Hiero
II
306?-215 BC
Golden wreath from
Amphipolis, Macedonia
Fourth century BC
Golden wreath from
Derveni, Macedonia
Fourth century BC
Golden
wreath from
Vergina, Macedonia
Fourth century BC
The wreath and the gold
nugget have equal weight
The wreath displaces more
water than the gold nugget
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In
the first century BC the Roman architect Vitruvius related a story
of how Archimedes uncovered a fraud in the manufacture of a golden
crown commissioned by Hiero II, the king of Syracuse. The crown
(corona in Latin) would have been in the form of a wreath,
such as one of the three pictured from grave sites in Macedonia.
Hiero would have placed such a wreath on the statue of a god or
goddess. Suspecting that the goldsmith might have replaced some
of the gold given to him by an equal weight of silver, Hiero asked
Archimedes to determine whether the wreath was pure gold. And because
the wreath was a holy object dedicated to the gods, he could not
disturb the wreath in any way. (In modern terms, he was to perform
nondestructive testing). Archimedes' solution to the problem, as
described by Vitruvius, is:
The
solution which occurred when he stepped into his bath and caused
it to overflow was to put a weight of gold equal to the crown,
and known to be pure, into a bowl which was filled with water
to the brim. Then the gold would be removed and the king's crown
put in, in its place. An alloy of lighter silver would increase
the bulk of the crown and cause the bowl to overflow.
Although theoretically
sound, this method has been criticized for several reasons. First,
in spite of Vitruvius' s description of it as "the result of a boundless
ingenuity", the method requires much less imagination than Archimedes
exhibits in his extant writings. Second, it does not make use of
Archimedes' Law of Buoyancy or his Law of the Lever. Third, and
most important, it would be difficult to implement Vitruvius's method
with the degree of measurement accuracy available to Archimedes.
The third
point needs some amplification. The largest known golden wreath
from Archimedes' time is the one pictured from Vergina. It has
a maximum rim diameter of 18.5 centimeters and a mass of 714 grams,
although some of its leaves are missing. For the purposes of illustration,
let us assume that Hiero's wreath weighed 1000 grams and that
a container with a circular opening of diameter 20 centimeters
was used. The opening would then have a cross-sectional area of
314 square centimeters. (All calculations are performed to three
significant figures.)
Because gold
has a density of 19.3 grams/cubic-centimeter, 1000 grams of gold
would have a volume of 1000/19.3 = 51.8 cubic-centimeters. Such
a quantity of gold would raise the level of the water at the opening
of the container by 51.8/314 = 0.165 centimeters.
Next, suppose
the dishonest goldsmith replaced 30% (300 grams) of the gold in
the wreath by silver. Silver has a density of 10.6 grams/cubic-centimeter
and so the gold-silver crown would have a volume of 700/19.3 +
300/10.6 = 64.6 cubic-centimeters. Such a crown would raise the
level of the water at the opening by 64.6/314 = 0.206 centimeters.
The difference
in the level of water displaced by the wreath and the gold is
thus 0.206 minus 0.165 centimeters, or 0.41 millimeters. This
is much too small a difference to accurately observe directly
or measure the overflow from considering the possible sources
of error due to surface tension, water clinging to the gold upon
removal, air bubbles being trapped in the lacy wreath, and so
forth. Additionally, the change in water level would be even less
than 0.41 millimeters if the wreath had a mass of less than 1000
grams, or if the diameter of the container opening were larger
than 20 centimeters, or if less than 30% of the gold were replaced
with silver.
A more imaginative
and practical technique to detect the fraud is the following,
which makes use of both Archimedes' Law of Buoyancy and his Law
of the Lever. Suspend the wreath from one end of a scale and balance
it with an equal mass of gold suspended from the other end. Then
immerse the suspended wreath and gold into a container of water.
If the scale remains in balance then the wreath and the gold have
the same volume, and so the wreath has the same density as pure
gold. But if the scale tilts in the direction of the gold, then
the wreath has a greater volume than the gold, and so its density
is less than that of gold. It must then be a alloy of gold and
some lighter material.
To check the
practicality of this technique let us again assume a 1000-gram
wreath which is an alloy of 70% gold and 30% silver. Because its
volume is 64.6 cubic centimeters, it displaces 64.6 grams of water.
(Water has a density of 1.00 gram/cubic-centimeter.) Its apparent
mass in water is thus 1000 minus 64.6 grams, or 935.4 grams. Next,
1000 grams of pure gold has a volume of 51.8 cubic centimeters,
and so its apparent mass in water is 1000 minus 51.8 grams, or
948.2 grams. Thus, when both ends of the scale are immersed in
water, there is an apparent mass of 935.4 grams at one end and
an apparent mass of 948.2 grams at the other end, an imbalance
of 12.8 grams. Scales from Archimedes' time could easily detect
such an imbalance in mass. Additionally, sources of error arising
with Vitruvius's method (surface tension and clinging water) would
not arise with this scale method.
It should
be remarked that the scale method still works if the masses of
the wreath and the gold are not equal. One simply adjusts their
distances from the fulcrum of the scale until the scale balances
before dipping them into the water. The two methods
described above can be summarized as follows: Under our assumptions
(a 1000-gram wreath consisting of 700 grams of gold and 300 grams
of silver) the difference in volume between the wreath and 1000
grams of pure gold is 12.8 cubic-centimeters. Vitruvius's method
attempts to detect this volume difference by detecting an equal
volume of displaced water. Now, 12.8 cubic-centimeters of water
would form a cube of 2.34 centimeters on each side and would be
easily detected in that form. But when 12.8 cubic-centimeters
of water is spread over a container opening large enough to accomodate
the wreath (in our example, an opening of area 314 square-centimeters)
it translates to a vertical displacement of only 0.41 millimeters.
Such a displacement of water level is too small to accurately
measure by sight or through an overflow measurement. The scale
method essentially translates this 12.8-cubic-centimeter volume
difference into a 12.8-gram imbalance on a scale, and such an
imbalance is easily detectable with ancient scales.
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