Speaker 1: Chapter nineteen, Part three of a short account of the
Speaker 1: history of mathematics by W. W. Rowseball. This is a
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Speaker 1: This is a reading by Paul King PJK dot Scripts
Speaker 1: dot mit dot edu. A short account of the history
Speaker 1: of mathematics by W. W. Rowsball, Chapter nineteen, Mathematics of
Speaker 1: Recent Times, Part three. The consideration of algebraical, trigonometrical, elliptic,
Speaker 1: hyper elliptic and other special kinds of functions pave the
Speaker 1: way for a theory of functions which promises to prove
Speaker 1: a most important and far reaching branch of mathematics. To
Speaker 1: a large extent, this is the work of living mathematicians,
Speaker 1: and therefore outside the limits of this chapter. I will
Speaker 1: content myself by referring to the following writers. First, I
Speaker 1: may mention Kosche, who gave the general elementary theory of functions,
Speaker 1: and Lieuville, who wrote chiefly on doubly periodic functions. Their
Speaker 1: investigations were extended and connected in the work of Brio
Speaker 1: and Bouquet, and further been developed by Hermite Next, I
Speaker 1: may refer to the researches on the theory of algebraic functions,
Speaker 1: which have their origin in Riemond's paper of eighteen fifty.
Speaker 1: Schwartz has established accurately certain theorems of which proofs given
Speaker 1: by Riemond were open to objection. Klein has connected Riemond's
Speaker 1: theory of functions with the theory of groups and have
Speaker 1: written on automorphic functions. Henri point Carey, professor in Paris,
Speaker 1: born at Nancy in eighteen fifty four, has also written
Speaker 1: on automorphic functions and on the general theory with special
Speaker 1: applications to differential equations. Finally, I may refer to the
Speaker 1: work of Weierstrauss and Mittaghlefer. Of these, Karl Weierstrauss has
Speaker 1: created a large part of the modern theory of functions,
Speaker 1: and in particular has constructed the theory of uniform analytical functions.
Speaker 1: And Magnus Gustav mitteg Leffler born at Stockholm in eighteen
Speaker 1: forty six and now professor there has greatly developed the
Speaker 1: theory of analytical functions, a subject on which Hermite has
Speaker 1: also written. In connection with these researches, Paul Emil Appel
Speaker 1: professor in Paris, born at Strasburg in eighteen fifty eight.
Speaker 1: C Emir Picard of Paris Edouard Goursat of Paris has
Speaker 1: written on special branches of the theory. As textbooks. I may
Speaker 1: mention Doctor Forsiss Theory of Functions of a complex variable, Cambridge,
Speaker 1: eighteen ninety three and Karl Numont's Vorslungen Uberriman's Theory der
Speaker 1: Absersion integral second edition, Leipezig, eighteen eighty four. The theory
Speaker 1: of numbers may be considered as a higher arithmetic, and
Speaker 1: the theory of elliptical and Abelian functions as a higher trigonometry.
Speaker 1: The theory of higher algebra, including the theory of equations,
Speaker 1: has also attracted considerable attention and was a favorite subject
Speaker 1: of study of the three mathematicians Kosche, Hamilton, and de Morgan,
Speaker 1: whom I propose to mention next. Though the interests of
Speaker 1: these writers were by no means limited to this subject. Koschi,
Speaker 1: the first of these mathematicians, is the best representative of
Speaker 1: the French school of analysis in this century. Augustine Luis Kosche,
Speaker 1: who was born in Paris on August twenty first, seventeen
Speaker 1: eighty nine, and died at Sioux on May twenty fifth,
Speaker 1: eighteen fifty seven. Was educated at the Polytechnic School, the
Speaker 1: nursery of so many French mathematicians of that time, and
Speaker 1: adopted the profession of a civil engineer. His earliest mathematical
Speaker 1: paper was won on polyhedra in eighteen eleven. Legendre thought
Speaker 1: so highly of it that he asked Koshi to attempt
Speaker 1: the solution of an analogous problem which had baffled previous investigators,
Speaker 1: and his advice was justified by the success of Kochi
Speaker 1: in eighteen twelve. Memoirs on Analysis and the Theory of Numbers,
Speaker 1: presented in eighteen thirteen, eighteen fourteen and eighteen fifteen, shewed
Speaker 1: that his ability was not confined to geometry alone. In
Speaker 1: one of these papers he generalized some results which had
Speaker 1: been established by Gauss and Legendre. In another of them,
Speaker 1: he gave a theorem on the number of values which
Speaker 1: an algebraical function can assume when the literal constance it
Speaker 1: contains are interchanged. It was the latter theorem that enabled
Speaker 1: Abel to shew that in general, an algebraic equation of
Speaker 1: a degree higher than the fourth cannot be solved by
Speaker 1: the use of purely algebraical expressions. To Koshe and Gauss,
Speaker 1: we owe the scientific treatment of series, which have an
Speaker 1: infinite number of terms, and the former established general rules
Speaker 1: for investigating the convergency and divergency of such series. It
Speaker 1: is only a few works of an earlier date that
Speaker 1: contain any discussion as to the limitations of the series employed.
