Review: A Manual of Greek Mathematics

Review: A Manual of Greek Mathematics

A Manual of Greek Mathematics by Thomas L. Heath

My rating: 4 of 5 stars

In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science.


As a supplement to my interest in the history of science, I figured that I ought to take a look into the history of mathematics, since the two are quite intimately related. This naturally led me to the Greeks and to Sir Thomas L. Heath, who remains the most noteworthy translator, divulgator, and commentator in English eighty years after his death. This book is likely the best single volume you can get on the subject, as it covers all of the major mathematicians in some detail while giving a complete overview.

It is also reasonably accessible (“reasonably” being the operative word). Certainly it is no work of popular math in the modern sense; it is not pleasure reading, and Heath assumes a certain amount of knowledge on the reader’s part. A thorough knowledge of algebra and geometry is assumed, and a few words in ancient Greek are not translated. What is more, large sections of the book are essentially extended summaries and explications of Greek treatises, which makes them almost impossible to read without the original text alongside. Personally I would certainly have appreciated more spoon-feeding, as it was quite difficult for me to prevent my eyes from glazing over.

The book is divided primarily by subject-matter and secondarily by chronology. Heath introduces us to notation, fractions, and techniques of calculation, and then on to arithmetic. Geometry, of course, dominates the book, as it was the primarily form of Greek mathematical thought. Heath summarizes the contributions to geometry by Pythagoras and his followers, and the scattered mathematicians we know of in the years between Thales and Euclid. Once Euclid appears, he writes his famous Elements, which encapsulates the entire subject and which rendered many previous works obsolete. After Euclid we come to the divine Archimedes and the great Apollonius, who put the capstone on the tradition. Ptolemy (among others) made great advances in trigonometry, while Diophantus made strides in algebra (as well as inspired Fermat).

Heath’s account of these mathematicians is largely internal, meaning that he is focused on the growth of their ideas rather than anything external to the science. Reading this convinced me—as if further evidence was needed—that I do not have the moral fiber or intellectual temper to appreciate mathematics. Heath writes admiringly of the works of Euclid and Archimedes, finding them not only brilliant but beautiful. While I can normally appreciate the brilliance, the beauty normally escapes me. Ratios, volumes, lines, and equations simply do not make my heart beat.

Indeed, the questions that I find most fascinating are those that are hardly touched upon in this book. Most important, perhaps, is this: What aspect of a culture or a society is conducive to the development of pure mathematics? Though claims of Greek specialness or superiority seem antiqued at best nowadays, it is true that the Greeks made outstanding contributions to science and math; while the Roman contribution to those fields—at least on the theoretical side—is close to nil. The mathematics of Ancient Egypt amount to techniques for practical calculations. Admittedly, as Otto Neugebauer wrote about in his Exact Sciences of Antiquity, the Babylonians had quite advanced mathematics, allowing them to solve complex polynomials; they also had impressive tabulations of the heavenly motions.

Even so, it was the Greeks who created science and math in the modern sense, by focusing on generality. That is, rather than collect data or develop techniques for specific problems, the Greeks were intent on proving theorems that would hold in every case. This also characterizes their philosophy and science: a rigorous search after an absolute truth. This cultural orientation towards the truth in the most general, absolute form seems quite historically special. It arose in one fairly limited area, and lasted for only a few centuries. Most striking is the Greek disdain of the practical—something that runs from Pythagoras, through Plato, to Archimedes.

Of the top of my head, here are some possible factors for this cultural development. The Greek economy was based on slavery, so that citizens often could afford to disdain the practical. What is more, the Greek political model was based on the city-state—a small, close-knit community with limited expansionist aims and thus with limited need for great infrastructure or novel weapons. The relative lack of economic, political, or military pressure perhaps freed intellectuals to pursue wholly theoretical projects, with standards that arose from pure logic rather than necessity. Maybe this seems plausible; but I am sure many other societies fit this description, not just the Greeks. The development of culture is something that we do not fully understand, to say the least.

This has taken me quite far afield. In sum, this book is an excellent place to start—either by itself, or as a companion to the original Greek works—if you are interested in learning something about this astounding intellectual tradition. That the Greeks could get so far using geometry alone—that is, without variables or equations—is a testament to human genius and persistence.



