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Readings for Calculus presents selections on the history of calculus and of mathematics, on the nature of mathematics and its applications, on the learning of calculus, and on the place of calculus and mathematics in society.
Mathematics texts tend to lead students to a preoccupation with getting right the answer rather than considering the richness of the material. This volume can be used as a supplement to the calculus text, showing students that there is more to calculus (and mathematics) than getting answers to problems that agree with the answers in the back of the textbook. Students learn that Newton was a person who lived in a definite time with a life outside of mathematics, that mathematics has much in common with art, and that mathematics is a human creation that has an effect on people.
Students do not have to real all of the selections. Readings gives teachers a source from which they can select one or two items for use during a semester. Some can be used as the base for class discussion (a rare feature of mathematics classes), and others might provide material for student essays (also rare, but also valuable). Some of the selections might by used as enrichment for the student who wants to probe further.
The selections, some published for the first time, have short introductions and added exercises and problems. The exercises and problems range from technical exercises, which can be assigned to insure careful reading of the selection, to unanswerable questions, which can provoke speculation or argument.
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 is 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 through 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 he? 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 he 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 had not. Why is the world the way it is? What is knowledge? How should a citizen behave? Why is the square of a hypotenuse of a right triangle equal to the sum of the squares on the two sides? The Greeks were not satisfied with the answers that had served up to them: "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 has. When Archimedes thought, he 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 pi lies between 3 10/71 and 3 1/7. How many people today know the value of pi that closely? How many think it is equal to 3 1/7? He found how 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 has 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 though that he was doing was solving some isolated problems about areas and volumes. But the method that he used, though based on ideas about levers, was that of dividing areas and volumes up to 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 not 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 changes in ways that are hard to imagine. However, it didn't happen. Archimedes lived at the time when Greek mathematics was 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 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.