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 PJK dot scripts thought I might
Speaker 1: dot edu forward slash p kJ. A short account of
Speaker 1: the history of mathematics by W. W. Rousbaul Chapter eighteen,
Speaker 1: Libaniz and the Mathematicians of the first half of the
Speaker 1: eighteenth century, Part three. Laplace Pierre Simon la Place was
Speaker 1: born at Beaumont an Ogue in Normandy on March twenty third,
Speaker 1: seventeen forty nine, and died at Paris on March fifth,
Speaker 1: eighteen twenty seven. He was the son of a small
Speaker 1: cottager or perhaps a farm laborer, and owed his education
Speaker 1: to the interest excited in some wealthy neighbors by his
Speaker 1: abilities and engaging presence. Very little is known of his
Speaker 1: early years, for when he became distinguished, he held himself
Speaker 1: aloof from both his relatives and from those who had
Speaker 1: assisted him. A similar pettiness of character marked many of
Speaker 1: his actions. It would seem that from a pupil he
Speaker 1: became an usher in the school at Beaumont, but having
Speaker 1: procured a letter of introduction to Lette de Lambert, he
Speaker 1: went to Paris to push his fortune. A paper on
Speaker 1: the principles of mechanics excited de Lambert's interest, and on
Speaker 1: his recommendations, a place in the military school was offered
Speaker 1: to Laplace. Secure of a competency, Laplace now threw himself
Speaker 1: onto the original research, and in the next seventeen years
Speaker 1: seventeen seventy one to seventeen eighty seven he produced much
Speaker 1: of his original work in astronomy. This commenced with a
Speaker 1: memoir read before the French Academy in seventeen seven three
Speaker 1: which he shewed that the planetary motions were stable and
Speaker 1: carried the proof as far as the cubes of the
Speaker 1: eccentricities and inclinations. This was followed by several papers on
Speaker 1: points in the integral calculus, finite differences, differential equations, and astronomy.
Speaker 1: During the years seventeen eighty four to seventeen eighty seven
Speaker 1: he produced some memoirs of exceptional power. Prominent among those
Speaker 1: is one read in seventeen eighty four and reprinted in
Speaker 1: the third volume of the mechanic Celeste, in which he
Speaker 1: completely determined the attraction of a spheroid on a particle
Speaker 1: outside it. This is memorable for the introduction into an
Speaker 1: analysis of spherical harmonics or Laploce's coefficients, and also for
Speaker 1: the development of the use of potential, a name first
Speaker 1: given by Green in eighteen twenty eight. If the coordinates
Speaker 1: of two points be r meure u omega and our
Speaker 1: prime uprime omega prime, and our prime is not less
Speaker 1: than r, the reciprocal of the distance between them can
Speaker 1: be expanded in powers of R divided by our prime,
Speaker 1: and the respected coefficients are Laplace coefficients. Their utility arises
Speaker 1: from the fact that every function of the coordinates of
Speaker 1: a point on a sphere can be expanded in a
Speaker 1: series of them. It should be stated that the similar
Speaker 1: coefficients for space of two dimensions, together with some of
Speaker 1: their properties, had been previously been given by Legendre in
Speaker 1: a paper sent to the French Academy in seventeen eighty three.
Speaker 1: Legentre had good reason to complain of the way in
Speaker 1: which he was treated in this manner. This paper is
Speaker 1: also remarkable for the development of the idea of the potential,
Speaker 1: which was appropriated from Lagrange, who had used it in
Speaker 1: his memoirs of seventeen seventy three seventeen seventy seven n.
Speaker 1: Seventeen eighty. Laplastiwd that the potential always satisfies the differential
Speaker 1: equation Dell squared V equals the second partial derivative of
Speaker 1: the voltage with respect to X plus the second partial
Speaker 1: derivative of the voltage with respect to y plus the
Speaker 1: second partial derivative of the voltage with respect to z
Speaker 1: equals zero, And on this result his subsequent work on
Speaker 1: attractions was based. The quantity Dell squared v had been
Speaker 1: termed the concentration of V, and its value at any
Speaker 1: point indicates the excess of the value of V over
Speaker 1: its mean value in the neighborhood of the point. Laplace's equation,
Speaker 1: or the more general form Dell squared v equals negative
Speaker 1: four pi row appears in all branches of mathematical physics.
Speaker 1: According to some writers, this follows at once from the
Speaker 1: fact that Dell squared is a scalar operator, or the
Speaker 1: equation may represent analytically some general law of nature which
Speaker 1: has not yet been reduced to words, or possibly I
Speaker 1: have sometimes thought it might be regarded by a Kantian
Speaker 1: as the outward sign of one of the necessary forms
Speaker 1: through which all phenomena are perceived. This memoir was followed
Speaker 1: by another unplanetary Inequalities, which was presented in three sections
Speaker 1: in seventeen eighty four, seventeen eighty five, and seventeen eighty six.
Speaker 1: This deals mainly with the explanation of the great inequality
Speaker 1: of Jupiter and Saturn. Laplace chewed by general considerations that
Speaker 1: the mutual action of two planets could never largely affect
Speaker 1: the eccentricities and inclinations of their orbits, and that the
Speaker 1: peculiarities of the Jovian system were due to the near
Speaker 1: approach to commensurability of the mean motions of Jupiter and Saturn.
