In 1787, the Italian conjurer, Signior Falconi, featured a prediction of “the Combination or Arrangements of four different Numbers” made by an audience member. (See Item 8 in his November 20th, 1787, broadside, reproduced in Milbourne Christopher's Illustrated History of Magic, 1973, p. 55.) By January 14, 1794, the feat had grown to the prediction of “the sum of six columns of numbers written by as many spectators,” (see “The Fabulous Falconi” by Christopher in The Sphinx, Vol. 40 No. 9, Nov. 1941, p. 269.
The earliest example of the Addition Test in print seems to be one by H. B. Wilton in The Somatic Conjuror, 1870, p. 32. The sum of three gentlemen's numbers appears written on the performer's forearm.
Frank W. Thomas uses the idea as one component of a book test methodology in his “Confessions of a Mind Reader” series in Ellis Stanyon's Magic, Vol. 2 No. 10, July 1902, p. 76. Two more tests, framed as predictions, are given by David P. Abbott in Behind the Scenes with the Mediums, 1907 p. 160. Both involve number switches (one being on the flyleaf of a book).
Another early example is “A Good Slate Trick” in Elbiquet's A Text Book of Magic, 1913, in which the slate plays a part in producing the total. The selected numbers are written on a sheet of paper that is switched in a flap card-box before being added.
In The Magician Monthly, Vol. 11 No. 3, Feb. 1915, p. 49, there appears “Tip for the 'Sum Trick'” (presumably by Will Goldston) with a switching method involving a pack of postcards. The opening sentence leads with, “In working any kind of sum trick…” which clearly suggests that it is, by then, an established and well-known plot. This is amply underscored in The Magician Monthly, Vol. 11 No. 4, July 1915, p. 126, wherein H. C. Mole describes “The 'Daily Mail' Trick,” a newspaper test that involves a switched-in total as part of its method. On this point, Mole writes: “This requires no description; each performer has his own way of working…”
The above is based in part on research provided by Max Maven.