> I would very much like to know what Swedenborg wrote about geometrically
> interpreting negative results of differentiation. Most authors give two
> examples (Euler); but l'Hopital gave a rule.
By l'Hospital's rule of negative results of differentiation I suppose you
mean his discussion of how the value of the subtangent PT has an effect on
which side of the origin of x the point T on the diameter is situated
(Section II, Proposition I).
The answer is that Swedenborg doesn't bother at all about that problem.
Swedenborg is not famous for being a good mathematician, I suppose he is
more well-known for his interpretation of the bible. It is though remarkable
that he presents some theory about the differential- and integral-calculus.
When he presents the differentials for a curve defined by polar-coordinates,
he discusses three cases when the fixed point (the pole) is
1. on the curve
2. "inside" the curve
3. "outside" the curve
He is defining the "differential triangle" in these three cases. Then he
shows (almost like Reyneau) the formula for the element of the area between
the curve and the diameter.
I have not seen this division into three cases in any other contemporary
elementary book. Perhaps Wolffs Anfangsgrunde ... (thank you D. N.) has got
a presentation like that. I have not met the book, but I will try to find
it.
Best greetings
Staffan Rodhe