""It is a rainy day, but notwithstanding that, the troops must be reviewed; "" that is, the rainy day not opposing or preventing. That, in this case, is a substitute for the whole first clause of the sentence. It is to that clause what a relative is to an antecedent noun, and which may be used in the place of it; notwithstanding which, that is, the rainy day.
""Christ enjoined on his followers not to publish the cures he wrought; but notwithstanding his injunctions, they proclaimed them."" Here, notwithstanding his injunctions, is the case independent or absolute; the injunctions of Christ not opposing or preventing.
This word answers precisely to the latin non obstante, and both are used with nouns or with substitutes for nouns, for sentences or for clauses of sentences. So in the Latin phrase, hoc non obstante, hoc may refer to a single word, to a sentence or to a series of sentences.