10 



KNOW^LEDGE ♦ 



[November 1, 1888. 



Now, if a mathematician were set such a problem, he 

 would have no other resource than to deal with it by direct 

 trial. Of course he would not try every number from 1 

 upwards to 36,083. lie would know that, if the number 

 can be di\-ided at all, it must be divisible by a number less 

 than 190; for any greater divisor would go, exactly, some 

 smaller number of times into 36,083 ; and that smaller 

 number would itself be a divisor. He would see that the 

 number is not even, and therefore cannot be divided by 

 2, 4, 6, or any even number. The number Ls not divisible 

 by 3 ; for, according to a well-known rule, if' it were, the 

 sum of its digits would be so divisible ; therefore he would 

 dismiss 3, 9, 15, and all numbers divisible by 3 not already 

 dismissed. So with 5 (for the number does not end with a 

 5) ; so with 7, by trLal : 11, 13, 17, and so on. But he 

 would have to try many numbers of two and three figures 

 by actual division before he had completed his proof that 

 36,083 has no divisors. Probablj' (for I must confess I 

 have not tried) he would require about a quarter of an hour 

 of calculation before he could be confident that 36,083 is a 

 prime number. Here however was a child, eight years old, 

 who, to all appearance, completed the work Immediately the 

 number was proposed ! 



The next feat was of the same nature, but very much 

 more difhcult ; indeed, it taxed the young calculator's 

 powere more than any other feat he accomplished. Fermat, 

 a mathematician who gave great attention to the theory of 

 numbers, had been led, by reasoning which need not here 

 be considered, to the conclusion that if the number 2 be 

 multiplied into itself 31 times (that is, raised to the thirty- 

 second power), and 1 added, the result will be a prime 

 number. The resulting number is 4,294,967,297. The 

 celebrated mathematician Euler succeeded, however, after a 

 great deal of labour (and, if the truth must be told, after a 

 great waste of time), in showing that this number is divisible 

 by 641. The number was submitted to Zerah Oolburn,who 

 was of course not informed of Euler's prior dealings with 

 the problem, and, ofter tJie lapse of aume wteJcs, the child- 

 calculator discovered the result which the veteran Swiss 

 mathematician had achieved with much gi-eater labour. 



Before proceeding to inquire how Colburn achieved these 

 wonders, we must consider what was learned about his 

 processes. He was not very communicative — doubtless 

 because the faculty he possessed was not accompanied by 

 commensurate clearness of ideas in other matters. In fact, 

 we might as reasonably expect to find a chUd of eight years 

 competent to explain processes of calculating, however easily 

 efifected, as to find him able to explain how he breathed or 

 spoke. One answer which he made to a mathematician 

 who pressed him more than othei-s to describe his method 

 was clever, though the mathematician was certainly not 

 to be ridiculed for ti-ying to get the true explanation of 

 Colburn 's seemingly mysterious powers — " God," said the 

 child, " put these things into my head, and I cannot put 

 them into youi-s." 



Some things, however, he explained as far as he could. 

 He did not seem able to multiply together at once two 

 numbers which both contained many figures. He would 

 decompose one or other into its factors, and work with 

 these separately. For instance, being asked to multiply 

 4,395 by itself, he ti-eated 4,395 as the product of 293 and 

 15, first multiplying 293 by itself, and then multiplying the 

 product twice by 15. On being asked to multiply 999,999 

 by itself, he treated it, in like manner, as the product of 

 37,037 and 27, getting the correct result. In this case 

 probably a mathematician would have got the start of him, 

 by treating 999,999 as a million less one, whence, by a well- 

 known rule, its square is a mLUion millions and one, less two 

 millions, or 999,998,000,001. "On being interrogated," 



proceeds the account, " as to the method by which he ob- 

 tained these results, the boy constantly declared that he did 

 not know how the answere came into his mind. In the act 

 of multiplying two numbers together, and in the raising of 

 powers, it was e^'ident (alike from the facts just stated and 

 from the motion of his lips) that some operation was going 

 forward in his mind, yet the operation could not, from the 

 readiness with which the answers were furnished, have been 

 at all allied to the usual modes of procedure." 



Baily, after discussing the remarkable feats of Zerah 

 Colburn, expressed the opinion that they indicate the exist- 

 ence of properties of numbere, as yet undiscovered, some- 

 what analogous to those on which the system of logarithms 

 is based. "And if," says Carpenter (quoting Baily), "as 

 Zerah grew older, he had become able to make known to 

 others the methods by which his results were obtained, a 

 real advance in knowledge might have been looked for. 

 But it .seems to have been the case with him, as with 

 George Bidder and other ' calculating boys,' that with 

 the general culture of the mind this special power faded 

 away." 



With all respect for a mathematician so competent to 

 judge on such matters as Francis Baily, I must say his 

 explanation seems to me altogether insuflicient. So far 

 from the properties of logarithms illustrating the feats of 

 Zerah Colburn, they illustrate the power of mathematical 

 development in precisely the opposite direction. The system 

 of logarithms enabled the calculator to obtain results more 

 quickly than of old, not by the more active exercise of his 

 own powers of calculation, but by employing results 

 accumulated by the labours of others. Its gi-eat advan- 

 tage, and the quality which causes every mathematician to 

 be grateful to the memory of Napier of Merchiston, resides 

 in the fact that, by taking advantage of a well-known pro- 

 perty of numbers, tables of moderate dimensions serve a 

 great number of purposes which by any ordinary plan of 

 tabulation would require several volumes of great size. If 

 it were possible for a calculator to use as readily a set of 

 tables equal in bulk to five volumes of the " London 

 Directory " as he now uses a book of logarithms, and if such 

 volumes could be as easily and as cheaply produced, tables 

 much more labour-saving than the books of logarithms 

 could be constructed. But of course such sets of volumes 

 would be practically useless if they could be produced, and 

 it would be impossible either to find calculators to form the 

 tables 01- printers and publishers to bring them out. Now, 

 of all processes by which mathematical calculation can be 

 carried out, no two can be more unlike than mental arith- 

 metic on the one hand, and the use of tables, of whatever 

 kind, on the other. Napier invented his system to reduce 

 as far as possible the mental effort in calculation, making 

 the calculator employ results collected by others ; young 

 Colburn's success depended on mental readiness, and he was 

 so far from using the results obtained by others, that he 

 did not even know the ordin.iiy methods of arithmetic. A 

 man of Napier's way of thinking would be the last to trust 

 to mental calculation ; whereas, if Colburn had retained his 

 skill until he had acquired power to explain his method, he 

 would have been the last to think of such a help to calcula- 

 tion as a table of logarithms. Napier strongly urged the 

 advantage of aids to calculation ; Colburn would scarcely 

 have been able to imagine their necessity. 



Nor is it at all likely — we could even say it is not possible 

 — that properties of numbers exist through the knowledge 

 of which what Colburn did could be commonly done. The 

 mathematicians who have dealt with theory of numbers 

 have been too numerous and too skilful, and have worked 

 too diligently in their field of research, to overlook such 

 properties, if they existed. Besides, it is scarcely reason- 