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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: On the Soul (Aristotle)

Review: On the Soul (Aristotle)
De Anima (On the Soul)

De Anima by Aristotle

My rating: 5 of 5 stars

As I have lately been making my way through Aristotle’s physical treatises, I have often observed that many of Aristotle’s errors stem from his tendency to see the physical world as analogous to a biological organism. So it is a pleasure to finally see Aristotle back on his home territory—living things. While Aristotle’s work in proto-physics and proto-chemistry is interesting mainly from a historical perspective, this work is interesting in its own right; in just a hundred pages, Aristotle manages to assemble a treatise on the fundamentals of life.

The first thing the modern student will notice is that Aristotle means something quite different by ‘soul’ than how we normally understand the word. The word ‘soul’ has come to mean an immaterial, specter-like wraith, the spiritual core of one’s personality—trapped, only temporarily, in a body; and this view has, over the years, caused problems for philosophers and theologians alike, for it remains to be explained how an immaterial spirit could move a material body, or how a material body could trap an immaterial spirit. Aristotle avoids these awkward questions. What he means is quite different.

Aristotle begins by observing that all forms of behavior, human or animal, require a body. Even supposedly ‘mental’ states, such as anger, love, and desire, all have concomitant physical manifestations: an angry man gets red in the face, a man in love stares at his beloved, and a man who desires alcohol tries to get it. From this, Aristotle quickly concludes that all the Pythagorean and Platonic talk of the transmigration of souls is silly; a soul needs a body, just as a body needs a soul. Furthermore, a specific soul doesn’t need just any body, but it needs its specific body. Soul and body are, in other words, codependent and inseparable. In Aristotle’s words, “each art must use its tools, each soul its body.”

This still leaves the question unanswered, what is a soul? Aristotle answers that the soul is the form of the body. Alright, what does that mean? Keep this in mind: when Aristotle says ‘form’, he is not merely talking about the geometrical shape of the object, but means something far more general: the form, or essence, of something is that by which it is what it is. Here’s an example: the form of a bowl is that which makes a bowl a bowl, as opposed to something else like, say, a plate or a cup. In this particular case, the form would seem to be the mere shape of the object; isn’t the thing that makes a bowl a bowl its shape? But consider that there is no such thing as a disembodied bowl; for a bowl to be a bowl, it must have a certain shape, be within a certain size range, and be embodied in a suitable material. All of these qualifications, the shape, size, and material, Aristotle would include in the ‘form’ of an object.

So the soul of living things is the quality (or qualities) that differentiate them from nonliving things. Now, the main difference between animate and inanimate objects is that animate objects possess capacities; therefore, the more capacities a living thing has, the more souls we must posit. This sounds funny, but it’s just a way of speaking. Plants, for Aristotle, are the simplest forms of living beings; they only possess the ‘vegetative soul’, which is what makes them grow and develop. Animals possess additional souls, such as that which allows them to sense, to desire, to imagine, and—in the case of humans—to think. The ‘soul’, then, is a particular type of form; it is a form which gives its recipient a certain type of capability. Plants are only capable of growth; animals are capable of growing, of moving, and of many other things.

Aristotle sums up his view in a memorable phrase: “From all this it is obvious that the affections of soul are enmattered formulable essences.” These capacities cannot be ‘enmattered’ in just anything, but must be embodied in suitable materials; plants are not made of just anything, but their capacities for growth always manifest themselves in the same types of material. Aristotle sums up this point with another memorable phrase: “soul is an actuality or formulable essence of something that possesses the potentiality of being besouled.”

So an oak tree is made of material with the potentiality of being ‘besouled’, i.e., turned into a living, growing oak tree. Conversely, a life-sized statue of an oak tree made of bronze would still not be an oak tree, even if it shared several aspects of its form with a real oak tree. It isn’t made of the right material, and thus cannot possess the vegetative soul.

I have given a somewhat laborious summary of this because I think it is a very attractive way of looking at living things. It avoids all talk of ‘ghosts in the machine’, and concentrates on what is observable. (I should note, however, that Aristotle thought that ‘mind’, which is the faculty of reason, is immaterial and immortal. Nobody’s perfect.)

