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: Epitome of Copernican Astronomy & Harmonies of the World

Review: Epitome of Copernican Astronomy & Harmonies of the World

Epitome of Copernican Astronomy and Harmonies of the WorldEpitome of Copernican Astronomy and Harmonies of the World by Johannes Kepler

My rating: 3 of 5 stars

The Earth sings MI, FA, MI so that you may infer even from the syllables that in this our domicile MIsery and FAmine obtain.

Thomas Kuhn switched from studying physics to the history of science when, after teaching a course on outdated scientific models, he discovered that his notion of scientific progress was completely mistaken. As I plow through these old classics in my lackadaisical fashion, I am coming to the same conclusion. For I have discovered that the much-maligned Ptolemy produced a monument of observation and mathematical analysis, and that Copernicus’s revolutionary work relied heavily on this older model and was arguably less convincing. Now I discover that Johannes Kepler, one of the heroes of modern science, was also something of a crackpot.

The mythical image of the ideal scientist, patiently observing, cataloguing, calculating—a person solely concerned with the empirical facts—could not be further removed from Kepler. Few people in history had such a fecund and overactive imagination. Every new observation suggested a dozen theories to his feverish mind, not all of them testable. When Galileo published his Siderius Nuncius, for example, announcing the presence of moons orbiting Jupiter, Kepler immediately concluded that there must be life on Jupiter—and, why not, on all the other planets. Kepler even has a claim of being the first science-fiction writer, with his book Somnium, describing how the earth would appear to inhabitants of the moon (though Lucian of Samothrace, writing in the 2nd Century AD, seems to have priority with his fantastical novella, A True Story). This imaginative book, by the way, may have contributed to the accusations that Kepler’s mother was a witch.

In reading Kepler, I was constantly reminded of a remark by Bertrand Russell: “The first effect of emancipation from the Church was not to make men think rationally, but to open their minds to every sort of antique nonsense.” Similarly, the decline in Aristotle’s metaphysics did not prompt Kepler to reject metaphysical thinking altogether, but rather to speculate with wild abandon. But Kepler’s speculations differed from the ancients’ in two important respects: First, even when his theories are not testable, they are mathematical in nature. Gone are the verbal categories of Aristotle; and in comes the modern notion that nature is the manifestation of numerical harmonies. Second, whenever Kepler’s theories are testable, he tested them, and thoroughly. And he had ample data with which to test his speculations, since he was bequeathed the voluminous observations of his former mentor, Tycho Brahe.

At its worst, Kepler’s method resulted in meaningless numerical coincidences that explained nothing. As many a statistician has learned, if you crunch enough numbers and enough variables, you will eventually stumble upon a serendipitous correlation. This aptly describes Kepler’s use of the five Platonic solids to explain planetary orbits; by trying many combinations, Kepler found that he could create an arrangement of these regular solids, nested within one another, that mostly corresponded with the size of the planets’ orbits. But what does this explain? And how does this help calculation? The answer to both of these questions is negative; the solution merely appeals to Kepler’s sense of mathematical elegance, and reinforced his religious conviction that God must have arranged the world harmoniously.

Another famous example of this is Kepler’s notion of the “harmonies of the world.” By playing with the numbers of the perihelion, aphelion, orbital lengths, and so forth, Kepler assigns a melodic range to each of the planets. Mercury, having the most elongated orbit, has the biggest range; while Venus’s orbit, which most approximates a perfect circle, only produces a single note. Jupiter and Saturn are the basses, of course, while Mars is the tenor, Earth and Venus the altos, and Mercury the soprano. He then suggests (though vaguely) that there are beings on the sun, capable of sensing this heavenly music. (The composer Laurie Spiegel created a piece in which she recreates this music; it is not exactly Bach.) Once more, we naturally ask: What would all this speculation on music and harmonies explain? And once more, the answer is nothing.

Kepler’s writing is full of this sort of thing—torturous explorations of ratios, data, figures, which strike the modern mind as ravings rather than reasoning. But the fact remains that Kepler was one of the great scientific geniuses of history. He was writing in a sort of interim period between the fall of Aristotelian science and the rise of Newtonian physics, a time when the mind of Europe was completely untethered to any recognizable paradigm, free to luxuriate in speculation. Most people in such circumstances would produce nothing but nonsense; but Kepler managed to invent astrophysics.

