"The above refers only to hyper-space of two dimensions. Naturally there arises the question whether there are different kinds of hyper-space of three or more dimensions. Riemann showed that there are three kinds of hyper-space of three dimensions having properties analogous to the three kinds of hyper-space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surfaces, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface; a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface."
Quote Details
Added by wikiquote-import-bot
Unverified quote
0 likes
Original Language: English
Available Languages (1)
Sources
Imported from EN Wikiquote
https://en.wikiquote.org/wiki/Non-Euclidean_geometry
Revision History
No revisions have been submitted for this quote.
Categories
Non-Euclidean geometry
91 quotes on TrueQuotesView all quotes by Non-Euclidean geometry →
Related Quotes
"The creation of non-Euclidean geometry showed... that mathematics could no longer be regarded as a body of unquestion…"
"Edwin Abbott Abbott"
"In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious onc…"
"Unlike those of science, the conclusions of mathematics had always regarded as deduced from basic truths. ...the very…"
"What was the effect of non-Euclidean geometry on the future progress of mathematics? ...Mathematics passed from serfd…"
"Even the mathematicians of the late nineteenth century did not take non-Euclidean geometry seriously for physical app…"
"... went considerably beyond Sacherri in deducing propositions under the hypotheses of the acute and obtuse angles. T…"
"It is no wonder that no contradiction was found under the hypothesis of the acute angle, for... the geometry develope…"
"Non-Euclidean geometry was the most weighty intellectual creation of the nineteenth century, or, at worst, might have…"
"Euclid’s Elements"