11
The Mathematicians Who Never Were: Fiction
and Humor
Let me see: four times five is twelve, and four times six is thirteen,
and four times seven is—oh dear! I shall never get to twenty at
that rate! However, the Multiplication Table don’t signify.
DOES IT SIGNIFY? FULL OF TERMS THAT OUTSIDERS DON’T UNDERSTAND, FULL of things that don’t exist in the real world (lines with no breadth, points with no size), full of apparent nonsense yet full of ambition to change the world: mathematics has long been a rich field for fiction, humour and satire. Fictional treatments of mathematics often focus on individual mathematicians, and these “mathematicians who never were” are much on display in this chapter. They include desert-island autodidacts, astronomers both admirable and absurd, and fantastic visions of the mathematicians of the distant past or of other worlds.
This riot of characters, and the other reflections on mathematics chosen for this chapter, articulate some serious concerns, too. Could mathematics threaten the stability of society? Could it distract us from the real business of life? Damage our health? Our minds? A reverse side to the last chapter’s optimism about mathematics is on view here, and we hear the voices of individuals who were genuinely concerned about what mathematics would and could do to the world.
But it’s not all gloom. These fictions show the capacity of mathematics to ennoble, to intrigue, to liberate, and of course to entertain. And, finally, they allow us to wonder: what might mathematics and mathematicians be like in the future?
If, for example, a Selenite is destined to be a mathematician 
his brain grows continually larger, at least so far as the portions engaging in mathematics are concerned; they bulge ever larger and seem to suck all life and vigour from the rest of his frame. His limbs shrivel, his heart and digestive organs diminish, his insect face is hidden under its bulging contours. His voice becomes a mere stridulation for the stating of formula; he seems deaf to all but properly enunciated problems. The faculty of laughter, save for the sudden discovery of some paradox, is lost to him; his deepest emotion is the evolution of a novel computation. (H.G. Wells, The First Men in the Moon (London, 1901), Chapter 24.)
For some different possibilities, read on.
Spider-Men and Lice-Men
Margaret Cavendish, Duchess of Newcastle, a versatile writer and the only woman to attend a meeting of the Royal Society in the seventeenth century, included the description of a “Blazing-World” in her Observations upon Experimental Philosophy (1666). In it she cast herself as the empress of an imaginary world and poked fun at various learned groups under the guise of “ant-men,” “licemen,” “bear-men,” and so on. The mathematical and scientific experts come off particularly badly, with the experimental philosophers ordered to break their telescopes because they cause so much trouble, the mathematicians told that their writings are incomprehensible, and the society of geometers “dissolved” because they have involved themselves too much in impossible scientific tasks.
Margaret Cavendish (1623–1673), The Description of a new World, called The Blazing-World. Written By the Thrice Noble, Illustrious, and Excellent Princesse, the Duchess of Newcastle (London: 2nd edition, 1688), pp. 15–16, 25–29, 55–56.
The rest of the Inhabitants of that World were men of several different sorts, shapes, figures, dispositions, and humors ; some were Bear-men, some Worm-men, some Fish- or Mer-men, otherwise called Sirens; some Bird-men, some Fly-men, some Ant-men, some Geese-men, some Spider-men, some Lice-men, some Fox-men, some Ape-men, some Jackdaw-men, some Magpie-men, some Parrot-men, some Satyrs, some Giants, and many more, which I cannot all remember. And of these several sorts of men, each followed such a profession as was most proper for the nature of their species, which the Empress encouraged them in, especially those that had applied themselves to the study of several Arts and Sciences, for they were as ingenious and witty in the invention of profitable and useful Arts as we are in our world—nay, more—and to that end she erected Schools, and founded several Societies. The Bear-men were to be her Experimental Philosophers, the Bird-men her Astronomers, the Fly-, Worm- and Fish-men her Natural Philosophers, the Ape-men her Chymists, the Satyrs her Galenic Physicians, the Fox-men her Politicians, the Spider- and Lice-men her Mathematicians, the Jackdaw-, Magpie- and Parrot-men her Orators and Logicians, the Giants her Architects, etc.
Figure 11.1. Margaret Cavendish, inventor and empress of the “Blazing-World.” (Margaret Cavendish, Playes (1662), frontispiece. © Princeton University Library. Rare Books Division, Department of Rare Books and Special Collections, 17th-752.)
To avoid hereafter tedious disputes, and have the truth of the Phenomena of Celestial Bodies more exactly known, 
commanded the Bear-men, which were her Experimental Philosophers, to observe them through such Instruments as are called Telescopes, which they did according to her Majesty’s Command. But these Telescopes caused more differences and divisions amongst them than ever they had before: for some said they perceived that the Sun stood still, and the Earth did move about it, others were of opinion that they both did move, and others said, again, that the Earth stood still, and the Sun did move. Some counted more Stars than others; some discovered new Stars never seen before; some fell into a great dispute with others concerning the bigness of the Stars; some said the Moon was another World like their Terrestrial Globe, and the spots therein were Hills and Valleys, but others would have the spots to be the Terrestrial parts, and the smooth and glossy parts the Sea.
At last, the Empress commanded them to go with their Telescopes to the very end of the Pole that was joined to the World she came from, and try whether they could perceive any Stars in it; which they did, and, being returned to her Majesty, reported that they had seen three Blazing-Stars appear there, one after another in a short time, whereof two were bright, and one dim. But they could not agree neither in this observation: for some said, It was but one Star which appeared at three several times, in several places; and others would have them to be three several Stars, for they thought it impossible that those three several appearances should have been but one Star, because every Star did rise at a certain time, and appeared in a certain place, and did disappear in the same place. Next, It is altogether improbable, said they, That one Star should fly from place to place, especially at such a vast distance, without a visible motion, in so short a time, and appear in such different places, whereof two were quite opposite, and the third side-ways. Lastly, If it had been but one Star, said they, it would always have kept the same splendor, which it did not, for, as above mentioned, two were bright, and one was dim.
After they had thus argued, the Empress began to grow angry at their Telescopes, that they could give no better Intelligence; for, said she, now I do plainly perceive that your Glasses are false Informers, and instead of discovering the Truth, delude your Senses. Wherefore I Command you to break them, and let the Bird-men trust only to their natural eyes, and examine Celestial Objects by the motions of their own Sense and Reason.
The Bear-men replied that it was not the fault of their Glasses, which caused such differences in their Opinions, but the sensitive motions in their Optic organs did not move alike, nor were their rational judgments always regular.
To which the Empress answered that if their Glasses were true Informers, they would rectify their irregular Sense and Reason. But, said she, Nature has made your Sense and Reason more regular than Art has your Glasses; for they are mere deluders, and will never lead you to the knowledge of Truth. Wherefore I command you again to break them; for you may observe the progressive motions of Celestial Bodies with your natural eyes better then through Artificial Glasses.
The Bear-men, being exceedingly troubled at her Majesty’s displeasure concerning their Telescopes, kneeled down, and in the humblest manner petitioned that they might not be broken; for, said they, we take more delight in Artificial delusions than in Natural truths. Besides, we shall want Employments for our Senses, and Subjects for Arguments; for, were there nothing but truth, and no falsehood, there would be no occasion to dispute, and by this means we should want the aim and pleasure of our endeavours in confuting and contradicting each other; neither would one man be thought wiser than another, but all would either be alike knowing and wise, or all would be fools. Wherefore we most humbly beseech your Imperial Majesty to spare our Glasses, which are our only delight, and as dear to us as our lives.
