Speaker 1: Chapter nineteen, Part two of A short account of the
Speaker 1: history of mathematics by W. W. Rowse Ball. This is
Speaker 1: a LibriVox recording. All LibriVox recordings are in the public domain.
Speaker 1: For more information or to volunteer, please visit LibriVox dot org.
Speaker 1: This is a reading by Paul King p J K
Speaker 1: dot scripts that I might t dot e d U
Speaker 1: forward slash p k J A short account of the
Speaker 1: history of mathematics by W. W. Rous Ball, Chapter nineteen,
Speaker 1: Mathematics of Recent Times, Part two, di Grey. One of
Speaker 1: Gauss's pupils to whom I may here allude is lejeun Derichree,
Speaker 1: who is generally known for his exposition of the discoveries
Speaker 1: of Jacoby, who was his father in law and of Gauss,
Speaker 1: rather than for his own original investigations, valuable, though some
Speaker 1: of these are Peter Gustave le Jeundr Crey was born
Speaker 1: at Dhurin on February thirteenth, eighteen o five and died
Speaker 1: at Guttingen on May fifth, eighteen fifty nine. He held
Speaker 1: successively professorships at Breslau and Berlin, and on Gauss's death
Speaker 1: in eighteen fifty five, was appointed to succeed him as
Speaker 1: professor of the Higher Mathematics at Gottingen. He intended to
Speaker 1: finish Gauss's incomplete works, for which he was admirably fitted,
Speaker 1: but his early death prevented this. He produced, however, several
Speaker 1: memoirs which have considerably facilitated the comprehension of some of
Speaker 1: Gauss's more abstruse methods. Of de Regree's original work, the
Speaker 1: most celebrated is that on the Determination of Means with
Speaker 1: applications to the distribution of prime numbers. The researches of
Speaker 1: Gauss on the theory of numbers were continued or supplemented
Speaker 1: by Jacobi, who firsts proved the law of cubic reciproc
Speaker 1: discussed on the theory of residues, and in his Canon
Speaker 1: Arithmeticus gave a table of residues of prime roots Eisenstein.
Speaker 1: This subject was next taken up by Ferdinand Gotthold Eisenstein,
Speaker 1: a professor at the University of Berlin, who was born
Speaker 1: at Berlin on April sixteenth, eighteen twenty three and died
Speaker 1: there on October eleventh, eighteen fifty two. The theory of
Speaker 1: numbers may be divided into two main divisions, namely the
Speaker 1: theory of congruences and the theory of forms. The solution of
Speaker 1: the problem of the representation of numbers by binary quadratic
Speaker 1: forms is one of the great achievements of Gauss, and
Speaker 1: the fundamental principles upon which the treatment of such questions
Speaker 1: rest were given by him in the Disquisition Arithmetic k.
Speaker 1: Guss there added some results relating to ternary quadratic forms,
Speaker 1: but the general extension from two to three indeterminates was
Speaker 1: the work of Eisenstein, who, in his memoir Nua Theorem
Speaker 1: de Hoharen Arithmetic, defined the ordinal and generic characters of
Speaker 1: ternary quadratic forms of an uneven determinant, and in the
Speaker 1: case of definite forms, assigned the weight of any order
Speaker 1: or genus. But he did not consider forms of an
Speaker 1: even determinant, nor give any demonstration of his work. Eisenstein
Speaker 1: also considered the theorems relating to the possibility of representing
Speaker 1: a number as a sum of squares. En shewed that
Speaker 1: the general theorem was limited to eight squares. The solutions
Speaker 1: in the cases of two, four, and six squares may
Speaker 1: be obtained by means of elliptic functions, but the cases
Speaker 1: in which the numbers of squares is uneven involved the
Speaker 1: special processes peculiar to the theory of numbers. Eisenstein gave
Speaker 1: the solution in the case of three square. He also
Speaker 1: left a statement of the solution he had obtained in
Speaker 1: the case of five squares, but his results were published
Speaker 1: without proofs and apply only to numbers which are not
Speaker 1: divisible by a square. Among Eisenstein's other investigations, I single
Speaker 1: out for special mention the remarkable rule he enunciated by
Speaker 1: means of which it is possible to distinguish whether a
Speaker 1: given series represents an algebraical or transcendental function. Henry Smith.
