Calculus in Ars Magica? A big issue in answering the question is: What do you mean by calculus?
Roughly speaking, one can think of two main components to calculus as taught in nodern schools:
- the notion of functions and
- the handling of infinitesimals (and infinites).
From the perspective of a modern high-school student 1 is very easy, and is taught first as an "introduction" to calculus; whereas 2. is the tough part where the "real meat" of calculus is.
This view is misleading.
In fact, (2) was understood at least 2400 years ago by greek philophers and mathematicians predating Aristotle (in particular, Eudoxus); they already had the notion of infinite series, and the rough idea of Riemann/Lebesgue integration -- which they readily applied to compute the lengths of, and areas under, complex curves. In this sense, I would disagree with Ouroboros when he states that Aristotle, by positing that the infinitely large and small are beyond the graps of men, blocked the advancement of calculus. In fact, it was quite the opposite. As far as I can tell Aristotle, echoing the prevailing views of mathematicians of his day, was saying that the infinitely large and small are tricky to handle directly (and instead best handled as infinite sets of progressively larger/smaller but finite approximations -- which is exactly the modern approach).
What seemed to be lacking in antiquity was (1). The very term "function" is first introduced by Leibniz, and the concept goes through a lot of revisions (and fallacies) in the 1700s and 1800s until we get to the modern view of it in the early 1900s. And here's the catch. Getting a "rough" idea of function is easy, and with it you can do basic calculus, which the ancient greeks did. Oreme (late 1300s) certainly has the idea when he talks about temperature of a metal rod as a function of the specific point on the rod, but I would argue that anyone who's studied quantitative phenomena where some of the variables are not spatial (e.g. at least one of them is time) intuitively has it: when/where quantity x has this value, quantity y has this other value, e.g. x multiplied by itself. But without a rigourous definition -- in fact, more rigorous than what was available to Leibniz and Newtown, as can be seen by the criticisms of e.g. Fourier -- the more advanced forms of calculus are not readily accessible and it's really easy to incur in deep fallacies.
To summarize, this is how I'd play it.
A high score in Artes Liberales already gives you access to "basic calculus".
Integration is covered by Artes Liberales (Geometry), and in this regard most of what a high school student can do without logarithms or exponentials can be done with a sufficiently high roll -- e.g. computing the area under a parabola, or the length an elliptical path.
The basics of definitions, theorems and proofs, as well as of functions, are covered by Artes Liberales (Logic) -- there's quite a lot of room for improvement here, but it's tough, tough work with very little to show for it in practice (the equivalent of minor breakthroughs might each provide a +1, or -1 botch die, to other "scientific" rolls).
