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Definition of: Surgeon General
1. Chief officer of the Medical Department in the United States Army or Navy.
2. Chief medical officer of the United States Public Health Service.
Since this title is analogous to a military rank (general), the word General is the noun and Surgeon is an adjective describing the type of general; general is not the adjective describing the type of surgeon, i.e. one who performs general surgery.
Another way to look at this is we are collecting generals (or more correctly vice-admirals since that is the rank) not surgeons.
This is not the same usage as for Attorneys General; that usage collects attorneys, not generals.