What gives Kepler a claim to this title was his conception of a scientific law (though he did not put it as such). Astronomers from Ptolemy to Copernicus used schemes to predict planetary movements; but there was no one underlying principle which could explain everything. Kepler’s relentless search for numerical coincidences led him to statements that unified observations of all the planets. These are now known as Kepler’s Laws.

The first of these was the seemingly simple but revolutionary insight that planets orbit in ellipses, with the sun at one of the foci. It is commonly said that previous astronomers preferred circles for petty metaphysical reasons, seeing them as perfect. But there were other reasons, too. Most obviously, the mathematics of shapes inscribed in circles was well-understood; this was the basis of trigonometry.

Yet the use of circles to track orbits that, in reality, are not circular, created some problems. Thus in the Ptolemaic system the astronomer used one circle (the eccentric) for the distance, and another, overlapping circle (the equant) for the speed. When these were combined with the epicycles (used to explain retrogression) the resultant orbits, though composed of perfect circles, were anything but circular. Kepler’s use of ellipses obviated the need for all these circles, reducing a complicated machinery into a single shape. It was this innovation that made the Copernican system so much more efficient than the Ptolemaic one. As Owen Gingerich, a Copernican scholar, has said: “What passes today as the ‘Copernican System’ is in detail the Keplerian system.”

Yet the use of ellipses, by itself, would not have been so useful were it not for Kepler’s Second Law: that planets sweep out equal areas in equal times of their orbits. For when a planet is closest to the sun (at perihelion) it is moving its fastest; and when it is furthest (at aphelion) it is slowest; and this creates a constant ratio (which is the result of the conserved angular momentum of each planet). Ironically, of the two, Ptolemy was closer than Copernicus to this insight, since Ptolemy’s much-maligned equant (the imaginary point around which a planet travels at a constant speed) is a close approximation of the Second Law. Even so, I think that Kepler moved far beyond all previous astronomy with these insights, jumping from observed and analyzed regularities to general principles.

Kepler’s Third Law seemed to have excited the astronomer the most, since he even includes the exact date at which he made the realization: “… on the 8th of March in this year One Thousand Six Hundred and Eighteen but unfelicitously submitted to calculation and rejected as false, finally, summoned back on the 15th of May, with a fresh assault undertaken, outfought the darkness of my mind.” This law states that, for every planet, the ratio of the orbital period squared to the orbital size cubed, is constant. (For the orbital size Kepler used half the major axis of the ellipse.)

While it is no doubt striking that this ratio is almost the same for every planet (this is because the planet’s mass is negligible compared with the sun’s), it is difficult to completely sympathize with Kepler’s excitement, since the resultant law is not useful for predicting orbits, and its significance was only explained much later by Newton as a derivable conclusion from his equations. Kepler, being the man he was, used this mathematical constant to fuel his metaphysical speculations.

However much, then, that Kepler’s theories may strike us nowadays as baseless, crackpot theorizing, he must be given a commanding place in the history of science. The reason I cannot rate this collection any higher is that Kepler is extremely tiresome to read. In his more lucid moments, his imaginative energy is charming. But much of the book consists of whole paragraphs of ratio after ratio, shape after shape, number after number, and so it is easy to get lost or bored. Since I have a decent grasp of music theory, I thought I might be able to get something out of his Harmonies of the World, but I found even that section mostly opaque, swirling in obscure and impenetrable reasoning.

The great irony, then, is that Kepler’s writings can strike the modern-day reader as far less “scientific” than Ptolemy’s; but perhaps we should expect such ironies from a man who helped to inaugurate modern science, but who made his living casting horoscopes.

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Review: On the Revolutions of Heavenly Spheres

Review: On the Revolutions of Heavenly Spheres

On the Revolutions of Heavenly SpheresOn the Revolutions of Heavenly Spheres by Nicholas Copernicus

My rating: 4 of 5 stars

And though all these things are difficult, almost inconceivable, and quite contrary to the opinion of the multitude, nevertheless in what follows we will with God’s help make them clearer than day—at least for those who are not ignorant of the art of mathematics.

