Review: The Works of Archimedes

Review: The Works of Archimedes

The Works of Archimedes by Archimedes

My rating: 5 of 5 stars

In fact, how many theorems in geometry which have seemed at first impracticable are in time successfully worked out!

Many of the most influential and ingenious books ever written possess the strange quality of being simultaneously exhilarating and quite boring. Unless you are among that rare class of people who enjoy a mathematical demonstration more than a symphony, this book will likely possess this odd duality. I admit this is the case for me. Reading this book was a constant exercise in fighting the tendency for my eyes to glaze over. But I am happy to report that it is worth the trouble.

Archimedes lived in the 3rd century BCE, somewhat after Euclid, in Syracuse on the island of Sicily. Apart from this, not much else can be said with certainty about the man. But he is the subject of many memorable stories. Everybody knows, for example, the story of his taking a bath and then running through the streets naked, shouting “Eureka!” We also hear of Archimedes using levers to move massive boats, and claiming that he could move the whole earth if he just had a place to stand on. Even his death is the subject of legend. After keeping the invading Romans at bay using ingenious weapons—catapults, cranes, and even mirrors to set ships afire—Archimedes was killed by a Roman soldier, too preoccupied with a mathematical problem to care for his own well-being.

True or not, good stories tend to accumulate around figures who are worthy of our attention. And Archimedes is certainly worthy. Archimedes did not leave us any extended works, but instead a collection of treatises on several topics. The central concern in these different works—the keystone to Archimedes’s method—is measurement. Archimedes set his brilliant mind to measuring things that many have concerned impossible to reckon. His work, then, is an almost literal demonstration of the human mind’s ability to scan, delimit, and calculate things far outside the scope of our experience.

As a simple example of this, Archimedes established the ratios between the surface areas and volumes of spheres and cylinders—an accomplishment the mathematician was so proud of that he apparently asked for it to be inscribed on his tombstone. Cicero describes coming across this tombstone in a dilapidated state, so perhaps this story is true. Archimedes also set to work on giving an accurate estimation of the value of pi, which he accomplished by inscribing and circumscribing 96-sided polygons around a circle, and calculating their perimeters. If this sounds relatively simple to you, keep in mind that Archimedes was operating without variables or equations, in the wholly-geometrical style of the Greeks.

Archimedes’s works on conoids, spheroids, and spirals show a similar preoccupation with measurement. What all of these figures have in common is, of course, that they are composed of curved lines. How to calculate the areas contained by such figures is not at all obvious. To do so, Archimedes had to invent a procedure that was essentially equivalent to the modern integral calculus. That is, Archimedes used a method of exhaustion, inscribing and circumscribing ever-more figures composed of straight lines, until an arbitrarily small gap remained between his approximations and what he was attempting to measure. To employ such a method in an age before analytic geometry had even been invented is, I think, an accomplishment difficult to fully appreciate. When the calculus was finally invented, about two thousand years later, it was by men who were “standing on the shoulders of giants.” In his time, Archimedes had few shoulders to stand on.

The most literal example of Archimedes’s concern with measurement is his short work, The Sand Reckoner. In this, he attempts to calculate the number of grains of sand that would be needed to fill up the whole universe. We owe to this bizarre little exercise our knowledge of Aristarchus of Samos, the ancient astronomer who argued that the sun is positioned at the center of the universe. Archimedes mentions Aristarchus because a heliocentric universe would have to be considerably bigger than a geocentric one (since there is no parallax observed of the stars); and Archimedes wanted to calculate the biggest universe possible. He arrives at a number is quite literally astronomical. The point of the exercise, however, is not in the specific number arrived at, but in formulating a way of writing very large numbers. (This was not easy in the ancient Greek numeral system.) Thus, we partly owe to Archimedes our concept of orders of magnitude.

Archimedes’s contributions to natural science are just as significant as his work in pure mathematics. Indeed, one can make the case that Archimedes is the originator of our entire approach to the natural sciences; since it was he who most convincingly demonstrated that physical relationships could be described in purely mathematical form. In his work on levers, for example, Archimedes shows how the center of gravity can be found, and how simple principles can explain the mechanical operation of counterbalancing weights. Contrast this with the approach taken by Aristotle in his Physics, who uses wholly qualitative descriptions and categories to give a causal explanation of physical motion. Archimedes, by contrast, pays no attention to cause whatever, but describes the physical relationship in quantitative terms. This is the exact approach taken by Galileo and Newton.

Arguably, the greatest masterpiece in this collection is On Floating Bodies. Here, Archimedes describes a physical relationship that still bears his name: the relationship of density and shape to buoyancy. While everyone knows thpe story of Archimedes and the crown, it is possible that Archimedes’s attention was turned to this problem while working on the design for an enormous ship, the Syracusia, built to be given as a present to Ptolemy III of Egypt. This would explain Book II, which is devoted to finding the resting position of several different parabolas (more or less the shape of a ship’s hull) in a fluid. The mathematical analysis is truly stunning—so very far beyond what any of his contemporaries were capable of that it can seem even eerie in its sophistication. Even today, it would take a skilled physicist to calculate how a given parabola would rest when placed in a fluid. To do so in ancient times was simply extraordinary.

Typical of ancient Greek mathematics, the results in Archimedes’s works are given in such a way that it is difficult to tell how he originally arrived at these conclusions. Surely, he did not follow the steps of the final proof as it is presented. But then how did he do it? This question was answered quite unexpectedly, with the discovery of the Archimedes Palimpsest in the early 1900s. This was a medieval prayer book that contained the remains of two previously unknown works of Archimedes. (Parchment was so expensive that scribes often scraped old books off to write new ones; but the faded impression of the original work is still visible on the manuscript.) One of these works was the Ostomachion, a collection of different shapes that can be recombined to form a square in thousands of different ways (and it was the task of the mathematician to determine how many).

