Galileo was a huge admirer of the ancient Greek mathematician and engineer Archimedes of Syracuse (c.287-212BC). In his earliest published work, *La bilancetta* (1586), Galileo says that anyone ‘who has read and understood the very subtle inventions of this divine man in his own writings… most clearly realizes how inferior all other minds are to that of Archimedes.’ Archimedes surviving writings cover a wide range of topics, from abstruse problems in pure geometry, through theoretical mechanics, to the mathematical analysis of some intriguing puzzles. What was it that impressed Galileo so much?

**Contents**

- I. Eureka! The case of the crooked crown
- II. How did this help? The lightness of silver
- III. Just one more thing. Would it work?
- IV. On floating bodies: a matter of principle
- V. Stability: the tip of the iceberg
- VI. Archimedes, military engineer

**I. Eureka! The case of the crooked crown**

In the popular imagination Archimedes is probably best known for his working methods. As every skoolboy kno, ^{1} Archimedes discovered something important in his bath. Whereupon, ‘transported with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek “Eureka, eureka [I have found it, I have found it]”.’

What had Archimedes found? The story was first recorded by the Roman architect and engineer Vitruvius (c.75-after 15 BC) in his tome *On Architecture* (*De architectura*, Bk.9). Apparently Hiero II (306 – 215BC), King of Syracuse, had given a goldsmith a carefully weighed quantity of gold with which to make a crown. The goldsmith, according to Vitruvius, duly returned ‘an exquisitely finished piece of handiwork, and it appeared that in weight the crown corresponded precisely to what the gold had weighed.’ Hiero, however, came to suspect that some of the gold had been replaced with cheaper silver. How to tell without damaging the fine crown? The king requested Archimedes to consider the matter.

The latter, while the case was still on his mind, happened to go to bathe and, on getting into a tub, observed that the more his body sank into it the more water ran out over the tub. ^{2} As this pointed out the way to explain the case in question, without a moment’s delay, and transported with joy, he jumped out of the tub and rushed home naked…’ as described above.

**II. How did this help? The lightness of silver**

But how did the overflow from the bath help to solve the question of the crown? Archimedes knew that silver is not only cheaper than gold but also lighter, i.e. less dense.

Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the mass of silver into it. As much water ran out as was equal in bulk [volume] to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.

After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.

In other words, the crown’s density was somewhere between the density of pure gold and that of pure silver, and was therefore made of a mixture of gold and silver, not of pure gold. Case closed! Collapse – well, actually, probably painful death – of stout party.

**III.
Just one more thing. Would it work?**

This looks fine in theory, but would this test actually work in practice? Gold is almost twice the density of silver, so a pure gold crown would be almost half the volume of one made of pure silver. Nevertheless, even supposing quite heavy (and costly) crowns of maybe 1 lb (or, say, 500g) in weight, the difference in volume will only be about 20 ml (i.e. about 1 tablespoonful for those of a culinary disposition). But the goldsmith may have replaced only a small proportion of the gold with silver, say ¼, so the difference in volume will be correspondingly less. Maybe only 4 or 5 ml, or about a teaspoonful or less? Even with Vitruvius’ refinement of refilling the vessel of water (rather than trying to catch the overflow) this will be a rather rough and ready test. No jury would convict.

Galileo
didn’t like Vitruvius’ method either, but he had a much more
fundamental objection. This method, he says, ‘seems a thing, so to
speak, very crude and far from refinement (*esquisitezza*)….
And lacking that exactness that is required in mathematical matters.’
In other words, from what Galileo knew of Archimedes’ writings, it
was unworthy of the great geometer. So, it ‘made me think several
times how, by means of water, one could exactly determine the mixture
of two metals. And at last, after having carefully gone over all that
Archimedes demonstrates in his books *On
Floating Bodies* and *On
the Equilibrium of Planes*,
a method came to my mind which very accurately solves our problem. I
think it probable that this method is the same that Archimedes
followed, since, besides being very accurate, it is based on
demonstrations found by Archimedes himself.’

