Here is a drawing of the type of compass quadrant bearings used in the plat. For simplicity, it shows degree bearings - the surveyor ran lines in degrees-minutes-seconds. So for example the South line at the bottom of the plat shows S 88 41' 32" E, that's 1deg 18' 28" south of due East, or 91 deg 18' 28" Azimuth.
To be official, everything would be tied into that SW section corner - the aluminum capped marker. So, the surveyor found that corner, then ran the south line at S 88 41' 32"E to the SE corner, then turned and ran the east line, etc., He monumented the corners for the SW1/4SW1/4 Section 19 with a nail in the Old Beechwood Road and two rebar posts. From that information, he identified the road R/W boundaries and consequently your property lines.
I found a couple of references on line to this type of notation and zillions of references to azimuth directions (0-360 degrees). Azimuths are the common notation used in map and compass training today, and are much less confusing. The surveyor used it because the original survey used it, and possibly some technical accuracy reasons I don't know about.
This discussion brings back memories of locating forest section corners for the State in the 1970's using a staff compass like this one. Used correctly they are quite accurate, including magnetic declination settings, a level bubble and aiming sights. I think it could be read in at least 1/4 degrees.