Eudoxus thought that by superimposing this figure-8 loop on a third, underlying west-to-east motion, he could simulate the retrograde motion of the planets. Half the time, the hippopede would also be moving west to east, so the combined motion would be west to east as well—this would be prograde, or direct, motion. Even much of the rest of the time, the hippopede would not be moving enough in the opposite direction to counteract the general west-to-east translation. Only when the hippopede was moving nearly as fast as possible, east to west, would there be a resulting backward slide, and this backward slide Eudoxus identified as retrograde motion.
It was a clever bit of explanation, but there were a number of problems with it. First of all, if it were correct, then all of the retrograde loops should have been symmetrical, and that wasn't so. Secondly, and more seriously, all the planets should remain at the same brightness throughout their orbits, and they certainly did not. Mars, in particular, is dozens of times brighter at some times than at others. For these reasons, Eudoxus's hippopede was eventually replaced, first by Ptolemy's theory of deferents and epicycles, equants and eccentrics, and 1,400 years thereafter by Copernicus and the heliocentric theory.
* * * *
The hippopede re-entered science, though, in a completely unexpected way—a way that was only opened up by the advent of accurate timekeeping.
For millennia, humans kept track of time by noting the general location of the Sun. One might speak of leaving for town at sunrise, or of returning when the Sun was a hand's breadth above the horizon, and so forth. The Sun's motion was sufficiently constant to provide a convenient basis for telling time.
At some point, it became expedient to divide both the day and the night into portions, and the Babylonians chose to divide them both into 12 equal parts called “hours,” from an ancient Greek word meaning “time of day.” Twelve was a useful number, in that a quarter, or a third, or a half of a day or night all came out to a whole number of hours. These hours could be labelled on a sundial, so the moving shadow of a stylus, or gnomon, would mark out the advancing hours—at least, during the daytime.
Unfortunately, all of the daytime hours were equal to each other, and all of the nighttime hours were also equal, but the daytime hours were not the same length as the nighttime hours. Instead, they were longer in summer (naturally) and shorter in winter. The explanation for this was in the changing height of the Sun. It rose higher in the sky in summer, and more of its circular path was then above the horizon, so naturally the 12 daytime hours took longer to pass. In the winter, exactly the opposite was true: the Sun did not get very high at all in the sky, even at its peak. Less of its circular path was above the horizon, so the 12 daytime hours took less time to pass.
Eventually, other devices for telling time were developed that did not depend on the slightly variable nature of the Sun's path: for instance, hourglasses, or burning candles. With the introduction of these timekeepers, the variations in the daytime and nighttime hours became quite troublesome. It was tedious to have to change candles or hourglasses with each month. How much easier it would be to replace the inconstant hours with 24 equal ones. The only inconvenience was that sunrise and sunset would take place at slightly different hours throughout the year, but that could easily be accounted for.
Then, in 1656, the Dutch astronomer and physicist Christiaan Huygens (1629-1695) developed the first pendulum clock. Galileo had had the idea previously, while watching a chandelier sway back and forth in a cathedral, but had never followed through on a design. Huygens was the first to overcome the physical obstacles to building a clock based on the principle of the pendulum, and he ushered in the era of precision timekeeping.
Huygens's clock was also the first to be accurate to minutes a day, and the clock face gained another hand. Later clocks were even accurate to seconds, and now was discovered an interesting discrepancy. The moment that the Sun crosses the meridian—an imaginary north-south line in the sky—is called local noon, after an old word meaning the ninth (daytime) hour of the day. (This was midafternoon, but later was moved back earlier, to midday.) By all rights, the time between local noon on two successive days should be exactly 24 hours. But as measured by these accurate clocks, the interval between two consecutive local noons was sometimes a few seconds long; at others, a few seconds short. If we set a clock exactly to noon when the Sun was at local noon on one day, then the next day, the Sun would reach local noon, not at 12:00 exactly, but perhaps at 11:59:58, or at 12:00:10. These discrepancies added up, so that at various times of the year, the Sun was as much as a quarter of an hour “early” or “late.” The errors repeated in a cycle of length one year, year after year.
