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<v Speaker 1>Chapter nineteen, Part one of a Short Account of the

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<v Speaker 1>History of Mathematics by W. W. Rowsball. This is a

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<v Speaker 1>LibriVox recording. All LibriVox recordings are in the public domain.

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<v Speaker 1>For more information or to volunteer, please visit LibriVox dot org.

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<v Speaker 1>This is a reading by Paul King p J K

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<v Speaker 1>dot scripts that I might t dot E d U

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<v Speaker 1>forward slash p K J. A Short account of the

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<v Speaker 1>History of mathematics by W. W. Rowsball, Chapter nineteen. Mathematicians

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<v Speaker 1>of Recent Times. It is evidently impossible for me to

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<v Speaker 1>discuss adequately the mathematicians of the age in which we live,

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<v Speaker 1>especially as I purposefully exclude from this work and any

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<v Speaker 1>detailed reference to living writers. I make therefore no attempt

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<v Speaker 1>to give a complete history of this period. But as

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<v Speaker 1>a sort of appendix to their preceding chapters, I had

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<v Speaker 1>a few notes on some of the more striking features

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<v Speaker 1>in the history during this century of pure mathematics, in

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<v Speaker 1>which I include theoretical dynamics and astronomy. But except for

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<v Speaker 1>a few allusions, I shall not discuss the applications of

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<v Speaker 1>mathematics to physics. These notes are brief, and in many

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<v Speaker 1>cases consists merely of a list of the names of

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<v Speaker 1>some of those to whom the development of any branch

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<v Speaker 1>of the subject is chiefly due, and an indication of

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<v Speaker 1>that part of it to which they have directed most attention.

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<v Speaker 1>I would refer to anyone who wishes for more details

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<v Speaker 1>to the invaluable catalog which has been compiled by the

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<v Speaker 1>Royal Society of London, and which contains a list under

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<v Speaker 1>the names of the authors of all the scientific papers

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<v Speaker 1>contributed during this century to journals and learned societies. In

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<v Speaker 1>only a few cases do I add any account of

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<v Speaker 1>the life and works of the mathematicians mentioned. Even with

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<v Speaker 1>these limitations, it has been very difficult to put together

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<v Speaker 1>a connected account of the mathematicians of recent times. And

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<v Speaker 1>I wish to repeat explicitly that I do not suggest,

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<v Speaker 1>nor do I wish my readers to suppose, that my

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<v Speaker 1>notes on a subject give the names of all the

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<v Speaker 1>chief writers who have studied it. In fact, the quality

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<v Speaker 1>of matter produced has been so enormous that no one

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<v Speaker 1>can expect to do more than make himself acquainted with

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<v Speaker 1>the work in some small department. As an illustration of

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<v Speaker 1>this remark, I must say that I have reason to

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<v Speaker 1>believe that something like fifteen thousand separate scientific memoirs are

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<v Speaker 1>now published every year by the different societies and journals

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<v Speaker 1>of Europe and America. Most of the treatises on the

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<v Speaker 1>history of mathematics omit all references to the work produced

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<v Speaker 1>during this century. The chief exceptions which I am acquainted

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<v Speaker 1>are a short dissertation by h. Hankell entitled z the

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<v Speaker 1>and Wiklang der Mathematique in de Lenzigne de junten in

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<v Speaker 1>Turberlingen eighteen eighty five. The eleventh and twelfth volumes of

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<v Speaker 1>Marie Histoire des Science, in which there are notes on

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<v Speaker 1>mathematicians who were born in the last century. Gerhartz Gislichte

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<v Speaker 1>der Mathematique in Deutschland, published in Munich eighteen seventy seven,

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<v Speaker 1>and a de scour on the professors at the Sorbonne

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<v Speaker 1>by Hermit in the Bulletende des Science Mathematique in eighteen ninety,

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<v Speaker 1>pages six to thirty six. A few histories of the

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<v Speaker 1>development of a particular subjects have been written, such as

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<v Speaker 1>those by the late Isaac Todd Hunter on the theories

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<v Speaker 1>of attraction and the calculus of probabilities. While annual volumes

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<v Speaker 1>of the British Association contain a number of reports on

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<v Speaker 1>the progress in several different branches of modern mathematics. A

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<v Speaker 1>few similar reports, and notably one in eighteen fifty seven

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<v Speaker 1>by J. Bertrand the development of mathematical analysis, have been

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<v Speaker 1>presented by the French Academy. I have found these authorities

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<v Speaker 1>and these reports useful, but I have derived most assistance

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<v Speaker 1>in writing this chapter from the obituary notices in the

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<v Speaker 1>proceedings of various learned societies, foreign as well as British.

