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**Session 35 - HAD III: From Hamlet to Crop Circles.**
*Oral session, Wednesday, January 07*
*Monroe, *

The late Lord Soli Zuckerman, science advisor to several British governments, encouraged the author, an astronomer, to test the theory that all crop circles were made by hoaxers. Within the hundreds of formations in Southern England he saw a thread of surprising historical content at the intellectual level of College Dons. One diagram in celestial mechanics involved triple conjunctions of Mercury, Venus and Mars every 67 2/3 years.

Ptolemy's fourth musical scale, tense diatonic, occurred in the circles during the period 1978-88. Starting on E, Ptolemaic ratios make our perfect diatonic scale of white notes on the keyboard of the piano or church organ. For separated circles the ratio was given by diameters, and for concentric circles it was diameters squared.

A series of rotationally symmetric figures began in 1988 which combined Ptolemy's ratios with Euclid's theorems. In his last plane theorem, Euclid (Elements 13,12) proved that the square on the side of an equilateral triangle is 3 times the square on the circumcircle radius -- diatonic note G. From the 1988 figure one can prove the square on the side is 16/3 times the square on the semi-altitude, giving note F(3). Later rotational figures over the next 5 years led to diatonic ratios for the hexagon, square and triangle. They gave with the exactness of Euclidean theorems the notes F, C(2) and E(2), and they are the only regular polygons to do so. Although these 4 crop theorems derive from Euclid, they were previously unknown as a set in the literature, nor had the Ptolemaic connection been published.

Professional magazines asked the readers to provide a fifth theorem that would generate the above 4 theorems, but none was forthcoming. Ultimately the cicle makers showed knowledge of this generating theorem using a 200-ft design at Litchfield, Hampshire.

After 1993, rotationally symmetric geometries continued to appear, but with much more complicated patterns. One design showed 6 crescent moons in a hexagon with cusps set on 2 concentric circles defining the note A(2). Here the mathem- atical level required application of Ptolemy's famous theorem of chords to confirm the A(2) ratio of exactly 10/3. The chords were the side of a hexagon joined to the side of a pentagon. We confirm Zuckerman's suggestion that there is a strong thread of expertise in the phenomenon worthy of scientific interest, and it spans a 20-year period. He asks: Why do they use a wheat field, and "how do they maintain their hidden identities?" Their type of knowledge rests in the past, and is not frequently found in the contemporary educational system.