In 1879, Charles Dodgson, better known as Lewis Carroll, published an odd little book called *Euclid and his Modern Rivals* (available for free at the Internet Archive). Though it takes the form of a play, it is a defense of Euclid's *Elements* as the best textbook for geometry. Carroll's introduction lays out his purpose and why he went about it the way he did. His words on writing for a non-scientific audience still sound particularly relevant.

The object of this little book is to furnish evidence, first, that it is essential, for the purpose of teaching or examining in elementary Geometry, to employ one textbook only; secondly, that there are strong a priori reasons for retaining, in all its main features, and specially in its sequence and numbering of Propositions and in its treatment of Parallels, the Manual of Euclid; and thirdly, that no sufficient reasons have yet been shown for abandoning it in favour of any one of the modern Manuals which have been offered as substitutes.It is presented in a dramatic form, partly because it seemed a better way of exhibiting in alternation the arguments on the two sides of the question; partly that I might feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and a little more acceptable to unscientific readers.

In one respect this book is an experiment, and may chance to prove a failure: I mean that I have not thought it necessary to maintain throughout the gravity of style which scientific writers usually affect, and which has somehow come to be regarded as an 'inseparable accident' of scientific teaching. I never could quite see the reasonableness of this immemorial law: subjects there are, no doubt, which are in their essence too serious to admit of any lightness of treatment—but I cannot recognise Geometry as one of them. Nevertheless it will, I trust, be found that I have permitted myself a glimpse of the comic side of things only at fitting seasons, when the tired reader might well crave a moment's breathing-space, and not on any occasion where it could endanger the continuity of a line of argument.

Pitying friends have warned me of the fate upon which I am rushing: they have predicted that, in thus abandoning the dignity of a scientific writer, I shall alienate the sympathies of all true scientific readers, who will regard the book as a mere

jeu d'esprit, and will not trouble themselves to look for any serious argument in it. But it must be borne in mind that, if there is a Scylla before me, there is also a Charybdis—and that, in my fear of being read as a jest, I may incur the darker destiny of not being read at all.In furtherance of the great cause which I have at heart—the vindication of Euclid's masterpiece—I am content to run some risk; thinking it far better that the purchaser of this little book should

readit, though it be with a smile, than that, with the deepest conviction of its seriousness of purpose, he should leave it unopened on the shelf.

Carroll's book was a salvo in the "math wars" of the day, the subject of which was how best to teach geometry. Euclid was the standard textbook in private schools that taught mathematics, but many people found it wanting. To them, it was dry, formal, and obscure, lending itself to rote learning with no understanding. To improve the geometry curriculum, the anti-Euclid Association for the Improvement of Geometrical Teaching was formed in 1871. In stage directions, Carroll skewers the association:

Enter a phantasmic procession, grouped about a banner, on which is emblazoned in letters of gold the title 'Association for the Improvement of Things in General.' Foremost in the line marches Nero, carrying his unfinished 'Scheme for lighting and warming Rome'; while among the crowd which follow him may be noticed—Guy Fawkes, President of the 'Association for raising the position of Members of Parliament.'... Afterwards enter, on the other side, Sir Isaac Newton's little dog 'Diamond,' carrying in his mouth a half-burnt roll of manuscript. He pointedly avoids the procession and the banner, and marches past alone, serene in the consciousness that he, single-pawed, conceived and carried out his great 'Scheme for throwing fresh light on Mathematical Research,' without the aid of any Association whatsoever.

Calling someone "Nero" must have been the 1879 version of calling him or her "Hitler." Carroll's defense is entertaining, sarcastic, and curmudgeonly. I don't think he was right, but it was interesting to get his perspective and think about how to define almost undefinable concepts such as line, direction, and angle.

But what stood out to me the most was that while Carroll spent plenty of time on the parallel postulate, he made no mention in his book of non-Euclidean geometry. It's not that the parallel postulate was not discussed; several sections include extensive explorations of the topic, and he included two related tables in his text. Table I was a list of propositions relating to lines, "of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms." In other words, he collects axioms and theorems about lines that don't rely on the parallel postulate. Table II is a list of "eighteen Propositions, of which no one is an undisputed Axiom, but all are real and valid Theorems, which, though not deducible from undisputed Axioms, are such that, if any one be admitted as an Axiom, the rest can be proved." Here we have statements that are equivalent to the parallel postulate. But he never talked about alternatives to the parallel postulate.

Mathematician Harold Scott MacDonald Coxeter wrote the introduction to the 1973 Dover edition of Carroll's book and also noticed the omission of anything about non-Euclidean geometry. He wrote:

In these days of enlightenment we find it difficult to realize that, 100 years ago, Professors Arthur Cayley of Cambridge and W.K. Clifford of London may well have been the only Englishmen who understood the philosophical revolution that had been instigated by Gauss, Bolyai and Lobachevsky, some 50 or 60 years earlier. One is tempted to speculate on what might have happened if Cayley or Clifford had met Dodgson and convinced him that there is a logically consistent 'hyperbolic' geometry in which the 'absolute' propositions in Table I still hold while all the statements in Table II are false.... In hisSylvie and Bruno Concludedthe real projective plane is represented as a "Purse of Fortunatus" made by sewing together three square handkerchiefs. The same easy style and fertile imagination, applied to the infinite hyperbolic plane, would surely have produced a thrilling exploration of this new Wonderland.

After *Euclid and His Modern Rivals* was published, Carroll did learn a bit about non-Euclidean geometry, but according to Robin Wilson's book *Lewis Carroll in Numberland, *he always believed that the parallel postulate had to be true. Like Coxeter, I wonder what interesting stories Carroll might have told about hyperbolic geometry if he had had a little more time to get used to the idea.