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The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes

David Darling
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Warren Johnson
, on

This is a curious book. The introduction says "If you want a comprehensive, academic dictionary of mathematics, look elsewhere. If you want rigor and proof, try the next shelf. Herein you will find only the unusual and the outrageous, the fanciful and the fantastic: a compendium of the mathematics they [those villains!] didn't teach you in school." The second sentence is spot on, and the author is to be commended for putting in the first sentence, but since the word "only" in the third sentence is not accurate, I fear that some people will still buy this book (for $40) thinking that it is a dictionary of mathematics. It is certainly a dictionary of something, with alphabetical entries from abacus to zonohedron. The subtitle, From Abracadabra to Zeno's Paradoxes, enhances the effect.

There are, in fact, entries for most common mathematical terms here, though the author is not very interested in many of them. The entry on ordinary differential equation has 16 words, 5 of which are "Compare with partial differential equation", which gets 29 words. There is a slightly longer entry for differential equation, which says (among other things) that "if only nth powers of the derivatives are involved, the equation is said to have degree n." The entry for abstract algebra reads as if written by someone who never took the course, though the entry for group is better. The binomial theorem is never fully stated, though it could be pieced together from the brief entries for binomial theorem and binomial coefficient. In the latter, the author writes m choose n when he means n choose m. There is no entry for Stirling's formula, though it is mentioned in the entries for π and e.

Quite a few of the entries have no obvious connection with mathematics, for example those on Jorge Luis Borges, John Cage, Ernst Florens Friedrich Chladni, John Dee and Lord Edward Plunkett Dunsany, all of whom get more space than Cauchy. I got tired of looking for these pretty quickly, but catch-22 is here too, and so is swastika, apparently because it is a 20-sided polygon.

The author is clearly very fond of puzzles and recreational mathematics, especially recreational number theory; thus Frederick Schuh gets an entry, but Issai Schur does not. He also likes exotic plane curves, games (backgammon gets half a page, with no attempt to justify its inclusion in what is ostensibly a mathematics book), optical illusions and extremely large numbers. (Ron Graham, who could be the subject of a very good book, gets 1/3 the space of his eponymous number.) If you share these predilections, and if you are not too pedantic, then you would probably enjoy this book. I didn't care for it much myself.

There are some embarrassing mistakes. Arthur Cayley's name is correct in his own entry, but he is called George in Cauchy's entry on the previous page. Erwin Schrodinger's name is correct at the top of page 34, but he is called Wernher in the entry for William Rowan Hamilton. Jacobi's name is correct in his entry, but he is called Charles in the entry for God. Gödel gets another l in the heading of page 135. Solomon Golomb's name is correct in his entry, but he is called Simon near the top of page 272. The author uses "loose" when he means lose in the entry for Jacobi. The entry on triangular numbers says that every triangular number is a perfect number. Somewhere in the book — I can't find it now — e is said to be about 2.712, though a good value is given in the entry for e. I suppose that Avogadro's constant is here because it's very large, but Avogadro is misspelled. The entry for Abel says that Galois died in a sword fight, but the duel was fought with pistols.

The author has a Ph.D in astronomy, so I find it a bit surprising that he seems to know so little about special functions. The entry on them has only 35 words, and one of the examples is Lagrange polynomials, which presumably means either Legendre or Laguerre polynomials. (Neither Legendre nor Laguerre gets an entry; Legendre is perhaps the best mathematician without one, though Eisenstein doesn't have one either, and Schur is another candidate.) The brief entry on Hermite says that he studied a class of differential equations now known as Hermite polynomials, thus confusing an equation with its solution.

The book is for the most part well-written, but not uniformly so. The entry for calculus of variations has two sentences, the first being "Calculus problems, especially differentiation and maximization, that involve functions on a set of functions of a real variable." The entry for partition number begins "A number that gives the number of ways of placing n indistinguishable balls into n indistinguishable urns." Characteristically, the author is interested (just barely) in the number of partitions, but not at all interested in partitions as such — there is no entry for partition.

The book is very weak on enumeration. Stirling numbers are not mentioned at all, though Bell numbers get an entry. The entry for Catalan numbers is remarkably short for this sort of book — evidently the author has never looked at Stanley's Enumerative Combinatorics or at his web site. Neither Euler numbers nor Eulerian numbers are mentioned. The entry on Bernoulli numbers would be improved by inserting "nonzero" in the sentence "The first few Bernoulli numbers are...."

Ramanujan's name is rendered twice as "Ramanujan, Srinivasa Aaiyangar." The first three words of the obituary by P.V. Seshu Aiyar and R. Ramachandra Rao are Srinivasa Ramanujan Aiyangar, though the title is just Srinivasa Ramanujan. In his outstanding biography of Ramanujan, Robert Kanigel (whose name is misspelled in the references) uses Srinivasa Ramanujan Iyengar once, explaining where each name comes from, and then stops using Iyengar (and for that matter Srinivasa). I don't think G.H. Hardy uses the third name at all, and Berndt only very rarely, e.g., when talking about Ramanujan's father. In any event, I don't know where the spelling Aaiyangar comes from. I also don't know what the source is for the assertion that Ramanujan's letters (to Baker and Hobson, though the author does not say so) were returned unopened. Kanigel doesn't say this (although he doesn't rule it out), and neither do Berndt or Hardy. It seems the least likely of Kanigel's three possibilities (letter ignored, discouraging reply) to me.

I am also uncomfortable with the sentence "But even in cases where [Ramanujan] arrived at conclusions already known, he'd often travel an original route, and, in many cases, almost purely by intuition." For one thing, it is not one of the author's better contributions to English prose style. More important, I think that the last four words are liable to give a false impression. We still do not know how Ramanujan found many of his theorems, but it should be emphasized that, however formidable his intuition may have been, it was the kind of intuition that comes from doing lots of very difficult calculations. If you're bothered by this sort of thing, then this is definitely not the book for you.

Warren Johnson ( teaches at Connecticut College.



Mathematics Entries A to Z.


Solutions to Puzzles.

Category Index.