The Copernican Revolution has become the prime exemplar of all the great transformations in our knowledge of the world—a symbol of scientific advance, the paradigmatic clash of reason and religion, a shining illustration of how cold logic can beat out old prejudices. Yet reading this groundbreaking book immediately after attempting Ptolemy’s Almagest—the Bible of geocentric astronomy—reveals far more similarities than differences. Otto Neugebauer was correct in calling Copernicus’s system an ingenious modification of Hellenistic astronomy, for it must be read against the background of Ptolemy in order to grasp its significance.

The most famous section of De revolutionibus was, ironically, not even written by Copernicus, but by the presumptuous Andreas Osiander, a Lutheran theologian who was overseeing the publication of the book, and who included a short preface without consulting or informing Copernicus. Knowing that Copernicus’s hypothesis could prove controversial (Luther considered it heretical), Osiander attempted to minimize its danger by asserting that it was merely a way of calculating celestial positions and did not represent physical reality: “for it is not necessary that the hypotheses should be true, or even probable; but it is enough if they provide a calculation which fits the observations.”

Though this assertion obviously contradicts the body of the work (in which Copernicus argues at length for the reality of the earth’s movement), and though Copernicus and his friends were outraged by the insertion, it did help to shield the book from censure. And arguably Osiander was being a good and true Popperian—believing that science is concerned with making accurate predictions, not in giving us “the truth.” In any case, Osiander was no doubt correct in this assertion: “For it is sufficiently clear that this art is absolutely and profoundly ignorant of the causes of the apparent irregular movements.” Neither Ptolemy nor Copernicus had any coherent explanation of what caused the orbits of the planets, which would not come until Einstein.

After this little interpolation, Copernicus himself wastes no time in proclaiming the mobility of the earth. In retrospect, it is remarkable that it took such a long stretch of history for the heliocentric idea to emerge. For it instantly explains many phenomena which, in the Ptolemaic system, are completely baffling. Why do the inner planets (Venus and Mercury) move within a fixed distance of the sun? Why does the perigee (the closest point in the orbit) of the outer planets (Mars, Jupiter, Saturn) occur when they are at opposition (i.e., when they are opposite in the sky from the sun), and why does their apogee (the farthest point) occur when they are in conjunction (when they are hidden behind the sun)? And why do the planets sometimes appear to move backwards relative to the fixed stars?

But putting the earth in orbit between Venus and Mars neatly and instantly explains all of these mysteries. Mercury and Venus always appear a fixed distance from the sun because they are orbiting within the earth’s orbital circle, and thus from our position appear to go back and forth around the sun. Mars, Jupiter, and Saturn, by contrast, can appear at any longitudinal distance from the sun because their orbits are outsider ours; but if Mars’ orbit were tracked from Jupiter, for example, it would, like Venus and Mercury, appear to go back and forth around the sun. Also note that Mars will appear to go “backwards” from earth when earth overtakes the red planet, due to our planet’s shorter orbital period. And since Mars will be closest to us when it is on the same side of the sun as earth (opposition from the sun), and furthest from us when it is on far side of the sun (conjunction with the sun), this also explains the apogee and perigee positions of the outer planets.

This allows Copernicus to collapse five circles—one for each of the planets, which were needed in the Ptolemaic system to account for these anomalies—into one circle: namely, the earth’s orbit. The advantages are palpable.

Nevertheless, while I think the benefits of putting the planets in orbit around the sun are obvious, perhaps even to a traditionalist, it is not obvious why Copernicus should put the earth in motion around the sun rather than the reverse. Indeed, this is exactly what the eminent astronomer Tycho Brahe did, several generations later. For it makes no observational difference whether the sun or the earth is in motion. And in the Aristotelian physics of the time, the former solution makes a great deal more sense, since the heavens were supposed to be constituted of the lightest elements and the earth of the heaviest elements. So how could the heavy earth move so quickly? What is more, there is no concept of inertia or gravity in Aristotelian physics, and so no explanation for why people would not fly off the earth if it were in rapid motion.

