Reprinted with permission
from the December 1992 issue of SHARE INTERNATIONAL magazine.

MUSIC OF THE SPHERES?

Interview with Gerald S. Hawkins
by Monte Leach

Gerald S. Hawkins earned a Ph.D. in radio astronomy with Sir Bernard
Lovell at Jodrell Bank, England, and a D.Sc. for astronomical research
at the Harvard-Smithsonian Observatories. His undergraduate degrees
were in physics and mathematics from London University. Hawkins'
discovery that Stonehenge was built by neolithic people to mark the
rising and setting of the sun and moon over an 18.6-year cycle
stimulated the new field of archaeoastronomy. From 1957 to 1969 he was Professor of Astronomy and Chairman of the Department at Boston
University, and Dean of the College at Dickinson College from 1969 to
1971. He is currently a commission member of the International
Astronomical Union, and is engaged in research projects in
archaeoastronomy and the crop circle phenomenon.

Monte Leach: How did you get interested in the crop circle phenomenon?

Gerald Hawkins: Many years ago, I had worked on the problem of
Stonehenge, showing it was an astronomical observatory. My friends and colleagues mentioned that crop circles were occurring around
Stonehenge, and suggested that I have a look at them.

I began reading Colin Andrews' and Pat Delgado's book, Circular Evidence. I found that the only connection I could find between
Stonehenge and the circles was geographic. But I got interested in
crop circles for their own sake.

ML: What interested you about them?

GH: I was very impressed with Andrews' and Delgado's book. It
provided all the information that a scientist would need to start an
analysis. In fact, Colin Andrews has told me that that's exactly what
they intended to happen. I began to analyse their measurements
statistically.

ML: What did you find?

GH: The measurements of these patterns enabled me to find simple
ratios. In one type of pattern, circles were separated from each other,
like a big circle surrounded by a group of so-called satellites. In
this case, the ratios were the ratios of diameters. A second type of
pattern had concentric rings like a target. In this case, I took the
ratios of areas. The ratios I found, such as 3/2, 5/4, 9/8, 'rang a
bell' in my head because they are the numbers which musicologists call
the 'perfect' intervals of the major scale.

ML: How do the ratios correspond with, for instance, the notes on a
piano that people might be familiar with?

GH: If you take the note C on the piano, for instance, then go up to
the note G, you've increased the frequency of the note (the number of vibrations per second), or its pitch, by 1 1/2 times. One and one-half
is 3/2. Each of the notes in the perfect system has an exact ratio --
that is, one single number divided by another, like 5/3.

ML: If we were going to go up the major scale from middle C, what
ratios would we have?

GH: The notes are C, D, E, F, G, A and B. The ratios are 9/8, 5/4,
4/3, 3/2, 5/3, 15/8, finishing with 2, which would be C octave.

ML: How many formations did you analyse and how many turned out to
have diatonic ratios relating to the major scale?

GH: I took every pattern in their book, Circular Evidence. I found
that some of them were listed as accurately measured and some were
listed as roughly or approximately measured. I finished up with 18
patterns that were accurately measured. Of these, 11 of them turned out
to follow the diatonic ratios. Colin Andrews has since given me
accurate measurements for one of the circles in the book that had been discarded because it was inaccurate. That one turned out to be
diatonic as well. We finished up with 19 accurately measured
formations, of which 12 were major diatonic.

The difficulty of hitting a diatonic ratio just by chance is
enormous. The probability of hitting 12 out of 19 is only 1 part in
25,000. We're sure, 25,000 to 1, that this is a real result.

ML: Could this in some way be a 'music of the spheres', so to speak?

GH: I am just a conventional scientist analyzing this mathematically.
One has to report that the ratios are the same as the ratios of our own
Western invention -- the diatonic ratios of the (major) scale. We have
only developed this diatonic major scale in Western music slowly
through history. These are not the ratios that would be used in
Japanese music, for instance.

But I am not calling the crop circles 'musical'. They just
follow the same mathematical relationships.

