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The best estimate of Archimedes’ year of birth is 287 BC and a good estimate of the year of Hiero “gaining the royal power in Syracuse” is 265 BC. This would make Archimedes around 22 years of age when, according to Vitruvius, he solved the golden crown problem.

Another genius of Archimedes’ rank, Isaac Newton, was 22 and 23 years of age during the two plague years of 1665-1666 when he made his most fundamental discoveries. As Newton later wrote about those years: “I was in the prime age for invention & minded Mathematicks & Philosophy more than at any time since.”

One more genius of Archimedean stature, Galileo, also deserves mention here. Galileo was age 22 when he wrote a little tract in 1586 entitled La Bilancetta, or The Little Balance. In this tract he described an accurate balance for weighing things in air and in water, inspired by Vitruvius’s account of Archimedes’ golden crown problem.

•The English translation is by Morris Hicky Morgan in Vitruvius: The Ten Books on Architecture, Harvard University Press, Cambridge, 1914, pages 253-254.
•The Latin text is from Vitruvius’s De Architetura libri decem published by Daniele Barbaro in Venice in 1567 (see pages 270-1).
•The woodcut is by Peter Flotner (1490-1546) from a German translation of Vitruvius (Vitruvius Teutsch) published by Johannes Petreius in Nurnberg in 1548. The following enlarged images are available:

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THE IMPOSSIBILITY OF PLEASURE ACCORDING TO EPICURUS

According to the story, Archimedes, as he was washing, thought of a way to compute the proportion of gold in King Hieron’s crown by observing how much water flowed over the bathing-stool. He leapt up as one possessed, crying heureka! ( “I’ve found it”). After repeating this several times, he went his way.

(Translation by William Baxter, 1859. The color illustration is from a wall painting in the Stanzino delle Matematiche of the Galleria degli Uffizi (Florence, Italy) painted by Giulio Parigi (1571-1635) in the years 1599-1600.)

 Gelon, son of Hiero A COMMENTARY ON THE FIRST BOOK OF EUCLID’S ELEMENTS by Proclus (AD 410-485) BOOK II, CHAPTER III Tradition has it that Gelon made the same remark when, without destroying the crown that had been made, Archimedes discovered the weight of each of its component materials.
Gelon was the son of Hiero II, for whom the crown was made. The remark mentioned ,“From this day forth we must believe everything that Archimedes says,” was made by Hiero after Archimedes demonstrated one of his machines to him.

(Translation by Glenn R. Morrow (1895-1976) in Proclus: A Commentary on the First Book of Euclid’s Elements, Princeton University Press, Princeton, 1970, Page 51.)

“EUREKA!”

Retold by James Baldwin (1841-1925)

THERE was once a king of Syracuse whose name was Hiero. The country over which he ruled was quite small, but for that very reason he wanted to wear the biggest crown in the world. So he called in a famous goldsmith, who was skillful in all kinds of fine work, and gave him ten pounds of pure gold.

“Take this,” he said, “and fashion it into a crown that shall make every other king want it for his own. Be sure that you put into it every grain of the gold I give you, and do not mix any other metal with it.”

“It shall be as you wish,” said the goldsmith. “Here I receive from you ten pounds of pure gold; within ninety days I will return to you the finished crown which shall be of exactly the same weight.”

Ninety days later, true to his word, the goldsmith brought the crown. It was a beautiful piece of work, and all who saw it said that it had not its equal in the world. When King Hiero put it on his head it felt very uncomfortable, but he did not mind that—he was sure that no other king had so fine a headpiece. After he had admired it from this side and from that, he weighed it on his own scales. It was exactly as heavy as he had ordered.

“You deserve great praise,” he said to the goldsmith. “You have wrought very skillfully and you have not lost a grain of my gold.”

There was in the king’s court a very wise man whose name was Archimedes. When he was called in to admire the king’s crown he turned it over many times and examined it very closely.
 “ ‘Well, what do you think of it?’ asked Hiero. ”

“Well, what do you think of it?” asked Hiero.

“The workmanship is indeed very beautiful,” answered Archimedes, “but—but the gold—”

“The gold is all there,” cried the king. “I weighed it on my own scales.”

“True,” said Archimedes, “but it does not appear to have the same rich red color that it had in the lump. It is not red at all, but a brilliant yellow, as you can plainly see.”

“Most gold is yellow,” said Hiero; “but now that you speak of it I do remember that when this was in the lump it had a much richer color.”

“What if the goldsmith has kept out a pound or two of the gold and made up the weight by adding brass or silver?” asked Archimedes.

“Oh, he could not do that,” said Hiero; “the gold has merely changed its color in the working.” But the more he thought of the matter the less pleased he was with the crown. At last he said to Archimedes, “Is there any way to find out whether that goldsmith really cheated me, or whether he honestly gave me back my gold?”

“I know of no way,” was the answer.

But Archimedes was not the man to say that anything was impossible. He took great delight in working out hard problems, and when any question puzzled him he would keep studying until he found some sort of answer to it. And so, day after day, he thought about the gold and tried to find some way by which it could be tested without doing harm to the crown.

One morning he was thinking of this question while he was getting ready for a bath. The great bowl or tub was full to the very edge, and as he stepped into it a quantity of water flowed out upon the stone floor. A similar thing had happened a hundred times before, but this was the first time that Archimedes had thought about it.

“How much water did I displace by getting into the tub?” he asked himself. “Anybody can see that I displaced a bulk of water equal to the bulk of my body. A man half my size would displace half as much.

“Now suppose, instead of putting myself into the tub, I had put Hiero’s crown into it, it would have displaced a bulk of water equal to its own bulk. All, let me see! Gold is much heavier than silver. Ten pounds of pure gold will not make so great a bulk as say seven pounds of gold mixed with three pounds of silver. If Hiero’s crown is pure gold it will displace the same bulk of water as any other ten pounds of pure gold. But if it is part gold and part silver it will displace a larger bulk. I have it at last! Eureka! Eureka!”

Forgetful of everything else he leaped from the bath. Without stopping to dress himself, he ran through the streets to the king’s palace shouting, “Eureka! Eureka! Eureka!” which in English means, “I have found it! I have found it! I have found it!”

The crown was tested. It was found to displace much more water than ten pounds of pure gold displaced. The guilt of the goldsmith was proved beyond a doubt. But whether he was punished or not, I do not know, neither does it matter.

The simple discovery which Archimedes made in his bath tub was worth far more to the world than Hiero’s crown. Can you tell why?

(The text and illustration were taken from Thirty More Famous Stories Retold by James Baldwin, (American Book Company, 1905). Baldwin was an American educator, editor, and author of more than fifty books. His retold stories in this book and in Fifty Famous Stories Retold (American Book Company, 1896) were extremely popular and molded many impressionable minds in the early twentieth century. An enlarged image of the illustration (95 kilobytes, 1153 x 1781 pixels, black & white gif) is available.)

HIS BATH GAVE HIM THE TIP-OFF

ARCHIMEDES was asked to check the suspected presence of silver alloy in the king’s gold crown. 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. So delighted was Archimedes with his solution that he leaped from his bath and ran through the streets of Syracuse crying “Eureka!” Presumably you won’t be in your bath when you read NBC’s new facts, but we would not be surprised to hear you, too, shout “Eureka!”

Enlarged Images:
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(This illustration and text are from a magazine advertisement for NBC, probably dating from the 1940’s. It was found among the files of the Print and Picture Department of the Free Library of Philadelphia. The word “eureka!” is coming out of Archimedes’ mouth.)