Interesting Medieval Mathematics Lecture/Activity Ideas?

You really ought to mention some of the mathematical achievements of the $14$th century philosopher Nicolas Oresme: he worked with fractional exponents; he was the first to prove that the harmonic series diverges; he gave in essence a formula for the sum of a geometric series with arbitrary first term and ratio $\frac1n$ for integers $n\ge 2$; and he came close to inventing Cartesian coordinates, using his version to prove that the distance travelled in a given period by an object moving under constant acceleration is equal to the distance travelled in the same period by an object moving at a constant speed equal to that of the first object at the midpoint of the period. There’s a fair bit of information available on the web; the summary here should be helpful.

I’d replace the abacus, which in the form in which we think of it was little used in mediæval Europe, with its mediæval equivalent, the counting board or counting table.

When I took a course in mathematical history, the only real achievement of the medieval mathematics (according to the professor teaching the course) was the following:

$$\sum_{n=1}^\infty\frac1n=\infty$$

The proof is due to Oresme, who gave a very nice proof of the following flavor:

$$1+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\ldots\geq\\1+\frac12+\frac14+\frac14+\frac18+\frac18+\frac18+\frac18+\ldots=\\1+\frac12+\frac12+\frac12+\ldots\geq1+1+1+\ldots=\infty$$

You can read about Oresme, and the proof on this Wikipedia page.

You can find a list of mathematicians from this period here and here, with links to biographies.

Alcuin of York wrote Problems to Sharpen the Young around 800 A.D., which is a nice collection of problems of recreational mathematics. Problem 18 is very well known, and I believe this is the oldest reference we have for this problem.

Leonardo Pisano (who acquired the name Fibonacci some 600 years after his death) has some neat number theory in his Book of Squares. He starts with the result $1+3+\dots+(2n-1)=n^2$, and develops from this a number of methods for solving Diophantine equations. The last problem that he solves is to find numbers $a,b,c$ such that the three numbers $a+b+c+a^2$, $a+b+c+a^2+b^2$, and $a+b+c+a^2+b^2+c^2$ all are squares. His solution: $a=35$, $b=144$, $c=360$.

Levi ben Gerson used mathematical induction in France in the early 1300's, giving proofs of formulas such as $C_k^n={n(n-1)\cdots(n-k+1)\over 1\cdot 2\cdots k}$, of $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$, and of the associate law of multiplication for any number of factors. (Reference: Katz.)

The oldest surviving copies of the texts of Euclid and Archimedes were written in the Byzantine empire in the ninth and tenth century. Obviously, there were interest in mathematics in this part of Europe, though what has survived is mostly commentaries on older texts.