If you are simply relying on gravity to keep that basement dry you are likely to experience the same problem in the future.
In a house as old as yours there is no saying if there is a decent foundation drainage (french drain) in place, if it was properly installed or if it is still working properly after all these years. These old fashioned drainage systems are known to fail overtime.
The only way to tell would be digging out the whole foundation to inspect the drain tile, which should be buried by the footing, assuming you do have one in place.
Then you will need to figure out where it is discharging, which, based on what you describe can be either tied to the city rainwater system or you may have no discharge line whatsoever.
Builders didn't know back then what we know now about proper drainage.
Anyway, if you really want to have a basement dry and suitable to be refinished, so that you will never again have to replace and rebuild it, consider upgrading the drainage system.
That can be done by installing an interior drain tile, along the internal perimeter of the basement and a good sump pump system.
This system will do exactly what a conventional french drain is supposed to do: intercept and collect ground water, divert it to a sump pump that will discharge as far away from the foundation as possible.
You will not need to dig out your foundation and will probably spend a fraction of what you would to pay to have a conventional french drain installed.
Here's some information on how to prepare a basement for finishing