Speaker 1: Further developments of these theorems on planetary motion were given
Speaker 1: in his two memoirs of seventeen eighty eight and seventeen
Speaker 1: eighty nine. It was on these data that de Lambre
Speaker 1: computed his astronomical tables. The year seventeen eighty seven was
Speaker 1: rendered memorable by Laplace's explanation and analysis of the relation
Speaker 1: between the lunar acceleration and the secular changes in the
Speaker 1: eccentricity of the Earth's orbit. This investigation completed the proof
Speaker 1: of the stability of the whole Solar system on the
Speaker 1: assumption that it consists of a collection of rigid bodies.
Speaker 1: All the memoirs above alluded to were presented to the
Speaker 1: French Academy, and they are printed in the memoir Presente
Speaker 1: pals di versavant. Laplace now set himself the task to
Speaker 1: write a work which should offer a complete solution of
Speaker 1: the great mechanical problem presented by the Solar system and
Speaker 1: bring theory to coincide so closely with observation that empirical
Speaker 1: equations should no longer find a place in astronomical tables.
Speaker 1: The result is embodied in the Exposition System DuMond and
Speaker 1: the Mechanique Arrest. The former was published in seventeen ninety
Speaker 1: six and gives a general explanation of the phenomena with
Speaker 1: a summary of the history of astronomy, but omits all details.
Speaker 1: The nebular hypothesis was here enunciated. According to this hypothesis,
Speaker 1: the Solar system had spin evolved from a globular mass
Speaker 1: of incandescent gas rotating around an axis through its center
Speaker 1: of mass as it cooled, this mass contracted and successive
Speaker 1: rings broke off from its outer edge. These rings, in
Speaker 1: their turn cooled and finely condensed into the planets, while
Speaker 1: the sun represents the central core, which was still left.
Speaker 1: Certain corrections required by modern science were added by M. Rosch,
Speaker 1: and recently the theory has been discussed critically by R. Wolf.
Speaker 1: The arguments against the hypothesis are some in phase Regin
Speaker 1: Dumonde Paris, eighteen eighty four, where an ingenious modification of
Speaker 1: the hypothesis is proposed, by which the author attempts to
Speaker 1: explain the peculiarities of the axial rotation of Neptune and
Speaker 1: Uranus and the retrograde motion of the satellites of the
Speaker 1: latter planet. Perhaps modern opinion is inclined to attribute the
Speaker 1: separation of the various members of the planetary system to
Speaker 1: tidal friction rather than to the successive separation and condensation
Speaker 1: of nebular rings. But the subject is one of great
Speaker 1: difficulty according to the rule published by Titus of Wittenberg
Speaker 1: in seventeen sixty six, but generally known as Bode's law
Speaker 1: from the fact that attention was called to it by
Speaker 1: Johann elt Bode in seventeen seventy eight. The distances of
Speaker 1: the planets from the Sun are nearly in the ratio
Speaker 1: of the numbers zero plus four, three plus four, six
Speaker 1: plus four to five, twelve plus four, et cetera, and
Speaker 1: the n minus tooth term being two to the power
Speaker 1: n times three with four added. It would be an
Speaker 1: interesting fact if this could be deduced from either the
Speaker 1: nebula or the title hypothesis, but so far as I
Speaker 1: am aware, only one serious attempt to do so has
Speaker 1: been made, and the conclusion was that the law was
Speaker 1: not sufficiently exact to be more than a convenient means
Speaker 1: of remembering the general result. The substance of Laplace's hypothesis
Speaker 1: had been published by Kant in seventeen fifty five in
Speaker 1: his Algamina nat Chogaschista, but it is probable that Laplace
Speaker 1: was not aware of this. This historical summary procured for
Speaker 1: its author the honor of admission to the forty of
Speaker 1: the French Academy. It is commonly esteemed one of the
Speaker 1: masterpieces of French literature, though it is not altogether reliable
Speaker 1: for the later periods of which it treats. The full
Speaker 1: analytical discussion of the solar system is given in Mechaniques Arrest,
Speaker 1: published in five volumes, Volumes one and two in seventeen
Speaker 1: ninety nine, volume three in eighteen oh two, Volume four
Speaker 1: in eighteen oh five, and volume five in eighteen twenty five.
Speaker 1: An analysis of the contents is given in the English Cyclopedia.
Speaker 1: The first two volumes contain methods for calculating the motions
Speaker 1: of the planets, determining their figures, and resolving title problems.
Speaker 1: The third and fourth volumes contain the application of these methods,
Speaker 1: and also several astronomical tables. The fifth volume is mainly historical,
Speaker 1: but it gives as appendices the results of Laplace's latest researches.
