9 February 1991
In the Summer of 1990 an assistant at the Bodleian Library in Oxford, England, was repairing the binding of a Jacbean psaltery. She noticed that the covers had been stiffened during an an earlier repair by gluing several layers of paper together. Using a new experimental technique, she managed to dissolve the glue and separate the sheets without destroying their contents. The papers turned out to be a hitherto unknown manuscript by Jonathan Swift, part of a draft of Gulliver's Travels: an episode on the flying Island of Laputa that does not appear in the published version. Speculation is rife among Swiftean scholars: was the passage excised by his editor, or did the author omit it for his own reasons? Mathematical Recreations has been fortunate to obtain a copy of the document, and invites readers to judge the material for themselves.
Now that my proficiency in the Laputan tongue had attained tolerable proportions, the king, assisted by several flappers, invited me to inspect his private academy, modelled upon the much larger one in Lagado, on the mainland of Balnibarbi. I made haste to accept, out of curiosity as much as politeness, for I had heard many tales about the academy and its professors, all strange, and some, I judged, incredible.
There was one man who had for sixteen years been working upon what he named a great-grandfather clock. It was in the manner of a grandfather clock, but with a double pendulum --- the better, so its maker assured me, to tell the time. Originally the intention had been to suspend the second pendulum from the end of the first, in as simple a manner as possible. However, certain fine points of the theory not being borne out in practice, the inventor had perforce added a spring here, a counterbalancing weight there, so that over the sixteen-year period of development the machine's complexity had greatly exceeded that originally envisioned (Fig.1). I asked how accurately the clock performed, and was told that it was correct twice each day.
1 Mechanism of the great-grandfather clock. Without the springs and elastic it would have exactly four equilibrium positions. Do at least that many exist if they are present?
The clockmaker became quite friendly when I complimented him on this excellent standard of performance, and he shewed me the many mathematickal calculations that determined the machine's design. While I am unable to recall them all, one has remained stubbornly in my memory. A fact central to the successful operation of the clock was that its pendulums should on occasion be at rest, so that the combined action of the multifarious forces, springs, elastic threads, ropes, and sundry paraphernalia with which it was bedecked, should come to perfect balance. The simplicity of the original design, he informed me, made such calulations both elegant and complete: the pair of pendulums could balance in precisely four positions.
I understood at once that if both pendulums were to hang vertically downwards, then they would remain forever at rest in that position. However, I begged to dispute the existence of three other such positions. At this the clockmaker called in several of his pupils --- ill-clad creatures with sour dispositions who carried out his bidding only reluctantly and with many mistakes --- to assist in his demonstrations. With the expenditure of much effort, and several whippings, he lead me to understand that a second such position was possible, with both pendulums pointing vertically upward. I admitted that such a configuration had not occurred to me, and upon being informed that in principle such an arrangement could be made to balance, I observed that in principle a monk may balance an eel on the end of his nose, but that such behaviour is seldom seen either in the fishmarkets or the monasteries. But he persisted, describing the arrangement as a thelmin frole, which after consultation with mathematickal acquaintances I now translate as 'unstable equilibrium', though at the time I believed him to have said that it was 'fiendishly unlikely', a sentiment with which I had hastened to agree. Having understood this, I was quick to deduce the remaining two equilibria, in each of which one pendulum hangs vertically downward, the other vertically upward; but in one it is the first pendulum that hangs down and the second that points up, whereas in the other, their positions are reversed.
The sole obstacle to completion of the project, he said, was to establish the existence of four or more comparable configurations for the actual apparatus, springs and weights and all. The precise positions were unimportant: all that was required was that they should exist. But the mathematicks was proving impenetrable, and he despaired of ever attaining a successful conclusion. At that point I was rescued by a messenger from the king, who bade me repair to the kitchens for a meal of cycloid pudding and roast cones, before continuing my visit.
There was a professor of astronomy, who sought to capture moonbeams in a jar, to light buildings. I enquired why he did not also try to capture sunbeams, which were both more powerful and more common, but he became angry, saying that sunbeams would work only in the daytime, when it was light in any case.