The Empress at last consented to their request, but upon condition that their disputes and quarrels should remain within their Schools, and cause no factions or disturbances in State, or Government. The Bear-men, full of joy, returned their most humble thanks to the Empress, and to make her amends for the displeasure which their Telescopes had occasioned, told her Majesty that they had several other artificial Optic-Glasses, which they were sure would give her Majesty a great deal more satisfaction. Amongst the rest, they brought forth several Microscopes, by the means of which they could enlarge the shapes of little bodies, and make a Louse appear as big as an Elephant, and a Mite as big as a Whale.
The Empress having hitherto spent her time in the Examination of the Bird-, Fish-, Worm-, and Ape-men, etc., and received several Intelligences from their several employments, at last had a mind to divert herself after her serious discourses, and therefore she sent for the Spider-men, which were her Mathematicians, the Lice-men, which were her Geometricians, and the Magpie-, Parrot- and Jackdaw-men, which were her Orators and Logicians.
The Spider-men came first, and presented her Majesty with a table full of Mathematical points, lines and figures of all sorts of squares, circles, triangles, and the like, which the Empress, notwithstanding that she had a very ready wit, and quick apprehension, could not understand, but the more she endeavoured to learn, the more was she confounded. Whether they did ever square the circle, I cannot exactly tell, nor whether they could make imaginary points and lines, but this I dare say: That their points and lines were so slender, small and thin, that they seemed next to Imaginary. The Mathematicians were in great esteem with the Empress, as being not only the chief Tutors and Instructors in many Arts, but some of them excellent Magicians and Informers of Spirits, which was the reason their Characters were so abstruse and intricate that the Empress knew not what to make of them. There is so much to learn in your Art, said she, that I can neither spare time from other affairs to busy myself in your profession, nor, if I could, do I think I should ever be able to understand your Imaginary points, lines and figures, because they are Non-beings.
Then came the Lice-men, and endeavoured to measure all things to a hair’s breadth, and weigh them to an Atom; but their weights would seldom agree, especially in the weighing of Air, which they found a task impossible to be done; at which the Empress began to be displeased, and told them, that there was neither Truth nor Justice in their Profession, and so dissolved their society.
In the Court of Lilliput
Not the work of Jonathan Swift, but that of an imitator, possibly Mrs. Eliza Heywood (1693?–1756), author and actress, this utopian fantasy displays an imagined attitude toward mathematics even more severe than that in Margaret Cavendish’s Blazing-World and illustrates vividly the notion that mathematics could be politically and even morally dangerous. We have met several times in this volume the notion that mathematics might be mistaken for magic or conjuring, that it might cause moral or social disturbance or merely be a dangerous misuse of resources; but this is the only occasion on which we see it, for any of those reasons, actually outlawed.
Gulliver, Captain, Memoirs of the court of Lilliput. Written by Captain Gulliver. (London, 1727), pp. 121–122, 126–127, 128–132, 134–135.
At the farther end of my Apartment I saw some little confused Spots and Lines drawn athwart each other in a Mathematical manner, which, though I had lived here for many Months, I had never observed before 
.
The Figures I saw as I approached increased my Wonder. I never saw in England a Globe more exactly drawn; I do not believe the nicest Mathematician could have found fault with the smallest Line. How, said I to myself, can these People have so just a notion of the Position of the World, yet imagine there are no parts of it habitable but that they possess, and the small Island of Blefuscu?
should have been tempted to have believed that Painting had been done by an European Hand, if the size of the Temple in which it was had not convinced me it could only be of service to a Lilliputian Race.
told me that had I been a Lilliputian born, or had lived among them long enough to be acquainted with their Laws, to have brought those Figures to light would have drawn on me some very severe Punishment, but as I was a Stranger, and had been guilty only through Ignorance, ’twas probable I might obtain Pardon from the Emperor, if he should happen to know it. However, he advised me to conceal what I had done, and erase the Figures, if by any means I could, so as they might not be seen by any that came to visit me.
begged he would inform me of the Reasons which made me guilty, if he could possibly do it without becoming so himself. .
This place, said he, which is now allotted for your Apartment, was formerly a Temple, and the most magnificent one in the whole Kingdom; this Painting that you have so miraculously discovered was done by the greatest Artist of his Time, from a Draught given him by the first and perhaps the greatest Philosopher, Mathematician, and Geographer that ever the World produced. He divided the Globe by straight and oblique Lines, in the manner you see it here deciphered, foretold the Change of Weather, counted the number of the Stars, and prefixed certain Times for the rising and setting of the Sun in such and such Seasons of the Year. He was greatly applauded for the success of his Labours, and our History informs us that never Man received more substantial Proofs of Esteem and Admiration. This brought the Art or Study of Mathematics so much in fashion, that all our young Nobility and Gentry bent their Minds this way, and presently after rose up a number of imaginary Proficients. Vanity, and a desire of broaching new Opinions, and rendering themselves remarkable, made everyone affect to have made new Discoveries in the Regions of the Air.
Vast Treatises were in a short time composed, and different Systems were every day set forth, some as distant from all Probability as they all were from one another. Every one had a particular Set of Followers, who appeared so well convinced of the Truth of what they professed that they declared themselves ready to endure Martyrdom for the Conviction of the rest. This puzzled the Minds of the People so much that they knew not to which to give Credit, and frequently occasioned Distinctions among them, to the ruin of many a noble Family.
For which reason, and also that the immoderate Application to Philosophy took our Youth from the more useful Studies of War, Politics, and Mechanism, Golbasto Momarin Eulame Guclo, the Father of our present Emperor, made an Edict, strictly prohibiting the use of Mathematics for the future, except in such Branches of it as were necessary for Navigation, or for Weight and Measure, with a Penalty of two thousand Gredulgribs (each of which in Gold is as big as a Silver Penny 
English Money) affixed to the Conviction of the Crime after the Publication of this Order; and if the Delinquent was found incapable of paying such a Fine, his Life must answer for his Fault. All the Books of Argument relating to this Science were immediately burned, all the Paintings of it demolished, or plastered over, as you see it was here, and the same Punishment allotted for anyone who should conceal the one, or by any means preserve the other, as for him who should be guilty either by writing or painting a new one of the same kind. By this means, added he, Astronomy, Geography, and many other Branches of this noble Science, are entirely lost, or lie dormant in the Breasts of those who dare not transmit them to their Posterity.
Although no Man is a greater Admirer of these kind of Studies than myself, yet I confess I could not avoid thinking it very prudent in the Government of Lilliput to suppress them, when they began to encroach on the practical and more useful Business of Mankind while they live in the world. For, on mature Consideration, what is it to us by what means it first received its Formation, or how it is since influenced and directed, if we are provided with all things needful in it? And how void of Reason must we appear to a disinterested Observer, to lose that time in visionary Speculations, which is too little to be employed in the endeavour of acquiring what alone can defend us from Insults, and Contempt, and the want of those Necessaries, without which Life is so far from being desirable, that it becomes a Burden? In the midst of these more serious Reflections, I could not forbear laughing at a sudden Thought which just then came into my Head: that if such a Law were put in force in England, what a loss our Ladies would be at, for the Amusements they meet with in having their Fortunes told.
again strictly charged me to blot out the Picture, as soon as I had enough considered it to be able to retain as much of it in my Mind as I thought would be of service either to the Amendment of my Morals, or Satisfaction of my Curiosity. I assured him, that I would obey him .