Copernicus takes a brief stab at answering these obvious counterarguments, even offering a primitive notion of inertia: “As a matter of fact, when a ship floats on over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement which is the image of their own, and they think on the contrary that they themselves and all the things with them are at rest.” Even so, it is obvious that such a brief example does not suffice to refute the entire Aristotelian system. Clearly, a whole new concept of physics was needed if the earth was to be in motion, one which did not arrive until Isaac Newton, born nearly two hundred years after Copernicus. It took a certain amount of boldness, or obtuseness, for Copernicus to proclaim the earth’s motion without at all being able to explain how the heaviest object in the universe—or so they believed—could hurtle through space.

In structure and content, De revolutionibus follows the Algamest pretty closely: beginning with mathematical preliminaries, onward to the orbits of the sun (or, in this case the earth), the moon, and the planets—with plenty of tables to aid calculation—as well as a description of his astronomical instruments and a chart of star locations, and finally ending with deviations in celestial latitude (how far the planets deviate north and south from the ecliptic in their orbits). Copernicus was even more wedded than Ptolemy to the belief that celestial objects travel in perfect circles, which leads him to repudiate Ptolemy’s use of the equant (the point around which a planet moves at a constant speed). The use of the equant upset Copernicus’s sense of elegance, you see, since its center is different from the planet’s actual center of orbit, thus requiring two overlapping circles.

Copernicus’s own solution was an epicyclet, which revolves twice westward (clockwise, from the celestial north pole) for each rotation eastward on the deferent. And so, ironically, though Ptolemy is sometimes mocked for using epicycles, Copernicus followed the same path. I also find it amusing that the combined effect of these circular motions, in both Ptolemy and Copernicus, added up to a non-circular orbit; clearly nature had different notions of elegance than these astronomers. In any case, it would have to wait until Kepler that it was realized that the planets actually follow an ellipse.

Perhaps the greatest irony is that Copernicus’s book is not any easier to use than Ptolemy’s as a recipe book for planetary positions. Now, it is far beyond my powers to even attempt such a calculation. But in his Very Short Introduction to Copernicus (which I recommend), Owen Gingerich takes the reader through the steps to calculation the position of Mars on Copernicus’s birthday: February 19, 1473. To do this you needed the radix, which is a root position of the planet recorded at a specified time; and you also need the planet’s orbital speed (the time needed for one complete orbit, in this case 687 days). The year must be converted into sexigesimal (base 60) system, and then converted in elapsed Egyptian years (which lack a leap year), in order to calculate the time elapsed since the date of the radix’s position (in this case is January 1st, 1 AD). Then this sexigesimal number can be looked up in Copernicus’s tables; but this only gives us the location of Mars with respect to the sun. To find out where it will appear in the sky, we also need the location of earth, which is another tedious process. You get the idea.

I read the bulk of this book while I was on vacation in rural Canada. Faced with the choice between relaxation or self-torture, I naturally chose the latter. While most of my time was spent scratching my head and helplessly scratching the page with a pencil, the experience was enough to show me—as if I needed more demonstration after Ptolemy—that astronomy is not for the faint of heart, but requires intelligence, patience, and care.

There was one advantage to reading the book on vacation. For it is the only time of year when I am in a place without light pollution. The stars, normally hiding behind street lights and apartment buildings, shone in the hundreds. I would have seen even more were it not for the waxing moon. But this did give me the opportunity to get out an old telescope—bought as a birthday present for a cousin, over a decade ago—and examine the moon’s pitted surface. It is humbling to think that even such basic technology was years ahead of Copernicus’s time.

Looking at the brilliant grey circle, surrounded by a halo of white light, I felt connected to the generations of curious souls who looked at the same moon and the same stars, searching for answers. So Copernicus did not, in other words, entirely spoil my vacation.

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Review: Almagest

Review: Almagest
The Almagest: Introduction to the Mathematics of the Heavens

The Almagest: Introduction to the Mathematics of the Heavens by Ptolemy

My rating: 4 of 5 stars

… it is not fitting even to judge what is simple in itself in heavenly things on the basis of things that seem to be simple among us.