Speaker 1: It is said that Laplace, who was present when Koshi
Speaker 1: read his first paper on the subject, was so impressed
Speaker 1: by the illustrations of the danger of employing such series
Speaker 1: without a rigorous investigation of their convergency, that he put
Speaker 1: on side on one side the work on which he
Speaker 1: was then engaged, and denied himself to all visitors, in
Speaker 1: order to see if any of the demonstrations given in
Speaker 1: the earlier volumes of the Mechanique Celeste were invalid, and
Speaker 1: he was fortunate enough to find that no material errors
Speaker 1: had been thus introduced. The treatment of series and of
Speaker 1: the fundamental conceptions of the calculus in most of the
Speaker 1: textbooks then current, was based on Euler's works, and to
Speaker 1: any one trained to accurate habits of thought, was not
Speaker 1: free from objection. It is one of the chief merits
Speaker 1: of Koshe that he placed those subjects on a logical foundation.
Speaker 1: On the restoration. In eighteen sixteen, the French Academy was purged,
Speaker 1: and in spite of the indignation and scorn of French
Speaker 1: scientific society, Kosche accepted a seat which was procured for
Speaker 1: him by the expulsion of Munge. He was also at
Speaker 1: the same time made professor at the Polytechnic, and his
Speaker 1: lectures there on algebraic and elisons, the calculus and the
Speaker 1: theory of curves were published as textbooks on the Revolution.
Speaker 1: In eighteen thirty he went into exile and was first
Speaker 1: appointed professor at Tehennes, where he soon moved to Prague
Speaker 1: to undertake the education of the Comte de Chambaud. He
Speaker 1: returned to France in eighteen thirty seven, and in eighteen
Speaker 1: forty eight, and again in eighteen fifty one, by special
Speaker 1: dispensation of the Emperor, was allowed to occupy a chair
Speaker 1: of mathematics without taking the oath of allegiance. His activity
Speaker 1: was prodigious, and from eighteen thirty to eighteen fifty nine
Speaker 1: he published in the transactions of the Academy of the
Speaker 1: Comptes Rendeus over six hundred original memoirs and about one
Speaker 1: hundred and fifty reports. In most of them the feverish
Speaker 1: haste with which they were thrown off as too visible,
Speaker 1: and many are marred by obscurity, repetition of old results,
Speaker 1: and blunders. Among the more important of his resist searches
Speaker 1: are the discussion of tests for the convergency of series
Speaker 1: and the determination of the number of real and imaginary
Speaker 1: roots of any algebraic equation, his method of calculating these
Speaker 1: roots approximately, his theory of the symmetric functions of the
Speaker 1: coefficiens of equations of any degree, his a priori valuation
Speaker 1: of quantity less than the least difference between the roots
Speaker 1: of an equation, and his papers on determinants in eighteen
Speaker 1: forty one, which did a great deal to bring them
Speaker 1: into general use. Koshi also did something to reduce the
Speaker 1: art of determining definite integrals to a science, but this
Speaker 1: branch of the integral calculus still remains without much system
Speaker 1: or method. The rule for finding the principal values of
Speaker 1: integrals was enunciated by him, and the calculus of residues
Speaker 1: was his invention. His proof of Taylor's theorem seems to
Speaker 1: have originated from a discussion of the double periodicity of
Speaker 1: elliptic functions. The means of shewing a connection between the
Speaker 1: different branches of a subject by giving imaginary values to
Speaker 1: independent variables is largely due to him. He also gives
Speaker 1: a direct analytical method for determining planetary inequalities of long
Speaker 1: period and to physics, he contributed a memoir on the
Speaker 1: quantity of light reflected from the surfaces of metals, as
Speaker 1: well as other papers on optics. Argande I may mention
Speaker 1: here the name of Jean Robert Argaunde, who was born
Speaker 1: at Geneva on July twenty seconds, seventeen sixty eight and
Speaker 1: died circa eighteen twenty five. In his essay issued in
Speaker 1: eighteen o six, he gave a geometrical representation of a
Speaker 1: complex number and applied it to shew that every algebraic
Speaker 1: equation has a root. This was prior to the memoirs
Speaker 1: of Gaussen Kosche on the same subject, but the essay
Speaker 1: did not attract much attention when it was first published.
