A Note on Augustus De Morgan

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I recently had the good fortune to stumble across a copy of Augustus De Morgan's A Budget of Paradoxes in a used bookstore in Belfast, Maine. It was in a shelf of "History of Ideas" books (a volume of essays by A. O. Lovejoy, usually considered the founder of that discipline, was next to it ). I knew just enough about De Morgan to know he was a mathematician, and to wonder at the shelf location. Taking the first volume from the shelf for a look, I was immediately taken by the idea that this was a book about "wrong books". And the assignment to "History of Ideas" was spot-on. Like many of my favorite books, not only is the book interesting in and of itself, but it also provides leads into areas I am not familiar with 1.

A Budget of Paradoxes is based on a column De Morgan wrote for The Athenĉum 2 in the 1860's. It is primarily a discussion of books of wrong-headed ideas, with emphasis on the mathematical and a particular emphasis on various attempts to square the circle. Showing Newton to be wrong about something or other was also a favorite. By paradoxer, De Morgan means someone who is promoting an idea that is opposed to the accepted canon; in mathematics the cause is usually ignorance of that which has come before, the opposite of Newton's "standing on the shoulders of Giants". Arrangement is roughly chronological, though by the middle of the second volume, when he is up to his current times there are changes to a subject-based approach. The first part is more like a conventional, heavily annotated bibliography, but it moves to more review-like articles at the end.

A sample entry from A Budget of Paradoxes
Circle squarers
A second sample entry from A Budget of Paradoxes
Newton puzzlers
A third sample entry from A Budget of Paradoxes
And someone simply way ahead of their time

De Morgan produced one of the earliest bibliographies of mathematics (see Arithmetical Books below), and all of the books he discusses have full citations. His annotations range from a brief discussion of the nature of the author's errors to more discursive essays; the further along in the series the more discursive he becomes; the discursions are my favorite parts of the book. An example below.

In an entry that begins discussing probabilities De Morgan, writing about Buffon, uses the familiar heads or tail tests to show how, as the number of experiments increases, the accumulating results approach the expected average. He then goes on to another experiment of Buffon's that, based on the same general principle, can be used to approximate the value of pi.

I now come to the way in which such considerations have led to a mode in which mere pitch-and-toss has given a more accurate approach to the quadrature of the circle than has been reached by some of my paradoxers. What would my friend in No. 14 have said to this? The method is as follows: Suppose a planked floor of the usual kind, with thin visible seams between the planks. Let there be a thin straight rod, or wire, not so long as the breadth of the plank. This rod, being tossed up at hazard, will either fall quite clear of the seams, or will lay across one seam. Now Buffon, and after him Laplace, proved the following: That in the long run the fraction of the whole number of trials in which a seam is intersected will be the fraction which twice the length of the rod is of the circumference of the circle having the breadth of a plank for its diameter. In 1855 Mr. Ambrose Smith, of Aberdeen, made 3,204 trials with a rod three-fifths of the distance between the planks: there were 1,213 clear intersections, and 11 contacts on which it was difficult to decide. Divide these contacts equally, and we have 1,218 ½ to 3,204 for the ratio of 6 to 5π, presuming that the greatness of the number of trials gives something near to the final average, or result in the long run: this gives π = 3.1553. [2nd edition, volume 1 page 383.]

De Morgan goes on to discuss other experiments with a greater number of trials and that give closer approximations. No doubt because of my lack of mathematics this on first or second reading seemed astonishing. A little research shows this is known as Buffon's Needle.

A Budget of Paradoxes. Reprinted, with the author's additions, from The Athenaeum.

  • London: Longmans, Green & Company, 1872. Vii, 511pp. Full text at HathiTrust
  • Second edition edited by David Eugene Smith. Chicago and London: The Open Court Publishing Co., 1915. 2 volumes: viii, 402; 387pp. Full text at HathiTrust  Volume 1  Volume 2
  • New York: Dover, 1954. Seen as two volumes bound in one, a reprint of the second edition, with a new introduction by Ernest Nagel.
  • Volume 1 only of the second edition as The Encyclopedia of Eccentrics. Chicago: Open Court Publishing Company, 1974. 414pp.
  • The second edition is preferable since the editor (David Eugene Smith, a notable writer of mathematical books himself) has added many footnotes, providing background on the many individuals referenced and translations of the numerous foreign quotations, essential for those who do not have languages. An example: De Morgan briefly mentions the notion at the basis of Pascal's Wager, thinking so little of it that he originally doesn't even remember it as associated with Pascal. In an update3 De Morgan provides more detail and, in fact, traces it back to Arnobius in the 3rd century AD, who he quotes (in Latin, translated by Smith in footnote) in convincing detail and with the summary that "Really Arnobius seems to have gotten as much out of the notion, in the third century, as if he had been fourteen centuries later, with the arithmetic of chances to help him". [2nd edition volume 2, page 73]

    The book is no longer in print from any of the above; it has since been reprinted by publishers of historical reprints intended for libraries, and, like almost every other book, shows as in print from a number of print-on-demand groups.

    Decades earlier than the paradoxes, De Morgan published an annotated bibliography of mathematics books.

    Arithmetical Books from the Invention of Printing to the Present Time : being brief notices of a large number of works drawn up from actual inspection.

  • London: Taylor and Walton, 1847. Xxviii, 124pp. Full text at HathiTrust
  • This is sometimes referred to as the first bibliography on a scientific basis; perhaps because he examined every book described in the main bibliography; he regularly cross references earlier lists of books. There is also an appendix of names at the end that represent citations of apparent publications that he has not seen.

    Arrangement is by year and place. The introduction contains an interesting discussion of problems of bibliographical description.

    A sample from Arithmetrical Books
    Blundevile is complimented
    Another sample from Arithmetrical Books
    Jacob Welsh taken to task

    Some of the annotations are much larger than the examples above; the above is typical for books that are not particularly distinctive. In the introduction to this catalogue de Morgan wrote: The most worthless book of a bygone day is a record worthy of preservation. Like a telescopic star, its obscurity may render in unavailable for most purposes; but it serves, in hands which know how to use it, to determine the places of more important bodies.

    And that would be a good motto for any bibliography or library.

    1. For example, I think of Open Court Publishing, in Chicago, as primarily a publisher of philosophy and comparative religion. But with a little effort I have identified at least four of De Morgan's books that they published in the early 1900's. What else did they publish in mathematics at that time?
    2. I find it impressive that a weekly, generally considered a "literary magazine", published columns as demanding as De Morgan's. And with all the typographical appeal of an early Bohn's Library volume. Clearly the demand was for content.
    3. The Budget of Paradoxes was published in book form after De Morgan's death. The ms. had been partially assembled and De Morgan had written occasional amendments and insertions for individual columns. The final preparation was by his widow, Sophia Elizabeth Frend De Morgan. In the published book, additions to the original columns are in brackets.