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The Budget of Paradoxes

A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: a paradox is something which is apart from general opinion, either in subject matter, method, or conclusion.

Many of the things brought forward would now be called crotchets, which is the nearest word we have to old paradox. But there is this difference, that by calling a thing a crotchet we mean to speak lightly of it; which was not the necessary sense of paradox.

Thus in the 16th century many spoke of the earth's motion as the paradox of Copernicus and held the ingenuity of that theory in very high esteem, and some I think who even inclined towards it. In the seventeenth century the depravation of meaning took place, in England at least."

How can the sound paradoxer be distinguished from the false paradoxer?  De Morgan supplies the following test:

The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself. . . . New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception.

The Budget consists of a review of a large collection of paradoxical books which De Morgan had accumulated in his own library, partly by purchase at bookstands, partly from books sent to him for review, partly from books sent to him by the authors. He gives the following classification: squarers of the circle, trisectors of the angle, duplicators of the cube, constructors of perpetual motion, subverters of gravitation, stagnators of the earth, builders of the universe. You will still find specimens of all these classes in the New World and in the new century.  De Morgan gives his personal knowledge of paradoxers.

I suspect that I know more of the English class than any man in Britain. I never kept any reckoning: but I know that one year with another- and less of late years than in earlier time - I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand. Nobody knows how they swarm, except those to whom they naturally resort. They are in all ranks and occupations, of all ages and characters. They are very earnest people, and their purpose is bona fide, the dissemination of their paradoxes. A great many - the mass, indeed - are illiterate, and a great many waste their means, and are in or approaching penury. These discoverers despise one another.

A paradoxer to whom De Morgan paid the compliment which Achilles paid Hector - to drag him round the walls again and again - was James Smith, a successful merchant of Liverpool. He found  that pi = 3 1/8.

His mode of reasoning was a curious caricature of the reductio ad absurdum of Euclid.  He said let pi = 3 1/8 , and then showed that on that supposition, every other value of pi must be absurd; consequently pi = 3 1/8 is the true value.  The following is a specimen of De Morgan's dragging round the walls of Troy:

Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper, he informs me, with reference to my obstinate silence, that though I think myself and am thought by others to be a mathematical Goliath, I have resolved to play the mathematical snail, and keep within my shell. A mathematical snail !

This cannot be the thing so called which regulates the striking of a clock; for it would mean that I am to make Mr. Smith sound the true time of day, which I would by no means undertake upon a clock that gains 19 seconds odd in every hour by false quadrative value of pi. But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell, and put me hors de combat. The confusion of images is amusing:

Goliath turning himself into a snail to avoid pi = 3 1/8

and James Smith, Esq., of the Mersey Dock Board: and put hors de combat by pebbles from a sling. If Goliath had crept into a snail shell, David would have cracked the Philistine with his foot. There is something like modesty in the implication that the crack-shell pebble has not yet taken effect; it might have been thought that the slinger would by this time have been singing - And thrice [and one-eighth] I routed all my foes, And thrice [and one-eighth] I slew the slain.