Speaker 1: Laplace's own investigations embodied in it are so numerous and
Speaker 1: valuable that it is regrettable to have to add that
Speaker 1: many results are appropriated from writers with scanty or no acknowledgment,
Speaker 1: and the conclusions, which have been described as the organized
Speaker 1: result of a century of patent toli oil are generally
Speaker 1: mentioned as if they were due to Laplace. And it
Speaker 1: is said, for I have not looked up into the
Speaker 1: matter myself, that the praise which he lavishes on Newton
Speaker 1: and Clairout is only the cloak under which he appropriates
Speaker 1: the work of other less known writers. The Mechanique seirest
Speaker 1: is by no means easy reading. Billotte, who assisted Laplace
Speaker 1: in revising it for the press, says that Laplace himself
Speaker 1: was frequently unable to recover the details in the chain
Speaker 1: of reasoning, and if satisfied that the conclusions were correct,
Speaker 1: he was content to insert the constantly recurring formula iritiz avoir.
Speaker 1: The best tribute to the excellency of the work is
Speaker 1: that it left very little for his successors to add.
Speaker 1: It is not only the translation of the Principia into
Speaker 1: the language of the differential calculus, but it also completes
Speaker 1: part of which Newton had been unable to fill in
Speaker 1: the details. M Tisseran's recent work may be considered as
Speaker 1: a continuation of Laplace's treatise. Laplace went in state to
Speaker 1: beg Napoleon to accept a copy of his work, and
Speaker 1: the following account of the interview is well authenticated and
Speaker 1: so characteristic of all parties. Concerned that I quoted in full.
Speaker 1: Someone had told Napoleon that the book contained no mention
Speaker 1: of the name of God. Napoleon, who was fond of
Speaker 1: putting embarrassing questions received it with the remark, Monsieur la Place,
Speaker 1: they tell me you have written this large book on
Speaker 1: the system of the universe and have never even mentioned
Speaker 1: its creator. Laplace, who though most supple of politicians, was
Speaker 1: as stiff as a martyr on every point of his philosophy,
Speaker 1: drew himself up and answered bluntly, Geneve Pavessois de setre pothees.
Speaker 1: La Napoleon, greatly amused, told this reply to La Grange,
Speaker 1: who exclaimed, ah, sit in beer, hypothe expric bo could chous.
Speaker 1: In eighteen twelve, Laplace issued is theory analytique de probabilities.
Speaker 1: The theory is stated to be only common sense expressed
Speaker 1: in mathematical language. The method of estimating the ratio of
Speaker 1: the number of favorable cases to the whole number of
Speaker 1: possible cases has been indicated by Laplace in a paper
Speaker 1: written in seventeen seventy nine. It consists in treating the
Speaker 1: successive values of any function as the coefficients in the
Speaker 1: expansion of another function with reference to a different variable.
Speaker 1: The latter is therefore called the generating function of the former.
Speaker 1: Laplace then choose how by means of interpolation, these coefficients
Speaker 1: may be determined from the generating function. Next, he attacks
Speaker 1: the converse problem, and from the coefficiency he finds the
Speaker 1: generating function. This is offered by the solution of an
Speaker 1: equation in finite differences. The method is cumbersome, and, in
Speaker 1: consequence of the increased power of analysis, is now rarely used.
Speaker 1: A summary of Laplace's reasoning is given in the article
Speaker 1: on probability in his Encyclopedia Metropolitana. This treatise includes an
Speaker 1: exposition of the method of lease squares, which is a
Speaker 1: remarkable testimony to Laplace's command over the process of analysis.
Speaker 1: The method of lease squares for the combination of numerous
Speaker 1: observations had been given empirically by Gauss and Legendre, but
Speaker 1: the fourth chapter of this work contains a formal proof
Speaker 1: of it, on which the whole of the theory of
Speaker 1: errors has been since spaced. This was effected only by
Speaker 1: a most intricate analysis specially invented for the purpose, but
Speaker 1: the form in which it is presented as so meager
Speaker 1: and unsatisfactory that, in spite of the uniform accuracy of
Speaker 1: the results, it was at one time question whether Laplace
Speaker 1: had actually gone through the difficult work he so briefly
Speaker 1: and often incorrectly indicates. In eighteen nineteen, Laplace published a
Speaker 1: popular account of his work on probability. This book bears
Speaker 1: the same relation to his theory the probabilities that the
Speaker 1: Si stem DuMond does to the mechanique Cereste. Amongst the
Speaker 1: minor discoveries of Laplace in pure mathematics, I may mention
Speaker 1: his discussion, simultaneously with Van Dermonde, of the general theory
Speaker 1: of determinants in seventeen seventy two, his proof that every
Speaker 1: equation of an even degree must have at least one
Speaker 1: real quadratic factor, his reduction of the solution of linear
Speaker 1: differential equations to definite integrals, and his solution to the
Speaker 1: linear partial equation of the second order. He was also
Speaker 1: the first to consider the difficult problems involved in the
Speaker 1: equations of mixed differences, and to prove that the solution
Speaker 1: of an equation and finite differences of the first degree
Speaker 1: and the second order might always be obtained in the
Speaker 1: form of a continued fraction. Besides these original discoveries, he
Speaker 1: determined in his theory of probabilities the values of a
Speaker 1: number of the more common definite integrals, and in the
Speaker 1: same book gave the general proof of the theorem enunciated
Speaker 1: by Lagrange for the development of any implicit function in
Speaker 1: a series by means of differential coefficients. In theoretical physics,
Speaker 1: the theory of capillary attraction is due to Laplace, who
Speaker 1: accepted the idea propounded by Hawkesby in the Philosophical Transactions
Speaker 1: for seventeen o nine that the phenomenon was due to
Speaker 1: a force of attraction which was insensible at sensible distances.
