Napier's original conceptualization of the logarithm was as a relationship between an arithmetic progression and a geometric progression; a point moving with zero acceleration and a point moving with negative acceleration. This is problem II in Book I of Maria Agnesi's Analyical Institutions. Agnesi uses ratios and proportions to describe the problem but shifts to fractions in her example solutions.
Does anybody have a historical reference to what might be called differential ratios? I'm thinking of ratios or proportions with a 'dt' term that were solved with non-arithmetic reasoning involving for example the mediant.
Attempts to find inchoate logarithms before Napier, for example in the work of Wallis, have as far as I know been fully discounted but the link between Napier's thinking and Agnesi's problem is so obvious one imagines that a ratio formulation of the logarithm might have existed.
Thanks for comments and insight.