13. Absolute Space. The sixteenth century also
initiated descriptive and projective geometry (J. L.
Coolidge, Chs. 5 and 6), and when, much later, in the
nineteenth century, projective geometry was fully
developing, its unfolding was part of the creation of
many novel structures, Euclidean and other (see sec.
16, below). In the seventeenth century there were
remarkable achievements by Gérard Desargues, Blaise
Pascal, and others. But after that there was a long
period of very slow advance, and non-Euclidean ge-
ometry, for instance, presented itself only in the nine-
teenth century, although, by content and method, the
eighteenth century was just as ready for it.
This retardation may have been caused in part by
Isaac Newton's insistence on the Euclidean character
of his absolute Space (for other such retardations
caused by Newton see Bochner [1966], pp. 346f.). In
Newton's Principia, the program was to erect a mathe-
matical theory of mechanics, based on the inverse
square law of gravitation, from which to deduce the
three planetary laws of Kepler and Galileo's parabolic
trajectory of a cannon ball, all in one. Newton suc-
ceeded in this endeavor, but virtually every step of
his reasoning required and presupposed that his under-
lying space be Euclidean. Newton was keenly aware
of this prerequisite, and following a general philo-
sophical trend of his age, he endowed his Euclidean
background space with extra-formal features of physi-
cal and metaphysical uniqueness and theological
excellence, by which it became “absolute.” These
extra-formal features are not needed for the deductions
of the main results, and Newton discourses on these
features in supplementary scholia only (Bochner [1969],
Ch. 12).
In support of his contention that there is an absolute
space, Newton adduces two arguments (experiment
with two globes, and, more importantly, with the rota-
ting bucket) which physicists find arresting even today,
although the arguments do not demonstrate that there
is space which is absolute in Newton's own sense. In
the Victorian era, the physicist-philosopher Ernst Mach
in his The Science of Mechanics... (Die Mechanik
in ihrer Entwicklung; many editions and translations),
which was composed from a post-Comtean positivist
stance of his age, was quite critical of Newton's argu-
ments and conclusions (Jammer, Ch. 5, esp. pp.
140-42); but a recent reassessment by Max Born leads
to a balanced appraisal of importance (Born, pp.
78-85).
The opposition to absolute space by philosophers
began immediately with the Leibniz-Clarke corre-
spondence (see Introduction, above), and has not quite
abated since. Yet the Encyclopédie of Diderot and
d'Alembert, under the heading Espace (1755), pro-
nounced the debate sterile: “cette question obscure est
inutile à la Géométrie & à la Physique” (Jammer, pp.
137f.).
As a background for mechanics, Newton's Euclidean
space eventually evolved a variant of non-Euclidean
structure out of itself (Bochner [1966], pp. 192-201,
338). In fact, one hundred years after the Principia,
Louis de Lagrange in his Mécanique analytique (1786),
when analyzing a mechanical system of finitely many
mass points with so-called “constraints,” introduced de
facto a multidimensional space of so-called “gener-
alized coordinates” (or “free parameters”) as a sub-
space of a higher-dimensional space. Implicitly, though
not at all by express assertion or even awareness,
Lagrange endowed this space with the non-Euclidean
Riemannian metric which the imbedding in the higher
dimensional Euclidean space is bound to induce.
Analysts in the nineteenth century knew this part
of the Lagrangian mechanics extremely well. This may
help to explain why, for instance, Carl Jacobi showed
no reaction of surprise at the news of the Bolyai-
Lobachevsky non-Euclidean hyperbolic geometry
around 1830; nor, apparently, did William Rowan
Hamilton ever mention it, or even Bernhard Riemann,
who should have felt “urged” to speak about it in his
great memoir on general “Riemannian” geometry, in
which non-Euclidean spherical geometry is adduced
as a particular case.