I also find Aristotle metaphysical views attractive. True to his doctrine of the golden mean, he places equal emphasis on matter and form. He occupies an interesting middle-ground between the idealism of Plato and the materialism of Democritus. In order for a particular thing to be what it is, it must both have a certain form—which is embodied in, but not reducible to, its matter—and be made of the ‘right’ types of matter. Unlike Plato’s ideals, which reside in a different sphere of reality, existing as perfect essences devoid of matter, Aristotle’s forms are inherent in their objects, and thus are neither immaterial nor simply the matter itself

The treatise ceases to be as interesting as it progresses, but there are a few gems along the way. He moves on to an investigation of the five senses, and, while discussing sight, has a few things to say about light. Aristotle defines light as the quality by which something transparent is transparent; in other words, light is the thing that can be seen through transparent things. I suppose that’s a respectable operational definition. Aristotle also considers the idea that light travels absurd; nothing could go that fast:

Empedocles (and with him all others who used the same forms of expression) was wrong in speaking of light as ‘traveling’ or being at a given moment between the earth and its envelope, its movement being unobservable by us; that view is contrary both to the clear evidence of argument and to the observed facts; if the distance traversed were short, the movement might have been unobservable, but where the distance is from extreme East to extreme West, the draught upon our powers of belief is too great.

Aristotle also has a few interesting things to say about sense:

By a ‘sense’ is meant what has the power of receiving into itself the sensible forms of things without the matter. This must be conceived of as taking place in the way in which a piece of wax takes on the impress of a signet-ring without the iron or gold; we say that what produces the impression is a signet of bronze or gold, but its particular metallic constitution makes no difference: in a similar way the sense is affected by what is colored or flavored or sounding, but it is indifferent what in each case the substance is; what alone matters is what quality is has, i.e. in what ratio its constituents are combined.

So we don’t take in the matter of a bowl through our eyes, but only its form. All of our senses, then, are adapted for observing different aspects of the forms of objects. Thus, Aristotle concludes, all knowledge consists of forms; when we learn about the world, we are mentally reproducing the form of the world in our minds. As he says: “It follows that the soul is analogous to the hand; for as the hand is a tool of tools [i.e. the tool by which we use tools], so the mind is the form of forms [i.e. the form by which we apprehend forms].” (Notice how deftly Aristotle wields his division of everything into matter and form; he uses it to define souls, to define senses, and then to define knowledge. It is characteristic of him to make so much headway with such seemingly simple divisions.)

For a long time, I was perplexed that Aristotle was so influential. I was originally repulsed by his way of thinking, put off by his manner of viewing the world. His works struck me as alternately pedantic, wrongheaded, or obvious. How could he have exerted such a tremendous influence over the Western mind? Now, after reading through much more Aristotle, this is no longer perplexing to me; in fact, I often find myself thinking along his lines, viewing the world through his eyes. It takes, I believe, a lot of exposure in order to really develop a sympathy for Aristotle’s thought; but with its emphasis on balance, on growth, on potentiality, it succeeds in being a very aesthetically compelling (if often incorrect) way of viewing things.

This piece represents, to me, Aristotle at his best. It is a grand synthesis of philosophy and biology, probably not matched until William James’s psychological work. Unlike many gentlemanly philosophers who shut themselves in their studies, trying to explain human behavior purely through introspection, Aristotle’s biologically rooted way of seeing things combines careful observation—of humans and nonhumans alike—with philosophical speculation. It is a shame that only the logic-chopping side of Aristotle was embraced by the medievals, and not his empirical outlook.

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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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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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Review: The Rise and Fall of the Dinosaurs

Review: The Rise and Fall of the Dinosaurs

The Rise and Fall of the Dinosaurs: A New History of a Lost World by Stephen Brusatte

My rating: 4 of 5 stars


Like so many people, I went through a dinosaur phase as a child. It was almost inevitable. Growing up on the Upper West Side, I could visit the Museum of Natural History nearly every week. Natural selection has overcome many engineering problems—flight, sight, growth, digestion—and it has certainly not failed in its ability to awe little boys. I picked up this book to finally learn something about these ancient beasts.

Any fair evaluation of this book must conclude that it does its job: it summarizes new discoveries about dinosaurs in accessible prose. Brusatte goes through the entire chronology of the group, from their beginnings as unremarkable reptiles which emerged after the great Permian-Triassic Extinction, to their gradual rise, growth, spread, and diversification, and finally to their eventual end—wiped out by an asteroid.

There are many interesting tidbits along the way. Dinosaurs had the efficient lungs we find in modern birds, which are able to extract oxygen during the inhale and exhale. They also had primitive feathers, which looked more like hairs. Indeed, modern birds are dinosaurs in the strict sense of the word. I was particularly surprised to learn that Tyrannosaurus Rex lived and hunted in groups; and that they achieved their massive size extremely quickly—growing several pounds a day for years on end.

