Today, a 2000-year-old engineer speaks to us about
freedom and creativity. The University of Houston's
College of Engineering presents this series about the
machines that make our civilization run, and the
people whose ingenuity created them.
A colleague stopped by my
office the other day. "What do you know about Hero
of Alexandria?" he asked. Well, his name says he
was from Alexandria. We know he was alive in AD 62
because he wrote about a solar eclipse that year.
All we do know comes from his books on engineering
and science. He's best known for what we call
Hero's turbine -- a steam-driven whirligig that
worked a little like a modern jet engine. He did
mechanics, astronomy, and math.

"I'm doing a long calculation," my friend said. "I
have to take lots of square roots. By programming
Hero's method I can make my calculation run very
fast." With that, he had my attention.

You see, Hero invented feedback control devices.
His self-filling wine bowl, for example, had a
hidden float valve that automatically sensed the
level of wine in a bowl. When guests ladled out
wine, the bowl mysteriously refilled itself.

That's called a closed feedback loop. Egyptian
inventors had been making gadgets like that for 300
years before Hero. They'd made feedback-controlled
water-clocks and lamp-fillers.

By the time Hero lived, Rome had annexed Egypt, and
intellectual freedom gave way to imperial
authority. Hero was the end of an inventive age.
Not 'til the 1600s was another feedback device
invented. Feedback meant letting go of control --
letting a machine make its own decisions.
Authoritarian minds don't like that.

But Hero's square root method also works that way.
Suppose Hero wanted the square root of 5. He'd
guess an answer and then do the following process a
few times: Divide 5 by the guess, add the guess,
then divide that sum by 2. That gave an improved
guess.

Suppose I guess that the square root of 5 is 1.
Hero's process gives 3. Then I guess 3 and get
2.33. I guess 2.33 and get 2.24 -- already accurate
within 0.1 percent. The result converges very
quickly. Four tries usually get you all the
accuracy you need.

Hero had set up a do-loop for what we call an
iterative solution. The same person who worked with
feedback control loops now gives us a wonderful
computer do-loop. In either case we start a process
and then let it pass out of our hands.

Hero's story has a moral: we have no freedom
without letting go of control. Hero, this Egyptian
engineer whose only autobiography is his creative
product, was the last breath of a Golden Age in
North Africa. It'd be 1600 years before math,
mechanics, and human affairs would catch up with
that fragile inventive freedom we sell so easily
for security -- that we sell to satisfy our need to
stay in control.

I'm John Lienhard at the University of Houston,
where we're interested in the way inventive minds
work.

(Theme music)

For the best brief overview of Hero (or Heron) of
Alexandria, I recommend the Encyclopaedia
Britannica. I am also grateful to N.
Shamsundar, UH Mechanical Engineering Department, for
bringing Hero's square root algorithm to my
attention. He makes the interesting point that all
existing computer algorithms for square roots derive
from Hero's method. He just found he could speed his
particular calculation by programming it himself and
truncating the computation sooner.
For more on Hero's mechanics, see:

Mayr, O., The Origins of Feedback
Control. Cambridge, MA: MIT Press, 1970,
Chapter II.

Drachmann, A.G., The Mechanical Technology of
Greek and Roman Antiquity. Copenhagen:
Munksgaard, 1963.

Hero's square root algorithm goes as follows:
Suppose you want the square root of N. You guess an
initial value, Xo, and calculate a better value,
X1, with the averaging formula,

X1 = (Xo + N/Xo)/2

Then you use X1 as your next guess and calculate a
better guess, X2:

X2 = (X1 + N/X1)/2

Thus to get the square root of 5, I guess 1, so (1
+ 5/1)/2 = 3.

Next I guess 3, so (3 + 5/3)/2 = 2.333...

On the third trial, I guess 2.333... This time
(2.333... + 5/2.333...)/2 = 2.2381...

The value of the square root of 5, accurate to five
significant figures, is 2.2361, so I'm now within
0.0020 of the correct value, or less than 1/10th of
one percent in error. One more trial, and the
result is accurate to within six significant
figures.

The Engines of Our Ingenuity is
Copyright © 1988-1997 by John H.
Lienhard.

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