Could I have taken it down, and brought it to England with me, I should have thought it a greater Treasure than all the Wealth of America; but that was impossible, not only on the account of the King’s express Command but also because it was painted on the Wall, which there was no removing without pulling down the Fabric, and by that means betraying the Theft.
Automathes
An intriguing take on the desert island story, John Kirkby’s novel depicts an individual whose shipwreck preserves (remarkably enough) little more than himself, a few books, and some mathematical instruments. He succeeds in teaching himself a range of skills including a good deal of mathematics, prompting reflections on the universality of mathematics and the possibility of devising an idiosyncratic form of it.
In one sense this book represents the converse of the various accounts we saw in Chapter 10 of the desirability of a mathematical education: for Automathes the acquisition of mathematical learning is simply unavoidable.
John Kirkby (c. 1705–1754), The capacity and extent of the human understanding; exemplified in the extraordinary case of Automathes; a young nobleman, who was accidentally left in his infancy, upon a desolate island, and continued nineteen years in that solitary state, separate from all Human Society. A narrative Abounding with many surprizing Occurrences, both Useful and Entertaining to the Reader. (London, 1745), pp. 165, 171–174, 177–187.
I was more fortunate in opening a Till in the End of the Chest, in which were deposited some Books and White Paper, a few Black-lead Pencils, Pen, and an Ink-Standish, a Pocket Magnifying-glass, a Seaman’s Scale, and a Case of Mathematical Instruments
The Books were three in Number viz. a Treatise of Divinity, a Piece of History, and a large System of the Mathematics. 
ad it not been for the Cuts interspersed here-and-there, especially in the History, and the Geometrical Schemes, which abounded in the Mathematical Book, I believe I should have for ever laid them aside, without father Examination. But these, being more intelligible than the Letters, convinced me of their Design, and were afterwards a Direction for me to hold the Book. And the Mathematical Volume became of great Use to instruct me in the Principles of that Science, though without the least Knowledge of a Letter contained in it.
This taught me the Use of the Ruler, Compasses, and Brass Semicircle,° from their exact Resemblances upon the Paper with various ocular Applications of them; by which means I presently learned several manual Operations, such as to measure a 
Line by equal Parts, to describe a circle, to bisect a Line, erect and let fall a Perpendicular, to inscribe a regular Polygon of any proposed Number of Sides in a Circle, and to measure the Angles or Openings of two intersecting Lines by the equal Divisions of an Arc.
I had also a frequent Sight of the nine Digits or Figures, both in the Book and upon the Instruments, and, observing the same Figures always to denote the same Numbers, I became at length perfectly acquainted with their Use, and likewise discovered how these alone, with the Character called a Cipher, were adapted to express all Numbers imaginable, by the Variation of their Places; A Contrivance, the exceeding Ingenuity of which, did not a little please me.
But though I had this Success with the Figures, yet it was impossible for me to have the like with the Letters of the Alphabet, which were put to signify Things I have not the least Notion of.
I presently found out a Way, by Help of the Black-lead Pencils, to impress several Marks of my Skill in geometry upon the unwritten Paper, in which I made continual Improvements.
I was discovered to have learned a Sort of Arithmetic peculiar to myself, and my Necessities had also taught me to perform several Problems in Geometry.
Upon the highest Part of the Hill, directly behind my Cottage, a little Mount reared up its Head, in form of the lower Frustum of a Cone, overlooking the whole Country around, whose Summit made an horizontal Plain, of not more than ten Paces in Circumference. This I took the Pains to cover with a smooth Cap of Clay, erecting a Stile near the Centre of about a Yard in Height, which was my Contrivance to trace out the Points of Shadow, according to the different Times of the Day, and Seasons of the Year, that I might attain a more perfect Knowledge of the Motions of the Sun. But here I had many Experiments before I could answer my Design, till at last I found out a way to make my Stile out of a Piece of the Copper Skillet, drilling an Hole through the Top to admit the Rays of the Sun; by which Invention I both obtained a more certain Perpendicular to my Plane, and a truer Method of marking the Points of Shadow from a lucid Point, without any perceptible Penumbra.
This daily Practice of marking out the Points of Shadow, presently suggested to me a Meridian Line from the Foot of the Stile, in which I saw every Day’s shortest (or Mid-day) Shadow constantly to fall. And I observed that all Lines drawn from the same Centre, making equal Angles on both Sides the Meridian, must every Day coincide with the Shadow of the Stile, the same Spaces of Time before and after Noon, and that the Length of the Mid-day Shadow alone was sufficient to mark out the annual Access and Regress of the Sun. I made me therefore a Circle from the Foot of the Stile as a Centre, to as great a Circumference as my Plane would admit, the Northern Limb of which I divided into five equal Arcs, of twenty Degrees apiece, on both Sides the Meridian, with Lines drawn from the Centre to each; and thus I had the Angles of Shadow marked out for an Hundred Degrees, on each Side the Meridian. But I presently perceived the Insufficiency of this to measure Time, which was the End I aimed at by it, from the greater or lesser Disproportion between these Angles, and the like Arcs of the Sun’s course, according as they were more or less distant from the Meridian. And this I had no Remedy for, till after much Study I invented a Time-teller out of one of my Boxes.
I chose out the compacter of these, which, after I had emptied of what it contained into the Chest, I endeavoured to make as tight as possible to contain Water; and, when I had brought it to my Mind, I made a Hole in the Bottom, fitting it with a Stopple. Afterwards I proceeded, with all the Care and Exactness I was Master of, to make another horizontal Plane, close by the Fountain, drawing a Circle, and erecting a perpendicular Stile in the Center, both of the same Dimensions with that on the Mount. And having found the Meridian Line, I placed my Box in a right horizontal Position across the Outlet of the Fountain, filling it with Water. Then waiting till the Sun came upon the Meridian, I immediately let out the Water, and, as soon as the Box was empty, marked down the Shadow upon the Circle, filling it again out of the Fountain, and placing it in the same Situation. All which was done with such Expedition, that the Box was scarce empty before it was full again, and the Orifice was kept continually running. This I kept repeating till the Setting of the Sun, still marking down the Points of Shadow, at the End of every Box-full thus run out, and laying the same Extent with my Compasses upon the Circle to the Eastward, which the Shadow made to the Westward. And by that means I obtained a pretty exact Dial within my Enclosure, for almost Ninety Degrees on both sides the Meridian, or from near six in the Morning to six in the Evening, which I afterwards increased at my Leisure, as the Days grew longer.
When I had brought this to as great a Perfection as I was able, I expunged all my Lines of Shadow, save the Meridian, upon the Mount, and laid down the same with these in their Places, having there a more commodious Situation for my intended Observations of the Heavenly Bodies. Out of the Lid of the Water-box I formed a Quadrant, of about two Foot Radius, graduated as I had seen in the Book, which, upon Occasion, I applied perpendicularly along the Meridian (with the Limb towards the Stile, and the Centre upon the Extremity of the lucid Ray) to measure the different Altitudes of the Sun in its Mid-day Situation. The same also served me to take the Meridian Altitudes of the Moon, which I performed by leaning along the Side of the Mount, where I made a Place to fix my Feet, and drawing the Centre to my Eye fixed against the North Edge of the Plane. And I was so intent upon these two great Luminaries, that I seldom carried my Observations of this Sort to the Stars, perceiving none of them, except four or five, but what always kept the same Situation one to another, so that whenever I obtained the true Altitude of one, I could refer all the rest to it in their Order.
