I know this is a silly little question, but it’s bothered me since the day I learned the terms.
Given the relative ranking in the military of lieutenant and major (with major, of course, being superior), why is it that a lieutenant general outranks a major general? Was there a particular reason this counterintuitive (to me) sequence developed?
Are you sure? It seems to me that in both lieutenant general and sergeant-major general, general was at least originally the adjective. They are the “general” version of the lieutenant and sergeant-major ranks.
Agree with this: “general” was always an adjective modifying the preceding noun (after the French, or possibly Spanish fashion, I suppose). Thus from top to bottom, “captain-general,” “lieutenant-general,” “sergeant-major general” in which general indeed means “of all the individual parts, overall”. This is probably a left over from the time when individual captains recruited and supplied a feudal army with its troops. Since each of these individuals deserved the appelation captain, the officer above them all (and there was usually a fairly flat hierarchy, without the division/corps structure of modern armies) was the general captain, or, indeed, the captain-general. And below him, logically, were the lieutenant-general, and the sergeant-major general.
Which is not to minimize the latter’s role: the sergeant-major general is very well acquainted, too, with matters mathematical, and understands equations both the simple and quadratical.