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Amount comes from Old French amonter ‘to amount to’, from amont ‘uphill’, from Latin ad montem ‘to the mountain’, ultimately from Proto-Indo-European *men- ‘to stand out’.

1. Proto-Indo-Europeanc. 4500–2500 BCE*men-¹‘to stand out, project’reconstructed
2. Proto-Indo-Europeanc. 4500–2500 BCE*móntis¹reconstructed
3. Proto-Italicc. 1500–700 BCE*monts¹reconstructed
4. Latinc. 75 BCE–200 CEmōns¹‘mountain’
5. Latinc. 75 BCE–200 CEad montem¹‘to the mountain’
6. Old Frenchc. 842–1400a mont¹‘toward a mountain or heap’
7. Old Frenchc. 842–1400amont¹‘uphill, upward’
8. Old Frenchc. 842–1400amonter¹‘to amount to’
9. Middle Englishc. 1150–1500amounten¹‘to mount up to’
10. Englishearliest quotation 1590amount¹

1. ¹ Wiktionary, English amount
2. Earliest quotation in Wiktionary’s entry: 1590, Edmund Spenser, Book I, Canto X: “So up he rose, and thence amounted straight.”.