This problem is inspired by this MO question, which I thought was very interesting.
What is the oldest open problem in TCS?
Clearly this question needs some clarification.
First, what is TCS? I think the existence of odd perfect numbers is not TCS. I would say that Hilbert's tenth problem is TCS. I think problems like "Can we construct X with a ruler and compass" are also TCS, since they are asking for an algorithm in a restricted model of computation. There may be no rigorous way to define what a TCS problem is, but use your judgment. Perhaps one test is "If this gets solved, would it most likely appear in STOC/FOCS? Would the researcher who solved it most likely be a theoretical computer scientist?"
Second, what is "oldest"? I mean oldest by date. The stated date should also be verifiable, but I don't think this should be too hard.
Third, what is an "open problem"? By "open problem", I mean a problem that was specifically considered open at some time. Perhaps it appeared at the end of a paper in the open problems section, or maybe there is evidence that some people worked on it and failed, or maybe there are incorrect proofs in the literature, which suggest that it has been studied. An example of something that doesn't fit this criteria: "The greeks studied objects X and Y. Z is clearly an intermediate object, surely they wondered if it can be constructed." If there's no literature on Z from that time period, then it is not an open problem from that time period.
Fourth, what do I mean by "problem"? I mean a specific "yes/no" question, and not something like "Characterize all the objects X with property Y", because such questions often do not have a satisfactory answer. Quite often there will be disagreement as to whether the question has been resolved. Let's not get into such questions here. If it is not a yes/no question, but it is clear that it is really open, that's fine too. (In case this isn't clear, by "problem" I mean a formally stated problem. Please do not convert some folk legend about gambling in the 16th century to a question about BPP and PSPACE.)
Lastly, since this is not a big-list question, please post an answer only if you think it is older than the answers already posted (or if you think the answers posted do not satisfy some other condition -- like they are not TCS, or they are not open). This isn't an indiscriminate collection of old open problems.