The Net Advance of Physics RETRO:
Part One, Chapter Three

SOURCE: Journal of Natural Philosophy, Chemistry, and the Arts 34, 291 [1813].

The words of William Saint are in bold.

Vindication of the Claims of the American Boy to extraordinary Talents and original Discovery.

In a Letter from Mr. W. SAINT.

To Mr. Nicholson.


In reading your last number, I was struck with surprise (in common, it should seem, with most of your readers) to find that you had inserted a letter from the Morning Chronicle which purported to give an account of the manner by which the American boy performs his calculations with such wonderful celerity.

Now I am persuaded, Sir, that, had you had sufficient leisure to examine into the merits of that letter, and into the claims of its author to the important discovery which he affects to have made, you would not have given publicity, (and, what is of still greater consequence, your sanction) to a statement so little calculated to effect the object of its author, which was "to reduce the child to what he really is -- a very clever boy, but no prodigy."

Your insertion of this letter, after the very excellent account you gave of the boy in a former number, has tended to produce a belief in the minds of such of your readers as are unaccustomed to abstruse calculations, that what this child does may likewise be effected by any other boy of good abilities, and thus a prejudice may be excited against this youthful and astonishing calculator, which may prove equally injurious to his own fame, and to his father's pecuniary interest. I have, therefore, to request, Sir, that you will assist me in my efforts to vindicate the reputation of this extraordinary boy, by inserting in your next number, if convenient, the following remarks on the letter alluded to, in which I have endeavoured to show, that Mr. A. H. E. has not succeeded in discovering the method by which this boy performs his calculations with such surprising celerity.

In the application of M. Rallier's method to the extraction of the cube root, Mr. A. allows, that "the result is ambiguous where the number proposed terminates with an even digit, or with a 5 ;" he proceeds, however, to explain how the difficulty may be removed with respect to the even digit, though I think I may safely challenge him to produce a single instance of a child from six to eight years of age, who would be able to comprehend the method, much less to apply it with facility and rapidity. Be this as it may, it is confessed by Mr. A. that the case of numbers ending with 5 is one which "can deceive," and I accordingly expected to find that Mr. A. had given the boy various examples of this ambiguous case, and that he had uniformly found the boy incapable of answering such questions correctly, or that he had obtained from him an acknowledgement that such questions were beyond the reach of his powers to answer. Yet nothing of this kind is mentioned by Mr. A. who leaves us totally in the dark upon the very point which would have cleared up the difficulty.

Are we to imagine, then, that Mr. A. though aware of the importance of putting such questions, for the purpose of ascertaining whether M. Rallier's method was employed or not, yet omitted to ask them ? Or, if he did ask questions of this kind, and received wrong answers, (which must have been the case if the boy employed the method alluded to,) how is it that he has neglected to avail himself of the statement of this circumstance, so materially affecting his claims to a discovery which he evidently considers to be an important one ?

But allow me, Sir, to examine the merits of this rule in its application to the square root. Let us suppose the boy was requested to extract the square root of the number 42436 ; here it is obvious the first figure of the root would be 2, and the last either 4 or 6 ; --- if 4 be taken, then 4 or 9 would be found to be the middle figure ; but if 6 be used, then 0 or 5 would be the middle figure ; hence there would be no fewer than four different roots obtained by M. Rallier's method, of which four the boy could not possibly know the correct one, and he might assign either 206, 256, 244, or 294 for the root of the required number.

This is no particular example, selected for the purpose of exhibiting M. Rallier's rule in the most unfavourable point of view ; for it will be found upon trial, that had any other number been proposed, four different results would have been obtained by this rule ; and that if a number ending with 5 had been proposed, no less than ten different results would have been produced, since all square numbers ending with 5 will likewise terminate with 25, as I have shown in your Philosophical Journal, No. 99, [ Journal of Natural Philosophy, Chemistry, and the Arts 22, 291 (1809)] where may also be seen some other curious properties relating to square numbers. It is manifest, therefore, that, if the boy adopted this method, he would not only make "many more errors in the computing by extraction of the square than in that of the cube root" ; but that he would, in most cases, fail three times out of four ; and, in some cases, nine times out of ten.

Any of your readers may satisfy themselves respecting this ambiguity, by referring to a table of square numbers, where they will find that the first 25 square numbers contain all the varieties of the two terminating figures of such numbers ; and that the squares of all numbers equally above and below 25 ; as of 24 and 26 ; or of 23 and 27, &c. will have their two last figures the same.

(This property may not have been noticed by your readers in general, but those of them who are but slightly acquainted with mathematics may satisfy themselves of its truth and universality ; for since the difference of the squares of the sum, and difference of any two numbers is equal to four times the product of those numbers, it is manifest that the difference of the squares of two numbers of the form 25 + a, and 25 - a, would be of the form 100a ; that is, this difference would be some exact multiple of 100 ; and therefore two such squares could not differ in their units and tens places of figures, viz. in their two last digits ; hence, then (since the two last figures only are used in M. Rallier's method) would arise the ambiguity which I have stated. It will be easily seen, that what I have shown of numbers of the form 25 + a, and 25 - a, is equally true of the general formulas 25n + a and 25n - a.)

Having proved, that M. Rallier's rule is only of partial utility in the extraction of the cube root, and of little or no use in the square root, I think it would be extremely unfair to conclude, that either this method, or one very similar to it, is adopted by the boy.