By this frequent Practice I learned the gradual Increase and Decrease of Day and Night, according to the Access and Regress of the Sun, as also the Number, nearly, of diurnal Revolutions, which he took up in finishing his annual Course. I determined the menstrual Periods of the Moon, and from the exact Conformity I beheld between the Changes of her Face, and her Distances from the Sun, I could, at any time, compute her Age, by a Circle contrived for that Purpose. And from the frequent Experience of Eclipses, both of the Sun and Moon, I also arrived to a true Notion of their Causes, being perfectly instructed in the borrowed Light of the latter from the former. So far proceeded my Astronomy.
And these constant Exercises in the Description of Lines, Circles, and Angles, brought me likewise to an Acquaintance with several Truths in speculative Geometry; for which Purpose, after my Paper was spent, I expunged the Lines out of my Dial by the Fountain, and drew all my Schemes upon that, inserting and blotting out at Pleasure. I perceived the Proportionality of the 
Sides of similar Figures, and the Equality between all the alternate (as well as between all the opposite) Angles made by the Intersection of the same Line with any Number of Parallels. I learned the Equality of the Angles in a Triangle to a Semicircle, and of any outward Angle in the same to the two inward opposite ones. I found out the 
Proportion which any Angle, at the Circumference of a Circle, bears to one at the Centre, standing upon the same Base. I discovered the Equality of the Areas of all Parallelograms upon the same Base between the same Parallels, which put me also upon studying the Proportions between other dissimilar Surfaces; and I likewise attained to the Knowledge of Pythagoras’s Theorem of the Equality between the Square of the longest Side in a right-angled Triangle, and the Sum of the Squares of the two shortest.
And as Lines led me to consider the Surfaces of which they were the Limits, so did these Surfaces bring me to an Acquaintance with the Solids bounded by them, for the right Understanding of which, I found great Helps from the actual Formation of several regular Bodies out of Clay, though the Draughts themselves, which I found of these in the Book, were adapted in the best Manner to assist the Imagination, as being delineated according to the exactest Rules of Perspective, and everywhere skilfully shaded. I observed the 
, which similar Surfaces, and the 
, which similar Solids, had 
their Altitudes, or Sides; and as I naturally pitched upon the Square for the Mensuration of the former, so did I apply the Cube for the Standard of these latter. I learned the chief Properties of the Sphere, the Cone, and the Cylinder, and became intimately conversant with the five regular Bodies. I formed an Ellipse from the oblique Section of a Cylinder, and had some Notion of the other Apollonian Curves,° from the actual Sections of a Cone.
I also made some Progress in the more abstract Theory of Numbers, as well as in the more sensible Speculation of Magnitude. I was necessarily led to that prime Distinction of Number, into Integer and 
, according as the Quantities they were brought to express were discontinu
, or continu, and I considered them likewise in respect of their Composition as Even and Odd, Prime and Composite, and compared them also together, as Commensura
and Incommensura. I deduced the principal Truths which belong to the Doctrine of Proportions, and arrived to some Skill in Progressions, both Arithmetical and Geometrical. But yet I would have you take notice that I attribute all those high Acquirements in Numbers, as well as Magnitudes, rather to the Assistance of the Book, than my own Invention alone. For when I had attained to the Knowledge of the numeral Figures, as you have already heard, I seldom failed, through the whole Work, to fish out the Author’s intentions by them, though I understood not the Meaning of a Word he wrote.
Notes
Brass Semicircle: Kirkby probably means something like a modern protractor for measuring angles.
Apollonian Curves: the conic sections; that is, the ellipse, parabola, and hyperbola produced by slicing a cone at various angles.
The Loves of the Triangles
John Frere’s spoof of Erasmus Darwin’s botanical poem The Loves of the Plants sheds a pleasing sidelight on the popular image of mathematics at the end of the eighteenth century; like Cavendish and the anonymous writer in Fun, Frere picks up on the incomprehensibility of mathematical jargon to outsiders. Frere was a diplomat and a poet and translator.
John Hookham Frere (1769–1846), The Loves of the Triangles. A Mathematical and Philosophical Poem. In The anti-Jacobin, 1798; republished in The Works (London, 1872), vol. I, pp. 88–93.
Stay your rude steps, or e’er your feet invade
The Muses’ haunts, ye sons of War and Trade!
Nor you, ye legion fiends of Church and Law,
Pollute these pages with unhallow’d paw!
Debased, corrupted, grovelling, and confined,
No Definitions touch your senseless mind;
To you no Postulates prefer their claim,
No ardent Axioms your dull souls inflame;
For you, no Tangents touch, no Angles meet,
No Circles join in osculation sweet!
For me, ye Cissoids, round my temples bend
Your wandering curves; ye Conchoids extend;
Let playful Pendules quick vibration feel,
While silent Cyclois rests upon her wheel;
Let Hydrostatics, simpering as they go,
Lead the light Naiads on fantastic toe;
Let shrill Acoustics tune the tiny lyre;
With Euclid sage fair Algebra conspire;
The obedient pulley strong Mechanics ply,
And wanton Optics roll the melting eye!
I see the fair fantastic forms appear,
The flaunting drapery, and the languid leer;
Fair sylphish forms—who, tall, erect, and slim,
Dart the keen glance, and stretch the length of limb;
To viewless harpings weave the meanless dance,
Wave the gay wreath, and titter as they prance.
Such rich confusion charms the ravish’d sight,
When vernal Sabbaths to the Park invite.
Mounts the thick dust, the coaches crowd along,
Presses round Grosvenor Gate th’impatient throng;
White-muslined misses and mammas are seen,
Link’d with gay cockneys, glittering o’er the green:
The rising breeze unnumber’d charms displays,
And the right ankle strikes th’ astonished gaze.
But chief, thou Nurse of the Didactic Muse,
Divine Nonsensia, all thy soul infuse;
The charms of Secants and of Tangents tell,
How Loves and Graces in an Angle dwell;
How slow progressive Points protract the Line,
As pendent spiders spin the filmy twine;
How lengthen’d Lines, impetuous sweeping round,
Spread the wide Plane, and mark its circling bound;
How Planes, their substance with their motion grown,
Form the huge Cube, the Cylinder, the Cone.
Figure 11.2. One Evening, as he viewed the sky, / Through his best tube, with curious eye. . . . (Combe, vol. 2, facing p. 38. © Princeton University Library. Rare Books Division, Department of Rare Books and Special Collections, 3686.7.332.)
Master Senex the Astronomer
William Combe is best known for the “Doctor Syntax” series of comic poems, an illustrated satire upon ideas of the “picturesque” which came toward the end of his long career as a writer, principally of satire and history. Syntax’s quixotic misadventures filled four installments, of which The English Dance of Death, from which this extract is taken, was the second. The writing is basically light-hearted, but “the astronomer” does not come across at all well.
William Combe (1742–1823), “The Astronomer” in The English Dance of Death (London, 1815), vol. 2, pp. 38–42.
He, who with care and much ado,
Has changed one blade of grass to two;
He, who an acre too has ploughed,
And with good seed that acre sowed;
He, who to the Earth has given
A Tree, to rear its boughs to Heaven,
And, with a chaste and loving wife,
Gives but a single babe to life,
Has, as ’tis said, by one whose name
Stands foremost on the roll of Fame,
Performed, in philosophic view,
All that a Man’s required to do.
This done, each social claim is paid,
And when in Earth his bones are laid,
The sculptured stone may truly tell
That he has lived and acted well.
