# De Moivre's Equation The Most Dangerous Equation - Trial of the Pyx

#### randomletters

##### New Member
I don't understand the Trial of the Pyx example in the following (http://press.princeton.edu/chapters/s8863.pdf).

De Moivre's Equation (http://math.stackexchange.com/questions/253486/name-of-de-moivres-equation)

Excerpt:
In 1150, a century after the Battle of Hastings, it was recognized that
the king could not just print money and assign to it any value he chose.
Instead the coinage’s value must be intrinsic, based on the amount of
precious materials in its makeup. And so standards were set for the
weight of gold in coins—a guinea should weigh 128 grains (there are
360 grains in an ounce). It was recognized, even then, that coinage
methods were too imprecise to insist that all coins be exactly equal
in weight, so instead the king and the barons, who supplied the London Mint (an independent organization) with gold, insisted that coins
when tested* in the aggregate [say one hundred at a time] conform
to the regulated size plus or minus some allowance for variability
[1/400 of the weight] which for one guinea would be 0.32 grains and
so, for the aggregate, 32 grains). Obviously, they assumed that variability decreased proportionally to the number of coins and not to its square root.

If the variability were too great, it would mean that there would be an unacceptably large number of too heavy coins produced that could be collected, melted down, and recast with the extra gold going into the pockets of the minter. By erroneously allowing too much variability, the Mint could stay within the bounds specified and still make extra money by collecting heavier than average coins and reprocessing them.

My Question: I am confused how assuming proportionality would allow for greater variability. The way I read the equation, if i replace the square root of n with n in the denominator I will get a smaller value and therefore my assumption of proportionality would lead to a lower allowable threshold for the aggregate weight of the coins.

I may be misunderstanding how to map the equation to this example. I was mapping 1/400 to the standard deviation of the averages of the samples.

Thanks!

#### rogojel

##### TS Contributor
hi,
I think the point the author makes is that by assuming proportionality with n the king accepted tests that had a higher then necessary variability and thus allowed the Mint to make money by melting down heavy guinea pieces.

Had he known that the variation is proportional to the square root of n his control limits would have been stricter resulting in tighter controls for the Mint.

BTW Neal Stephensons Baroque Trilogy is describing this trick of remelting heavier coins in several places.

regards
rogojel