EPISODES FROM THE EARLY HISTORY OF MATHEMATICS by Asger Aaboe First, let's get straight the point of that famous anecdote. That's the one about how the king gave Archimedes the problem of determining if his new crown was pure gold or if it was part gold and part some cheaper metal. Snipping off a sample of the crown was not allowed. The story goes that the solution came to Archimedes one day while he was in the bath, and that it so excited him that he ran down the street naked yelling, "Eureka, eureka!" that is, "I have found it, I have found it!" To those of us who were not brought up in the Greek culture of the third century B. C., the surprising part of the story is Archimedes' nakedness: just think of some famous scholar running naked down the street today! But Archimedes does not live today, and when and where he did live-more than two millennia ago, on Sicily, where Greeks dominated the natives-male nakedness was common and not to be remarked on. The surprising part of the story was that Archimedes, that dignified and renowned scholar and scientist, would be running down the street yelling his head off. Greek men of the time did not behave in such a manner. With that out of the way, we can go on to more important things about Archimedes. It is hard to know where to begin, since there were so many of them. Ask any mathematician who were the three greatest mathematicians who ever lived and the answer is 95% certain to be "Archimedes, Newton, and Gauss." There is no question about it. Euler-a superb technician who bubbled over with ideas, but his ideas were ... smaller. Leibniz-well, he wasn't really a mathematician, was be? Fermat-great, but a great might-have-been. Galois, Hilbert, Lagrange-no, there is no one else who can be mentioned in the same breath. On occasion, I have a doubt or two about Newton, but never about Gauss, and certainly never about Archimedes. One reason for his greatness was the time in which be lived. His dates are 287-212 B. C., not too long after the time when, on the rocky peninsula of Greece, people started to ask questions that their ancestors bad not. Why is the world the way it is? What is knowledge? How should a citizen behave? Why is the square on the hypotenuse of a right triangle equal to the sum of the squares on the other two sides? The Greeks were not satisfied with the answers that had served up to then: "The king says so, that's why," "That's the way it has always been," "It is the will of the gods," "Just because," or "Shut up and get back to work." They wanted reasons. To convince a Greek, you had to think. Yelling, though it helped, was not enough. The human race started to think. Parts of it, that is: at no time, including right now, has the entire race been thinking, but at all times since then at least some of it bas. When Archimedes thought, be did not have all that many other thinkers to look back on, so his achievements were all the greater. The following excerpt tells some of what Archimedes did in mathematics. He found that the value of :rt lies between 3 10/71 and 3 1/7. How many people today know the value of :rt that closely? How many think it is equal to 3 1/7? He found bow many grains of sand it would take to fill the entire universe. How many people today would say that there is no such number, or that the number of grains would be infinite? He trisected the angle, using a compass and a straightedge that bas two scratches on it. He constructed a regular heptagon. He did a lot. He also invented calculus, almost. It is only hindsight that lets us say that, since all Archimedes thought that be was doing was solving some isolated problems about areas and volumes. But the method that be used, though based on ideas about levers, was that of dividing areas and volumes up into many small pieces, finding the areas and volumes of the pieces, and adding them up. He was able to find the surface area of a sphere and several other results that we now get by integration. If his ideas had been carried further as, almost two thousand years later, the ideas of Fermat, Barrow, Cavalieri and others would be carried further by Newton and Leibniz, then we might have had calculus two thousand years earlier and the history of the race would have been changed in ways that are hard to imagine. However, it didn't happen. Archimedes lived at the time when Greek mathematics was at its absolute never-to-be-equaled peak, in its golden age at the time when the gold was shining most brightly. After the time of Archimedes, Greek 11 12 READINGS FOR CALCULUS mathematics went downhill (though with occasional bumps upward, as a roller coaster does not go directly from top to bottom), until towards its end five hundred years later, all that was being written were commentaries on the works of the giants of the past. Opportunities to take the work of Archimedes further did not arise. Also, mathematicians were not as thick on the ground then as they are now, or as they were in the seventeenth century when calculus was developed. The number of people skilled in mathematics at any one time in the Greek world was very, very small. You might need more than your fingers and toes to number them all, but you and a few friends would have more than enough digits for the job. Thus, it was easy for ideas to get lost, and that is what happened to Archimedes' method of finding areas and volumes. He had no successors, and the possibility of calculus in the ancient world died with him. 1. Archimedes' Life about 287 B. C. In the preface to his book The Sand­ reckoner, be speaks of his father Pbeidias, the astrono­ In weightiness of matter and elegance of style, no mer, who is otherwise unknown. It is said that Archi­ classical mathematical treatise surpasses the works of medes studied in Alexandria, then the centre of learning, Archimedes. This was recognized already in antiquity; and it is certain that he had friends among the Alexan­ thus Plutarch says of Archimedes' works; drian mathematicians, as we Jearn from his prefaces; but be spent most of his life in Syracuse where be was a It is not possible to find in all geome­ friend and, as some even say, a relation of the reigning try more difficult and intricate ques­ bouse. He spent his life pursuing interests which tions, or more simple and lucid expla­ extended from pure mathematics and astronomy to nations. Some ascribe this to his mechanics and engineering. Indeed, it was his more genius; while others think that incredi­ practical achievements that caught the public fancy. If ble effort and toil produced these, to we may trust the stories about him, he was not above all appearances, easy and unlaboured adding a dramatic touch to his demonstrations; thus results. Plutarch tells, in Dryden's stately translation: Plutarch, who lived in the second half of the first Archimedes, however, in writing to century A. D., writes this in his Lives of the Noble King Hiero, whose friend and near Grecians and Romans, more specifically his life of relation he was, had stated that given Marcellus. Marcellus was the general in charge of the the force, any given weight might be Roman arn1y that besieged, and ultimately took, the moved, and even boasted, we are told, Greek colony of Syracuse on Sicily during the second relying on the strength of demonstra­ Punic War (218-201 B. C.). Archimedes' ingenious tion, that if there were another earth, war-machines played an important role in the defense of by going into it could remove this. Syracuse, and for this reason Plutarch writes about him Hiero being struck with amazement at at some length. this, and entreating him to make good Archimedes introduces each of his books with a this problem by actual experiment, dedicatory preface where be often gives some back­ and show some great weight moved ground for the problem he is about to treat. These by a small engine, he fixed according­ prefaces contain precious information for the historian ly upon a ship of burden out of the . of mathematics, and they even throw some light on king's arsenal, which could not be Archimedes' life. There are, furthermore, scattered drawn out of the dock without great references to him in the classical literature and so be labour and many men; and, loading becomes the Greek mathematician about whom we have her with many passengers and a full the most biographical information, even though it is freight, sitting hilnself the while far precious little. off, with no great endeavour, by only Archimedes was killed in 212 B. C. during the sack holding the head of the pulley in his of Syracuse that ended the Roman siege. Since he is hand and drawing the cords by de­ said to have reached the age of 75 years, he was born grees, he drew the ship in a straight AABOE, ARCHIMEDES 13 line, as smoothly and evenly as if she that, as Archimedes was carrying to had been in the sea.