I also appreciated Brusatte’s descriptions of the methods that paleontologists use—new statistical techniques for analyzing fossils, or piecing together ancient ecosystems, or determining rates of evolutionary change. Nowadays paleontologists to not merely look for old bones, but they study living animals to make hypotheses about the speed, strength, and size of these extinct creatures. One researcher even studied fossils under a microscope to deduce the color of the feathers from the indentations. Brusatte also covers some of the history of dinosaur research, which is surprisingly colorful—especially the tragic life of the Baron Franz Nopcsa von Felső-Szilvás.

So the book undoubtedly accomplishes its goal. My only complaint is the style. When Brusatte sticks to the science, he is clear and engaging. But whenever he chooses to embellish the story—which is rather too often—the prose becomes strained and grating. Here is a description of a seagull that opens his chapter on birds:

When the sun breaks through for a moment, I catch a glint reflected in its beady eyes, which start to dance back and forth. No doubt this is a creature of keen senses and high intelligence, and it’s onto something. Maybe it can tell that I’m watching. Then, without warning, it yawns open its mouth and emits a high-pitched screech—an alarm to its compatriots, perhaps, or a mating call. Or maybe it’s a threat directed my way.

In fairness, I did enjoy his description of what the dinosaurs would have experienced in the first few minutes after the asteroid impact.

More irksome, however, were the thumbnail sketches of his colleagues, which are interspersed throughout the book. I would have understood the necessity of these passages if Brusatte were introducing a researcher who would play an important role in the book. Yet inevitably these researchers were introduced with fanfare only to be immediately dropped. What is more, Brusatte always focuses on the quirkiest aspects of these researchers, in a superficial attempt at coolness; and he also makes sure to tell us that he is one of their best friends.

In one particularly aggravating example, Brusatte describes one researcher’s fashion (“leopard-print Lycra, piercings, and tattoos”), ethnicity (“half-Irish, half-Chinese”), hobbies (“raving and even occasionally DJ-ing in the trendy clubs of China’s suddenly hip capital”), and conversation style (“delivering caustic one-liners one moment, speaking in eloquent paragraphs about politics the next”). Does this add anything of value to the book?

These stylistic irritations mar what is otherwise an excellent popular book about dinosaurs. And since these offending passages do not add anything to the substance of the book, my advice is just to skip on until he gets back on the subject of dinosaurs—a topic which brings out the best in Brusatte.

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Review: The Copernican Revolution

Review: The Copernican Revolution
The Copernican Revolution: Planetary Astronomy in the Development of Western Thought

The Copernican Revolution: Planetary Astronomy in the Development of Western Thought by Thomas S. Kuhn

My rating: 5 of 5 stars

There are few phrases more annoying or more effective than “I told you so.”

This is my second encounter with Thomas Kuhn, and again I emerge deeply impressed. To do justice to an event so multifaceted as the Copernican Revolution a scholar must have a flexible mind; and Kuhn is fully equal to the task. He moves seamlessly from scientific data, to philosophical analysis, to historical context, and then back again. The result is a book that serves as an admirable introduction to the basics of astronomy and a thorough overview of the Copernican Revolution, while raising intriguing questions about the nature of scientific progress.

Kuhn first makes an essential point: that the conceptual schemes of science serve both a logical and a psychological function. Their logical function is to economically organize the data (in this case, the position and movement of heavenly objects); their psychological function is to make people feel at home in the universe. Belief is only necessary for this second function. A scientist can use a conceptual scheme perfectly well without believing that it represents how the universe ‘truly is’; but people have an obvious and, apparently, near-universal need to understand their place in, and relation to, the cosmos. Thus, scientists throughout history have insisted on the truth of their systems, despite the history of science being littered with the refuse of abandoned theories (to use Kuhn’s expression). Even if this belief cannot be justified philosophically, however, it does provide a powerful emotional impetus to scientific activity.

Another question Kuhn raises is when and why scientists decide that an old paradigm is unsustainable and a new one is required. For centuries astronomers in the Muslim and Western worlds worked within the basic approach laid down by Ptolemy, hoping that small adjustments could finally remove the slight errors inherent in the system. During this time, the flexibility of the Ptolemaic approach—allowing for fine-tuning in deferents, equants, and epicycles—was seen as one of its strengths. Besides, the Ptolemaic astronomy was fully integrated within the wider Aristotelian science of the age; and this science blended perfectly with common everyday notions. The fact that the Ptolemaic science broke down is attributable as much, or more, to factors external to the science as to those internal to it. Specifically, with the Renaissance came the rediscovery of Neoplatonism, with its emphasis on mathematical harmonies—something absent from Aristotelianism—as well as its strain of sun-worship.

