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Preface
What is Flatland, and why should it be annotated?

Flatland is a work of scientific fantasy written by the English clergyman and headmaster Edwin Abbott Abbott and published in 1884. It is a charming, slightly pedestrian tale of imaginary beings: polygons who live in the two-dimensional universe of the Euclidean plane. Just below the surface, though, it is a biting satire on Victorian values- especially as regards women and social status- and an accomplished and original piece of scientific popularization about the fourth dimension. And, perhaps, an allegory of a spiritual journey.

It deserves to be annotated because- just as Euclid's plane is embedded in the surrounding richness of three-dimensional space- so Flatland is embedded in rich veins of history and science. Investigating these surroundings have led me to such diverse items as The Good Grave Guide to Hampstead Cemetery, phrenology, ancient Babylon, Karl Marx, the suffragettes, the Indian Mutiny of 1857, the Gregorian calendar, Mount Everest, the mathematician George Boole and his five remarkable daughters, the Voynich manuscript, H.G.Wells's The Time Machine, the 'scientific romances' of Charles Hinton, spiritualism, and Mary Shelley's Frankenstein.

I first read Flatland in 1963 as an undergraduate newly arrived at the University of Cambridge (England) to study mathematics. I enjoyed it, added it to my science fiction collection- it fits a broad definition of the genre- and forgot about it. Years later, I re-read it, and the idea of a modern sequel began to form in my mind. I wasn't the first person to think of that, or to do it, but recent advances in science and mathematics made it easy for me to invent a new scenario. The result was Flatterland, whose genesis I have related in its own preface. While Flatterland was being readied for publication, my editor Amanda Cook at Perseus Books came up with the idea of a companion volume- a republication of the original Flatland, but with added annotations.

I started with the idea that I would focus mainly on the mathematical concepts that Flatland uses or alludes to, so the writing ought to be simple and straightforward. But when I started looking into the life and times of its author, his associates, and the scientific and cultural influences that led up to the writing of Abbott's unique book, I was hooked. My amateur-historian investigations led into ever more fascinating byways of Victorian England and America, and I began to rediscover many things that are no doubt well known to Abbott scholars, but are far from common currency.

At first, I was concerned that I might not be able to lay hands on the necessary material. But a glance at one of the more obvious and accessible sources- Abbott's entry in the Dictionary of National Biography- brought to light a curious coincidence. Abbott's professional life revolved around the City of London School- he is its most famous Headmaster, a post that he took up in 1865. Now, there exists in London an institution called Gresham College. It was founded in 1597 with a legacy from Sir Thomas Gresham (1518/19-1579), originator of Gresham's Law ('bad money drives out good') and founder of the Royal Exchange in 1566-68. Gresham was a philanthropist, and his will instructed the Mercer's Company ( one of the livery companies created by King Richard II) and the City of London to 'permit and suffer seven persons by them from time to time to be elected and appointed... sufficiently learned to read... seven lectures.' The College has no students- only the general public- and until recently it appointed seven professors, in Astronomy, Divinity, Geometry, Law, Music, Physic, and Rhetoric. To these have been added an eighth: Commerce.

Anyway, between 1994 and 1998 I was the Gresham Professor of Geometry. The first such was Henry Briggs (1561-1630, appointed in 1596), inventor of 'natural' logarithms; others include Isaac Barrow (1630-1677, appointed 1662), who recognized that differentiation and integration, the two basic operations of calculus, are mutually inverse; Robert Hooke (1635-1703, appointed 1664), who discovered the law of elasticity named after him, suggested that Jupiter rotates, and laid the early foundations of crystallography; and Karl Pearson (1857-1936, appointed 1890), one of the founders of statistics. The College is still funded by the Mercer's Company and the City of London. The City also has a long interest in the City of London School, and as a Gresham Professor I had lectured at its sister insitution, the City of London School for Girls (founded 1894). So I had an easy introduction to Abbott's professional home. The City of London School had been badly damaged in World War II and had moved to new premises; I wrote a letter asking whether, despite that, it still had any Abbott documents, pictures, or other information. In response, Head Porter Barry Darling sent me a history of the School (City of London School by A.E. Douglas-Smith) which contained extensive information about Abbott, and invited me to visit and look through the School's archives.

