credits: arxiv/From Euclid to Riemann

The purpose of this essay is to trace the historical development of geometry while focusing on how we acquired mathematical tools for describing the shape of the universe." More specfically, our aim is to consider, without a claim to completeness, the origin of Riemannian geometry, which is indispensable to the description of the space of the universe as a generalized curved space.

and even thought that length, area, and volume belong to different categories, thus lumping them together makes no sense. Under such shackles (and being devoid of symbolic algebra), the "equality," addition/subtraction," and the "large/small relation" for two figures in the same category were defined by means of geometric operations. Specifically, they considered that two polygons (resp. polyhedra) are "equal" if they are scissors-congruent ; i.e., if the first can be cut into finitely many polygonal (resp. polyhedral) pieces that can be reassembled to yield the second.

Any two polygons with the same numerical area are scissors-congruent as shown independently by W. Wallace in 1807, Farkas Bolyai in 1832, and P. Gerwien in 1833.

Gauss questioned whether this is the case for polyhedra in two letters to his former student C. L. Gerling dated 8 and 17 April, 1844 (Werke, VIII, 241{42). In 1900, Hilbert put Gauss's question as the third problem in his list of the 23 open problems at the second ICM. M. W. Dehn, a student of Hilbert, found two tetrahedra with the same volume, but non-scissors-congruent (1901).

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Long ago, Eudoxus developed an idea similar to Dedekind's in essence. Before
that, Greek geometers had tacitly assumed that two magnitudes α β, are always
commensurable; i.e., there exist two natural numbers m; n such that mα = nβ .

Thus, with the discovery of incommensurable magnitudes which results from the
Pythagorean
Theorem, the issue arose as to how to define equality of two ratios.

The impeccable definition attributed to Eudoxus is :

- "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order" (Book V).

In modern terminology, this is expressed as

- Given four magnitudes α β γ δ , the two ratios α:β and γ:δ are said to be the same if for all natural numbers m, n, it be the case that according as mα >=< nβ , so also is mγ >=< nδ .

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Newton >>>

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Leibniz claimed that

- space is merely relations between objects, thereby no absolute location in space,
- and that time is order of succession.

on the basis of "the principle of sufficient reason" and "the principle of the identity of indiscernibles,".

To be specific, his principle of sufficient reason made him assert that
nothing happens without a reason, and that all reasons are *ex hypothesi*
God's reasons.

The period that begins with Newton and Leibniz corresponds to
the commencement of the close relationship between mathematics and physics.

From then on, both disciplines have securely influenced each other.

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