I was shown something of which the king was very proud, a geodetickal survey of the entire island of Laputa. The Surveyor Royal was a ruddy-visaged person of huge girth, who ever carried with him a plumbline and bob as a badge of office. His task, he told me, was to catalogue every hill, every valley, and every pass upon the island. I enquired as to the precise definition of these terms, wishing to apprehend the exact nature of the enterprise. Was an anthill, for example, accounted as a hill? He said proudly that it was; that a hill was any prominence whose height exceeded that of its immediate vicinity: it was a blugnibb floom, or 'local maximum'. A valley, correspondingly, was a blignubb floom, or 'local minimum'. The concept of a pass was more subtile: a place that was a local maximum in one direction but a local minimum in another, akin to the saddle of a horse, and thus known as a zadul,which is Laputan for 'horse', though my mathematickal acquaintances tell me that the term 'saddle' is now more favoured in this country (Fig.2).
2 The object of the Royal Geodetickal Survey: to count the hills, valleys, and passes on the island of Laputa.
The enumeration of these features, he explained, was performed to the utmost precision --- so that, for example, if a small anthill were to appear upon the side of a larger anthill, then they would be counted as two hills together with a pass separating them. By this reckoning, he said proudly, there were exactly one thousand, two hundred, and sixty-seven hills in Laputa; and one thousand, five hundred, and six valleys. At this juncture I interposed a comment, that there must perforce be two thousand, seven hundred, and seventy-one passes.
Nay, quoth he: the Royal Geodetickal Survey has mapped precisely one thousand, nine hundred, and forty-four passes.
Why then, I replied: some have been omitted, for there is a mathematickal relationship between the three numbers, thus: if H be the number of hills, V the number of valleys, and P the number of passes, then always
H+V-P = 2.
Thus, I informed him, there must be a deficit in the number of passes, amounting to the sum of eight hundred and twenty-seven; always presuming that the number of hills and valleys should be correct, in which assumption the circumstances did not lead me to place much confidence. The proof of it, I informed the Royal Surveyor, is both general and conclusive [Editor's note: see BOX 1].
BOX 1 Gulliver's Formula
The theorem to be proved is that for any closed, smooth surface, the numbers H of local maxima, V of local minima, and P of saddlepoints, always satisfy the equation
H+V-P = 2.
The idea of the proof is to deform the surface, continually reducing these numbers, in such a way that the expression H+V-P remains unchanged. The deformation consists of a series of moves, in each of which either a hill or a valley is merged with a neighbouring pass, so that both disappear. The process continues until all passes have been eliminated, after which there can remain only one hill and one valley (since if there are two hills, or two valleys, there must be a pass somewhere 'between' them).
Merging a hill and a pass decreases both H and P by 1, so H+V-P remains unchanged. Similarly merging a valley and a pass decreases both V and P by 1, so H+V-P again remains unchanged.
3 Deforming a surface to reduce the number of hills, valleys, and passes, by cancelling them in pairs.
Fig.3 shows the effect of a sequence of such moves. When the deformation is completed, H = V = 1 and P = 0, so H+V-P = 1+1-0 = 2. Since H+V-P is unchanged throughout the sequence of deformations, the value of H+V-P at the beginning must also be 2.
I was surprised that such a simple remark could cause such argumentation. The room was lined with cupboards and shelves of crudely hewn but stout oak, and from these the Surveyor Royal produced more than two score maps in support of his contention. He paraded his subordinates by the dozen to swear to the accuracy of their methods, instruments, and observations. However, piece by piece, discrepancies began to appear, and shortly the Surveyor Royal announced that owing to a small oversight the number of passes must be augmented slightly, to become two thousand, seven hundred, and seventy two; the numbers of hills and valleys remaining unchanged.
I applauded his diligence, but ventured to remark that there remained a discrepancy of one. Either he had overestimated the number of passes, or a hill or a valley had been overlook'd. He protested that one cannot, by its nature, overlook a hill; but admitted that by the same token virtually any valley might be overlook'd. Then he became greatly excited, and took me before yet another professor of the academy, an historian. I confess that less likely a personage for such an office I had never before clapp'd eyes on, for he could scarce remember his own name from one second to the next, and he was perpetually forgetting where he had placed his eyeglasses. However, by dint of great application on his part and even greater patience on mine, a semblance of a tale began to unfold.
