Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably geometric) intuition behind the use of the word "slope" for this?
This is an addition to Angelo's comment. Given a vector bundle $E$, you can consider its HarderNarasimhan filtration and assign to each element of the filtration a point in the degreerank plane. The HNpolygon is the polygon obtained by connecting the dots. S.S.Shatz discussed the behaviour of the HNpolygon under specialisation in The decomposition and specialisation of algebraic families of vector bundles I beleive this is where the term orginated, see also AtiyahBott, section 7 (p.565). Addendum: You can gain minor visual gratification from looking at the degreerank plane as follows. If $F\subset E$ is a subbundle, then $$ \deg \underline{Hom}(F,E)= \textrm{rk}F\deg E\textrm{rk}E \deg F= \left \begin{array}{cc} \deg E & \deg F\\\ \textrm{rk} E & \textrm{rk} F\\\ \end{array} \right. $$ Also, $F$ destabilises $E$ exactly when the above determinant has negative sign. 


slope'' is justified by the fact that it measures the relative growth of polyvectors (along orbits of the KontsevichZorich cocycle) inside flat subbundles of the Hodge bundle over the moduli space of Abelian differentials with respect to their
counterparts'' inside the tautological subbundle. – Matheus Oct 3 '12 at 17:11