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Macmillan Childrens Publishing Group

Timing of Biological Clocks

Arthur T. Winfree

Henry Holt and Co.


The Timing of Biological Clocks

Chapter One
Time Zones
... it was Thursday, which was a great cause of wondering to us, since with us it was only Wednesday.
9 JULY 1522
A hardy little ship rides at anchor off the Canary Islands in midsummer of 1522. Fernão de Magalhaes' (Magellan's) westward expedition to circumnavigate the earth has limped home, its captain buried long ago in a world previously unknown to Europeans, its original crew of hundreds reduced by hardship and disaster to 31 ailing men, all aged beyond their years. As he pens almost the last of three years' daily entries, a warm offshore breeze ruffles the pages of Antonio Pigafetta's logbook:
In order to see whether we had kept an exact account of the days, we charged those who went ashore to ask what day of the week it was, and they were told by the Portuguese inhabitants of the island that it was Thursday, which was a great cause of wondering to us, since with us it was only Wednesday. We could not persuade ourselves that we were mistaken; and I was more surprised than the others, since having always been in good health, I had every day, without intermission, written down the day that was current.
Thus did sixteenth-century Europeans first encounter a phenomenon that excited them with the wonder we feel today about the intertwining of space and time in relativity. They eventually came to accept that this quirk of time was a consequence of traversing space, just as they had to accept that you can sail west and return from the east: everybody knew it in theory, but few were quite prepared to find that it had become a commercial reality.
As global commerce expanded, the paradox of the vanishing day became more and more practically annoying. More than three centuries after Pigafetta's return, the young Charles Dodgson (the future Lewis Carroll, author of Alice in Wonderland) suggested what to do about the paradox. A clergyman's son, he amused himself, family, and friends by writing a humorous periodical called The Rectory Umbrella. About 1850 it carried the following whimsical summary of Pigafetta's Paradox: 1
Half of the world, or nearly so, is always in the light of the sun: as the world turns round, this hemisphere of light shifts round too, and passes over each part of it in succession.
Supposing on Tuesday it is morning at London; in another hour it would be Tuesday morning at the west of England; if the whole world were land we might go on tracing Tuesday morning, Tuesday morning all the way round, till in 24 hours we get to London again. But we know that at London 24 hours after Tuesday morning it is Wednesday morning. Where then, in its passage round the earth, does the day change its name? Where does it lose its identity?
Practically there is no difficulty in it, because a great part of its journey is over water and what it does out at sea no one can tell: and besides there are so many different languages that it would be hopeless to attempt to trace the name of any one day all round. But is the case inconceivable that the same land and the same language should continue all round the world? I cannot see that it is: in that case either there would be no distinction at all between each successive day, and so week, month etc. so that we should have to say "the Battle of Waterloo happened to-day, about two million hours ago," or some line would have to be fixed, where the change should take place, so that the inhabitant of one house would wake and say "heigh ho! Tuesday morning!" and the inhabitant of the next, (over the line,) a few miles to the west would wake a few minutes afterwards and say "heigh ho! Wednesday morning!" What hopeless confusion the people who happened to live on the line would always be in, it is not for me to say. There would be a quarrel every morning as to what the name of the day should be. I can imagine no third case, unless everybody was allowed to choose for themselves, which state of things would be rather worse than either of the other two.
Thus today we have the international date line, relegated like an embarrassing error correction in 1884 to a meridian as far as possible from Greenwich, England. We also have astronauts circling the globe in 90minutes, whose travels show us Pigafetta's paradox in its most extreme form. If an astronaut were to reset his wristwatch every 4 minutes as he crosses another time zone from west to east, he would gain nearly a full day in the 90 minutes it takes to get from Hawaii back to Hawaii. But when he crosses that line in the middle of the Pacific, he nominally pops back 24 hours into "yesterday," so he arrives back over Honolulu 90 minutes later today, not tomorrow. Each new date starts along the date line as it rolls through midnight. The international date line leads an expanding crescent of tomorrow around the east side of the earth into the dawn light, around through noon and dusk, and back to midnight. When the date line reaches the midnight point it starts the next day. Sothe citizens of Tafahi, in the Tonga Islands, are among the first to report back to work after each weekend. Meanwhile, their neighbors on Savai'i, in nearby Western Samoa, 150 miles to the east, greet the same moment as a Sunday morning, exactly as Lewis Carroll foresaw.
By this device, Europeans accommodated intellectually to the fact that the earth is a rotating ball. Merely intellectual accommodation was sufficient for that era, but later the fact also turned out to have physiological consequences that could not be dealt with so easily. Pigafetta was not their discoverer. At the end of his trip he remained exactly one day behind the rest of Europe, there being no way known at the time for him to catch up other than circumnavigating the globe again in the opposite direction. This full day's lag may have bewildered Pigafetta's mind but it did not affect his body: a gradually acquired shift in time by exactly one day has no known consequences for our physiological routines. Pigafetta functioned normally on local time even if his log showed the wrong day.
No individual noticeably felt the effects of shifting time zones until about 400 years later, when in 1931 Wiley Post flew eastward around the world in eight days. It turns out that your body cannot adjust to changing time zones much faster than two hours a day, as though your skin can travel at arbitrary speed but your insides are limited to about 100 miles an hour. By flying across time zones much faster than that, Post discovered that he had an internal clock; he recognized the adverse effects of time zone displacement on his flying proficiency, and he struggled to evade them. He was the first man to experience jet lag, that disconcerting sensation of time travelers that their organs are strewn across a dozen time zones while their empty skins still forge boldly into the future, be that tomorrow or yesterday on the local calendar.
Post had a clock, or we might say, was a clock because he evolved on this planet. We live on a rotating planet. We grew up here, straight out of the warm Archeozoic seas where molecules first assembled into genes, and genes into species. For three billion years, life here has grown and adapted, passing from cell to cell innumerable times in unbroken descent, generation after generation. We and all other living things are the aggregate of all the changes made in that descent. All the while, we've felt the sky brighten and darken again and again while the planet relentlessly rotated: a trillion cycles of brightness and dark, of warmth and chill, never missing a beat, always felt deep in the chemical essence of what we are. We are well adapted to the pervasive monotony of sunrise and sunset, to the steady tone of a planet tirelessly spinning.
What would a trillion cycles sound like? Like high C for 400 years. Little wonder, then, that we've grown used to it, that we harmonize deeply with that unending note. Little wonder, too, if, as we first set foot beyond our home and venture out among the stars, we still hum with the pitch of our homeland. Anyone we meet out there will know where we came from, not just by our carbon and our water and the colors we see best, but also by the approximately 24-hour pitch of all that we do.
Keeping Time with the Earth
Physiological time, like local time on a rotating world, also has a circular character. Resetting any clock, internal or external, by one or several full cycles has no observable effect. But resetting a biological clock within a cycle does have physiological consequences, as the phenomenon of jet lag shows. Shifts within the cycle are called shifts in phase, the position of a repetitive process in its cycle (for example, a phase of the moon).
Quite apart from the new experience of jet travel, some means to reset the phase of a biological clock is made necessary by the slight mismatchbetween that clock's native period and the period of the earth's rotation. A discrepancy of an hour or so one way or the other is usual among the many species whose internal clocks have been timed with care. The human body clock, for example, has a period close to 25 hours. An hour's discrepancy is only 4 percent of 24 hours, a seemingly reasonable tolerance. This close adherence to the period of the earth's days has prompted the name circadian for this class of biological clocks, from the Latin circa (about) and dies (a day).
There is nothing so absurd but some philosopher has said it.


