Why did Holmes assume that Dr. Mortimer is a country practitioner?
Watson assumes that Dr. Mortimer is a country doctor on account of his walking stick. As he perceptively observes, Dr. Mortimer's waking stick has seen better days. Though originally a very handsome looking piece of wood it's been knocked about rather a bit, with the thick-iron ferrule—the metal ring or cap on the tip of a walking stick—worn down.
As the walking stick has clearly been used so often this suggests to Watson that Dr. Mortimer does a great deal of visiting on foot, and it's hard to image a town practitioner doing that. So, he concludes, the good doctor is a country practitioner.
Holmes congratulates Watson on correctly concluding that Mortimer is indeed a country practitioner. However, he quickly points out that his companion arrived at his conclusions the wrong way. Watson was right about Mortimer's being a country practitioner who walks a lot, but that's about it. Holmes goes on to tell Watson how he himself figured out that Mortimer is a country doctor.
The walking stick contains the inscription 'C.C.H.' which Holmes concludes stands for 'Charing Cross Hospital'. This indicates the walking stick was given to Mortimer as a going-away present. Furthermore, Mortimer could not have been on the hospital's staff as such a position would be reserved for those with an established London practice. And someone with an established London practice would not "drift into the country" as Holmes puts it.
At best, Mortimer could only have been a house-surgeon or a house-physician, little more than a medical student, just the kind of unambitious, low-status medical practitioner who would seek a new life in the country.
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