# Were integrals really called solution curves (or vice versa)?

For some reason I recall hearing that around the time Euler wrote his Calculus books (1768-1770), or even before then, what we call integrals now were called solution cuvres (or even possibly the other way around since my memory on this is seriously hazy). Can I get some confirmation or refutation on this?

They are still so called in some contexts, although "integral curves" is more common. If $dy/dx=X(x)$ then $y$ is both an integral of $X$ and a solution to the differential equation. Such dual use does already appear in Euler's Institutionum Calculi Integralis. For example, he writes in §11 (in Bruce's translation):
"Now even if here only a single variable quantity $x$ appears, yet in fact two are considered; for the other is that function itself, the differential of which we take to be $Xdx$; which if we designate with the letter $y$, becomes $dy=Xdx$ or $dy/dx=X$, thus as here generally is the ratio of the differentials $dy:dx$ proposed, which is equal to $X$, and thus there becomes $y=\int Xdx$."