But what says Science to the Rule
Thus taught in simple Nature’s school:
That Science which pursues her way,
Through gloomy night, or glaring day,
Creation’s every work explores,
Digs deep for all the hidden stores
Which the Earth’s darksome caves contain,
And dives within the watery main,
Expatiates through the fields of air,
And sees the storms engendered there,
Or boldly bids her daring eye
Explore the wonders of the sky;
While Genius, to no spot confined,
That brightest offspring of the mind,
Ranges at will, through Space and Time,
In every age, in every clime,
And oft, its glorious toil to crown,
Creates new Systems of its own.
—Such are the classes that embrace
Man’s social, cultivated Race;
And, as each acts the part assigned,
It helps, in due degree, to bind,
By harmonising, just control,
The general order of the whole.
Now Master Senex, who was bred
To guide into the youthful head,
Not that poor Two and Two make Four,
Or that three Twenties form Threescore,
But the nice, calculating play
Of Decimals and Algebra,
With Problems and the curious store
That’s found in Mathematic Lore,
He always felt himself at home
When ’mong the Stars he chose to roam,
And, for a frisk, would sometimes stray
Delighted in the Milky Way.
Would bask in the Meridian Noon,
And clamber Mountains in the Moon.
He would the Comet’s course pursue,
And tell, with calculation due,
How many million miles it posted,
While a small Leg of Mutton roasted,
And how many a thousand years
Will pass before it re-appears.
—He never for one moment thought
But of the Sciences he taught;
Him never did the Fancy seize
Of ploughing land, or planting trees;
Nor was the sober Sage beguil’d
To be the Father of a Child.
A Sister, an old saving Elf,
Who was as barren as himself,
Added a figure to the scene,
And dressed his meat, and kept him clean.
One Evening, as he viewed the sky
Through his best tube, with curious eye,
And ’mid the azure wilds of air
Pursued the progress of a Star,
A Figure seemed to intervene,
Which in the sky he ne’er had seen,
But thought it some new planet given,
To dignify his views of Heaven.
“O this will be a precious boon!
Herschel’s Volcanoes in the Moon,°
Are nought to this,” Old Senex said,
“My Fortune is for ever made.”
—“It is, indeed,” a voice replied;
The Old Man heard it—terrified,
And, as Fear threw him to the ground,
Through the long tube Death gave the wound.
Though Senex died, no thunder rolled,
No lightning flashed, no tempests growled;
Nor did the Pleiades descend,
In rain, to weep their faithful friend;
Nor would the Moon in sorrow shroud
Her silver light within a cloud;
Nay, not a single sigh was given
By any Star that shines in Heaven.
Note
Herschel’s Volcanoes in the Moon: William Herschel (1738–1822), the famous astronomer, published a paper on the mountains on the moon in the Philosophical Transactions of the Royal Society in 1780.
An Ode to the Mathematics
Alfred Domett had a varied career as a New Zealand politician and administrator, including a period as premier in 1862–1863. In youth and retirement he published volumes of verse in England; this poem is dated in 1830, when he was nineteen. Seemingly directed solely at the author’s own mathematical incapacity, it touches the theme of the incomprehensibility of mathematics which we have met elsewhere, without reaching the harshness of Combe or even Cavendish in condemning the subject itself.
Alfred Domett (1811–1887), “Ode to the Mathematics: A Cambridge Ebullition,” in Poems (London, 1833), pp. 52–53.
Ye Mathematics! over which I pore
Full stolidly—yet to my sorrow find
I cannot fix upon your crabbèd lore
My scape-grace, wandering, weak, wool-gathering mind—
Oh, are yet not in language plain, a bore,
For luckless wight like me a plague refined—
Ye intellectual catacombs—where drones
Of many an age have piled up musty bones!
We are old foes—yet can’t, it seems, be loosed
From one another, though we tug the chain
Like coupled hounds; I have so oft abused
And railed at you, and yet returned again
To be by your dark mysteries confused,
That at my fate I smile, and friendship fain
Would offer—foes are well-nigh friendly (trust ’em),
Whose regular abuse becomes a custom.
They say you lead to grand results, and Science
Makes you unto her heaven a Jacob’s ladder;
So clouded though, we cannot see the sky hence,
And black-gowned students are a vision sadder,
Nor promise half so much for what they spy hence,
As did the white-robed angels Jacob had a
Glimpse of; but be that as it may, you lead to
Things greater far than I can e’er give heed to.
You teach the stars—their courses—like Silenus,°
Teach what the world is set a-going by,
And all the eccentricities of Venus;
You compass earth and ocean, land and sky,
Teach us to argue and to squabble, wean us
From base delights (no doubt) to pure and high;
You teach mankind all, all that can ennoble ’em;
Meantime I’m staggered by this plaguy problem!
Note
Silenus: the mythological companion to the wine god Dionysos variously appeared in myths as tutor to Dionysos, adviser to King Midas, and a commentator on earthly rulers.
“Some Veritable Urania”
Augusta Jane Evans, popular novelist of the American South, produced some magnificent depictions of women intellectuals. Here, in what is surely one of the best such passages, she describes one heroine’s devotion to astronomy.
Augusta J. Evans (1835–1909), Macaria: or, Altars of Sacrifice (Richmond: 2nd edition, 1864), pp. 77–79.
She gave no intimation of having heard him.
“Irene, my child, it is one o’ clock.”
Without looking up she raised her hand toward the clock on the mantle, and answered, coldly:
“You need not sit up to tell me the time of night; I have a clock here. Go to sleep, Uncle Eric.”
He rested his shoulder against the door-facing, and, leaning on his crutches, watched her.
She sat there just as he had seen her several times before, with her arms crossed on the table, the large celestial globe drawn near, astronomical catalogues scattered about, and a thick folio open before her. She wore a loose wrapper, or robe de chambre, of black velvet, lined with crimson silk and girded with a heavy cord and tassel. The sleeves were very full, and fell away from the arms, exposing them from the dimpled elbows, and rendering their pearly whiteness more apparent by contrast with the sable hue of the velvet, while the broad round collar was pressed smoothly down, revealing the polished turn of the throat. The ivory comb lay on the table, and the unbound hair, falling around her shoulders, swept over the back of her chair and trailed on the carpet. A miracle of statuesque beauty was his queenly niece, yet he could not look at her without a vague feeling of awe, of painful apprehension; and, as he stood watching her motionless figure in its grand yet graceful pose, he sighed involuntarily. She rose, shook back her magnificent hair, and approached him. Her eyes, so like deep, calm azure lakes, crossed by no ripple, met his, and the clear, pure voice echoed through the still room.
“Uncle Eric, I wish you would not sit up on my account; I do not like to be watched.”
“Irene, your father forbade your studying until this hour. You will accomplish nothing but the ruin of your health.”
“How do you know that? Do statistics prove astronomers short-lived? Rather the contrary. I commend you to the contemplation of their longevity. Good-night, Uncle; starry dreams to you.”
“Stay, child; what object have you in view in all this laborious investigation?”
“Are you sceptical of the possibility of a devotion to science merely for science’s sake? Do my womanly garments shut me out of the Holy of Holies, debar me eternally from sacred arcana, think you? Uncle Eric, once for all, it is not my aim to
‘—brush with extreme flounce
The circle of the sciences.’
