Ivars Peterson's MathTrek

October 13, 1997

# Fractions, Cycles, and Time

In our efforts to measure and understand the universe in which we live, we often find ourselves dealing with "messy" numbers. Our tendency is to replace those inconveniently long strings of digits by rough approximations. In the everyday world, we say that a inch is about two and a half centimeters, light travels at a speed of nearly 300,000 kilometers per second, and the number pi is close to 22/7 or 3.14.

In ancient times, people had to confront awkward numbers in astronomical contexts when they compared the motions of the sun and moon. The unfailing, daily passages of the sun across the sky and the corresponding movements of the stars at night represented one highly predictable cycle. The periodic changes in the moon's appearance and position represented another cycle.

Closer observations over months and years revealed subtle shifts in these patterns. The sun, for instance, doesn't rise at precisely the same point on the horizon every day. The location of sunrise drifts back and forth along the horizon. These recurring excursions define a longer cycle tied to the changing seasons. Similarly, particular stars rise at different locations along the horizon and, at certain times, disappear from the sky for lengthy periods. These movements also have a definite rhythm attuned to the seasons.

In modern terms, taking the day as the standard unit of measurement, the seasons recur every 365.242199 days (a year), while the period of the moon's phases is 29.530588 days (a month). And to be really precise, we must also note that these numbers decrease each century by 1 or 2 in the last decimal place because tidal friction is slowing Earth's rotation and making the day longer. Indeed, official timekeepers add a second every year or so to keep their clocks in sync with Earth's rotation rate.

The ancients didn't use decimals, but they could represent these cycles with remarkable precision by considering ratios. The Athenian astronomer Meton (5th century B.C.), for example, noted that 235 months very nearly equals 19 years. The so-called Metonic cycle is still used to determine the Jewish calendar and set the date of Easter.

The following table gives the error when various numbers of months are compared with the corresponding numbers of years. The listed entries represent successive improvements in the accuracy of the ratio of months to years used to approximate a cycle. Each line is obtained by adding a certain multiple of its predecessor to the one before that. For example, to get 99 months in line 5, you add two times 37 (fourth line) to 25 (third line).

No. of months No. of yearsError Multiplier
1 month 0 year +29.530588 12
12 months 1 year -10.875143 days 2
25 months 2 years +7.780302 days 1
37 months 3 years -3.094841 days 2
99 months 8 years +1.590620 days 1
136 months 11 years -1.504221 days 1
235 months 19 years +0.086399 days 17
4131 months 334 years -0.035438 days

Meton's approximation is off by just 2 hours 4.4 minutes! And it's bettered only by comparing 4,131 months with 334 years.

Now consider the successive fractions (number of months divided by the corresponding number of years): 12/1, 25/2, 37/3, 99/8, 136/11, 235/19, 4131/334. Those fractions, in turn, can be written in the following manner:

Such expressions are known as continued fractions. They can be used in designing gear trains, including those that might be found in a planetarium to simulate the relative motion of the sun and moon around Earth.

The so-called Antikythera mechanism, apparently constructed in the first century B.C., recovered in 1900 from a Mediterranean shipwreck, and analyzed just a few decades ago, is one of the most striking examples of such engineering in the ancient world. It contained a system of gears whose gear ratios corresponded to well-known astronomical cycles involving the moon, including the Metonic cycle. The mechanism was clearly a type of analog computer, using fixed gear ratios to make calculations displayed as pointer readings on a dial.

The Antikythera mechanism -- the sole survivor of what was undoubtedly a long tradition of astronomical automata -- served primarily as an elegant simulation of the heavens. It was a tabletop monument to Greek and Alexandrian astronomy. Such ingenious devices also illuminated the intimate link between mathematics and astronomy, especially the role of number in astronomical prediction.

By demonstrating an ability to predict the movements of the moon, the rising and setting times of stars, and the changes of the seasons, astronomers could please their rulers while contemplating the subtleties of the evident mathematical order displayed in the heavens.