everybody! I graduated with a degree in mathematics but switched careers and work as a software developer since then. I still have to satisfy that itch from time to time and refresh some long lost knowledge of mine or to get into a field I have neglected previously.
So I've picked up the talk "On the Hypotheses which lie at the Bases of Geometry" by Riemann but was immediately hit by a notion barrier. The cornerstone of the talk is the notion in the title of this post I have no idea what it means. To give a context, the first sentence using this term reads
"The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked."
I know (hopefully) the meanings of the individual words but the meaning still eludes me. If I'm not mistaken the word magnitude can be used in a very general meaning of a value (whatever that means) of a quantity (whatever that is). Like in
"As Kepler demonstrated, the non-uniform elliptical planetary orbit defines the magnitude of action within an orbit, as equal areas, rather than the arbitrary mathematical magnitudes of equal arcs or equal angles."
And if I'm not mistaken, the word magnitude is also often a synonym for a vector.
There is also the sentence "... space is only a particular case of a triply extended magnitude." hinting on the meaning having something to do with dimension.
Can anybody help me here, please?