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Updated: May 25, 2025
It would make prediction wholly vain, and entangle truth in a totality of things which is unique at every instant, and never can recur. The principles of mathematics are as clearly postulates. In Euclidean geometry we assume definitions of 'points, 'lines, 'surfaces, etc., which are never found in nature, but form the most convenient abstractions for measuring things.
It would be better to say that space is infinite in relation to practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry. Another example of an infinite continuum is the plane. On a plane surface we may lay squares of cardboard so that each side of any square has the side of another square adjacent to it.
By positing the point as the unit from which to start, and deriving our conception of the plane from the point, we constitute Euclidean space. By starting in the manner described above, with the plane as the unit, and conceiving the point from it, we constitute polar-Euclidean space.
Some of them have been more or less fully worked out, while in certain instances all that has been done is to show that they are mathematically conceivable. Among these there is one which in all its characteristics is polarically opposite to the Euclidean system, and which is destined for this reason to become the space-system of levity.
At the middle of the forehead horizontally subdivide this upper quoin, and then you have two almost equal parts, which before were naturally divided by an internal wall of a thick tendinous substance. *Quoin is not a Euclidean term. It belongs to the pure nautical mathematics. I know not that it has been defined before.
As, however, he glanced to-day over the pages of Part Three, "The Origin and Nature of the Affects," he felt somehow out of tune with this bloodless vivisection of human emotions, this chain of quasi-mathematical propositions with their Euclidean array of data and scholia, marshalling passions before the cold throne of intellect.
Without it the following reflection would have been impossible: In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry.
Accordingly Euclidean metrical geometry in space is completely established and lengths in the spaces of different time-systems are comparable as the result of definite properties of nature which indicate just that particular method of comparison. The comparison of time-measurements in diverse time-systems requires two other axioms.
Deservedly or not, Eliezer Mann was called "the Hebrew Socrates"; and many a Maskil in his study of mathematics turned for guidance to Manoah Handel of Brzeszticzka, Volhynia, author and translator of several scientific works, who rendered seven Euclidean propositions into Hebrew. Polyglots they were compelled to be by force of circumstances.
When enough abstract logical properties of such relations have been enumerated to determine the resulting kind of geometry, say, for example, Euclidean geometry, it becomes unnecessary for the pure geometer in his abstract capacity to distinguish between the various relations which have all these properties. He considers the whole class of such relations, not any single one among them.
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