Speaker 1: The part which deals with the action of a solid
Speaker 1: on a liquid and the mutual actions of two liquids
Speaker 1: was not worked out thoroughly, but ultimately was completed by Gauss.
Speaker 1: Newman later filled in a few details. In eighteen sixty two,
Speaker 1: Lord Kelvin Sir William Thompson shoot that if we assume
Speaker 1: the molecular constitution of matter, the laws of capillary attraction
Speaker 1: can be deduced from the Newtonian law of gravitation. Laplace,
Speaker 1: in eighteen sixteen was the first to point out explicitly
Speaker 1: why Newton's theory of vibratory motion gave an incorrect value
Speaker 1: for the velocity of sound. The actual velocity is greater
Speaker 1: than that calculated by Newton in consequence of the heat
Speaker 1: developed by the sudden compression of air, which increases the
Speaker 1: elasticity and therefore the velocity of sound transmitted. Laplace's investigations
Speaker 1: and practical physics were confined to those carried on by
Speaker 1: him jointly with Lovoisier in the year seventeen eighty two
Speaker 1: and seventeen eighty four on the specific heat of various bodies.
Speaker 1: Laplace seems to have regarded analysis as merely a means
Speaker 1: of attacking physical problems, though though the ability with which
Speaker 1: he invented the necessary analysis is almost phenomenal, As long
Speaker 1: as his results were true, he took but little trouble
Speaker 1: to explain the steps by which he arrived at them.
Speaker 1: He never studied elegance or symmetry in his processes, and
Speaker 1: it was sufficient for him if he could, by any
Speaker 1: means solve the particular question he was discussing. In these
Speaker 1: respects he stands in marked contrast to his great contemporary Lagrange.
Speaker 1: It would have been well for Laplace's reputation if he
Speaker 1: had been content with his scientific work, But above all
Speaker 1: things he coveted social fame. The skill and rapidity with
Speaker 1: which he managed to change his politics as occasion required
Speaker 1: would be amusing if they had not been so servile.
Speaker 1: As Napoleon's power increased, Laplace abandoned his Republican principles, which,
Speaker 1: since they had faithfully read reflected the opinions of the
Speaker 1: party in power, had themselves gone through numerous changes, and
Speaker 1: begged for the First Council to give him the post
Speaker 1: of Minister of the Interior. Napoleon, who desired the support
Speaker 1: of men of science, accepted the offer, but a little
Speaker 1: less than six weeks saw the close of Laplace's political career.
Speaker 1: Napoleon's memorandum on the subject as as follows. Geometre de
Speaker 1: premier range la Place natal da asse montre administraturplus cux
Speaker 1: mediocre des sent premier travaill news raconneum cunu zetien trompe
Speaker 1: la place ne sees santon coun question sus saint vere
Speaker 1: table pronte de vieux charche de sabtili te partu nave
Speaker 1: cux desi de problematique er porte enfant la sprite's infinima
Speaker 1: petit juscud dans la administracion. Although Laplace was expelled from office,
Speaker 1: it was desirable to retain his allegiance. He was accordingly
Speaker 1: raised to the Senate, and to the third volume of
Speaker 1: the Mechanique Celeste he prefixed a note that, of all
Speaker 1: the truths therein contained, the most precious to the author
Speaker 1: was the declaration he thus made of his devotion toward
Speaker 1: the peacemaker of Europe in copies sold after the restoration.
Speaker 1: This was struck out in eighteen fourteen. It was evident
Speaker 1: that the Empire was falling. Laplace hastened to tender his
Speaker 1: services to the Bourbons, and on the restoration was rewarded
Speaker 1: with the title of Marquis. The contempt that his more
Speaker 1: honest colleagues felt for his conduct in the matter may
Speaker 1: be read in the pages of Pauluis Courier. His knowledge
Speaker 1: was useful on the numerous scientific commissions on which he served,
Speaker 1: and probably accounts for the manner in which his political
Speaker 1: insincerity was overlooked. But the pettiness of his character must
Speaker 1: not make us forget how great were his services to science.
Speaker 1: That Laplace was vain and selfish is not denied by
Speaker 1: his warmest admirers. His conduct to the benefactors of his
Speaker 1: youth and his political friends was ungrateful and contemptible, while
Speaker 1: his appropriation of the results of those who were comparatively
Speaker 1: unknown seems to be well established and is absolutely indefensible.
Speaker 1: Of those whom he thus treated, three subsequently rose to
Speaker 1: distinction le Gentre and Froyer in France and Young in England,
Speaker 1: and never forgot the injustice of which they had been
Speaker 1: the victims. On the other side, it may be said
Speaker 1: that on some questions he shoot independence of character, and
Speaker 1: he never concealed his views on religion, philosophy, or science,
Speaker 1: however distasteful they might be to the authorities in power.
