628 



SCIENCE 



[N. S. Vol. XXVI. No. 671 



MATHEMATICAL PRODIGIES'^ 

 "When scarcely three years old Gauss, 

 according to an anecdote told by himself, 

 followed mentally a calculation of his 

 father's relative in regard to the wages of 

 some workmen, who were to be paid for 

 overtime in proportion to their regular 

 wages, and, detecting a mistake in the 

 amount, he called out "Father, the reckon- 

 ing is wrong, it makes so much," naming 

 the exact amount. The calculations were 

 repeated and it turned out that the child 

 was correct, while all who witnessed the 

 performance were greatly surprised. He 

 retained an extraordinary ability for 

 mental calculations throughout life and 

 remembered the first few decimals of the 

 logarithms of all numbers, so that he was 

 able to use the data of a logarithmic table 

 in his mental calculations, and hence he 

 possessed a mental slide rule— a unique 

 possession. 



Gauss was not only one of the greatest 

 mental calculators on record, but he ex- 

 celled equally in all branches of pure and 

 applied mathematics. At the age of 

 twenty he discovered the first rigorous 

 proof of the fundamental theorem of alge- 

 bra, which affirms that every algebraic 

 equation has as many roots as its degree, 

 and at the age of twenty- four he published 

 his great work on the theory of numbers 

 under the title " Disquisitiones Arithme- 

 ticse." Later in life he turned his atten- 

 tion principally to applied mathematics — 

 especially to astronomy and geodesy — and 

 he is generally regarded as the last of the 

 great niathematicians who was preeminent 

 in nearly all branches of mathematical 

 knowledge of his day. He considered 

 mathematics the queen of the sciences and 

 number theory the queen of mathematics. 

 "While Gauss was both a great mental 



' Read before the Summer School students, Uni- 

 versity oi Illinois, August 5, 1907. 



calculator and a great mathematician, and 

 was a real mathematical prodigy, we pro- 

 ceed to consider several who were merely 

 arithmetical prodigies and seemed to have 

 very little general mathematical ability. 

 The greatest of these is Dase, who was born 

 at Hamburg in 1824, and "seems to have 

 been little more than a human calculating 

 machine, able to carry on enormous calcu- 

 lations in his head, but nearly incapable of 

 understanding the principles of mathe- 

 matics, and of very limited ability outside 

 his chosen field." His extraordinary 

 ability in mental calculation is evidenced 

 by the fact that he was able to multiply 

 mentally two numbers, each of which con- 

 tained one hundred figures. It took him 

 eight and three quarter hours to perform 

 this feat, which stands in a class by itself, 

 as no other arithmetical prodigy is known 

 to have been able to multiply mentally two 

 numbers each consisting of more than 

 thirty-nine figures. Two forty-figure num- 

 bers, Dase was able to multiply mentally in 

 forty minutes, while he would multiply 

 two eight-figure numbers in less than one 

 minute. 



What is most surprising about this 

 greatest mental calculator on record is that 

 he was stupid in mathematics. Petersen 

 is said to have tried in vain for six weeks 

 to get the first elements of mathematics 

 into his head, and other eminent mathe- 

 maticians found that he had very little 

 mathematical ability. Fortunately he was 

 advised by some of the leading mathe- 

 maticians of his day to turn his extraor- 

 dinary ability to scientific uses instead of 

 going around the country giving public 

 exhibitions, a career upon which he had 

 entered at the age of fifteen. He calcu- 

 lated many useful tables and was engaged 

 on an extensive factor table at the time 

 of his death. The ease and speed with 

 which he could count the number of books 



