A Combined Miniature Gnomon
Compound Dioptra

c. 35.7 N. Lat. 10 AM 2/27/2000
Gnomon, Miniature Gnomon & Compound Dioptra

Relativity, at least in the Cartesian sense, implies the relating of centers or origins together with their arbitrary systems of axes, rays or vectors. Since all points in the universe of discourse are related within each of the individual systems then the systems can be related to each other, at least mathematically. We pick a center and a system carefully to accomplish, as easily as possible, some objective. It is not that one system and center is inherently superior to any other in some Platonic ideal, it is essentially a pragmatic choice. The device pictured above enjoys two different centers and two different coordinate systems.

In astronomy, the great invention of the Age of Enlightenment was a helically centered solar system although the idea had been present since the fourth century BCE. Their real invention was the development of Cartesian/Polar coordinates where every point could be named by an ordered triple with the sun at the center. It is this system that is taught or at least hinted at in our schools today. For many questions this is the ideal system to provide simple answers. But, by embracing it as the only acceptable, correct, system the layman has lost contact with the observable universe. There is little cognitive understanding on the occasions where the sky is seen, either at night or during the day. The gnomon and compound dioptra has provided me with a wealth of surprises that help bring theory and observation together. It makes the sky a richer place, a place well worth observing.

Local Noon, 2/28/2000
Miniature Gnomon with attached recording sheet
(the compound dioptra has been removed)

My first gnomon was a pointed wooden fence post standing about five feet above ground. The shadow plane was reasonably flat gravel and I had a hard time determining the end point of the shadow so I was amazed at how accurate my observations were. My first major surprises came with the shadow geometry including the true east - west line of the equinox which I hadn’t anticipated. I am still puzzling over the the pattern of change of the shadow of the full moon when the moon is due south. Also how to observe Alexander Thom’s major and minor lunar standstills from megalithic Britain which also seem to have been recognized in the designs of the Great Houses and Sun Dagger of Chaco Canyon. The broader implications of the gnomon as a system with its center at a particular point (latitude, longitude) and polar coordinates for observation points did not occur to me until I built the compound dioptra.

The inspiration for the compound dioptra came from James Evans’ book, The History and Practice of Ancient Astronomy (Oxford University Press, 1998). In it he describes the early Greek (6 th C. BCE to 200 CE) astronomical principles. The two that apply here are that the astronomical distances are so great that the Earth can be considered a point in the center of the celestial sphere and that there is no measurable parallax for star positions from anywhere on Earth. (Parallax was not measured until many centuries later.) The earliest dioptra (c.300 BCE) was a device for measuring the angle, in parts of a “Sign”, between two celestial objects. The Babylonian degree had not arrived in Greece at that time and they used the Sign as a measure. It was one twelfth of a great circle, a double hour or 30 degrees. The other compound dioptra, discussed here, focused one sighting tube at the North Celestial Pole while the other rotated around it following the selected celestial object as the earth or heavenly orb rotated. The Greeks knew that either rotation was possible and that observation could not determine which but thought a stationary earth more likely. We fall into their dichotomistic error in insisting that it is the Earth that rotates. Either system is equally valid but for our computational purposes a rotating earth is far more useful. For celestial observation, a rotating heavens is more intuitive and probably superior.

9 AM , 2/29/2000
Compound Dioptra showing meridian & equatorial protractors

As I worked on how to join the two sighting tubes so that one would stay focused on Polaris and the second rotate around it I came to realize that the join point represented the center of the Earth, the north tube represented the axis of the Earth and the other a heavenly scanner. This was a first level of abstraction and the basis of the Greek celestial spheres of antiquity. The center was a point sized Earth and the coordinated were (right accession, declination) with the fiduciary point as the vernal equinox. My first observations with the dioptra seemed to have little relation to those from the gnomon and took me a good deal of rethinking to relate the two.

In constructing the miniature gnomon a 10cm high gnomon works well at 35N on a 9 by 15 inch “ground” (mine is 12” x 24”) and the radii at 10 degree sun elevation intervals out from the gnomon help see the shadow position at a glance but the marked papers with alignment holes allow for more serious observations over time.

In constructing the compound dioptra I used a 3 foot north sighting tube and an elevation screw for accurate alignment. The protractors measure distance from the celestial equator and the angle of rotation of the tube. The hardest part of the construction is the inserting of the bolt joining the two tubes. Since no artifacts nor pictures have come down to us, the design is my own.

Shooting the Sun, 11:40 AM, 2/29/2000 (8 E of Meridian, 5 S of Equator)

In conclusion, even if a theory is understood its implications can come as a surprise. There is a joy and a fascination to be found in active observation when it is coupled with intuition and theory. For me the effort observation takes is well worth it and I take this opportunity to recommend the rethinking of focused training together with a little serious observation involving counting and measuring.

m.h.webster, 2000 a.d. page #