At the time when Hardy wrote his address he was under the impression that Waring first stated his theorem [that every positive integer is a sum of at most 4 positive squares, 9 positive cubes, 19 positive biquadrates, and so on] in the third edition (1782) of his Meditationes Algebraicae [pp. 349- 350], but in a letter of Jan. 4, 1921, he writes me that a correspondent has called his attention to its appearance in an earlier edition. On examining the three editions I fail to find it in the first (1762), but find it in the second (1770) [pp. 204-205], and in the third (1782), as indicated. These references are given also in Dickson's History of the Theory of Numbers, vol. II, pp. xviii and 717 in connection with his elaborate history of Waring's problem.
[Am I right in supposing (as I have done in numbering the editions above) that Waring, when he came to publish the third edition, treated the first part (pp. 1-65) of his Miscellanea Analytica (1762) as the first edition of his Meditationes Algebraicae, its material being reproduced in the editions of 1770 and 1782; or was there another edition of the latter between those of 1770 and 1782? The view which I have taken agrees with a statement given in an old manuscript note on the fly-leaf of a copy of the 1770 edition of the Meditationes Algebraicae in the library of the University of Illinois.]