In 1884 Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions, the first ever book that could be described as ‘mathematical fiction’. Ian Stewart, author of Flatterland and The Annotated Flatland, introduces the strange tale of the geometric adventures of A. Square.

Edwin Abbott Abbott, who became Headmaster of the City of London School at the early age of 26, was renowned as a teacher, writer, theologian, Shakespearean scholar, and classicist. He was a religious reformer, a tireless educator, and an advocate of social democracy and improved education for women. Yet his main claim to fame today is none of these: a strange little book, the first and almost the only one of its genre: mathematical fantasy. Abbott called it Flatland, and published it in 1884 under the pseudonym A. Square.

On the surface — and the setting, the imaginary world of Flatland, is a surface, an infinite Euclidean plane — the book is a straightforward narrative about geometrically shaped beings that live in a two-dimensional world. A. Square, an ordinary sort of chap, undergoes a mystical experience: a visitation by the mysterious Sphere from the Third Dimension, who carries him to new worlds and new geometries. Inspired by evangelical zeal, he strives to convince his fellow citizens that the world is not limited to the two dimensions accessible to their senses, falls foul of the religious authorities, and ends up in jail.

The story has a timeless appeal, and has never been out of print since its first publication. It has spawned several sequels and has been the subject of at least one radio programme and two animated films. Not only is the book about hidden dimensions: it has its own hidden dimensions. Its secret mathematical agenda is not the notion of two dimensions, but that of four. Its social agenda pokes fun at the rigid stratification of Victorian society, especially the low status of women, even the wives and daughters of the wealthy.

Flatland’s inhabitants are triangles, squares, and other geometric figures. In the planar world’s orderly hierarchy, one’s status depends on one’s degree of regularity and how many sides one has. An isosceles triangle is superior to a scalene triangle (all sides different) but inferior to an equilateral triangle. But all triangles must defer to squares, which in turn defer to pentagons, hexagons, right up to the pinnacle of Flatland society, the Priestshood. Referred to as ‘circles’, the priests are actually polygons with so many sides that no one can distinguish them. The sons of squares are usually pentagons, the grandsons hexagons, so there is a general progression along the greasy pole (there is no ‘up’ in Flatland).

But what of the wives and daughters? Flatland’s women are mere line segments, actually very thin triangles, whose social standing is zero. Their intelligence is little greater. They are required by law to wiggle from side to side so that they can be seen, and to emit loud cries so that they can be heard, because a collision with a woman is as fatal as one with a stiletto. Abbott took a bit of stick from some of his female contemporaries, who failed to appreciate his irony. But we know from his own life, including his daughter’s education, that he did a lot to improve the status of women and to ensure they received the same level of education as men.

Abbott wasn’t particularly good at, or keen on, mathematics, but his book tackled an issue of great interest in Victorian times, the notion of four (or more) dimensions. This idea was becoming fundamental in science and mathematics, and it was also being invoked in areas like theology and spiritualism, because an invisible extra dimension was just the place to locate God, the spirit world, or ghosts. Charlatans like the American medium Henry Slade were exploiting conjuring tricks to claim access to the Fourth Dimension. Legitimate hyperspace philosophers were speculating about the role that additional dimensions might play in illuminating the human condition.

Flatland approaches this topic by way of a dimensional analogy, widely used ever since, and not entirely Abbott’s own invention. The difficulties facing a three-dimensional Victorian attempting to grasp the geometry of four dimensions are similar to those facing A. Square attempting to grasp the geometry of three. Among Abbott’s sources for this analogy were frequent social encounters with eminent scientists such as the physicist John Tyndall, whom he met at George Eliot’s house in 1871. Tyndall may have told Abbott about the work of Hermann von Helmholtz, who gave public lectures on non-Euclidean geometry using the image of an imaginary two-dimensional creature living on a mathematical surface. Another likely source is the outrageous Charles Howard Hinton, who wrote his own book about a two-dimensional world in his 1907 An Episode of Flatland: How a Plane Folk Discovered the Third Dimension.

Abbott’s mathematico-literary legacy is a series of Flatland spin-offs: Dionys Burger’s Sphereland, Rudy Rucker’s short story ‘Message Found in a Copy of Flatland’ and his novel The Fourth Dimension, Alexander Keewatin Dewdney’s The Planiverse, and my own Flatterland. But what he was really trying to tell his readers was more subtle. Just as a humble square can transcend his plane world and aspire to the Third Dimension, so the women and the lower classes of Victorian England could transcend the confines of their stratified society and aspire to a higher plane of existence. More than 120 years later, it is a message that has lost none of its urgency.

Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick and the author of numerous popular mathematics books, including Flatterland and The Annotated Flatland. His most recent book is Mathematics of Life.

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Between 1617 and 1621 the English physician and polymath Robert Fludd published his masterwork Utriusque Cosmi, a book split into two volumes and packed with over 60 intricate engravings. Urszula Szulakowska explores the philosophical and theological ideas behind the extraordinary images found in the second part of the work.

Robert Fludd was a respected English physician (of Welsh origins) employed at the court of King James I of England. He was a prolific writer of vast, multi-volume encyclopaedias in which he discussed a universal range of topics from magical practices such as alchemy, astrology, kabbalism and fortune-telling, to radical theological thinking concerning the inter-relation of God with the natural and human worlds. However, he also proudly displayed his grasp of practical knowledge, such as mechanics, architecture, military fortifications, armaments, military manoeuvres, hydrology, musical theory and musical instruments, mathematics, geometry, optics and the art of drawing, as well as chemistry and medicine. Fludd used the common metaphor for the arts as being the “ape of Nature,” a microcosmic form of the manner in which the universe itself functioned.

Fludd’s most famous work is the History of the Two Worlds (Utriusque Cosmi … Historia, 1617-21) published in five volumes by Theodore de Bry in Oppenheim. The two worlds under discussion are those of the Microcosm of human life on earth and the Macrocosm of the universe (which included the spiritual realm of the Divine).

Fludd himself was a staunch member of the Anglican Church. He was educated in the medical profession at St. John’s College in Oxford. In 1598-1604/ 5 he set out for an extended period of travel on the continent. He spent a winter with some Jesuits, a Roman Catholic order deeply opposed to Protestantism who, nevertheless, tutored Fludd on magical practices. Fludd, however, always claimed to have worked out the theological and magical systems in his first volume of the Utriusque Cosmi … Historia, concerning the Macrocosm (1617), during his undergraduate days at Oxford. In this work Fludd devised a lavishly illustrated cosmology, based on the chemical theory of Paracelsus, in which the materials of the universe were separated out of chaos by God who acted in the manner of a laboratory alchemist.

Fludd was a deeply convinced adherent of the medical and magical practice of the German doctor, surgeon and radical theologian, Theophrastus Paracelsus von Hohenheim. This loyalty led Fludd into severe conflicts with the established British medical profession. His later publications described a medical practice, almost devoid of chemical remedies but which depended almost solely on prayer and the use of the name of Jesus on the model of the first apostles of Christ. This devotional medicine was supported by a theology derived from the secret mystical teaching of Judaism, the kabbalah, which Fludd employed in a Christianised form derived from the ideas of the German philosopher, Johannes Reuchlin.

In his medicinal incantations Fludd used the Hebrew form of the name of Jesus which, he claimed, possessed immense magical potency. He equated Jesus Christ with the kabbalistic angel Metattron, the heavenly form of the Jewish Messiah, (UCH, 2 1621: 2-5). He was said to be the soul of the world, pervading it through-out (“anima mundi”), or Anthropos (UCH, 2 1621, Tract II, Sect I: 8-9). Fludd states that “Hochmah” (Wisdom in the kabbalistic Tree of Life) is the same as the “Verbum” (the “Word” in the Christian Gospel of John who is identified with Jesus Christ). The Christian “Word” is the same as the first letter of the Hebrew alphabet, “Aleph. This Christian Messiah is the most potent medicine for all human ills. The “Verbum”, or Metattron-Christ-Messiah, is the form of God himself, residing in the the sun.

In Fludd’s medical theories the operation of the aerial nitre, or quintessence, in the human body was of critical importance to health. It was said to originate in the tabernacle of the aerial spirit which was the sun. The essence of the aerial nitre was celestial light. It was breathed in by the lungs and carried to the heart, where it was separated from the air and dispersed as the vital spirit through-out the body. In his short text, the “Tractatus de Tritico” (the “Tractate on Wheat”) , which re-appeared in his Anatomiae Amphiteatrum (1623) and also in the Philosophia Moysaica (1638), Fludd described the distillation of the aerial nitre from the wheat using the heat and light of the sun’s own rays. Fludd claimed that this distilled spirit was the universal panacea, whose generative celestial fire had been drawn out of the sun.

