What is the correct spelling of the name of the brother to the Swiss mathematician Johann Bernoulli? I've seen it spelt both ways, which one is the correct one?
It is simply the difference between German and Latin spelling (German Wikipedia calls him Jakob).
The correct spelling (by definition) is how the person wrote his name himself. And here is how:
(Jacobi in Latin means "by Jacob"). Of course it is possible that it was spelled Jacob in Latin and Jakob in German. But all mathematics was written in Latin at that time. (He could probably write Jakob in his financial or legal documents).
There is an extensive Latin correspondence between Leibniz (addressed, by the way, as "Leibnitz") and Jacob Bernoulli, in which Bernoulli writes his name "Jacobus" or declensions thereof. A letter to Clüver written in French is signed with "Jacques Bernoulli".
An astronomical treatise published in German in 1681 bears the name Jacob.