3 4 6 D O C . 2 2 0 N O N - E U C L I D E A N G E O M E T R Y

Published in Die Neue Rundschau 36, no. 1 (1925): 16–20. For a manuscript, see [5 036]. The man-

uscript is dated “März 1924.”

[1]For earlier discussions of the historical foundations of geometry, see Einstein 1917a (Vol. 6,

Doc. 42), pp. 1–3, and Einstein 1921c (Vol. 7, Doc. 52).

[2]See Torretti 1978 for a historical overview of the development of the foundations of geometry

in the nineteenth century.

[3]Farkas Bolyai (1775–1856), his son János Bolyai (1802–1860), and Nikolai Lobachevsky

(1792–1856). Whereas Euclid’s parallel axiom can be formulated as “For a given line l and a point P

not on l, there is only one line through P that does not intersect l,” the modified version by János

Bolyai and Lobachevsky states that there is more than one such line. The resulting non-Euclidean

geometry later became known as “hyperbolic geometry”; it has constant negative curvature. For a his-

torical overview and reprints (in translation) of the relevant publications by Bolyai and Lobachevsky,

see Bonola 1912.

[4]Both of the standpoints Einstein describes here can also be found in Einstein 1921c (Vol. 7,

Doc. 52) and in Einstein 1924n (Doc. 321). However, in these texts Einstein does not link the first

standpoint to Helmholtz. See Giovanelli 2013 and Howard 2014 for a discussion of the two stand-

points and of these documents.

[5]In the manuscript, “mechanische” is missing.

[6]Hermann von Helmholtz (1821–1894), Professor of Physics at the University of Berlin, devel-

oped his ideas about the relation between geometry and physics in Helmholtz 1868, 1884. Einstein

read the latter in his Bern years, as part of the readings of the Olympia Academy.

[7]See Poincaré 1902 for the “conventionalism” of Henri Poincaré (1854–1912). This book, too,

was read in the Olympia Academy. For historical discussion of Poincaré’s conventionalism and Ein-

stein’s position, see Ben-Menahem 2006, especially chap. 3.

[8]In Einstein 1921c (Vol. 7, Doc. 52), p. 8, Einstein likewise states that, for the time being, con-

cepts such as that of a rigid body have to be taken as fundamental, even though a future theory is likely

to show them to be derived concepts.

[9]If one assumes, with Helmholtz, that all geometrical statements are statements about rigid phys-

ical bodies, one automatically excludes the possibility of spaces of nonconstant curvature, allowing

only Euclidean (zero curvature), hyperbolic (constant negative curvature), and elliptic (constant pos-

itive curvature) spaces. Einstein here warns that one may have to give up this idealization, pointing

out that general relativity in fact allows nonconstant curvature on an astronomically large scale, and

suggests that the same may be the case for the infinitely small. He explored this question in Einstein

1919a (Vol. 7, Doc. 17), in which he tries to make a connection between general relativity and the

constitution of elementary particles.

Bernhard Riemann (1826–1866) expounded his ideas about the foundations of geometry and its

relation to physics in Riemann 1867; see in particular his discussion of curvature in the infinitely large

and the infinitely small on p. 149.

[10]The elliptic geometry of Riemann is a counterpart to the Bolyai-Lobachevsky hyperbolic geom-

etry in the sense that in it there are no lines through P (see note 3) that do not intersect.

[11]Tullio Levi-Civita; see Levi-Civita 1917.

[12]See Weyl 1918, 1919, 1923, and Eddington 1921. See also Einstein 1923q (Doc. 75), note 8, for

Einstein’s initial skepticism about the unified theories of Weyl and Eddington.