“I take my heart, my intellect, my life, and offer all upon the altar of its penetralia. You men doubt women’s credentials for work like mine; but this intellectual bigotry and monopoly already trembles before the weight of stern and positive results which women lay before you—data for your speculations—alms for your calculation. In glorious attestation of the truth of female capacity to grapple with some of the most recondite problems of science stand the names of Caroline Herschel, Mary Sommerville, Maria Mitchel, Emma Willard, Mrs. Phelps, and the proud compliment paid to Madame Lepaute° by Clairaut and Lalande, who, at the successful conclusion of their gigantic computations, declared: ‘The assistance rendered by her was such that, without her, we never should have dared to undertake the enormous labor in which it was necessary to calculate the distance of each of the two planets, Jupiter and Saturn, from the comet, separately for every degree, for one hundred and fifty years.’ Uncle Eric, remember
“ ‘—Whoso cures the plague,
Though twice a woman, shall be called a leech;
Who rights a land’s finances is excused
For touching coppers, though her hands be white.’ ”
She took the volume she had been reading, selected several catalogues from the mass, and, lighting a small lamp, passed her uncle and mounted the spiral staircase leading to the observatory. He watched her tall form slowly ascending, and, in the flashing light of the lamp she carried, her black dress and floating hair seemed to belong to some veritable Urania—some ancient Egyptic Berenice.° He heard her open the glass door of the observatory, then the flame vanished, and the click of the lock fell down the dark stairway as she turned the key. With a heavy sigh the cripple returned to his room, there to ponder the singular character of the woman whom he had just left, and to dream that he saw her transplanted to the constellations, her blue eyes brightening into stars, her waving hair braiding itself out into brilliant, rushing comets. The night was keen, still, and cloudless, and, as Irene locked herself in, the chill from the marble tiles crept through the carpet to her slippered feet. In the centre of the apartment rose a wooden shaft bearing a brass plate, and to this a telescope was securely fastened. Two chairs and an old-fashioned oaken table, with curious carved legs, comprised the furniture. She looked at the small siderial clock, and finding that a quarter of an hour must elapse before she could make the desired observation, drew a chair to the table and seated herself. She took from the drawer a number of loose papers, and prepared the blank-book for registering the observation, then laid before her a slate covered with figures, and began to run over the calculation. At the close of fifteen minutes she placed herself at the telescope, and waited patiently for the appearance of a small star which gradually entered the field; she noted the exact moment and position, transferred the result to the register, and after a time went back to slate and figures. Cautiously she went over the work, now and then having recourse to pen and paper; she reached the bottom of the slate and turned it over, moving one finger along the lines. The solution was wrong; a mistake had been made somewhere; she pressed her palm on her forehead, and thought over the whole question, then began again. The work was tedious, the calculation subtle, and she attached great importance to the result; the second examination was fruitless as the first; time was wearing away; where could the error be? Without hesitation she turned back for the third time, and commenced at the first, slowly, patiently threading the maze. Suddenly she paused and smiled; there was the mistake, glaring enough, now. She corrected it, and working the sum through, found the result perfectly accurate, according fully with the tables of Leverrier by which she was computing. She carefully transferred the operation from slate to paper, and, after numbering the problem with great particularity, placed all in the drawer and turned the key. It was three o’ clock; she opened the door, drew her chair out on the little gallery, and sat down, looking toward the East. The air was crisp but still, unswayed by current waifs; no sound swept its crystal waves save the low, monotonous distant thunder of the Falls, and the deep, cloudless blue ocean of space glowed with its numberless argosies of stellar worlds. Constellations which, in the purple twilight, stood sentinel at the horizon, had marched in majesty to mid-heaven, taken reconnoissance thence, and as solemnly passed the opposite horizon to report to watching gazers in another hemisphere. “Scouts stood upon every headland, on every plain”; mercilessly the inquisitorial eye of science followed the heavenly wanderers; there was no escape from the eager, sleepless police who kept vigil in every clime and country; as well call on Böotes to give o’er his care of Ursa Major, as hopelessly attempt to thrust him from the ken of Cynosura.° From her earliest recollection, and especially from the hour of entering school, astronomy and mathematics had exerted an overmastering influence upon Irene’s mind. The ordinary text-books only increased her interest in the former science, and while in New York, with the aid of the professor of astronomy, she had possessed herself of all the most eminent works bearing upon the subject, sending across the Atlantic for tables and selenographic charts which were not to be procured in America.
Under singularly favorable auspices she had pursued her studies perseveringly, methodically, and, despite her father’s prohibition, indefatigably. He had indulged, in earlier years, a penchant for the same science, and cheerfully facilitated her progress by rearranging the observatory so as to allow full play for her fine telescope; but, though proud of her proficiency, he objected most strenuously to her devoting so large a share of her time and attention to this study, and had positively interdicted all observations after twelve o’ clock. Most girls patronize certain branches of investigation with fitful, spasmodic vehemence, or periodic impulses of enthusiasm; but Irene knew no intermission of interest, she hurried over no details, and, when the weather permitted, never failed to make her nightly visit to the observatory. She loved her work as a painter his canvas, or the sculptor the marble one day to enshrine his cherished ideal; and she prosecuted it, not as a mere pastime, not as a toy, but as a life-long labor, for the labor’s sake.
Notes
Caroline Herschel, Mary Sommerville, Maria Mitchel, Mrs. Phelps, and Madame Lepaute: Women scientists and astronomers of the late eighteenth and nineteenth centuries. Emma Willard was a pioneer of women’s higher education.
Urania: the muse of astronomy. Berenice II, queen of Egypt in the third century BC, was the subject of a famous story in which her hair, cut off and placed in the temple of Aphrodite, disappeared and was said to have been placed among the stars. The constellation coma berenices still appears on star maps.
Böotes and Ursa Major are (adjacent) constellations, traditionally the heavenly transformations of Arcas and his mother Callisto; Cynosura is the pole star.
Fun
Founded in 1861 to compete with Punch, the London magazine Fun appeared weekly with a mix of satire, parody, and political, literary, and other reportage until 1901; its best-known contributor was W. S. Gilbert. The second of these passages is a response to J. J. Sylvester’s Laws of Verse, mentioned in Chapter 10.
Fun, Saturday, March 28, 1863, p. 19; Saturday, September 10, 1870, p. 98.
Mathematical Problems
Required—
To sit comfortably in a chair, the back and seat of which are at right angles.
To know when skating whether a plane 
hath length and breadth by personal experience.
To find the centre of a family circle when there is cold meat for dinner, and a few female friends of some benevolent society coming to tea in the evening.
To find the centre of gravity when Sothern° is performing at the Haymarket.
To feel gratified on seeing in the mirror over the drawing-room mantelpiece after (as you thought) a telling entré, the segment of a circle beautifully imprinted on your brow from the rim of your hat.
To find that sticking the fish-hook into the fleshy part of your arm is anything but an acute angle.
To shave with your razor at a right angle without cutting your throat, or that, if you do so partially, you think it a right angle.
Mathematical Poetry
To the Editor of Fun.
Sir, I see that Professor Sylvester, the famous mathematician, has written a book, entitled The Laws of Verse, to prove that poetry is as exact a science as mathematics, and that any fellow who has passed the Pons Asinorum° can write an epic, and that after all an algebraic equation is very much the same thing as a sonnet. Hurrah! will be the cry of our lads, who will naturally prefer taking their Euclid in the form of Don Juan, while the children will promote “Dickery, dickery dock” over the multiplication table.
But no matter! I have turned my attention to the subject, and although I have not seen the learned professor’s work, I think I have hit on the method. Here is my first poem:—
Of course this equation has to be worked out, but I will not occupy your valuable space by so doing, as it is a thing which any schoolboy—especially if he knows mathematics remarkably well—will effect with ease.
Or, to put it geometrically instead of algebraically, let ABC be a given line. From the centre B at the distance DEF describe a circle, whose square shall be equal to the squares of AB, BC together with twice the rectangle contained by BE, DF.
