HISTORY OF SCIENCE AS AN ERROR BREEDER 443 



side. He then states, in substances, that the area of an equilateral 

 triangle is one-half of the product of the base and the altitude thus 

 obtained, and he calls this the "geometric rule" for finding such an 

 area. 



Thus far there is nothing surprising in this letter, and no one seems 

 to have claimed much credit for this part, but Gerbert then makes 

 some very inaccurate and foolish remarks about finding the area of 

 such a triangle by another rule, called the "arithmetic rule," and it is 

 just upon these remarks which exhibit a great lack of geometric in- 

 sight that the high claims of this letter have been based. The in- 

 accuracy of these remarks had been noted by M. Chasles in his well 

 and favorably known Apergu Historique, 1875, page 506, but notwith- 

 standing this fact various later mathematical authors, including all 

 those noted above, have called them correct in their general histories. 



In order to appreciate the crudeness of this "arithmetic rule," 

 which is equivalent to the formula l^a(a-)-l), it may be noted that in 

 the work of the Egyptian Ahmes, written about 1700 B. C, the area 

 of an isosceles triangle seems to have been found by multiplying one- 

 half the base by a side instead of by the altitude. This method has 

 been regarded as remarkable on account of its crudity, but when we 

 are told that more than two thousand years later the Roman surveyors 

 were taught to find the area of such a triangle by finding the product 

 of one-half of the numerical measure of the base and a number which 

 is even larger than the numerical measure of another side, there seems 

 to be sufficient ground for surprise even in a scientific matter. 



It must be admitted that the instances cited above are insufficient 

 to establish the fact that a general history of science is an unusually 

 favorable ground for the breeding and the propagation of scientific 

 errors. In fact, all that has been attempted here is to advance a few 

 reasons why one might suspect danger here, and to support these reasons 

 by illustrations which were assumed to be also of interest to the reader 

 on account of their unusual intrinsic, features. Perhaps a more con- 

 clusive argument in support of the thesis in question is furnished by 

 the fact that G. Enestrom noted more than two thousand desirable 

 changes relating to the general history of mathematics by M. Cantor, 

 to which reference was made. 



