## Note (2) on Hardy's inaugural lecture

R D Carmichael reviewed G H Hardy's Inaugural Lecture booklet in the Bulletin of the American Mathematical Society in June-July part of the 1921 volume. Carmichael gives the following note concerning the source of Waring's Problem:

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.]