Copernicus was one of those affected by the new current of Neoplatonism; and it is this, Kuhn argues, that ultimately made him dissatisfied with the Ptolemaic system and apt to place the sun at the center of his system. We often hear of science progressing as a result of new experiments and empirical discoveries; but no such novel observation played a role in Copernicus’s innovation. Rather, the source of Copernicus’s rejection of an earth-centered universe was its inability to explain why the planets’ orbits are related to the sun’s. His system answered that question. But this was only an aesthetic improvement. It did not lead to more accurate predictions—the essential task of astronomy—and, indeed, it did not even lead to more efficient calculations. The oft-reproduced image of the Copernican universe, consisting of seven concentric circles, is a simplification; his actual system used dozens of circles and was cumbersome and difficult to use.

But the most puzzling feature of Copernicus’s innovation is that it achieves qualitative simplification at the expense of rendering it completely incompatible with the wider worldview. Aristotelian physics cannot explain why a person would not fly off of a moving earth. And, indeed, the entire cosmological picture, such as that painted so convincingly by Dante, ceases to make sense in a Copernican universe. For centuries people had understood the earth as a midpoint between the fires of hell and the perfect heavens above. Now, hell was only metaphorically “below” and heaven only metaphorically “above.” Besides that, the universe had to be expanded to mystifying proportions; the earth became only a small and unimportant speck in an unimaginably vast space. Strangely, however, Copernicus seemed blind to most of these consequences of his innovation. A specialist concerned only with creating a harmonious system, his attempt to render it physically plausible or theologically palatable is, at best, half-hearted.

This leads to the irony that one of the greatest intellectual revolutions in history started with a man concerned with technical minutiae inaccessible to the vast majority of the public, who had access to no fundamentally new data, whose system was neither more accurate nor more efficient than its predecessor, and whose main concern was qualitative harmoniousness. Copernicus was no radical and had no notion of upsetting the established authority; he himself would likely have been appalled at the Newtonian universe that was the end result of this process.

Yet this simple innovation, once proposed, had ripple effects. Though the earth’s motion was near universally rejected as a fact, its use in a serious astronomical work kept it alive as an option. And this new option could not be laughed away when, in the next generation under Tycho Brahe, better observations and novel phenomena upset the Ptolemaic world order. The heavens could no longer be seen as perfect and unchanging when Brahe proved that supernovae and comets do not exhibit a parallax (as in, they do not to change location when the observer moves), and thus could not be atmospheric phenomena. Further, Brahe’s unprecedentedly accurate observations of the planets were incompatible with any Ptolemaic system.

This seems to be one of many cases in the history of science when novel observations followed, rather than preceded, a theoretical innovation. us
Granted, this incongruence led Brahe to propose his own earth-centered system, the Tychonic, rather than adopt a sun-centered universe. But this new system used Copernican mathematics, and embodied the Copernican harmonies. In any case it is hard to see how the Tychonic system could ever have been anything but a stopgap, since the jump from Ptolemy to Brahe was scarcely easier than the jump from Ptolemy to Copernicus. Besides, it struck many as dynamically implausible that everything in the universe would orbit the sun except the earth and the moon.

Kepler and Galileo were among those unconvinced by the Tychonic system. The two very different men were both of an independent turn of mind, and their work finally made the Copernican universe unequivocally superior. Kepler particularly made the decisive step with his three laws: that planets orbit in ellipses with the sun at a focus, that they sweep out equal areas in equal times, and that they orbit the sun in a ratio of the 3/2 power (the orbital axis to the orbital time). But in Kepler we find further ironies. Far from the dispassionate lover of truth, Kepler was a Neoplatonic mystic, bursting with occult hypotheses. Many parts of his work strike the modern reader as scarcely more rational than the ravings of a conspiracy theorist. Yet the hard core of Kepler’s astronomical work lifted Copernicanism into a league of its own for accuracy of prediction and efficiency of calculation. If the orbits of the planets were related to the sun in such simple, elegant ways, it was difficult to see how earth could be at the center of it all.