The fact that living things on this rotating planet harbor internal clocks is easy to believe. This is surely one reason that the pseudoscientific Biorhythm theory has gained widespread acceptance, even among well-educated people. Its basis is the claim that we all have cycles comparable to a woman's menstrual cycle, but much more precise: physical ability waxes and wanes every 23 days, emotional condition varies in a 28-day cycle, and intellectual performance may follow a weaker 33-day cycle.

The three periods are the same for everyone, regardless of age, sex, or medical condition. Each period is an exact integer number of days, as though some inner mechanism were counting day/night cycles. Otherwise the theory would have to be dismissed a priori: the delicate timing of two-and threefold coincidences would be drastically undermined over the course of a long life if the three periods were numbers such as 28.02 ± 0.04 days (depending on the individual, vicissitudes of health, travel, and so on) rather than 28 discrete clicks of the earth-sun clock. Anyway, the putative cycles start on the day of birth, whether chosen by the Caesarean surgeon or by the baby and mother in the usual way, whether or not the process spanned a midnight. Presumably the effective day of birth for Biorhythmic purposes must be offset by the cumulative number of date line crossings in a lifetime. Individuals who have missed a lot of days while exploring caves, living above the Arctic Circle, or working in submarines would presumably also need to make an appropriate correction to their actual birth dates. It is not clear what should be done if you move to a substantially different time zone.