In my abysmal ignorance, I had for years assumed that tracking the orbits of the sun and planets would be straightforward. All you needed was a starting location, a direction, and the daily speed—and, with some simple arithmetic and a bit of graph paper, it would be clear as day. Attempting to read Ptolemy has revealed the magnitude of my error. Charting the heavenly bodies is a deviously complicated affair; and Ptolemy’s solution must rank as one of the greatest intellectual accomplishments of antiquity—fully comparable with the great scientific achievements of European Enlightenment. Indeed, Otto Neugebauer, the preeminent scholar of ancient astronomy, went so far as to say:

One can perfectly well understand the ‘Principia’ without much knowledge of earlier astronomy but one cannot read a single chapter in Copernicus or Kepler without a thorough knowledge of Ptolemy’s “Almagest”. Up to Newton all astronomy consists in modifications, however ingenious, of Hellenistic astronomy.

With more hope than sense, I cracked open my copy of The Great Books of the Western World, which has a full translation of the Almagest in the 16th volume. Immediately repulsed by the text, I then acquired a students’ edition of the book published by the Green Lion Press. This proved to be an excellent choice. Through introductions, preliminaries, footnotes, and appendices—not to mention generous omissions—this edition attempts to make Ptolemy accessible to a diligent college student. Even so, for someone with my background to attain a thorough knowledge of this text, he would still require months of dedicated study with a teacher as a guide. For the text is difficult in numerous ways.

Most obviously, this book is full of mathematical proofs and calculations, which are not exactly my strong suit. Ptolemy’s mathematical language—relying on the Greek geometrical method—will be unfamiliar to students who have not read some Euclid; and even if it is familiar, it proves cumbrous for the sorts of calculations demanded by the subject. To make matters worse, Ptolemy employs the sexagesimal system (based on multiples of 60) for fractions; so his numbers all must be converted into our decimals for calculation. What is more, even the names of the months Ptolemy uses are different, bearing their Egyptian names (Thoth, Phaöphi, Athur, etc.), since Ptolemy was an Alexandrian Greek. Yet even if we put all these technical obstacles to the side, we are left with Ptolemy’s oddly infelicitous prose, which the translator describes thus:

In general, there is a sort of opacity, even awkwardness, to Ptolemy’s writing, especially when he is providing a larger frame for a topic or presenting a philosophical discussion.

Thus, even in the non-technical parts of the book, Ptolemy’s writing tends to be headache-inducing. All this combines to form an unremitting slog. So since my interest in this book was amateurish, I skimmed and skipped liberally. Yet this text is so rich that, even proceeding in such a dilettantish fashion, I managed to learn a great deal.

Ptolemy’s Almagest, like Euclid’s Elements, proved so comprehensive and conclusive when it was published that it rendered nearly all previous astronomical work obsolete or superfluous. For this reason, we know little about Ptolemy’s predecessors, since there was little point in preserving their work after Ptolemy summed it up in such magnificent fashion. As a result it is unclear how much of this book is original and how much is simply adapted. As Ptolemy himself admits, he owes a substantial debt to the astronomer Hipparchus, who lived around 200 years earlier. Yet it does seem that Ptolemy originated the novel way of accounting for the planets’ position and speed, which he puts forth in later books.

Ptolemy begins by explaining the method by which he will measure chords; this leads him to construct one of the most precise trigonometric tables from antiquity. Later, Ptolemy goes on to produce several proofs of spherical trigonometry, which allows him to measure distances on the inside of a sphere, making this book an important source for Greek trigonometry as well as astronomy. Ptolemy also employs Menelaus’ Theorem, which uses the fixed proportions of a triangle to establish ratios. (From this I see that triangles are marvelously useful shapes, since they are the only shape which is rigid—that is, the angles cannot be altered without also changing the ratio of the sides, and vice versa. This is also, by the way, what makes triangles such strong structural components.)