This is my best attempt at summarizing the most salient points of the book. But of course there is far more in here, most of it worthwhile. I particularly enjoyed Kuhn’s chapter on the oft-ignored medieval research into physics, such as the impetus theory in the work of Nicole Oresme. The only weak point of the book was the rather brief epilogue to Copernicus. In particular, I would have appreciated an entire chapter devoted to Newton, since it was his Principia that was, in Kuhn’s phrase, the “capstone” of the revolution. But on the whole I think this is a superlative book, serious yet accessible, informative while brief. Kuhn captures the reality of scientific progress, which is far less neat that we may like to believe. Most striking is how a revolution which was guided by many extra-logical considerations—the Neoplatonic belief in celestial harmonies, the desire for mathematical elegance, the weakening of the religious worldview, the need to feel at home in the universe—fueled a process which, taken as a whole, resulted in a science definitively better than the Ptolemaic system it replaced.

Kuhn makes no mistake about this. Here is what the reputed relativist has to say:

The last two and one-half centuries have proved that the conception of the universe which emerged from the Revolution was a far more powerful intellectual tool than the universe of Aristotle and Ptolemy. The scientific cosmology evolved by seventeenth-century scientists and the concepts of space, force, and matter that underlay it, accounted for both celestial and terrestrial motions with a precision undreamed of in antiquity. In addition, they guided many novel and immensely fruitful research programs, disclosing a host of previously unsuspected natural phenomena and revealing order in fields of experience that had been intractable to men governed by the ancient world view.

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Review: The Invention of Nature

Review: The Invention of Nature
The Invention of Nature: Alexander von Humboldt's New World

The Invention of Nature: Alexander von Humboldt’s New World by Andrea Wulf

My rating: 3 of 5 stars

Alexander von Humboldt was a remarkable man. Simultaneously a savant and an explorer, he knew everyone, studied everything, and did his best to travel everywhere. Andrea Wulf brings together the many seemingly divergent worlds that he bridged: the worlds of Thomas Jefferson, Simón Bolívar, Napoleon, Goethe, Charles Darwin, and even Isambard Kingdom Brunel. He left his fingerprints on the worlds of science, literature, art, and even politics. Yet today he is (or was, before Wulf) a fairly obscure figure in the English-speaking world.

Thus this book is not simply a biography, but an attempt at rehabilitation. Wulf wishes to restore Humboldt to his place of honor; and she does this by arguing that his influence has been fundamental and pervasive. But before she can deal with Humboldt’s reputation, she must first narrate the scientist’s own coming of age. Humboldt was one of these figures with seemingly boundless energy, who threw himself into his work with complete abandon. We watch the young Humboldt as he struggles with, and finally throws off, the expectations of his upbringing, and then dashes away to South America. Once he embarks on his voyage, it does not take a strong writer—which Wulf is—to make his story exciting. Humboldt’s own travelogues were bestsellers.

Humboldt emerges from his travels with a concept of nature which, Wulf argues, was revolutionary and which became extremely influential. Wulf identifies three new elements of Humboldt’s approach to nature: First, that nature cannot be understood without both the scientific and the poetic eye; analysis and sentiment are necessary to do justice to the natural world. Second, that the living world must be understood as a gestalt, with organisms depending on one another in an intimate set of relationships that boggles the intellect. And third, that scientists must think on a global scale if they wish to understand the complex interactions between plants, animals, and climates.

This is the meat of the book. Yet it is here that I began to shift from enchantment to disappointment. For Wulf does not do nearly enough work to convince the skeptical reader that Humboldt’s view of nature was so entirely new. I would have appreciated far more background on previous conceptualizations of the natural world. Without this, it is hard to tell where Humboldt was innovative. Further, Wulf is always rather vague with Humboldt’s actual scientific contributions. She elects to keep the narrative pace driving forward, which doubtless helped her sales; yet I would have appreciated an explanation of Humboldt’s thought in more detail, with a good deal more quoting of the man.

Conversely, Wulf could have greatly reduced the space devoted to the men Humboldt influenced. She has individual chapters for John Muir, Henry David Thoreau, Charles Darwin, George Perkins Marsh, and Ernst Haeckel—space that she uses as opportunities to prove her thesis that Humboldt’s writings were fundamental to their success. But I found the biographical detail for these men excessive, and her point overstated. She makes it seem as if these men owed their accomplishments—if not wholly, at least in large part—to Humboldt’s influence. But you cannot measure influence, and you cannot prove a counterfactual (what would they have done without Humboldt?). In any case, the point is entirely abstract without a more careful discussion of Humboldt’s ideas; lacking that, it is not possible to say where his influence begins or ends.