A day when two or three of these rhythms are passing their average levels (heading up or down) is said to require particular care: you are more accident prone. Days when two cycles are both negative or when one is cresting while the other is at trough are also alleged to be dangerous. The pattern of such days, starting from birth, is exactly the same for everyone. The dominant physical/emotional (male/female) part repeats exactly after 644 days. Including the 33-day intellectual cycle, the whole sequence of critical days repeats from a second effective birthday at age 58.2 years.

Biorhythm was heavily advertised and became a commercial success long before anyone checked to see whether it works. The notions behind this "science" originated just before the turn of the century in the mind of Wilhelm Fliess, a nose and throat surgeon in Berlin, author of a monographlengthily entitled in German "Relations Between the Nose and the Female Sex Organs ..." The cause was taken up shortly afterwards (precipitating rancorous priority disputes) by Hermann Swoboda, a psychologist at the University of Vienna. Fliess and Swoboda developed the numerology of the 23-day "male" cycle of physical ability and the 28-day "female" cycle of emotional condition (not the same as the more real but less exact menstrual cycle). The less important 33-day "intellectual" cycle was invented in the 1920s by Alfred Teltscher, an engineer in Innsbruck. There was a renaissance of popular interest in the United States during the 1930s, but it didn't survive World War II. Few living persons have read the authoritative source, Teltscher's The Rhythms of Life: Foundations of an Exact Biology, published in 1906. Martin Gardner, author of Fads and Fallacies in the Name of Science, reviewed Fliess's works for Scientific American's "Mathematical Games" section of July 1966, and dismissed them as arbitrary nonsense, dusty "masterpieces of Teutonic crackpottery." However, George S. Thommen took a more constructive approach by reviving this lore in the United States. His books Is This Your Day? (1964) and Biorhythm: Is This Your Day? (1969) have gone through numerous revisions and reprintings, and Thommen became president of a firm that sells charting kits and calculators to the gullible. Everyone has by now seen advertisements for "computerized" Biorhythm charting services. Casio Biorhythm calculators are commonly seen in the hands of well-dressed individuals in airports and office buildings.

