SOURCE: Journal of Natural Philosophy, Chemistry, and the Arts 34, 291 [1813].
The words of William Saint are in bold.
In a Letter from Mr. W.
SAINT.
To Mr. Nicholson.
SIR,
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,
W. SAINT.
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.