ISTOVEMBER 1, 1888.] 



♦ KNOVS^LEDGE ♦ 



again, the periods with which we have to deal are probably 

 short compared with those which may be expected (when 

 the laws of mental development come to be understood) to 

 separate the appearance of exceptionally great minds. We 

 carry back our thoughts to the last of the great ones in each 

 department of mental action : and even if we do not 

 exaggerate his relative elevation above his contemporaries, 

 as we are apt to do, or overlook (as we are equally apt to 

 do) the elevation of the great minds of our own time, we 

 still forget that, in the steady rising of the mighty tide of 

 meutal progress, the waves successively flowing in above 

 the tide-line may be separated in time by intervals of many 

 generations, and a greater wave may be followed by several 

 lesser ones, before another like itself, but riding on a higher 

 sea, flows higher still above the shore-line which separates 

 the unknown from the known. 



We may begin conveniently by considering some illustra- 

 tions of exceptional power in the form of mental activity 

 least likely to deceive us — aptitude in dealing with numbers. 

 It is well remarked by Dr. Carpenter, that this quality is so 

 completely a product of culture that we can trace pretty 

 clearly the history of its development. " The definite ideas 

 which we now form of munbers," he proceeds, "and of the 

 relations of numbers, are the products of intellectual opera- 

 tions based on experience. There are savages at the present 

 time who cannot count beyond five; and even among races 

 that have attained to a considerable proficiency in the arts 

 of life, the range of numerical power seems extremely low. 

 . . . The science of Arithmetic, as at present existing, may 

 be regarded as the accumulated product of the intellectual 

 ability of successive generations, each generation building 

 up some addition to the knowledge which it has received 

 from its predecessor. But it can scarcely be questioned by 

 any observant person that an aptitwJe for the apprehension 

 of numerical ideas has come to be embodied in the congenital 

 constitution of races which have long cultivated this bianch 

 of knowledge ; so that it is far easier to teach arithmetic to 

 the child of an educated stock than it would be to a young 

 Yanco of the Amazons, who, according to La Condamine, 

 can count no higher than three, his name for which is 

 Poettarrarorincoaroac." 



As an illustration of congenital aptitude for dealing with 

 numbers, Dr. Carpenter takes the case of Zerah Colburn : 

 and in this I shall follow him, though, as will jiresently 

 appear, I differ from him as to the significance of that case, 

 the true interpretation of which I believe to be firr simpler, 

 but to promise much less, than that adopted by Fi-ancis 

 Baily and quoted with approval by Carpenter. 



Let us first consider the facts of this remarkable case — 

 Zerah Colburn was the son of an American peasant or 

 small farmer. When he was not yet six years of age, ho 

 surprised his father by his readiness in multiplying numbers 

 and .solving other simple arithmetical problems. Ho was 

 brought to London in 1812, when only eight years old, and 

 his powers were tested by Francis Baily and other skilful 

 mathematicians. Fiom Carpenter's synopsis of the experi- 

 ments thus made the following account is taken, technical 

 expressions being as far as possible elinunated (or not used 

 until explained) : — 



lie would multiply any number loss than 10 into itself 

 successively nine times, giving the results (by actual 

 multiplication, not from memory) faster than the person 

 ap[)ointed to record them could set them down. He 

 multiplied 8 into itself fifteen times, or, in technical terms, 

 raisoil it to the sixteenth power ; and the result, consLsting 

 of fifteen digits, was right in every figure. He raised some 

 numbeis of two figures as high as the eighth power, but 

 found a dilficiUty in proceeding when the result contained 

 a i^rcat number of fiijures. 



So far there is nothing- which cannot be explained (or 

 which could not, if other facts did not render the explana- 

 tion invalid) by assuming that the child possessed simply 

 the power of multiplying mentally, with extreme rapidity 

 and correctness, but in the ordinary way.* But the next 

 test removes at once all possibility of explaining his work 

 as done in the ordinary manner. He was asked what number, 

 multiplied by itself, gave 106,929, and he answered 327, 

 before the oriylnal number could he writfeti down. This was 

 wonderful. But he next achieved a more wonderful feat 

 still, judging his work by the usual rules. He was asked 

 what number, multiplied twiceinto itself, gave 268,336,125 — 

 in other words, to find the cube root of that array of digits; 

 ivith equal Jacilit)/ and pi-omptness he replied 6-15. Now, 

 anyone acquainted with the process for finding the cube 

 root — even the most convenient form of the process, as 

 presented by Colenso and others — knows that the cube root 

 of a number of nine digits could not be correctly determined, 

 with pen and paper, in less than three or four minutes, if 

 so soon. If the computer had so perfect a power of cal- 

 culating mentally that he could proceed as safely as though 

 writing down every step, and as rapidly with each line as 

 Colburn himself in the simple processes before described, he 

 would yet need half a minute at least to get the correct 

 result. This, too, would imply such a power of mentally 

 picturing sets of figures that, even if it explained Colburn's 

 work, it would still be altogether marvellous, if not utterly 

 inexplicable. We know, however, that Colburn was not 

 following ordinary rules, but a method peculiar to himself. 

 In point of fact, he was so entirely ignorant of the usual 

 modes of procedure, that he could not perform on paper a 

 simple sum in multiplication or division. 



Let us proceed to further instances of his remarkable 

 power of calculation. 



On being asked how many minutes there are in forty- 

 eight years, he answered, before the question could be 

 written down, 25,228,800 ; which is correct, if the extra 

 days for leap years are left out of account. He immediately 

 after gave the correct number of seconds. 



We come next, however, to results which appear much 

 more surprising to the mathematician than any of the above, 

 because they relate to questions for which mathematicians 

 have not been able to provide any systematic method of 

 procedure whatever. He was asked to name two numbers 

 which, multiplied together, would give the number 2-17,4:83, 

 and he immediately named 941 and 263, which are the only 

 two numbers satisfying the condition. The same problem 

 being sot with re.spect to the number 171,395, he named 

 the following pairs of numbers : 5 and 34,279 ; 7 and 

 24,485 ; 59 and 2,905 ; 83 and 2,065 ; 35 and 4,897 ; 295 

 and 581 ; and, lastly, 413 and 415. (1 presume, as Mr. 

 Baily gives the pairs in this order, that they were so 

 announced by Colburn. The point is of some importance 

 in considering the explanation of liio boy's mental pro- 

 cedure.) The next feat was a wonderful one. He was 

 .asked to name a number which will divide 36,083 exactly, 

 and ho immediately replied that there is no such number; 

 in other words, he recognised this number as what is called 

 a prime number, or a number only divisible by itself and 

 l)y unity, just as readily and quickly as most people would 

 recogni,>;e 17, 19, or 23 as such a number, and a great deal 

 more quickly th.iu probablj' nine persons out of ten would 

 recognise 53 or 59 as such. 



• The account does not say whether he gave the figures 

 successively from right to left or from left to rislit. If he began 

 at. I lie left, ordinary multiplication would not cxjilain his success; 

 for no one, bowever skilful, could multiply a number of thirteen or 

 fourteen tigures by a number of one liguro so rapidly as to begin at 

 once to name the Itfthiuid dibits. 



