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“augment” in Latin

The English word augment itself goes back to Latin augeō, through augmentare.

Riddle & Arnold A Copious and Critical English–Latin Lexicon, 1864

augment
augere: adaugere. (The words are found in this connection and order), ampliticare et augere: anything with anything, augere or adangere aliquid aliqua re: addere aliquid alicui rei or ad aliquid (add or append anything to anything): amplificare (to make longer in compass): multiplicare (to make numerically greater).
INTRANS. augeri (of persons and things): crescere (of things).

Quick answer

What is the Latin word for augment?
The usual Latin words for “augment” are augeō, adaugeō, crēscō; also amplificō, multiplicō. The right choice depends on the sense; see the distinctions below.

Entry from J. E. Riddle & T. K. Arnold, A Copious and Critical English–Latin Lexicon (rev. C. Anthon, New York: Harper, 1864), digitised by Fergus J. P. Walsh, CC BY-SA 4.0; Latin markup adapted by latindictionary.co.