Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution
Viktor Blåsjö

I offer a revisionist interpretation of Galileo's role in the history of science. My overarching thesis is that Galileo lacked technical ability in mathematics, and that this can be seen as directly explaining numerous aspects of his life's work. I suggest that it is precisely because he was bad at mathematics that Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His reliance on these hands-on modes of research was not a pioneering contribution to scientific method, but a last resort of a mind ill equipped to make a contribution on mathematical grounds. Likewise, it is precisely because he was bad at mathematics that Galileo expounded at length about basic principles of scientific method. “Those who can't do, teach.” The vision of science articulated by Galileo was less original than is commonly assumed. It had long been taken for granted by mathematicians, who, however, did not stop to pontificate about such things in philosophical prose because they were too busy doing advanced scientific work.

§ INTRODUCTION

Galileo is overrated, I maintain. I shall systematically go through all his major achievements and all standard arguments as to his alleged greatness, and offer a critical counter-assessment. I go after Galileo with everything but the kitchen sink, but I hasten to add that assembling the case against him is ultimately a means to an end. The stakes are much higher than Galileo's name alone. Galileo is at the heart of fundamental questions: What is the relation between science, mathematics, and philosophy? Between ancient and modern thought? What is the history of our scientific worldview, of scientific method? Galileo is right in the thick of the action on all of these issues. Consequently, I shall use my analysis of Galileo as a fulcrum to articulate a revisionist interpretation of the history of early modern science more broadly. But let's start small, with a simple snapshot of Galileo at work.

§ MATHEMATICS

§.§ Cycloid

The cycloid is the curve traced by a point on a rolling circle, like a piece of chalk attached to a bicycle wheel (Figure <ref>). Many mathematicians were interested in the cycloid in the early 17th century, including Galileo. What is the area under one arch of the cycloid? That was a natural question in Galileo's time. Finding areas of shapes like that is what geometers had been doing for thousands of years. Archimedes for instance found the area of any section of a parabola, and the area of a spiral, and so on. Galileo wanted nothing more than to join their ranks. The cycloid was a suitable showcase. It was a natural next step following upon the Greek corpus, and hence a chance to prove oneself a “new Archimedes.”

There was only one problem: Galileo wasn't any good at mathematics. Try as he might, he could not for the life of him come up with one of those clever geometrical arguments for which the Greek mathematicians were universally admired. All those brilliant feats of ingenuity that Archimedes and his friends had blessed us with, it just wasn't happening for Galileo.

Perhaps out of frustration, Galileo turned to the failed mathematician's last resort since time immemorial: trial and error. Unable to crack the cycloid with his intellect, he attacked it with his hands. He cut the shape out of thick paper and got his scales out to have this instrument do his thinking for him. As best as he could gather from these measurements, Galileo believed the area under the cycloid was somewhere near, but not exactly, three times the area of the generating circle.[<cit.>.]

The cycloid.

This was no way to audition for the pantheon of geometers. Galileo was left red-faced when mathematically competent contemporaries solved the problem with aplomb while he was fumbling with his cutouts. These actual mathematicians proved that the cycloid area was in fact exactly three times the area of the generating circle,[Roberval in 1634. <cit.>, [Whitman , 1943], <cit.>.] even though Galileo had explicitly concluded the contrary on the basis of his little cardboard diorama.

When Galileo heard of others working on the cycloid challenge, he sought help on this “very difficult” problem from his countryman Bonaventura Cavalieri, a competent mathematician. “I worked on it fruitlessly,” lamented Galileo. “It needs the mind of a Cavalieri and no other,”[Galileo to Cavalieri, 24 February 1640, <cit.>. Cavalieri did not take up the problem—“I too left it aside” <cit.>—but Torricelli solved it soon thereafter.] he pleads, tacitly acknowledging his own unmistakably inferior mathematical abilities. It is interesting to contrast this with the very different reaction to the same problem by Galileo's contemporary René Descartes, the famous philosopher who was also a vastly better mathematician than Galileo. When Descartes heard of the problem he immediately wrote back to his correspondent that “I do not see why you attribute such importance to something so simple, that anyone who knows even a little geometry could not fail to observe, were he simply to look.”[Descartes to Mersenne, 27 May 1638, AT.II.135, <cit.>.] He then immediately goes on to give his own proof of the result composed on the spot. Descartes is not famous for his humility, but the fact of the matter is that a number of mathematicians solved the cycloid problem with relative ease, while Galileo was fumbling about with scissors and glue.

In the case of the cycloid, it is an unequivocal fact that Galileo used an experimental approach because he lacked the ability to tackle the problem as a mathematician. If Galileo could have used a more mathematical approach he would unquestionably have done so. It is my contention that what is so glaringly obvious in this case holds for Galileo's science generally. Galileo's celebrated use of experiments in science is not a brilliant methodological innovation but a reluctant recourse necessitated by his pitiful lack of mathematical ability.

The cycloid case also makes it clear why the mathematically able prefer geometrical proofs to experiments: the latter are hopelessly unreliable. By relying on experiments unchecked by proper mathematics, Galileo got the answer wrong, and not for the first time nor the last. “Do not think that I am relying on experiments, because I know they are deceitful,”[<cit.>. Oeuvres.XI.115.] said Huygens, and all other mathematicians with him. It had always been obvious that mathematics and science can be explored using experiment and observation. As Galileo says: “You may be sure that Pythagoras, long before he discovered the proof …, had satisfied himself that the square on the side opposite the right angle in a right triangle was equal to the squares on the other two sides”[Galileo, , OGG.VII.75, <cit.>.]—presumably by making numerical measurements on various concretely drawn triangles. But able mathematicians had always known that haphazard trial and error had to be superseded by rigorous demonstration for a treatise to be worth the parchment it is written on. This and only this is why you don't see experimental and numerical data defiling the pages of masterpieces of ancient mathematics and science such as those of Archimedes.[<ref>.]

In this case as in so many others, Galileo's inglorious contribution to the history of thought is to cut off mathematical reasoning at the training-wheels stage, and to air in public what true mathematicians considered unworthy scratch work. He experiments because his intellect comes up short. He cannot reach insights by reason, so he turns to more simplistic, hands-on methods. In physics this ignominious shortcoming has been mistaken for methodological innovation. But in the case of the cycloid its true colours are unmistakable. We see that it is a sign of failure rather than genius. Galileo's empiricism is not science being born; it is science being dumbed down.

§.§ Mathematicians versus philosophers

There are mathematicians and then there are other people. Mathematicians have always felt strongly about this. Profound respect for other mathematicians and borderline contempt for anyone else is a persistent theme in their writings. “I have no doubt that talented and learned mathematicians will agree with me,”[<cit.>.] says Copernicus with complete confidence when introducing his new sun-centered astronomical system in 1543, decades before Galileo was born. Copernicus promises to make everything “clearer than day—at least for those who are not ignorant of the art of mathematics.”[<cit.>.] As for non-mathematicians, who cares what those fools think? “If perchance there are certain idle talkers …, although wholly ignorant of mathematics, … [who] dare to reprehend and to attack my work; they worry me so little that I shall … scorn their judgements.”[<cit.>.] These are the kinds of people who “on account of their natural stupidity hold the position among philosophers that drones hold among bees.”[<cit.>.] “The studious need not be surprised if people like that laugh at us. Mathematics is written for mathematicians.”[<cit.>.]

Johannes Kepler—the best mathematical astronomer of Galileo's age—says the same thing. “Let all the skilled mathematicians of Europe come forward,”[<cit.>.] he implores at one point, confident that mathematical reason compels them to speak with one voice. Those who are not “skilled mathematicians,” on the other hand, might as well stay where they are, for they have no credibility as witnesses in scientific matters. Kepler says as much in a remark he labels “advice for idiots”: “But whoever is too stupid to understand astronomical science, … I advise him that, having dismissed astronomical studies and having damned whatever philosophical opinions he pleases, he mind his own business and betake himself home to scratch in his own dirt patch.”[Kepler, Astronomia Nova (1609), 6r, <cit.>.]

This is and always has been the worldview of mathematicians. My thesis in this essay consists in little more than taking them at their word. What does the history of science look like if we adopt the worldview of the mathematicians as a historiographic perspective? If we “let all the skilled mathematicians come forward” and leave the philosophers in their “dirt patch”? Many things do indeed become “clearer than day—at least for those who are not ignorant of the art of mathematics.”

If we adopt this perspective, virtually Galileo's entire claim to fame evaporates before our eyes. For one thing, mathematicians certainly did not need Galileo to tell them that Aristotle was wrong. Decades before Galileo's first book, Aristotle was already ridiculed by mathematicians. As a contemporary source notes, “these modern mathematicians solemnly declare that Aristotle's divine mind failed to understand [mathematics], and that as a result he made ridiculous mistakes.”[Colombe, c. 1611, <cit.>. OGG.III.253–254.] “And they are right in saying so,” noted Galileo in the margin next to this passage, “for he committed many and serious mathematical blunders.”[Galileo, <cit.>. OGG.III.253–254.]

Hence Descartes's judgement of Galileo's science: “He is eloquent to refute Aristotle, but that is not hard.”[Descartes to Mersenne, 11 October 1638, <cit.>.] This is essentially the thesis of my essay in so many words. It was also, I claim, the standard opinion of Galileo among mathematicians.

Galileo's supposed greatness can only be maintained by ignoring this unequivocal sentiment in the mathematical tradition. Galileo’s claim to fame rests on the assumption that everyone but him was a raving Aristotelian. Galileo himself went out of his way to ensure this framing. His two big books are both dialogues in which Galileo spends hundreds and hundreds of pages in back-and-forth squabbles with a fictive Aristotelian opponent. This is the contrast class Galileo wants us to use when evaluating his achievements. And no wonder. Refuting Aristotle is not hard, so of course Galileo can score some zingers against this feeble opposition. But that’s fish in a barrel, not a scientific revolution.

No other mathematician, ancient or modern, ever engaged with Aristotelian science to anywhere near the same extent as Galileo, let alone devoted the entirety of their main works to refuting it. As Kepler put it: “The foolish studies of humans have come to such a pitch of vanity that no one's work becomes famous unless he … either fortifies himself with the authority of Aristotle, or takes a stand in the battle against him, seeking to show off.”[Kepler, Astronomiae Pars Optica (1604), 29, <cit.>.] To the mathematicians this was obviously “foolish vanity” either way, because they considered it self-evident that Aristotelians were so ridiculous that one should simply dismiss them as so many “idiots” in one sentence of the introduction and leave it at that, just as Copernicus and Kepler do in their major books. Descartes did the same, calling followers of Aristotle “less knowledgable than if they had abstained from study.”[<cit.>.] Tycho Brahe—a prominent astronomer in the generation between Copernicus and Galileo—likewise lamented “the oppressive authority of Aristotle”: “Aristotle's individual words are worshipped as though they were those of the Delphic Oracle.”[Tycho Brahe, 1589, <cit.>.] Kepler once castigated a colleague with the ultimate insult: “your theory will attract lecturers and philosophers.” Lest anyone mistake this for a compliment, Kepler explained what he meant: “it will offer a way out to the enemies of the physics of the sky, the patrons of ignorance.”[Kepler to David Fabricius, 10 November 1608, <cit.>.]

Such was always the universal opinion of philosophers among mathematicians. Mathematically competent people were united and had nothing but complete contempt for Aristotelian philosophy and the like. By the time Galileo comes along and belabours this point it has been old news for hundreds of years. As Galileo himself says: “the philosophers of our times … philosophize … as men of no intellect and little better than absolute fools.”[Galileo, <cit.>.] Precisely. Which is why there is little value in writing several thick books hammering home this point and little else, which is what Galileo did. That’s just beating a dead horse as far as the mathematicians are concerned.

The standard view among modern scholars is very different: “Galileo's significance for the formation of modern science lies partly in his discoveries and opinions in physics and astronomy, but much more in his refusal to allow science to be guided any longer by philosophy.”[<cit.>.] But in this refusal Galileo is just following the mathematicians, who had always maintained that “only mathematics can provide sure and unshakeable knowledge” and that other “divisions of theoretical philosophy should rather be called guesswork than knowledge.”[Ptolemy, Almagest, I.1, <cit.>.] The only difference is that, while competent mathematicians took this for granted and got on with real science, Galileo stopped to dwell on the matter for hundreds of pages.