Speaker 1: An earlier demonstration that the square root of minus one
Speaker 1: indicates perpendicularity, due to Beuet, was published in the Philosophical
Speaker 1: Transactions for eighteen o six, and the idea was foreshadowed
Speaker 1: in a memoir by H. Kuhn in the Transactions for
Speaker 1: seventeen fifty of the Saint Petersburg Academy. Hamilton, in the
Speaker 1: opinion of some writers, the theory of quaternions will be
Speaker 1: ultimately esteemed by one of the great discoveries of this century.
Speaker 1: The discovery is due to Sir William Rowan Hamilton, who
Speaker 1: was born of Scotch parents in Dublin on August fourth,
Speaker 1: eighteen o five, and died there on September second, eighteen
Speaker 1: sixty five. His education, which was carried out at home,
Speaker 1: seems to have been singularly discursive. Under the influence of
Speaker 1: an uncle who was a good linguist, he first devoted
Speaker 1: himself to linguistic studies. By the time he was seven,
Speaker 1: he could read Latin, Greek, Friends and German with facility,
Speaker 1: and when thirteen he was able to boast that he
Speaker 1: was familiar with as many languages as he had lived years.
Speaker 1: It was about this time that he came across a
Speaker 1: copy of Newton's Universal Arithmetic. This was his introduction to
Speaker 1: modern analysis, and he soon mastered the elements of analytical
Speaker 1: geometry and the calculus. He next read the Principia and
Speaker 1: the four volumes then published of Laplace's Mechanique Celeste. In
Speaker 1: the latter he detected a mistake, and his paper on
Speaker 1: the subject, written in eighteen twenty three, placed him at
Speaker 1: once in the front rank of mathematicians. In the following
Speaker 1: year he entered at Trinity College, Dublin. His university careers
Speaker 1: unique for the chair of Astronomy, becoming vacant in eighteen
Speaker 1: twenty seven while he was yet an undergraduate. He was
Speaker 1: asked by the electors to stand for it, and was
Speaker 1: elected unanimously, it being understood that he should be left
Speaker 1: free to pursue his own line of study. Earliest paper,
Speaker 1: written in eighteen twenty three was on optics, and was
Speaker 1: published in eighteen twenty eight under the title of a
Speaker 1: Theory of Systems of Rays, which two supplements written in
Speaker 1: eighteen thirty one in eighteen thirty two were afterwards added.
Speaker 1: In the latter of these, the phenomenon of conical refraction
Speaker 1: is predicted. This was followed by a paper in eighteen
Speaker 1: twenty seven on the principle of varying action, and in
Speaker 1: eighteen thirty four and eighteen thirty five by memoirs on
Speaker 1: a general method in dynamics, the subject of theoretical dynamics
Speaker 1: being properly treated as a branch of pure mathematics. His
Speaker 1: Lectures on Quaternions was published in eighteen fifty two. Amongst
Speaker 1: his other papers, I may specially mention one on the
Speaker 1: form of the solution of the general algebraic equation of
Speaker 1: the fifth degree, which confirmed the conclusion arrived at by
Speaker 1: Abel that it cannot be expressed in terms of the
Speaker 1: more elementary operations and functions, one on fluctuating functions, one
Speaker 1: on the hodograph, and lastly on the numerical solution of
Speaker 1: differential equations e. His Elements of Quaternions were issued in
Speaker 1: eighteen sixty six. Of this, a competent authority says that
Speaker 1: the methods of analysis there given shoe as great an
Speaker 1: advance over those of analytical geometry, as the latter shued
Speaker 1: over those of Euclidean geometry. In more recent times a
Speaker 1: subject has been further developed by Tait. Hamilton was painfully
Speaker 1: fastidious on what he published, and he left an immense
Speaker 1: collection of manuscript books, which are now in the library
Speaker 1: of Trinity College, Dublin, and some of which it is hoped,
Speaker 1: will be ultimately printed Grassman. The idea of noncommutative algebras
Speaker 1: and of quaternions seemed to have occurred to Grassmann about
Speaker 1: the same time as to Hamilton. Hermann Gunther Grassmann born
Speaker 1: at stettin An April fifteenth, eighteen o nine and died
Speaker 1: there in eighteen seventy seven. He was a professor at
Speaker 1: the Gymnasium at Stettin. His researches on noncommutative algebras are
Speaker 1: contained in his Austio Anglire, first published in eighteen forty
Speaker 1: four and enlarged in eighteen sixty two. The scientific treatment
Speaker 1: of the fundamental principles of algebra initiated by Hamilton and
Speaker 1: Grassmann were continued by De Morgan and Buell, and subsequently
Speaker 1: was further developed by H. Henkel in his work on
Speaker 1: Complexes eighteen sixty seven and by G. Cantor on his
Speaker 1: Memoirs on the Theory of Irrationals eighteen seventy one. The
Speaker 1: discussion is, however, so technical, that I am unable to
Speaker 1: do more than allude to it. Grassmann also investigated the
Speaker 1: properties of homoloidal hyperspace. De Morgan Augustus de Morgan born
Speaker 1: in Madura, Madras in June eighteen o six, and died
Speaker 1: in London on March eighteenth, eighteen seventy one. Was educated
Speaker 1: at Trinity College, Cambridge, but in the then state of
Speaker 1: the law, was as a Unitarian ineligible to a fellowship.
Speaker 1: In eighteen twenty eight he became professor at the newly
Speaker 1: established University of London, which is the same institution as
Speaker 1: that now known as University College. There, except for five
Speaker 1: years from eighteen thirty one to eighteen thirty five, he
Speaker 1: taught continuously till eighteen sixty seven, and through his works
Speaker 1: and pupils, exercised a wide influence on English mathematicians of
Speaker 1: the present day. The London Mathematical Society was largely his creation,
Speaker 1: and he took a prominent part in the proceedings of
Speaker 1: the Royal Astronomical Society. He was perhaps more deeply read
Speaker 1: in the philosophy and history of mathematics than any of
Speaker 1: his contemporaries, but the results are given in scattered articles
Speaker 1: as well, which well deserved collection and republication. A list
Speaker 1: of these is given in his life, and I have
Speaker 1: made considerable use of some of them in this book.
Speaker 1: The best known of his works are the Memoirs of
Speaker 1: the Foundation of Algebra, Cambridge Philosophical Transactions, Volumes eight and
Speaker 1: nine is treatise on the differential calculus, published in eighteen
Speaker 1: forty two, a work of great ability and noticeable for
Speaker 1: the rigorous treatment of infinite series, and his articles on
Speaker 1: the calculus of functions on the theory of probabilities in
Speaker 1: the Encyclopedia Metropolitana. The article on the calculus of functions
Speaker 1: contains an investigation of the principles of symbolic reasoning, but
Speaker 1: the applications deal with the solution of functional equations rather
Speaker 1: than with the general theory of functions. The article on
Speaker 1: the theory of probabilities gives a clear analysis of the
Speaker 1: mathematics of the subject to the time at which it
Speaker 1: was written. Besides those above named, I may mention the
Speaker 1: following who have written on the subjects of higher algebra,
Speaker 1: the theory of forms and the theory of equations. Joseph
Speaker 1: Ludwig Rabbe, who in eighteen thirty two discussed the tests
Speaker 1: for the convergency of series, a subject also discussed later
Speaker 1: by Joseph Louis Francois Bertrand, Secretary of the French Academy
Speaker 1: born in Paris in eighteen eight twenty two. Couhmer URIs
Speaker 1: Deni of Pisa and A. Pringsheim of Munich, and on
Speaker 1: the researches of the above writers see the Bulletin of
Speaker 1: the New York Mathematical Society, October eighteen ninety two, pages
Speaker 1: one to ten. George Boule born at Lincoln in November second,
Speaker 1: eighteen fifteen, died at Cork on December eighth, eighteen sixty four,
Speaker 1: who invented a system of noncommunative algebra, and from whose
Speaker 1: memoirs on linear transformations, part of the theory of covariance
Speaker 1: has developed. Everist galois one of the most original and
Speaker 1: powerful mathematicians of the century, born at Paris on October
Speaker 1: twenty six, eighteen eleven, and then killed in a duel
Speaker 1: on May thirtieth, eighteen thirty two, at the early age
Speaker 1: of twenty, whose writings are mainly concerned with the theory
Speaker 1: of equations and substitution groups. On his investigations see Liauville's
Speaker 1: Journal for eighteen forty six, Volume eleven, pages three eighty one,
Speaker 1: four forty four and the American Journal of Mathematics for
Speaker 1: eighteen ninety one, Volume thirteen pages one hundred nine to
Speaker 1: one hundred forty two. Carl Wilhelm Borchardt, Professor in Berlin,
Speaker 1: born there on February twenty second, eighteen seventeen, died there
Speaker 1: eighteen eighty, who in particular discussed generating functions in the
Speaker 1: theory of equations and er arithmetic geometric means. A collected
Speaker 1: edition of his works, edited by g Hettner, was issued
Speaker 1: at Berlin in eighteen eighty one. Eight Kayley, whose ten
Speaker 1: classical memoirs on quantics, binary and ternary forms, and researches