The notion clearly has appeal, and it has earnest supporters. Unfortunately, their evidence, largely anecdotal or based on inadequately controlled statistical surveys and unpublished sources, would not stand scrutiny. Nonetheless, a fair-minded person could persist in asking whether the Biorhythm theory might be true anyway; important discoveries are sometimes made intuitively or by diffuse folk wisdom before formal proof can be mustered. The airlines, the military, and others responsible for industrial accident prevention thought that a rigorous check might be worth the trouble. During the 1970s in a dozen independent published studies, close to 40,000 carefullydocumented suicides and accidents of diverse kinds were compared with the predictions of Biorhythm, according to birthdays. No correlation was found.2 As one team concluded, "Individuals do have good and bad days; their occurrence, however, is not predictable by the Biorhythm theory." People experience or report more trouble on the predicted days only if they have been primed to expect it on those days.
An hour in 24 seems negligible, but like the difference in speed between cars on a racetrack, it has rapidly cumulative effects. Some cars will fall behind and others will pass, even repeatedly, on a circular track. But living organisms are not racing the earth; what is needed is synchrony, so corrective measures must be taken to maintain it. If your inaccurate clock could not be reset, then you would have to travel continuously to stay in synchrony with your surroundings (40 miles an hour at the equator, to compensate for the mismatch of an hour a day). If the clock were dead accurate, you would be rooted in the time zone of its manufacture, the zone of your birth, or even perhaps in the time zone of your mother's birth, and of her mother's! Moreover, this perfect biological clock would have to be immune to every resetting influence: if it slowed down in cold weather, or jumped ahead or back in reaction to shocks, or stopped while the batteries were being recharged, or just gained a second every day, it would not long be useful. Nobody wants a clock like that, including Nature.
Without the means to reset their 25-hour internal clock--a capacity that many sightless individuals and some with normal vision seem to have lost--nontravelers would drift in and out of step with the 24-hour world. If the discrepancy remained always an hour in 24, then synchrony would recur fleetingly every 24 days. Improving the internal clock's match to the external period does not eliminate the problem; it merely slows the inevitable drifting in and out of synchrony. With adiscrepancy of only 1 minute in 24 hours, an imperturbable internal clock would drift through the same cycle, regaining exact synchrony once in every 1440 days. To maintain synchrony between our internal clock and the earth's rotation requires more than a close match between the two periods: it requires some cue, a cadence caller by which our clocks may be reset daily to agree with local time where we live.
This capacity for phase resetting is the essence of any biological clock's utility. By resetting on cue, it keeps our insides in the right time zone, in phase with local time, adjusting for any mismatch due to travel east or west and the discrepancy between our clock's period and the earth's.
But what is the cue? Not much can yet be said with assurance about humans. But about every other organism that has been examined, the answer is clear, and might have been guessed. The signal must be inflexibly connected to the rotation of the earth, perfectly dependable in its daily appearance, and utterly unmistakable to the organism. One good choice would be light, detected with such sensitivity that no amount of overcast could disguise the hour of sunrise. And most species' internal clocks do indeed react more sensitively to light as a timing cue than to anything else. Continuous illumination at the intensity of moonlight, in fact, is enough to arrest the progress of the circadian cycle in fruit flies and fungi in my laboratory. The clocks in mammals are not that sensitive, and in humans they seem very much less so, but for almost all species tested, watching for light is the best way to learn what time it is.
Phase resetting of circadian clocks is the main theme of this book. The story of these clocks inevitably melds into the study of dynamical systems: of chemical oscillators, of the equations governing unstable equilibria, of the geometry of cycles in multidimensional state spaces (see Chapter 8). To understand body clocks, however, it is not necessary first to have a course in geometrical dynamics. The same phenomena that drive our bodies' daily rhythms can be seen acting in single cells, and much of what is securely known and apparently universal about biological clocks concerns only timing relationships. The geometrical principles involved are plainly laid out on the earth's surface.
Patterns of Timing in Space
A curious principle of timing that turns up everywhere in biological contexts seems to have escaped even the prescient pen of Lewis Carroll.The modern astronaut evades jet lag by neglecting to continually reset his wristwatch and his body clocks. Should he attempt to reset, he would still excuse himself from what seemed a bewildering conundrum to Pigafetta by glib reference to the international date line. If you must reset, then you need only accept the line, and some problems go away. Yet there remains something peculiar about this line. What kind of line is it? Evidently it is not a circle since it was encountered only once in a flight from Hawaii to Hawaii. So it ends somewhere. What if I go to the end and walk around it, cross over, walk around the end and cross over again, and so on? Can I ratchet myself into the remote future or past, a day at a time? Unfortunately not. The international date line has its endpoints at the poles, the two mirror-image points where the boundaries of all the time zones converge. The time zones mark different phases of local time, and the orderly convergence of all time zones and phases marks a phase singularity. In walking around an endpoint of the date line, I also walk backwards around the phase singularity: what is done in stepping across the line is undone in walking around its endpoint.
