Ebook: Development of Mathematics in the 19th Century

K l e i n on n i n e t e e n t h c e n t u r y m a t h e m a t i c s
vorlesungen uber die entwicklung der mathematik im 19. Jahrhunderl
by felix klein. Teil i. Berlin, springer, 1926. Xiii+385 pp.

twenty years ago, during a series of walks in the forest about hahnen-
klee, in t h e h a r t z mountains, the conversation between klein and a com-
panion covered, as would naturally be the case, a wide range. Three state-
ments, however, impressed his listener* very strongly. One was political:
"there was a time when we looked up to england socially, politically, and
as a naval power,—but t h a t is a thing of the past. " the second was political
and of military significance: "america has no standing army today;
twenty-five years hence she will have a large one. " it is not strange t h a t
his auditor wondered at the real significance of these two statements by
a man of klein's vision and prominence. The third remark was in response
to a statement to the effect t h a t he of all men was the one to write a history
of mathematics in the 19th century. "i am too old," was the reply, "it
needs a young man who could devote years to its preparation. " when it
was urged t h a t he had seen the development and had taken part in it as
few if any others living had done, he remarked, "no, all t h a t i could do
would be to give a few lectures on the great events, but i am too much occu-
pied to prepare even these. " ten years later, when the war was on, and
his family had been sorely stricken, he gave these very lectures in his home
in göttingen, before a small group of listeners anxious to receive from a
master t h a t which only a master could give.
t h e lectures have been edited by professors courant and neugebauer
and are published as band x x i v of die grundlehren der mathematischen
wissenschaften, a recent series, already well known to all mathematical
students. Rightly did they say in their vorwort: "diese vorlesungen sind
die reife frucht eines reichen lebens inmitten der wissenschaftlichen
ereignisse, der ausdruck iiberlegener weisheit und tiefen historischen
sinnes, einer hohen menschlichen kultur und einer meisterhaften gestal-
tungskraft; sie werden sicherlich auf alle mathematiker und physiker und
weit über diesen kreis hinaus eine grosse wirkung ausüben. "
the work is divided into eight chapters. The first naturally begins with
the founder of the modern german school of pure and applied mathematics,
—gauss. It considers his work in the applied field with respect to astron-
omy, geodesy, and physics, the last in connection with a. Von humboldt
and wilhelm weber. In pure mathematics the attention is given chiefly
to the number theory, forms, and the function theory, with a succinct
statement as to the claim t h a t gauss is entitled to the award of priority
* t h e reader may suspect the identity of the "listener," who modestly
conceals it. The editors may state at least t h a t the conversation was actual
and t h a t this report is a first-hand report.

in t h e discovery of non-euclidean geometry, although not to priority of
publication (pp. 57-60).
chapter ii relates to france and the ecole polytechnique in t h e first
decade of the century. In this klein pays high tribute to the work of men
like fourier, cauchy, poncelet, monge, and galois. Of cauchy he speaks
as one "der sich mit seinen glanzenden leistungen auf allen gebieten der
mathematik fast neben gauss stellen kann. "
chapter i i i concerns t h e founding of crelle's journal and the rise of
pure mathematics in germany. In this period the names mentioned as
most prominent are those of abel, jacobi, moebius, plücker, and steiner.
chapter iv considers the development of algebraic geometry by moe-
bius, plücker, and steiner, with reference to lagrange, chasles, cayley,
sylvester, salmon, beltrami, and clifford, as well as to his own country-
men,—riemann, hesse, and grassmann.
chapter v deals with mechanics and mathematical physics in germany
and england before 1880,—the period of hamilton, thomson (kelvin),
and maxwell in england, of gibbs in america, and of franz neumann
in germany.
chapter vi is devoted to the development of the theory of functions of
a complex variable, chiefly at the hands of riemann and weierstrass, but
with mention of the influence of dirichlet, h. A. Schwarz, fuchs, c.
neumann, kovalevski, and others.
chapter vii is taken up with the study of algebraic forms, and chapter
v i i i with t h a t of the theory of groups and automorphic functions, with
special reference to the work of galois, c. Jordan, hermite, riemann, and
poincaré.
it is proposed, in band ii, to treat chiefly of the theory of invariants
and of relativity.
all who knew professor klein with any degree of intimacy are aware
of his broadmindedness and catholicity of spirit. His love for science did
not permit him to allow political prejudices to warp his judgment as to the
tribute due to scholarship beyond the boundaries of his own country. It
speaks well for his judicial spirit to observe that, of the scientists mentioned,
nearly half were not of german nationality. Nearly a fourth were french,
about an eighth were british, and a fair number were from italy, scan-
dinavia, russia, and our own country. It was natural t h a t the work of
the germans should have been best known to him, and it was entirely fair
t h a t he should have ranked them higher than others of the last half of the
century, not alone numerically b u t on the score of achievement.

David eugene smith