What is the original source of "fiber" in mathematics? Maybe this question is not relevant to mathematics, but I am very curious. I am studying Abstract Algebra these days. 
I learn that the word fibre with definition: $f^{-1}({y}) = \{x \in X | f(x) = y\}$ for $\forall y \in Y$, where $f: X \rightarrow Y$ is a map.
And the definition of normal fiber in Wikipedia states:

Fiber or fibre (from the Latin fibra) is a natural or synthetic substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials.

Does fiber in mathematics have any relationship with the normal defintion? Or are they somewhat alike? Why do they define the inverse image with fibre?
Can anyone help me out?

Without having a source, it appears plausible to me that the term fiber comes from projection-like maps $f:X\to Y$ where $X$ can be thought of as looking somehow like $Y\times[0,1]$ (a cylinder over $Y$, or more generally, a fiber bundle).
Consider this picture:

The preimage $f^{-1}(y)$ is a "line over $y$". If you look at many such points $y$ and their preimages simultaneously, then you can get the impression that $X$ is a thick bundle of these lines, or fibers, tightly packed together (like a rope).
