Antikythera mechanism 4 - Gear ratios and continued fractions

Gear ratios and continued fractions

It is interesting to speculate how the first century B.C. designers of the Antikythera Mechanism were able to discover the excellent rational approximation 254/19 = 13.36842105 to the astronomical ratio 13.368267.. . The error is 0.00015, which corresponds to one part in 86,000.

The most economical explanation is that in keeping records, early astronomers were struck by the almost exact duplication of the pattern of equinoxes and solstices (sun) and phases of the moon in a 19-year cycle. Nineteen years almost exactly matches 235 lunar-phase cycles ("synodic months"), which correspond to 235+19=254 revolutions of the moon with respect to the stars. It picks up an extra one each year from its trip with us around the sun.

But part of the answer comes from the astronomical ratio itself, which turns out to be one of those numbers that can be very well approximated by rationals. This is manifest in its continued fraction expansion:

13.368267.. = [13, 2, 1, 2, 1, 1, 17, ...]

1
= 13 + ------------------------
1
2+ ---------------------
1
1+ ------------------
1
2+ ---------------
1
1+ ------------
1
1+ --------
1
17+ ----
etc


Stopping the process after the last 1 gives the "continuant" 254/19 used in the Antikythera Mechanism. Continuing with the the 17 gives the next continuant, 4465/334. The large increase in the denominator comes from the 17.
Here is a useful fact from the theory of continued fractions:

Suppose pn/qn and pn+1/qn+1 are consecutive continuants, and that L is the number represented by the continued fraction. Then,

[*This equation can't be transcribed here, because it uses programming symbols for HTML. To view this equation see,

http://www.ams.org/new-in-math/cover/kyth3.html ]

This means on the one hand that the error in any continuant is less than one over the product of its denominator with the denominator of the next continuant. So the approximation 254/19 is guaranteed to have an error less than 1/(19x334) = .0001576 just from the continued fraction expansion of the astronomical ratio. On a different planet things would have gone otherwise. If that 17 had been a 1, then 254/19 would still be a continuant, but the denominator of the next continuant would only be 30. The other side of the same analysis guarantees an error of at least .00088.