The son of Charles Kaye, a solicitor, he was educated at Eton College and at the Royal Military College, Addiscombe.
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Arthur William, the younger son of William Garnett of Westmoreland, inspector-general of inland revenue, was born 1 June 1829, and educated at Addiscombe Military Seminary, where he obtained his first commission in 1846, and proceeded to India in 1848 as a lieutenant of the Bengal engineers.
In 1825 he went on to study at the East India Military College at Addiscombe near Croydon where his mathematical studies continued under the guidance of Jonathan Cape a tutor at the college who was also a Fellow of Trinity College, Cambridge.