Speaker 1: One of the most original and powerful mathematicians of the
Speaker 1: school founded by Gauss was Henry Smith. Henry John Stephen
Speaker 1: Smith was born in London on November two, eighteen twenty six,
Speaker 1: and died at Oxford on February ninth, eighteen eighty three.
Speaker 1: He was educated at Rugby and at Balliol College, Oxford,
Speaker 1: of which a latter society he was a fellow, and
Speaker 1: in eighteen sixty one he was elected civilian Professor of
Speaker 1: Geometry at Oxford. He resided till his death. The subject
Speaker 1: in connection with which Smith's name will be always specially
Speaker 1: remembered is the theory of numbers, and to this he
Speaker 1: devoted the years from eighteen fifty four to eighteen sixty four.
Speaker 1: The results of his historical researches were given in his report,
Speaker 1: published in parts in the Transactions of the British Association
Speaker 1: from eighteen fifty nine to eighteen sixty five. This report
Speaker 1: contains an account of what had been done on the
Speaker 1: subject to that time, together with some additional matter. The
Speaker 1: chief outcome of his own original work on the subjects
Speaker 1: is included in two memoirs printed in the Philosophical Transactions
Speaker 1: for eighteen sixty one and eighteen sixty seven, the first
Speaker 1: being on linear indeterminate equations and Congruences, and the second
Speaker 1: on the orders and genera of ternary quadratic forms. In
Speaker 1: the latter memoir, demonstrations of Eisenstein's results on their extension
Speaker 1: to ternary quadratic forms of an even determinant were supplied,
Speaker 1: and a complete classification of ternary quadratic forms was given. Smith, however,
Speaker 1: did not confine himself to the case of three indeterminates,
Speaker 1: but succeeded in establishing the principles on which the extension
Speaker 1: to the general case of n indeterminates depends and obtained
Speaker 1: the general formulae, thus affecting the greatest advance made in
Speaker 1: the subject since the publication of Gauss's work. In the
Speaker 1: account of his Methods and Results, which appeared in the
Speaker 1: Proceedings of the Royal Society, Smith remarked that the theorems
Speaker 1: relating to the representation of numbers by four squares and
Speaker 1: other simple quadratic forms are deducible by a uniform method
Speaker 1: from the principles there indicated, as also are the theorems relating
Speaker 1: to the representation of numbers by six and eight squares.
Speaker 1: He then proceeded to say that as the series of
Speaker 1: theorems relating to the representation of numbers by sums of
Speaker 1: square's ceases for the reason assigned by Eisenstein, when the
Speaker 1: number of squares surpasses eight, it was desirable to complete it.
Speaker 1: The results for even squares were known. The principal theorems
Speaker 1: relating to this case of five squares had been given
Speaker 1: by Eisenstein, but he had considered only those numbers which
Speaker 1: are not divisible by a square, and he had not
Speaker 1: considered the case of seven squares. Smith here completed the
Speaker 1: annunciation of the theorems for the case of five squares,
Speaker 1: and added the corresponding theorems for the case of seven squares.