“It is evident that Galileo admired Archimedes as his mentor,” while “no philosopher had ever taken the slightest interest in Archimedes.”[<cit.>.] This was the one thing that set Galileo apart from the philosophical hordes of his day. “What was novel in Galileo's approach, when contrasted with that of his Aristotelian opponents, was … his faith in the relevance of mathematics.”[<cit.>.] It is true that Galileo got this much right. He had a crush on the right person. Insofar as Galileo ever accomplished anything, it stems from his infatuation with Archimedes. His only regret was that his own mathematical abilities were so vanishingly small compared to those of his hero, as he frankly admitted: “Those who read his works realise only too clearly how inferior are all other minds compared with Archimedes', and what small hope is left of ever discovering things similar to those he discovered.”[Galileo, <cit.>. OGG.I.215–216.]

This is Galileo's unique position in the history of science. He is the worst of the mathematicians and the best of the philosophers. He is too incompetent to do mathematics but competent enough to admire it. Relative to the mathematicians he is a dud who has to spend his time spelling out obvious philosophical points because he lacks the ability to do real mathematics. Relative to the philosophers he is a revelation and a watershed, articulating for the first time in philosophical prose many central tenets of the modern scientific worldview.

Evaluating Galileo thus comes down to understanding the historical relation between mathematics and philosophy. If you believe that mathematicians from Greek times onwards considered it obvious that mathematics encompassed an expansive, daringly creative, empirically informed, interconnected study of all quantifiable aspects of the world, and that philosophers talking about mathematics and science are basically just derivative commentators, then Galileo is nobody. If you believe that mathematics before Galileo was a technocratic enterprise that was intrinsically compartmentalised and limited to traditionally defined specific tasks, and that the empirical-mathematical method of modern science was a great “conceptual” revelation coming from philosophy, then you may indeed be inclined to view Galileo as the “father of modern science.” It is my contention that the former is the right way to understand the history of scientific thought.

[Galileo kneeling before muses.]Galileo kneeling before personifications of Astronomy, Perspective, and Mathematics (far side, with compasses, hiding her face in shadows and averting her eyes, perhaps symbolising Galileo's uneasy relationship with mathematics). Frontispiece for Opere di Galileo Galilei, Bologna, 1656.

§.§ Professor

Galileo was a “Professor of Mathematics” for some twenty years. But we must not let the title fool us. The position had nothing to do with the research frontier in the field. In modern terms it is more comparable to that of high school teacher. Galileo taught basic and practical courses,[<cit.>, <cit.>.] and his lectures were unoriginal and usually cribbed from standard sources.[<cit.>.] His mathematical lectures went no further than elementary Euclidean geometry. He also had to teach a basic astronomy course “mainly for medical students, who had to be able to cast horoscopes.”[<cit.>.] They “needed it to determine when not to bleed a patient” and the like.[<cit.>.] Perhaps Galileo didn't mind, for he seems to have been quite open to astrology judging by the fact that he cast horoscopes for his own family members and friends without renumeration.[[Rutkin , 2018], <cit.>, <cit.>, [Kollerstrom , 2001].] Alas, he did not enjoy much success as an astrologer: “In 1609, Galileo … cast a horoscope for the Grand Duke Ferdinand I, foretelling a long and happy life. The Duke died a few months later.”[<cit.>.]

As soon as Galileo had made a name for himself he eagerly resigned his lowly university post in favour of a court appointment. This freed him from teaching duties and boosted his finances, but Galileo also had an additional demand:
I desire that in addition to the title of “mathematician” His Highness will annex that of “philosopher”; for I may claim to have studied more years in philosophy than months in pure mathematics.[Galileo, negotiating his Tuscan court appointment in 1610, <cit.>.]
This is traditionally taken as a request for a kind of promotion: in addition to being a great mathematician, Galileo also wanted recognition in philosophy, which in some circles was considered more prestigious and in any case included what today is called science (then “natural philosophy”). But I think a more literal reading is in order. Galileo is not only declaring himself a philosopher; he is also confessing his ignorance in mathematics. Taken literally, his statement implies that he could not have spent more than two or three years on pure mathematics—which indeed sounds about right considering his documented mediocrity in this field.

§.§ Sector

[Galileo's sector.]Galileo's “geometric and military compass”: a slide-rule-style calculation device.

For his teaching, Galileo developed an instrument—his “geometric and military compass” or sector (Figure <ref>)—that could be used to facilitate certain computations, somewhat like a slide rule. Is this a great testament to his mathematical genius? Should Galileo be credited with inventing “the first mechanical computing device,”[<cit.>.] as one sensationalistic headline has it? Obviously not, since far more sophisticated mechanical computing devices were developed in antiquity (such as the astrolabe and the Antikythera mechanism, although they were devised for different problems). Nor is it clear to what extent, if any, Galileo's instrument was original in its time. “It has always been supposed that Galileo took over a calculating device already in service,”[<cit.>.] and various at least partial precursors are documented,[[Drake , 1976], [Galileo , 1978], <cit.>.] but it is hard to say anything definitive since very few instruments from this era have survived. One recent study declines to pronounce on Galileo's degree of originality, noting only that, at any rate, “students did not necessarily buy the compass because it was better or more original than other similar devices one could buy in Germany, France, or England. … They trusted Galileo the way the residents of a certain neighborhood trust the local baker.”[<cit.>.] In any case it is undisputed that a “very similar instrument,” developed in England “entirely independently of Galileo's work,” was extensively described in print eight years before Galileo published anything about his device.[Drake, <cit.>. Thomas Hood, The Making and Use of the Geometricall Instrument, Called a Sector, London, 1598.]

Galileo is upfront about the target audience of his sector: it is “suited to … first youthful studies”[<cit.>.] and an aid “for those who cannot easily manage numbers.”[<cit.>.] The fact that the latter group arguably includes Galileo himself is perhaps not coincidental. Ancient Greek folklore has it that some king found geometrical study too onerous and demanded to be brought to its fruits by an easier route, only to be reprimanded that “there is no royal road to geometry.” Many a mathematician has shared this beautifully egalitarian sentiment. But not Galileo, who leapt at any opportunity to kiss the feet of princes. “Not entirely improper was the request of that royal pupil who sought … an easier and more open road,”[<cit.>.] Galileo declares in his instrument manual. While Galileo encouraged this shortcut mindset, more committed mathematicians took a principled stance to the contrary: “The true way of Art is not by Instruments, but by demonstration: … it is a preposterous course of vulgar Teachers, to beginne with Instruments, and not with the Sciences, and so in stead of Artists, to make their Schollers onely doers of tricks, and as it were jugglers.”[William Oughtred, To the English gentrie, and all others studious of the mathematicks (1634), 27–28.] The divide remains as current as ever. Anyone familiar with modern mathematics classrooms have observed this unmistakeable relationship: the weaker the student (or teacher), the greater their reliance on calculators. From this point of view, Galileo's instrument can be seen as one more indication of that aversion to serious mathematical work that characterises his career as a whole.

§.§ Book of nature

Here is one of Galileo's most quoted passages:
This grand book, the universe, … stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it.[Galileo, Assayer (1623), <cit.>.]
Like so much else Galileo gets credit for, this idea had obviously been a truism among mathematicians for thousands of years. Pythagoras had put it succinctly more than two millennia before: “All is number.”[<cit.>.] Plato agreed, saying “God is always doing geometry.”[<cit.>, slightly paraphrased.] Mathematicians have lived by these words ever since.[<ref>.] It was obvious even to for instance Grosseteste, a medieval bishop, that “the usefulness of considering lines, angles, and figures is very great, since it is impossible to understand natural philosophy without them.”[Grosseteste, first half of 13th century, <cit.>, <cit.>.]

Galileo's version of this platitude is better read as a frank confession of his reactionary restriction to simplistic mathematics. “Triangles and circles” are hardly the cutting edge. Ellipses, for instance, are passed over in silence by Galileo, not just in this quote but throughout all his works, even though Kepler had already demonstrated many years before that planets move in ellipses about the sun.[Kepler, Astronomia Nova (1609).]
Unfortunately, for reasons that remain unclear or controversial, Galileo never did pay the proper attention to Kepler's writings and so ignored or neglected the elliptical nature of planetary orbits.[<cit.>]
There is nothing “unclear” about it. The reason Galileo ignored Kepler's writings is abundantly obvious: He didn't have anywhere near the mathematical ability to understand them. Kepler had already anticipated as much:
It is extremely hard these days to write mathematical books. … There are very few suitably prepared readers. … How many mathematicians are there who put up with the trouble of working through the Conics of Apollonius of Perga?[Kepler, Astronomia Nova (1609), 2r, <cit.>.]
Not many indeed, and certainly not Galileo. Yet Kepler saw no other option than to write mathematics all the same and “let it await its reader for a hundred years.”[<cit.>.]

Compared to Kepler's visionary science, Galileo's book of nature is a children's book sticking to baby mathematics like “triangles and circles.” The traditional way to read Galileo's phrase that without these it is “humanly impossible to understand” nature is to see it as stipulating a minimum knowledge requirement: if you don't know math, you're screwed. But it can also be read as a confession of an upper limit on Galileo's own understanding: anything beyond “triangles and circles” and I'm lost. This would also explain Galileo's equally reactionary rejection of “infinities and indivisibles, the former incomprehensible to our understanding by reason of their largeness, and the latter by their smallness.”[<cit.>, OGG.VIII.73.] Virtually all competent mathematicians in Galileo's time were eagerly exploring infinitesimal methods, prefiguring what we call calculus today. But Galileo found all of this “incomprehensible.” When he did try to dabble with such issues his arguments were “rejected immediately by Descartes as sophistic, and then on similar grounds by Bonaventura Cavalieri, Galileo's own disciple, perhaps the greatest expert on indivisibles in the early seventeenth century.”[<cit.>. See also <cit.>.] Such shortcomings undermine core parts of his physics, as we shall see in the next section.

§.§ Calculus of speeds

Galileo's “technique for dealing with problems involving infinitesimals … is incompatible with basic features of integration … and even leads, if explicated systematically, to absurd consequences.”[<cit.>.] The core issue is illustrated in Figure <ref>. Galileo tries to prove areas equal by “matching” the lines that make them up one by one:
The aggregate of all parallels contained in the quadrilateral [ABFG is] equal to the aggregate of those included in triangle AEB. For those in triangle IEF are matched by those contained in triangle GIA, while those which are in the trapezium AIFB are common. Since each instant and all instants of time AB correspond to each point and all points of time AB, from which points the parallels drawn and included within triangle AEB represent increasing degrees of the increased speed, while the parallels contained within the parallelogram represent in the same way just as many degrees of speed not increased but equable, it appears that there are just as many momenta of speed consumed in the accelerated motion according to the increasing parallels of triangle AEB, as in the equable motion according to the parallels of the parallelogram GB. For the deficit of momenta in the first half of the accelerated motion (the momenta represented by the parallels in triangle AGI falling short) is made up by the momenta represented by the parallels of triangle IEF.[<cit.>, OGG.VIII.208–209.]
The problem, as shown in Figure <ref>, is that this reasoning invites false results because “parallels” can be “matched” also for shapes that do not have the same area. “Neither Galileo's manuscripts nor his published texts indicate that he noticed this clearly nonsensical implication of the proof technique.”[<cit.>.] Many of his contemporaries were more alert, including Cavalieri, who “criticizes
Galileo for not having emphasized that the indivisibles have to be taken as equidistant.”[<cit.>. Bonaventura Cavalieri to Galileo, June 28, 1639. OGG.XVIII.67.] Cavalieri's reaction is the most charitable option: Galileo neglected to “emphasise” (or even mention, in fact) some implicit assumptions that rule out the false applications of his method of proof.

Galileo's dubious way of comparing two motions.

A better way of thinking about these kinds of problems is to view areas as composed of thin rectangles rather than lines. The correct conclusion of Figure <ref> works because the two areas can be sliced into thin rectangles that correspond to an equal rectangle in the other figure. The false conclusion of Figure <ref>, on the other hand, doesn't go through this way, because while each rectangle “fits” in a natural place in the other area (just as the lines do, as illustrated), the rectangles positioned this way do not cover the full figure, but rather leave gaps between them. Galileo's proof ought to have included some clarifications or precautions to specify in what way his method avoids false conclusions on these kinds of grounds.

If we look only at the examples in Figure <ref>, it may seem that we are interpreting Galileo quite maliciously. It is quite evident that the two triangles are equal, while the areas under the curved velocity diagram are not. Surely, we may be tempted to say, it is clear what Galileo must have meant, even if he may not have expressed it with great precision.