All of Fludd’s treatises were lavishly illustrated with extraordinary engravings, unique in their form and subject-matter, which have the visionary quality of a genuine spiritual seer and which exerted an influence on his contemporary occultists such as Michael Maier, Jacob Boehme and Johannes Mylius. Fludd himself designed these images and they were engraved by the artisans employed at his publishers. (Some of his own original drawings still exist for the first volume of the Utriusque Cosmi … Historia, 1617).

Fludd’s concepts of the creative and healing forces of light were illustrated by diagrams, the principles of light and darkness being represented by two intersecting cones, or pyramids. The base of the “pyramidis formalis” was placed in the Empyreum of God, signifying rays of divine light, while the base of the “pyramidis materialis” was located on the earth pointing upward towards God. Fludd described these diagrammatic forms as “pyramides lucis”, “cones of light,” claiming to have invented them himself, although they seem to be based on antique and medieval optical theory. Within the lozenge shape created by the intersection of the downward and upward pointing cones, Fludd placed the sun, since the nature of this sphere balanced the oppositions of spirit and matter, male and female, sulphur and mercury.

From the outset in the “Macrocosm” (UCH, 1617) Fludd’s cosmogony was based on the three generative principles which were those conceptualised by Paracelsus in his own alchemy, namely, those of light, darkness and water, from which emerged the three primary elements that constituted matter, that is, Salt from darkness (as the “prima materia”), Sulphur from light (as the soul) and Mercury from water (as the spirit). These, in turn, produced the four qualities of antique and medieval physics, the qualities of heat, cold, dryness and moistness. Fludd’s alchemical cosmology was explained by a series of intricate geometrical diagrams of great power and beauty.

In the third book of the “Macrocosm” Fludd also offered another interpretation of the structure of the universe, complementing his earlier alchemical visualisations, but expressed musically. He analysed what he claimed to be the “Musica Mundana”, the musical forms that pervaded and structure universal creation based on the musicology and mathematics of the ancient Greek, Pythagoras (UCH, 1, 1617, pp. 79-81). The fifth chapter of the “Musica Mundana” included an illustration of a cosmic lyre.

Fludd achieved a certain notoriety in his own time for his early support of the “Rosicrucian Manifestos” in his treatise the Apologia (1616) (expanded into the Tractatus Apologeticus, 1618). The two texts, which came to be called the Rosicrucian Manifestos, consisted of the Fama which appeared in 1614, followed by the Confessio in 1615. They were published anonymously in Kassel and they have been the subject of extensive debate in regard to their origins and authorship (see Donald R. Dickson, The Tessera of Antilia (Leiden: Brill, 1998), pp. 32-36). The authors never revealed themselves but they called for supporters to join the Rosicrucian movement, claimed to have been in existence for centuries. Those who responded by publishing letters to the Rosicrucians were a heterogeneous group of Protestants, mostly of a radical reformist character, as well as adherents to antique philosophy and magic (kabbalists, followers of Hermes Trismegistos – a mythical Egyptian sage, as well as practitioners of magic on the model of Cornelius Agrippa (a sixteenth century German sage) and of Paracelsus.v The Manifestos seem to have been written to counteract the re-conversionary activities of the Jesuit order in central Europe. Said to be supporters of a legendary figure of the fourteenth century, Christian Rozenkreutz, there never did exist in reality any such entity as the Rosicrucian Order in the seventeenth century. It could be said, however, that there did exist a popular Rosicrucian movement whose supporters developed the original ideas of the founders and in which Fludd’s influence was a dominating factor.

(All images in the public domain, courtesy of Deutsche Fotothek)

Urszula Szulakowska has been a lecturer in the History of Art at Sydney University, Queensland University, Bretton Hall College and the University of Leeds (1977-2011). Currently she is Visiting Research Fellow in the School of Fine Art, History of Art and Cultural Studies at the University of Leeds. She has published extensively on the history of art and alchemy including monographs: The Alchemy of Light (Brill: 2000), The Sacrificial Body and the Day of Doom (Brill: 2005) and Alchemy in Contemporary Art (Ashgate: 2010), as well as many learned articles and papers in scholarly sources.

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• Images from Utriusque Cosmi

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Dogs who shop, bury frogs, and take 800-mile solo round trips by rail – writer and broadcaster Frank Key gives a brief tour of the strange and delightful Dog Stories from The Spectator.