Speaker 1: This paper was the occasion of a dramatic incident in
Speaker 1: the history of mathematics. Fourteen years later, in ignorance of
Speaker 1: Smith's work, the demonstration and completion of Eisenstein's theorems for
Speaker 1: five squares were set by the French Academy as a
Speaker 1: subject of their Grand Prix des Science Mathematique. Wrote out
Speaker 1: the demonstration of his general theorem so far as was
Speaker 1: required to prove the results in the special case of
Speaker 1: five squares, and only a month after his death in
Speaker 1: March eighteen eighty three, the prize was awarded to him,
Speaker 1: another prize being awarded to H. Minkowski of Bonn. No
Speaker 1: episode could bring out in a more striking light the
Speaker 1: extent of Smith's researches than that a question of which
Speaker 1: he had given in the solution in eighteen sixty seven,
Speaker 1: as a corollary from the general formulae which governed the
Speaker 1: whole class of investigations to which it belonged, should have
Speaker 1: been regarded by the French Academy as one whose solution
Speaker 1: which is of such difficulty and importance as to be
Speaker 1: worthy of their great prize. It has also been a
Speaker 1: matter of comment that they should have known so little
Speaker 1: of contemporary English and German researches on the subject as
Speaker 1: to be unaware that the result of the problem they
Speaker 1: were proposing was then lying in their own library. Among
Speaker 1: Smith's other investigations, I may specially mentioned his geometrical memoir
Speaker 1: cirquecure problem quebique a bi quadratique, for which in eighteen
Speaker 1: sixty eight he was awarded the Steiner Prize of the
Speaker 1: Berlin Academy. In a paper which he contributed to the
Speaker 1: Atti of the Academia de Linsei for eighteen seventy seven,
Speaker 1: he established a very remarkable analytical relation connecting the modular
Speaker 1: equation of order N and the theory of binary quadratic
Speaker 1: forms belonging to the positive determinant N. In this paper,
Speaker 1: the modular curve is represented analytically by a curve in
Speaker 1: such a manner as to present an actual geometrical image
Speaker 1: of the complete systems of the reduced quadratic forms belonging
Speaker 1: to the determinant, and a geometrical interpretation is given to
Speaker 1: the ideas of class equivalence and reduced form. He was
Speaker 1: also the author of important papers in which he succeeded
Speaker 1: in extending to complex quadratic forms. Many of Gauss's investigations
Speaker 1: relating to real quadratic forms. He was led by researches
Speaker 1: on the theory of numbers to the theory of elliptic functions,
Speaker 1: and the results he arrived at, especially on the theory
Speaker 1: of the theta and omega functions, are of importance. The
Speaker 1: theory of numbers as treated today may be said to
Speaker 1: originate with Gauss. I have already mentioned very briefly the
Speaker 1: subject of the subsequent investigations of Jacobi Derichree Eisenstein and
Speaker 1: Henry Smith, among other mathematicians who have written on it.
Speaker 1: I may allude to the following Riemann, who investigated the
Speaker 1: distribution of primes. James Joseph Sylvester, civilian professor in the
Speaker 1: University of Oxford born in London on September third, eighteen fourteen,
Speaker 1: who has also written on the distribution of primes. Koshi,
Speaker 1: who in particular to discussed the expression of quadratic binomials.
Speaker 1: Joseph Lieuville, the editor from eighteen thirty six to eighteen
Speaker 1: seventy four of the well known journal, who was born
Speaker 1: at Saint Omer on March twenty fourth, eighteen o nine
Speaker 1: and died in eighteen eighty two, most of whose numerous
Speaker 1: investigations dealt with the representation of numbers by special forms.