But it gets worse. Galileo applies his “matching” technique again later, in a way that very much complicates the picture. He applies this method of proof when comparing the motion of a falling body to that of a ball rolling down an inclined plane (Figure <ref>). Galileo wants to show that time it takes the ball to complete the motion is proportional to the distance it has to travel: for example, if $AC=2AB$, then the rolling ball needs twice as long as the falling ball to reach the ground. Galileo tries to derive this result from the fact that the two balls have the same velocity when they are the same vertical distance from the starting point. This fact establishes a one-to-one correspondence or “matching” between the speeds of the falling and the rolling body. Galileo claims that the result follows from this:
If … parallels from all points of the line AB are supposed drawn as far as line AC, the momenta or degrees of speed at both ends of each parallel are always matched with each other. Thus the two spaces AC and AB are traversed at the same degrees of speed. But … if two spaces are traversed by a moveable which is carried at the same degrees of speed, then whatever ratio those spaces have, the times of motion have the same [ratio]. Therefore the time of motion through AC is to the time through AB as the length of plane AC is to the length of vertical AB, which was to be demonstrated.[<cit.>, OGG.VIII.216–217.]
But again the rules of this method is unclear. From Galileo's words, there seems to be nothing stopping us from applying the same proof, word for word, to the situation of a non-straight path, as in Figure <ref>. But this clearly produces false results, for in such cases the time it takes to complete the motion is no longer proportional to the distance travelled. So Galileo's proof technique can be used to prove false results again.

A falling ball compared to a ball rolling down an inclined plane.

Variant of Figure <ref>. The ramp along which the ball is rolling is no longer straight.

The example of Figure <ref> makes it much harder to argue that Galileo's method can be fixed by adding some common-sense implicit assumptions or restrictions. Galileo clearly thought his method could be applied even in cases where the time-velocity diagrams have different shape and different spacing of the matched lines, as in Figure <ref>. In terms of the area interpretation mentioned above, this reasoning is dependent on matching rectangles of one area to thinner rectangles of another area, but for Galileo's reasoning to work the ratio of the thicknesses of these rectangles must remain the same throughout. This is what sets the permissible conclusion of Figure <ref> apart from the impermissible conclusion of Figure <ref>. But note that this is a global property: it has to do with how all the slices relate to one another—the ratio must be the same for all pairs of rectangles. This seems to be directly incompatible with Galileo's text. Galileo quite clearly highlights only the local, pairwise matching of velocities and draws his conclusion from there. There is no indication in Galileo's text that he considers a constant-ratio requirement essential to the legitimate application of his proof technique.

Hence we see why scholars have argued that “closer inspection of [Galileo's] proof [of Figure <ref>] in the light of [his use of this technique in the case of Figure <ref>] indeed makes it clear that this proof also works equally well for nonuniformly accelerated motion,” and therefore leads to false results.[<cit.>.]

§ PHYSICS

§.§ Fall and weight

Galileo often portrays himself as defeating obstinate philosophers who would rather cling to the words of Aristotle than believe empirical evidence made clear through  experimental demonstration. Thus he imagines his enemies to say things like: “You have made me see this matter so plainly and palpably that if Aristotle's text were not contrary to it … I should be forced to admit it to be true.”[<cit.>.] It makes life easy for Galileo to pretend that his enemies are fools blind to facts. In reality, Galileo's “anti-Aristotelian polemics … were directed only at straw men.” Galileo concocted these caricature Aristotelians in order to “have fun with them, let them play the buffoon in his dialogues, and thus enhance his own image in the eyes of his readers.”[<cit.>.]
Galileo's ploy was well calculated. It tricks many of his readers to this day into believing the fairytale of Galileo the valiant knight singlehandedly fighting for truth in world beset by dogmatism. But “excessive claims for Galileo the dragon slayer have to be muted,”[<cit.>.] to say the least.

This is clearly seen in the case of a famous question: do heavy objects fall faster than lighter ones? Aristotle had answered: yes. Twice as heavy means twice as fast, according to Aristotle.[E.g. De Caelo, I.6: “A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement.”] Legend has it that Galileo shocked the world when he dropped some balls of differing weights from the tower of Pisa and revealed them to fall at the same speed. But the notion that it required some kind of radical conceptual innovation by a scientific genius like Galileo to realise that one could test the matter by experiment is ludicrous. Of course one can drop some rocks and see if it works: this much has been obvious to anyone since time immemorial. Indeed, Philoponus—an unoriginal commentator—had clearly and explicitly rejected Aristotle's law of fall by precisely such an experiment more than a thousand years before Galileo.[<cit.>.] So if you want to believe that experimental science and empirical verification was a radical new insight then this puts you in quite a pickle. If that was a revolution, then why is it found for the first time in this mediocre commentator from the 6th century? If people like Philoponus had the key to science, then why did they sit around and write commentary upon commentary on Aristotle? Their contributions to mathematics and science is otherwise zero. How likely is it that such elementary scientific principles eluded generations of the best mathematical minds the world has ever seen, in the age of Archimedes, only to be then discovered by derivative and subservient thinkers in an age where the pinnacle of mathematical expertise extended little further than the ability to multiply three-digit numbers?

Long before Philoponus, for that matter, Lucretius had clearly stated that, in the absence of air resistance, all objects fall at the same speed regardless of weight,[Lucretius, De rerum natura, II:225–239.] a result many still believe Galileo was the first to discover.[E.g. <cit.>.] As ever, Galileo gets credit for elementary ideas that are thousands of years old.

Nor was Galileo original in his own age. Have you ever heard anyone calling Benedetto Varchi “the father of modern science”? Yet here is his statement of “Galileo's” great insight, expressed two decades before Galileo was even born:
The custom of modern philosophers is always to believe and never to test that which they find written in good authors, especially Aristotle. … [But it would be] both safer and more delightful to otherwise and sometimes descend to experience in some cases, as for example in the motion of heavy bodies, in which both Aristotle and all other philosophers without ever doubting the fact, have believed and affirmed that according to how much a body is more heavy, by so much more [speed] will it descend, the test of which shows it not to be true.[Benedetto Varchi, 1544, <cit.>.]
The fact that run-of-the-mill humanists like Varchi were arguing this way long before Galileo shows what an obvious idea it was.

It is not clear whether Galileo did in fact carry out such an experiment from the tower of Pisa when he was teaching at the university there, as legend would have it.[<cit.>.] Personally I find the story plausible inasmuch as a field day throwing rocks surely had great appeal to a professor whose strong suit certainly wasn't pure mathematical theory. But be that as it may. It is in any case perfectly clear that many people carried out such experiments around that time, independently of Galileo. Simon Stevin, for example, certainly did, and published his results years before Galileo made his experiment, if indeed the latter ever took place. Stevin dropped lead balls of different weights from a height of 30 feet onto a surface that would make much noise, and noted that they banged to the ground in unison.[<cit.>.] Similarly, Mersenne was already dropping weights out out Parisian chamber windows before he heard of Galileo's work.[<cit.>, <cit.>.]

“To crown the comedy, it was an Aristotelian, Coresio, who in 1612 claimed that previous experiments had been carried out from too low an altitude. In a work published in that year he described how he had improved on all previous attempts—he had not merely dropped the bodies from a high window, he had gone to the very top of the tower of Pisa. The larger body had fallen more quickly than the smaller one …, and the experiment, he claimed, had proved Aristotle to have been right all the time.”[<cit.>.] Girolamo Borro, another Aristotelian philosopher, similarly investigated whether lead falls faster than wood. “We took refuge in experience, the teacher of all things,” he says, and hence “threw these two pieces of equal weight from a rather high window of our house at the same time. … The lead descended more slowly. … Not only once but many times we tried it with the same results.”[Girolamo Borro, De motu et levium (1575), <cit.>.] These philosophers were united in their emphasis on actually testing their theories practically and even in taking care to mitigate possible sources of error in the measurements. So much for Aristotelians hating experiments. On the contrary, appeal to experiments were commonplace long before Galileo.

But these people got the result wrong, you say. Maybe they didn’t experiment at all, or if they did they messed it up somehow. Well, so did Galileo. Galileo's own early experiments also produced the wrong results.[Just as they did for the cycloid, <ref>.] “If an observation is made, the lighter body will, at the beginning of the motion, move ahead of the heavier and will be swifter,”[Galileo, <cit.>.] Galileo reports, but if the fall is long enough the heavier body will eventually overtake it. Galileo devotes a full chapter to following this up, “in which the cause is given why, at the beginning of their natural motion, bodies that are less heavy move more swiftly than heavier ones.”[Galileo, <cit.>.] So when Galileo started experimenting on this he got the wrong result, and he also believed himself to a have a good theory “explaining” those false results. We now know that the true cause for the errant results was not a theoretical one like Galileo imagined, but more likely the pedestrian circumstance that we are not good at dropping one object from each hand at the same time. Modern experiments show that people tend to drop the lighter object sooner, “even though the subject `feels' that she has released both simultaneously.”[<cit.>.] This is why Galileo and others ended up thinking that lead balls started out slower than lighter bodies and only then picked up speed.

With experiments being so inconclusive, it is no wonder that Galileo relied more on a theoretical argument in his published account. His supporters would have us believe that “Galileo showed that Aristotle’s rule could be refuted by logic alone.”[<cit.>.] Let us listen to his argument, which is supposedly a “splendid and incontrovertible” model example of “cast-iron reductio ad absurdum reasoning.”[<cit.>.]
By a short and conclusive demonstration, we can prove clearly that it is not true that a heavier moveable is moved more swiftly than another, less heavy … If we had two moveables whose natural speeds were unequal, it is evident that were we to connect the slower to the faster, the latter would be partially retarded by the slower … But the two stones joined together make a larger stone …;  therefore this greater stone is moved less swiftly than the lesser one. But this is contrary to your assumption. So you see how, from the supposition that the heavier body is moved more swiftly than the less heavy, I conclude that the heavier moves less swiftly.[<cit.>, OGG.VIII.107–108.]
“From this we conclude that both great and small bodies, of the same specific gravity, are moved with like speeds.”[<cit.>, OGG.VIII.109.] Furthermore, “if one were to remove entirely the resistance of the medium, all materials would descend with equal speed.”[<cit.>, OGG.VIII.116.]

With this argument Galileo allegedly exposes a fundamental logical inconsistency in the Aristotelian theory of fall. But he doesn't. Aristotle is perfectly clear: heavier objects fall faster. So when you put the heavy and the light together they will fall faster. The inconsistency arises only when one inserts the additional assumption that when you put two bodies together the lighter will retard the heavier and slow it down. But there is no basis for this latter assumption in Aristotle. It is a fiction that Galileo has made up. Only by dishonestly misrepresenting the view he is trying to refute in this way is he able to draw his triumphant conclusion.

A more honest form of the argument, which doesn't depend on misrepresenting Aristotle, is the following, which Galileo knew but didn't include in his published treatises:
Two identical bricks would fall side by side; no doubt about that. If a piece of string was tied to them they still would. Shortening the string could not change that. Hence two bricks tied together end to end would fall at the same speed as either brick alone. Now throw away the string and glue the bricks together; no reason appears why this double brick of double weight should fall faster than two bricks tied together—or either one alone.[<cit.>, paraphrasing notes Galileo wrote at Pisa. This argument was known before Galileo. For example, a very similar argument appears in Benedetti, Diversarum speculationum mathematicarum et physicarum liber (1585), Chapter 10 <cit.>.]
So the real crux of the argument is the claim that two bricks held side by side should behave the same way in terms of fall whether they are glued together or not. This is not a bad argument, but it falls short of being a matter of “logic alone.” For instance, imagine you are taking a basketball free throw. You can choose between trying to hit the hoop with either two bricks glued together or two bricks merely held side by side as you throw them. Would you really say that “no reason appears” why nature should treat the two cases differently, so you might as well go with the loose bricks? I don’t think so. Then why should this assumption be accepted in the case Galileo describes? Of course these two cases are quite different, but my point is that even though the assumption about falling bricks may feel “obvious,” it is really dependent on experience. It makes little sense to claim that we know purely by a priori intuition how the bricks behave in the case of simple fall, yet that we do not have such a priori knowledge of other scenarios like that bricks thrown through a hoop. If one case is “logic alone” then why isn't the other? So, if we are being honest, we are back to having to rely on experiment after all. It is not “logic” that guarantees that bricks behave the same way whether they are tied together or not; it is experience.

Galileo indeed discusses the experimental side of the matter too in his treatise. He admits that actual experiments do not come out in accordance with his law because of air resistance. But, he says, the fit is much better than for Aristotle’s law. To make this point Galileo provides specific measurements of how much the slower ball lags behind the heavier one:
Aristotle says, “a hundred-pound iron ball falling from the height of a hundred braccia hits the ground before one of just one pound has descended a single braccio.” I say that they arrive at the same time. You find, on making the experiment, that the larger anticipates the smaller by two inches; that is, when the larger one strikes the ground, the other is two inches behind it. And now you want to hide, behind those two inches, the ninety-nine braccia of Aristotle, and speaking only of my tiny error, remain silent about his enormous one.[<cit.>, OGG.VIII.109. Of course the statement attributed to Aristotle is not a literal quote but rather an inference from his law of fall.]
But this is fake data. Galileo cooks the numbers to sound much more convincing in favour of his theory. The actual lag or difference between the two bodies is more than 20 times greater than the fake data Galileo reports in his published so-called masterpiece. “In no case could Galileo have consistently achieved the results he reported.”[<cit.>.]