Either the clocks were wrong, or the Sun's apparent motion across the sky was not as constant as previously thought (or both). We now know that it's the latter, and this repeating cycle is called “the equation of time” by astronomers. The Sun does not go at the same rate in right ascension (the astronomical version of longitude) all year long, but instead moves through lines of right ascension faster at some times, slower at others. At no point does it actually go the “wrong” way—it doesn't exhibit retrograde motion, in other words—but this variation is what causes the Sun to cross the meridian early or late. And if we plot the “location” of the Sun, with its northern and southern advances drawn along the vertical axis, and its earliness or lateness drawn along the horizontal axis, we get the figure drawn on my dad's globe, which is called an “analemma.” (See Figure 6.)
The word “analemma” is Greek for the pedestal of a sundial, and itself comes from the Greek verb analambanein, meaning “to take up, to resume, to repair,” so that the pedestal is something that supports the sundial upon it. Early on, “analemma” seems to have been extended to refer to a particular kind of sundial, in which only the height of the Sun was indicated, by measuring the size of the shadow cast by the sundial. Later, it was used for a number of meanings related to the height of the Sun; its latest meaning, and that with which we are interested here, is some kind of representation of the Sun's gradually changing path in the sky at the same time (noon by the clocks) each day.
It surely hasn't escaped your attention that the analemma and Eudoxus's hippopede share a certain resemblance, a resemblance that, as it turns out, is more than accidental. The hippopede results from the conjunction of two circular motions, and so does the analemma.
The apparent motion of the Sun is really due to two motions of the Earth. One is the Earth's orbit around the Sun. The Earth completes one revolution about the Sun in one year, and if that were the only motion that the Earth had, then we on the Earth would see the Sun appear to go around the Earth just once a year.
However, the Earth has a second motion: its rotation on its axis. It does so approximately once a day, and it is for that reason, mostly, that the Sun appears to revolve around the Earth once each day. Since these two motions have periods in approximately the ratio 365.25:1 (the number of days in a year), while the hippopede results from two motions with equal periods, you might think that the hippopede doesn't have much relevance to the analemma.
But you'd be wrong. As I mentioned, the Earth rotates on its axis only approximately once a day, and the Sun's apparent motion across the sky is only mostly due to this rotation. A tiny component is due to the first motion of the Earth, its orbital revolution. Since this revolution takes 365.25 times longer than the rotation, it contributes 1/365.25 as much to the Sun's apparent motion across the sky as does the Earth's rotation. Now, the Earth's rotation makes the Sun seem to move east to west, from dawn to dusk, but its orbital revolution appears to add a second component, from west to east. This second component very slightly counteracts the first, so that the 24-hour day is longer than you might expect based solely on rotation. In fact, the Earth actually rotates on its axis, with respect to the stars, every 23 hours, 56 minutes, and 3.5 seconds. This slightly shorter day is called the “sidereal day,” after a Latin word meaning “star,” since this is the time i
t takes for the Earth to rotate once relative to the stars. The extra four minutes each day is due to the Earth's orbit around the Sun, and is 1/365.25 of the 24-hour day.
In other words, if the Earth didn't revolve around the Sun, but only rotated in place, in defiance of the law of gravity, the Sun would appear to go once around the Earth in 23 hours, 56 minutes, and 3.5 seconds, instead of the customary 24 hours. And if we were to take a snapshot of the Sun every day at the same time by the clock, it would be 3 minutes and 56.5 seconds further along each day. After two days, it would be ahead (that is, further west) by 7 minutes and 53 seconds; after three days, by 11 minutes and 49.5 seconds; after four days, by 15 minutes and 46 seconds, and so forth.
How long would it take for this margin to extend to 24 hours, so that the Sun would once again be “on time,” on the meridian at noon? Why, as many times as 3 minutes and 56.5 seconds goes into 24 hours—and as we noted above, this interval is 1/365.25 of 24 hours, so it would take 365.25 days for the Sun to “lap” the 24-hour clock. A year, in other words. In short, if the Earth only rotated, and didn't revolve around the Sun, the Sun would appear to revolve around us every 23 hours, 56 minutes, and 3.5 seconds, but by taking snapshots of the Sun every 24 hours, which is just about four minutes longer, this motion would appear to be slowed down to just one revolution per year.