I was not, it seem'd, the first sea-captain to visit Laputa. One Captain Kidd, a pyrate by trade, was rumoured to have buried a considerable treasure somewhere upon the island, 'at the bottom of its deepest valley'. Despite repeated searches, no such treasure had been found; but perhaps my contention that a valley might have been overlook'd would resolve the mystery. I protested that my method pointed only to its existence: it did not commit itself to the precise situation of the missing valley. Upon thinking the matter over, however, I realised that the 'deepest valley' on Laputa could only be the very lowest point on the underside of the island, which was smooth and gently rounded, like a meat dish. Discreet enquiries confirmed that the Royal Geodetickal Survey had not included the underside of the island, and the discrepancy in valleys was resolved to my own satisfaction. I vowed to inspect the nethermost region of the island for myself, and pretended to dismiss the matter from my thoughts.
As a distraction I asked whether any natural stone arches existed on Laputa. After some hours the Professor of Geology was located. Indeed, he told me, there are many stone arches: the Arches Royal Monument is famous for having a great number. He did not know how many exactly. I returned to the Surveyor Royal to inform him that his observations must still be at fault. My proof that H+V-P = 2 had assumed the absence of holes in the island. On any surface that, like stone arches, possesses holes, the relation between the numbers H, V, and P must be different from the one that I had previously thought to have established. Now H+V-P = 2-2g where g is the number of holes [Editor's note: see BOX 2]. A truly accurate survey, establishing the number of hills, valleys, and passes without error, would make it possible to deduce the number g of stone arches. For example, if there were a million hills, a million valleys, and two million and twenty passes, then the relation 1000000+1000000-2000020 = 2-2g would hold, whence 2g = 22, so that there would exist precisely 11 stone arches. I do not believe that the Surveyor Royal was pleased by these revelations, but he pledged on the spot to repeat the survey in total accuracy, perhaps because His Majesty had begun to take a keen interest in the proceedings.
My mind had never been far from Capt'n Kidd's treasure. At the centre of Laputa there is a deep cavern, called Flandona Gagnole or 'the Astronomers' Cave', which I had been shewn earlier in my visit. Within this cavern is a lodestone of prodigious size, sustained upon an axle of adamant. By means of the lodestone, the island is made to rise and fall, and is conveyed to different parts of the monarch's dominions. It occurred to me that Capt'n Kidd might well have buried his treasure beneath the lodestone, this being as near as practicable to the island's nethermost point. I resolved to dig for it, and to this end secured a spade from the king's gardens. Regrettably, before my tunnel had been dug more than a few yards, I was apprehended, and imprison'd for violating a sacred place.
BOX 2 A topological refinement
The argument in BOX 1 assumes the surface is topologically equivalent to a sphere. However, a general surface is equivalent to a sphere with a number g of holes bored through it. The value of g is the genus of the surface. For any closed, smooth surface of genus g, the numbers H of local maxima, V of local minima, and P of saddles, satisfy the equation
H+V-P = 2-2g.
The proof is the same as in BOX 1, but the deformation ends with Fig.4. Here H = 1, V = 1, P = 2g, because each hole yields two passes. So H+V-P = 1+1-2g = 2-2g.
4 End of the deformation procedure for a surface of genus g.
There are many other theorems that relate multidimensional analogues of maxima, minima, and saddles to topology. They form an area of mathematics known as Morse Theory after its inventor, the American mathematician Marston Morse.
I lay in chains three days, and then was taken before the king, who belaboured me mightily for my infrigment upon his hospitality. At length I prevailed upon him to hear my plea of mitigation, and I revealed my deduction that a great pyrate treasure was to be found on the underside of Laputa. The king, eventually convinced of the soundness of my reasoning, then convened a meeting of the entire academy, the better to decide, how to go about finding the treasure.