Nevertheless it remains true that Galileo's law doesn't fare as poorly as that of Aristotle in this experiment. So Galileo has indeed managed to improve on a two-thousand-year-old claim, made passingly by a non-mathematician. Aristotle himself obviously did not intend his so-called law as quantitative science. He only introduces it very parenthetically as a stepping-stone toward making the philosophical point that there can be no such thing as an object of infinite weight. Aristotle has no interest in this “law” beyond drawing this isolated metaphysical conclusion from it.

Galileo's entire case rests on his readers considering Aristotle to be a great authority. No wonder he clings to this framing and uses it as the trope of his dialogues. If we admit the truth—that Aristotle's law had been refuted more than a thousand years before, and that the notion of relying on Aristotle for quantitative science would never have entered the mind of any mathematically competent person in Galileo's time—then what does Galileo have to show for himself? An unproven claim that doesn't even fit the fake data Galileo has specifically concocted for the purpose, let alone the many experiments that proved the opposite, including his own before he knew which way he was supposed to fudge the data. Galileo likes to portray himself as doing the world a great service by defeating the rampant Aristotelianism all around him. The truth is that he is rather doing himself a great service by pretending that these Aristotelian opinions are ever so ubiquitous, so that he can inflate the importance of his own contributions, the feebleness of which would be all too evident if he addressed actual scientists instead of straw men Aristotelians.

§.§ Law of fall

Here is “Galileo's” law of fall, familiar to everyone from high school physics: ignoring air resistance, the acceleration of a freely falling object is constant. Equivalently, the velocity it acquires during its fall is proportional to time, and the distance fallen is proportional to time squared. The equivalence of these three formulations of the law of fall is elementary and immediately suggests itself to any mathematically competent person. In modern terms we can express the three version of the law in calculus terms as $y''=g$, $y'=gt$, and $y=gt^{2}/2$, respectively, but the straightforward equivalence of these three relationships long predates the calculus.[Cf. <ref>.] Galileo set out to prove as much, but since he botched his treatment of calculus-style deductions,[<ref>.] “it is not Galileo but Descartes who was the first to conceptualize the relations of time, space, and velocity in a manner consistent with the conceptual framework of classical mechanics.”[<cit.>. Descartes to Huygens, 18 February 1643, AT.III.807–808.]

Motion on an inclined plane (without friction) is closely related to this, because a ball rolling down a plane will acquire the same speed as it would have in free fall through the same vertical distance.[As we would say today, in anachronistic terms, since all balls covering the same vertical distance trade in the same amount of potential energy, they get the same amount of kinetic energy out of it.] All of these laws for free fall and for the inclined plane were indeed clearly stated by Galileo.[In various places in both the  and the , conveniently compiled and quoted in <cit.>. OGG.VII: 47, 50–52, 225, 248, 252. OGG.VIII: 198, 202–205, 208–209.] The equivalence with the inclined plane is important since it makes empirical verification of the law much easier. A ball rolling down a slope is basically a slow-motion version of falling.
In a way it is surprising that the … law for the spontaneous descent of heavy bodies had not been recognized long before the 17th century. Measurements sufficient to put the law within someone's grasp are quite simple. Equipment for making them had not been lacking—a gently sloping plane, a heavy ball, and the sense of rhythm with which everyone is born.[<cit.>.]
Relying on the sense of rhythm means that no time-keeping device is necessary. One places small grooves or bells along the ramp at intervals that the ball ought to cover in equal times according to the law of fall to be tested. The ball will make a click every time it rolls across one of the markings, so hearing whether the law holds is as easy as telling when a musician is off beat.[<cit.>.] Nevertheless, it is “rather unlikely” that Galileo is truthful in his exorbitant claim “that he had obtained precise correspondence in hundreds of trials on inclined planes.”[<cit.>.]

Descartes seems to have arrived at the correct law of fall independently of Galileo. In fact, Descartes once held the mistaken belief that velocity is proportional to distance rather than to time, and he initially retained this belief even after reading Galileo's work, which he disliked.[<cit.>, <cit.>. <ref>.] The question of whether velocity is proportional to distance or to time was indeed a tricky and elusive one.[<cit.>, <cit.>, <cit.>.] Galileo too got it wrong for a long time,[For example in a letter to Sarpi, 1604, OGG.X.115.] and at one point still gets himself confused and in effect uses the wrong form even in his mature .[In the course of the argument regarding centrifugal whirling discussed in <ref>. <cit.> and <cit.> make the case that Galileo's error in effect amounts to assuming that speed in free fall is proportional to distance rather than to time.]

In any case, Beeckman independently discovered the correct law of fall,[<cit.>, <cit.>.] as did Harriot, both at about the same time as Galileo and completely independently of him. “In his mathematical analysis of [uniformly accelerated] motion, Harriot discovered that this actually includes two distinct possibilities: either the velocity of fall increases in proportion to the time elapsed or it increases in proportion to the space traversed. Harriot performed free fall experiments in order to decide between the two possibilities. … Harriot concluded that velocity increases in proportion to time and thus arrived at the law of time proportionality, which is correct also according to classical mechanics.”[<cit.>.] Of course he also immediately derived the equivalent form of the law that  distance fallen is proportional to time squared, in keeping with our observation that this equivalence is trivial and obvious.

§.§ Gravitational constant

All objects fall with the same acceleration, we have now concluded. But how fast is that exactly? What is this same universal acceleration that every object shares? The answer is well-known to any student of high school physics: the constant of acceleration is $g\approx 9.8$ m/sec$^2$. But Galileo messed this up. He gives values equivalent to less than half of the true value.[<cit.>.] According to his defenders, “clearly, round figures were taken here in order to make the ensuing calculation simple.”[<cit.>.] In other words, Galileo “used arbitrary data.”[<cit.>.] And that’s what the people trying to defend Galileo are saying. Isn’t the law of fall supposed to be one of Galileo’s greatest discoveries? Why did he use fake data? Why not use real data? It was readily doable. His contemporaries did it. Why not do a little work to get the details right when you are publishing your supposed key results in your mature treatise? Is that really too much to ask? Instead of reporting make-believe evidence with a straight face, as Galileo does.

Competent and serious readers were in disbelief at Galileo’s inaccurate data. They certainly did not think it was fine to “use arbitrary data” in order to get “round” numbers for simplicity. Nor did they think it was “clear” that this is what Galileo was doing, contrary to what Galileo’s defenders are forced to argue when they try to excuse his inexcusable behaviour. Mersenne put it clearly: “I doubt whether Mr Galileo has performed the experiment on free fall on a plane, since … the intervals of time he gives often contradict experiment.”[Mersenne, Harmonie universelle, I.122, <cit.>.] Being a serious and diligent scientist, Mersenne did the work to find the correct value, unlike Galileo. The same goes for Riccioli, another contemporary of Galileo, who likewise did his own detailed experiments and found that they did not at all agree with the data Galileo had made up.[<cit.>.]

Once again the historical record fails to conform to Galileo's narrative of himself as a novel master experimenter refuting dogmas everyone believes in. Instead, contemporaries like Mersenne were already doing free fall experiments of their own before hearing of Galileo's,[<ref>.] and immediately recognised Galileo's data as fraudulent. Galileo is not addressing actual scientists. To them, much of what he has to say is disappointingly shallow and lacks serious scientific follow-through. Galileo's claim to fame relies on us not knowing this, and buying into the fiction that everyone at the time was an Aristotelian who had never heard of such a thing as experiment and quantitative science.

§.§ Planetary speeds

Before praising Galileo for stating the correct law of fall, it is sobering to consider the foolish uses he put it to. One is what has been called his “Pisan Drop” theory of planetary speeds.[<cit.>.] The planets orbit the sun at different speeds. Mercury has a small orbit and zips around it quickly. Saturn goes the long way around in a big orbit and it is also moving very slowly. Galileo imagines that these speeds were obtained by the planets falling from some faraway point toward the sun, and then being somehow deflected into their circular orbits at some stage during this fall (Figure <ref>). That supposedly explains why the planets have the speeds they do.

[gray] (0,0) circle (5.2cm);
[gray] (0,0) circle (4.4cm);
[gray] (0,0) circle (3.6cm);
[gray] (0,0) circle (2.6cm);
[gray] (0,0) circle (1.8cm);
[gray] (0,0) circle (1.1cm);

compass style/.append style=<-;

[inner sep=0pt] (sun) at (0,0)

[inner sep=0pt] (mercury) at (7.5*cos(94),7.5*sin(94))
[inner sep=0pt] (venus) at (7.5*cos(92),7.5*sin(92))
[inner sep=0pt] (earth) at (7.5*cos(90),7.5*sin(90))
[inner sep=0pt] (mars) at (7.5*cos(88),7.5*sin(88))
[inner sep=0pt] (jupiter) at (7.5*cos(86),7.5*sin(86))
[inner sep=0pt] (saturn) at (7.5*cos(84),7.5*sin(84))

(7*cos(94),7*sin(94)) – (1.1*cos(94),1.1*sin(94));
(7*cos(92),7*sin(92)) – (1.8*cos(92),1.8*sin(92));
(7*cos(90),7*sin(90)) – (2.6*cos(90),2.6*sin(90));
(7*cos(88),7*sin(88)) – (3.6*cos(88),3.6*sin(88));
(7*cos(86),7*sin(86)) – (4.4*cos(86),4.4*sin(86));
(7*cos(84),7*sin(84)) – (5.2*cos(84),5.2*sin(84));

[Planetary speeds.]Galileo's erroneous theory that the orbital speeds of the planets are equal to the speeds they would have acquired through free fall if dropped from a common height.

Galileo describes it as follows in the . “Suppose all the [planets] to have been created in the same place … descending toward the [sun] until they had acquired those degrees of velocity which originally seemed good to the Divine mind. These velocities being acquired, … suppose that the globes were set in rotation [around the sun], each retaining in its orbit its predetermined velocity. Now, at what altitude and distance from the sun would have been the place where the said globes were first created, and could they all have been created in the same place? To make this investigation, we must take from the most skilful astronomers the sizes of the orbits in which the planets revolve, and likewise the times of their revolutions.” Using this data and “the natural ratio of acceleration of natural motion” (that is, the constant $g$), one can compute “at what altitude and distance form the center of their revolutions must have been the place from which they departed.” According to Galileo this shows that indeed all the planets were dropped from a single point and their orbital data “agree so closely with those given by the computations that the matter is truly wonderful.”[<cit.>.]

Galileo was so proud of this erroneous argument that he repeated it in his second major work, the , as well:
Our Author [Galileo] … may at some time have had the curiosity to try whether he could assign a determinate sublimity [i.e., distance from the sun] from which the bodies of the planets left from a state of rest, and were moved through certain distances in straight and naturally accelerated motion, and were then turned at the acquired speeds into equable motions. This might be found to correspond with the sizes of their orbits and the times of their revolutions. … Indeed, I seem to remember that he told me he had once made the computation, and also that he found it to answer very closely to the observations.[<cit.>, OGG.VIII.284.]
Galileo omits the details though. He has one of the characters in his dialogue say that “making these calculations … would be a long and painful task, and perhaps one too difficult for me to understand,” whereupon Galileo's mouthpiece in the dialogue confirms that “the procedure is indeed long and difficult.”[<cit.>.]

Actually there is nothing “difficult” about it, at least not to mathematically competent people. Mersenne immediately ran the calculations and found that Galileo must have messed his up, because his scheme doesn't work.[Marin Mersenne, Harmonie Universelle, II.103–107, <cit.>. Later Newton made the same observation, <cit.>.] There is no such point from which the planets can fall and obtain their respective speeds. Galileo's precious idea is so much nonsense, which evidently must have been based on an elementary mathematical error in calculation.

§.§ Moon fall

Galileo tried to compute how long it would take for the moon to fall to the earth, if it was robbed of its orbital speed. “Making the computation exactly,” according to himself, he finds the answer: 3 hours, 22 minutes, and 4 seconds.[<cit.>.] This is way off the mark because Galileo assumes that his law of fall—that is, constant gravitational acceleration—extends all the way to the moon. This assumption is erroneous; the force of gravity diminishes with distance according to Newton’s inverse-square law. Ironically, Galileo’s purpose with this calculation was to refute the claim of another scholar that the fall would take about six days, which is a much better value: in fact it would take the moon almost five days to fall to the earth.[<cit.>.] That’s Galileo, the great hero of quantitative science, in action for you: bombastically claiming to refute others with his “exact calculations,” only to make fundamental mistakes and err orders of magnitude worse than his opponents did.