Dogs Belonging to the Medici Family in the Boboli Gardens by Tiberio Di Tito (1573–1627)

Here is a puzzle:

[Feb. 2, 1895.] I venture to send you the following story I have lately heard from an eye-witness, and to ask whether you or any of your readers can throw any light upon the dog’s probable object. The dog in question was a Scotch terrier. He was one day observed to appear from a corner of the garden carrying in his mouth, very gently and tenderly, a live frog. He proceeded to lay the frog down upon a flower-bed, and at once began to dig a hole in the earth, keeping one eye upon the frog to see that it did not escape. If it went more than a few feet from him, he fetched it back, and then continued his work. Having dug the hole a certain depth, he then laid the frog, still alive, at the bottom of it, and promptly scratched the loose earth back into the hole, and friend froggy was buried alive! The dog then went off to the corner of the garden, and returned with another frog, which he treated in the same way. This occurred on more than one occasion; in fact, as often as he could find frogs he occupied himself in burying them alive. Now dogs generally have some reason for what they do. What can have been a dog’s reason for burying frogs alive? It does not appear that he ever dug them up again to provide himself with a meal. If, sir, you or any of your readers can throw any light on this curious, and for the frogs most uncomfortable, behaviour of my friend’s Scotch terrier, I should be very much obliged. – R. Acland-Troyte.

It appears in a curious volume entitled Dog Stories From The Spectator : Being Anecdotes Of The Intelligence, Reasoning Power, Affection And Sympathy Of Dogs, Selected From The Correspondence Columns Of The Spectator by J St Loe Strachey (1895). Strachey’s purpose, as given in his introduction, is to provide “no little entertainment for all who love dogs”, and in this he surely succeeds. Indeed, one may have no liking for dogs whatsoever, yet find many of these stories oddly compelling.

A Poodle after Matham by Wenceslas Hollar (1607-1677)

There are tales of syllogistic dogs, sermonising dogs, hospital dogs, parcel-carrying dogs, and purchasing dogs – that is, dogs which “understand the first principles of the science of exchange”. We learn of dogs with a sense of humour, dogs’ talent for friendship with hens, rabbits, and pigeons, dogs that foretell death, and dogs that recognise themselves in the mirror. By the time we get to the end of the book we may agree with Strachey that “a single story of a clever dog may amuse, but… if we have half a dozen illustrating the same form of intelligence, the value of the evidence is enormously increased”.

An added pleasure of the book is, of course, its age. This was a time when the correspondence columns of a general interest magazine were filled with letters written in formal, elegant, crafted prose. The sense of a lost world of good manners and civility could not be better expressed than in Strachey’s apology, worth quoting in full:

Before I conclude this Introduction, I should like to address a word of apology to the correspondents of the Spectator whose letters form the present volume. Though the copyright of the letters belongs to the editors and proprietors of the Spectator I should have liked to ask the leave of the various writers before republishing their letters. Physical difficulties have, however rendered this impossible. In the case of nearly half the letters the names and addresses have not been preserved. In many instances, again, only the names remain. Lastly, a large number of the letters are ten or twelve, or even twenty years old, and the writers may therefore be dead or out of England. Under these circumstances I have not made any effort to enter into communication with the writers before including their letters in this book. That their permission would have been given, had it been asked, I do not doubt. The original communication of the letters to the Spectator is proof that the writers wished a public use to be made of the anecdotes they relate. As long, then, as the letters are not altered or edited, but produced verbatim, I may, I think, feel assured that I am doing nothing which is even remotely discourteous to the writers.

Detail from Venus and Cupid with an Organist (1548) by Titian, depicting what looks to be some breed of Bichon in the bottom righthand corner.

As for that Scotch terrier fond of burying frogs, Frances Power Cobbe suggested an explanation a week later:

[Feb. 9, 1895.] I think I can explain the puzzle of the Scotch terrier and his interment of the frogs, for the satisfaction of your correspondent. A friend of mine had once a retriever who was stung by a bee, and ever afterwards, when the dog found a bee near the ground, she stamped on it, and then scraped earth over it and buried it effectually -presumably to put an end to the danger of further stings. In like manner, another dog having bitten a toad, showed every sign of having found the mouthful to the last degree unpleasant. Probably Mr. Acland-Troyte’s dog had, in the same way, bitten a toad, and conceived henceforth that he rendered public service by putting every toad-like creature he saw carefully and gingerly “out of harm’s way,” underground.

Frank Key is a writer and broadcaster best known for his self-published short-story collections and his long-running radio series Hooting Yard on the Air, which has been broadcast weekly on Resonance FM since April 2004.

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• Dog stories, from the “Spectator” (1896)
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