In a sincere, genuine, or warm-hearted manner; with genuine affection or earnestness.
From 'heart' plus the adverbial suffix '-ly' (from Old English '-lice,' meaning 'in the manner of'). This word is an archaic variant of 'heartily,' which became more standard in modern English.
Language evolves as people adopt slightly different pronunciations—'heartly' was once common, but 'heartily' won out, showing how words compete with each other and eventually one becomes the 'official' spelling while the other fades away.
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