One professor asserted that my plan, to tunnel downwards from the Astronomers' Cave, was sound in principle, but better carried out employing an instrument of his own devising, a carriage shaped like a corkscrew, drawn by ten thousand worms. However, before he had finished tethering the worms to the carriage, a great swarm of crows descended and devoured them. Another proposed constructing a wooden bridge, placed beneath the island like a sling, but when the trick was tried, the ropes were unable to sustain the weight. A third suggested reversing the lodestone, thus causing Laputa to roll on to its back like a beetle molested by a small boy. Upon it being pointed out that all the island's inhabitants, and their houses, cattle, dogs, cats, and wives would thereby be caused to fall off, a fourth opined that his secret recipe for a strong glue, made by boiling the legs of wasps, would provide the necessary security.
I enquired of the king how I might gain my freedom, and he promised that I would be released if I assisted him in some important task. I had noticed that it seem'd not to occur to any of the learned professors that the simplest solution would be to lower Laputa to within a few yards of the ground, and erect a ladder. I considered proposing this myself, to win my freedom, but became apprehensive lest I be sent to try it, and the island lowered upon my head by accident. Instead, I conceived a curious notion concerning the great-grandfather clock to which I was first introduced.
The possible configurations of the machine are defined by two angles, those of the two pendulums. These two angles naturally correspond to the points on a particular surface, namely a torus (Fig.5a), which I shall call the configuration space of the machine. This torus has one hole, hence its genus is g=1. Thus, however the torus is arranged in space, the number H+V-P always vanishes.
5 Configuration space of the great-grandfather clock, and its associated energy surface, with equilibrium positions marked.
When a spring is compressed, or a weight raised, it requires the expenditure of considerable quantities of energy. Thus, associated with any configuration of the mechanism of the great-grandfather clock, there is a mathematical quantity, the total energy. Imagine each point of the toroidal surface displaced to a height equal to the energy of the corresponding configuration, and let this be named the energy surface (Fig.5b). It is well known is staticks that equilibrium configurations correspond to positions of stationary energy, that is, to hills, valleys, and passes on this surface. Thus the total number of equilibria is H+V+P. Now, every finite surface must have a highest point, hence at least one hill, and a lowest, hence at least one valley, whence H is at least 1, and V is at least 1. Since H+V-P = 0, it follows that P = H+V, and therefore P is at least 2. Finally, the number of equilibria H+V+P is at least 1+1+2 = 4, which is the result required by the clockmaker.
My method does not determine where these equilibria are, but it successfully provides the desired lower bound for their number. I remark that the conclusion is quite independent of the arrangement of springs, ropes, weights, and other embellishments: it depends solely upon the number of holes in the configuration space. I hastened to explain my reasoning to the assembled professors, and after considerable and largely ill-informed debate my proof was adjudged sound, albeit outlandish. The king agreed to honour his promise, the lodestone was tilted appropriately, and Laputa was lower'd towards the ground. Soon it hung a few yards above Balnibarbi's dark soil, and a ladder was readied.
I was about to step on to the ladder, when the Surveyor Royal, harassed and red of visage, begged audience to inform the king that the Second Royal Geodetick Survey had been completed. The numbers, he said, scowling darkly in my direction, were:
Hills H = 1893
Valleys V = 1942
Passes P = 3816.
The king seem'd pleas'd to hear the news. By my reckoning, however, the number g of stone arches must perforce be given by
2-2g = 1893+1942-3816 = 19,
so that Laputa possesses minus eight-and-a-half arches. Before the king could become apprised of this fact, I bowed deep, took my leave, and descended the ladder, kissing the ground upon reaching it, even though it was inches deep in mud. Despite the fear of being crushed, I was tempted to inspect the underside of the island, but upon hearing a great commotion from above, I departed in haste towards the distant metropolis of Lagado.
FURTHER READING
P.A.Firby and C.F.Gardiner, Surface Topology, Ellis Horwood, Chichester 1982.
H.B.Griffiths, Surfaces, Cambridge University Press, Cambridge 1976.
Jonathan Swift, Gulliver's Travels and Other Writings, (ed. Louis A. Landa), Oxford University Press, Oxford 1976.