§.§ Resistance of the medium

As we have seen, Aristotelians were often as inclined to experiment as Galileo—a point obscured by Galileo's pretences to the contrary when it suited his purposes. Elsewhere it suited Galileo better to feign other straw men. “In one of the dialogues of Galileo, it is Simplicius, the spokesman of the Aristotelians—the butt of the whole piece—who defends the experimental method of Aristotle against what is described as the mathematical method of Galileo.”[<cit.>.]

Consider for example the question of the resistance of the medium (such as air or water) on a moving object. Aristotle stated a law regarding how a body moves faster in a rarer medium than in a dense one. Galileo, in an early text, criticises Aristotle for accepting this “for no other reason than experience”; instead one must “employ reasoning at all times rather than examples,” “for we seek the causes of effects, and these are not revealed by experience.”[<cit.>. OGG.I.260, 263.] Alas, Galileo's own law as to how resistance depends on density of the medium is also “incompatible with classical mechanics” as one study puts it—a polite, scholarly way of saying it's wrong.[<cit.>.]

Employing some more “reasoning” along the same lines, Galileo decided that air resistance doesn't really increase appreciably with speed: “The impediment received from the air by the same moveable when moved with great speed is not very much more than that with which the air opposes it in slow motion.”[<cit.>, OGG.VIII.277.] A surprising conclusion to modern bicyclists, among others. Yet “experiment gives firm assurance of this,”[<cit.>, OGG.VIII.277.] Galileo promises. Alas, once again “the statement is false, and the experiment adduced in its support is fictitious.”[Drake, <cit.>.] More fake data, in other words. This is quickly becoming a pattern.

§.§ Inertia

Newton's law of inertia says: “Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.”[Newton, Principia (1687), Law 1. <cit.>.] Is this important conception, so crucial to classical mechanics, due to Galileo? No. Even the most ardent defenders of Galileo “freely grant that Galileo formulated only a restricted law of inertia”[<cit.>.] and that “he neglected to state explicitly the general inertial principle” that everyone knows today, which was instead correctly “formulated two years after his death by Pierre Gassendi and René Descartes.”[<cit.>. Descartes, Principia philosophiae (1644), II.37, 39. On Gassendi, see <cit.>.]

The charitable interpretation of trying to attribute to Galileo some kind of “restricted law of inertia” is a murky business. According to one author who tries to do so, “in my opinion the essential core of the inertial concept lies in the ideas … of a body's indifference to motion or to rest and its continuance in the state it is once given. This idea is, to the best of my knowledge, original with Galileo.”[<cit.>.] You could very well argue that that’s not really inertia at all because it doesn’t involve the straightness of the direction of the motion, nor does it explicitly say that the motion keeps going at a perpetual uniform speed. It only focusses on indifference of motion versus rest and preservation of the state of motion. But it is a dangerous game to cherry-pick some aspects of inertia. With this logic, one could equally well argue that that Aristotle had “the essential core of the inertial concept,” for he already highlighted indifference of motion versus rest and preservation of the state of motion two thousand years before Galileo:
No one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than there? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.[Aristotle, Physics, IV.8, 215a.]
If this is inertia, then Aristotle was the pioneering near-Newtonian who conceived it, not Galileo.

This claim is rather isolated in Aristotle and didn’t really form part of a sustained and coherent physical treatment of motion comparable to how we use inertia today. Aristotle as usual is focussed on much more philosophical purposes. So you might say: that’s a one-off quote taken out of context which sounds much more modern than it really is.

Indeed. But then again the same could be said for Aristotle’s so-called law of fall that Galileo refuted with so much fanfare.[<ref>.] This too is only mentioned in passing very briefly and plays no systematic role in Aristotle’s thought. Yet Galileo takes great pride in defeating this incidental remark, and his modern fans applaud him greatly for it. So if we want to dismiss Aristotle’s inertia-like statement as insignificant, then, by the same logic, we ought to likewise dismiss all of Galileo’s exertions to refute his law of fall as completely inconsequential as well. If we argue that statements such as those of Aristotle don’t count as scientific principles unless they are systematically applied to explain various natural phenomena, then we would have to conclude that there was no Aristotelian science of mechanics at all. This, of course, would be a disastrous concession to make for advocates of Galileo’s greatness, since so much of Galileo’s claim to fame is based on contrasting his view with so-called “Aristotelian” science.

What about the rectilinear character of inertia? The thing keeps going straight. Was this key point grasped by Galileo? The following passage may appear to suggest as much:
A projectile, rapidly rotated by someone who throws it, upon being separated from him retains an impetus to continue its motion along the straight line touching the circle described by the motion of the projectile at the point of separation. … The projectile would continue to move along that line if it were not inclined downward by its own weight. … The impressed impetus, I say, is undoubtedly in a straight line.[<cit.>.]
In Newtonian terms this would be interpreted in terms of the law of inertia. But it hardly seems Galileo is thinking about it that way. He calls it “impetus.” What is “impetus” and why should we equate it with inertia? Will “impetus” run out? Is the motion caused by the “impetus” perpetual and uniform? In many other sources at the time, “loss of impetus by projectiles was likened to … the diminution of sound in a bell after it is struck, or heat in a kettle after it is removed from the fire.”[<cit.>.] This conception is perfectly compatible, to say the least, with what Galileo writes. In fact, “neither in the Discourses nor in the Dialogue does Galileo anywhere assert the eternal conservation of rectilinear motion.”[<cit.>.] On the contrary, he explicitly rejects it: “Straight motion cannot be naturally perpetual.”[<cit.>.] “It is impossible that anything should have by nature the principle of moving in a straight line.”[<cit.>.] It is easy to understand, then, why Galileo's defenders are so eager to insist on characterising “the essential core of the inertial concept” in a way that does not involve its rectilinear character, since Galileo clearly and explicitly rejected rectilinear inertia.

If there is any inertia in Galileo, it is horizontal rather than rectilinear inertia. For example:
To some movements [bodies] are indifferent, as are heavy bodies to horizontal motion, to which they have neither inclination … or repugnance. And therefore, all external impediments being removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest or to movement toward any part of the horizon. And it will remain in that state in which it has once been placed; that is, if placed in a state of rest, it will conserve that; and if placed in movement toward the west, for example, it will maintain itself in that movement. Thus a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping.[<cit.>.]
Motion in a horizontal line which is tilted neither up nor down is circular motion about the center; … once acquired, it will continue perpetually with uniform velocity.[<cit.>. See also <cit.>.]
Again, as with the sling and the projectile, one can debate whether this is inertia per se. In Newtonian mechanics too a hockey puck on a spherical ice earth would glide forever in a great circle, even though this is not inertial motion. But this agreement with Newtonian mechanics only holds if the object is prevented from moving downward, as the puck is by the ice, or the ship by the water. Galileo seems to have believed horizontal inertia to hold also for objects travelling freely through the air, which is not compatible with Newtonian mechanics. For example:
I think it very probable that a stone dropped from the top of the tower … will move, with a motion composed of the general circular movement and its own straight one.[<cit.>. See also <cit.>.]
Once again it is not entirely clear that this is supposed to represent inertia at all. It is conceivable that, in Galileo's conception, the circular motion itself is not a force-free, default motion, but rather a motion caused or contaminated by some force or other. Who knows? Galileo just isn't clear about these kinds of things.

In his discussion of projectile motion too Galileo seems to rely on a horizontal conception of inertia. He speaks unambiguously of “the horizontal line … which the projectile would continue to follow with uniform motion if its weight did not bend it downward.”[<cit.>.] But he does not make the same claim for projectiles fired in non-horizontal directions. It rather seems as if he studiously avoided committing himself on that point.

An argument against attributing circular or horizontal inertia to Galileo, on the other hand, is his discussion attempting to prove that the the earth's rotational speed, no matter how much it increased, could not cause objects on its surface to be whirled off into space.[As will be discussed further in <ref>.] Galileo uses some intricate arguments to try to prove this, which he could have avoided by a simple appeal to circular inertia if he truly believed in the latter.

In conclusion, “there is no general principle of inertia in Galileo's work.”[<cit.>.] He definitely never stated the correct law of inertia. He often spoke of what appears to be a kind of circular inertia, in particular for horizontal motion (that it to say, circular motion around the center of the earth). But altogether it is impossible to attribute to him any one consistent view on the matter. Newton and Descartes, like the good mathematicians that they are, state concisely and explicitly what the exact fundamental assumption of their theory of mechanics are. Their laws of inertia are crystal clear and specifically announced to be basic principles upon which the rest of the theory is built. Galileo never comes close to anything of this sort. He uses the casual dialogue format of his books to hide behind ambiguities. One moment he seems to be saying one thing, then soon thereafter something else, like an opportunist who doesn't have a systematically worked out theory but rather adopts whatever assumptions are most conducive to his goals in any given situation.

§.§ Relativity of motion

When teaching basic astronomy at Padua, Galileo explained to his students that Copernicus was undoubtedly wrong about the earth's motion. The earth doesn’t move, Galileo explained. Because, if the earth moved, a rock dropped from a tower would strike the ground not at its foot but some distance away, since the earth would have moved during the fall. In support of this claim, “Galileo observed that a rock let go from the top of a mast of a moving ship hits the deck in the stern.”[<cit.>.] This had indeed been reported as an experimental fact by people who actually carried it out.[<cit.>.]

Of course this is completely backwards and the opposite of Galileo’s later views that he is famous for. To be sure, these lectures do not necessarily say anything about Galileo’s personal beliefs. In all likelihood he simply taught the party line because it was the easiest way to pay the bills. But at least the episode does show that the simplistic narrative that “the experimental method” forced the transition from ancient to modern physics is certainly wrong. On the contrary, experimental evidence was among the standard arguments for the conservative view well before Galileo got into the game.

In his later works Galileo of course affirms the opposite of what he said in those lectures: the rock will fall the same way relative to the ship regardless of whether the ship is standing still or travelling with a constant velocity. He gives a very vivid and elaborate description of this principle:
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other.[<cit.>.]
Galileo's prose is as embellished with fineries as this little curiosity-cabinet of a laboratory that he envisions. But is it any good of an argument? Insofar as it is, the credit is perhaps due to Copernicus himself, who had already made much the same point a hundred years before.[“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 an image of their own, and they think on the contrary that they themselves and all the things with them are at rest. So it can easily happen in the case of the movement of the Earth that the whole world should be believed to be moving in a circle. Then what would we say about the clouds and the other things floating in the air or falling or rising up, except that not only the Earth and the watery element with which it is conjoined are moved in this way but also no small part of the air?” <cit.>.]
So Galileo's relativity argument, like so much else he says, is old news. The primary contribution of his version is literary ornamentation—adding some butterflies and whatnot, while saying nothing new in substance.

Today the “Galilean” principle of relativity says that the phenomena in the cabin cannot be used to distinguish between the ship being at rest or moving with constant velocity in a straight line. But Galileo clearly has another scenario in mind: he sees the ship as travelling along a great circle around the globe. This is the kind of motion he believes cannot be distinguished from rest, in keeping with his misconceived idea of horizontal inertia.[<ref>.] This principle of relativity—the actually “Galilean” one—is of course false.[<cit.>.]

In fact it’s even worse than that. Galileo’s purpose with this passage about the ship is to argue, erroneously, that his ship cabin experiment “alone shows the nullity of all those [arguments based on the motions of objects such as bird and clouds and falling objects relative to the earth] adduced against the motion of the earth.”[<cit.>.] That is to say, the rotation of the earth has no detectible mechanical effects, according to Galileo. But he is wrong. The earth's rotation is in fact mechanically detectible. The Foucault pendulum is a device that does behave differently due to the rotation of the earth than if the earth had been stationary, contrary to Galileo's claim that all local mechanical phenomena would be identical regardless of whether the earth was fixed or uniformly rotating, just as there is no way to detect the uniform motion of the ship from inside the cabin.

So the attribution of the principle of relativity of motion to Galileo in modern textbooks is doubly mistaken. First of all, relativity of motion and the idea of an inertial frame had been noted long before and was invoked by Copernicus to much the same end as Galileo. Moreover, Galileo’s principle is wrong in itself, because it’s about motion in a great circle, not in a straight line. And furthermore Galileo's purpose in introducing it is to draw another false conclusion from it, namely that the earth’s rotation is undetectable.