Is there perhaps some way of drawing the boundaries of time zones to avoid phase singularities? We might try by adopting another definition of time zone. The conventional time zones of cartography correspond closely to the time zones of city streetlight operation only on the equinoxes. At any other time of year there is a disk of darkness surrounding one pole and a disk of daylight surrounding the other, due to the tilt of the earth's axis that gives us our latitude-dependent seasons. When measured from the sun's daily maximum elevation, the time zones of convention do indeed have a phase singularity at the poles all the time, but the time zones of experienced sunrise or sunset terminate tangentially along the singular fringes of those seasonally expanding and contracting disks; they have a singular point only on each equinox. This attempt to evade the necessity of a singularity has only enlarged it! For topological reasons, convergence is inevitable, and a singular point is its minimum manifestation.
Time zones cannot be organized in arbitrary ways. The equator describes a closed-ring path in space along which the time zones must run through a full cycle, so that if you fly along the equator from one time zone to the next until you return to where you started, you pass through a full day. Your path cuts the globe into two territories, one hemisphere to the north and one to the south. Along that path, the equator, all the time zone lines enter each hemisphere and they don't come back out. So no matter how the zone boundaries may be drawn, somewhere inside they must end or come together. Whether or not the time zones converge neatly to a mere point, somewhere in each hemisphere the systematic cyclic pattern of phases must break down. There the time is ambiguous, indeterminate. What happens to a clock at such a place? Nothing, of course; it will keep its own time in violation of the pattern, for a pattern cannot be adhered to if it requires that a clock must indicate no definite time at all. (A confusion about this point resulted in circadian clock experiments being conducted at the South Pole in 1962 to see whether anything peculiar would happen. Nothing did.) We shall later examine phase singularities in which the clock must adhere to the pattern; its only choices then are to abandon the usual cycle of phases or to simply quit altogether.
The ultimate basis of this dilemma of clocks is geometric: it stems from the fact that longitude, like compass direction and periodic time, is represented by points on a circle, with no point on the circle distinguished as the beginning/end except by arbitrary convention. This feature contrasts starkly with the geometric underpinnings of the numbers we are compelled to use, so clumsily and inappropriately, to describe compass direction or periodic time or longitude. Every number represents a point on a line. To get from one number to another you must go either higher or lower: you can't take your choice as you can on a circle. If you keep going in one direction on a line, you will never get back to where you started, though you must on a circle. The numerical jump on a compass (360 degrees jumps to 0 degrees), a clock (12 or 24 hours jumps to 0), or a globe (-180 degrees jumps to +180 degrees at the date line) is a consequence of trying to adapt numbers for a purpose alien to their nature. This distinction between a circle and a line remains no matter how they may be distorted, so long as no cuts are made and no parts are glued together. The branch of geometry that deals with such properties is called topology, and the topological point of view unifies much of the study of biological clocks.
Color-coding Periodic Time
Another familiar quantity that takes its values on a circle is color, or more specifically one of the three qualities that make up any color: its hue (we neglect the other two attributes of color, its saturation and brilliance). Hues have names and an ordering: yellow is close to green is close to blue is close to violet is close to purple is close to red is close toorange is close to yellow is close to green is close to blue is close to violet is close to purple is close to red, and so on. We all learned that in grammar school; it is the "color wheel" lesson.
The physicist's spectrum does not close in a cycle. It reaches only from red through blue and has no "purple." The sensation of purple is elicited by no pure spectral color, but only by a mixture of the extremes, red and blue. It is not between them in the same sense as yellow is between them. Yet, going from blue through violet and purple to red, it is possible to finish the circuit without backing up or repeating a color, to back to where you started. The circle of perceived hues, as opposed to the visible range of spectral frequencies, thus constitutes a natural language for talking about periodic, cyclical things that must come back on themselves: angles about a point, time zones, phase in a cyclic process. Color is a natural language for describing the state of such systems. It will be our language for straightforwardly saying simple things about cycles that sound like riddles in the more familiar language of numbers.
The mere fact that our perception of hue is ordered on a ring implies inescapably that we must also be capable of a hueless sensation, a totally unsaturated color of ambiguous hue. We are indeed, and it is no big deal: it is just the series of achromatic "colors" white-gray-black (according to intensity). This sensation conveniently embodies the idea of a phase singularity being assignable to no point on the circle but at the same time being arbitrarily close to grays tinged with every hue on the circle.
Time Zones for the Tides
This book is about occasions when periodic timing is annihilated, about what happens when a biological clock is confronted with a singularity in its internal pattern of phases. Our understanding will advance by progressively generalizing the idea of time zones. As a first step away from the rigid simplicities of drawing meridian lines on a rotating ball, let us consider another kind of cycle with time zones and singularities.