Speaker 1: It should be also added that, towards the close of
Speaker 1: his life, and especially to the work of his pupils,
Speaker 1: Laplace was both generous and appreciative, and in one case
Speaker 1: suppressed a paper of his own in order that a
Speaker 1: pupil might have the sole credit of the discovery. Le
Speaker 1: Gendre Atrienne Marie Legendre was born at Toulouse on September eighteenth,
Speaker 1: seventeen fifty two and died at Paris on January tenth,
Speaker 1: eighteen thirty three. The leading events of his life are
Speaker 1: very simple, and may be summed up briefly. He was
Speaker 1: educated at the Mazarin College in Paris. Appointed the professor
Speaker 1: at the Military School in Paris in seventeen seventy seven.
Speaker 1: Was a member of the Anglo French Commission of seventeen
Speaker 1: eighty seven to connect Greenwich and Paris. Geodetically served on
Speaker 1: several of the public commissions from seventeen ninety two to
Speaker 1: eighteen ten. Was made a professor at the Normal School
Speaker 1: in seventeen ninety five, and subsequently held a few minor
Speaker 1: government appointments. The influence of Laplace was steadily exerted against
Speaker 1: his obtaining office or public recognition, and Legendre, who was
Speaker 1: a timid student, accepted the obscurity to which the hostility
Speaker 1: of his colleague condemned him. Le Gentre's analysis is of
Speaker 1: a high order of excellence, and is second only to
Speaker 1: that produced by Lagrange and Laplace, though it is not
Speaker 1: so original. His chief works are his geometry, his Theery
Speaker 1: de Numbre, and his calcul integral and his functions elliptique.
Speaker 1: These include the results of his various papers on these subjects.
Speaker 1: Besides these, he wrote a treatise which gave the rule
Speaker 1: for the method of least squares and two groups of memoirs,
Speaker 1: one on the theory of attractions and the other on
Speaker 1: geodetical operations. The memoirs on attractions are analyzed and discussed
Speaker 1: in Todd Hunter's History of the Theory of Attraction. The
Speaker 1: earliest of these memoirs, presented in seventeen eighty three, was
Speaker 1: on the attraction of spheroids. This contain the introduction of
Speaker 1: la Gendre's coefficients, which are sometimes called circular horizonal harmonics,
Speaker 1: and which are particular cases of Laplace's coefficients. It also
Speaker 1: includes the solution of a problem in which the potential
Speaker 1: is used. The second memoir was communicated in seventeen eighty
Speaker 1: four and is of the form of equilibrium of a
Speaker 1: mass of rotating liquid which is approximately spherical. The third,
Speaker 1: written in seventeen eighty six, is on the attraction of
Speaker 1: confocal elliptoids. The fourth is on the figure which a
Speaker 1: fluid planet would assume and its law of density. His
Speaker 1: papers on geodes are three in number and were presented
Speaker 1: to the Academy in seventeen eighty seven and seventeen eighty eight.
Speaker 1: The most important result is that by which a spherical
Speaker 1: triangle may be treated as a plane, provided certain corrections
Speaker 1: applied to the angles. In connection with this subject, he
Speaker 1: paid considerable attention to geodesics. The method of Lee squares
Speaker 1: was enunciated in his Nouvelle Methode, published in eighteen o six,
Speaker 1: to which supplements were added in eighteen ten and eighteen twenty.
Speaker 1: Gauss independently had arrived at the same result and had
Speaker 1: used it in seventeen ninety five, and published it in
Speaker 1: the Law of Facility in eighteen o nine. Laplace was
Speaker 1: the earlier writer to give a proof of it. This
Speaker 1: was in eighteen twelve. On the other books produced by
Speaker 1: Le Gendre, the one most widely known is his Ede
Speaker 1: Monte Gianmetri, which was published in seventeen ninety four and
Speaker 1: was generally adopted on the continent as a substitute for Euclid.