ML: You've established that there's a 25,000 to 1 chance that these
ratios are random occurrences. What about natural science processes?

ML: You're investigating the theory that it's done by hoaxers to see
if that makes sense?

GH: Yes, but now I've upgraded the investigation, because I've found
an intellectual profile. This means I've eliminated all natural
science processes, so I don't have to consider any of those any more.
The intellectual profile narrows it down.

ML: What have you found in terms of this intellectual profile?

GH: My mathematical friends have commented on my findings. The
suspected hoaxers are very erudite and knowledgeable in mathematics.
We have equated the intellectual profile, at least at the mathematics
level, as senior high school, first year college math major. That's
pushing it to a narrow slot. But there's more to this than just the
diatonic ratios.

ML: How so?

GH: The year 1988 was a watershed because that was when the first
geometry appeared. It is in Circular Evidence. These geometrical
patterns were quite a surprise to me. There are only a few of them.

ML: These are in addition to the circles you investigated in terms of
the diatonic ratios?

GH: The geometry is really 'the dog', and the diatonic ratios of the
circles are 'the tail.' That is, there is much more involved in the
geometry than in those simple diatonic ratios in the circles, although, interestingly, the diatonic ratios are also found in the geometry,
without the need for measurement. The ratio is given by logic -- mind
over matter.

ML: What did you find from these more complex patterns?

GH: Very interesting examples of pure geometry, or Euclidean geometry.

ML: You found Euclidean theorems demonstrated in these other patterns?

GH: These are plane geometry, Euclidean theorems, but they are not in
Euclid's 13 books. Everybody agrees that they are, by definition,
theorems. But there's a big debate now between people who say that
Euclid missed them, and those that say he didn't care about them -- in
other words, that the theorems are not important. I believe that
Euclid missed them, the reason being that I can show you a point in his
long treatise where they should be. They should be in Book 13, after proposition 12. There he had a very complicated theorem. These would
just naturally follow. Another reason why he missed them was that we
are pretty sure that he didn't know the full set of perfect diatonic
ratios in 300 BC.

ML: These are theorems based on Euclid's work, but ones that Euclid
did not write down himself. But they are widely accepted as fulfilling
his theorems on geometry?

GH: Only widely accepted after I published them. They were unknown.

ML: Based on your analysis of these crop circles, you discovered the
theorems yourself?

GH: Yes. A theorem, if you look it up in the dictionary, is a fact
that can be proved. The trouble is, first of all, seeing the fact, and
then being able to prove it. But there's no way out once you've done
that. The intellectual profile of the hoaxer has moved up one notch.
It has the capability of creating theorems not in the books of Euclid.

It does seem that senior high school students can prove these
theorems, but the question is, could they have conceived of them to put
them in a wheat field? In this regard, we've got a very touchy
situation in that there is a general theorem from which all of the
others can be derived. I stumbled upon it by luck and accident and
colleagues advised me to not publish it. None of the readers of
Science News [which published an article on this subject] could
conceive of that theorem. In a way, it does indicate the difficulty of conceiving these theorems. They may be easy to prove when you're told
them, but difficult to conceive.

ML: And I would assume that the readers of Science News,would be
pretty well versed in these areas.

GH: It's a pretty good cross-section. The circulation is 267,000. We
found from the letters that came in that Euclidean geometry is not part
of the intellectual profile of our present-day culture. But it is part
of the culture of the crop circle makers.

ML: What about the more recent formations?

GH: Now we enter the other types of patterns -- the pictograms, the insectograms. Exit Gerald S. Hawkins. I don't know what to do about
those.

ML: Your investigations leave off at the geometric patterns.

GH: The investigations are continuing, but I haven't gotten anywhere.
I see no recognizable mathematical features. I'm approaching it
entirely mathematically, because there is the strength of numbers.
There's the unchallengeability of a geometric proof of a theorem, for
example. The other patterns involve other types of investigation, such


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