In case that sounds confusing, it's like watching a car drive by you on the road. In reality, the car's wheels may be rotating very rapidly—let's say, 25 times a second. (That'd be one fast car, by the way—probably around 150 to 200 kilometers an hour!) But if you watch a film of the car, where the camera takes 24 frames per second, each frame catches the wheel when it has gone through 1 1/24 of a rotation. Since the eye can't tell the difference between 1 1/24 of a rotation and just 1/24 of a rotation, it appears as though the wheel is actually rotating at only 1/24 rotation per frame. That works out to one rotation every 24 frames—or once a second.
In much the same way, when we take our figurative snapshots of the Sun every 24 hours, the Earth's rotation, alone, makes the Sun appear to revolve around the Earth, once a year, from east to west, along a path called the celestial equator. Meanwhile, as described above, the Earth's orbital revolution, alone, makes the Sun appear to revolve around the Earth, once a year in the opposite direction, from west to east, along another path called the ecliptic. Both the celestial equator and the ecliptic are great circles. What's more, these two great circles are not the same, but because of the Earth's axial tilt, are instead inclined to one another by an angle of 23.4 degrees.
We therefore have an exact analogue of Eudoxus's hippopede, but this time applied to the apparent motion of the Sun throughout the year. These two motions combine to create the figure-8 shape of the analemma. Eudoxus could not possibly have known about this application of his theory, which was originally designed to account for the retrograde motion of the planets. As an explanation of that behavior, the hippopede was basically dead on arrival. Too bad that accurate clocks were not available in his day; otherwise, he might have found the right use for his geometric intuition.
* * * *
But one last objection remains: The analemma on the globe is not a symmetric figure-8 at all! Rather, it's smaller on the northern end, and larger on the southern end. Why is that?
That asymmetry is due to one further property of the Earth's orbit around the Sun: its eccentricity. The Earth's orbit is nearly circular, but not precisely so. It is actually an ellipse, and the Earth moves along that ellipse in accordance to Kepler's laws of planetary motion. (See “Music of the Ellipses.") As such, the Earth moves faster when it is closer to the Sun, and slower when it is further from the Sun, and this translates to a corresponding variation in the Sun's apparent west-to-east motion due to the Earth's revolution. Just how elliptical the orbit is, and the angle between the long axis of the orbit and the axis of the Earth, determine the contour of the analemma.
Incidentally, I'm not certain just why the analemma is specifically in the southern Pacific—perhaps because that's the least crowded part of the planet, cartographically speaking—or why it's needed on a globe at all. It does have some significance to sundial builders, since it can be used to correct for the equation of time, if the months of the year are marked out (as they are on my dad's globe) and one rotates the dial of the sundial according to the analemma. But it doesn't seem to need to be on a globe, and indeed, more modern globes now eschew the analemma in favor of a more extensive legend.
Here is a C program to compute and plot the analemma for various different orbital parameters. It's not tremendously user friendly, and can probably use some additional documentation. (It also uses the “system” call, which probably should be replaced with something in the “exec” family, if that means anything to you.) However, it uses the ideas presented in this essay, with the additional amendment that the eastward march due to the Earth's orbital revolution varies in speed because that orbit is elliptical. This approach is more accurate than programs where the effects on the equation of time of the two motions is added linearly (such as this one). That's reasonably accurate for small eccentricities and axial inclinations, but becomes noticeably inaccurate for extreme orbits.
Adapted from Astronomical Games, August 2002.
* * * *
Brian Tung is a computer scientist by day and avid amateur astronomer by night. He is an active member of the Los Angeles Astronomical Society and runs his own astronomy Web site. His previous publications in Strange Horizons can be found in our Archive.
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Mr. Muerte and the Eyeball Kid
By Sean Klein, illustration by Alain Valet
8/5/02
Lunchtime, Jimmy finds an eyeball in his stew. “Cool,” he says, balancing it on his spoon.
