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A Profitless Demonstration.
We find the following profundity in the Boston Transcript, where it appears as a communication:
[For the Transcript.] Mr. Editor: If the following process does not prove that 2 equals 1, I should like to know what it does prove? If there be fallacy in the reasoning, pray ask some of your algebraical friends where it lies.
Assume that x equals a,
Then multiply by x, x square equals ax,
And subtracting a square, x square — a square equals ax — a square,
Or, in factors (x plus a) multiplied by (x — a) is equal to a (x — a)
Therefore, dividing by (x — a); x plus a equals a.
But x equals a,
Therefore, by substituting, a plus a equals a,
Or, 2a equals a, [[a]]
Dividing by a 2 equals 1.
Yours, N’-EST-CE PAS?
Now if x equals a, then the equation x square — a square equals ax — a square amounts to the statement that nothing is equal to nothing; and no doubt it is. The fallacy of the “demonstration” lies in the beginning of the fourth line, where x plus a and x — a [[a]] are taken as factors of x square — a square, whereas the true factors are x alone and x — a. it is a mere blunder. If we put, as we should, x (x — a) equals a (x — a) and then divide by x — a we get only x equal to a, which is precisely the proposition with which Mr. N’est-ce pas started.
This demonstrating that two are one is only adapted to the case of man and wife, and is a needless piece of business altogether. To demonstrate that one is two, might indeed be a profitable and very convenient thing — especially to one who has a note in bank, and can get only within fifty per cent of the payment.
A more sensible proposition is that of Punch, who says that the new London cab, which holds three inside and a driver on top, demonstrate that three can go into one and have over.
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Notes:
This item was attributed to Poe by T. O. Mabbott. Mabbott's notes at the University of Iowa say, “Who but Poe on mirror staff would have gone into a mathematical proof of a puzzle was not puzzle.” The original puzzle appears in the Boston Transcript for February 4, 1845 (p. 2, col. 3). On February 6, 1845, The Boston Transcript notes, “We have received a'shower’ of answers,” printing one signed “G. M. W.” (p. 2, col. 3). Another unsigned algebraic puzzle is printed by the Transcript for February 10, 1845 (p. 2, col. 3).
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[S:0 - NYEM, 1845] - Edgar Allan Poe Society of Baltimore - Works - Criticism - A Profitless Demonstration (E. A. Poe ?, 1845)