By now I am convinced that Humboldt was an important and compelling figure in the history of science. But I am far from convinced that his late obscurity was a mere result of anti-German prejudice caused by the two World Wars, as Wulf claims in the Epilogue. Too many other German scientists and philosophers remained famous. Rather, I think Humboldt may have fallen into obscurity because it is difficult to do justice to the nature of his contribution. Unlike Darwin, he did not originate any major scientific theory that could unify a great many phenomena under a simple explanation. Humboldt’s major contributions seems to be perspectival: seeing nature as complex yet whole, as godless yet beautiful, as vast and inhuman yet spiritually refreshing. And it is difficult to work that into a textbook.



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Review: Treatise on Light

Review: Treatise on Light
Treatise on Light (Illustrated Edition)

Treatise on Light by Christiaan Huygens

My rating: 4 of 5 stars

But sound, as I have said above, only travels 180 toises in the same time of one second: hence the velocity of light is more than six hundred thousand times greater than that of sound.


This little treatise is included in volume 34 of the Great Books of the Western World, which I used to read Newton’s Principia and his Opticks. In this edition the Treatise comes out to about 50 pages, so I decided it was worth combing through. Christiaan Huygens is one of the relatively lesser known figures of the scientific revolution. But even a brief acquaintance with his life and work is enough to convince one that he was a thinker of gigantic proportion, in a league with Descartes and Leibniz. His work in mechanics prefigured Newton’s laws, and his detailed understanding of the physics of pendulums (building from Galileo’s work) allowed him to invent the pendulum clock. His knowledge of optics also improved the technology of telescope lenses, which in turn allowed him to describe the rings of Saturn and discover the first of Saturn’s moons, Titan.

Apart from all this, Huygens was the progenitor of the wave theory of light. This is in contrast with the corpuscular theory of light (in which light is conceived of as little particles), put forward 14 years later in Isaac Newton’s Opticks. Newton’s theory quickly became more popular, partially because of its inherent strength, and partially because it was Isaac Newton who proposed it. But Huygens’s wave theory was revived and seemingly confirmed in the 19th century by Thomas Young and Augustin-Jean Fresnel.

Essentially, Huygens’s idea was to use sound as an analogy for light. Just as sound consists of longitudinal waves (vibrating in the direction they travel) propagated by air, so light must consist of much faster waves propagated by some other, finer medium, which Huygens calls the ether. He conceives of a luminous object, such as a burning coal, as emitting circular waves at every point in its surface, spreading in every direction throughout a space.

Like Newton, Huygens was aware of Ole Rømer’s calculation of the speed of light. It had long been debated whether light is instantaneous or merely moves very quickly. Aristotle rejected the second option, thinking it inconceivable that something could move so fast. Little progress had been made since then, because making a determination of light’s speed presents serious challenges: not only is light several orders of magnitude faster than anything in our experience, but since light is the fastest thing there is, and the bearer of our information, we have nothing to measure it against.

This changed once astronomers began measuring the movement of the Jovian moons. Specifically, the moon Io is eclipsed by Jupiter every 42.5 hours; but as Rømer measured this cycle at different points in the year, he noticed that it varied somewhat. Realizing that this likely wasn’t due to the moon’s orbit itself, he hypothesized that it was caused by the varying distance of Earth to Jupiter, and he used this as the basis for the first roughly accurate calculation of the speed of light. Newton and Huygens both accepted the principle and refined the results.

Huygens gets through his wave theory, reflection, and refraction fairly quickly; and in fact the bulk of this book is dedicated to an analysis of Icelandic spar—or, as Huygens calls it, “The Strange Refraction of Icelandic Crystal.” This is a type of crystal that is distinctive for its birefringence, which means that it refracts light of different polarizations at different angles, causing a kind of double image to appear through the crystal. Huygens delves into a detailed geometrical analysis of the crystal, which I admit I could not follow in the least; nevertheless, the defining property of polarization eludes him, since to understand it one must conceive of light as a transverse, not a longitudinal, wave (that is, unlike a sound wave, which cannot be polarized). In the end, he leaves this puzzling property of the crystal for future scientists, but not without laying the groundwork of observation and theory that we still rely upon.

