Two New Sciences/A History of Free FallTwo New Sciences/A History of Free Fall by Galileo Galilei

My rating: 4 of 5 stars

But in what seas are we inadvertently engulfing ourselves, bit by bit? Among voids, infinities, indivisibles, and instantaneous movements, shall we ever be able to reach harbor even after a thousand discussions?

When most people think about the Copernican revolution, the name that comes most readily to mind—more even than that of Copernicus himself—is that of Galileo Galilei. It was he, after all, who fought most valiantly for the acceptance of the theory, and it was he who suffered the most for it—narrowly escaping the tortures of the Inquisition. It was also Galileo who wrote the most famous book to come out of the revolution: Dialogue Concerning the Two Chief World Systems, whose publication most directly resulted in Galileo’s punishment.

Some years ago I read and admired that eloquent work. But lately, after slogging my way through Ptolemy, Copernicus, and Kepler, I have come to look upon Galileo’s famous dialogue with more suspicion. For it was only through the work of Kepler that the Copernican system became unquestionably more efficient than the Ptolemaic as a method of calculating celestial movements; and though Kepler was a contemporary and a correspondent of Galileo, the Italian scientist was not aware of the German’s groundbreaking innovations. Thus the version of heliocentrism that Galileo defends is Copernicus’s original system, preserving much of the cumbrous aspects of Ptolemy—epicycles, perfect circles, and separate tables for longitude and latitude, etc.

Added to this, the most decisive advantages in favor of Copernicus’s system over Ptolemy’s—explaining why the planets’ orbits seem related to the sun’s—are given little prominence, if they are even mentioned. Clearly, a rigorous defense of Copernicanism would require a demonstration that it made calculating heavenly positions easier and more accurate; but there is nothing of the kind in Galileo’s dialogue. As a result, Galileo comes across as a propagandist rather than a scientist. But of course, even if his famous dialogue was pure publicity, Galileo would have a secure place in the annals of astronomy from his observations through his improved telescope: of the lunar surface, of the moons of Jupiter, of the rings of Saturn, of sunspots, and of the phases of Venus. But I doubt this would be enough to earn him his reputation as a cornerstone of the scientific revolution.

This book provides the answer. Here is Galileo’s real scientific masterpiece—one of the most important treatises on mechanics in history. Rather inconveniently, its title is easy to confuse with Galileo’s more famous dialogue; but in content Two New Sciences is an infinitely more serious work than Two Chief World Systems. It is also a far less impassioned work, since Galileo wrote it when he was an old man under house arrest, not a younger man in battle with the Catholic authorities. This inevitably makes the book rather more boring to read; yet even here, Galileo’s lucid style is orders of magnitude more pleasant than, say, Kepler’s or Ptolemy’s.

As in Two Chief World Systems, the format is a dialogue between Simplicio, Sagredo, and Salviati (though Galileo cheats by having Salviati read from his manuscript). Unlike the earlier dialogue, however, Simplicio is not engaged in providing counter-arguments or in defending Aristotle; he mostly just asks clarifying questions. Thus the dialogue format only serves to enliven a straightforward exposition of Galileo’s views, not to simulate a debate.

The book begins by asking why structures cannot be scaled up or down without changing their properties. Why, for example, will a small boat hold together if slid down a ramp, but a larger boat fall to pieces? Why does a horse break its leg it falls down, but a cat can fall from the same distance entirely uninjured? Why are the bones of an elephant proportionately so much squatter and fatter than the bones of a mouse? In biology this is known as the science of allometry, and personally I find it fascinating. The key is that, when increasing size, the ratio of volume to area also increases; thus an elephant’s bones must support far more weight, proportionally, than a mouse’s. As a result, inventors and engineers cannot just scale up contraptions without providing additional support—quite a counter-intuitive idea at the time.

Galileo next delves into infinities. This leads him into what is called “Galileo’s paradox,” but is actually one of the defining properties of infinite sets. This states that the parts of an infinite set can be equal to the whole set; or in other words, they can both be infinite. For example, though the number of integers with a perfect square root (4, 9, 16…) will be fewer than the total number of integers in any finite set (say, from 1-100), in the set of all integers there is an infinite number of integers with a perfect square roots; thus the part is equal to the whole. Galileo also takes a crack at Aristotle’s wheel paradox. This is rather dull to explain; but suffice to say it involves the simultaneous rotation of rigid, concentric circles. Galileo attempts to solve it by postulating an infinite number if infinitesimal voids in the smaller circle, and in fact uses this as evidence for his theory of infinitesimals.

As a solution to the paradox, this metaphysical assertion fails to do justice to its mathematical nature. However, the concept of infinitely small instants does help to escape from of the Zeno-like paradoxes of motion, to which Greek mathematics was prone. For example, if you imagine an decelerating object spending any finite amount of time at any definite speed, you will see that it never comes to a full stop: the first second it will travel one meter, the next second only half a meter, the next second a quarter of a meter, and so on ad infinitum. The notion of deceleration taking places continuously over an infinite number of infinitely small instants helped to escape this dilemma (though it is still unexplained how a thing can be said to “move” during an instant).

Galileo had need of such concepts, since he was writing long before Newton’s calculus and too early to be influenced by Descartes’s analytical geometry. Thus the mathematical apparatus of this book is Greek in form. Galileo’s calculations consist exclusively of ratios between lines rather than equations; and he establishes these ratios using Euclid’s familiar proofs. Consequently, his mechanics is relational or relativistic—able to give proportions but not exact quantities.

This did not stop Galileo from anticipating much of Newton’s system. He establishes the pendulum as an exemplar of continually accelerated motion, and shows that pendulums of the same length of rope swing at the same rate, regardless of the height from which they fall. He asserts that an object, once started in motion, would continue in motion indefinitely were it not for friction and air resistance. He recounts experiments of dropping objects of different masses from the same distance, and seeing them land at the same moment, thus disproving the Aristotelian assertion that objects fall with a speed proportional to their mass. (Unfortunately, there is scant evidence for the story that Galileo performed this experiment from the Leaning Tower of Pisa.) Galileo also makes the daring asserting that, in a vacuum, all objects would fall at the same rate.

There are still more riches to be excavated. Galileo asserts that pitches are caused by vibrating air, that faster vibrations causes higher pitch, and that consonant harmonies are caused by vibrations in regular ratios. He exhaustively calculates how the time and speed of a descending object would differ based on its angle of descent—straight down or on an inclined plane. He also shows that objects shot into the air, as in a catapult, descend back to earth in a parabolic arc; and he shows that objects travel the furthest when shot at 45 degrees. In an appendix, Galileo uses an iterative approach to find the center of gravity of curved solids; and in an added dialogue he discusses the force of percussion.

As you can see, this book is too rich and, in parts, too technical for me to appraise it in detail. I will say, however, that of all the scientific classics I have read this year, the modern spirit of science shines through most clearly in these pages. For like any contemporary scientist, Galileo assumes that the behavior of nature is law-like, and is fundamentally mathematical; and with Galileo we also see a thinker completely willing to submit his speculations to experiment, but completely unwilling to submit them to authority. Far more than in the metaphysical Kepler—who speculated with wild abandon, though he was a scientist of comparable importance—in Galileo we find a true skeptic: who believed only what he could observe, calculate, and prove. The reader instantly feels, in Galileo, the force of an exceptionally clear mind and of an uncompromising dedication to the search for truth.

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