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<v Speaker 1>I am also indebted to information kindly furnish me by

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<v Speaker 1>various friends, and if I do not further dwell on this,

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<v Speaker 1>it is only that I would not seem to make

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<v Speaker 1>them responsible for errors and omissions which they would have

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<v Speaker 1>avoided in their own works. A period of exceptional intellectual

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<v Speaker 1>activity in any subject is usually followed by one of

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<v Speaker 1>comparative stagnation, and after the deaths of Lagrange, Laplace, Legendre

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<v Speaker 1>and Poissin, of the French school, which had occupied so

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<v Speaker 1>prominent a position at the beginning of this century, ceased

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<v Speaker 1>for some years to produce much new work. Some of

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<v Speaker 1>the mathematicians whom I intend first to mention, gauss Abel

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<v Speaker 1>and Jacoby, were contemporaries of the later years of the

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<v Speaker 1>French mathematicians just named, but their writings appear to me

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<v Speaker 1>to belong to a different school, and thus are properly

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<v Speaker 1>placed at the beginning of a fresh chapter. There is

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<v Speaker 1>no mathematician of this century whose writings have had a

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<v Speaker 1>greater effect than those of Gauss, nor is it on

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<v Speaker 1>only one branch of the science that his influence has

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<v Speaker 1>left a permanent mark. I cannot therefore commence my account

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<v Speaker 1>of the mathematicians of recent times better than by describing

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<v Speaker 1>very briefly his more important researches. Gauss Carl Frederick Gauss

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<v Speaker 1>born at Brunswick on April twenty third, seventeen seventy seven

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<v Speaker 1>and died in Gautingen on February twenty third, eighteen fifty five.

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<v Speaker 1>His father was a bricklayer, and Gauss was indebted for

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<v Speaker 1>liberal education, much against the will of his parents, who

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<v Speaker 1>wished to profit by his wages as a laborer, to

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<v Speaker 1>the notice which his talents procured from the reigning Duke.

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<v Speaker 1>In seventeen ninety two he was sent to the Caroline College,

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<v Speaker 1>and by seventeen ninety five, professors and pupils alike admitted

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<v Speaker 1>that he knew all of the former could teach him.

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<v Speaker 1>It was while there that he investigated the method of

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<v Speaker 1>Lee squares and proved by induction the law of quadratic reciprocity.

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<v Speaker 1>Thence he went to Gunteingen, where he studied under Costner.

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<v Speaker 1>Many of his discoveries in the theory of numbers were

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<v Speaker 1>made while a student there. In seventeen ninety eight he

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<v Speaker 1>returned to Brunswick, where he earned somewhat a somewhat precarious

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<v Speaker 1>livelihood by private tuition. In seventeen ninety nine he Gauss

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<v Speaker 1>published his demonstration that every algebraical equation has a root

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<v Speaker 1>of the form A plus b I, a theorem of

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<v Speaker 1>which altogether he gave gave three distinct proofs. In eighteen

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<v Speaker 1>oh one, this was followed by Disquisition's ARITHMETICAI, which is

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<v Speaker 1>printed as the first volume of his collected works. The

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<v Speaker 1>greater part of it has been sent to the French

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<v Speaker 1>Academy in the preceding year, and has been rejected with

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<v Speaker 1>a sneer, which, even if the book had been as

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<v Speaker 1>worthless as the referees believed, would have been unjustifiable. Gauss

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<v Speaker 1>was deeply hurt, and his reluctance to publish his investigations

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<v Speaker 1>may be partly attributable to this unfortunate incident. The next discovery,

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<v Speaker 1>Gauss was in a totally different department of mathematics. The

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<v Speaker 1>absence of any planet in the space between Mars and Jupiter,

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<v Speaker 1>where Bode's law would have led observers to expect one,

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<v Speaker 1>had been long remarked, but it was not until eighteen

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<v Speaker 1>oh one that any one of the numerous group of

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<v Speaker 1>minor planets which occupy that space was observed. The discovery