Land-dwellers usually care a lot about sunrise and sunset. Living organisms mark time in cycles of 24 hours in step with the alternation of dark with light needed for vision or photosynthesis. To some organisms, the moon is as important as the sun, but not for light: concern for the rise and fall of the tides can dominate daily life as much as the alternation of light and darkness. You might feel this way about things if you were an intertidal crab, for example, or anyone else living in the marshy grasslands of the southeastern United States, or if you operate a fishing boat in shallow water, or even if you like to run on the beach every day. In fact, if you were to habitually start your day by watching the moon set, you would be following the innate period of your body clock much better than you do by sticking to a schedule based on sunrise (more on this later).
The tides are governed by both moon and sun, their relative proportions of influence varying from place to place. The moon-dominated tides roughly repeat every 12 hours and 26 minutes (almost an hour later every day). But their timing varies from place to place. Francis Bacon, near the end of the sixteenth century, had first framed the natural question: What is the pattern of timing of tidal crests? They can't be simultaneous everywhere if the oceans have only a fixed volume of water. By the early nineteenth century, there were plenty of tidal recording stations along the shores of all the world's oceans and on a few islands. In the early 1800s William Whewell (who also gave us the word physics) organized what may have been the first international geophysical collaboration to sketch on a chart of the North Sea a series of "cotidal lines," as he called them: the curves along which you find high tide now, the curve along which you'll find it an hour from now, two hours from now, and so on.
Because there are no cliffs of saltwater in the ocean, you might imagine that you could trace cotidal contours smoothly across the sea to match shoreline tidal data. And you can, up to a point, but then an unforeseen difficulty emerges: in a few strange places the contours converge to a point! As Whewell put it,3
It appears that we may best combine all the facts into a consistent scheme by dividing [the North Sea] into two rotary systems of tide-waves; --[in each] space the cotidal lines may be supposed to revolve round [a point] where there is no tide, for it is clear that at a point where all the cotidal lines meet, it is high water equally at all hours, that is, the tide vanishes.
That sounds like time zones and phase singularities again: the tides have their own kind of North and South poles, singularities where all the tidal time zones meet. The cotidal contours converge in clockwise or counterclockwise rotating circular order, much as the time zone boundaries converge to the opposite poles. At those points the lunar rhythm vanishes: the depth varies irregularly with the weather or follows some other frequency-component of the tide, with a daily period, for example. A moderately detailed map of the entire globe shows about eight major clockwise and eight major counterclockwise convergences of the dominant component, the lunar M2 tide, as it is called.
Dubbed amphidromic points, or amphidromes, these phase singularities organize the spatial pattern of tidal timing throughout the globe. It is not easy to grasp intuitively how and why they arise where they do,how they move seasonally, and so on. But the underlying physical principles are known with precision. To begin with, at least two of each kind must exist for the simple reason that the lunar tides resemble a pair of diametrically opposed gigantic bulges in the world ocean, held in place by the moon while the earth spins within. On a perfectly symmetric globe with an upright axis there would be a neutral, amphidromic point at the North Pole and a mirror-image point at the South Pole. Each hemisphere would have two contours of maximum tide, rotating clockwise around the North Pole and counterclockwise around the South Pole. But in fact, the earth's axis is tilted and its oceans are shallow and irregular, so the symmetry is spoiled. This does not eliminate the topological necessities. There must still be centers for a twofold clockwise rotation and a twofold counterclockwise rotation, although they could be manifest as four simple centers. Might the many additional mirror-pairs have arisen like eddies in our resonantly sloshing seas? Might some also be hidden within the borders of continental landmasses? Answers can be calculated in surprisingly accurate detail by taking into account the inertia and viscosity of water, the shapes of continents, and the slopes of seabeds along with some tidal observations. In fact, the map on the preceding page represents such a computation.4
The track of Magellan's epic voyage is marked in white on the map on pages 16 and 17; completely unaware, he traversed the basins of severalmajor amphidromes in the world's oceans. Because there are (including relatively minor ones) 10 clockwise and 9 counterclockwise amphidromic points south of his path, Antonio Pigafetta and colleagues experienced one extra cycle of the tides. But Pigafetta was not monitoring the tides and never noticed the extra ebb and flow.
He was monitoring the day/night cycle. Because of the single convergence of time zones south of the ship's course, the returning crew missed out exactly one complete cycle of day and night. Why then did they not notice it? A modern traveler would notice the progressive loss of major fractions of a day long before reaching home again because he would be carrying a timepiece locked to his original time zone. But Magellan's men carried only their biological clocks, and biological clocks have evolved for resettability. The first large-scale experiment probing the nature of these clocks may have been unwittingly conducted by Antonio Pigafetta and his shipmates while struggling westward back to Portugal through disastrous misadventures on the far side of the world. As they sailed around the world their biological clocks were reset day by day to show no discrepancy with local time. Imperceptibly, this systematic resetting brought them closer and closer to its culmination in Pigafetta's Paradox: the unnoticed vanishing of a full day from their calendar.

Copyright © 1987 by Scientific American Books, Inc.