Let us give some further examples of two-sided grammatical analysis:

• “à dessein” (purposedly), “à volonté” (at will), “à tort” (mistakenly): from an analytical standpoint, these are prepositions followed by a singular noun. From a synthetical viewpoint, they are adverbs (adverbial locutions).
• “à jamais” (forever): from an analytical standpoint, it is a preposition followed by an adverb. From a synthetical viewpoint, it is an adverb (adverbial locution).
• “à genoux” (on my/his/her/… knees), “à torrents” (in torrents): from an analytical standpoint, these are prepositions followed by a plural noun. From a synthetical viewpoint, they are adverbs (adverbial locutions).