Speaker 1: on noncommutative algebras, especially on matrices, will be found in
Speaker 1: the collected edition of his works. Sylvester from among those
Speaker 1: whose numerous memoirs I may in particular single out those
Speaker 1: on canonical forms, on the theory of contravariance, reciprocants of
Speaker 1: differential invariants, on the theory of equations, and that on
Speaker 1: Newton's rule, to which I may add that he has
Speaker 1: created the language and notation of considerable parts of the
Speaker 1: subjects on which he has written. Camille Jordan, who has
Speaker 1: written on the theory of substitutions in general and with
Speaker 1: special applications to differential equations. Sir George Gabriel Stokes, Lucasian
Speaker 1: professor in the University of Cambridge, born near Sligo, August thirteenth,
Speaker 1: eighteen nineteen, who has written on the critical values of
Speaker 1: the sums of the periodic series and on the summation
Speaker 1: of series Cambridge Philosophical Transactions, eighteen forty seven, Volume eight,
Speaker 1: pages five thirty three to five eighty three. See also
Speaker 1: below pages four ninety two and four ninety six. Eugene
Speaker 1: Netto of Strasburg, who has written on substitutions Hermite, who
Speaker 1: has in particular discussed the theory of associated contravariance in
Speaker 1: binary quantics, the theory of Turnery quantics, and who has
Speaker 1: applied elliptic functions to the solution of the quintic equation.
Speaker 1: Enrico Betty of Pisa, who died in eighteen ninety two
Speaker 1: and Brioschi, both of whom discussed binary quantics. Siegfried Heinrich
Speaker 1: Hnhold born at Engerberg on July sixteenth, eighteen nineteen, who
Speaker 1: developed symbolic methods, especially in connection with turnery quantics. This
Speaker 1: was done concurrently with, but independently of Keiley's work on
Speaker 1: the same subject. Paul Gorden, professor at Erlingen, who had
Speaker 1: discussed the theory of forms enchewed that there were only
Speaker 1: finite number of concomitants of quantics. An edition of his
Speaker 1: work on invariants, determinants and binary forms, edited by G. Kersteiner,
Speaker 1: was issued at Leipzig in three volumes eighteen eighty five,
Speaker 1: eighteen eighty seven and eighteen ninety three. Rudolph Friedrich Alfred
Speaker 1: Klebsch born at Knigsburg in eighteen thirty three, died in Gautigen,
Speaker 1: where he was professor in eighteen seventy two, who also
Speaker 1: independently investigated the theory of binary forms in some papers
Speaker 1: collected and published in eighteen seventy one. An account of
Speaker 1: his life and works as printed in The Mathematist Annelen
Speaker 1: eighteen seventy three, volume six, pages one ninety seven to
Speaker 1: two oh two and eighteen seventy four volume seven, pages
Speaker 1: one to fifty five. See also below. McMahon, who has
Speaker 1: written on the connection of symmetric functions, the derivation of
Speaker 1: invariants and the covariance from elementary algebra, and the concomitance
Speaker 1: of binary forms. Sophis Lae, professor at Leipzig, who has
Speaker 1: written on groups of continuous substitutions, differential invariants and complexes
Speaker 1: of lines. Klein, who has investigated the problem of discontinuous
Speaker 1: substitutions and polyhedra groups. And lastly Andrew Russell Forsyth, Fellow
Speaker 1: and lecturer of Trinity College, Cambridge, born at Glasgow on
Speaker 1: June eighteenth, eighteen fifty eight, who has developed a theory
Speaker 1: of invariants of differential equations, ternarians and quaternarians. No account
Speaker 1: of contemporary writings of this subject would be complete without
Speaker 1: a reference to the admirable textbooks produced by George Salmon,
Speaker 1: Provost at Trinity Trinity College, Dublin, born in eighteen nineteen
Speaker 1: in his Higher algebra, and by Joseph Alfred Serret, professor
Speaker 1: at the Sorbonne in Paris on August thirtieth, eighteen nineteen