Speaker 1: The later editions contain the elements of trigonometry and proofs
Speaker 1: of the irrationality of pie and pie squared. Indix on
Speaker 1: the difficult question of the theory of parallel lines was
Speaker 1: issued in eighteen o three and is bound up with
Speaker 1: most of the subsequent editions. Is The Ree de Numbre
Speaker 1: was published in seventeen ninety eight and appendices were added
Speaker 1: in eighteen sixteen and eighteen twenty five. The third edition,
Speaker 1: issued in two volumes in eighteen thirty, includes the results
Speaker 1: of his various later papers, and still remains a standard
Speaker 1: work on the subject. It may be said that he
Speaker 1: here carried the subject as far as was possible by
Speaker 1: the application of ordinary algebra, but he did not realize
Speaker 1: that it might be regarded as a higher arithmetic and
Speaker 1: so form a distinctive subject in mathematics. The law of
Speaker 1: quadratic reciprocity, which connects any two odd primes, is first
Speaker 1: proved in this book, but the result had been enunciated
Speaker 1: in a memoir of seventeen eighty five. Gauss called the
Speaker 1: proposition the gem of arithmetic, and no less than six
Speaker 1: separate proofs are to be found in his works. The
Speaker 1: theorem is as follows. If P be a prime and
Speaker 1: N be a prime to P, then we know that
Speaker 1: the remainder when the square root of nd to the
Speaker 1: power of P minus one is divided by P, is
Speaker 1: either plus one or minus one. Legendre denoted this remainder
Speaker 1: by N over P. When the remainder is plus one,
Speaker 1: it is possible to find a square number which, when
Speaker 1: divided by P leaves a remainder N. That is, N
Speaker 1: is a quadratic residue of P. When the remainder is
Speaker 1: minus one, there exists no such square number, and N
Speaker 1: is a non residue of P. The law of quadratic
Speaker 1: reciprocity is expressed by the theorem that of A N, B,
Speaker 1: B any odd primes, then the legentre symbol A over
Speaker 1: B times the legentre symbol B over A is the
Speaker 1: foreth root of minus one to the power of A
Speaker 1: minus one times B minus one. Thus, if B is
Speaker 1: a residue of A, then A is also residue of B,
Speaker 1: unless both the primes of A and B are of
Speaker 1: the form four times M plus three. In other words,
Speaker 1: if A and B be odd primes, then we know
Speaker 1: that the square root of A to the power of
Speaker 1: B minus one is defined as plus or minus one
Speaker 1: mod B, and the square root of B to the
Speaker 1: power of A minus one is defined as plus or
Speaker 1: minus one mod a. But Legendre's law and the two
Speaker 1: ambiguities will be either both positive or both negative unless
Speaker 1: A and B are both of the form four M
Speaker 1: plus three. Thus, if one odd prime be a non
Speaker 1: residue of another, then the latter will be a non
Speaker 1: residue of the former. Gauss and Coumer have subsequently proved
Speaker 1: similar laws of cubic and biquadratic reciprocity, and an important
Speaker 1: branch of the theory of numbers has been based on
Speaker 1: these researches. This work also contains the useful theorem by which,
Speaker 1: when it is possible an indeterminate equation of the second
Speaker 1: degree can be reduced to the form a x squared
Speaker 1: plus b y squared plus c z squared equals zero.
Speaker 1: Legentretu here discussed the forms of numbers which can be
Speaker 1: expressed as the sums of three squares, and he proved
Speaker 1: article four O four that the number of primes less
Speaker 1: than n is very approximately n divided by the quantity
Speaker 1: log n minus one point eight three sixty six. The
Speaker 1: Excelsist Cacute Integal was published in three volumes eighteen eleven,
Speaker 1: eighteen seventeen, and eighteen twenty six. Of these, the third
Speaker 1: and most of the first were devoted to elliptic functions,
Speaker 1: the bulk of this being ultimately included in the functions elliptiq.
Speaker 1: The contents of the remainder of the treatise are of
Speaker 1: a miscellaneous character. They include integration by series, definite integrals,
Speaker 1: and in particular an elaborate discussion of the beta and
Speaker 1: the gamma functions. The Preete de fonciens elliptique was issued
Speaker 1: in two volumes in eighteen twenty five and eighteen twenty six,
Speaker 1: and is the most important of le Gendre's works. A
Speaker 1: third volume was added a few weeks before his death
Speaker 1: and contains three memoirs of the researches of Abel and Jacoby.
Speaker 1: The genre's investigations have commenced with a paper written in
Speaker 1: seventeen eighty six on elliptic arcs. But here and in
Speaker 1: his other papers he treated the subject merely as a
Speaker 1: branch of the integral calculus. Tables of the elliptic integrals
Speaker 1: were constructed by him. The modern treatment of the subject
Speaker 1: is founded on that of Abel and Jacoby. The superiority
Speaker 1: of their methods was at once recognized by Legendre, and
Speaker 1: almost the last act of his life was to recommend
Speaker 1: those discoveries which which he knew would consign his own
Speaker 1: labors to comparative oblivion. This may serve to remind us
Speaker 1: of a fact which I wished to specially emphasize, namely
Speaker 1: that Gauss, able Jacobi, and some others of the mathematicians
Speaker 1: alluded to in the next chapter were contemporaries of the
Speaker 1: members of the French school. Foff I may here mention
Speaker 1: another writer who also made a special study of the
Speaker 1: integral calculus. This was Johann Friedrich Fauff born in Stuttgart
Speaker 1: on December twenty second, seventeen sixty five and died at
Speaker 1: Halle on April twenty first, eighteen twenty five, who was
Speaker 1: described by Laplace as the most eminent mathematician in Germany
Speaker 1: at the beginning of this century, a description which had
Speaker 1: it not been for Gauss's existence, would have been true enough.
Speaker 1: Fauf was the precursor of the German school, which, under
Speaker 1: Gauss and his followers, has largely determined the lines on
Speaker 1: which mathematics have developed during this century. He was an
Speaker 1: intimate friend of Gauss, and in fact the two mathematicians
Speaker 1: lived together at Helmstadt for the year after Gauss finished
Speaker 1: his university course in seventeen ninety eight. Fauf's chief work
Speaker 1: was his unfinished Disquisitions and elytiquet on the Integral Calculus,
Speaker 1: published in seventeen ninety seven, and his most important memoirs
Speaker 1: were either on the calculus or on differential equations. On
Speaker 1: the latter subject, his paper read before the Berlin Academy
Speaker 1: in eighteen fourteen, is still a standard authority the creation
Speaker 1: of modern geometry. While Euler, Lagrange, Laplace and le Gendre
Speaker 1: were perfecting analysis, the members of another group of the
Speaker 1: French mathematicians were extending the range of geometry by methods
Speaker 1: similar to those previously used by de Ragu and Pascal.
