**The Antikythera Mechanism I**
**With Java animations by Bill Casselman**

## 4. 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:

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 then guarantees an error of *at least* .00088.

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*

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