442 THE SCIENTIFIC MONTHLY 



especially by the Arabs in algebra and in trigonometry. In particular, 

 the work from which our modern term algebra is derived was com- 

 posed during this period, and the work of several well-known Hindu 

 writers appeared therein. Hence the reader seems justified for having 

 somewhat high expectations as regards the mathematical importance 

 of Gerbert's letter if it actually deserves being called "the first mathe- 

 matical paper of the Middle Ages." In fact, the non-mathematical 

 reader might be inclined to fear that the mathematical merits of this 

 letter were too great to lie within the limits of his comprehension. 



These expectations and fears are apt to be enhanced by the reading 

 of the accounts of Gerbert's letter in some of our most popular his- 

 tories of mathematics, including the ones already noted. Not only is 

 it stated here that this letter contained a correct explanation for the 

 difference of the results obtained by using two different formulas for the 

 determination of the area of an equilateral triangle, but some of the 

 other statements relating to this letter are sufficiently obscure and mis- 

 leading to arouse the suspicion that the subject treated therein might 

 possibly be difficult. In various instances the obscurity is increased 

 by the fact that figures of triangles which are not equilateral are given, 

 while the text relates to an equilateral triangle. This is done, for in- 

 stance, on page 249 of Giinther's, Geschichte der Mathematik, 1908, 

 as well as in the three editions of volume I of Cantor's well known 

 Vorlesungen ilber Geschichte der Mathematik, pages 744, 815 and 866, 

 respectively. 



From the preceding remarks the reader will naturally conclude 

 that the present writer does not believe that the letter in question merits 

 to be called "the first mathematical paper of the Middle Ages which 

 deserves this name," notwithstanding the fact that this epithet has been 

 applied to it by eminent authoritities. In fact, the present writer 

 believes not only that the letter does not merit this epithet but that it is 

 of so little mathematical importance as to make it appear ridiculous to 

 make such a claim for it. Moreover, he believes that other statements 

 made about this letter in well-known mathematical histories are strik- 

 ingly inaccurate. In order to establish the correctness of this 

 point of view, it is necessary to state just what is found in the part of 

 this letter which has been preserved, upon which our view of its 

 merits must be based. 



In view of the great claims made for this letter, the reader will 

 naturally be surprised to find that it deals with the very elementary 

 question of finding the area of an equilateral triangle, a question which 

 had been completely solved many centuries before. Gerbert gives 

 here the rule that the altitude of such a triangle can be found by sub- 

 tracting one-seventh from its side, which is a sufficiently close approxi- 

 mation for many purposes, since the altitude is a / 2 \/3, where a is the 