Speaker 1: died in eighteen eighty five, in his Coups delchebre Superieur,
Speaker 1: in which the chief discoveries of their respective of authors
Speaker 1: are embodied. An admirable historical summary of the theory of
Speaker 1: the complex variable is given in the vassolingen Uberg Dies
Speaker 1: complex in Zalen, Leipzig, eighteen sixty seven by H. Henkel,
Speaker 1: professor at Tubingen, born at Halle in eighteen thirty nine
Speaker 1: and died at Schremburg in eighteen seventy three. Before mentioning
Speaker 1: the creators of modern synthetic geometry, it will be convenient
Speaker 1: to call attention to two other divisions of pure mathematics,
Speaker 1: which have been greatly developed in recent years. But any
Speaker 1: sketch of the results arrived at or the methods by
Speaker 1: which they have been attained, would be so closely connected
Speaker 1: with the work of living mathematicians that I shall do
Speaker 1: little more than mention the names of the subjects. Analytical
Speaker 1: geometry has been studied by a host of modern writers,
Speaker 1: but I do not propose to describe their investigations, and
Speaker 1: I shall content myself by merely mentioning the names of
Speaker 1: the following mathematicians. James Booth born in the country of
Speaker 1: Leetrim on August twenty fifth, eighteen o six and died
Speaker 1: at Buckinghamshire on April fifteenth, eighteen seventy eight was one
Speaker 1: of the earliest writers in this century to devote himself
Speaker 1: to the development of analytical geometry. His chief results are
Speaker 1: embodied in his work entitled A treatise on some new
Speaker 1: geometrical methods. The researches of James McCulloch, professor in Dublin,
Speaker 1: born near Starbane in eighteen o nine and died in
Speaker 1: Dublin in October twenty fourth eve eighteen forty six, which
Speaker 1: includes some valuable discoveries in the theory of quadrics, will
Speaker 1: be found in this collected works edited by Jellet and Houghton, Dublin,
Speaker 1: eighteen eighty See also below eighteen page four ninety six.
Speaker 1: Julius Plucker, professor after eighteen thirty six in Bonn, born
Speaker 1: at Elberfield on July sixteenth, eighteen o one and died
Speaker 1: at Bonn on May twenty second, eighteen sixty eight, devoted
Speaker 1: himself chiefly to the study of algebraic curves of a
Speaker 1: geometry in which the line is the element in space,
Speaker 1: and the theory of congruences and complexes. His equations connecting
Speaker 1: the singularities of curves are well known. In eighteen forty
Speaker 1: seven he exchanged his chair for one of physics, and
Speaker 1: his subsequent researches were on spectra and magnetism. An account
Speaker 1: of his works was published by clubsch gelschafter Wizenschaften, Goettingen,
Speaker 1: eighteen seventy two, Volume sixteen. The majority of the memoirs
Speaker 1: on analytical geometry by Kayley and by Henry Smith's deal
Speaker 1: with the theory of curves and services. The most remarkable
Speaker 1: of those of Ludwig otto Hess born at Knigsburg on
Speaker 1: eight Pril twenty second, eighteen eleven and died at Munich,
Speaker 1: where he was professor in eighteen seventy four, or on
Speaker 1: the plain geometry of curves see the notice of these
Speaker 1: by FC. Klein, of those of Jean Gaston d'arboux, professor
Speaker 1: in Paris born at Nines at eighteen forty two, on
Speaker 1: the geometry of services, and those of Halfen on the
Speaker 1: singularities of surfaces and on tortuous curves. The singularities of
Speaker 1: curves and surfaces have also been considered by Hieronymus George Suthen,
Speaker 1: professor at Copenhagen born in eighteen thirty nine, and by
Speaker 1: Hermann Keesar Hannibal Schubert, professor at Hamburg, born at Potsdam
Speaker 1: in eighteen forty eight. The lectures of the latter have
Speaker 1: been published by F. Lindemann two volumes, Leipezig, eighteen seventy
Speaker 1: five and eighteen ninety one. Notether has discussed the theory
Speaker 1: of tortuous curves, and Klebsch has applied Abel's theorem to geometry.
