## The Net Advance of Physics RETRO:Weblog THE BOOK OF ZERAH COLBURN Part One, Chapter Two

The words of the anonymous author are in bold.

UPON CERTAIN READY PROCESSES FOR COMPUTATION, SUPPOSED TO HAVE BEEN INVENTED BY THE AMERICAN BOY EXHIBITED IN LONDON

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 73 = 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 :

• 32 × 3 = 27
• A = 7
• 33 = 27
of which the penultimate figure is 5.
[sic]

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, 42 = 16 and 62 = 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, A3 + 3A2B, &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 72 × 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.]