Chesham exhibited in 1777 at the Royal Incorporated Society of Artists in Piccadilly an engraving The Death of Richard III, after John James Barralet, and in the following year The Death of William Rufus.
In 1780 he exhibited with the Society of Artists at Spring Gardens Inside of the Chapter House at Margam and View of the Abbey Church at Llanthony.
In 1779–80 he engraved several views of various places in the United Kingdom, after Paul Sandby, for Michael Angelo Rooker's Copper Plate Magazine.
These two plates are well engraved in the style and method brought into fashion by François Vivares and his school.
Chesham also engraved after his own design a large plate Moses striking the Rock; after Giovanni Battista Ciprianil he engraved an allegorical figure of Britannia; and after Robert Dodd, The Naval Victory gained by Admiral Parker in 1781.