Measure the height of the ceiling from the floor of the basement to the base of the joists in the potions of the ceiling where no sagging is present. Use a tape measure to do this.
Measure the length of sag in the floor. Put on a pair of safety goggles and then use a circular saw to cut a 4-by-4-inch beam to match the measured length.
Place two of the 8-foot adjustable posts beneath the sagging area, placing each beneath one of the outside joists of the sagging area. Raise the posts until they are a little more than 4 inches from the base of the floor joists.
Place the cut 4-by-4 on top of the adjustable posts so that it spans the joists beneath the sagging joists. Raise the posts the remaining distance to the ceiling until the 4-by-4 is against the base of the joists, spanning the sag.
Raise the floor slowly over the next few months by screwing the jack under the 4-by-4 a turn or two a month. The jack is the adjustable top portion of the posts. Each turn of the screw on the posts will raise the height of the jack slightly, pushing upward on the joists. The slow speed of the raising process is necessary to prevent cracking the floor joists as you raise the floor. When the floor no longer sags, you can then proceed to reinforce the joists to provide additional support for the floor before removing the posts. The easiest method of reinforcing the joist is to add a second joist alongside the first, spanning the length of the joist. Gluing and then screwing the second joist to the previously sagging joist adds strength along the entire joist length.