Speaker 1: Ernest Edward Coomer born at Soroux on January twenty ninth,
Speaker 1: eighteen ten, and until recently professor at Berlin, to whom
Speaker 1: we owe the conception of the so called ideal primes
Speaker 1: which are required in the treatment of complexes, and which
Speaker 1: he applied to the problem of Fermat's equation, and whose
Speaker 1: paper on hypergeometric series may rank with that Bygaus is
Speaker 1: a classical memoir on the subject. Leopold Kronecker, professor in Berlin,
Speaker 1: born at lie Niitz on December seventh, eighteen twenty three
Speaker 1: and died at Berlin on December twenty ninth, eighteen ninety one,
Speaker 1: most of whose investigations on this branch of mathematics were
Speaker 1: on nary on quadratic forms. On his investigations generally see
Speaker 1: the Bulletin of the New York Mathematical Society, Volume one,
Speaker 1: pages one seventy three to one eighty four. See also
Speaker 1: below page four sixty nine. Charles Hermite, professor in Paris,
Speaker 1: born in Lorraine on December twenty fourth, eighteen twenty two,
Speaker 1: who wrote on ternary forms. Julius Wilhelm Richard Dedekind born
Speaker 1: at Brunswick on October sixth, eighteen thirty one, whose more
Speaker 1: important researches, given in an appendix to his edition of
Speaker 1: Dirichley's writings, are on ideal primes. See also below page
Speaker 1: four ninety three. Panotige Chebyshev, formerly professor at the University
Speaker 1: of Saint Petersburg, born in Russia in eighteen twenty one,
Speaker 1: who has written on the number of primes between given limits,
Speaker 1: a problem also considered by Legendre de Richley and Riemonn
Speaker 1: and James Whitebread. Lee Glacier, a fellow and tutor of
Speaker 1: Trinity College in Cambridge, born at Lewisham on November fifth,
Speaker 1: eighteen forty eight, from whose numerous papers I may single
Speaker 1: out those relating to prime numbers, those on functions of
Speaker 1: a number which are formed from its real or complex divisors,
Speaker 1: and on those on the possible divisors of numbers of
Speaker 1: a given form. Finally, I may mention that the problem
Speaker 1: of the partition of numbers, to which Euler paid considerable attention,
Speaker 1: has in recent times attracted the attention of Arthur Cayley,
Speaker 1: Sadleian professor in the University of Cambridge, born in Richmond,
Speaker 1: Surrey on August sixteenth, eighteen twenty one, of Sylvester and
Speaker 1: of Perry Percy. Alexander McMahon, professor at Woolwich and a
Speaker 1: major in the English Artillery, born at Malta on September
Speaker 1: twenty sixth, eighteen fifty four. Interest and problems connected with
Speaker 1: the theory of numbers seems recently to have flagged, and
Speaker 1: possibly it may be found hereafter that the subject is
Speaker 1: approached better on other lines. The theory of functions of
Speaker 1: double and multiple periodicity is another subject to which much
Speaker 1: attention has been paid during this century. I have already
Speaker 1: mentioned that as early as eighteen o eight Gauss had
Speaker 1: discovered the theta functions and their chief properties, but his
Speaker 1: investigations remained for many years concealed in note books, and
Speaker 1: it was to the researches made between eighteen twenty and
Speaker 1: eighteen thirty by Abel and Jacoby that the modern development
Speaker 1: of the subject is due. The treatment of it has
Speaker 1: been completely superseded that used by Legendre, and they are
Speaker 1: justly reckoned as the creators of this branch of mathematics.
Speaker 1: Abel Nils Henrik Abel born at Findo in Norway on
Speaker 1: August fifth, eighteen o two and died at Arundel on
Speaker 1: April sixth, eighteen twenty nine, at the age of twenty six.