Ptolemy gets down to business by analyzing the sun’s motion. This is tricky for several reasons. For one, the sun does not travel parallel to the “fixed stars” (so called because the stars do not position change relative to one another), but rather at an angle, which Ptolemy calculates to be around 23 degrees. We now know this is due to earth’s axial tilt, but for Ptolemy it was called the obliquity of the ecliptic (the angle of the sun’s path). Also, the angle that the sun travels through the sky (straight overhead or nearer the horizon) is determined by one’s latitude; this also determines the seasonal shifts in day-length; and during these shifts, the sun rises on different points on the horizon. To add to these already daunting variables, the sun also shifts in speed during the course of the year. And finally, Ptolemy had to factor in the procession of the equinoxes—the ecliptic’s gradual westward motion from year to year.

The planets turn out to be even more complex. For they all exhibit anomalies in their orbits which entail further complications. Venus, for example, not only speeds up and slows down, but also seems to go forwards and backwards along its orbit. This leads Ptolemy to the adoption of epicylces—little circles which travel along the greater circle, called the “deferent,” of the planet’s orbit. But to preserve the circular motion of the deferent, Ptolemy must place the center (called the “eccentric”) away from earth, in empty space. Then, Ptolemy introduces another imaginary circle, around which the planet travels with constant velocity: and the center of this is called the “equant,” which is also in empty space. Thus the planet’s motion was circular around one point (the eccentric) and constant around another circle (the equant), neither of which coincide with earth (so much for geocentric astronomy). In addition to all this, the orbit of Venus is not exactly parallel with the sun’s orbit, but tilted, and its tilt wobbles throughout the year. For Ptolemy to account for all this using only the most primitive observational instruments and without the use of calculus or analytic geometry is an extraordinary feat of patience, vision, and drudgery.

Even after writing all this, I am not giving a fair picture of the scope of Ptolemy’s achievement. This book also includes an extensive star catalogue, with the location and brightness of over one thousand stars observable with the naked eye. He argues strongly for earth’s sphericity (so much for a flat earth) and even offers a calculation of earth’s diameter (which was 28% too small). Ptolemy also calculates the distance from the earth to the moon, using the lunar parallax (the difference in the moon’s appearance when seen from different positions on earth), which comes out the quite accurate figure of 59 earth radii. And all of this is set forth in dry, sometimes baffling prose, accompanied by pages of proofs and tables. One can see why later generations of astronomers thought there was little to add to Ptolemy’s achievement, and why Arabic translators dubbed it “the greatest” (from which we get the English name).

A direct acquaintance with Ptolemy belies his popular image as a metaphysical pseudo-scientist, foolishly clinging to a geocentric model, using ad-hoc epicycles to account for deviations in his theories. To the contrary, Ptolemy scarcely ever touches on metaphysical or philosophical arguments, preferring to stay in the precise world of figures and proofs. And if science consists in predicting phenomena, then Ptolemy’s system was clearly the best scientific theory around for its range and accuracy. Indeed, a waggish philosopher might dismiss the whole question of whether the sun or the earth was at the “center” as entirely metaphysical (is it falsifiable?). Certainly it was not mere prejudice that kept Ptolemy’s system alive for so long.

Admittedly, Ptolemy does occasionally include airy metaphysical statements:

We propose to demonstrate that, just as for the sun and moon, all the apparent anomalistic motions of the five planets are produced through uniform, circular motions; these are proper to the nature of what is divine, but foreign to disorder and variability.

Yet notions of perfection seem hard to justify, even within Ptolemy’s own theory. The combined motions of the deferent and the epicycle do not make a circle, but a wavy shape called an epitrochoid. And the complex world of interlocking, overlapping, slanted circles—centered on imaginary points, riddled with deviations and anomalies—hardly fits the stereotypical image of an orderly Ptolemaic world.

It must be said that Ptolemy’s system, however comprehensive, does leave some questions tantalizingly unanswered. For example, why do Mercury and Venus stay within a definite distance from the sun, and travel along at the same average speed as the sun? And why are the anomalies of the “outer planets” (Mars, Jupiter, Saturn) sometimes related to the sun’s motion, and sometimes not? All this is very easy to explain in a heliocentric model, but rather baffling in a geocentric one; and Ptolemy does not even attempt an explanation. Even so, I think any reader of this volume must come to the conclusion that this is a massive achievement—and a lasting testament to the heights of brilliance and obscurity that a single mind can reach.

