Tag: Science & Math

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: the Martian Chronicles

Review: the Martian Chronicles

Crónicas marcianas by Ray Bradbury

My rating: 4 of 5 stars


I am not sure that I am in the best position to judge this short story collection, since the circumstances of my reading it were far from optimal. I downloaded the audio version to pass the time on a long drive, and I decided on the Spanish translation since my co-pilot would have fallen asleep otherwise. I soon discovered that listening to a foreign-language book while navigating mountain roads is not conducive to careful appreciation (or careful driving).

This is much closer to a short-story collection than to a conventional novel, but Bradbury blurred the lines a bit by adding some connecting passages to his stories (originally published separately). It is really only the setting and a vague sense of chronology that connects the separate chapters. And despite his post-facto additions, Bradbury did not achieve full consistency in his Martian world. This is not a problem, however, since I think the inconsistency adds to the stories rather than detracts. The final effect is much like an episodic TV show, which can invent itself anew with each iteration.

Bradbury has become known as a science-fiction writer; and yet these stories may be more accurately described as “anti-science fiction.” He has little interest in the details of technology, cosmology, or space travel, and even less interest in making his stories plausible or realistic. Indeed, Bradbury is not merely uninterested, but positively worried about what the future may bring. For Bradbury, Mars is not the fourth planet from the sun—with its own moons, its unique geology, its practical challenges—but a kind of parallel world where his fears can play out. Much like The Twilight Zone, these stories have one consistent message: “Be careful what you wish for.” Where other people saw the dawning of the space age, Bradbury saw only an extension of human idiocy beyond the clouds.

Arguably, this is quite a conservative message—anti-science, anti-technology, anti-change—but it also resonated with me. I remember being a little kid and contemplating the wonders that the future would bring: flying cars, tourism to the moon, miracle cures. Nowadays, this mood of optimism seems very distant. New technologies, rather than filling us with wonder, are prompting second-thoughts: automation that reduces job opportunities, face-recognition technology that only extends the surveillance state, or the unknown threat of artificial intelligence. And when I think of space travel, rather than imagining the next glorious phase of humanity’s ascent, two buffoons come to mind: Elon Musk (with his SpaceX) and Donald Trump (with his Space Force).

Well, I do not want to get too gloomy in a book review. My point is that Bradbury’s stories may indeed contain a valuable lesson: be careful what you wish for.



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Review: On the Heavens (Aristotle)

Review: On the Heavens (Aristotle)

On the Heavens by Aristotle

My rating: 4 of 5 stars


This is quite a charming little book. In it, one can find the description of an entire way of viewing the natural world. Aristotle moves on from the abstract investigations of the Physics to more concrete questions: Is the earth a sphere or flat? What are the fundamental constituents of matter? Why do some things fall, and some things rise? Is the earth the center of everything? Aristotle’s answers, I’m afraid, have not stood the test of time; such, it appears, is the risk of all science—obsolescence.

The reader is immediately presented with a beautiful piece of Aristotelian reasoning. First, the good philosopher reminds us that “the perfect is naturally prior to the imperfect, and the circle is a perfect thing.” Circular motion, therefore, is more perfect than simple up-and-down motion like we see on earth; and since we do not find bodies whose natural motion is circular on earth, and since nature always strives towards perfection, it follows that there must be bodies not on earth which naturally move in a circular fashion. Again, since none of the earth-bound elements—fire, water, air, and earth—exhibit natural (i.e. unforced) circular motion, it follows that the heavenly bodies must be composed of something different; and this different substance (let us call it aether), since is exhibits the most perfect motion, must be itself perfect.

In Aristotle’s words:

… we may infer with confidence that there is something beyond the bodies that are about us on this earth, different and separate from them; and that the superior glory of its nature is proportionate to its distance from this world of ours.

Everything below the moon must be born and pass away; but the heavenly bodies abide forever in their circular course. Q.E.D.

In his physical investigations, it seems that Aristotle was not especially prescient. For example, he argues against “the Italian philosophers known as the Pythagoreans… At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre.” Not so, says Aristotle; the earth is the center. He also argues against Democritus’s atomic theory, which posits the existence of several different types of fundamental particles, which are intermingled with “void,” or empty spaces in between them.

To be fair, Aristotle does think that the earth is round; he even includes an estimation of the earth’s circumference at 400,000 stadia, which is, apparently, somewhere around 40,000 miles. (The current-day estimate is about 24,000 miles.) Aristotle also thinks that “heavy” objects tend toward the earth’s surface; but puzzlingly (for the modern reader), he doesn’t think this has anything to do with the pull of the earth, but instead thinks it has something to do with earth’s position in the center of all things. In his words: “If one were to remove the earth to where the moon now is, the various fragments of earth would each move not towards it but to the place in which it now is.”

Then Aristotle launches into his investigation of the elements. As aforesaid, Aristotle posits four sublunary elements: earth, water, fire, and air. Earth is the heaviest, followed by water, and then air; and fire is the lightest. Aristotle believes that these elements have “natural” motions; they tend toward their proper place. Earth tries to go downward, towards the center of the planet. Fire tries to go upward, towards the stars. Aristotle contrasts this “natural” motion with “unnatural” or “violent” motion, which is motion from an outside source. I can, of course, pick up a piece of earth, thereby thwarting its natural tendency towards its proper place on the ground.

The elements naturally sort themselves into order: we have earth on the bottom, then water floating on top, then the air sitting on the water, and fire above the air. (Where all that fire is, I can’t say.) There are some obvious difficulties with this theory. For example, how can boats float? and birds fly? This leads Aristotle to a very tentative definition of buoyancy, with which he ends the book:

… since there are two factors, the force responsible for the downward motion of the heavy body and the disruption-resisting force of the continuous surface, there must be some ratio between the two. For in proportion as the force applied by the heavy thing towards disruption and division exceeds that which resides in the continuum, the quicker will it force its way down; only if the force of the heavy thing is the weaker, will it ride upon the surface.

The more one reads Aristotle, the more one grasps just how much his worldview was based on biology. The key word of his entire philosophy is entelechy, which simply means the realization of potential. We can see this clearly in his definition of motion: “The fulfillment of what exists potentially, in so far as it exists potentially, is motion.” That’s a mouthful, but think of it this way: the act of building a house can be thought of as the expression of the potential of a house; the physical house in progress is the partially actualized house, but the building itself is the potential qua potential.

It is easy to see how Aristotle might get interested in the expression of potentialities from investigating living things. For what is an egg but a potential chicken? What is a child but a potential man? This idea of fully realizing one’s potential is at the basis of his ethics and his physics; just as fire realizes its potential for moving upwards, so do citizens realize their potential through moderation. Aristotle’s intellectual method is also heavily marked by one who spent time investigating life; for it is the dreary task of a naturalist to catalogue and to categorize, to investigate the whole by looking at the parts.

While this mindset served him admirably in many domains, it misled him in the investigation inanimate matter. To say that chickens grow from eggs as an expression of potential is reasonable; but to attribute the downward motion of rocks as an expression of their potential sounds odd. It is as if you asked somebody why cars move, and they responded “because it is the nature of the vehicle”—which would explain exactly nothing. But it is difficult not to be impressed by Aristotle; for even if he reached the wrong conclusions, at least he was asking the right questions.

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