Speaker 1: His memoirs on elliptic functions, which were originally published in
Speaker 1: Krell's Journal, treat the subject from the point of view
Speaker 1: of the theory of equations and algebraic forms, a treatment
Speaker 1: to which his research has naturally led him. The important
Speaker 1: and very general result known as Abel's theorem, which was
Speaker 1: subsequently applied by Riemann to the theory of transcendental functions,
Speaker 1: was sent to the French Academy in eighteen twenty eight, but,
Speaker 1: mainly through the action of Koshi, was not published for
Speaker 1: several years. The name of Abelian function has been given
Speaker 1: to the higher transcendence of multiple periodicity, which were first
Speaker 1: discussed by Abel. He criticized the use of infinite series,
Speaker 1: but I do not know that the results lead to
Speaker 1: any definite rules for testing convergency. As illustrating his fertility
Speaker 1: of ideas, I May in passing notice his celebrated demonstration
Speaker 1: that it is impossible to solve a quintic equation by
Speaker 1: means of radicals. This theorem was the more important since
Speaker 1: it definitely limited a field of mathematics which had previously
Speaker 1: attracted numerous writers. I should add that this theorem had
Speaker 1: been enunciated as early as seventeen ninety eight by Pallo Raffini,
Speaker 1: an Italian physician practicing at Modena, but I believe that
Speaker 1: the proof he gave was deficient in generality. Jacoby Karl
Speaker 1: Gustav Jacob Jacobi born of Jewish parents at Potsdam on
Speaker 1: December tenth, eighteen o four and dine at Berlin on
Speaker 1: February eighteenth, eighteen fifty one. Was educated at the University
Speaker 1: of Berlin, where he obtained a degree of Doctor of
Speaker 1: Philosophy in eighteen twenty five. In eighteen twenty seven he
Speaker 1: became extraordinary professor of mathematics at Knigsburg, and in eighteen
Speaker 1: twenty nine was promoted to be an ordinary professor. This
Speaker 1: chair he occupied till eighteen forty two, when the Prussian
Speaker 1: government gave him a pension and he moved to Berlin,
Speaker 1: where he continued to live until his death in eighteen
Speaker 1: fifty one. Jacobi's most celebrated investigations are those on elliptic functions,
Speaker 1: the modern notation in which is due to him, and
Speaker 1: the theory of which he established simultaneously with Abel, but
Speaker 1: independently of him. These are given in his treatise Fundamental
Speaker 1: Nova theire Functionionum Ellipticarum, Kanigsburg, eighteen twenty nine and in
Speaker 1: some later papers in Krell's journal. The correspondence between la
Speaker 1: Jeh and Jacobi on elliptic functions has been reprinted in
Speaker 1: the first volume of Jacobi's collected works. Jacob, like Abel,
Speaker 1: recognized elliptic functions were not merely a group of theorems
Speaker 1: on integration, but that there were types of a new
Speaker 1: kind of function, namely one of double periodicity. Hence, he
Speaker 1: paid particular attention to the theory of the theta function.
Speaker 1: The following passage in which he explains this view is
Speaker 1: sufficiently interesting to deserve textual reproduction. Equo communi versam k
Speaker 1: fingipo test amplicatur perio datticatam and eleticam elusit functiones ellipticas
Speaker 1: non alis ad numerari de ber transendantibus k quibusdam god
Speaker 1: dant eligantis fortas pluribus elas otte marioribus said speciam quadnam
Speaker 1: is ines perfecti et absoluti. Among Jacobi's other investigations, I
Speaker 1: may specially single out his papers on determinants, which did
Speaker 1: a great deal to bring them into general use, and
Speaker 1: particularly his invention of the Jacobian, that is, of the
Speaker 1: functional determinant formed by the n squared partial differential coefficients
Speaker 1: of the first order of n given functions of n
Speaker 1: independent variables. I ought also to mention his papers on
Speaker 1: Abelian transcendance, his investigations on the theory of numbers, his
Speaker 1: important work on the theory of partial differential equations, his
Speaker 1: development of the calculus of variations, and his numerous memoirs
Speaker 1: on the planetary theory and other particular dynamical problems, in
Speaker 1: the course of which he also extended the theory of
Speaker 1: differential equations. Most of the results of the researches last
Speaker 1: named are included in his Valusungen uber Dynamic, edited by Klebsch, Berlin,
Speaker 1: eighteen sixty six. Riemann Jeorg Friedrich Bernhard Riemann was born
Speaker 1: at Bersens on September seventeenth, eighteen twenty six, and died
Speaker 1: at Selaska on July twentieth, eighteen sixty six. He studied
Speaker 1: at Gattigen under Gauss and subsequently at Berlin under Jacobi
Speaker 1: dirichle Steiner and Eisenstein, all of whom were professors there
Speaker 1: at the same time. His earliest paper, written in eighteen
Speaker 1: fifty was on algebraic functions of a complex variable and
Speaker 1: on at The recent investigations of Schwartz, Klein, and Pointerrey
Speaker 1: are largely based to those that refer very briefly below.
