Title: The Vitruvius Tale of Archimedes and the Golden Crown
ArXiv ID: 1108.2204
Date: 2011-08-11
Authors: Amelia Carolina Sparavigna
📝 Abstract
The paper discusses the tale that we can find in "The Architecture" by Vitruvius, on a method used by Archimedes to determine the percentage of gold and silver in a crown. The method is based on the immersion of bodies, allowing the evaluation of their volume in the case of irregular shapes. The measurement, as reported in "The Architecture", seems to be difficult to realize. But, using a vessel for a water-clock, the approach that Vitruvius described is possible. Here the discussion and experiments.
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case, is the Vitruvius' description useful for measurement? First of all, let us see what Vitruvius is writing in the third chapter, "of the method of detecting silver when mixed with gold" [1]. "Charged with this commission (to determine whether the crown had silver inside or not), he (Archimedes) by chance went to a bath, and being in the vessel, perceived that, as his body became immersed, the water ran out of the vessel. Whence, catching at the method to be adopted for the solution of the proposition, he immediately followed it up, leapt out of the vessel in joy, and, returning home naked, cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, Eureka, (I have found it out). After this, he is said to have taken two masses, each of a weight equal to that of the crown, one of them of gold and the other of silver. Having prepared them, he filled a large vase with water up to the brim, wherein he placed the mass of silver, which caused as much water to run out as was equal to the bulk thereof. The mass being then taken out, he poured in by measure as much water as was required to fill the vase once more to the brim. By these means he found what quantity of water was equal to a certain weight of silver. He then placed the mass of gold in the vessel, and, on taking it out, found that the water which ran over was lessened, because, as the magnitude of the gold mass was smaller than that containing the same weight of silver. After again filling the vase by measure, he put the crown itself in, and discovered that more water ran over then than with the mass of gold that was equal to it in weight ; and thus, from the superfluous quantity of water carried over the brim by the immersion of the crown, more than that displaced by the mass, he found, by calculation, the quantity of silver mixed with the gold, and made manifest the fraud of the manufacturer." Let us see now the calculations in Ref. 3, which is questioning the Vitruvius' description. This reference assumes the method described by Vitruvius is based on a simple immersion of bodies. It is supposed that the Hiero's crown weighed 1000 grams. Gold has a density of 19.3 grams/cm 3 , 1000 grams of gold would have a volume of 51.8 cm 3 . As an example, the reference continues considering that the goldsmith replaced a 30% (300 grams) of the gold in the crown by silver. The density of silver is of 10.5 grams/cm 3 . As a consequence, the goldsilver crown has a volume of 64.8 cm 3 . In fact, in this example, the difference of volumes is more than 10 cm 3 . Ref.3 supposes a vessel used in the experiment, with a circular opening with a diameter of 20 cm. The opening has then a cross-sectional area of 314 cm 2 . The solid gold immersed in the vessel would raise the level of water at the opening by 0.165 cm. The crown instead would raise the level of the water at the opening by 0.206 cm. The difference in the level of water, displaced by the crown and the solid gold is of 0.41 mm. Of course, this difference is too small to be directly observed. Other sources of error are the surface tension, the water remaining on the objects after removal and air bubbles. After this discussion, this method of measurement is discharged. In fact a simple immersion in a vessel is not suitable for measurements. But Vitruvius is not describing a simple immersion method, but a method based on the overflow of water at the rim of the vessel (labra) and replacement of water in the vessel, evaluated by means of some small measuring vessels (sextario mensus, for the Latin text, see Ref.4). If we imagine a classic marble vessel with a smooth rim, the overflow of the water is difficult to control. But Archimedes had surely a more suitable device, to measure the water displaced by immersion. In my opinion, it is the vessel used for the water-clock (see Fig. 1), that is, a vessel with a hole near the rim. Let us remember that a water-clock, or clepsydra, is a device in which time is measured by the flow of water. Water clocks, along with sundials, are assumed to be the oldest time-measuring devices, "with the only exceptions being the vertical gnomon and the day-counting tally stick" [5]. For sure, the simplest form of water-clocks existed in Babylon and Egypt, around the 16th century BC. The Greeks improved the water clock design. Let us assume that Archimedes used a vessel with a hole. I have tried to repeat what is described by Vitruvius in the following manner. I used two transparent plastic cups, one with a hole, with a plastic ring glued about it. It simulates the vessel of a water-clock. Then I used a plumb (400 grams) and a weight of a balance of 100 grams for immersions. The initial levels of water in the two cups is shown in Fig. 3A. The level in the right cup is determined by its hole. The level in the left cup is marked on it by a pen. The plumb (red) is inserted in the cup with the hole; with a slow immersion of the object, the
