A famous cartoon shows Einstein at a board, trying out one possibility after
another: E = mc^{1}, E = mc^{2}, E = mc^{3}.... But he didn't
really do it that way, arriving at the squaring of c by mere chance, of
course. So why did the conversion factor turn out to be c^{2}?
The story of how an equation with a "squared" in it came to be plucked from all
other possibilities takes us to France in the early 1700s—to a woman who,
in her outspoken brilliance, was out of step with her time.
Emilie du Châtelet (17061749)
As a girl, Emilie de Breteuil lived with her family overlooking the Tuileries
gardens in Paris, in an apartment with 30 rooms and 17 servants. But although
her brothers and sisters turned out as might be expected, Emilie was different,
as her father wrote: "My youngest flaunts her mind, and frightens away the
suitors."
Despite her father's fears, Emilie had many suitors. At the age of 19, she
chose one of the least objectionable courtiers as a husband. He was a wealthy
soldier named du Châtelet who would conveniently be on distant campaigns
much of the time. It was a pro forma arrangement, and in the custom of the
time, her husband accepted her having affairs while he was away.
When she was a 27yearold mother of three, du Châtelet began perhaps the
most passionate affair of her life—a true partnership of heart and mind.
Her lover, the writer Voltaire, recounted later, "In the year 1733 I met a
young lady who happened to think nearly as I did." She and Voltaire shared deep
interests: in political reform, in the fun of fast conversations, and, above
all, in advancing science as much as they could.
Correcting Voltaire and improving on Newton
Together, du Châtelet and Voltaire turned her husband's château at
Cirey, in northeastern France, into a base for scientific research with a
library comparable to that of the Academy of Sciences in Paris, as well as the
latest laboratory equipment from London.
When they engaged in their teasing, mock battling, it wasn't the case of a
widely read man deciding when to let his young lover win. Du Châtelet was
the real investigator of the physical world, and the one who decided that
there was one key question that had to be turned to now: what is energy?
Most people felt energy was already sufficiently understood. Voltaire had
covered the seemingly ordained truths in his own popularizations of Newton: an
object's energy is simply the product of its mass times its velocity, or
mv^{1}. If a fivepound ball is going 10 mph, it has 50 units of
energy. But du Châtelet knew there was a competing, albeit highly
theoretical view proposed by Gottfried Leibniz, the great German natural
philosopher and mathematician. For Leibniz, the important factor was
mv^{2}.
A weighty test
Du Châtelet and her colleagues found the decisive evidence in the recent
experiments of Willem 'sGravesande, a Dutch researcher who'd been letting
weights plummet onto a soft clay floor. If the simple E = mv^{1} was
true, then a weight going twice as fast as an earlier one would sink in twice
as deeply. One going three times as fast would sink three times as deep. But
that's not what 'sGravesande found. If a small brass sphere was sent down twice
as fast as before, it pushed four times as far into the clay. It if was
flung down three times as fast, it sank nine times as far into the
clay.
Du Châtelet deepened Leibniz's theory and then embedded the Dutch results
within it. Now, finally, there was a strong justification for viewing
mv^{2} as a fruitful definition of energy.
End of a partnership
Du Châtelet was one of the leading interpreters of modern physics in
Europe as well as a master of mathematics, linguistics, and the art of
courtship. But there was one thing she couldn't control. In April of 1749, she
wrote to Voltaire, "I am pregnant and you can imagine ... how much I fear for
my health, even for my life ... giving birth at the age of forty." She didn't
rage at the clear incompetence of her era's doctors; she just said to Voltaire
that it was sad leaving before she was ready.
She survived the birth the next fall, but infection set in, and within a week
she died. Voltaire was beside himself: "I have lost the half of myself—a
soul for which mine was made."
Einstein's new take on an old formula
Over time, physicists became used to multiplying an object's mass by the square
of its velocity (mv^{2}) to come up with a useful indicator of its
energy. If the velocity of a ball or rock was 100 mph, then they knew that the
energy it carried would be proportional to its mass times 100 squared. If the
velocity is raised as high as it could go, to 670 million mph, it's almost as
if the ultimate energy an object will contain should be revealed when you look
at its mass times c squared, or its mc^{2}.
This isn't a proof, of course, but it seemed so natural, so "fitting," that
when the expression mc^{2} did suddenly appear within Einstein's more
detailed calculations, it helped make more plausible his startling conclusion
that the seemingly separate domains of energy and mass could be connected, and
that the symbol c—the speed of light—was the bridge.

As a teenager, Emilie du Châtelet pored through Descartes' analytic
geometry. As a grown woman, she was one of Newton's greatest interpreters.
 
Voltaire, perhaps the most renowned intellectual of the Enlightenment movement,
wrote that du Châtelet had "a soul for which mine was made."
 
This plate of diagrams is from du Châtelet's Institutions
physiques, her elaboration on the ideas of Leibniz. She finished a major
commentary on Newton just before her death.
 