Speaker 1: In eighteen fifty four, Riemann wrote his celebrated memoir on
Speaker 1: the Hypotheses on which geometry is founded. This was succeeded
Speaker 1: by his memoirs on elliptic functions and the theory of numbers.
Speaker 1: He also wrote on physical subjects. The question of the
Speaker 1: truth of the assumptions usually made in our geometry had
Speaker 1: been considered by J. Sacheri as long ago as seventeen
Speaker 1: thirty three, and in more recent times had been discussed
Speaker 1: by Nikolai Avenovich Lobachevsky, professor at Khan born at Ninji
Speaker 1: Novogorod in seventeen ninety three, and died at Kassen on
Speaker 1: February twelfth, eighteen fifty six, in eighteen twenty six and
Speaker 1: again in eighteen forty, by Gauss in eighteen thirty one
Speaker 1: and in eighteen forty six, and by Johann Bolyai born
Speaker 1: at Klausenberg in eighteen o two and died at Masmaros
Speaker 1: Wazerheli in eighteen sixty and in eighteen thirty two in
Speaker 1: the appendix to the first volume of his father's Tentamen.
Speaker 1: But Riemon's memoir of eighteen fifty four attracted general attention
Speaker 1: to the subject of hypergeometry, and the theory has been
Speaker 1: since extended and simplified by various writers, notably by Eugenio Beltrami,
Speaker 1: professor at Pavia, born at Cremonia in eighteen thirty five,
Speaker 1: and by Hermann Ludwig Ferdinand von Helmholtz, professor at Berlin,
Speaker 1: born at Potsdam on August thirty one, eighteen twenty one.
Speaker 1: The subject is so technical that I confine myself to
Speaker 1: a bare sketch of the argument from which the idea
Speaker 1: is derived that a space of two dimensions should have
Speaker 1: the geometrical properties with which we are made familiar in
Speaker 1: the study of elementary geometry, it is necessary that it
Speaker 1: should be possible at any place to construct a figure
Speaker 1: congruoned to a given figure, and this is so only
Speaker 1: if the product of the principal radii of curvature at
Speaker 1: every point in space or surface be a constant. There
Speaker 1: are three species of surfaces which possess this property, namely,
Speaker 1: one spherical surfaces where the product is positive, two plane
Speaker 1: surfaces which lead to Euclidean geometry where it is zero,
Speaker 1: and three what Beryltrami has called pseudospherical surfaces where it
Speaker 1: is negative. Moreover, if any service be bent without dilation
Speaker 1: or contraction, the measure of curvature remains unaltered. Thus, these
Speaker 1: three species of surfaces are types of three kinds on
Speaker 1: which congruent figures can be constructed. For instance, a plane
Speaker 1: can be rolled into a cone, and the system of
Speaker 1: geometry on a conical surface is similar to that on
Speaker 1: a plane. These kinds of space of two dimensions are
Speaker 1: distinguished from one another by a simple test through a
Speaker 1: point of spherical space. No geodedic line a geodetic line
Speaker 1: being defined as the shortest distance between two points, can
Speaker 1: be drawn parallel to a given gen decline through a
Speaker 1: point of Euclidean or plane space, one and only one
Speaker 1: geoddic line i e. Is straight line can be drawn
Speaker 1: parallel to a given geod decline. Through a point of
Speaker 1: pseudospherical space, more than one geodedic line can be drawn
Speaker 1: parallel to a given geod decline, But all these lines
Speaker 1: form a pencil whose vertical angle is constant. It may
Speaker 1: be thought that we have a demonstration that our space
Speaker 1: is plain, since through a given point we can draw
Speaker 1: only one straight line parallel to a given straight line.
