Modern Math Begins

Modern Mathematics is a term by no means well defined. Algebra

cannot be called modern, and yet the theory of equations has

received some of its most important additions during the nineteenth

century, while the theory of forms is a recent creation. Similarly

with elementary geometry; the labors of Lobachevsky and Bolyai

during the second quarter of the century threw a new light upon the

whole subject, and more recently the study of the triangle has added

another chapter to the theory. Thus the history of modern

mathematics must also be the modern history of ancient branches,

while subjects which seem the product of late generations have root

in other centuries than the present.

 

How unsatisfactory must be so brief a sketch may be inferred from a

glance at the Index du Rep’ertoire Bibliographique des Sciences

Math’ematiques (Paris, 1893), whose seventy-one pages contain the

mere enumeration of subjects in large part modern, or from a

consideration of the twenty-six volumes of the Jahrbuch ”uber die

Fortschritte der Mathematik, which now devotes over a thousand pages

a year to a record of the progress of the science.footnote{The

foot-notes give only a few of the authorities which might easily be

cited. They are thought to include those from which considerable

extracts have been made, the necessary condensation of these

extracts making any other form of acknowledgment impossible.}

 

The seventeenth and eighteenth centuries laid the foundations of

much of the subject as known to-day. The discovery of the analytic

geometry by Descartes, the contributions to the theory of numbers by

Fermat, to algebra by Harriot, to geometry and to mathematical

physics by Pascal, and the discovery of the differential calculus by

Newton and Leibniz, all contributed to make the seventeenth century

memorable. The eighteenth century was naturally one of great

activity. Euler and the Bernoulli family in Switzerland,

d’Alembert, Lagrange, and Laplace in Paris, and Lambert in Germany,

popularized Newton‘s great discovery, and extended both its theory

and its applications. Accompanying this activity, however, was a too

implicit faith in the calculus and in the inherited principles of

mathematics, which left the foundations insecure and necessitated

their strengthening by the succeeding generation.

 

The nineteenth century has been a period of intense study of first

principles, of the recognition of necessary limitations of various

branches, of a great spread of mathematical knowledge, and of the

opening of extensive fields for applied mathematics. Especially

influential has been the establishment of scientific schools and

journals and university chairs. The great renaissance of geometry is

not a little due to the foundation of the ’Ecole Polytechnique in

Paris (1794-5), and the similar schools in Prague (1806), Vienna

(1815), Berlin (1820), Karlsruhe (1825), and numerous other

cities. About the middle of the century these schools began to exert

a still a greater influence through the custom of calling to them

mathematicians of high repute, thus making Z”urich, Karlsruhe,

Munich, Dresden, and other cities well known as mathematical centers.

 

In 1796 appeared the first number of the Journal de l’'Ecole

Polytechnique. Crelle’s Journal f”ur die reine und angewandte

Mathematik appeared in 1826, and ten years later Liouville began the

publication of the Journal de Math’ematiques pures et appliqu’ees,

which has been continued by Resal and Jordan. The Cambridge

Mathematical Journal was established in 1839, and merged into the

Cambridge and Dublin Mathematical Journal in 1846. Of the other

periodicals which have contributed to the spread of mathematical

knowledge, only a few can be mentioned: the Nouvelles Annales de

Math’ematiques (1842), Grunert’s Archiv der Mathematik (1843),

Tortolini’s Annali di Scienze Matematiche e Fisiche (1850),

Schl”omilch’s Zeitschrift f”ur Mathematik und Physik (1856), the

Quarterly Journal of Mathematics (1857), Battaglini’s Giornale di

Matematiche (1863), the Mathematische Annalen (1869), the Bulletin

des Sciences Math’ematiques (1870), the American Journal of

Mathematics (1878), the Acta Mathematica (1882), and the Annals of

Mathematics (1884).footnote{For a list of current mathematical

journals see the Jahrbuch ”uber die Fortschritte der Mathematik. A

small but convenient list of standard periodicals is given in Carr’s

Synopsis of Pure Mathematics, p. 843; Mackay, J. S., Notice sur le

journalisme math’ematique en Angleterre, Association franc{c}aise

pour l’Avancement des Sciences, 1893, II, 303; Cajori, F., Teaching

and History of Mathematics in the United States, pp. 94, 277;

Hart, D.~S., History of American Mathematical Periodicals, The Analyst,

Vol. II, p. 131.} To this list should be added a recent venture,

unique in its aims, namely, L’Interm’ediaire des Math’ematiciens

(1894), and two annual publications of great value, the Jahrbuch

already mentioned (1868), and the Jahresbericht der deutschen

Math-e-ma-tik-er-Vereinigung (1892).

%% Are those the correct hyphenation points?

 

To the influence of the schools and the journals must be added that

of the various learned societiesfootnote{For a list of such

societies consult any recent number of the Philosophical

Transactions of Royal Society of London. Dyck, W., Einleitung zu dem

f”ur den mathematischen Teil der deutschen

Universit”atsausstellung ausgegebenen Specialkatalog, Mathematical

Papers Chicago Congress (New York, 1896), p. 41.} whose published

proceedings are widely known, together with the increasing

liberality of such societies in the preparation of complete works of

a monumental character.

 

The study of first principles, already mentioned, was a natural

consequence of the reckless application of the new calculus and the

Cartesian geometry during the eighteenth century. This development

is seen in theorems relating to infinite series, in the fundamental

principles of number, rational, irrational, and complex, and in the

concepts of limit, contiunity, function, the infinite, and the

infinitesimal. But the nineteenth century has done more than

this. It has created new and extensive branches of an importance

which promises much for pure and applied mathematics. Foremost among

these branches stands the theory of functions founded by Cauchy,

Riemann, and Weierstrass, followed by the descriptive and

projective geometries, and the theories of groups, of forms, and of

determinants.

 

The nineteenth century has naturally been one of specialization. At

its opening one might have hoped to fairly compass the mathematical,

physical, and astronomical sciences, as did Lagrange, Laplace, and

Gauss. But the advent of the new generation, with Monge and Carnot,

Poncelet and Steiner, Galois, Abel, and Jacobi, tended to split

mathematics into branches between which the relations were long to

remain obscure. In this respect recent years have seen a reaction,

the unifying tendency again becoming prominent through the theories

of functions and groups.footnote{Klein, F., The Present State of

Mathematics, Mathematical Papers of Chicago Congress (New York,

1896), p. 133.}

Related posts:

1. Beginning of Modern Math
2. Algebra
3. Adjacent