I'm currently going through Barrow's theorem for a geometrical interpretation of the Fundamental Theorem of Calculus and there's some notation I can't get my head around. Suppose the following diagram:
The author makes statements like: $DT = R\frac{DF}{DE}$ or $\frac{LF}{LK} = \frac{DF}{DT}$
I've been looking around for a while for an explanation of what it means to divide two line segments by one another but I have found nothing. My problem is: there's been no coordinate system defined, so how can we stick a result on the division? And furthermore, all these segments are in a definite relationship but their lengths are arbitrary so I don't understand how we could get a definite result out of that division without first defining a coordinate system and name some variables.
Looking at the figure you can see that $${|LF|\over|LK|}={|DF|\over|DT|}\ .$$ I suggest you interpret $DT$, $DF$, and $DE$ in the same way as segment lengths. Finally $R$ is the length of the segment drawn to the right.
It is difficult to say more without knowing what the curves in the figure mean.