All together, this little treatise is a deeply impressive work of science: combining sophisticated mathematical modeling with careful experimentation to reach surprising new conclusions. Huygens illustrates perfectly the rare mix of gifts that a scientist must have in order to be successful: a sharp logical mind, careful attention to detail, and a creative imagination. The world is full of those with only one or two of these qualities—brilliant mathematicians with no interest in the real world, obsessive recorders and cataloguers with no imagination, brilliant artists with no gift for logic—but it takes the combination to make a scientist of the caliber of Huygens.



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Review: A Very Short Introduction to Galileo

Review: A Very Short Introduction to Galileo
Galileo: A Very Short Introduction

Galileo: A Very Short Introduction by Stillman Drake

My rating: 4 of 5 stars

There is not a single effect in Nature, not even the least that exists, such that the most ingenious theorists can ever arrive at a complete understanding of it.

One of the most impressive aspects of the Very Short Introduction series is the range of creative freedom allowed to its writers. (Either that, or its flexibility in repurposing older writings; presumably a version of this book was published before the VSI series even got off the ground, since its author died in 1993.) This is a good example: For in lieu of an introduction, Stillman Drake, one of the leading scholars of the Italian scientist, has given us a novel analysis of Galileo’s trial by the Inquisition.

Admittedly, in order to contextualize the trial, Drake must cover all of Galileo’s life and thought. But Drake’s focus on the trial means that many things one would expect from an introduction—for example, an explanation of Galileo’s lasting contributions to science—are only touched upon, in order to make space for what Drake believed was the crux of the conflict: Galileo’s philosophy of science.

Galileo Galilei was tried in 1633 for failing to obey the church’s edict that forbade the adoption, defense, or teaching of the Copernican view. And it seems that he has been on trial ever since. The Catholic scientist’s battle with the Catholic Church has been transformed into the archetypical battle between religion and science, with Galileo bravely championing the independence of human reason from ancient dogma. This naturally elevated Galileo to the status of intellectual heroe; but more recently Galileo has been criticized for falling short of this ideal. Historian of science, Alexandre Kojève, famously claimed that Galileo hadn’t actually performed the experiments he cited as arguments, but that his new science was mainly based on thought experiments. And Arthur Koestler, in his popular history of astronomy, criticized Galileo for failing to incorporate Kepler’s new insights. Perhaps Galileo was not, after all, any better than the scholastics he criticized?

Drake has played a significant role in pushing back against these arguments. First, he used the newly discovered working papers of Galileo to demonstrate that, indeed, he had performed careful experiments in developing his new scheme of mechanics. Drake also points out that Galileo’s Dialogue Concerning the Two Chief World Systems was intended for popular audiences, and so it would be unreasonable to expect Galileo to incorporate Kepler’s elliptical orbits. Finally, Drake draws a hard line between Galileo’s science and the medieval theories of motion that have been said to presage Galileo’s theories. Those theories, he observes, were concerned with the metaphysical cause of motion; whereas Galileo abandoned the search for causes, and inaugurated the use of careful measurements and numerical predictions in science.

Thus, Drake argues that Galileo never saw himself as an enemy of the Church; to the contrary, he saw himself as fighting for its preservation. What Galileo opposed was the alignment of Church dogma with one very particular interpretation of scripture, which Galileo believed would put the church in danger of being discredited in the future. Galileo attributed this mistaken policy to a group of malicious professors of philosophy, who, in the attempt to buttress their outdated methods, used Biblical passages to make their views seem orthodox. This was historically new. Saint Augustine, for example, considered the opinions of natural philosophers entirely irrelevant to the truth of the Catholic faith, and left the matter to experts. It was only in Galileo’s day (during the Counter-Reformation) that scientific theories became a matter of official church policy.

Drake’s conclusion is that Galileo’s trial was not so much a conflict between science and religion (for the two had co-existed for many centuries), but between science and philosophy: the former concerned with measurement and prediction, the latter concerned with causes. And Drake notes that many contemporary criticisms of Galileo—leaving many loose-ends in his system, for example—mirror the contemporary criticisms of his work. The trial goes on.

Personally I found this book fascinating and extremely lucid. However, I am not sure it exactly fulfills its promise as an introduction to Galileo. I think that someone entirely new to Galileo’s work, or to the history and philosophy of science, may not get as much out of this work. Luckily, most of Galileo’s own writings (translated by Drake) are already very accessible and enjoyable.



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