A week later, I was ushered into a small, rather disorganized, room, lined with shelves and crammed to the celing with old books, magazines, photographs, and bound volumes of letters. On the top shelf, tucked away in one corner, was an almost complete collection of Abbott's books, including a rare first edition of Flatland. (I knew that a second revised edition had followed hard on the heels of the first, because the preface to that second edition says so. What had he changed? Now I could find out.) I went away with a stack of photocopies, and three framed photographs of Abbott at various stages of his career, loaned to me for copying. I had his obituary in the School magazine, a review of Flatland in the same journal, samples from geometry texts used by the School in Abbott's day, extracts from his publications- even a copy of his letter of resignation.

Of course, the Abbott scholars had been there before me- but even so, I felt like Sherlock Holmes hot on the trail of Moriarty.

Other sources now came into their own. I could surf the net, because I had some idea of what to look for. Entering 'Flatland' into Yahoo turned up thousands of sites about off-road vehicles, but 'Edwin Abbott Abbott' was much more helpful. An article by Thomas Banchoff (the leading expert on Abbott, currently working on a biography) explained the crucial connection to Charles Howard Hinton, whose wild but ingenious speculations about the fourth dimension undoubtedly inspired Abbott's fable. A conversation with a colleague, Bruce Westbury, in the Warwick University Mathematics Common Room, put me on to the four-dimensional mathematics of Alicia Stott Boole. As a science fiction aficionado I already knew that H.G. Wells had used four-dimensional geometry in The Time Machine; now the web turned up a brilliant historical survey by the science fiction author Stephen Baxter, and another by James Beichler, linking Wells to Hinton. Rudy Rucker's The Fourth Dimension opened up dozens of further leads... and so it went.

What is the purpose of an annotated edition? Martin Gardener, in the classic among all such books, The Annotated Alice: the Definitive Edition, says: 'I see no reason why annotators should not use their notes for saying anything they please if they think it will be of interest, or at least amusing.' Which is exactly my feeling. So I have pursued trails wherever they led, and reported anything that seems to fit the overall story. The most extreme case is a series of associations that links Abbott to Mary and Percy Bysshe Shelley, Lord Byron, Augusta Ada Lovelace, Charles Babbage, Sir Edward ffrench Bromhead, George Boole, Mary Boole, Charles Howard Hinton, Alicia Stott Boole, and the Dutch mathematician Peiter Schoute- with a side branch to the science fiction writer H G. Wells.

Something important emerges from such chains of connections: Victorian England was a tightly-knit society. The intellectuals all knew each other socially, traded and stole each other's ideas, and married each other's sons and daughters. It was an exciting period of scientific and artistic discovery, for the staid and repressive attitudes of the Victorian era were sowing the seeds of their own destruction. Abbott knew many of these people- most of them more colourful than he was-and they influenced his thinking in profound ways. It's been fun ferreting out their stories. For example, along the way I discovered that I once held the same job as Abbott's mathematics teacher- but 146 years later.

As a strictly amateur historian, I know that I will have made some mistakes, misinterpreted some events, or left out some vital items of information that are well known to all the experts. This happens with any book: it is virtually impossible to track down all the relevant documentation, all the names, all the dates. (Oh, yes: I've been moderately obsessive about giving dates for almost everything and everybody- except for minor figures- because the timing is so crucial in this kind of investigation. When I don't know a date- and sometimes when nobody does- I've either put a question mark or omitted it.) So I invite anyone who has constructive criticisms, useful observations, wild theories, or new information, to e-mail them to me. I can't promise you a reply- though I'll do my best- but I do promise that I'll take note of anything I think is interesting. And when (I'm sure it will be 'when') it is time to prepare a new edition, I'll make the necessary changes.

However, I also promise that nearly everything I say is true- or, if it's an opinion, plausible. I've tried to do my historical and scientific homework. I hope you'll come to agree with me that there is so much more to Flatland than meets the eye, even if it is a world of only two dimensions.