NOVEMBEB 8, 1907] 



SCIENCE 



629 



in a case, the number of sheep in a herd, 

 etc., was almost more surprising than his 

 ability as a mental calculator. 



AjQother well-known mental calculator, 

 having even less mathematical ability than 

 Dase, is Buxton, who remained illiterate 

 through life, although his father had some 

 education. He had a wonderful memory 

 for numbers and could call off long num- 

 bers from right to left or from left to 

 right with equal facility. On one occa- 

 sion he squared mentally a thirty-nine- 

 figure number in two and a half months. 

 He was extremely slow and in this respect 

 resembled a negro by the name of Tom 

 Puller who is known as the Virginia cal- 

 culator. Although entirely illiterate he 

 was able to reduce mentally years and 

 months to seconds and could multiply two 

 nine-figure numbers. 



Darboux has called attention to an in- 

 fant prodigy, of interest both because 

 it relates to a man who afterwards be- 

 came a very prominent mathematician 

 and also because it is not included in the 

 lists of mathematical prodigies which have 

 recently appeared in this country.^ Joseph 

 Bertrand was born in Paris in 1822 and 

 was such a delicate child that his parents < 

 did not expect him to arrive at manhood, 

 and hence his early education was partly 

 neglected. At the age of four he was sick 

 for a long time and overheard the lessons 

 which were given his brother in the same 

 room. He knew the letters of the alphabet, 

 but nothing more. When he was convales- 

 cent his parents brought him a book to 

 look at the pictures, and he relates, in his 

 account of his childhood, that he remem- 

 bers distinctly how he shocked his parents 



' Scripture, " Arithmetical Prodigies," American 

 Journal of Psychology, Vol. IV. (1891), p. 1; 

 Mitchell, " Mathematical Prodigies," ibid., Vol. 

 XVIII. (1907), p. 61. Bertrand was a mathe- 

 matical prodigy, but he can not be classed among 

 the arithmetical prodigies. 



by reading the text fluently. His fright- 

 ened father snatched the book from him 

 and commanded that under no pretext 

 should he be allowed to do any work. 



The manner in which he learnt ele- 

 mentary algebra and elementary geometry 

 is still more extraordinary. We repro- 

 duce his own account : 



At the age of nine I had the great misfortune 

 to lose my father, who, during the last part of his 

 life, resided with my uncle who directed then a 

 school preparing for I'Ecole Polytechnique. The 

 students, the youngest of whom was twice my age, 

 loved me very much and I was very happy ia 

 their midst. I was assiduous at their recreations 

 and often followed them to their classes. The 

 teachers regarded me with astonishment but paid 

 little attention to me. The students observed 

 that I understood the work and when a demon- 

 stration appeared difficult, the first one who 

 noticed me would run after me, take me up in his 

 arms, and, placing me on a chair so that I could 

 reach the blackboard, made me repeat the dem- 

 onstration. 



At the age of sixteen he entered I'Ecole 

 Polji;echnique, and, as the examiner knew 

 that he had already passed the examina- 

 tion for the doctor's degree in science, 

 he gave him some very difficult questions. 

 From one of the answers it appeared that 

 Bertrand had never opened a table of 

 logarithms. The examiner considered this 

 answer an impertinence but gave him the 

 highest grade. At I'Ecole Polytechnique 

 Bertrand says that he was a problem for 

 his companions. He always received the 

 highest grades but he was ignorant of 

 some of the simplest things. For instance, 

 he did not know what words were called 

 adverbs, as he had never prepared a lesson 

 in literature or in science and no teacher 

 had ever asked him to make any calcula- 

 tion of any kind. 



Bertrand 's extraordinary youth gave 

 rise to many marvelous stories. Fortu- 

 nately, he wrote a brief account of his 

 early life when he was elected in 1884 to 