Speaker 1: The most eminent of those who created modern synthetic geometry
Speaker 1: was Poncelts, but the subject is also associated with the
Speaker 1: names of Monge and er Carnant. Its development in more
Speaker 1: recent times is largely due to Steiner, Vonstadt and Cremona. Monge.
Speaker 1: Gaspard Moinge was born at Bonn on May tenth, seventeen
Speaker 1: forty six and died at Paris on July twenty eighth,
Speaker 1: eighteen eighteen. He was a son of a small peddler
Speaker 1: and was educated in the schools of the Oratorians, one
Speaker 1: of which he subsequently became an usher. A plan of
Speaker 1: Bonn which he had made fell into the hands of
Speaker 1: an officer who recommended the military authorities to admit him
Speaker 1: to their training school at Messiere. His birth, however, precluded
Speaker 1: his receiving commission in the army, but his attendance at
Speaker 1: an annex of the school where surveying and drawing where
Speaker 1: tut was tolerated, though he was told that he was
Speaker 1: not sufficiently well born to be allowed to attempt problems
Speaker 1: which required calculation. At last his opportunity came a plan
Speaker 1: of a fortress having to be drawn from data supplied
Speaker 1: by certain observations. He did it by a geometrical construction.
Speaker 1: At first the officer in charge refused to receive it,
Speaker 1: because etiquette required that not less than a certain time
Speaker 1: should be used in making such drawings. But the superiority
Speaker 1: of the method over that then taut was so obvious
Speaker 1: that it was accepted, and in seventeen sixty eight Mange
Speaker 1: was made professor, on the understanding that the results of
Speaker 1: his descriptive geometry were to be a military secret confined
Speaker 1: to the officers above a certain rank. In seventeen eighty
Speaker 1: he was appointed to a chair of mathematics in Paris,
Speaker 1: and this, with several provincial appointments which he held, gave
Speaker 1: him a comfort income. The earliest paper of any special
Speaker 1: importance which he communicated to the French Academy, was one
Speaker 1: in seventeen eighty one, in which he discussed the lines
Speaker 1: of curvature drawn on a surface. These had been first
Speaker 1: considered by Euler in seventeen sixty and defined as those
Speaker 1: normal sections whose curvature was a maximum or a minimum.
Speaker 1: Monge treated them as the locusts of those points in
Speaker 1: the surface as which successive normals intersect, and thus obtained
Speaker 1: the general differential equation. He applied his results to the
Speaker 1: central quadrits in seventeen ninety five. In seventeen eighty six
Speaker 1: he published his well known work on statics. Monge eagerly
Speaker 1: embraced the doctrines of the revolution. In seventeen ninety two
Speaker 1: he became Minister of the Marine and assisted the Committee
Speaker 1: of the Public Safety in utilizing science for the defense
Speaker 1: of the republic. When the terrorists obtained power, he was
Speaker 1: denounced and only escaped the guillotine by a hasty flight.
Speaker 1: On his return in seventeen ninety four, he was made
Speaker 1: professor at the short lived Normal School, where he gave
Speaker 1: lectures on descriptive geometry. The notes of these were published
Speaker 1: under aregulation alluded to above. In seventeen ninety six he
Speaker 1: went to Italy on the Roving Commission, which was sent
Speaker 1: with orders to compel the various Italian towns to offer
Speaker 1: any pictures, sculpture, or other works of art that they
Speaker 1: might possess as a present or in lieu of contributions
Speaker 1: to the French Republic for removal to Paris. In seventeen
Speaker 1: ninety eight he accepted a mission to Rome, and after
Speaker 1: executing it, joined Napoleon in Egypt. Thence after the naval
Speaker 1: and military victories at England, he escaped to France. He
Speaker 1: was then made professor at the Polytechnic School, where he
Speaker 1: gave lectures on descriptive geometry. These were published in eighteen
Speaker 1: hundred in the form of a text book entitled Geometry Descriptive.
Speaker 1: This work contains prepositions on the form and relative position
Speaker 1: of geometrical figures deduced by the use of transversals. The
Speaker 1: theory of perspective is considered, and this includes the art
Speaker 1: of representing in two dimensions geometrical objects which are of
Speaker 1: three dimensions, a problem which Mage usually solved by the
Speaker 1: aid of two diagrams, one being the plan and the
Speaker 1: other being the elevation. Monge also discussed the question as
Speaker 1: to whether, if in solving a problem, certain subsidiary quantities
Speaker 1: introduced to facilitate the solution become imaginary, the validity of
Speaker 1: the solution is thereby impaired, and es shewed that the
Speaker 1: result would not be effected on the restoration. He was
Speaker 1: deprived of his offices and honors, a degradation which preyed
Speaker 1: on his mind and which he did not long survive.
