HISTORY OF SCIENCE AS AN ERROR BREEDER 441 



the founders of the calculus" have an important historical significance. 

 One might almost say that such incomplete assertions constitute the 

 meat of a general history of science. When it is remembered that 

 they are supposed to represent only first approximations to the truth 

 they can not be regarded in the light of actual error breeders, even 

 if they are apt to have this effect temporarily, especially on beginners. 



The significance of many statements in the history of science grows 

 with the growth of the reader's breadth of knowledge relating to the 

 subject involved. Statements which at one period of his scientific de- 

 velopment would have appeared satisfactory may not appear so at a 

 later period. It is, of course, impossible for an author of a general 

 history of science to provide for these various stages of development, 

 but there are instances where such authors have failed to provide a 

 satisfactory account for any of their readers irrespective of the stage 

 of their scientific development. While such instances are compara- 

 tively rare and do not necessarily impair seriously the works in which 

 they appear they may serve to illustrate the general topic under con- 

 sideration and are unusually interesting in themselves. Hence we give 

 here one such instance relating to mathematics during medieval times. 



On page 314 of HankeFs Geschichte der Mathematik, 1874, it is 

 stated that "the first mathematical paper of the Middle Ages which 

 deserves this name is a letter of Gerbert to Adelbold, bishop of 

 Utrecht." In view of the fact that we are now living in an age of 

 numerous scientific papers, this remark by Hankel should be of great 

 interest at the present time. Such interest is reflected in the fact that 

 F. Cajori quotes this remark in the two editions of his History of 

 Mathematics, 1894 and 1919, respectively, as well as in the two editions 

 of his History of Elementary Mathematics, 1896 and 1917, respectively. 

 Hence the American reader of the history of science is seriously ex- 

 posed to the danger of assuming that the said remark by Hankel repre- 

 sents a well-established historical fact. 



Such an assumption does not imply that the said letter by Gerbert, 

 who died as Pope Sylvester II in 1003, contained any new mathematical 

 results, since there are now expository mathematical papers as well 

 as research papers. It does not imply that the letter in question was 

 long, since there are now many brief mathematical papers as well as 

 long ones. It does, however, imply that this letter was superior to 

 the many other mathematical writings which had appeared during the 

 four or five centuries which had then elapsed since the beginning of 

 the Middle Ages. 



It is true that such a superiority even in so long a period of years 

 does not imply very much, since this particular period was unusually 

 barren as regards mathematical developments. Notwithstanding this 

 comparative barrenness, it includes a few noteworthy oases created 