The other was the Method, which is Archimedes’s account of how he made his geometrical discoveries. Apparently, he did so by clever use of weights and balances, imagining how different shapes could be made to balance one another. His method of exhaustion was also a crucial component, since it allowed Archimedes to calculate the areas of irregular shapes. A proper Greek, Archimedes considered mechanical means to be intellectually unsatisfactory, and so re-cast the results obtained using this method into pure geometrical form for his other treatises. If it were not for the serendipitous discovery of this manuscript, and the dedicated work of many scholars, this insight into his method would have been forever lost to history.

As I hope you can see, Archimedes was a genius among geniuses, a thinker of the rarest caliber. His works are exhilarating demonstrations of the power of the human mind. And yet, they are also—let us admit it—not the most exciting things to read, at least for most of us mere mortals. Speaking for myself, I would need a patient expert as a guide if I wanted to understand any of these works in detail. Even then, it would be hard work. Indeed, I have to admit that, on the whole, I find mathematicians to be a strange group. For the life of me I cannot get excited about the ratio of a sphere to a cylinder—something that Archimedes saw as the culmination of his entire life.

Archimedes is the very embodiment of the man absorbed in impractical pursuits—so obsessed with the world of spirals and curves that he could not even avoid a real sword thrust his way. And yet, if subsequent history has shown anything, it is that these apparently impractical, frigid, and abstract pursuits can reveal deep truths about the universe we live in—much deeper than the high-flown speculations of our philosophers. I think this lesson is worth suffering through a little boredom.



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Review: Calculus (Kline)

Review: Calculus (Kline)

Calculus: An Intuitive and Physical Approach by Morris Kline

My rating: 5 of 5 stars

Morris Kline’s book, Mathematics for the Nonmathematician, is my favorite book on the discipline. Kline showed an amazing ability to explain mathematical concepts intuitively, and to situate them within a sensible human context. In his hands, math was not simply a series of equations or deductive proofs, but an integral aspect of our civilization: a crucial tool in our species’ attempt to understand and manipulate the world. The book changed my view of the subject.

So when I found that Kline had written a book on the calculus, I knew that I had to read it. Calculus represents the furthest I have ever gone with mathematics in my formal schooling. By the time that I graduated high school, I was a problem-solving machine—with so many rules of algebra, trigonometry, derivation, and integration memorized that I could breeze through simple exercises. Yet this was a merely mechanical understanding. I was like a well-trained dog, obeying orders without comprehension; and this was apparent whenever I had to do any problems that required deeper thinking.

In time I lost even this, leaving me feeling like any ordinary mathematical ignoramus. My remedial education has been slow and painful. This was my primary object in reading this book: to revive whatever atrophied mathematical skills lay dormant, and to at least recover the level of ability I had in high school. Kline’s text was perfect for this purpose. His educational philosophy suits me. Rather than explain the calculus using formal proofs, he first tries to shape the student’s intuition. He does this through a variety of examples, informal arguments, and graphic representation, allowing the learner to get a “feel” for the math before attempting a rigorous definition.

He justifies his procedure in the introduction:

Rigor undoubtedly refines the intuition but does not supplant it. . . . Before one can appreciate a precise formulation of a concept or theorem, he must know what idea is being formulated and what exceptions or pitfalls the wording is trying to avoid. Hence he must be able to call upon a wealth of experience acquired before tackling the rigorous formulation.

This rings true to my experience. In my first semester of university, when I thought that I was going to study chemistry, I took an introductory calculus course. It was divided into lectures with the professor and smaller “recitation” classes with a graduate student. In the lectures, the professor would inevitably take the class through long proofs, while the grad student would show us how to solve the problems in the recitatives. I inevitably found the professor’s proofs to be pointless, and soon decided to avoid them altogether, since they confused me rather than aided me. I got an A-minus in the class.

Though Kline forgoes the rigor one would expect in formal mathematics, this book is no breezy read. It is a proper textbook, designed to be used in a two-semester introductory course, complete with hundreds of exercises. And as fitting for such a purpose, this book is dry. Gone are the fascinating historical tidbits and gentle presentation of Kline’s book on popular mathematics. This book is meant for students of engineering and the sciences—students who need to know how to solve problems correctly, or planes will crash and buildings will collapse. But Kline is an excellent teacher in this context, too, and explains each concept clearly and concisely. It was often surprisingly easy to follow along.

The exercises are excellent as well, designed to progress in difficulty, and more importantly to encourage independent thinking. Rather than simply solving problems by rote, Kline encourages the student to apply the concepts creatively and in new contexts. Now, I admit that the sheer amount of exercises taxed my patience and interest. I wanted a refreshment, and Kline gave me a four-course meal. Still, I made sure to do at least a couple problems per section, to check whether I was actually understanding the basic idea. It helped immensely to have the solutions manual, which you can download from Dover’s website.

In the end, I am very glad to have read this book. Admittedly this tome did dominate my summer—as I plowed through its chapters for hours each day, trying to finish the book before the start of the next school year—and I undoubtedly tried to read it far too quickly. Yet even though I spent a huge portion of my time with this book scratching my head, getting questions wrong, it did help to restore a sense of intellectual confidence. Now I know for sure that I am still at least as smart as I was at age 18.

And the subject, if often tedious, is fascinating. Learning any branch of mathematics can be intensely satisfying. Each area interlocks with and builds upon the other, forming a marvelous theoretical edifice. And in the case of the calculus, this abstract structure contains the tools needed to analyze the concrete world—and that is the beauty of math.

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