Speaker 1: Among more recent textbooks are Klebsch's vordusunjen ubur Geometry edited
Speaker 1: by F. Lindemann, and Selman's conic Sections Geometry of three
Speaker 1: dimensions and Higher Plane Curves, in which the chief discoveries
Speaker 1: of these writers are embodied. Finally, I may allude to
Speaker 1: the extension of the subject matter of analytical geometry by
Speaker 1: the introduction of the ideas of space of n dimensions
Speaker 1: and the writings of Grassmann in eighteen forty four. In
Speaker 1: eighteen sixty two, among those who have extended the range
Speaker 1: of analysis, including the calculus and differential equations are whom
Speaker 1: it is difficult to place in any of the preceding
Speaker 1: categories are the following, whom I place in alphabetical order,
Speaker 1: Appel bertrand Buel Koshi d'arbou scythe who has written on
Speaker 1: Faff's problem and is also the author of the standard
Speaker 1: English treatise on differential equations. Robinius Lazarus Fuchs, Professor at
Speaker 1: Berlin born in Prussia eighteen thirty three, Halfen Jacobi, Jordan Kanigsberger,
Speaker 1: Sophie Kuloweski, professor at Stockholm born on December twenty seventh,
Speaker 1: eighteen fifty three and died eighteenth, eighteen ninety one. See
Speaker 1: the bulletin Descience Mathematique, Volume fifteen, pages two twelve to
Speaker 1: two twenty lye pointerre Riemann, who wrote on the theory
Speaker 1: of partial differential equations, Schwartz, Sylvester and Weierstrauss, who has
Speaker 1: developed the calculus of variations. The writers I have mentioned
Speaker 1: above mostly concerned themselves with analysis. I will next describe
Speaker 1: some of the more important works produced in this century
Speaker 1: on synthetic geometry. Modern synthetic geometry may be said to
Speaker 1: have had its origin in the works of mange in
Speaker 1: eighteen Hudred Carnelt in eighteen o three, and Poncelet in
Speaker 1: eighteen twenty two, but these only dimly foreshadowed the great
Speaker 1: extension it was to receive in Germany, of which Steiner
Speaker 1: and von Staudt are perhaps the best known exponents. Steiner
Speaker 1: Jacob Steiner, the greatest geometrician since the time of Apollonius,
Speaker 1: was born at Utzendorf on March eighteenth, seventeen ninety six,
Speaker 1: and died at Berne on April first, eighteen sixty three.
Speaker 1: His father was a peasant, and the boy had no
Speaker 1: opportunity to learn reading and writing till the age of fourteen.
Speaker 1: He subsequently read to Heidelberg and thence to Berlin, supporting
Speaker 1: himself by giving lessons. His system Metische enter Whichelungen was
Speaker 1: published in eighteen thirty two and at once made his reputation.
Speaker 1: It contains a full discussion of the principle of duality
Speaker 1: and of the projective and homographic relations of rose pencils,
Speaker 1: et cetera, based on metrical properties. By the influence of Krell,
Speaker 1: Jacobi and the van Humboldts, who were impressed by the
Speaker 1: power of this work, a chair of geometry was created
Speaker 1: for Steiner at Berlin, and he continued to occupy it
Speaker 1: till his death. The most important of his other researches
Speaker 1: are contained in papers which appeared originally in Krell's journal
Speaker 1: and are embodied in his Systematist Geometrie Volume one edited
Speaker 1: by C. F. Geyser, volume two by H. Schrozer. These
Speaker 1: relate chiefly to the properties of algebraic curves, and surfaces,
Speaker 1: pedals and roulettes, and maxima and minima. The discussion is
Speaker 1: purely geometrical. Steiner's works may be considered as the classical
Speaker 1: authority on recent synthetic geometry. Von Stadt, a system of
Speaker 1: pure geometry quite distinct from that expanded by Steiner, was
Speaker 1: proposed by Karl George Christian von Stadt born at Ruthingburg
Speaker 1: on January twenty fourth, seventeen ninety eight and died in
Speaker 1: eighteen sixty seven, who held the chair of mathematics at Erleengen.
Speaker 1: In his Geometry Grlage, published in eighteen forty seven, he
Speaker 1: constructed a system of geometry built up without any reference
Speaker 1: to number or magnitude. But in spite of its abstract form,
Speaker 1: he succeeded by means of it alone in establishing the
Speaker 1: non metrical projective properties of figures, discussed imaginary points, lines
Speaker 1: and planes, and even obtained a geometrical definition of a number.
