These four illustrations depict the Moon's orbit around the Earth for one,
five, ten and sixty years respectively. The last illustration shows a rather
thick doughnut around the Earth. One can readily see why it is so difficult
to accurately describe the Moon's orbit!
We will correct only for the three greatest anomalies. These are the Great
±.6.58° equaling 26.322 minutes and the
Projection (which contains two corrections) at ±2.464° which
translates into about ±9.857 minutes in time. So
the total maximum error involved is 36.179 minutes.
All of the remaining anomalies amount to a combined 9.442 minutes. It is no
accident that the dials on the
Schwilgué and Festo clocks are large to take advantage of this information.
The combined total of all anomalies comes to ±45.621 minutes from a simple
rolling moon dial. Our Moon rise and set dial
is fairly small at just over 3½" or 10cm in
diameter so any further corrections would be barely noticeable at this
scale. Even thirty six minutes will be hard to discern. This is really more
of an exercise to incorporate a classic complication rather than important
additional accuracy in the dial.
These four photos show the first concrete example of what Buchanan is
proposing for the Moon corrections. They show an alternative way of
applying a differential gear system and are based on Antide Janvier's method
to simulate the erratic motions of celestial bodies and that were later employed
by Jean-Baptiste Schwilgué. I have always been fascinated by
this differential gear arrangement and wanted to have it incorporated into
the panoply of mechanical contrivances within the machine.
The first two photos show the slant wheel in its two extreme
positions. In this model Buchanan is mimicking the Great Anomaly in
connection with the 23.50 tilt of the
earth’s axis. The next two photos show the same wheel set with a pair of
dials denoted in degrees mounted to the slant wheel as well as the input
wheel. In this way one can see how the slant wheel moves in relation to the
input wheel. Since we did a Breguet style differential for the equation
work I suggested we use this design for the Moon’s motion.
That relationship as
represented by the Great Anomaly is plotted on two graphs. The first one
above shows the differential changes as a change in radial degrees over one
complete rotation. It plots the difference between the pointer on the drive
wheel which turns at a constant rate and the readout of the slant wheel,
Buchanan calls this the “error from the true pointer”. That true pointer is
the one mounted to the drive wheel as seen in the fourth photo above of the
The second graph is the
same information, but plotted as the mathematical function. This is not
unlike the graphs one would get plotting the equation of time which is the
difference between the mean solar time, or the time we read off our clocks,
and the position of the Sun in the sky. The physical cam used to represent
this function in a clock has a similar kidney shape as well as the graphical
illustration that looks like a sine wave for the mathematical function. This
similarity to the equation of time function is no accident as the Earth’s
tilt is one of the two components used to determine the equation of time,
the second is the eccentricity of its orbit, which is also contained within the
fourth anomaly known as the Great Anomaly, 2nd term.
Illustrated above is the initial
design layout for the Moon complication. Here we have made manifest the two
largest anomalies represented by the two differential wheel cages 3 and 4.
These represent the First or Great Anomaly and the second the Moon’s
evection, the acceleration and deceleration as it approaches and move away
from the Sun. Note that the angle in the first differential is in error at 9.29°
but should be 6.29°.
The interesting thing
in Deryck’s design is that he takes Janvier’s slant wheel differential one
step further. In Janvier’s application the areas represented by each
square box remains stationary, that is the wheels ‘G’ and ‘L’ rotate and
move from left to right, but the entire structure including the rest of the
wheel works rotate on their axis but do not otherwise move. In this design,
the boxes should be seen as rotating cages, not unlike a tourbillion and
each rotate around arbors ‘J’ and ‘T’. The first differential represented by
box ‘4’ supplies its output to the next differential, box ‘3’. These all
rotate along with the Moon once every 29.53 days. Each cage will be
approximately 3 inches or 7.5 cm in diameter. I did discover later that Schwigué did
use this rotating cage design. As the saying goes “There’s nothing new under
the sun”, see photo below. Although we have employed a few novel mechanical ideas in other
areas of the clock.
