THE NECKLACE OF DEMOCRITUS
Giorgio Carboni, May 1999
Tranlsation edited by Ron Wickersham, Santa Rosa, California, USA
The atomic theory of matter was first proposed by Leucippus, a Greek philosopher who lived in the 5th century before Christ. At this time the Greeks were trying to understand the way matter is made. According to Anassagora, it is possible to subdivide matter in smaller and smaller parts, and he proposed that this process can be continued with no limit. In Anassagora's view, you can always divide a bit of substance into two parts, and each of these parts is also divisible into two parts, and so on--no matter how small each part gets there is no problem dividing it again into even smaller parts. But according to Leucippus, eventually you arrive at small particles which can not be further subdivided. Leucippus called these indivisible particles atoms. Leucippus's atomic theory was further developed by his disciple, Democritus, (the subject of our story) who concluded that infinite divisibility of a substance belongs only in the imaginary world of mathematics and should not be applied to physics because he believed that in the real world matter is composed of discrete particles.
Now, imagine that between Democritus and his master Leucippus an argument arose on the real dimensions of these particles. We do not know if a dispute of this nature really took place between these two philosophers, but nothing prohibits us from imagining it. Both men believe that a limit will be reached when matter cannot be further subdivided, and Leucippus believes that this limit will be reached after dividing very few times: he takes the position that atoms are relatively large, almost big enough to be visible to the naked eye, but Democritus maintains instead that atoms are much, much, smaller. Extremely small, Democritus argues. But how small?
Now, Democritus is turning a grain of salt between his fingers, seemingly absorbed in his reflections on the problem at hand. This behavior is common enough for a philosopher, moreover in his world lacking television sets, it usually happened during those times. But what is he thinking? If we peer into his mind, we see him thinking that, "If it were possible to align the individual atoms of this grain end to end along a line, then I could demonstrate their smallness. In fact, the smaller each atoms is, the more of them there are and so the row gets longer. Therefore the length of the row would be indicative of their actual dimension". In particular, every time you halve the size of the particles which compose it, the length of the row you obtain increases four times (see diagram).
Democritus wondered how he could take apart the atoms in his grain of salt and line them up in a row, a problem made particularly difficult if he is correct that they are very, very small because of the resulting huge number of them. Here it is necessary to make another effort of imagination and Democritus addresses Hephaestus (Vulcan in Latin), the artisan God who has his workshop in the bowels of the volcano, Mt. Etna. In truth, the Greek philosophers did not love the gods very much and avoided resorting to their supernatural powers to explain reality, but taking a grain of salt apart into it's individual atoms and stringing them in a row was a technical problem out of the capacity of a common mortal (the reader forgives me).
Democritus goes to Hephaestus, and describes the nature of the problem: "... therefore, beloved Hephaestus, I give you this grain of salt. It's volume is exactly a cubic millimeter. Would you, please, align all the atoms side by side assembling them into one necklace? I shall present the necklace to the wife of Leucippus, so that when he sees it, he will understand how small the atoms are. Only a God can complete an enterprise like this, and among all the gods, only you would be able to do it". Hephaestus answers: "Actually, we gods observe with interest you mortal men, and often your problems also provoke arguments between us. The problem of the dimensions of atoms has not lacked to raise controversies on Mount Olympus. You must know that we gods have much knowledge, and while each of us is an expert in his field only, no one wants to appear less knowledgeable than another...so I am also anxious to find how long this necklace will be. Come back tomorrow and take the necklace I'll prepare for you!"
Therefore the two parted. That night, Democritus tried hard to sleep... "How long will the necklace be?" he wondered "meters, tens of meters, hundred of meters or even kilometers? Heaven knows!... " Poor Democritus, he hasn't the faintest idea! "I hope that it is at least 200 meters long!" he sighed. Unfortunately, Hephaestus did not deliver the necklace to him the next day, nor the next, not even many days following, engaged as he was to manufacture crews for gods and heroes and to look for his wife Aphrodite (Venus in Latin) who was rather libertine in spirit. It is for this reason that since that time, long ago, our poor Democritus has not been able to fall asleep.
Now it's hundreds and thousands of years later, and we mortal men have
accumulated much knowledge of physics and chemistry. Today we do know the
dimensions of atoms. You can find the sizes of atoms in a chemistry handbook such as the
"CRC Handbook of Chemistry and Physics". But reading the dimension from a book
doesn't always give us the perception of how large these particles are... or better, how
very, very small. Follow the reasoning of Democritus and demonstrate how tiny real atoms
are by calculating the length of his necklace made from a cube of table salt (pure NaCl)
exactly one millimeter on a
side, then send me the result:
soon I receive your answers, I will send an e-mail to him with the
solution to his question, so he will finally be able to fall asleep.
In reality, it is not possible to make a single row of Sodium and Chlorine atoms sitting side by side in a row as this arrangement would be unstable. However, this calculation does serve to demonstrate just how tiny atoms really are. Anyway, make your calculation as if it were possible to maintain these atoms in a row, as this is the necklace of atoms Democritus was contemplating.
Before looking at the answer on the following page, at least look up the dimensions of the atoms of salt or make your own estimate or better calculate the length of the necklace. Let's compare your answer to the result Hephaestus whispered to us.
To the Solution of the Problem