Question

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Does anybody have a historical reference to what might be called differential ratios?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.

Cheers, Scott

Answer 1

The most famous historical reference to differential ratios that I know of is Euler's Institutiones calculi differentialis vol. 1, caput 3. Starting on p. 64 you can find a lot of differential ratios like

$\frac{dx + dx^2}{dx} = 1$

or

$a\sqrt{dx} + bdx = a\sqrt{dx}$.