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<v Speaker 1>was made made by Piazzi of Palermo, and was the

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<v Speaker 1>more interesting as its announcement occurred simultaneously with a publication

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<v Speaker 1>by Hegel, in which he severely criticized astronomers for not

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<v Speaker 1>paying more attention to philosophy, a science, said he, which

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<v Speaker 1>would not at once have shewn them that there could

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<v Speaker 1>not possibly be more than seven planets, a study of

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<v Speaker 1>which would therefore have prevented an absurd waste of time

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<v Speaker 1>in looking for what, in the nature of things could

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<v Speaker 1>never be found. The new planet was named Cerrees, but

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<v Speaker 1>it was seen under conditions which appeared to render it

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<v Speaker 1>almost impossible to forecast its orbit. The observations were fortunately

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<v Speaker 1>communicated to Gauss. He calculated its elements, and his analysis

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<v Speaker 1>proved to him to be the first of theoretical astronomers,

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<v Speaker 1>no less than the greatest of arithmeticians. The attention excited

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<v Speaker 1>by these investigations procured him, in eighteen o seven the

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<v Speaker 1>offer of a chair at Saint Petersburg, which he declined.

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<v Speaker 1>In the same year he was appointed director to Guntagen

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<v Speaker 1>Observatory and professor of astronomy there. These offices he retained

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<v Speaker 1>to his death, and after his appointment he never slept

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<v Speaker 1>away from his observatory, except on one occasion, where he

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<v Speaker 1>attended a scientific conference at Berlin. His lectures were singularly

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<v Speaker 1>lucid and perfect in form and it is said that

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<v Speaker 1>he used here to give the analysis by which he

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<v Speaker 1>had arrived at his various results, and which it is

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<v Speaker 1>so conspicuously absent from his published demonstrations. But for fear

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<v Speaker 1>his auditors should lose the threat of his discourse, he

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<v Speaker 1>never willingly permitted them to take notes. I have already

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<v Speaker 1>mentioned Gauss's publications in seventeen ninety nine, eighteen oh one,

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<v Speaker 1>and eighteen o two. For some years after eighteen o

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<v Speaker 1>seven his time was almost wholly occupied by work connected

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<v Speaker 1>with his observatory. In eighteen oh nine he published at

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<v Speaker 1>Hamburg his Theorea Motus corporum celestium, a treatise which contributed

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<v Speaker 1>largely to the improvement of practical astronomy and introduced the

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<v Speaker 1>principle of curve linear triangulation. And on the same subject,

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<v Speaker 1>but connected with observations in general, we have his memoir

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<v Speaker 1>therea Comnetonis observationionim erobinus minimus Obnoxia, with a second part

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<v Speaker 1>and a supplement. Somewhat later, he took up the subjects

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<v Speaker 1>of geodesi, acting from eighteen twenty one to eighteen forty

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<v Speaker 1>eight as scientific advisor to the Danish and Hanoverian governments

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<v Speaker 1>for the survey then in progress. His papers of eighteen

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<v Speaker 1>forty three in eighteen sixty six Ubergedestada de houngueo desi,

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<v Speaker 1>contain his researches on the subject. Gauss's researches on electricity

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<v Speaker 1>and magnetism date from about the year eighteen thirty. His

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<v Speaker 1>first paper on the theory of magnetism, entitled intensitest vis

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<v Speaker 1>megnektica terrestris adventsuram absolatum ravokata, was published in eighteen thirty three,

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<v Speaker 1>a few months Afterwards, he, together with Reber, invented the

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<v Speaker 1>declination instrument and the Bifiller magnetometer. In the same year

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<v Speaker 1>that they erected at Gottingen a magnetic observatory free from iron,

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<v Speaker 1>as Humboldt and Arago had previously done on a smaller scale,

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<v Speaker 1>where they made magnetic observations, and in particular schwed that

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<v Speaker 1>it was possible and predictable to send telegraph signals. In

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<v Speaker 1>connection with this observatory, Gauss founded the association called the

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<v Speaker 1>Magnetische Veren, with the object of securing continuous observations at

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<v Speaker 1>fixed times. The volumes of their publications Restata ulstur Boobachtungen

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<v Speaker 1>des magnetician Vere for eighteen thirty eight and eighteen thirty

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<v Speaker 1>nine contained two important memoirs by Gauss, one on the

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<v Speaker 1>general theory of earth magnetism and on the other on

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<v Speaker 1>the theory of forces attracting according to the inverse square

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<v Speaker 1>of the distance. Like Poisson, he treated the phenomenon in

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<v Speaker 1>electrostatics as due to attractions and repulsions between imponderable particles.