Anyhow, here is the poem:
I never knew a wild kodoo
Analysis, Zitella,
Tobacco-box, or kangaroo,
Or infantile umbrella
Could dance a jig
Like the learned pig,
Or even a Tarantella!
For why, I have never visited
The coasts of Willy-Nilly,
Or the cataract-head of Slugabed,
Or the billowy shores of Chili.
Nor sailed a spoon
To the crescent moon
With a cargo of picalilly!
So I tear my hair in blind despair,
And blister my eyes with peaches,
In anti-macassar trousers,°
And my soul I vex
With “n + x,”
And mathematical speeches.
This, if not according to Professor Sylvester, at least according to Euclid is mathematical verse, for it is composed of “lines, having length without breadth.”
Yours,
A Ninny to the Power of Nine.
Notes
Sothern: Edward Askew Sothern (1826–1881), comic actor. The Haymarket was (and is) a major London theatre.
Pons Asinorum: Proposition 5 of Book 1 of Euclid’s Elements: the angles opposite the equal sides of an isosceles triangle are equal.
In anti-macassar trousers: “There’s some mistake here, owing to the difficulty of ascertaining the value of the symbol 
” (footnote appearing in the original).
A Sight of Thine Interior
Edwin Abbott Abbott, Anglican priest, teacher, and writer, wrote works of theology, grammar, and history and edited the Essays of Francis Bacon, as well as serving for more than two decades as the inspiring headmaster of the City of London School. But he is best known as “A. Square,” the author of the science fiction classic Flatland, which recounts the adventures of an inhabitant of a two-dimensional world given glimpses of both higher and lower dimensions. The book combined sharply pointed satire and Christian teaching with serious mathematics and has inspired a multitude of other works: adaptations, sequels, and films.
In this passage, A. Square, intoxicated by his view of a three-dimensional world, begins to speculate about worlds with still more dimensions. His audacity does not go down very well, and in his own world it will eventually land him in prison.
A. Square [Edwin A. Abbott (1838–1826)], Flatland: A Romance of Many Dimensions (London, 1884), pp. 85–89.
The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me.
“Pardon me,” said I, “O Thou Whom I must no longer address as the Perfection of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior.”
SPHERE. “My what?”
I. “Thine interior: thy stomach, thy intestines.”
SPHERE. “Whence this ill-timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty?”
I. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me—O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend—some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed.
SPHERE. Pooh! Stuff! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland.
I. Nay, gracious Teacher, deny me not what I know it is in thy power to perform. Grant me but one glimpse of thine interior, and I am satisfied for ever, remaining henceforth thy docile pupil, thy unemancipable slave, ready to receive all thy teachings and to feed upon the words that fall from thy lips.
SPHERE. Well, then, to content and silence you, let me say at once, I would show you what you wish if I could; but I cannot. Would you have me turn my stomach inside out to oblige you?
I. But my Lord has shown me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines in Spaceland, and the intestines of every solid living creature, even of the noble and adorable Spheres.
SPHERE. But where is this land of Four Dimensions?
I. I know not: but doubtless my Teacher knows.
SPHERE. Not I. There is no such land. The very idea of it is utterly inconceivable.
I. Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot see that other higher Spaceland now, because we have no eye in our stomachs. But, just as there was the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there was close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant?
In One Dimension, did not a moving Point produce a Line with two terminal points?
In Two Dimensions, did not a moving Line produce a Square with four terminal points?
In Three Dimensions, did not a moving Square produce—did not this eye of mine behold it—that blessed Being, a Cube, with eight terminal points?
And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points?
Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord’s own words—“strictly according to Analogy”?
Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”?
O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a Fourth Dimension; but, if I am right, my Lord will listen to reason.
I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer.
SPHERE (AFTER A PAUSE). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business.
I. I was certain of it. I was certain that my anticipations would be fulfilled. And now have patience with me and answer me yet one more question, best of Teachers! Those who have thus appeared—no one knows whence—and have returned—no one knows whither—have they also contracted their sections and vanished somehow into that more Spacious Space, whither I now entreat you to conduct me?
SPHERE (MOODILY). They have vanished, certainly—if they ever appeared. But most people say that these visions arose from the thought—you will not understand me—from the brain; from the perturbed angularity of the Seer.
I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space with a wake of its own—shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth—
How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his commands of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through Space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness—which was now to become my Universe again—spread out before my eye. Then a darkness. Then a final, all-consummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace-Cry of my approaching Wife.
Scenes in the Life of Pythagoras
A development of the twentieth century was the light-hearted take on the “greats” of the mathematical past, and few were more irreverent than that of the remarkable schoolboy Nigel Molesworth, the superb comic creation of Geoffrey Willans and Ronald Searle. Willans was a modestly successful author of (mostly) humorous nonfiction, Searle one of the great cartoonists and illustrators of the twentieth century, his work ranging from harrowing depictions of conditions in World War II prisoner-of-war camps to the well-known belles of St. Trinian’s. Figure 11.3 shows his take on classical geometry.
The spelling is that of the original.
Geoffrey Willans (1911–1958) and Ronald Searle (b. 1920), Molesworth (London, 1992; originally published 1953), pp. 45–46.
Pythagoras as a mater of fact is at the root of all geom. Instead of growing grapes figs dates and other produce of greece Pythagoras aplied himself to triangles and learned some astounding things about them which hav been inflicted on boys ever since.
Whenever he found a new thing about a triangle Pythagoras who had no shame jumped out of his bath and shouted “Q.E.D.” through the streets of athens its a wonder they never locked him up.
To do geom you hav to make a lot of things equal to each other when you can see perfectly well that they don’t. This agane is due to Pythagoras and it formed much of his conversation at brekfast.
PYTHAGORAS (HELPING HIMSELF TO PORRIDGE): Hmm. I see the sum of the squares on AB and BC = the square on AC.
WIFE: Dear dear.
PYTHAGORAS: I’m not surprised, not surprised at all. I’ve been saying that would come for years.
WIFE: Yes dear.
PYTHAGORAS: Now they’ll hav to do something about it. More tea please. There’s another thing—the day is coming when they’re going to have to face the fact that a strate line if infinitely protracted goes on for ever.
Figure 11.3. A few lazy parrallelograms basking on Mount Olympus. Pythagoras stalking them. (Willans and Searle, p. 45.)
WIFE: Quite so.
PYTHAGORAS: Now take the angle a, for xsample.
(His wife sudenly looses control and thro the porridge at him. Enter Euclid: another weed and the 2 bores go off together)
Bao Suyo
It seems appropriate to end this anthology with a passage set in the distant future. Kim Stanley Robinson’s Mars trilogy envisages the colonization of Mars over a period of more than two centuries, beginning in the 2020s, and the passage reproduced here about Bao Suyo, “the first queen of physics,” takes place in the twenty-third century. (Sax Russell, the other character in this extract, is one of the—now extremely elderly—original Martian colonists.) If it bears a relationship with the extracts from science fiction of the seventeenth (The Blazing-World) and eighteenth (Automathes) centuries we have seen, it also draws something from the other discussions and representations of mathematics and mathematicians—Kanigel’s Ramanujan, for instance—that make up this book and brings together some of the particular concerns of the twentieth century: the odd beauty of mathematics and the abiding fear that it may turn out to be a “house of cards.”
Kim Stanley Robinson (b. 1952), Blue Mars (London, 1996), pp. 432–438. Reprinted by permission of HarperCollins Publishers Ltd. © 1996 Kim Stanley Robinson.
