170 year old home with sagging 2nd floors
Excellent description as to existing condition and what is occurring on site. 

You have several issues to deal with: 1) due to the age and amount of sag, you will not be able to jack it level, 2) cutting the sag out and sistering another joist will not solve the sag on top, 3) installing a beam will require a post and footing at each end, 4) solutions will be limited to the height of the ceiling. 

Code requires a minimum of 40 lbs. per square foot live load (furniture and people) plus I’d guess about 15 lbs. per square foot dead load (joists, flooring, ceiling finish, etc.) for a total load of 55 lbs. per square foot. 

When retrofitting older buildings, we like to “evenly distribute” loads rather than create point loads. That is to say, adding joists to span the room rather than adding a beam will probably create fewer problems. ...but here are two options:

Option 1: Add Joists: 2x8’s can’t span 19’, so I’d use 2x10’s and hold them down to the bottom of the sag. (It’s okay to notch them at the ends to sit on the plate.) However, 2x10’s spanning 19’ will only support about 60 lbs.per square foot, (depending on the species and grade,) so you’ll need to install them 12” o.c. (Be careful about creating squeaks between the new floor joists and existing subfloor. I’d recommend you nail or screw the subfloor to the new joists.)

Option 2: Add Beam: I’d recommend not trimming the bottom sag off the existing joists if possible...just sister a joist to make the bottom level. The existing 2x8 joists can support about 80 - 100 lbs. per square foot at 24” o.c. and spanning about 9’. So, I’d sister 2x4’s or 2x6’s to the bottom and make the bottom level. For a beam to span 19’ and support the floor, you’d need to support about 520 lbs. per linear foot. You’d need a 6x14 (depending on the species and grade) or a glu-lam beam. Option 2 does not seem plausible because you’ll reduce the height in the room by 14” or so and create about a 5,000 lbs. point load at each end.