§.§ Centrifugal force

Galileo wished to refute this ancient argument: “The earth does not move, because beasts and men and buildings” would be thrown off.[<cit.>.] Picture an object placed at the equator of the earth, such as a rock lying on the African savanna.
Imagine this little rock being “thrown off” by the earth’s rotation. In other words, the rock takes the speed it has due to the rotation of the earth, and shoots off with this speed in the direction tangential to its motion. The spectacle will be rather underwhelming at first: since the earth is so large, the tangent line is almost parallel to the ground, and since the speed of the rock and of the earth are the same they will keep moving in tandem. So rather than shooting off into the air like a canon ball, the rock will slowly begin to hover above the ground, a few centimeters at a time. For instance, Figure <ref> shows how far the rock has gotten ten minutes after being thrown off the earth. As we can see, the deviation of the tangential path from the curved surface of the earth is only now becoming noticeable in relation to the size of the earth.

Tangential motion of an object if gravity stopped acting on it. The object was located on the equator of the earth. The arrow shows how far it has gone in ten minutes.

Of course this is not what happens to an actual rock, because gravity is pulling it back down again. The rock stays on the ground since gravity pulls it down faster than it rises due to the tangential motion. How can we compare these two forces quantitatively? Since we know the size and rotational speed of the earth, it is a simple task (suitable for a high school physics test) to calculate how much the rock has risen after, say, one second. This comes out as about 1.7 centimeters. We need to compare this with how far the rock would fall in one second due to gravity. Again, this is a standard high school exercise (equivalent to knowing constant of gravitational acceleration $g$). The answer is about 4.9 meters. This is why the rock never actually begins to levitate due to being “thrown off”: gravity easily overpowers this slow ascent many times over.

But of course this conclusion depended on the particular size and speed and mass of the earth. We could make the rock fly by spinning the earth fast enough. For example, if we run the above calculations again assuming that the earth rotates 100 times faster, we find that, instead of rising a measly 1.7 centimeters above the ground in one second, the rock now soars to 170 meters in the same time. The fall of 4.9 meters due to gravity doesn’t put much of dent in this, so indeed the rock flies away.

These things were calculated correctly in Galileo's time.[By Mersenne. <cit.>.] But Galileo, alas, gets all of this horribly wrong. Even though we are supposed to celebrate Galileo as the discoverer of the law of fall, it is apparently too much to ask that he work out this very basic application of it. As noted,[<ref>.] Galileo did not offer a serious estimate for the constant of gravitational acceleration $g$, unlike his contemporaries. Therefore he did not have the quantitative foundations to carry out the above analysis, which high school students today can do in five minutes.

Worse yet, Galileo maintains that no such analysis is needed in the first place, because he can “prove” that the rock will never be thrown off regardless of the rotational velocity. “There is no danger,” Galileo assures us, “however fast the whirling and however slow the downward motion, that the feather (or even something lighter) will begin to rise up. For the tendency downward always exceeds the speed of projection.”[<cit.>.] He even offers us “a geometrical demonstration to prove the impossibility of extrusion by terrestrial whirling.”[<cit.>.]

Galileo's so-called “demonstration” is shown in Figure <ref>.[<cit.>. The errors in Galileo's argument have been analysed by [Chalmers & Nicholas , 1983], [Hill , 1984].] It is indeed a qualitative argument that rules out all possible cases of centrifugal projection, regardless of the rotational speed of the earth $V$, the radius of the earth $R$, or the magnitude of gravitational acceleration $g$. It is true, as Galileo says, that the ratio $v(t)/h(t)$ goes to $\infty$ as $t$ goes to zero. But this is obviously comparing apples to oranges, namely a velocity with a distance. The relevant comparison is between $h(t)$ and the distance $d(t)$ covered by free fall in this time. Galileo evidently felt that since in small time intervals $v(t)$ is overwhelmingly larger than $h(t)$, then $d(t)$ must surely be larger than $h(t)$ as well. But this is false. Instead,

\[ \lim_{t\rightarrow 0} \frac{d(t)}{h(t)} = \lim_{t\rightarrow 0} \frac{gt^{2}/2}{R-\sqrt{R^{2} -V^{2}t^{2}}} = \frac{gR}{V^{2}}.\]

In other words, $d(t)$ does not always overpower $h(t)$, as Galileo mistakenly believes. Rather, whether $d(t)$ is greater or smaller than $h(t)$ for small $t$ depends on the specific parameters of the situation in question. A strong gravitational acceleration $g$, or a big radius of the rotational path $R$, makes it easier for the object to “catch up” with the surface of the earth, while a big rotational speed $V$ makes it harder. Whether the object catches up with the surface or flies away depends on the relation between these parameters.

[Impossibility of centrifugal projection.]Galileo's “proof” that centrifugal projection can never hurl objects off the earth. If gravity stops acting on an object at $A$, it would move inertially in the tangential direction $AB$. Since inertial motion has uniform speed, it would reach the equally spaced points $AFHK$ in equal time intervals. If the object had instead been dropped from rest, it would have acquired a certain downward speed in those same time intervals. These speeds are represented in the diagram by $FG$, $HI$, $KL$. Since the velocity acquired in free fall is proportional to time, $AGILE$ is a straight line. The slope of the line depends on the magnitude of gravitational acceleration, but for the purposes of this argument this value does not matter; in other words, we could just as well consider the speeds to be determined by some other line $AD$. The impossibility of centrifugal projection follows, according to Galileo, from the fact that as we consider smaller and smaller time intervals (that is to say, as we zoom it at $A$), the distance $h(t)$ required to catch up with the earth shrinks very rapidly to zero, while the speed $v(t)$ acquired from fall shrinks only linearly to zero. Therefore, says Galileo, the speed of fall $v(t)$ will, for some small enough $t$, be more than enough to cover the distance $h(t)$ and then some. In other words, the object will never get off the ground.

Galileo's claim to fame as a “mathematiser of nature” is certainly done no favours by this episode. He doesn’t know how to quantify his own law of fall, and doesn't understand basic implications of it. His physical intuition is categorically wrong on a qualitative level, and worse than that of the ancients he is trying to refute (whose stance was quite reasonable and would be accurate if the earth was spinning faster). He even offers a completely wrongheaded geometrical “proof” that the ancients' conception is impossible, even though so-called “Galilean” physics leads to the opposite conclusion in an elementary way.

§.§ Circular path of fall

A rock dropped from the top of a tower falls in a straight line to the foot of the tower. But its path of fall is not actually straight if we take into account the earth's rotation. Seen from this point of view—that is to say, from a vantage point that doesn't move with the rotation of the earth—what kind of path does the rock trace? Galileo answers, erroneously, that it will be a semicircle going from the top of the tower to the center of the earth (Figure <ref>):
If we consider the matter carefully, the body really moves in nothing other than a simple circular motion, just as when it rested on the tower it moved with a simple circular motion. … I understand the whole thing perfectly, and I cannot think that … the falling body follows any other line but one such as this. … I do not believe that there is any other way in which these things can happen. I sincerely wish that all proofs by philosophers had half the probability of this one.
[<cit.>, OGG.VII.191.]
This is “so obviously false (and besides incompatible with his own theory of uniformly accelerated motion of falling bodies) that one may wonder that Galileo did not see it himself.”[<cit.>.] Once again Galileo doesn't understand basic implications of his own law. Mersenne readily spotted Galileo's error, whereupon Fermat observed that the path should be a spiral (Figure <ref>), not a semicircle.[<cit.>, [Engelberg & Gertner , 1981], <cit.>.] This would be the right answer given Galileo's assumptions, namely that the path is generated by composing uniform angular motion with uniformly accelerated radial motion toward the center of the earth.[As stated in <cit.> and again later when he admitted Fermat's correction <cit.>.] This implies that the path of fall is $r=r_{0}-a\theta^{2}$ in polar coordinates, which is indeed a spiral. This is still not the true path of fall, since Galileo's assumption that his law of fall remains unchanged in the interior of the earth is itself false. But I am not concerned here with criticising Galileo on such anachronistic grounds. Much worse is the fact that he got the wrong answer even if we grant his own assumptions.

[Circular path of fall.]Circular path of fall of a rock dropped from a tower, according to Galileo. “$AB$ [is the radius of] the terrestrial globe. Next, prolonging $AB$ to $C$, the height of the tower $BC$ is drawn. … The semicircle $CIA$ …, along which I think it very probable that a stone dropped from the top of the tower $C$ will move, with a motion composed of the general circular one [due to the rotation of the earth] and its own straight one [due to gravity].” <cit.>, OGG.VII.191.

[Spiral path of fall.]Corrected version of the path of fall of Figure <ref>. From Galileo's assumptions it follows that the path should be a spiral rather than a semicircle.

When his embarrassing error was pointed out to him, Galileo replied that “this was said as a jest, as is clearly manifest, since it is called a caprice and a curiosity.”[Galileo to Pierre Carcavy, 5 June 1637, OGG.XVII.89, <cit.>.] Some defence this is! Far from offering exonerating testimony, Galileo actually openly pleads guilty to the main charge: namely, that his science is a joke. But in reality “it is hard to believe that Galileo had really meant his solution of the trajectory of the falling body to be merely a joke.”[<cit.>.] If Galileo truly meant his argument to be taken merely in jest, then why did he say that he “considered the matter carefully” and “sincerely wished that all proofs by philosophers had half the probability of this one” and so on? Many of Galileo's errors come with these kinds of bombastic claims where Galileo is editorialising about how remarkably convincing his own arguments are. Isn’t that convenient? Throw out a bunch of half-baked guesses, and when they turn out right you can claim credit for stating it with such confidence while a more responsible scientist may have been exercising prudent caution. And when the guesses turn out wrong, you can apparently just write it off as a “joke” and pretend that that was what you intended all along, even though you published it with all those extremely assertive phrases right in the middle of your big definitive book on the subject. It’s easy to be “the father of science” if you can count on posterity to play along with this double standard.

§.§ Projectile motion

Pick up a rock and throw it in front of you. The path of its motion makes a parabola. That’s Galileo’s great discovery, right? Well, not really. Galileo does claim this but he doesn’t prove it. Even Galileo’s own follower Torricelli acknowledged this. The result is “more desired than proven,”[Torricelli, 1644, <cit.>.] as he says, very diplomatically. And the reason why Galileo doesn’t prove it is a revealing one. It is due to a basic physical misunderstanding.

The right way to understand the parabolic motion of projectiles like this is to analyse it in terms of two independent components: the inertial motion and the gravitational motion. If we disregard gravity, the rock would keep going along a straight line forever at exactly the same speed. That’s the law of inertia. But gravity pulls it down in accordance with the law of fall.[<ref>.] The rock therefore drops below the inertial line by the same distance it would have fallen below its starting point in that amount of time if you had simply let it fall straight down instead of throwing it. A staple fact of elementary physics is that the resulting path composed of these two motions has the shape of a parabola.

Galileo does not understand the law of inertia, and that is why he fails on this point. If the projectile is fired horizontally, such as for instance a ball rolling off a table, then Galileo does prove that it makes a parabola. He proves it the right way, the way just outlined, by composition of inertial and gravitational motion.[<cit.>, OGG.VIII.269, 272–273.]

But if you throw the rock at some other angle, not horizontally, then Galileo doesn’t dare to give such an analysis. “Although [Galileo's] Discorsi takes it for granted that the trajectory for oblique projection is a parabola, no derivation of this proposition is presented.”[<cit.>.] “At the point in the systematic treatment of projectile motion in the Discorsi where oblique projection is actually dealt with and correctly stated to yield a parabolic trajectory, there is simply a gap in the argumentation, and no derivation is offered for this claim.”[<cit.>.]

Galileo's failure is quite clearly due to his not daring to believe in uniform inertial motion in any other direction than along the horizontal. He seems to fear that the law of inertia is perhaps not true for such motions. He equivocates and never takes a clear stand, because he is unsure whether it is the case or not. He is worried that the rectilinear component of the projectile's motion should be seen not as uniform but rather as gradually slowing down, like a ball struggling to roll up a hill or an inclined plane. In the latter case the trajectory is still a parabola, though not an “upright” one. See Figure <ref>.

[Projectile motion.]Left: Correct conception of projectile motion. The dots indicate uniform inertial motion in the firing direction. Right: Erroneous conception of projectile motion, as drawn by Galileo in unpublished manuscripts. The dots indicate a decelerating motion in the firing direction, as if the projectile was struggling to ascend the incline. In both cases the rectilinear motion is composed with an independent vertical motion according to the law of fall. Based on <cit.>.