Suppose, however, Sir, that it were possible for the boy to have answered such questions as related merely to the square and cube roots of numbers by the help of the above rule, still this will not explain the method by which he multiplies four figures by four, or by which he ascertains the factors of any number, however large, with a rapidity that has astonished some of the first mathematicians in the country.

I am aware, indeed, that Mr. A. refers to another memoir of M. Rallier, on prime and and composite numbers, and I regret, in common with most of your readers, that he has not given us so much as a single hint respecting the method employed in this second memoir, though he says "it is probably the one pursued by the boy to find prime numbers, and to resolve numbers into their factors." Without knowing myself, however, what this method may be, I cannot think that it has been adopted by the boy, for several reasons ; first, because it has been known for nearly fifty years, secondly, that none of the mathematicians who have seen the boy (except Mr. A.) have considered any of the known methods of operating with prime and composite numbers, as sufficient to account for the rapidity with which the calculations have been performed ; and thirdly, that the method itself could never have fallen into disrepute, but would have been adopted not only by every mathematician, but by every teacher of arithmetic in the most obscure country villages, if it had been of such inestimable utility as to have enabled boys of only six years of age to have performed such astonishing calculations !

Again, Sir, Mr. A. made no new discovery when he found that the boy, in extracting the square or cube root of any proposed number, made use only of the two first and two last figures. This curious and singular fact had been known for many months to several eminent mathematicians who had visited the boy, and who were soon convinced, from the quickness and accuracy of his answers, and from the power which he possessed of correcting himself whenever he committed an error, that M. Rallier's method was not the one he employed, even in the extraction of roots, much less in ascertaining the factors of large numbers, which he does with a rapidity and apparent facility, astonishing to those who have been long acquainted with the method alluded to, and who, notwithstanding their years of practice in abstruse calculations, find, that they themselves cannot perform such operations, neither by that method, nor by any other yet made public !

What, then, shall we think of Mr. A.'s claims to the discovery of the "modus operandi ?"

Mr. A. might have spared himself the trouble of suggesting an alteration in the intermediate figures of any perfect cube, which may be proposed to the boy, since such intermediate figures need not be mentioned at all ; for it is well known that, in a company of upwards of one hundred persons, amongst whom were some of the first literary and scientific characters in the kingdom, the following question was distinctly and unequivocally put to the child. --- "Can you tell the root of a perfect cube number by means of the two first and two last digits only ?" He answered "Yes :" and that the company might be satisfied that he clearly understood the nature of the question, it was put to him again in the following manner :

"If a number of 12 figures be taken (which shall be a perfect cube) and the two first and the two last figures only be named to you, can you tell the cube root of the whole number ?"

To this he also replied, Yes. He was then tried by various examples, which he answered with a facility and correctness that excited the wonder and admiration of every one present.

Now, Sir, was there in all this any appearance of a wish to deceive ? any desire to conceal any thing ? any fear expressed by the boy lest the various questions which were put to him might lead to a detection of his method? No, Sir, all was fair, frank, open, and ingenuous !

But I am persuaded, Sir, that what I have stated must be sufficient to convince any unprejudiced person, that Mr. A. has not succeeded in discovering the method by which the child performs his operations ; and I am therefore led to hope that I may thus counteract the tendency which, the publication of Mr. A.'s letter in your Journal may probably have had to injure both the boy and his father.

I am, Sir, with the warmest wishes for the success of your Journal,

Your most humble Servant,


Lower Close, Norwich, March 13th, 1813.

Who was "W. Saint"? He had been a mathematics professor at the Royal Military Academy (one of the few places in Georgian England other than Cambridge and Oxford that offered advanced tuition in the exact sciences), but by 1813 had become a schoolmaster in Norwich, where he would live in narrow circumstances, battling constant ill-health, until his death six years later at the age of 39. When he mentioned "every teacher of arithmetic in the most obscure country villages" above, he spoke of his own profession.

In a letter to The Monthly Repository of Theology and General Literature, XIV, 493 (1819) (responding to an obituary stressing Saint's "genius" and his insistance on the education of his daughter), one of his friends comments:

"Having his time very much occupied with his pupils, he was not able to write so often as he wished, but when he did possess a little interval of leisure, he employed it in writing us letters so long ... that they might be called pamphlets more properly than letters. One of these which is now before me, dated April 16, 1813, consists of not less than forty-four pages octavo. Blessed, forever blessed, be the memory of this generous, kind-hearted, excellent person! ... In a small and unhealthy body there was a soul of very fine and eminent powers, acute, sagacious, penetrating, judicious, and discriminating. Mathematical demonstrations of a very abstruse kind he went over with all the ease and spirit of a consumate master, seeing his way before him with a perspicuity truly admirable, and coming to his conclusion with the most complete accuracy and correctness ... Upon religious topics there was certainly some degree of reserve, and as I believed I knew the cause of it, I was averse to press him ..."

Why was Saint sensitive about religion? Perhaps because he was the student (and in fact the biographer) of John Fransham, one of the great Eighteenth Century English eccentrics. Fransham was a mathematician, philosopher, and designer of utopias who lived in poverty, sometimes working as a weaver. Depending on which version of his legend one accepts (and Saint accepted neither, insisting on his teacher's "Christian character"), he was also either a rationalist disciple of Spinoza or an avowed neo-Pagan who none-too-secretly worshipped the Greek gods.