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Scientific AmericanSixty-one years ago W. W. Norton and Company published Lancelot Hogben's Mathematics for the Millions, one of the truly great popularizations of this century. It has sold more than 600,000 copies and is still selling thousands of copies a year. With the publication of Jan Gullberg's Mathematics: From the Birth of Numbers, Norton seems to have done it again. The book is an enthusiastic and utterly amazing popularization that promises to be in print for decades. The two books have much in common. Both were written by nonmathematicians: Hogben was a zoologist and Gullberg a surgeon. Neither was an American, but each ended up being published by an American company. Both saw mathematics as a social equalizer and essential for helping the masses to escape control by society's "clever people" -- mathematicians, scientists and government officials. Both authors richly appreciated the historical development of mathematics. And both books are big: Hogben at 649 pages and Gullberg at 1,093 pages. Hogben cuts a broad mathematical swath, from the development of counting to calculus and statistics. Gullberg covers nearly all that and much more. When Hogben's classic was published in 1937, it was praised for its illustrations. The number and variety of pictures in Gullberg's opus dwarf those of Hogben. Both books are serious, although Gullberg also makes generous use of humor. But here is a sharp contrast: Hogben claimed to have written his book in six weeks; Gullberg took 10 years for his. An itinerant surgeon (or "Dr. Sawbones" as he sometimes referred to himself), Gullberg seemed to be drawn to small towns, each a bit off the beaten track. In the years that he worked on the book, he lived in Angola, Ind.; Forks, Morton, Moses Lake and Sequim in Washington State; Moab, Utah; Møsjen and Rjukan in Norway; and Karlskoga, Sweden. He claimed that his peripatetic way of life was crucial to the gathering of data and the development of ideas. Certainly his book project provided something of an anchor during that period. Gullberg died at age 62 in May 1998 of the consequences of a stroke suffered a few days earlier while at work at a hospital near his home in Nordfjordeid, Norway. Fortunately, he lived long enough to see his book become a publishing success. The scope of the book is huge-numbers and language, combinatorics, logic, set theory, theory of equations, geometry, trigonometry, fractals, sequences and series, calculus, harmonic analysis, probability and differential equations. Few mathematicians would undertake to write a survey so ambitious. A full quarter of the tome is devoted to calculus topics. Given the tremendous influence of calculus since its invention more than 300 years ago, perhaps it deserves that much space. Bits of humor and other ancillary material are present in the calculus chapters, though not with the same frequency as in other chapters. The book's large pages are loaded with eye-catching graphics and illustrations, frontispieces from original works, cartoons and jokes. The illustrations reveal much about Gullberg's remarkably broad and deep curiosity about mathematics, his appreciation of mathematics in history, his fondness for languages (he was fluent in English, Swedish, Norwegian, French and German, and he could read Greek and Latin) and his humorous side.
The opening illustration is a photograph of a sculpture by Swedish sculptor Axel Ebbe. Depicting a young man emerging from an imposing mass of stone, it is entitled Breaking Away from the Darkness of Ignorance, to which Gullberg forcefully adds, "and then there was Mathematics."Within the first five pages, as the author explores "The Origins of Reckoning," we encounter lovely whimsical drawings by his wife, Ann, and twin sons, Kalin and Kamen, then nine years old, and apt quotations from Thomas Hobbes, Umberto Eco and Carl von Linné (Linnaeus). And he never lets up throughout the almost 1,100 pages-history, brief profiles, more than 1,000 technical illustrations done by his son Pär, and excerpts from an astonishingly wide range of sources, from daily newspapers to Newton to Winnie the Pooh. He is even bold enough to include some of his own verse and jokes, most of which are quite good.
The illustrations alone make the book worth the price-and at less than five cents per page, it is a bargain. Gullberg searched through an amazingly large number of classics and reproduced from them a wonderful collection. Among them are the frontispiece of Gottfried Leibniz's Dissertatio de arte combinatoria (1666); Pascal's original triangle from his Traité du triangle arithmetique (1665); Johannes Kepler's drawings of 13 semiregular polyhedra in his Harmonices mundi (1619); the frontispiece and table of contents of the first printed textbook on calculus, Analyse des infiniment petits by L'Hospital (1696); brief selections from John Napier's A Description of the Admirable Table of Logarithms with a Declaration of the Most Plentiful, Easy, and Speedy Use Thereof in Both Kinds of Trigonometry, as Also in All Mathematical Calculations; and a map of Königsberg circa 1740 with its seven bridges that inspired the famous problem of the same name (solved by Leonhard Euler in 1736), which helped to launch graph theory.
A third of the way through, Gullberg presents a chapter entitled "Overture to the Geometries." He begins the chapter with a chilling quotation from the book of Roman civil law (circa A.D. 650) that serves as a reminder that mathematicians have not always been held in high regard: "The study and teaching of the science of geometry are in the public interest, but whosoever practices the damnable art of mathematical divination shall be put to the stake." In the same chapter, and on a decidedly lighter note, we find instructions on "How to Catch Elephants-An Elephantasy," which appeared in the Copenhagen newspaper Berlingske Tidende. By the time we have finished the chapter, we have been treated to wonderfully readable sections on history, geometric abstraction, perspective and projection, form and shape, survey of geometries, topology and Euclidean and non-Euclidean geometries. The next chapter continues with more geometry, which is followed by chapters on trigonometry (plane and spherical), hyperbolic fractions, analytic geometry, vector algebra and fractals.
Gullberg gives the reader plenty of motivation and a multitude of worked-out examples, which are especially helpful. He derives many familiar formulas and provides many proofs, but he is careful not to overwhelm the reader with detail. He was particularly keen on reading the masters, as is reflected in the opening remarks to his extensive "Works Cited" section: "The strength of those who have gone before should not be gleaned solely from histories or random quotations. True inspiration and power lie in the original works." Not only did Gullberg write Mathematics: From the Birth of Numbers, he built the book using his Macintosh Plus. He set the type and lovingly designed the pages, deciding where to place each illustration, drawing and cartoon. His son Pär, now a computer graphics specialist, provided vital technical assistance, including the computer-generated illustrations. After the manuscript was reviewed by several specialists in different areas of mathematics, corrected, copy edited and the pages put into final form by Gullberg, all the publisher had to do was start the presses. The volume is remarkably clean for a book of this size and scope. Typos and mathematical glitches can be found, but they are rare. Few of us, whether amateurs like Gullberg or professional mathematicians, could write (or even read) a manuscript of this size without occasional lapses.
About a year ago, when I asked Gullberg why he wrote Mathematics: From the Birth of Numbers, he responded: "So that people might develop confidence in doing mathematics and an appreciation for how long the development of mathematics has taken." In the same interview, when asked about his target group, he said, "I can't write for a target group-only for myself-as a means of exploring mathematics and its history." Readers of Scientific American, students, teachers of mathematics and all others who care about my favorite subject will want to own this book. It is an important reference and a book that is plain fun to dip into. If a family is to have only one mathematics book on the reference shelf, then this is the one. It is a great present for anyone who likes mathematics. But if you buy this extraordinary tome for someone, there is considerable danger that you will end up keeping it for yourself-and then you will need to buy another copy.