In his final account, Galileo correctly “postulated upright parabolas for all angles of projection. Galileo's reasoning for this shape is, however, untenable in classical mechanics. What is more, Galileo was unable to derive it from the consideration of two component motions.”[<cit.>.]
“Galileo was … confronted with a contradiction between the inclined-plane conception of projectile motion and his claim that the trajectory is an upright parabola for all angles of projection, a contradiction he was never able to resolve.”[<cit.>.] Since he only trusted the horizontal case, Galileo tried to analyse other trajectories in terms of this case. To this end he assumed, without justification, that a parabola traced by an object rolling off a table would also be the parabola of an object fired back up again in the same direction.[<cit.>, OGG.VIII.296. <cit.>, <cit.>.] In other words, “he takes the converse of his proposition without proving or explaining it,”[Descartes to Mersenne, 11 October 1638, AT.II.387. <cit.>.] as Descartes—a mathematically competent reader—immediately pointed out.

Instead, “it was Galileo's disciples who first derived the parabolic trajectory for oblique projection, although they present it merely as an explication of Galileo's Discorsi,” which it is not.[<cit.>.] Indeed, “even before Galileo's Discorsi appeared in print, Bonaventura Cavalieri published a derivation of the parabolic trajectory that is consistent with classical mechanics and is not restricted to horizontal projection.”[<cit.>.] Cavalieri was a good mathematician, unlike Galileo. He was also Galileo's countryman and in some sense “disciple,” and was very generous in deferring credit to Galileo.

The failures of Galileo's treatment of projectile motion confirms his misconception that inertia is limited to horizontal motion, which, as we have seen, was already independently suggested by other passages.[See <ref>.] Some have tried to argue that “if Galileo never stated the law [of inertia] in its general form, it was implicit in his derivation of the parabolic trajectory of a projectile.”[<cit.>.] This would have been a very good argument if Galileo had treated parabolic trajectories correctly. But he didn't, so the evidence goes the other way: Galileo's bungled treatment of parabolic motion is yet more proof that he did not understand inertia.

Even apart from the above errors and omissions, the mathematical details of Galileo's presentation of projectile motion are very clumsy. Galileo's “calculations are unnecessarily complicated, and were greatly simplified by Torricelli in … 1644, a complete revision and enlargement … which … makes Galileo's demonstrations and procedures obsolete.”[<cit.>.] Once again Galileo's text bears the marks of an amateur mathematician, in other words. And once again his followers almost immediately cleaned up his mess in more mathematically able works that were full of deference to Galileo. “While … inspired by veneration of Galileo, Torricelli is more logical in his treatise.”[<cit.>.] Hence later mathematicians who used Torricelli's better but reverential account rather than Galileo's original for the mathematical details could easily be left with a much more flattering impression of the mathematical quality of “Galileo's” theory than if they had studied Galileo's own treatise in detail. Perhaps it is not so strange, then, that posterity got a bit confused and attributed much more to Galileo than he actually earned.

§.§ Practical ballistics

Galileo made no theoretical use of his theory of projectile motion. In particular, he made no connection between this theory and the motion of the planets, the moon, or comets—a huge missed opportunity. Instead Galileo erroneously claimed that his theory would be practically useful for people who were firing cannons. “He seems to have written [this theory] only to explain the force of cannon shots fired at different elevations,”[Descartes to Mersenne, 11 October 1638, <cit.>.] as Descartes put it. Descartes correctly denied that Galileo's idealised theory of projectile motion would correspond to practice. Galileo was less prudent. “In many passages Galileo remarks that the theory of projectiles is of great importance to gunners. He made little or no distinction between his theory and useful ballistics; he believed—though without experiment—that he had discovered methods sufficiently accurate within the limitations of military weapons to be capable of direct application in the handling of artillery.”[<cit.>.] This belief, however, was completely wrong. “Galilean ballistics were not and could not be the fruit of experimental method; experiment … at once reveals that projectiles do not move in a parabola.”[<cit.>.] A contemporary of Galileo put the matter to experimental test:
I was astonished that such a well-founded theory responded so poorly in practice. … If the authority of Galileo, to which I must be partial, did not support me, I should not fail to have some doubts about the motion of projectiles, [and] whether it is parabolical or not.[Giovan Battista Renieri to Torricelli, 2 August 1647, <cit.>.]
In reply, Torricelli “stated outright that his book was written for philosophers, not gunners.” “According to Torricelli [the Galilean theory] had no connection with practical gunnery or with real projectiles.”[<cit.>.] He also attributed this view to Galileo, but “Torricelli's assertion that neither he nor Galileo ever believed that the science of motion had anything to do with practical affairs … is contradicted by Galileo's letters.”[<cit.>.] “Here Galileo himself must be charged with confusion.”[<cit.>.] This is evident for example from the extensive tables that Galileo included in his big book: ballistic range tables based on his theory. These long tables make no sense other than as a practical guide for firing cannons.[<cit.>.] So clearly Galileo thought his theory was practically viable, which it is absolutely not.

§.§ Catenary

Left: The catenary, or shape of a hanging chain, which Galileo erroneously believed to be a parabola. Right: The catenary (dotted) compared to a parabola (solid) of equal arc length between the same endpoints.

The shape of a hanging chain (Figure <ref>) looks deceptively like a parabola. It is not, but Galileo fell for the ruse: “Fix two nails in a wall in a horizontal line … From these two nails hang a fine chain … This chain curves in a parabolic shape.”[<cit.>, OGG.VIII.186.] More competent mathematicians proved him wrong: Huygens demonstrated that the shape was not in fact parabolic.[[Bukowski , 2008].] Admittedly, his proof is from 1646, which is four years after Galileo's death. So one may consider Galileo saved by the bell on this occasion, since he was proved wrong not by his contemporaries but only by posterity. It is not fair to judge scientists by anachronistic standards. On the other hand Huygens was only seventeen years old when he proved Galileo wrong. So another way of looking at it is that a prominent claim in Galileo's supposed masterpiece of physics was debunked by a mere boy less than a decade after its publication.

In any case, Galileo thus ascribed to the catenary the same kind of shape as the trajectory of a projectile. He considered this to be no coincidence but rather due to a physical equivalence of the forces involved in either case.[<cit.>, OGG.VIII.309.] Indeed, Galileo made much of this supposed equivalence and “intended to introduce the chain as an instrument by which gunners could determine how to shoot in order to hit a given target.”[<cit.>.]

Galileo also tried to test experimentally whether the catenary is indeed parabolic. To this end he drew a parabola on a sheet of paper and tried to fit a hanging chain to it. His note sheets are preserved and still show the holes where he nailed the endpoints of his chain.[<cit.>.] The fit was not perfect, but Galileo did not reject his cherished hypothesis. Instead of questioning his theory, he evidently reasoned that the error was due merely to a secondary practical aspect, namely the links of the chain being too large in relation to the measurements. Therefore he tried it with a longer chain, and found the fit to be better. In this way he evidently convinced himself that he was right after all.[<cit.>.]

The catenary case thus undermines two of Galileo's main claims to fame. First it brings his work on projectile motion into disrepute. The composition of vertical and horizontal motions that we are supposed to admire in that case looks less penetrating and perceptive when we consider that Galileo erroneously believed it to be equivalent to the vertical and horizontal force components acting on a catenary. Secondly, Galileo's reputation as an experimental scientists par excellence is not helped by the fact that his experiments in this case led him to the wrong conclusion, apparently because his love of his pet hypothesis led him to a biased interpretation of the data and a sweeping under the rug of an experimental falsification.

One may add a further complication to this account. While Galileo clearly and unequivocally stated that the catenary is a parabola in the passage quoted above, later in the same work he returns to the topic again and now speaks in less definitive terms, seemingly saying that the fit is approximate only:
The cord thus hung, whether much or little stretched, bends in a line that is very close to parabolic. … The similarity is so great that if you draw a parabolic line in a vertical plane surface … and then hang a little chain from the extremities of the base of the parabola thus drawn, you will see by slackening the little chain now more and now less, that it curves and adapts itself to the parabola and the agreement will be the closer, the less curved and the more extended the parabola drawn shall be.[<cit.>, OGG.VIII.310.]
So the catenary is expressly not a parabola, only close to it, which is the correct view. But then again Galileo immediately goes on to add that “then with a chain wrought very fine, one might speedily mark out many parabolic lines on a plane surface.”[<cit.>, OGG.VIII.310.] So now we are back to the erroneous view again. The insertion of the qualifier that the chain be “very fine” might be taken to suggest that while a typical “cord” will not be perfectly parabolic, an ideal chain with infinitesimally small links would be, which is not true.

Galileo may well have left the matter ambiguous on purpose. We can be absolutely sure that, if the catenary had in fact turned out to be equal to the parabola, then Galileo would certainly have been widely praised for having announced this fact. The passages about approximate fit could then be written off as having to do with practical limitations only, while the unequivocal statements that the catenary is parabolic could be taken as Galileo's core theoretical claim. On the other hand, now that we know that the catenary is not in fact a parabola, Galileo's defenders can twist his words to say that that is what he meant all along too: it is the parts about the deviation from a perfect fit that are the prescient and wise ones worthy of a pioneer of science, while the clear statements suggesting perfect fit are just simplifications for pedagogical purposes. Galileo left his text so ambiguous that he could always be construed as being “right” regardless of whether the facts of the matter turned out one way or the other.

§.§ Pendulum

“With regard to the period of oscillation of a given pendulum, [Galileo] asserted that the size of the arc [i.e., the height of the starting position of the pendulum] did not matter, whereas in fact it does.”[<cit.>. <cit.>, OGG.VIII.139. <cit.>.] Galileo's allegedly experimental report on pendulums in the  is clearly fabricated—or “exaggerated,” to use the diplomatic term.[<cit.>.] Mersenne did the experiment and rejected Galileo's claim.[Mersenne, Les nouvelles pensées de Galilée (1639), 72–73, <cit.>.] Galileo's friend Guidobaldo del Monte did the same, but when he told Galileo of his error Galileo rejected the experiment and insisted in his claim.[<cit.>, OGG.X.97–100.] Instead of admitting what experiments made by sympathetic and serious scientists showed, Galileo preferred to defend his false theory with “conscious deception”[<cit.>.] (more commonly known as lying).

§.§ Brachistochrone

The brachistochrone problem. The ball rolls down the fastest on a cycloidal ramp, not a circular ramp as Galileo thought. Points of the same shade correspond to the same moment in time. Based on [Sanchis , 2014].

The brachistochrone problem asks for the path along which a ball rolls down the quickest from one given point to another. Galileo believed himself to have proved that the optimal curve was a circular arc: “From the things demonstrated, it appears that one can deduce that the swiftest movement of all from the terminus to the other is not through the shortest line of all, which is the straight line, but through the circular arc.”[<cit.>, OGG.VIII.263. <cit.>.] Actually the fastest curve is not a circle but a cycloid. This was only proved in the 1690s, using quite sophisticated calculus methods.[<cit.>.] We cannot blame Galileo for not possessing advanced mathematical tools developed only half a century after his death. Nevertheless it is one more addition to his sobering pile of erroneous assertions about various physical problems. We are supposed to celebrate him for being the first to discover the parabolic path of projectile motion, and conveniently forget that at the same time he was wrong on the brachistochrone, wrong on the catenary, wrong on the shape of the isochrone pendulum, etc. With all these errors stacking up, one starts to wonder whether the one thing he got right was much more than luck. Throw up enough half-baked guesses and something's bound to stick.

§.§ Floating bodies

Archimedes studied the science of floating bodies almost two thousand years before Galileo. Archimedes's work is excellent and spot on correct. Galileo had trouble understanding the most basic aspects of it, however. Archimedes's fundamental finding is that the weight of a floating body is equal to the weight of however much water would fit into the space occupied by the part of the floating body that is below the water level. On the open sea, an equivalent way of phrasing this is that the weight of a floating body equals the weight of the water it displaces. In a small container such as a bucket or a bowl, however, Archimedes's principle cannot be phrased in terms of displaced water in this way. Figure <ref> explains why.

[Archimedes' principle.]Archimedes' principle. In open water, such as an ocean, the rise in water level when a body is submerged is negligible. Therefore the submerged volume of the body is equal to the volume of water displaced. In a closed container, however, this equality does not hold, since the water level changes appreciably when a body is submerged.

Galileo, in his early work on floating bodies, got himself confused on precisely this point. He made the elementary blunder of using the formulation of the law in terms of displaced volume of water for containers. Thus his work was “rooted in the false assumption that the volume of water that is displaced when a solid is immersed is equal to the total volume of the solid immersed whereas it is only equal to the volume of that part of the solid under the initial level of the water.” It is baffling how “the `Father of Experimental Science' could make such a mistake after some twenty years of University teaching.”[<cit.>.]

Some scholars, however, have raved that “the whole of the Scientific Revolution is encapsulated” in how Galileo handled the matter, for in this very case we can “watch our modern conception of common sense (as applied to science …) being born.”[<cit.>.] On this reading, Galileo didn't make a dumb mistake, as I claim. Instead he “discovered” an “anomaly”:[<cit.>.]
A block of wood floating in a tank displaces less than its own weight in water. … Archimedes’ principle did not apply. … Indeed, the actual water in the tank could weigh less than the block of wood it was lifting—which, according to Archimedes’ principle as traditionally understood, was impossible. (You can do this yourself by putting a small amount of water inside a wine cooler and then floating a bottle of wine in it.)[<cit.>.]
As a result of this “discovery”:
Boldly, [Galileo] had gone back, reanalysed Archimedes’ principle, and revised it. He had then tested his new theory with a quite different experiment. This back and forth movement between theory and evidence, hypothesis and experiment has come to seem so familiar that it is hard for us to grasp that Galileo was doing something fundamentally new. Where his predecessors had been doing mathematics or philosophy, Galileo was doing what we call science.[<cit.>.]
I beg to differ. Galileo may have “discovered” an “anomaly” in “Archimedes’ principle as traditionally understood,” but only because people were much too stupid to read Archimedes. The erroneous formulation of Archimedes's principle is obviously not due to Archimedes, and the foolish confusion about it would never occur to anyone who had actually read Archimedes's work with understanding. It is true that Galileo used experiment to uncover a mistake, but that mistake would not have occurred to a mathematically competent person in the first place. Galileo's use of experiment in this case was not a new innovation leading to progress in the theory of hydrostatics; rather it was a way for the mathematically inept to correct a misconception that arose from not having understood the mathematical theory.

It is self-evident that a theory of how floating bodies behave can be tested with a bucket of water and some blocks of wood. This was not a “fundamentally new” idea “born” with Galileo. It is “common sense” not only to us but to anyone who ever lived. It was just as obvious to Archimedes and the other ancient Greeks as it is to us. Archimedes did not neglect such experiments because he lacked the modern scientific method. Rather, he did not discuss such experiments because his sophisticated mathematical treatment went miles beyond the kinds of baby steps that occupied Galileo. It is a fact that Archimedes's theory of hydrostatics is excellent and correct, even when it goes into highly specific and nontrivial claims (see <ref>). It makes no sense to imagine that he somehow achieved this without scientific and experimental methods. It makes perfect sense, however, that he was more interested in the advanced mathematical theory he was developing than in explaining how childishly simple experiments can be used to avoid blockheaded misunderstandings such as that committed by Galileo.

§.§ Tides

At the end of his famous Dialogue, Galileo lists what he considers to be his three best arguments for proving that the earth moves around the sun. One of these is his argument “from the ebbing and flowing of the ocean tides,”[<cit.>.] or high and low tidal water. Galileo believed the tides were caused by the motion of the earth. This is truly one of his very worst theories, even though he was ever so proud of it.

First things first. How do the tides work? As we know today, “the ebb and flow of the sea arise from the action of the sun and the moon,” as Newton proved.[<cit.>.] The moon, and to a lesser extent the sun, pull water towards them, causing our oceans to bulge now in one direction, then the other (Figure <ref>). This was clearly understood already in Galileo's time. Kepler explained it perfectly, and many others proposed lunar-attraction theories of the tides.[<cit.>, <cit.>.] In fact, the lunisolar theory of tides is found already in ancient sources, including the causal role of the sun and moon, and descriptions of the effects in extensive and accurate detail.[<cit.>.]

The correct theory of tides.

Galileo, however, got all of this completely wrong. Why should “the tides of the seas follow the movements of the fireballs in the skies,”[Kepler to Herwart, 9–10 April 1599, <cit.>.] as Kepler had put it? Galileo considered the very notion “childish” and “occult,” and declared himself “astonished” that “Kepler, enlightened and acute thinker as he was, … listened and assented to the notion of the Moon's influence on the water.”[<cit.>, <cit.>.]
There are many who refer the tides to the moon, saying that this has a particular dominion over the waters … [and] that the moon, wandering through the sky, attracts and draws up toward itself a heap of water which goes along following it.[<cit.>.]
Yes, many indeed believed such things. And they were right. But Galileo would have none of it. This theory is not “one which we can duplicate for ourselves by means of appropriate devices.” How indeed could we ever “make the water contained in a motionless vessel run to and fro, or rise and fall”? Certainly not by moving some heavy rock located thousands of miles away. “But if, by simply setting the vessel in motion, I can represent for you without any artifice at all precisely those changes which are perceived in the waters of the sea, why should you reject this cause and take refuge in miracles?”[<cit.>.] That’s Galileo’s objection to the lunar theory of tides: It’s hocus-pocus. It assumes the existence of mysterious forces that we cannot otherwise observe or test. Proper science should be based on stuff we can do in a laboratory, like shaking a bowl of water.

[Torus.]Galileo's theory of the tides can be understood in terms of a torus.

Galileo's theory of tides.

Galileo's own theory of the tides can be described as follows (Figure <ref>). Picture a torus (Figure <ref>), laying flat on the ground, filled halfway with water. This represents the water encircling the globe of the earth. Now spin the torus in place, around its midpoint, like a steering wheel. This represents the rotation of the earth. For symmetry reasons it is clear that the water will remain equally distributed around the full circle. Because there is no reason for the surface of the water to become higher or lower in one place of the tube than another. Every part is rotating equally, everything is symmetrical, and therefore no asymmetrical distribution of water could arise from this process.

But now picture the torus sitting on a merry-go-round. The represents the earth orbiting the sun. And also at the same time it is still spinning around its own midpoint like before. Now there is asymmetry in the configuration. The torus has one part facing inward toward the middle of the merry-go-round, and one part facing outward. As far as the rotation of the torus around its midpoint is concerned, these two parts of the torus are moving in opposite directions. But as far as the rotation of the merry-go-round is concerned they are both going the same way. So one part of the torus spins along with the orbital motion, and one part against the orbital motion. Therefore one part of the water moves faster than the other. It is boosted by the merry-go-round rotation helping it along in the direction it was already going, while the other part of the torus is slowed down; the merry-go-round cancelling its efforts by going in the opposite direction.

So you have fast-moving water and slow-moving water. The fast-moving water will catch up with the slow water and pile up on top of it, creating a high tide. And the space it vacated will not be replenished because the slow water behind it isn’t keeping up. Hence the low tide. In Galileo's words:
Mixture of the annual and diurnal motions causes the unevenness of motion in the parts of the terrestrial globe. … Upon these two motions being mixed together there results in the parts of the globe this uneven motion, now accelerated and now retarded by the additions and subtractions of the diurnal rotation upon the annual revolution.[<cit.>.]
Thus, in fact, “the flow and ebb of the seas endorse the mobility of the earth.”[<cit.>.]

Unfortunately, Galileo's theory is completely out of touch with even the most rudimentary observational facts about tidal waters. High and low water occur six hours apart. In the lunisolar theory this is explained very naturally as an immediate consequence of its basic principles. The rotation of the earth takes 24 hours. There’s a wave of high water pointing toward the moon, basically (give or take a bit of lag in the system), and then another high water mark diametrically opposite to it on the other side of the earth. So that’s two highs and two lows in 24 hours, so 6 hours apiece.

Galileo's theory, on the other hand, implies that high and low water should be twelve hours apart rather than six, as he himself says: “there resides in the primary principle no cause of moving the waters except from one twelve-hour period to another.”[<cit.>.] So Galileo immediately find himself on the back foot, having to somehow talk himself out of this obvious flaw of his theory. To this end he alleges that “the particular events observed [regarding tides] at different times and places are many and varied; these must depend upon diverse concomitant causes,”[<cit.>.] such as the size, depth and shape of the sea basin, and the internal forces of the water trying to level itself out.[<cit.>.] The fact that everyone could observe two high and two low tides per day Galileo thus wrote off as purely coincidental:
Six hours … is not a more proper or natural period for these reciprocations than any other interval of time, though perhaps it has been the one most generally observed because it is that of our Mediterranean.[<cit.>.]
Galileo even mistakenly believed that he had data to prove his erroneous point, namely that tides twelve hours apart are “daily observed in Lisbon.”[<cit.>, <cit.>. OGG.V.388–389.] That is not in fact how tides behave in Lisbon. Others corrected Galileo on this point, which is why he did not include this claim in the published Dialogue. But that did not stop him from publishing the rest of his tidal theory without change.

The fact that Mediterranean and Atlantic tidal periods are identical is obviously a huge problem for Galileo's theory. The lunar theory of the tides immediately and correctly explains why these two periods are equal, and why they are both six hours specifically. According to Galileo's theory, both the equality and the six-hour period are purely coincidental. Any reasonable scientist interested in objectively evaluating these two hypotheses would find these facts significant. And Galileo clearly understood this perfectly well, which is why he was so keen to seize upon the false Lisbon data. Yet, in the published Dialogue, Galileo simply suppresses these inconvenient facts. Ever the opportunist, Galileo is very happy to highlight the importance of certain data when he believes it proves his point, only to then pretend that that data doesn't exist when it turns out it proved the opposite of what he wanted.

There is a further complication involved in Galileo's theory, which “caused embarrassment to his more competent readers.”[<cit.>.] The inclination of the earth's axis implies that the effects Galileo describe should be strongest in summer and winter. Unfortunately for Galileo’s theory, the reverse is the case. Actually the tides are most extreme in spring and fall because they receive the maximum effects of the sun’s gravitational pull.[Drake, <cit.>.] Galileo got himself confused on this point because he was again relying on false data. Galileo—the self-declared enemy of relying on textual authority, who often mocked his opponents for believing things simply because it said so in some book—was the one in this case who found in some old book the claim that tides are greatest in summer and winter, took this for fact and derived this supposed effect from his own theory.[<cit.>.]

The embarrassing mismatch between Galileo's theory and basic facts is on display in another episode as well:
Galileo … attacked [those who] postulated that an attractive force acted from the Moon on the ocean for failing to realize that water rises and falls only at the extremities and not at the center of the Mediterranean. [But his opponents] can hardly be blamed for failing to detect this [so-called] phenomenon: it only exists as a consequence of Galileo’s own theory.[<cit.>.]
In other words, Galileo was so biased by his wrongheaded theory that he used its erroneous predictions as “facts” with which to attack those who were actually right.

But all of the above is not yet the worst of it. There is an even more fundamental flaw in Galileo's theory: it is inconsistent with the principle of relativity he himself espoused. Think back to his scenario of the scientist locked in a cabin below deck of a ship that may or may not be moving.[<ref>.] Galileo's conclusion on that occasion was that no physical experiment could detect whether the ship was moving or not. But the torus tidal simulation, if it really worked as Galileo claimed, certainly could detect such a motion. If we put the torus on the floor of the cabin and spun it, one part would be spinning with the direction of motion of the ship, and another part against it. Hence high and low water should arise, by the same logic as in Galileo's tidal theory. If the ship stood still, on the other hand, no such effect would be observed, of course. So we have a way of determining whether the ship is moving, which is supposed to be impossible. And indeed it is impossible. But if that’s so then Galileo’s theory of the tides cannot possibly work because it is inconsistent with this principle.

This objection against Galileo’s theory was in fact raised immediately already by contemporary readers:
They draw attention to a difficulty raised by several members about the proposition you make that the tides are caused by the unevenness of the motion of the different parts of the earth. They admit that these parts move with greater speed when they [go] along [with] the annual motion than when they move in the opposite direction. But this acceleration is only relative to the annual motion; relative to the body of the earth as well as to the water, the parts always move with the same speed. They say, therefore, that it is hard to understand how the parts of the earth, which always move in the same way relative to themselves and the water, can impress varying motions to the water.[Jean-Jaques Bouchard to Galileo, 1633, <cit.>.]
That is to say, picture the earth moving along its orbit and also rotating around its axis at the same time. Hit pause on this animation and mark two diametrically opposite spots on the equator. Then hit play, let it run for a second or two, and then pause it again. Now, compare the new positions of the two marked spots with their original position. One will have moved further than the other. But that’s in a coordinate system that doesn’t move with the earth. That type of inequality of speed is irrelevant. What is needed to create tides is a different kind of inequality of speed of the water. Namely, a difference in speed relative to the earth itself and to the other water. Tides arise when a fast-moving part of the water catches up with a slow-moving part of the water. But that is to say, these waters are fast and slow in their speed of rotation about the earth. So inequality of speed in a coordinate system centred on the earth. But no inequality of this type arises from the motion of the earth about the sun. Galileo had no solution to this accurate objection.