I found this write up on beams @
http://myconco.com/ComEngProb.html
And this is what he said
Think twice before cutting beams
It's easy to pull out a saw and cut off the top corner of a beam that must be kept beneath a roofline. But if too much cross section is removed, shear forces can cause the beam to split and eventually to fail. For solid-sawn beams, you should leave at least half the width of the beam above the supporting wall and confine the length of the tapered cut to no more than three times the original width of the beam. If you don't have room to leave this much cross section, your best bet is to lower the beam (set it in a pocket) or have a tapered beam engineered.
Using this and asuming a house this old will have 2x4 rafters with that birdmouth, figure out the largest beam that could be installed and tell us it would be big enough to carry the unknown load.
Ok, so 2x4 rafters, 2x4 walls, 6/12 roof pitch;
The full length of the header is about 12', so 4x12 DF #2 and better, (11-1/2"), or 4x10 DF select structural.
2-1/2" between top plate and top edge of the rafter.
At 12" the uncut height of the hdr. is 8-1/2", at 15-1/2" the uncut height of the hdr. is 10-1/2."
CJ at 16"OC falls on the first load point.
By the way, I do not agree with the author of the link you presented, because I learned from working with a civil who provides building inspectors for 11 municipalities, plan check for 20+, provided the structural engineering for the federal facility in Santa Ana CA. and carries a 3 million dollar performance bond.
While the site offers a lot of useful information, I'll stick with what I know.