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<v Speaker 1>In electrodynamics he arrived in eighteen thirty five at a

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<v Speaker 1>result equivalent to that given by W. E. Weber in

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<v Speaker 1>eighteen forty six, namely that the attraction between two electrified

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<v Speaker 1>particles E and E prime, whose distance apart as are,

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<v Speaker 1>depends on their relative motion and position according to the

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<v Speaker 1>formula ee prime r to the power of minus two

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<v Speaker 1>multiplied by the quantity one plus r r umlaut minus

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<v Speaker 1>one half r squared quantity squared times c to the

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<v Speaker 1>minus two. However, held that no hypothesis was satisfactory which

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<v Speaker 1>rested on a formula and was not a consequence of

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<v Speaker 1>a physical conjecture, and as he could not frame a

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<v Speaker 1>plausical physical conjecture, he abandoned the subject. Such conjectures were

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<v Speaker 1>proposed by Riemann in eighteen fifty eight and by C.

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<v Speaker 1>Newman and E. Betty in eighteen sixty eight, but Helmholtz

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<v Speaker 1>in eighteen seventy, eighteen seventy three and eighteen seventy four

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<v Speaker 1>chewed that they were untenable. A simpler view, which regards

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<v Speaker 1>all electric and magnetic phenomenon as stresses and motions of

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<v Speaker 1>a material elastic medium, have been outlined by Michael Faraday

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<v Speaker 1>and was elaborated by James Clerk Maxwell, the latter by

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<v Speaker 1>the use of generalized coordinates was able to deduce the

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<v Speaker 1>consequences and the agreement with experiment as close. These and

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<v Speaker 1>other electric theories were classified and critically discussed in a

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<v Speaker 1>memoir by J. J. Thompson in eighteen eighty five. Gauss's

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<v Speaker 1>researches on optics, including systems of lenses, were published in

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<v Speaker 1>eighteen forty in his De hok Tricia unto Sohungen. From

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<v Speaker 1>this sketch, it will be seen that the ground covered

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<v Speaker 1>by Gauss's researches was extraordinarily wide. I will now mention

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<v Speaker 1>very briefly some of his most important discoveries in peer mathematics.

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<v Speaker 1>His most celebrated work in pere mathematics is Dedisguisizioni's ARITHMETICAI,

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<v Speaker 1>which has proved the starting point for several interesting investigations

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<v Speaker 1>on the theory of numbers. This treatise and the Genres

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<v Speaker 1>Tierri de Nombre remains standard works on the theory of numbers.

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<v Speaker 1>But just as in his discussion of elliptic functions, le

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<v Speaker 1>Gendre failed to rise to the conception of a new

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<v Speaker 1>subject and confused himself to regarding their theory as a

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<v Speaker 1>chapter in the integral calculus. So he treated the theory

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<v Speaker 1>of numbers as a chapter in algebra. G also, however,

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<v Speaker 1>realized that the theory of discrete magnitudes or higher arithmetic, was

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<v Speaker 1>of a different kind from that of continuous magnitudes or algebra,

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<v Speaker 1>and he introduced a new notation and new methods of

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<v Speaker 1>analysis of which subsequent writers have generally availed themselves, In particular,

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<v Speaker 1>the disquisitiones ARITHMETICI introduced the modern theory of congruences of

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<v Speaker 1>the first and second orders, and to this Gauss reduced

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<v Speaker 1>intermediate analysis. In it also he discussed the solution of

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<v Speaker 1>binomial equations of the form x to the power n

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<v Speaker 1>equals one. This involves the celebrated theorem that the only

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<v Speaker 1>regular polygons which can be constructed by elementary geometry are

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<v Speaker 1>those in which the number of sides is two to

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<v Speaker 1>the power of m multiplied by two to the power

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<v Speaker 1>of n plus one, where m and n are integers

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<v Speaker 1>and two to the power n plus one is a prime,

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<v Speaker 1>a discovery he had made in seventeen ninety sive he

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<v Speaker 1>developed the theory of ternary quadratic forms involving two indeterminates.