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Review: Dialogue Concerning the Two Chief World Systems

Review: Dialogue Concerning the Two Chief World Systems

Dialogue Concerning the Two Chief World SystemsDialogue Concerning the Two Chief World Systems by Galileo Galilei

My rating: 4 of 5 stars

I should think that anyone who considered it more reasonable for the whole universe to move in order to let the earth remain fixed would be more irrational than one who should climb to the top of your cupola just to get a view of the city and its environs, and then demand that the whole countryside should revolve around him so that he would not have to take the trouble to turn his head.

It often seems hard to justify reading old works of science. After all, science continually advances; pioneering works today will be obsolete tomorrow. As a friend of mine said when he saw me reading this, “That shit’s outdated.” And it’s true: this shit is outdated.

Well, for one thing, understanding the history of the development of a theory often aids in the understanding of the theory. Look at any given technical discipline today, and it’s overwhelming; you are presented with such an imposing edifice of knowledge that it seems impossible. Yet even the largest oak was once an acorn, and even the most frightening equation was once an idle speculation. Case in point: Achieving a modern understanding of planetary orbits would require mastery of Einstein’s theories—no mean feat. Flip back the pages in history, however, and you will end up here, at this delightful dialogue by a nettlesome Italian scientist, as accessible a book as ever you could hope for.

This book is rich and rewarding, but for some unexpected reasons. What will strike most moderns readers, I suspect, is how plausible the Ptolemaic worldview appears in this dialogue. To us alive today, who have seen the earth in photographs, the notion that the earth is the center of the universe seems absurd. But back then, it was plain common sense, and for good reason. Galileo’s fictional Aristotelian philosopher, Simplicio, puts forward many arguments for the immobility of the earth, some merely silly, but many very sensible and convincing. Indeed, I often felt like I had to take Simplicio’s side, as Galileo subjects the good Ptolemaic philosopher to much abuse.

I’d like to think that I would have sensed the force of the Copernican system if I were alive back then. But really, I doubt it. If the earth was moving, why wouldn’t things you throw into the air land to the west of you? Wouldn’t we feel ourselves in motion? Wouldn’t canon balls travel much further one way than another? Wouldn’t we be thrown off into space? Galileo’s answer to all of these questions is the principal of inertia: all inertial (non-accelerating) frames of reference are equivalent. That is, an experiment will look the same whether it’s performed on a ship at constant velocity or on dry land.

(In reality, the surface of the earth is non-inertial, since it is undergoing acceleration due to its constant spinning motion. Indeed the only reason we don’t fly off is because of gravity, not because of inertia as Galileo argues. But for practical purposes the earth’s surface can be treated as an inertial reference frame.)

Because this simple principle is the key to so many of Galileo’s arguments, the final section of this book is trebly strange. In the last few pages of this dialogue, Galileo triumphantly puts forward his erroneous theory of the tides as if it were the final nail in Ptolemy’s coffin. Galileo’s theory was that the tides were caused by the movement of the earth, like water sloshing around a bowl on a spinning Lazy Susan. But if this was what really caused the tides, then Galileo’s principle of inertia would fall apart; since if the earth’s movements could move the oceans, couldn’t it also push us humans around? It’s amazing that Galileo didn’t mind this inconsistency. It’s as if Darwin ended On the Origin of Species with an argument that ducks were the direct descendants of daffodils.

Yet for all the many quirks and flaws in this work, for all the many digressions—and there are quite a few—it still shines. Galileo is a strong writer and a superlative thinker; following along the train of his thoughts is an adventure in itself. But of course this work, like all works of science, is not ultimately about the mind of one man; it is about the natural world. And if you are like me, this book will make you think of the sun, the moon, the planets, and the stars in the sky; will remind you that your world is spinning like a top, and that the very ground we stand on is flying through the dark of space, shielded by a wisp of clouds; and that the firmament up above, something we often forget, is a window into the cosmos itself—you will think about all this, and decide that maybe this shit isn’t so outdated after all.

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