There are 2 titles associated with this firm.

Cottle, Joseph. Alfred; by Joseph Cottle. Third edition. London: William Button and Son, 1816.
Neale, Hannah. Sacred history, in familiar dialogues, for the instruction of youth: with maps. To which is added, the history of the Jews, from the time of Nehemiah to the destruction of Jerusalem. By Miss H. Neale, author of 'Amusement Hall,' 'Britannus and Africus,' &c. &c. With a recommendatory preface, by the Rev. John Ryland, D.D. In two volumes. London: Baldwin, C. Cradock, and W. Joy, W. Simpkin and R. Marshall, Sherwood, Neely, and Jones, John Hatchard [190 Piccadilly], Thomas Gardiner and Son, Thomas Hamilton, William Darton, Joseph Harvey, and Samuel Darton, William Button and Son, William Fenner, William Oliphant, John Waugh, and Innes, 1817.