Speaker 1: Most of his miscellaneous papers are embodied in his works
Speaker 1: Applicacion de la Jebra a la Geometrie, published in eighteen
Speaker 1: oh five and Applicacion den de len aalis alla Giometri,
Speaker 1: the fourth edition of which, published in eighteen nineteen, was
Speaker 1: revised by him just before his death. It contains, among
Speaker 1: other results, his solution of a partial differential equation of
Speaker 1: the second order. Carnot Lazar Nicholas Marie Carnau born at
Speaker 1: Nolas on May thirteenth, seventeen fifty three and died at
Speaker 1: Magdeburg on August twenty second, eighteen twenty three. Was educated
Speaker 1: at Burgundy and obtained a commission at the Engineer Corps
Speaker 1: of Conde. Although in the army. He continued his mathematical studies,
Speaker 1: which he felt great interest. His first work, published in
Speaker 1: seventeen eighty four, was on machines. It contained a statement
Speaker 1: which foreshadows the principle of energy is applied to a
Speaker 1: falling way, and the earliest proof of the fact that
Speaker 1: kinetic energy is lost in the collision of bodies. In
Speaker 1: the outbreak of the revolution in seventeen eighty nine, he
Speaker 1: threw himself into politics. In seventeen ninety three he was
Speaker 1: elected on the Committee of Public Safety, and the victories
Speaker 1: of the French army were largely due to his powers
Speaker 1: of organization and enforcing discipline. He continued to occupy a
Speaker 1: prominent place in every successive form of government till seventeen
Speaker 1: ninety six, when having opposed to Napoleon's Coup detais, he
Speaker 1: had to fly from France. He took refuge in Geneva,
Speaker 1: and there in seventeen ninety seven issued La Metaphysik de
Speaker 1: cac infinitesimal. In eighteen o two he assisted Napoleon, but
Speaker 1: his sincere republican convictions were inconsistent with the retention of office.
Speaker 1: In eighteen o three he produces Geometrid de position. This
Speaker 1: work deals with projective rather than descriptive geometry. It also
Speaker 1: contains an elaborate discussion of the geometrical meaning of negative
Speaker 1: roots of an algebraical equation. In eighteen fourteen he offered
Speaker 1: his services to fight for France, though not for the Empire,
Speaker 1: and on the restoration he was exiled. Poncelet Jean Victor
Speaker 1: Poncelet born at Metz in July first, seventeen eighty eight
Speaker 1: and died at Paris on December twenty second, eighteen sixty seven,
Speaker 1: held a commission in the French Engineers, having been made
Speaker 1: a prisoner in the French retreat from Moscow in eighteen twelve.
Speaker 1: He occupied his enforced leisure by writing the Trete de
Speaker 1: propertis projaxion de figures, published in eighteen twenty two, which
Speaker 1: was long one of the best known textbooks on modern geometry.
Speaker 1: By means of projection, reciprocation and homologous figures, he established
Speaker 1: all the chief properties of conix and quadrits. He also
Speaker 1: treated the theory of polygons his Treatise on the Practical
Speaker 1: Mechanics in eighteen twenty eight, his Memoir on Water Mills
Speaker 1: in eighteen twenty six, and his report on the English
Speaker 1: machinery and tools exhibited at the International Exhibition held in
Speaker 1: London in eighteen fifty one deserve mention. He contributed numerous
Speaker 1: articles to Krell's journal. The most valuable of these deal
Speaker 1: with the explanation of imaginary solutions and geometrical problems by
Speaker 1: the aid of the doctrine of continuity. The development of
Speaker 1: mathematical physics. It will be noticed that Lagrange, Laplace and
Speaker 1: le Genre mostly occupied themselves with analysis, geometry and astronomy.
Speaker 1: I am inclined to regard Cosche and the French mathematicians
Speaker 1: of the present day as belonging to a different school
Speaker 1: of thought to that considered in this chapter, and I
Speaker 1: placed them amongst modern mathematicians. But I think that Fourier,
Speaker 1: Poissin and the majority of their contemporaries are the lineal
Speaker 1: successors of Lagrange and Laplace. If this view is correct,
Speaker 1: it would seem that the latter members of the French
Speaker 1: school devoted themselves mainly to the application of mathematical analysis
Speaker 1: to physics. Before considering these mathematicians, I may mention the
Speaker 1: distinguished English experimental physicists who were their contemporaries and whose
Speaker 1: merits have only recently received an adequate recognition. Chief among
Speaker 1: these are Cavendish and Young Cavendish. The Honorable Henry Cavendish
Speaker 1: born at Nice on October tenth, seventeen thirty one, and
Speaker 1: died at London on February twenty fourth, eighteen ten. His
Speaker 1: tastes for scientific research and mathematics were formed at Cambridge,
Speaker 1: where he resided from seventeen forty nine to seventeen fifty three.
Speaker 1: He created experimental electricity and was one of the early
Speaker 1: writers to treat chemistry as an exact science. I mention
Speaker 1: him here on account of its experiment in seventeen ninety
Speaker 1: eight to determine the density of the Earth by estimating
Speaker 1: its attraction as compared with that of two given lead balls.
Speaker 1: The result is that the mean density of the Earth
Speaker 1: is about five and a half times that of water.
Speaker 1: This experiment was carried out in accordance with a suggestion
Speaker 1: which had been first made by John Mitchell, a fellow
Speaker 1: of Queen's College, Cambridge, who had died before he was
Speaker 1: able to carry it into effect. End of Part thirty
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