496 



SCIENCE. 



l\. S. Vol. XVII. No. 430 



first in a positive direction, and after train- 

 ing show a little increase of positive asym- 

 metry, with the exception of the strength- 

 weight index, in which the skewness de- 

 creases to nearly perfect symmetry. 



The relation between capacity for modi- 

 fication and the initial position in the scale 

 can be determined only after calculation 

 of coefficients of correlation, and for this 

 purpose correlation tables are noAV being 

 constructed. 



The relation between the amount of 

 modification and the length of time of 

 training has been studied in only one series 

 of measurements, that of the strength of 

 legs. The measurements were plotted for 

 every second month, that is, October, De- 

 cember, February, April and June. The 

 magnitude of the mean was found to in- 

 crease during each succeeding period, 

 rapidly at first and then more and more 

 slowly. The increase amounted during 

 the first period to 20 kilos, during the sec- 

 ond to 8.8 kilos, third to 6.4 kilos, and 

 fourth to 0.37 kilos. 



GiLMAN A. Drew, 

 Secretary {Eastern Branch). 



University of Maine. •T"^ ' 



SOME FUNDAMENTAL DISCOVERIES IN 



MATHEMATICS.* 



The oldest extensive work on mathe- 

 matics which has been deciphered was 

 written by an Egyptian named Ahmes be- 

 tween 1700 and 2000 B.C. It bears the 

 following title : ' Direction for obtaining a 

 knowledge of all dark things * * * all 

 secrets which are contained in the things,' 

 and claims to be modeled after writings' 

 which were then old. The first part is 

 devoted to a table in which every fraction 

 whose numerator is 2 and whose denom- 

 inator is any odd number from 5 to 99 is 

 resolved into the sum of fractions with 



* Read before the Science Association of Stan- 

 ford University, November 5, 1902. 



unity as a common numerator. The fol- 

 lowing are examples : 



I = I + "iVi f = T + iVl T!T = A + ^^ + Tl ¥■ 



As this table is constructed according to 

 no general rule, it is probable that it is a 

 collection of results obtained by mathe- 

 maticians during a long period of years. 

 In fact some of these numbers are found 

 in a mathematical papyrus which is many 

 hundred years older than the work of 

 Ahmes. This table, therefore, furnishes 

 one of the many evidences of the fact that 

 the early development of ma'thematics is 

 largely based upon experiments. Com- 

 prehensive rules and theorems are a much 

 later product. 



From a modern point of view it might 

 be said that the theory of arithmetical 

 progression marked the highest point 

 reached by Ahmes in arithmetic. He 

 solves linear algebraic equations involving 

 one unknown and considers the area of a 

 circle equivalent to a square whose side is 

 eight ninths of the diameter. This is 

 equivalent to calling - == 3.1605, which is 

 a much closer approximation than many 

 later nations employed.* To find the area 

 of an isosceles triangle he multiplied the 

 base by half of one of the equal sides in- 

 stead of by half the altitude. This in- 

 accuracy seems to be due to the fact that 

 the Egyptians did not know how to ex- 

 tract the square root of a number, and 

 hence they could not find the exact area 

 of such a triangle from its sides. 



While the work of Ahmes is of the great- 

 est interest to the mathematical historian, 

 yet it contains few facts of sufficient gen- 

 erality or beauty to be classed among the 

 fundamental discoveries in mathematics. 

 It .emphasizes rules rather than thought. 

 In fact, it is practically confined to prob- 

 lems and answers, with the verifications of 



* Cf. I. Kings, ch. 7, v. 23 and II. Chronicles, 

 ch. 4, V. 2. 



