Archimedes, Letter to Eratosthenes (Third Century B.C.E.)and

Marcus Vitruvius Pollio, Archimedes’ “Eureka!” Moment (c. 30–20 B.C.E.)

Scientific innovation blossomed during the Hellenistic period. Alexander the Great had ordered the city of Alexandria built, and it became the capital of the Hellenistic kingdom of the Ptolemies. Scientists flocked to the city’s royally funded library and research institute, including the mathematician Archimedes of Syracuse (287–212 B.C.E.). Here is where he most likely met the library’s director, Eratosthenes of Cyrene (c. 285–194 B.C.E.), a pioneer in mathematical geography. Upon returning home, Archimedes remained in contact with Eratosthenes, thereby promoting the exchange of ideas typical of the intellectual scene at the time. Archimedes included the letter excerpted below in the preface to his treatise Method. Discovered in 1906, it is unusual in its focus when compared to works by other classical Greek geometers. As Archimedes writes to Eratosthenes, he did not simply want to present his discoveries as finished products, in this case certain geometric theorems; he wanted to pull back the veil on the steps of analysis he took to arrive at them. The second document recounts a story about Archimedes’ thinking in action included in a first-century B.C.E. text. For its author, the Roman architect Vitruvius, the tale embodied the wisdom past thinkers had to offer contemporary civilization. Together, the two documents illuminate the thrill of discovery through measurement, observation, and experimentation that helped to lay the foundation of later Western scientific thought.

From The Works of Archimedes, ed. T. L. Heath (Mineola, NY: Dover Publications, Inc., 2002), 12–13; and Vitruvius: Ten Books on Architecture, trans. Ingrid D. Rowland (New York: Cambridge University Press, 1999), 108.

Archimedes

Archimedes to Eratosthenes greeting.

I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to discover the proofs, which at the moment I did not give. . . .

. . . The proofs then of these theorems I have written in this book and now send to you. Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer [of mathematical inquiry], I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. . . .

Vitruvius

As for Archimedes, although in his limitless wisdom he discovered many wonderful things, nonetheless, of all of them, one in particular, which I shall now describe, seems to convey his boundless ingenuity. It is no surprise that Hieron,¹ after he had obtained immense kingly power in Syracuse, decided, because of the favorable turn of events, to dedicate a votive crown of gold to the immortal gods in a certain shrine. He contracted for the craftsman’s wages, and he [himself] weighed out the gold precisely for the contractor. This contractor completed the work with great skill and on schedule; it was approved by the king, and the contractor seemed to have used up the furnished supply of gold. Later, charges were leveled that in the making of the crown a certain amount of gold had been removed and replaced by an equal amount of silver. Hieron, outraged that he should have been shown so little respect, and not knowing by what method he might expose the theft, requested that Archimedes take the matter under consideration on his behalf.

Now Archimedes, once he had charge of this matter, chanced to go to the baths, and there, as he stepped into the tub, he noticed that however much he immersed his body in it, that much water spilled over the sides of the tub. When the reason for this occurrence came clear to him, he did not hesitate, but in a transport of joy he leapt out of the tub, and as he rushed home naked, he let one and all know that he had truly found what he had been looking for—because as he ran he shouted over and over in Greek: “I found it! I found it!” (Eurêka! Eurêka!)

On the basis of this discovery he is said to have made two masses whose weight was equal to that of the crown: one of gold and one of silver. When he had done this, he filled a large vessel to the brim with water, into which he sank the mass of silver. Whatever amount of silver was submerged, that much water spilled out. Then, once the mass had been taken out, he poured back the missing amount of water so that it would be level with the brim in the same way as before, using a one-sextarius pitcher [= 1/2 liter] as a measure. From this procedure he discovered that a certain weight of silver corresponded to a certain measure of water. Once he had tried this, then in the same fashion he sank the gold mass into the full vessel, and when he had removed it, replacing the water by the same method, he discovered that not so much of the water had been lost, and less was required to replace it, as much less as a mass of gold will be smaller in body than a mass of silver which has the same weight. After this, once he had filled the vessel yet again, the crown itself was sunk into the water, and he discovered that more water was required to replace the crown than to replace the mass of gold of equal weight, and because there was more water in the crown’s place than in the place of the mass of gold, he detected, by deduction, the mixture of silver in the crown and the contractor’s flagrant theft.

DISCUSSION QUESTIONS