**IV.
On floating bodies: a matter of principle**

Archimedes’ treatise *On Floating Bodies* is concerned with bodies that sink as well as those that float.^{3} In the rigorous style of Euclidean geometry, Archimedes’ treatise starts from a few simple axioms, which are then developed into ever more complex theorems. One of the early propositions (I.7, Boyer 137) is known as Archimedes’ ‘Principle of Buoyancy’, namely that:

A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.

I think most of us would accept the first part without too much argument. The second is less obvious, or maybe not obvious at all. But suppose that you take a brick, say, and weigh it when it is immersed in water; buoyed up by the water, it will weigh less than it does when weighed in air. How much less? Imagine an identically brick-shaped volume of water within the larger body of water. The water-brick, if weighed on its own in air, would have weight, and would fall. But when in the water it will not move, neither down nor up, so the surrounding water must be holding it up with a force equal to its weight. And that upthrust would be the same if the brick-shaped space were an actual brick, whether made of clay or gold or whatever. So, any heavy body weighed in water will be lighter by the weight of the volume of water that the body displaces. QED.^{4}

##
**V. Stability: the tip of the iceberg**

Clearly
Galileo was delighted to work out what was almost certainly the truth
behind Vitruvius’ story – although actually Galileo was very far
from being the first person to do so. Archimedes’ principle and the
basic process described above – weigh something in air, weigh it
again in water – provides a neat, quick, simple and very precise
way of measuring the relative density of any object heavier than
water. Galileo promptly designed a clever little ‘hydrostatic’
balance, his *bilancetta*,
to make the process even simpler. (I’ll come back to that in
another post very soon.)

But
Archimedes himself wrote nothing about any such practical
applications. The bulk of his treatise *On
Floating Bodies* is devoted to proving a range
of often extraordinarily complex theorems about the equilibrium
positions of various floating bodies of different shapes and
densities. A chunk cut off a paraboloid^{5}, anyone? Will it float
base up (like a boat, say), or tip up (like an iceberg)? Archimedes
is your man. In developing and proving his theorems Archimedes uses
the cumbersome (but rigorous) methods of traditional Greek geometry.
There was no modern algebra or calculus to make his task easier, let
alone computer modelling. Areas and volumes are often estimated by
the aptly named ‘method of exhaustion’; a conclusion may be
proved by *reductio ad absurdum*,
that is, by eliminating all alternatives. It is like watching someone
perform brain surgery with an axe.

I used to think that this was the purest of pure mathematics, mathematics for its own sake, with no menial thought of practical application. But then I remembered Carl Boyer’s playful claim^{6}: ‘Archimedes could well have taught a theoretical course on naval architecture’. And I also remembered the ‘Mary Rose’, and the many other ships that have capsized unexpectedly. Either way, Archimedes’ mechanical writings gave Galileo a model for a new style of mathematical physics, precise and powerful, which he used in his own subsequent studies of mechanics and motion. And Galileo explicitly sets his work in the great Venetian ship-building yard, the Arsenal .

##
**VI. Archimedes, military engineer**

Ironically, in antiquity Archimedes was probably most famous, not so much as a brilliant geometer, but rather as a military engineer. His home city of Syracuse, on the eastern shore of Sicily, was caught up in the Punic Wars[LINK], the long-running conflict between Carthage and Rome for control of the Mediterranean. According to the Roman historian Plutarch[LINK] (c.46 – c.120 AD), Archimedes master-minded the construction of a whole series of machines and devices to help defend his city: catapults, huge claws to crush enemy ships, burning glasses to set them on fire.

Eventually, however, Syracuse did fall to the Romans, and Archimedes himself was killed in the subsequent pillage of the city.

- With apologies to Geoffrey Willans and Ronald Searle’s ‘Down with Skool’ (London, 1953) which contains some fine Searle drawings of Pythagoras stalking a parallelogram, but sadly not one of Archimedes.
- We have to imagine that the tub was full to the brim.
- A later volume in my
*Galileo, Detective*series will probably be entitled*On bodies that float in water*, after Galileo’s later treatise of that title. Probably set in Venice, of course. - i.e. Quod erat demonstrandum, That which was to be proved.
- the solid that you get if you spin a parabola about its axis
- Boyer, History of Mathematics