Speaker 1: Chapter nineteen, Part one of a Short Account of the
Speaker 1: History of Mathematics by W. W. Rowsball. This is a
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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. Rowsball, Chapter nineteen. Mathematicians
Speaker 1: of Recent Times. It is evidently impossible for me to
Speaker 1: discuss adequately the mathematicians of the age in which we live,
Speaker 1: especially as I purposefully exclude from this work and any
Speaker 1: detailed reference to living writers. I make therefore no attempt
Speaker 1: to give a complete history of this period. But as
Speaker 1: a sort of appendix to their preceding chapters, I had
Speaker 1: a few notes on some of the more striking features
Speaker 1: in the history during this century of pure mathematics, in
Speaker 1: which I include theoretical dynamics and astronomy. But except for
Speaker 1: a few allusions, I shall not discuss the applications of
Speaker 1: mathematics to physics. These notes are brief, and in many
Speaker 1: cases consists merely of a list of the names of
Speaker 1: some of those to whom the development of any branch
Speaker 1: of the subject is chiefly due, and an indication of
Speaker 1: that part of it to which they have directed most attention.
Speaker 1: I would refer to anyone who wishes for more details
Speaker 1: to the invaluable catalog which has been compiled by the
Speaker 1: Royal Society of London, and which contains a list under
Speaker 1: the names of the authors of all the scientific papers
Speaker 1: contributed during this century to journals and learned societies. In
Speaker 1: only a few cases do I add any account of
Speaker 1: the life and works of the mathematicians mentioned. Even with
Speaker 1: these limitations, it has been very difficult to put together
Speaker 1: a connected account of the mathematicians of recent times. And
Speaker 1: I wish to repeat explicitly that I do not suggest,
Speaker 1: nor do I wish my readers to suppose, that my
Speaker 1: notes on a subject give the names of all the
Speaker 1: chief writers who have studied it. In fact, the quality
Speaker 1: of matter produced has been so enormous that no one
Speaker 1: can expect to do more than make himself acquainted with
Speaker 1: the work in some small department. As an illustration of
Speaker 1: this remark, I must say that I have reason to
Speaker 1: believe that something like fifteen thousand separate scientific memoirs are
Speaker 1: now published every year by the different societies and journals
Speaker 1: of Europe and America. Most of the treatises on the
Speaker 1: history of mathematics omit all references to the work produced
Speaker 1: during this century. The chief exceptions which I am acquainted
Speaker 1: are a short dissertation by h. Hankell entitled z the
Speaker 1: and Wiklang der Mathematique in de Lenzigne de junten in
Speaker 1: Turberlingen eighteen eighty five. The eleventh and twelfth volumes of
Speaker 1: Marie Histoire des Science, in which there are notes on
Speaker 1: mathematicians who were born in the last century. Gerhartz Gislichte
Speaker 1: der Mathematique in Deutschland, published in Munich eighteen seventy seven,
Speaker 1: and a de scour on the professors at the Sorbonne
Speaker 1: by Hermit in the Bulletende des Science Mathematique in eighteen ninety,
Speaker 1: pages six to thirty six. A few histories of the
Speaker 1: development of a particular subjects have been written, such as
Speaker 1: those by the late Isaac Todd Hunter on the theories
Speaker 1: of attraction and the calculus of probabilities. While annual volumes
Speaker 1: of the British Association contain a number of reports on
Speaker 1: the progress in several different branches of modern mathematics. A
Speaker 1: few similar reports, and notably one in eighteen fifty seven
Speaker 1: by J. Bertrand the development of mathematical analysis, have been
Speaker 1: presented by the French Academy. I have found these authorities
Speaker 1: and these reports useful, but I have derived most assistance
Speaker 1: in writing this chapter from the obituary notices in the
Speaker 1: proceedings of various learned societies, foreign as well as British.
Speaker 1: I am also indebted to information kindly furnish me by
Speaker 1: various friends, and if I do not further dwell on this,
Speaker 1: it is only that I would not seem to make
Speaker 1: them responsible for errors and omissions which they would have
Speaker 1: avoided in their own works. A period of exceptional intellectual
Speaker 1: activity in any subject is usually followed by one of
Speaker 1: comparative stagnation, and after the deaths of Lagrange, Laplace, Legendre
Speaker 1: and Poissin, of the French school, which had occupied so
Speaker 1: prominent a position at the beginning of this century, ceased
Speaker 1: for some years to produce much new work. Some of
Speaker 1: the mathematicians whom I intend first to mention, gauss Abel
Speaker 1: and Jacoby, were contemporaries of the later years of the
Speaker 1: French mathematicians just named, but their writings appear to me
Speaker 1: to belong to a different school, and thus are properly
Speaker 1: placed at the beginning of a fresh chapter. There is
Speaker 1: no mathematician of this century whose writings have had a
Speaker 1: greater effect than those of Gauss, nor is it on
Speaker 1: only one branch of the science that his influence has
Speaker 1: left a permanent mark. I cannot therefore commence my account
Speaker 1: of the mathematicians of recent times better than by describing
Speaker 1: very briefly his more important researches. Gauss Carl Frederick Gauss
Speaker 1: born at Brunswick on April twenty third, seventeen seventy seven
Speaker 1: and died in Gautingen on February twenty third, eighteen fifty five.