Speaker 1: This is not so, for it is conceivable that our
Speaker 1: means of observation do not permit us to say with
Speaker 1: absolute accuracy whether two lines are parallel. Hence, we cannot
Speaker 1: use this as a means to tell whether our space
Speaker 1: is plain or not. A better test can be deduced
Speaker 1: from the proposition that in any two dimensional space of
Speaker 1: uniform curvature, the sum of the angles of a triangle,
Speaker 1: if it differ from two right angles, will differ by
Speaker 1: a quantity proportional to the area of the triangle. Hence,
Speaker 1: it may happen possibly that although for triangles such as
Speaker 1: we can measure, the difference is imperceptible, yet for triangles
Speaker 1: which are millions of times bigger. There would be a
Speaker 1: sensible difference. If space be spherical or pseudospherical, its extent
Speaker 1: is finite. If it be plain, its extent is infinite.
Speaker 1: In regard to pseudospherical space, I should add that its
Speaker 1: extent may be infinite if it be constructed in space
Speaker 1: of four dimensions. In the preceding sketch of the foundations
Speaker 1: of non Euclidean geometry, I have assumed tacitly that the
Speaker 1: measure of a distance remains the same everywhere cline assume
Speaker 1: that if this be not the case, and if the
Speaker 1: law of the measurement of distance be properly chosen, we
Speaker 1: can obtain three systems of plane geometry analogous to the
Speaker 1: three systems mentioned above. These are called, respectively, elliptic, parabolic,
Speaker 1: and hyperbolic geometries. The above refers to only to hyper
Speaker 1: space of two dimensions. Naturally, there arises the question whether
Speaker 1: there are different kinds of hyperspace of three or more dimensions.
Speaker 1: Remontchute that there are three kinds of hyperspace of three
Speaker 1: dimensions having properties analogous to the three kinds of hyperspace
Speaker 1: of two dimensions already discussed. These are differentiated by the
Speaker 1: test whether at every point no geodetical surfaces, or one
Speaker 1: geodetical surface or a facisculus of geodetical surfaces can be
Speaker 1: drawn parallel to a given surface, a geodiitical surface being
Speaker 1: defined as such that every geodetic line joining two points
Speaker 1: on it lies wholly on the surface. I returned now
Speaker 1: to Riemon's other investigations in multiply periodic functions. It is
Speaker 1: hardly too much to say that he and his memoir
Speaker 1: in Bouchardt's Journal for eighteen fifty seven did for the
Speaker 1: Abelian functions what abel had done for the elliptic functions.
Speaker 1: And it is this, perhaps that will constitute one of
Speaker 1: his chief claims to future distinction. In the theory of numbers,
Speaker 1: Riemond's short tract of eight pages on the number of
Speaker 1: primes which lie between two given numbers affords a striking
Speaker 1: instance of his analytical powers. Legendre had previously shewn that
Speaker 1: the number of primes less than n is very approximately
Speaker 1: n divided by the quantity log n minus one point
Speaker 1: eight three sixty six. But Riemon went further, and this
Speaker 1: tract and a memoir by Chebyshev contains nearly all that
Speaker 1: had been done yet in connection with a problem of
Speaker 1: so obvious of a character that has suggested itself to
Speaker 1: all who has considered the theory of numbers, and yet
Speaker 1: which overtacks the powers even of Lagrange and Gauss. Among
Speaker 1: others than those already named, I may mention the following
Speaker 1: who have written on elliptic and Abelian functions. Johann Jorg Rosenheian,
Speaker 1: professor in Knigsburg, born there on June ten, eighteen sixteen
Speaker 1: and died in eighteen eighty seven, who wrote in eighteen
Speaker 1: eighty four on the hyper elliptic double theta function and
Speaker 1: functions of two variables with four periods. Adolf Goeppel born
Speaker 1: at Rostock in September eighteen twelve and died at Berlin
Speaker 1: in March eighteen forty seven, who discussed hyper elliptic functions
Speaker 1: see Krell's Journal, Volume thirty five, eighteen forty seven, pages
Speaker 1: three thirteen three. Karl Weierstrauss, professor in Berlin, born at
Speaker 1: Austenfeld on October thirty first, eighteen fifteen, whose earlier researches
Speaker 1: related to the theta functions, which he treated under a
Speaker 1: modified form in which they are expressible in powers of
Speaker 1: the modulus. At a later period he developed a method
Speaker 1: for treating all elliptic functions in a symmetrical manner, a
Speaker 1: process to which he was naturally led by his researches
Speaker 1: on the general theory of functions. In this theory, the
Speaker 1: theta functions are independent of the form of their space boundaries.
