hearers on until almost unknown to themselves,
Speaker 1: they were themselves extending the bounds of the subject. Above all,
Speaker 1: he pressed on his pupils the advantage of always using
Speaker 1: general methods expressed in a symmetrical notation. His lectures on
Speaker 1: the differential calculus form the basis of his Theire de
Speaker 1: Functio Analytique, which was published in seventeen ninety seven. This
Speaker 1: work is the extension of an idea contained in a
Speaker 1: paper he had sent to the Berlin Memoirs in seventeen
Speaker 1: seventy two, and its object is to substitute for the
Speaker 1: differential calculus as a group of theorems based on the
Speaker 1: development of algebraic functions in series. A somewhat similar method
Speaker 1: had been previously used by John Landon in his Residual Analysis,
Speaker 1: published in London in seventeen fifty eight. Lagrange believed that
Speaker 1: he could thus get rid of those difficulties connected with
Speaker 1: the use of infinitely large or infinitely small quantities which
Speaker 1: philosophers professed to see in the usual treatment of the
Speaker 1: differential calculus. The book is divided into three parts. Of these,
Speaker 1: the first treats of the general theory of functions and
Speaker 1: gives an algebraic proof of Taylor's theorem, the validity of
Speaker 1: which is, however, to open the question. The second deals
Speaker 1: with applications to geometry, and the third with applications to mechanics.
Speaker 1: Another treatise on the same lines was his lessence Le
Speaker 1: Calcure de function, issued in eighteen o four. These works
Speaker 1: may be considered as the starting point for the researches
Speaker 1: of Koshi and Jacoby. At a later period, Lagrange reverted
Speaker 1: to the use of infinitesimals in preference to the founding
Speaker 1: the differential calculus on a study of algebraic forms, and
Speaker 1: in the preface to the second edition of the Mechanique,
Speaker 1: which was issued in eighteen eleven, he justifies their use
Speaker 1: and concludes by saying that when we have grasped the
Speaker 1: spirit of the infinitesimal method, and have verified the exactness
Speaker 1: of its results, either by the geometrical method of prime
Speaker 1: and ultimate ratios or by the analytical method of derived functions,
Speaker 1: we may employ infinitely small quantities as a sure and
Speaker 1: valuable means of shortening and simplifying our proofs. His resolution
Speaker 1: des equasion nu meric, published in seventeen ninety eight, was
Speaker 1: also the fruit of his lectures at the Polytechnic. In
Speaker 1: this he gives the method of approximating to the real
Speaker 1: roots of an equation by means of continued fractions and
Speaker 1: enunciates several other theorems. In a note at the end,
Speaker 1: he shews how Fermat's theorem that a to the power
Speaker 1: of p minus one minus one is defined as zero
Speaker 1: mod p, where p is prime and a is prime
Speaker 1: to p, combined with a certain suggestion due to Gauss,
Speaker 1: may be applied to give the complete algebraical solution of
Speaker 1: any binomial equation. He also here explains how the equation
Speaker 1: whose roots are the squares of the differences of the
Speaker 1: roots of the original equation, may be used so as
Speaker 1: to give considerable information so as to the position and
Speaker 1: nature of those roots. The theory of the planetary motions
Speaker 1: had formed the subject of some of the most remarkable
Speaker 1: of Lagrange's Berlin papers. In eighteen oh six a subject
Speaker 1: was reopened by Poissan, who, in a paper read before
Speaker 1: the French Academy, she shewed that Lagrange's formula led to
Speaker 1: certain limits for the stability of the orbits. La Grange,
Speaker 1: who was present, now discussed the whole subject afresh, and,
Speaker 1: in a memoir communicated to the Academy in eighteen oh eight,
Speaker 1: explained how, by the variation of arbitrary constance, the periodical
Speaker 1: and secular inequalities of any system of mutually interacting bodies
Speaker 1: could be determined. In eighteen ten, Lagrange commenced the thorough
Speaker 1: revision of the Mechanique an letique, but he was able
Speaker 1: to complete only about two thirds of it before his death.
Speaker 1: In appearance, he was of medium height and slightly formed,
Speaker 1: with pale blue eyes colorless complexion. In character, he was
Speaker 1: nervous and timid. He detested controversy, and to avoid it
Speaker 1: willingly allowed others to take the credit for what he
Speaker 1: had himself done. Lagrange was, above all a student of
Speaker 1: pure mathematics. He sought and obtained far reaching abstract results,
Speaker 1: and was content to leave the applications to others. Indeed,
Speaker 1: no considerable part of the discoveries of his great contemporary
Speaker 1: Laplace consists of the applications of the Lagrangian formulae to
Speaker 1: the facts of nature. For example, Laplace's conclusions on the
Speaker 1: velocity of sound and the secular acceleration of the moon
Speaker 1: are implicitly involved in Lagrange's results. The only difficulty in
Speaker 1: understanding Lagrange is that of the subject matter and of
Speaker 1: the extreme generality of his processes. But his analysis is
Speaker 1: as lucid and luminous as it is symmetrical and ingenious.
Speaker 1: A recent writer, speaking of verg Lagrange, says truly that
Speaker 1: he took a prominent part in the advancement of almost
Speaker 1: every branch of pure mathematics. Like Theophantus and Fermat, he
Speaker 1: proposed a special genius for the theory of numbers, and in
Speaker 1: this subject he gave solutions of most of the problems
Speaker 1: which had been proposed by Fermat and added some theorems
Speaker 1: of his own. He created the calculus of variations. To him, too,
Speaker 1: the theory of differential equations is indebted for its position
Speaker 1: as a science, rather than as a collection of ingenious
Speaker 1: artifices for the solution of particular problems. To the calculus
Speaker 1: of finite differences, he contributed the formula of interpolation, which
Speaker 1: bears his name. But above all he impressed a mechanics which,
Speaker 1: it will be remembered. He considered a part of pure
Speaker 1: mathematics that generality and completeness towards which his labours invariably tended.
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