This is a photo of the drive to the Moon on
Schwilgué's astronomical cathedral clock in Strasbourg. The two silver
horizontal structures in the back and foreground is his rotating cage.
Within the cage is one slant wheel differential combined with a second pair
of interactive differentials in the middle portion of the device. Note the
sector gear just to the left of the center nested differentials just before
the last two large output wheels. The gears are being driven from right to
left. Just this one mechanical unit displays a constellation of complex and
visually stunning mechanical achievements a tour de force in wheel work
design. We owe much to the masters of the past.
Here the complex mathematical formulas are depicted.
This drawing shows one
of the two anomaly correction, slant wheel differentials. The two slightly lenticular, biconvex circles on the front
elevation section of the drawing, left, are showing the two wheels that are
slanted in relation to that elevation and the rest of the wheel work.
In this photo are shown
the areas where the wheel assemblies will be located. The twin anomaly
differentials are anticipated to each be just over 3” in diameter and 2 ¼”
long (7.5 cm, 6 cm). The pair with their associated drive gearing will be 5”
(12.5 cm) wide. This will nicely balance the density seen in the calendar
mechanism on the left hand side of the machine.
The first drawing
depicts the gear trains for the moon’s phases as well as its cycling
around the dial. Next is an overlay with paper disks that depict the wheel
train controlling the sun rise and set shutters.
Here we have a completed drawing with both the moon and sun gearing
represented respectively drive wheels all revolve around the dial's axis
according to the input of the two anomaly differentials. Their input is the
first small red wheel at 12 o’clock
(1). That wheel is fixed and moves the second large red wheel
upon which is mounted a nested pair of wheels
which are fixed together. The largest of the pair
(3), meshes with the center
wheel (5), which is fixed. The
smaller of the pair (4), meshes
with wheel (6) and then to
(8) delivers rotation to the
Moon globe through a pair of bevel wheels.
The rotation of the input wheel feed from the anomaly differentials, by
rotating the large red wheel upon which the rest of the moon mechanism is
mounted induces rotation of the nested pair of wheels engaged with the fixed
center wheel; thus causing rotation throughout the rest of the wheel train
through to the moon globe.
These gears are represented by the calculator shown above for a four
stage wheel set to obtain the accuracy of the moon’s rotation to seven
decimal places. The other thing the calculator does besides giving
extraordinary accurate results is that it allows Buchanan to perform a trial
and error exercise with various numbers and tooth counts almost instantly.
Thus he could try out a five stage to see how it might look vs. a four but
still retaining the correct results. It not only saves time but allows for a
much better design both mathematically as well as esthetically.
The sun itself revolves around the dial once per day and is mounted to
the center green wheel (a). That
wheel drives the next four wheels
(b, c, d, e) which then turn a pair of cams
(f, f’). These cams each in turn
have roller follower arms that rotate upon their edges and are attached to
set of sector gears. Those in turn control the two shutters for the sun rise
and set horizon. Only one set of sector gears is shown as they are
superimposed upon each other in this view.
Front and side elevation of the Sun and Moon rise/set dial
This is how the pair of
differentials will fill out behind the dial work. They are drawn to the same
scale as the dial work, These will reach almost to the first strike hammer
mechanism as shown by the diagramed box in a prior photo above.
Buchanan sent the
drawing I requested to see how far the slant wheels as designed would
project beyond the dials. In this front elevation the anomaly differential is
about 2” in diameter. We later decided that it would be esthetically more
advantageous to expand that wheel to 3” (7.75 cm). This will allow more
space to be taken up by this device, yes there still is a bit of limited
real estate left in this machine! But more importantly, the larger wheels
combined with a delicately thin cage design will make the entire pair of
differential wheel presentation look more delicate and appear to ‘float’ as
they rotate within their turning cages.
A schematic showing the full assemblage for the Sun and Moon rise/set dial
The proposed design for the Sun and Moon rise/set dial. The following
information and thus complications can be read from this dial and thus add
16 complications to the astronomical clock: