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[Science] is written in this grand book - I mean the
universe - which stands open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is
written.It is written in the language of mathematics,
and its characters are triangles, circles, and other
geometrical figures, without which it is humanly impossible to understand a single word of
it; without these, one is wandering about in a dark labyrinth.


Let us now turn to an example of scientific reasoning in situ. I have chosen certain episodes in Galileo's life for two reasons: First, this story is one of the most famous and fascinating in the history of science. Secondly, it provides excellent material with which to illustrate the ways in which real life scientific practice does and does not conform to thehypothetico deductive model.
The controversy over the acceptability of the Copernican theory involved at least four separable debates. As you study this case it will be helpful to keep the following four topics in mind:


1.The Astronomical Dispute: What were the competing models of the
universe? What was the evidence for and against each?
2.The Dispute in 'Physics: What were the competing theories of motion? What was the evidence for and against each?
3. The Religious Dispute : What were the competing theories about the proper relationship between the Bible and science? What were the arguments on each side?
4.The Methodological Debate: Was Galileo introducing new scientific methods as well as new scientific theories?
Although the story I tell below is intended to be roughly correct and certainly not seriously misleading, at times I have oversimplified things slightly. And since there is an ever-growing body of historical information about this period, my story (and the secondary sources on which I relied) may very well be out-of-date at some points.


I.Life Begins at Forty-Five 1
Prior to his famous telescopic observations, Galileo's scientific career had not been anything extraordinary. After brief studies at a monastery, Galileo studied medicine and then mathematics at the University of Pisa. In 1589, he gained the chair of mathematics there. In 1591 he moved to the University of Padua.
At that time mathematics included not only Euclidean geometry but also quantitative sciences such as astronomy. Most discussions of subjects which we would include under physics took place within philosophy departments. One concern of Galileo (and other anti- Aristotelians) was to introduce mathematical methods into the study of motion. When Galileo later moved to court at Florence in 1610 he insisted that his title be "mathematician and philosopher to the grand duke of Tuscany."
During this period Galileo gave lectures on Ptolemaic astronomy. He also knew about the Copernican system and wrote a letter to Kepler in 1597 in which he expressed his sympathy towards it. However, he did not make his sentiments public, although Kepler urged him to.
At this time Galileo was much more interested in mechanics than in astronomy. While at Pisa he wrote, but did not publish, a treatise on motion (called De Motu) in which he criticized the Aristotelian account of the motions of

1. Based on Stillman Drake's article on Galileo in The Dictionary of 'Scientific Biography.

falling bodies and projectiles. Galileo's own positive account of motion in this early work was a variant of the Medieval impetus theory. It was only later that he arrived at a theory which resembles the modern account.
Galileo also invented several useful practical instruments - a proportional compass for surveyors, a pendulum device for timing pulses in hospitals, and a clever little balance to be used for assaying metals according to their density. In 1606 someone stole his idea for the proportional compass and so Galileo pressed charges. Following the custom of the times Galileo also wrote a pamphlet denouncing the plagiarist: "Difesa . . . contro alle calunnie & imposture di Baldessar Capra." At the time when Galileo heard about the telescope (subsequently he sold the idea to the Venetian government) this pamphlet was his only published work.


II. Aristotelian Cosmology and Physics
Although the Aristotelian world-view had been criticized and revised in important ways during the Middle Ages, (Footnote 2) it was the traditional Aristotelian cosmology and physics which Galileo always set up as the chief opponent. And to a large extent, people in the Church and University establishments were Aristotelians.
According to Aristotle, the universe is finite. It is convenient to divide phenomena into two classes: sub lunar (or terrestrial) and celestial. Below the moon everything is composed of four elements - earth, air, fire, and water. Each element has associated with it a natural propensity for motion. Fire and air have levity and tend to go up (away from the center of the earth). Earth and water are heavy and tend to go down. Thus the upward motion of smoke (composed largely of the element air) and the downward motion of a cannon ball (largely earth) are natural motions requiring no further explanation. Cannon balls fall faster than cork balls because they are heavier (they have a larger percentage of the element earth in them). All objects move faster as they get closer to their natural place. Th us smoke goes faster and faster as it flees away from the earth and cannon balls go faster as they near the center of the earth.


(1) Aristotle died in 322 B.C.
(2 This section is written with apologies to historians of Medieval Science.
In addition to these natural motions, there are also so-called "violent" motions. All horizontal motions, such as the flight of an arrow, are violent. Vertical motions are also violent if they are in an unnatural direction (e.g., when we throw a ball straight up). Whereas natural motions happen spontaneously, violent motions have to be forced to occur. They always require a source of motive power, such as the hand and arm of the person throwing the ball or the "animal soul" of a wiggling worm.
The speed of violent motions increases with the strength of the motive force and decreases with resistance. For example, a sledge will go faster if it is pulled by two horses instead of one and slower if it is pulled through mud instead of on beaten ground.
One problem for the Aristotelian was to explain why projectiles, such as an arrow or ball, continued to move once they ceased to be in contact with the source of motive power. One proposal was that air was set in motion by the original action of the bowstring or arm and somehow continued to


propel the projectile. Another more ingenious solution went roughly as follows. As the projectile moved forward, there was a tendency for a vacuum to form in its wake. However, since nature abhors a vacuum, air would swarm in to fill the empty space, thus hitting the rear of the projectile and propelling it onward.
According to Aristotle, things in the celestial domain behaved quite differently. Heavenly bodies were made out of a fifth element (called the


"quintessence") and in this sphere there was no generation or corruption or change of any kind. The natural motion for bodies made of the fifth element was circular. The planets, stars, sun and moon were embedded in transparent crystalline spheres all of which were inter-nested like a graduated series of embroidery hoops. The outermost sphere (called the primum mobile) provided the dominant 24 hour circular motion shared by all bodies in the celestial system, although each planet, etc., also had its own proper motion, too.
A popular analogical model which was used for pedagogical purposes in the Middle Ages was the following: Imagine a round solid wheel rotating on its axis. Suppose that there are also circular grooves on the wheel populated by marching ants. Here the wheel corresponds to the primum mobile.
which carries the stars around every 24 hours and the ants correspond to the sun, moon and planets. An ant's total motion is compounded of two parts - the basic motion of the wheel (shared by all ants) and its own proper motion as it walks along the wheel.
III. Ptolemaic (1) Astronomy (2)
The simple concentric sphere model of the universe described above gave a rough, qualitative account of what we can observe in the sky, but it didn't get the details right. In particular, it failed to explain the retrograde motion of the planets - the fact that at certain times the planets appear to move backwards.
In order to obtain a more accurate theoretical modelling of what we actually observe in the sky, Ptolemy introduced various geometrical devices, the most famous being the epicycle. If we were to develop our ants-on- the wheel analogy, we would have to imagine the ants moving along the groove on a little Tilt-a-whirl!
The proper motion of a planet moving on an epicycle can be diagrammed as follows:

[See Kuhn figure 19, p. 61]

By a judicious adjustment of the sizes and velocities of the big circle (called the deferent) and the little circle (the epicycle) one could hope to reproduce both the velocity and duration of the retrograde motion. Note that on this model, the planet is closer to the earth when it is in retrograde motion and hence we should expect it to appear biggest and brightest
(1)Ptolemy flourished in 127-51 A.D. His book on astronomy was called the Almagest.
(2)For more details see T. S. Kuhn The Copernican Revolution.
at this time. This effect is in fact observed, and is especially dramatic in the case of Mars.
Although the epicycle was a useful geometrical device for "saving the phenomena" it was difficult to make a realistic physical model of it. (Some made the deferent into a hollow tube and had a solid epicycle rolling around in it like a marble.
Ptolemy himself sometimes treated his theory simply as a useful calculating device or instrument and did not claim that it was a true physical description.


IV.The Medieval Impetus Theory (1)

During the Middle Ages, there was much piece-meal criticism of Aristotle's natural philosophy. We will mention only a few of the revisions in his theory of motion. In order to handle the problem of projectile motion, it was suggested that as they were hurled a certain degree of motive force was impressed on them. This impressed force or impetus kept them moving until it was used up in combatting the resistance of the medium.
Impetus was analogous to heat - it takes effort to raise the temperature of a body, but once it is heated up it will stay hot until the heat dissipates into a cooler environment.
The impetus theory explained natural motion as the result of a constant tendency (or conatus) of a body to move towards its natural place. Falling bodies speed up because the conatus continues to act as it falls thus giving the body more and more impetus.
When we throw a body upward it moves more and more slowly until its remaining impetus upward just balances the conatus downward. At that moment it is stationary; then the conatus takes over and it falls slower and slower to the ground.

Footnote 1:

One important contributor was Nicole Oresme, 1323-82.
Medieval philosophers also proposed a quantitative account of the motion of falling bodies.
Let y be the velocity of a falling body and x be the time elapsed. Since triangle ABC is equal in area to the rectangle 1/2 AB.AC, we see that the distance traversed by a uniformly accelerated body is the same as that covered by a body moving at the given initial and final velocities.
The "Mean Speed Theorem", as it was called, provided a simple method for integrating under a curve and as such was a quite legitimate piece of mathematics. However, medieval philosophers had no way of knowing whether their curve described motion ofany important motions in nature such as bodies in free fall because they had not checked in detail the behavior of falling bodies.
Actually it is rather difficult to do a direct experimental test of the Mean Speed Theorem because bodies fall so rapidly. (A ball dropped from the top of a ten-story building takes only about three seconds to hit the ground.)

Galileo later measured distances and times for balls rolling down
inclined planes and this provided an indirect test of the Mean Speed Theorem.


V.Copernican (1) Theory
In De revolutionibus orbium caelestium, published just after his death in 1543, Copernicus put forward a detailed heliocentric system of the universe, Like Ptolemy's system it was constructed out of circles (Kepler introduced elliptical orbits in 1609-1619). It was superior to Ptolemy's account
in two major respects. First, it gave more accurate predictions as to exactly where the heavenly bodies would be seen at any given time. This improved accuracy was not due to any intrinsic superiority of the Copernican system, but arose simply because he had used more up-to-date observations in fixing the various orbital parameters. The second advantage of the new
(1) For a charming account of the personalities as well as the scientific achievements of the characters in this story, see Arthur Koestler's The Sleepwalkers. Koestler calls Copernicus (1473-1543) "the timid canon".


system was the fact that it was supposedly simpler. Although Copernicus used at least as many circles as Ptolemy did (hence the overall simplicity of the new system was hardly greater), his theory did have one impressive feature: It was not necessary to introduce epicycles to explain the existence of retrograde motion. The qualitative aspects of the retrograde motions of both the superior and inferior planets were a natural result of the basic geometry of the situation. Since the earth was moving around the sun with all the other planets, it was relatively easy to see that sometimes they might appear to be moving backwards - for example, when the earth passed the outer planets which were moving more slowly.
There were some other technical qualitative advantages to Copernicus' system which appealed to astronomers. However, it presented real problems for the physicists.
In his introductory chapter, Copernicus tried to suggest a modification of the Aristotelian doctrine of natural motions. But his system required that the earth have two "natural" motions. One, the yearly revolution around the sun, wasn't so bad - at least the other planets also moved this way. But the daily rotation around its axis caused all sorts of problems. None of the other heavenly bodies were observed to spin. And if the earth was whirling around like a great wheel, why didn't things fly off like mud from the rim of a spinning wheel? Why weren't there terrible winds?


Copernicus hinted at a couple of possible answers but he didn't work them out in detail.Neither did he offer any arguments for either of them:
"Perhaps the contiguous air contains an admixture of earthy or watery matter and so follows the same natural law as the Earth, or perhaps the air acquires motion from the perpetually rotating Earth by propinquity and absence of resistance[De revolutionibus, Section 8]
Even though the Copernican system desperately needed the foundations which only a new physics could provide, it might still have been taken as a serious new cosmological conjecture had it not been for its cautious preface.
The story of the publication of De revolutionibus is a very complicated one, full of unknowns and ironies. A few of the facts are these. It is almost certain that Copernicus would never have gotten around to publishing
anything had not Rheticus, a young enthusiastic Lutheran astronomer and mathematician, heard about his heliocentric ideas and literally seduced Copernicus into writing them up.
Rheticus took the finished manuscript from Copernicus' house in Frauenburg up on the Baltic Sea down to Nuremberg and was intending to see it through publication but had to leave town unexpectedly. (It seems that he got into trouble because of his liking for what the Germans call "the Italian perversion".)
In any case, another Lutheran, this one a theologian called Osiander, took over responsibility for the printing. Although Osiander was sympathetic to the Copernican system, he knew that Luther opposed it and so he added a preface to the reader in which he proposed that the heliocentric system not be construed as a realistic description of the universe but merely as a useful device for making astronomical calculations:
"For these hypotheses need not be true or even probable ...as far as hypotheses are concerned,
no one expect anything certain from astronomy, which cannot furnish it, lest
he accept as the truth ideas conceived for another purpose [i.e., as mere calculating aids], and depart from this study a greater fool than when he entered it. Farewell."
Osiander's Preface accomplished what he intended it to. Copernicus' system became popular as a basis for making calendars and star charts. But for a long time it had little impact on pure science.


VI.Galileo's Telescopic Observations (1)
In 1609 Galileo heard about the newly invented telescope and designed one which was good enough for astronomical observations. By March, 1910, he had already made a series of discoveries which refuted or at least seriously undermined several features of Aristotle's cosmology.
First of all, he "observed" (we will return later to the question of the reliability of Galileo's interpretations of what he saw) that the moon


Footnote 1. For Galileo's own account (including diagrams) , see his 1610 "Siderius Nuncius," translated in S. Drake, Discoveries and Opinions of Galileo.
had mountains. This was inconsistent with Aristotle's claim that the heavenly bodies were perfect and suggested that some of them might be made of stuff similar to the earth.
Secondly, he "observed" (again there are some problems about interpretation) that Jupiter had four moons (he called them "Medicean stars" in order to gain points with the Venetian Duke). This discovery argued against the claim that Jupiter was carried along by an invisible crystalline sphere. (Tycho Brahe had reached a similar conclusion in 1577 when he observed a comet move freely through several places where spheres were supposed to be.)
The moons revolving around Jupiter also showed conclusively that there was more than one center of motion in the universe. This was important because Copernicus had the moon moving around the earth as the earth in turn moved around the sun. The Jupiter-four moons system showed that such a motion was possible. It did not of course prove that the earth-moon system actually worked in a similar manner.
Galileo also determined the composition of the Milky Way. This discovery had no direct relevance to the debate over the Copernican system. However, it did indicate that Aristotle didn't get everything right and also that the Universe was bigger than had been previously suspected.
In 1543 Copernicus had pointed out two important areas in which his system and Ptolemy's made different predictions. One concerned the phases of Venus. On Copernicus' account, if Venus shone by reflected light, it should appear to wax and wane. According to the Ptolemaic system it should always appear crescent shaped. Since Venus always appears round, some Ptolemains concluded that it must generate its own light as do the stars and the sun.
In 1610 (but not in time to be reported in The Starry Messenger Galileo observed that Venus did indeed have phases, the timing and apparent magnitudes of which were just as predicted by the Copernican system.
This discovery provided a decisive refutation of the Ptolemaic system. Unfortunately, the other major new prediction of the Copernican system, stellar parallax, told against it and for a geocentric system. If the earth is in motion, a line between an observer on earth and a fixed star does not quite stay parallel the year around. Therefore, each star should seem to shift its position slightly with respect to the pole of the stellar sphere.
However, stellar parallax could not be detected - even with the new telescope. (It was eventually observed in 1838.) Defenders of Copernicus could explain this away by postulating that the stars were much farther away than had been thought, but this seemed like a rather ad-hoc move since there was no reason to believe it except that it would save the Copernican theory from refutation.
The observations of the sunspots around 1612 by Galileo and others showed that Aristotle was wrong in claiming that the heavenly bodies were immutable. It was not clear exactly what or where the sunspots were, but they surely came and went in a most imperfect fashion!


VII.Galileo's Dialogo
In 1632 Galileo published his Dialogo sopra i due Massimi Sistemi del Mondo; Tolemaico, e Copernico. His strategy had two parts. First, he wished to show that it was possible that the earth moved. To do so he had to answer all the physical arguments against Copernicus, e.g., that birds would get left behind, etc. essentially what was required was a new physics of inertial motion.
Secondly, he wanted to show that the earth actually moved. His major argument here was his theory of the tides (which, as we will see, many historians of science find embarrassingly mistaken).
Before looking at these arguments in any detail, the significance of the title must be pointed out. Galileo speaks of two world systems, but in so doing he omits a third possibility, the very one which was most popular in the early 17th century. Tycho Brahe, a very good Danish astronomer, who invented many new instruments and had by far the most accurate astronomical data available at that time, had proposed a third alternative which seemed to many to be the ideal compromise. It was geocentric - so there were no problems about birds getting blown away and furthermore it explained the absence of stellar parallax.


But in Tycho's system all the planets revolved around the
sun - so, unlike the Ptolemaic system, it made the right predictions about the phases of Venus.
Galileo, unlike his contemporaries, never took Tycho's system seriously. For one thing, he considered it to be very inelegant - it seems rather clumsy to have all the planets carried around the earth by the sun. But more importantly, he recognized that if his theory of the tides was correct, it would refute all geostatic systems, Ptolemaic, Tychonic, or what have you. It all hinged on his theory of the tides.
Galileo's Dialogo is a masterpiece of both polemics and popular scientific writing. There are three protagonists: Simplicio is a very likable but fundamentally stupid Aristotelian. Salviati is the slick expert who often refers in reverential tones to a learned Academician (obviously Galileo). The moderator is Sagredo, a kind of Dick Cavett character personable, alert and determined to keep both sides honest. (Unfortunately, Sagredo does not know about the Tychonic system.) Almost all of the discussion is non-technical. Galileo's quantitative theory of motion came later in the Discorsi.
From the beginning Galileo attacks a naive reliance on observation and common sense reasoning. He points out that as we walk along the street at night the moon appears to run along behind us like a cat on the rooftops. Likewise as a ship floats along a canal, the shore sometimes appears to be moving instead. A tower in the distance appears to be a continuous translucent streak.
But all of these appearances are deceiving. The observations suitable for science have to be based on correct theories and good instruments. For example, observations of size (such as in the case of the tower) have to be corrected by the laws of perspective. Many observations with the naked-eye can be improved by using the telescope. And observations of relative motion alone can never tell us which object is actually at rest.
Galileo also extols the use of what are sometimes misleadingly called "thought experiments." For example, in criticizing the Aristotelian claim that heavier bodies fall faster, he not only reports on experiments done by dropping balls from towers, but also argues as follows: Suppose Aristotle
were right. Now imagine two identical cannon balls with strings attached falling side by side. Now suppose the strings become knotted. We now have a composite body which weighs twice as much as the separate parts. It follows on Aristotle's account that they should immediately start falling faster. But that is absurd. Therefore, Aristotle is wrong.
(Because Galileo criticized naive observation and relied on thought experiments, some historical commentators have concluded that he was not an empiricist. However, this may only show that he was a sophisticated empiricist. It hinges in part on what is meant by "absurd" in the above argument. Do we conclude that the cannon balls would not speed up when tied together because of some a priori metaphysical principle such as "no effect without a cause"? Or is it because we have lots of experience which indicates that a change in velocity requires some force to be applied?)
Galileo argues that the birds would not get left behind if the earth were moving in a variety of ways. He points out that flies in the cabin of a ship share the ship's motion and do not have to fly all the way from Venice to Constantinople. Likewise, if a ball is dropped from the mast of a moving ship it lands at the foot of the mast, not behind it.
In the fourth and final section of the book Galileo switches from merely arguing that it is possible that the Copernican system is true and tries to prove that it is true. Here the claims that the ebbing and flowing of the tides is caused by a combination of the daily rotation and yearly revolution of the earth. Roughly, the theory goes like this: Consider a given point on the earth's surface. During the night the two motions add up so that water accelerates. During the day the daily and yearly motions partly cancel out so the water slows down.
This theory, which Santillana calls "Galileo's folly" and Koestler labels as an idée fixe, is unsatisfactory for two reasons. First of all it violates Galileo's own ideas about motion. Relative to the earth, the water does not speed up or get left behind. It travels along with the earth just as the air does. Galileo's theory of the tides is inconsistent with his own physics.
Secondly, it predicts that there should be a high tide once a day. However, tides are generally observed to occur about every twelve hours. Galileo explained this discrepancy away by vague talk about the major tide bouncing back and forth in the sea bed. It was not a good concluding section for an otherwise brilliant book.