We may as well take these together, for it is only the use of “salt” in these phrases that is interesting.
Because of the origin of the word, one would expect to be able to find that either of the expressions could be traced to ancient Latin, but actually the first of them, the earliest, goes back only to the first quarter of the nineteenth century.
“Salt,” in these expressions, is no more than an etymological play upon the source of the word “salary,” perhaps a source that was not understood before 1800. In the days of the Roman legions, a soldier received a part of his pay in the form of a salarium, a salary, which was actually an allowance for the purchase of salt (Latin, sal).
Salt was not so easily obtainable in those times, but even then the Roman generals knew that this mineral was essential to health and vigor.
A soldier who was not worth or did not earn this small allowance was worthless indeed.