“Eat it,” Frankie says.
“Yeah, eat it,” Denny says. He always agrees with what Frankie says. If Frankie jumped off a cliff, Denny would follow.
Only Peter remains silent.
The boys sit at a plastic table in the cafeteria's corner. The other kids leave them alone. The cafeteria lady stands amongst the tables like a sentinel, ready to avert disaster, or at least a food fight.
“Miracle Man would eat it,” Frankie says.
“No he wouldn't,” Jimmy says.
“Would so."
“Would not. Miracle Man is a wuss."
“Mr. Muerte,” Denny says, “would eat an eyeball."
Mr. Muerte is Jimmy's favorite superhero. Mr. Muerte wears black and is rarely seen in tights and a cape. He fights the Midnight Stranger and sometimes Zeus ex Machina, and never anyone lame like the Sandbagger or Al K. Traz. Mr. Muerte is supernatural. Mr. Muerte is immortal. Mr. Muerte is cool.
Jimmy moves the spoon and makes the eyeball rock in its brown bed of sauce.
“I dare you to eat it,” Frankie says.
“Double dare,” Denny says.
Frankie peers across the table, leaning forward over his forgotten lunch. “Double dog dare."
Jimmy studies the eyeball. It stares back at him. It's green if you ask Frankie, but blue if you ask Jimmy. Denny would say it's green because of how he feels about Frankie. Peter's eyes are brown and today they have a shadowy, haunted look like he hasn't been sleeping.
Peter likes Mr. Muerte too. He's the one who told Jimmy that Mr. Muerte is cool. Peter told Jimmy a lot of things. He told him that Miracle Man is lame. He told him to watch The Six Million Dollar Man on television long before all the other kids started running in slow motion. He told him about Fizzies and the Doberman that ate that guy's fingers. He told him about real rock and roll and they listened to the Beach Boys together, Elton John and Pink Floyd, records they borrowed from Peter's older brother when he wasn't home. Frankie and Denny like Elton John and the Beach Boys, but they think Pink Floyd is lame.
Peter doesn't tell Jimmy to eat the eyeball. But Jimmy can't deny Frankie's double dog dare. If he does, then Frankie and Denny will tell everyon
e that he's a wuss. Jimmy looks to Peter, but Peter only stares.
Jimmy doesn't want to be a wuss.
He chews the eyeball like he'd chew a small round potato. It's not rubbery, and nothing squirts in his mouth—and it doesn't taste too bad. It's actually a bit mushy, like a vegetable that's been cooked too long. No worse than anything else served at the school cafeteria.
Frankie's scream cuts through the cafeteria noise: “Eeeewww!” Two girls at the next table over stop talking. They look at the boys as if they are lepers escaped from the colony. Kids at the other tables turn to look. Denny joins the yelling: “Grooosss!” Frankie announces, “Jimmy ate an eyeball!” Denny has his hands clasped at his throat, his eyes crossed, and his tongue hanging out. He's wobbling like a punching bag clown that just took one in the kisser. The girls wrinkle their noses and go back to eating. “Eyeball! Eyeball!” Denny yells. The other kids return to their lunches and interrupted conversations. Frankie quiets down and Denny follows.
“From now on,” Frankie says, “your superhero name is ‘The Eyeball Kid.’”
* * * *
The boys have secret identities. Jimmy was Captain Justice, but now he's the Eyeball Kid. Frankie is the Crimson Streak because he runs fast. Everyone always wants him on their team for capture the flag. Denny is the Phantom Avenger. Last week he was the Mysterious Avenger. And the week before he was Captain Mystery, but he had to change it because Jimmy was Captain Justice. Peter was Matter Master until he got sick. Then he was the Radioactive Kid. Now he's the Silver Specter. He's been the Silver Specter for a while now.
* * * *
In school, Jimmy and Peter sit next to one another. When Peter got sick, his parents came to class and emptied his desk. He did his work at home: spelling lists, book reports, the math book full of multiplication and division problems. Peter was good at spelling, but not as good at math. Jimmy helped him with the fractions last year.
Strange Horizons, August 2002 Page 5