Interesting in a different way was the fact that one of the leading theorists in this new stage of development was working right there in Da Vinci, part of the impressive group Sax was sitting in on. Her name was Bao Suyo. She had been born and raised in Dorsa Brevia, her ancestry Japanese and Polynesian. She was small for one of the young natives, though still half a metre taller than Sax. Black hair, dark skin, Pacific features, very regular and somewhat plain. She was shy with Sax, shy with everyone; she even sometimes stuttered, which Sax found extremely endearing. But when she stood up in the seminar room to give a presentation, she became quite firm in hand if not in voice, writing her equations and notes on the screen very quickly, as if doing speed calligraphy. Everyone in these moments attended to her very closely, in effect mesmerized; she had been working at Da Vinci for a year now, and everyone there smart enough to recognize such a thing knew that they were watching one of the pantheon at work, discovering reality right there before their eyes.
The other young turks would interrupt her to ask questions, of course—there were many good minds in that group—and if they were lucky, off they all would all go together, mathematically modelling gravitons and gravitinos, dark matter and shadow matter—all personality and indeed all persons forgotten. Very productive, exciting sessions; and clearly Bao was the driving force in them, the one they relied on, the one they had to reckon with.
It was a bit disconcerting. Sax had met women in math and physics departments before, but this was the only female mathematical genius he had ever even heard of, in all the long history of mathematical advancement, which, now that he thought of it, had been a weirdly male affair. Was there anything in life as male as mathematics had been? And why was that?
Disconcerting in a different way was the fact that areas of Bao’s work were based on the unpublished papers of a Thai mathematician of the previous century, an unstable young man named Samui, who had lived in Bangkok brothels and committed suicide at the age of twenty-three, leaving behind several “last problems” in the manner of Fermat, and insisting to the end that all of his math had been dictated to him by telepathic aliens. Bao had ignored all that and explained some of Samui’s more obscure innovations, and then used them to develop a group of expressions called advanced Rovelli-Smolin operators, which allowed her to establish a system of spin networks that meshed with superstrings very beautifully. In effect this was the complete uniting of quantum mechanics and gravity at last, the great problem solved—if it were true. And true or not, it had been powerful enough to allow Bao to make several specific predictions in the larger realms of the atom and the cosmos; and some of these had since been confirmed.
So now she was the queen of physics—the first queen of physics—and experimentalists in labs all over were online to Da Vinci, anxious to have more suggestions from her. The afternoon sessions in the seminar room were invested with a palpable sense of tension and excitement; Max Schnell would start the meeting, and at some point call on Bao; and she would stand and go to the screen at the front of the room, plain, graceful, demure, firm, pen flying over the screen as she gave them a way to calculate precisely the neutrino mass, or described very specifically the ways strings vibrated to form the different quarks, or quantized space so that gravitinos were divided into three families, and so on; and her colleagues and friends, perhaps twenty men and one other woman, would interrupt to ask questions, or add equations that explained side-issues, or tell the rest of them about the latest results from Geneva or Palo Alto or Rutherford; and during that hour, they all knew they were at the centre of the world.
And in labs on Earth and Mars and in the asteroid belt, following her work, unusual gravity waves were noted, in very difficult, delicate experiments; particular geometric patterns were revealed in the fine fluctuations in the cosmic background radiations; dark matter WIMPs and shadow matter WISPs were being sought out; the various families of leptons and fermions and leptoquarks were explained; galactic clumping in the first inflation was provisionally solved; and so on. It seemed as if physics might be on the brink of the Final Theory at last. Or at least in the midst of the Next Big Step.
Given the significance of what Bao was doing, Sax felt shy about speaking to her. He did not want to waste her time on trivial things. But one afternoon at a kava party, out on one of the arc balconies overlooking Da Vinci’s crater lake, she approached him—even more shy and stumbling than he was—so much so that he was forced into the very unusual position of trying to put someone else at ease, finishing sentences for her and the like. He did that as best he could, and they stumbled along, talking about his old Russell diagrams for gravitinos, useless now he would have thought, though she said they still helped her to see gravitational action. And then when he asked a question about that day’s seminar, she was much more relaxed. Yes, clearly that was the way to put her at ease; he should have thought of it immediately. It was what he liked himself.
After that, they got in the habit of talking from time to time. He always had to work to draw her out, but it was interesting work. And when the dry season came, in the autumn helionequinox, and he started going out sailing again from the little harbour Alpha, he asked her haltingly if she would like to join him, and they stuttered their way through a deeply awkward interaction, which resulted in her going out with him the next nice day, sailing in one of the lab’s many little catamarans.
When day sailing Sax stayed in the little bay called the Florentine, southeast of the peninsula, where Ravi Fjord widened but before it became Hydroates Bay. This was where Sax had learned to sail, and where he still felt best acquainted with the winds and currents. On longer trips he had explored the delta of fjords and bays at the bottom end of the Marineris system, and three or four times he had sailed up the eastern side of the large Chryse Gulf, all the way to Mawrth Fjord and along the Sinai Peninsula.
On this special day, however, he confined himself to the Florentine. The wind was from the south, and Sax tacked down into it, enlisting Bao’s help at every change of tack. Neither of them said much. Finally, to get things started, Sax was forced to ask about physics. They talked about the ways in which strings constituted the very fabric of spacetime itself, rather than being replacements for points in some absolute abstract grid.
Thinking it over, Sax said, “Do you ever worry that work on a realm so far beyond the reach of experiment will turn out to be a kind of house of cards—knocked over by some simple discrepancy in the math, or some later, different theory that does the job better, or is more confirmable?”
“No,” Bao said. “Something so beautiful as this has to be true.”
“Hmmm,” Sax said, glancing at her. “I must admit I’d rather have something solid crop up. Something like Einstein’s Mercury—a known discrepancy in the previous theory, which the new theory resolves.”
“Some people would say that the missing shadow matter fits that bill.”
“Possibly.”
She laughed. “You need more, I can see. Perhaps some kind of thing we can do.”
“Not necessarily,” Sax said. “Although it would be nice, of course. Convincing, I mean. If something were better understood, so that we could manipulate it better. Like the plasmas in fusion reactors.” This was an ongoing problem in another lab at Da Vinci.
“Plasmas might very well be better understood if you modelled them as having patterns imposed by spin networks.”
“Really?”
“I think so.”
She closed her eyes—as if she could see it all written down, on the inside of her eyelids. Everything in the world. Sax felt a piercing stab of envy, of—loss. He had always wanted that kind of insight; and there it was, right in the boat beside him. Genius was a strange thing to witness.
And they went back to talking about the new results from CERN; about weather; about the sailboat’s ability to point to within a few degrees of the wind. And then the following week she went out with him again, on one of his walks on the peninsula’s seacliffs. It was a great pleasure to show her a bit of the tundra. And over time, taking him through it step by step, she managed to convince him that they were perhaps coming close to understanding what was happening at the Planck level. A truly amazing thing, he thought, to intuit this level, and then make the speculations and deductions necessary to flesh it out and understand it, creating a very complex powerful physics, for a realm that was so very small, so very far beyond the senses. Awe-inspiring, really. The fabric of reality. Although both of them agreed that just as with all earlier theories, many fundamental questions were left unanswered. It was inevitable. So that they could lie side by side in the grass in the sun, staring as deeply into the petals of a tundra flower as ever one could, and no matter what was happening at the Planck level, in the here and now the petals glowed blue in the light with a quite mysterious power to catch the eye.