Speaker 1: These views were further elaborated in his Bertrages Geometry Derlage
Speaker 1: eighteen fifty six to eighteen sixty This geometry is curious
Speaker 1: and brilliant, and has been used by Colemann as the
Speaker 1: basis of his graphical statistics. Among other works on pure geometry,
Speaker 1: I may refer to the introduzzione ado una terria geometoca
Speaker 1: della cuva Piane eighteen sixty two and its continuation Prelimieri
Speaker 1: de una Teoria Geometrica della superficie by Luigi Cremona of
Speaker 1: the Polytechnic School at Rome. As usual textbooks, I may
Speaker 1: mention M. Challes's treetege Geometri superieur eighteen fifty two, J.
Speaker 1: Steiner's voresunjen Ubercyntetitia Geometri eighteen sixty seven and L. Cremona's
Speaker 1: ele Monte Geometri Projective translated into English by C. Luzendorff, Oxford,
Speaker 1: eighteen eighty five and Thomas Reyes Geometri Delage, three Volumes,
Speaker 1: third edition. I shall conclude the chapter with a few
Speaker 1: notes more or less discursive on branches of mathematics of
Speaker 1: a less abstract character and concerned with problems that occur
Speaker 1: in nature. Closely connected with the subject of modern geometry
Speaker 1: is a science of graphics, in which rules are laid
Speaker 1: down for solving various problems by the aid of the
Speaker 1: drawing board. The modes of calculation which are permissible are
Speaker 1: considered in modern projective geometry. This method of attacking questions
Speaker 1: has been hitherto applied chiefly to problems in mechanics, elasticity,
Speaker 1: and electricity, and it is especially useful in engineering, and
Speaker 1: in that subject, an average draftsmen ought to be able
Speaker 1: to obtain approximate solutions of most of the equations differential
Speaker 1: or otherwise with which he is likely to be concerned,
Speaker 1: which will not involve errors greater than would have to
Speaker 1: be allowed for in any case in consequence of our
Speaker 1: imperfect knowledge of the structure of the materials employed. The
Speaker 1: theory may be said to have originated with Poncelette's work,
Speaker 1: but I believe that it is only within the last
Speaker 1: twenty years that systematic expositions of it have been published.
Speaker 1: Among the best known of such works I may mention
Speaker 1: are Grafiche Statique by C. Culman Zurich eighteen seventy five,
Speaker 1: recently edited by w Ritter, and the Lizioni di Statistica
Speaker 1: Graphica by A. Favaro Padua eighteen seventy seven French translation
Speaker 1: annotated by P. Terrier in two volumes eighteen seventy nine
Speaker 1: to eighty five, and the Calcolo Graphico by L. Cremona, Milan,
Speaker 1: eighteen seventy nine. English translation by T. H. Ber Oxford
Speaker 1: eighteen eighty nine, which is largely founded on Mobius's work.
Speaker 1: Las Statique Graphique by M. Levy, Paris, four volumes, eighteen
Speaker 1: eighty six to eighty eight and Las Statistica Graphica by C. Serotti, Milan,
Speaker 1: eighteen eighty eight. The general character of these books will
Speaker 1: be sufficiently illustrated by the following note on the concepts
Speaker 1: of Culman's work. Culman commences with a description of the
Speaker 1: geometrical representation of the four fundamental processes of addition, subtraction, multiplication,
Speaker 1: and division, and proceeds to evolution and involution, the latter
Speaker 1: being affected by the use of equilangular spirals. He next
Speaker 1: shows how the quantities considered such as volumes, moments, and
Speaker 1: moments of inertia, may be represented by straight lines. Thence
Speaker 1: deduces the laws for combining forces, couples, et cetera, and
Speaker 1: then explains the construction and use of the ellipse and
Speaker 1: ellipsoid of inertia, the neutral axis, and the kern. The
Speaker 1: remaining and larger part of the book is devoted to
Speaker 1: shewing how geometrical drawings made up on these principles give
Speaker 1: the solutions of many practical problems connected with arches, bridges, frameworks,
Speaker 1: earth pressure on walls and tunnels, et cetera. The subject
Speaker 1: has been treated during the last twenty years by numerous writers,
Speaker 1: especially in Italy and Germany, and applied to a large
Speaker 1: number of problems. But as I stated at the beginning
Speaker 1: of this chapter that I should, as far as possible,
Speaker 1: avoid discussions of the works of living authors, I content
Speaker 1: myself with a bare mention of the subject. End of
Speaker 1: section thirty six. Recording by Paul King, Mississauga, Ontario, p
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