Speaker 1: His father was a bricklayer, and Gauss was indebted for
Speaker 1: liberal education, much against the will of his parents, who
Speaker 1: wished to profit by his wages as a laborer, to
Speaker 1: the notice which his talents procured from the reigning Duke.
Speaker 1: In seventeen ninety two he was sent to the Caroline College,
Speaker 1: and by seventeen ninety five, professors and pupils alike admitted
Speaker 1: that he knew all of the former could teach him.
Speaker 1: It was while there that he investigated the method of
Speaker 1: Lee squares and proved by induction the law of quadratic reciprocity.
Speaker 1: Thence he went to Gunteingen, where he studied under Costner.
Speaker 1: Many of his discoveries in the theory of numbers were
Speaker 1: made while a student there. In seventeen ninety eight he
Speaker 1: returned to Brunswick, where he earned somewhat a somewhat precarious
Speaker 1: livelihood by private tuition. In seventeen ninety nine he Gauss
Speaker 1: published his demonstration that every algebraical equation has a root
Speaker 1: of the form A plus b I, a theorem of
Speaker 1: which altogether he gave gave three distinct proofs. In eighteen
Speaker 1: oh one, this was followed by Disquisition's ARITHMETICAI, which is
Speaker 1: printed as the first volume of his collected works. The
Speaker 1: greater part of it has been sent to the French
Speaker 1: Academy in the preceding year, and has been rejected with
Speaker 1: a sneer, which, even if the book had been as
Speaker 1: worthless as the referees believed, would have been unjustifiable. Gauss
Speaker 1: was deeply hurt, and his reluctance to publish his investigations
Speaker 1: may be partly attributable to this unfortunate incident. The next discovery,
Speaker 1: Gauss was in a totally different department of mathematics. The
Speaker 1: absence of any planet in the space between Mars and Jupiter,
Speaker 1: where Bode's law would have led observers to expect one,
Speaker 1: had been long remarked, but it was not until eighteen
Speaker 1: oh one that any one of the numerous group of
Speaker 1: minor planets which occupy that space was observed. The discovery
Speaker 1: was made made by Piazzi of Palermo, and was the
Speaker 1: more interesting as its announcement occurred simultaneously with a publication
Speaker 1: by Hegel, in which he severely criticized astronomers for not
Speaker 1: paying more attention to philosophy, a science, said he, which
Speaker 1: would not at once have shewn them that there could
Speaker 1: not possibly be more than seven planets, a study of
Speaker 1: which would therefore have prevented an absurd waste of time
Speaker 1: in looking for what, in the nature of things could
Speaker 1: never be found. The new planet was named Cerrees, but
Speaker 1: it was seen under conditions which appeared to render it
Speaker 1: almost impossible to forecast its orbit. The observations were fortunately
Speaker 1: communicated to Gauss. He calculated its elements, and his analysis
Speaker 1: proved to him to be the first of theoretical astronomers,
Speaker 1: no less than the greatest of arithmeticians. The attention excited
Speaker 1: by these investigations procured him, in eighteen o seven the
Speaker 1: offer of a chair at Saint Petersburg, which he declined.
