Photo credit: The Fitzwilliam Museum
I previously submitted a suggestion to Art Detective that this painting might be of Bishop George Berkeley, a friend of and sitter to John Smibert (see attached composite). However, I have since reconsidered this possibility and would like to suggest instead that this is a portrait by Sir Godfrey Kneller of George Granville, 1st Lord Lansdowne (1666–1735).
The attached composite image depicts four engravings of Granville, together with the image under consideration. The first of the engravings decorates the start of Cooke's 1797 edition of 'The Poetical Works of the Right Hon, Lord Lansdowne...'.
On page 251 of Volume 2 of the Rev. Mark Noble's 'A Biographical History of England...' (1806), there is a list of several of the then-known images of George Granville (see attachment). Two refer to engravings by George Vertue taken from a painting by Godfrey Kneller. https://bit.ly/2Hi3EnV
On page 29 of 'A Description of the Villa of Horace Walpole ... at Strawberry Hill, near Twickenham' (1774), there is an entry for a portrait of George Granville, Lord Lansdow [sic], in red, together with a print of the same subject by George Vertue, after Godfrey Kneller.
On the 14th May 1842, at Horace Walpole's Strawberry Hill sale (Day 18, Lot 127), a watercolour portrait miniature of Granville was sold. A copy of the print can be seen here: https://bit.ly/2AUelHx
I am obviously aware of the absence of the hand that is featured in three of the engravings, but this limb might have been added by the engravers to give more interest to the subject. Otherwise, the open shirt, the jacket, the hat and the general facial expression all seem to echo the clothing worn by and the attitude of the man in this Fitzwilliam Museum portrait.
1. The painting and the 'oval engraving' relate to each other in 'facial structure'. They appear to be the same man, Lonsdown.
2. The 3 square engravings relate to the Oval engraving through facial features and also stated to be: George Granville or Lord Lansdowne
3. The 3 square engravings match the painting in 'general pose and attire', with the exception of the missing hand. They do not prove to be the 'same man' as in the painting.
Conclusion: Through the engravings a good case can be made that the 'Unknown Gent' is George Granville. And being that the painting is a 'fine example of realism' it is possibly the work of Godfrey Kneller or a competing artist who may have copied Kneller's original.
The handling of everything but the face, especially below the neck, is rather crude, and the face is not really first-rate, so assuming this is Kneller, I would favor "after" or "studio of" or "style of" Kneller. Here's a Kneller for comparison:
However, the attribution to Smibert seems quite plausible, although this is still "style of" Kneller.
And by the way, Smibert should not be listed on Art UK as "American," but rather as Scottish-American or simply Scottish.
Although Smibert was born in Scotland and only went to America at the age of 40, I believe he is generally known as an American artist. Presumably he became an American citizen? Art UK follows Getty ULAN, which lists him as American, but our entry records that he was born in Scotland.