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THE BOOK OF ZERAH COLBURN

Part One, Chapter Two

SOURCE:
*Journal of Natural Philosophy, Chemistry, and the Arts 34*, 193
[1813].

The words of the anonymous author are in **bold**.

**
**

* From the *Morning Chronicle* of Feb 17, last.*

SIR,

I shall make no apology for troubling you on a subject, which, though generally esteemed dry and abstruse, has at present acquired, from particular circumstances, considerable interest.

There is a boy in town, who is exhibited as a curiosity, from the facility with which he performs several difficult arithmetical operations. It is pretended that this is a gift, and that he has had no instructions to enable him to do this.

Now, Sir, as there are easy methods of solving these questions, which are not, I believe, generally known, I shall simply state them to the public, that this matter may, if necessary, be further investigated ; and that this boy may be reduced to what he really is --- a very clever boy, but no prodigy.

In extracting the cube root where it consists of three figures,
it is well known that the first figure of the root may be obtained
by a simple inspection of the number of millions, and the last
figure, by observing the final figure of the number whose root
is proposed to be extracted ; if then, the middle figure could be
found, we should have the root. To find this, square the final
figure of the root so previously obtained ; multiply this square
by 3, call *A* the last figure of this product. Now cube the last
figure of the root, subtract its penultimate digit from the
penultimate digit of the number given (adding ten to this
last, if it be the smaller of the two), call the result *B*.

Then that number, which being multiplied into *A*, produces
a number terminating with the figure B, is the middle figure
of the root.

An example or two will make it manifest : suppose
377,933,067 to be proposed; here 7 is the first figure,
(as 7^{3} = 343, the nearest cube below 377) and 3 is the last
figure ; since the cube of 3 terminates with 7, the last figure
of the number. Now to find the middle figure :

- 3
^{2}× 3 = 27 *A*= 7- 3
^{3}= 27

Now the penultimate figure of the number is 6, and 6 - 2 = 4 = *B*.
And since 2 × 7 (or 2 × *A*) = 14,
the last figure of which is 4
or *B*, the middle figure of root is 2, and root is 723.

This rule, I should add, becomes ambiguous in all cases
where the number proposed terminates with an even digit, or
with a 5 ; thus, in 41,421,736 *A* = 8 and *B* = 2.

Now, as either 4 × 8 = 32 or 9 × 8 = 72, it follows that, according to the rule, either 4 or 9 might be the middle figure, and either 346 or 396 the root ; but as the cube of 396 is nearly equal that of 400, or 64 millions, it appears on inspection of the number proposed, that 346 must be the true answer. No error would, therefore, be produced by this ambiguity. Indeed, the only cases of ambiguity which can deceive, are in numbers terminating with 5.

The rule for the square root differs only in these particulars ;
to determine *A* take the simple power of the last figure of
the root, and instead of 3, multiply by 2. To determine *B*,
subtract the penultimate figure of the square instead of the
cube of the last figure of the root. In all other respects, the
two rules exactly agree. In the case of square, there is,
however, an ambiguity which does not exist in the cube. It
happens, that the final figure of a square number gives two figures
which may terminate the root ; as for instance, 4^{2} = 16 and
6^{2} = 36. If, therefore, a square number terminate with 6,
its root may terminate with either 4 or 6, and, therefore, more
mistakes will occur in the application of the rule. I believe
this coincides with the fact ; since the boy makes many more
errors in the extraction of the square, than in that of the cube
root.

The principles of these rules, and the rules themselves, or a
very slight modification of them, have been known so long
ago as the year 1768; in that year, M. Rallier des Ourmes
published two memoirs on the subject. They are to be found in
Pp. 485 and 550 of the fifth volume of "Sçavans Etrangers."
They are entitled *Methode Nouvelle, &c.* or
*A New Method of
dividing, when the dividend is a multipile of the divisor, and of
extracting the roots of perfect powers.* See page 550. His method
only takes the last figures into account. In the extraction of the
higher powers, this is undoubtedly the easier way. The second
is, *Methode facile, &c,* or *An easy Method of discovering all the
prime numlers contained in an unlimited series of odd numbers
in succession, and at the same time, the simple divisors of
those which are not primes.*

This latter memoir is probably
the method pursued by the boy to find prime numbers, and
to resolve numbers into their factors. Of the method of M.
Rallier, he himself says, "*In a word, we do not hesitate to
assert from experiment, that by this method, in a single day,
and in the way of amusement, computations may be effected,
which by the old methods, would require months of severe
labour.*"

I will only now add this observation. As the
above rules depend upon the two or three first, and the two last
figures of any number, it follows that the change of the
intermediate ones cannot affect the result. If it should have
occured to anyone, as it has to me, *to have altered any of
these,* and yet to have obtained the true result ; it will, I think,
not be unfair to conclude, that either of these very methods,
or some similar to them in principle, are those adopted.

Let me add, that I have no doubt, but that any clever boy would, in a week's time, learn to apply those given above with the utmost facility.

I am, Yours, &c.

A. H. E.

*The following is from the same respectable daily Journal
of the 18th.*

SIR,

I agree with your correspondent A. H. E. that the young American is a very clever boy, but no prodigy, as one visit to him has convinced me.

The ambiguity of the cases A. H. E.
mentions, in extracting the cube root, may be readily cleared
by any one conversant in figures in a few seconds, by finding
*B* in the common formula for the cube root, which is the cube
of the binomial *A* + *B* ;
namely, *A ^{3} + 3A^{2}B,* &c. which is, no
doubt, perfectly well known to A. H. E. -- though to some of
your readers, who may be interested in this matter, it may not
be so familiar. For such the following directions may be useful.

The first fig. of the root being known by inspection, take its cube from the millions given, then the remainder being divided by the first two digits (for they will be sufficient) of thrice the square of the said first figure, will immediately shew which of the ambiguous figures should be taken for the second figure of the root.

Thus, if the proposed number be
465,484,375, here the first and last digits of the root are 7 and 5 ;
*A* = 5 and *B* = 5 ; any odd number, therefore, multiplied by
*A* will give *B* ; but if the cube of 7, that is 343, be taken from 465,
and the remainder 122 be divided by 14 (the first two digits
of 7^{2} × 3) it will be instantly seen that 9 is too great, and 5 is
manifestly too little ; there only remains 7, therefore, for the
second digit of the root. The same method will easily clear
the ambiguity when the proposed cube ends with an even
digit.

I am, &c.

**
O.
**

[*Interested readers can find a more modern, and probably more
intelligible, description of essentially this method for extracting cube-roots
posted at
Ask Dr. Math.*]

- Next Chapter:
*Vindication of the Claims of the American Boy to extraordinary Talents and original Discovery, by Mr. W. Saint.* - Table of Contents