It’s at least debatable whether following in the 1826 example is a preposition. The OED’s 1947 citation is:
The prologue was written by the company following an incident
witnessed by them during anti-Jewish demonstrations following the
hanging of two British soldiers in Palestine.
In that example, following can be replaced by at least one of the three terms provided in the OED’s definition: as a sequel to, in succession to and after. Can we, equally, instead of 'Following the order observed in the report', say 'As a sequel to / in succession to / after the order observed in the report'? Possibly, but that may not give the meaning intended in the original. What the writer probably meant was 'Following the order observed in the report, the first branch of study we notice is Engineering . . .' If so, the way it has come out has following as the -ing form of the verb follow. As such it is a dangling participle, wrongly hooked up to the first branch of study.