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(2012-11-19)   Horology
Approximating astronomical ratios with gears.

Horologists are primarily concerned with the measurement of mean solar time  (there are  approximately  86400 atomic seconds in a mean solar day).  To design a mechanical watch that would keep both mean solar time and sidereal time, we need a good approximation of their ratio.  The value at  J2000.0  was approximately:

1.002737909350795   =   [ 1; 365, 4, 7, 1, 2, 1, 361 ... ]

This  continued fraction expansion  (CFE)  is clearly  begging  to be trunctated just before the last term quoted above.  This yields:

[ 1; 365, 4, 7, 1, 2, 1 ]   =   46879 / 46751   =   1.00273790934953263...

Remarkably, although the actual ratio does vary, the above rational approximation will remain  optimal  for more than a century  (well beyond the warranty period of any mechanical clock or watch).  For a watchmaker, the problem is that the denominator of that fraction is  prime  and can only be produced by a wheel with at least that many teeth  (clockmakers call them  leaves).  Making a wheel with  46751  leaves is out of the question...

So, we have to settle for somehing less.  Here are good approximations to  46879 / 46751  that can be attained with wheels that have no more than  n  teeth.  The error in the last column is expressed in seconds per day.

Approximating  46879 / 46751   with
at most  m  wheels of  n  teeth or fewer :

m	n	Numerator	Denominator	Error
3	46751	(7 . 37) . 181	46751	0
2	46879	(7 . 37 . 181)	46751	0

Space Traveller's Watch  by  George Daniels (1926-2011) with the help of Henry Daniels, FRS (1912-2000).
(In this video, George Daniels misquotes the above ratio as "1.002737942", which it ain't.)

(2012-11-19)  Daniels Coaxial Escapement 
The most notable innovation of the 20^th Century in mechanical horology.

Escapement   |   Co-axial escapement
 
Daniels Chronograph (2:27)  presented by its maker,  George Daniels (1929-2011).