Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

II. |

II. |

II. |

VI. |

VI. |

VI. |

VI. |

III. |

I. |

VI. |

VI. |

I. |

VI. |

VI. |

VI. |

VI. |

VI. |

IV. |

IV. |

II. |

IV. |

V. |

III. |

VI. |

III. |

III. |

V. |

VI. |

III. |

III. |

VI. |

VI. |

VI. |

V. |

V. |

VII. |

V. |

I. |

I. |

V. |

VI. |

VII. |

III. |

III. |

III. |

VII. |

III. |

I. |

III. |

VI. |

II. |

VI. |

I. |

V. |

III. |

I. |

VII. |

VII. |

II. |

VI. |

V. |

V. |

I. |

II. |

II. |

IV. |

V. |

V. |

V. |

II. |

II. |

V. |

V. |

IV. |

Dictionary of the History of Ideas | ||

*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

which non-Euclidean spherical geometry is adduced

as a particular case.

Dictionary of the History of Ideas | ||