Speaker 1: In the same year he was appointed director to Guntagen
Speaker 1: Observatory and professor of astronomy there. These offices he retained
Speaker 1: to his death, and after his appointment he never slept
Speaker 1: away from his observatory, except on one occasion, where he
Speaker 1: attended a scientific conference at Berlin. His lectures were singularly
Speaker 1: lucid and perfect in form and it is said that
Speaker 1: he used here to give the analysis by which he
Speaker 1: had arrived at his various results, and which it is
Speaker 1: so conspicuously absent from his published demonstrations. But for fear
Speaker 1: his auditors should lose the threat of his discourse, he
Speaker 1: never willingly permitted them to take notes. I have already
Speaker 1: mentioned Gauss's publications in seventeen ninety nine, eighteen oh one,
Speaker 1: and eighteen o two. For some years after eighteen o
Speaker 1: seven his time was almost wholly occupied by work connected
Speaker 1: with his observatory. In eighteen oh nine he published at
Speaker 1: Hamburg his Theorea Motus corporum celestium, a treatise which contributed
Speaker 1: largely to the improvement of practical astronomy and introduced the
Speaker 1: principle of curve linear triangulation. And on the same subject,
Speaker 1: but connected with observations in general, we have his memoir
Speaker 1: therea Comnetonis observationionim erobinus minimus Obnoxia, with a second part
Speaker 1: and a supplement. Somewhat later, he took up the subjects
Speaker 1: of geodesi, acting from eighteen twenty one to eighteen forty
Speaker 1: eight as scientific advisor to the Danish and Hanoverian governments
Speaker 1: for the survey then in progress. His papers of eighteen
Speaker 1: forty three in eighteen sixty six Ubergedestada de houngueo desi,
Speaker 1: contain his researches on the subject. Gauss's researches on electricity
Speaker 1: and magnetism date from about the year eighteen thirty. His
Speaker 1: first paper on the theory of magnetism, entitled intensitest vis
Speaker 1: megnektica terrestris adventsuram absolatum ravokata, was published in eighteen thirty three,
Speaker 1: a few months Afterwards, he, together with Reber, invented the
Speaker 1: declination instrument and the Bifiller magnetometer. In the same year
Speaker 1: that they erected at Gottingen a magnetic observatory free from iron,
Speaker 1: as Humboldt and Arago had previously done on a smaller scale,
Speaker 1: where they made magnetic observations, and in particular schwed that
Speaker 1: it was possible and predictable to send telegraph signals. In
Speaker 1: connection with this observatory, Gauss founded the association called the
Speaker 1: Magnetische Veren, with the object of securing continuous observations at
Speaker 1: fixed times. The volumes of their publications Restata ulstur Boobachtungen
Speaker 1: des magnetician Vere for eighteen thirty eight and eighteen thirty
Speaker 1: nine contained two important memoirs by Gauss, one on the
Speaker 1: general theory of earth magnetism and on the other on
Speaker 1: the theory of forces attracting according to the inverse square
Speaker 1: of the distance. Like Poisson, he treated the phenomenon in
Speaker 1: electrostatics as due to attractions and repulsions between imponderable particles.
Speaker 1: In electrodynamics he arrived in eighteen thirty five at a
Speaker 1: result equivalent to that given by W. E. Weber in
Speaker 1: eighteen forty six, namely that the attraction between two electrified
Speaker 1: particles E and E prime, whose distance apart as are,
Speaker 1: depends on their relative motion and position according to the
Speaker 1: formula ee prime r to the power of minus two
Speaker 1: multiplied by the quantity one plus r r umlaut minus
Speaker 1: one half r squared quantity squared times c to the
Speaker 1: minus two. However, held that no hypothesis was satisfactory which
Speaker 1: rested on a formula and was not a consequence of
Speaker 1: a physical conjecture, and as he could not frame a
Speaker 1: plausical physical conjecture, he abandoned the subject. Such conjectures were
Speaker 1: proposed by Riemann in eighteen fifty eight and by C.
Speaker 1: Newman and E. Betty in eighteen sixty eight, but Helmholtz
Speaker 1: in eighteen seventy, eighteen seventy three and eighteen seventy four
Speaker 1: chewed that they were untenable. A simpler view, which regards
Speaker 1: all electric and magnetic phenomenon as stresses and motions of
Speaker 1: a material elastic medium, have been outlined by Michael Faraday
Speaker 1: and was elaborated by James Clerk Maxwell, the latter by
Speaker 1: the use of generalized coordinates was able to deduce the
Speaker 1: consequences and the agreement with experiment as close. These and
Speaker 1: other electric theories were classified and critically discussed in a
Speaker 1: memoir by J. J. Thompson in eighteen eighty five. Gauss's
Speaker 1: researches on optics, including systems of lenses, were published in
Speaker 1: eighteen forty in his De hok Tricia unto Sohungen. From
Speaker 1: this sketch, it will be seen that the ground covered
Speaker 1: by Gauss's researches was extraordinarily wide. I will now mention
Speaker 1: very briefly some of his most important discoveries in peer mathematics.
Speaker 1: His most celebrated work in pere mathematics is Dedisguisizioni's ARITHMETICAI,
Speaker 1: which has proved the starting point for several interesting investigations
Speaker 1: on the theory of numbers. This treatise and the Genres
Speaker 1: Tierri de Nombre remains standard works on the theory of numbers.
