"Deep into" something means far down in something, particularly a body of water or liquid
- "The chef plunged the meat deep into the fat
- The pearl divers go deep into the Agean.
But this phrase is nowe often used as a metaphor, ratehr than literally.
He is deep into Ruskin
means that he has been studying Ruskin and is seriously interested in the works or thought of that author.
One could also say "she is deeply into gold" meaning she has invested a lot in gold.
"Deep into the nineteenth century" is a somewhat unusual phrase. Here it clearly means some time significantly later than 1800, but not as late as 1900. It could easily mean anywhere from say 1840 to 1890, but one cannot be sure how the author intends it.
"Well into the nineteenth century" has exactly the same meaning, and is a far more common phrase.
"Deep into", "well into", "well along" and "late" are all relatice in such uses, of course. "deep into the night" refers to a period of hurs, perhaps midnight, perhaps 5 in the morning. "Well into the nineteenth century" means a period of decades. In any case "Well into X" means a point significantly beyond the start of X, but short of its end.
"Far on" is more often used of a physical distance: "They went far on along the road". Again it means significantly past the start, but not all the way to the end. "far along" can be use in the same way,
The original example
Deep into the nineteenth century, Hippocrates and most of those who followed him also believed that natural processes must not be interfered with.
is also a little odd it its use of "Hippocrates and most of those who followed him". Hippocrates himself, of course, did not believe anything in the nineteenth century; he was long dead by that time. I think this would be better phrased as:
Well into the nineteenth century, most of those who followed Hippocrates also believed that natural processes must not be interfered with.