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If the joists are only spanning 10 - 12 feet then they must be pretty small to be sagging and bouncing that much. Could be a sign of rot or insects. If you don't need the space in basement (assuming rot or insect problem has been investigated and/or fixed) then you can go with posts(jacks). If you need the space to be post-free then you can add a girder beam or just sister (double)every joist or two. If you use your jacks then get a nice beam chunk that will span all the affected joists at their mid span point. Put a post (jack) about 1/3 of the way in from each end. That way, as long as your beam is sized correctly, two jacks will hold whole floor section up. Basically this is installing a girder beam in a fashion. If you don't cut slab, dig out and pour proper footings then you might use steel plates or another beam lying on the consrete to spread the load. The posts would rest on it and spread the load. Pad would have to be fairly large or just use a beam on the floor as a sort of "grade beam". Posts would bear on this (bottom sets on it). the only other thing is to be very sure to attach everything very securely. I would use nails or probably screws (at least) to hold every piece of wood and post (jack). Assume that all will fall apart and down otherwise. I might use use "sipson" (or is it simpson) ties, available at any bldg. ctr. to fully secure, especially in earthquake zone.
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