Speaker 1: But just as in his discussion of elliptic functions, le
Speaker 1: Gendre failed to rise to the conception of a new
Speaker 1: subject and confused himself to regarding their theory as a
Speaker 1: chapter in the integral calculus. So he treated the theory
Speaker 1: of numbers as a chapter in algebra. G also, however,
Speaker 1: realized that the theory of discrete magnitudes or higher arithmetic, was
Speaker 1: of a different kind from that of continuous magnitudes or algebra,
Speaker 1: and he introduced a new notation and new methods of
Speaker 1: analysis of which subsequent writers have generally availed themselves, In particular,
Speaker 1: the disquisitiones ARITHMETICI introduced the modern theory of congruences of
Speaker 1: the first and second orders, and to this Gauss reduced
Speaker 1: intermediate analysis. In it also he discussed the solution of
Speaker 1: binomial equations of the form x to the power n
Speaker 1: equals one. This involves the celebrated theorem that the only
Speaker 1: regular polygons which can be constructed by elementary geometry are
Speaker 1: those in which the number of sides is two to
Speaker 1: the power of m multiplied by two to the power
Speaker 1: of n plus one, where m and n are integers
Speaker 1: and two to the power n plus one is a prime,
Speaker 1: a discovery he had made in seventeen ninety sive he
Speaker 1: developed the theory of ternary quadratic forms involving two indeterminates.
Speaker 1: He also investigated the theory of determinants, and it was
Speaker 1: on Gauss's results that Jacobi based his researches on that subject.
Speaker 1: The theory of functions of double periodicity had its origin
Speaker 1: in the discoveries of Abel and Jacoby, which I described later.
Speaker 1: Both these mathematicians arrived at the theta functions, which play
Speaker 1: so large a part in the theory of the subject. Gauss, however,
Speaker 1: had independently and a deed at a far earlier date,
Speaker 1: discovered these functions and their chief properties, having been led
Speaker 1: to them by certain integrals which occurred in the determinacio
Speaker 1: atraxiones to evaluate, which he invented the transformation now associated
Speaker 1: with the name of Jacoby, though Gauss at a later
Speaker 1: time communicated the fact that Jacoby he did not publish
Speaker 1: his researches. They occur in a series of notebooks of
Speaker 1: a date not later in eighteen o eight, and are
Speaker 1: included in his collected works. Of the remaining memoirs in
Speaker 1: pure mathematics, the most remarkable are those in the theory
Speaker 1: of bi quadratic residues, wherein the notion of complex numbers
Speaker 1: of the form A plus bi was first introduced into
Speaker 1: the theory of numbers, in which are included several tables
Speaker 1: and notably one of the number of the classes of
Speaker 1: binary quadratic forms, that of relating to the proof of
Speaker 1: the theorem at every numerical equation has a real or
Speaker 1: imaginary route, that on the summation of series, that on
Speaker 1: the hypergeometric series, which contains a discussion of the gamma function,
Speaker 1: and lastly one on interpolation. His introduction of rigorous tests
Speaker 1: for the convergency of infinite series especially noticeable. Finally, we
Speaker 1: have the important memoir on the conformal representation of one
Speaker 1: surface upon another, in which the results given by Lagrange
Speaker 1: for surfaces of revolution are generalized for any surfaces. In
Speaker 1: the theory of attractions, we have a paper on the
Speaker 1: attraction of homogeneous ellipsoids, the already mentioned memoir of eighteen
Speaker 1: thirty nine Algermaine lesstes in betsiung hofzi in verkiten verhetnes
Speaker 1: des quadrits der enfherroung on the theory of forces attracting
Speaker 1: according to the inverse square of the distance, and the
Speaker 1: memoir determinaccio Etchaciones, in which it is shewn that the
Speaker 1: secular variations which the elements of the orbit of a
Speaker 1: planet experience from the attraction of another planet, which disturbs
Speaker 1: it are the same as if the mass of the
Speaker 1: disturbing planet were distributed over its orbit into an elliptic
Speaker 1: ring in such a manner that equal masses of the
Speaker 1: ring would correspond to the arcs of the orbit described
Speaker 1: in equal times. The great masters of modern anounce us
Speaker 1: are Lagrange, Laplace, and Gauss, who were contemporaries. It is
Speaker 1: interesting to note the marked contrast in their styles. Lagrange
Speaker 1: is perfect both in form and matter. He is careful
Speaker 1: to explain his procedure, and though his arguments are general,
Speaker 1: they are easy to follow. Laplace, on the other hand,
Speaker 1: explains nothing is absolutely indifferent to style, and if satisfied
Speaker 1: with his that his results are correct, is content to
Speaker 1: leave them either with no proof or with a faulty one.
Speaker 1: Gauss is as exact and elegant as Lagrange, but even
Speaker 1: more difficult to follow than Laplace, for he removes every
Speaker 1: trace of the analysis by which he reached his results
Speaker 1: and studies to give a proof which, while rigorous, shall
Speaker 1: be as concise and synthetical as possible. End of Part
Speaker 1: thirty four. Recording by Paul King, Mississauga http PJK dot
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