440 THE SCIENTIFIC MONTHLY 



time. It is here where the breeding of errors makes its appearance. 

 The historian is greatly tempted to state interesting and striking facts. 

 He finds that many such facts have been emphasized by the specialists, 

 but he naturally fails at times in his efforts to interpret the language 

 of the specialists. The next historian who tries to interpret the words 

 of this earlier historian frequently misses the correct interpretaion 

 still more, and hence statements conveying an entirely false notion 

 tend to creep into the general histories of science. 



As pure mathematics is the most exact science it may perhaps be 

 assumed that the history of this subject is the most accurate among the 

 histories of the various sciences. At any rate, it seems desirable to 

 illustrate some of the preceding observations by examples of such an 

 elementary type that they can be easily understood by all. The history 

 of mathematics furnishes many such examples since some of its early 

 permanent developments belong to a period when the mathematical 

 specialists dealt with questions which all could easily understand. 

 These specialists had not then raised themselves to great intellectual 

 heights by standing on the shoulders of other specialists, who, in turn, 

 stood on the shoulders of earlier specialists in almost endless suc- 

 cession. 



To furnish a striking but somewhat extreme illustration of the fact 

 that the mathematical historian is apt to repeat statements which he does 

 not fully understand, it may be noted here that on page 165 of the 

 third edition of Cantor's well-known Vorlesungen iiber Geschichte der 

 Mathematik, 1907, it is stated that the Greeks used the term epimorion 

 to denote the ratio n/(n-\-l), and that "Archytas had already stated 

 and proved the theorem that if an epimorion, a//? is reduced to its 

 lowest terms, which may be called (i/v, then v==p-\-l. It is evident that 

 this remark is practically meaningless, for if a/(3 is an epimorion then 

 it is obviously already in its lowest terms according to the defini- 

 tion of the term epimorion just noted, which seems itself to be incorrect. 

 Notwithstanding this obvious lack of clearness, the statement appears 

 again on page 53 of the second edition of Cajori's History of Mathe- 

 matics, 1919, in spite of the fact that G. Enestrom had in the meantime 

 directed attention to its inaccuracy in volume 8 of the Bibliotheca 

 Mathematica, 1917-1918, page 174. 



An important feature of the history of science is that many state- 

 ments made therein are intended to be true only in a general way, 

 while others are supposed to be exact, and the reader has frequently to 

 decide for himself to which of these two classes a particular state- 

 ment is supposed to belong. For instance, one can usually not deter- 

 mine accurately who was the founder of a large subject since steps 

 towards its development were commonly taken by a number of different 

 men. On the other hand, such statements as "Newton and Leibniz were 