Speaker 1: Leopold Kroniker, who wrote on elliptic functions, Francesco Brioschi of Rome,
Speaker 1: who wrote on elliptic and hyper elliptic functions. Henry Smith,
Speaker 1: who discussed the transformation theory, the theta and omega functions,
Speaker 1: and certain functions of the modulus. Kayley, who was the
Speaker 1: first work out in eighteen forty five the theory of
Speaker 1: doubly infinite products and determine their periodicity, and who has
Speaker 1: written at length on the connection between the researches of
Speaker 1: Legendre and Jacobi. His later writings have dealt mainly with
Speaker 1: the theory of transformations and the modular equation. Cayley's collected
Speaker 1: works are now being issued by the University of Cambridge.
Speaker 1: The researches of Hermite are mostly concerned with the transformation theory,
Speaker 1: the higher development of the theta functions and the connection
Speaker 1: between the methods and results of Weierstrauss and Jacobi. The
Speaker 1: transformation of the double theta function has been also considered
Speaker 1: by Leo Koenigsberger, present professor at Heidelberg, born in Prussia
Speaker 1: eighteen thirty seven. See his lectures published at Leipzig in
Speaker 1: eighteen seventy four. The investigations of George Henri Haffan, an
Speaker 1: officer in the French Army born at Roan on October
Speaker 1: thirtieth ve eighteen forty four and died in Paris on
Speaker 1: May twenty first, eighteen eighty nine, are largely founded on
Speaker 1: weyer Strauss's work. A sketch of Helfen's life and works
Speaker 1: given in Leauville's Journal for eighteen eighty nine, pages three
Speaker 1: forty five to three fifty nine, and in the comptess
Speaker 1: Rendus eighteen ninety, volume one hundred and ten, pages four
Speaker 1: eighty nine to four ninety seven. See also below pages
Speaker 1: four eighty one to four eighty two. Felix Christian Klein
Speaker 1: born in eighteen forty nine, now professor in Guttingen, has
Speaker 1: written on Abelian functions, elliptic modular functions and hyper elliptic functions. Finally, H. A. Schwartz,
Speaker 1: formerly of Guttingen, now in Berlin, born eighteen forty five. H. W.
Speaker 1: Webber of Marburg, m nauthor of Erlengen, W. Stahl of
Speaker 1: eixas Chappelle. FG. Frobenius, now of Berlin and formerly of Zurich.
Speaker 1: And Glacier have written on various branches of the theory,
Speaker 1: and doctor Glacier has in particular developed the theory of
Speaker 1: the zeta function. The textbook by Briotte and Bouquet contains
Speaker 1: a clear account of the elliptic functions as it exists
Speaker 1: at present, developed from the point of view of the
Speaker 1: complex variable. Albert Briot was born at Saint Hippolyte in
Speaker 1: eighteen seventeen, occupied a chair at the Sorbonne in Paris,
Speaker 1: and died in eighteen eighty two. Jean Claude Bouquet was
Speaker 1: born in eighteen nineteen and died in Paris in eighteen
Speaker 1: eighty five. End of section thirty five of A Short
Speaker 1: Account of the History of mathematics by W. W. Rousball.
Speaker 1: Recording by Paul King, Mississauga, Ontario, PJK dot scripts dot
Speaker 1: MIT dot edu forward slash p kJ