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<v Speaker 1>He also investigated the theory of determinants, and it was

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<v Speaker 1>on Gauss's results that Jacobi based his researches on that subject.

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<v Speaker 1>The theory of functions of double periodicity had its origin

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<v Speaker 1>in the discoveries of Abel and Jacoby, which I described later.

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<v Speaker 1>Both these mathematicians arrived at the theta functions, which play

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<v Speaker 1>so large a part in the theory of the subject. Gauss, however,

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<v Speaker 1>had independently and a deed at a far earlier date,

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<v Speaker 1>discovered these functions and their chief properties, having been led

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<v Speaker 1>to them by certain integrals which occurred in the determinacio

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<v Speaker 1>atraxiones to evaluate, which he invented the transformation now associated

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<v Speaker 1>with the name of Jacoby, though Gauss at a later

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<v Speaker 1>time communicated the fact that Jacoby he did not publish

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<v Speaker 1>his researches. They occur in a series of notebooks of

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<v Speaker 1>a date not later in eighteen o eight, and are

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<v Speaker 1>included in his collected works. Of the remaining memoirs in

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<v Speaker 1>pure mathematics, the most remarkable are those in the theory

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<v Speaker 1>of bi quadratic residues, wherein the notion of complex numbers

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<v Speaker 1>of the form A plus bi was first introduced into

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<v Speaker 1>the theory of numbers, in which are included several tables

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<v Speaker 1>and notably one of the number of the classes of

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<v Speaker 1>binary quadratic forms, that of relating to the proof of

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<v Speaker 1>the theorem at every numerical equation has a real or

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<v Speaker 1>imaginary route, that on the summation of series, that on

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<v Speaker 1>the hypergeometric series, which contains a discussion of the gamma function,

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<v Speaker 1>and lastly one on interpolation. His introduction of rigorous tests

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<v Speaker 1>for the convergency of infinite series especially noticeable. Finally, we

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<v Speaker 1>have the important memoir on the conformal representation of one

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<v Speaker 1>surface upon another, in which the results given by Lagrange

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<v Speaker 1>for surfaces of revolution are generalized for any surfaces. In

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<v Speaker 1>the theory of attractions, we have a paper on the

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<v Speaker 1>attraction of homogeneous ellipsoids, the already mentioned memoir of eighteen

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<v Speaker 1>thirty nine Algermaine lesstes in betsiung hofzi in verkiten verhetnes

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<v Speaker 1>des quadrits der enfherroung on the theory of forces attracting

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<v Speaker 1>according to the inverse square of the distance, and the

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<v Speaker 1>memoir determinaccio Etchaciones, in which it is shewn that the

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<v Speaker 1>secular variations which the elements of the orbit of a

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<v Speaker 1>planet experience from the attraction of another planet, which disturbs

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<v Speaker 1>it are the same as if the mass of the

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<v Speaker 1>disturbing planet were distributed over its orbit into an elliptic

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<v Speaker 1>ring in such a manner that equal masses of the

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<v Speaker 1>ring would correspond to the arcs of the orbit described

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<v Speaker 1>in equal times. The great masters of modern anounce us

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<v Speaker 1>are Lagrange, Laplace, and Gauss, who were contemporaries. It is

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<v Speaker 1>interesting to note the marked contrast in their styles. Lagrange

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<v Speaker 1>is perfect both in form and matter. He is careful

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<v Speaker 1>to explain his procedure, and though his arguments are general,

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<v Speaker 1>they are easy to follow. Laplace, on the other hand,

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<v Speaker 1>explains nothing is absolutely indifferent to style, and if satisfied

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<v Speaker 1>with his that his results are correct, is content to

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<v Speaker 1>leave them either with no proof or with a faulty one.

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<v Speaker 1>Gauss is as exact and elegant as Lagrange, but even

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<v Speaker 1>more difficult to follow than Laplace, for he removes every

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<v Speaker 1>trace of the analysis by which he reached his results

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<v Speaker 1>and studies to give a proof which, while rigorous, shall

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<v Speaker 1>be as concise and synthetical as possible. End of Part

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<v Speaker 1>thirty four. Recording by Paul King, Mississauga http PJK dot

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<v Speaker 1>scripts at it M I T Dot E du forward

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<v Speaker 1>slash p k J