Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution
GGalileo, Ignoramus: Mathematics versus Philosophy in theScientific Revolution
Viktor Blåsjö
Abstract
I offer a revisionist interpretation of Galileo’s role in the history of science. My overarching thesis isthat Galileo lacked technical ability in mathematics, and that this can be seen as directly explainingnumerous aspects of his life’s work. I suggest that it is precisely because he was bad at mathematicsthat Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His relianceon these hands-on modes of research was not a pioneering contribution to scientific method, but a lastresort 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 scientificmethod. “Those who can’t do, teach.” The vision of science articulated by Galileo was less original thanis commonly assumed. It had long been taken for granted by mathematicians, who, however, did notstop to pontificate about such things in philosophical prose because they were too busy doing advancedscientific work.
Contents a r X i v : . [ m a t h . HO ] F e b Before Galileo 79
Galileo is overrated, I maintain. I shall systematically go through all his major achievements and all stan-dard arguments as to his alleged greatness, and offer a critical counter-assessment. I go after Galileo witheverything but the kitchen sink, but I hasten to add that assembling the case against him is ultimately ameans to an end. The stakes are much higher than Galileo’s name alone. Galileo is at the heart of fun-damental questions: What is the relation between science, mathematics, and philosophy? Between ancientand modern thought? What is the history of our scientific worldview, of scientific method? Galileo is rightin the thick of the action on all of these issues. Consequently, I shall use my analysis of Galileo as a fulcrumto articulate a revisionist interpretation of the history of early modern science more broadly. But let’s startsmall, with a simple snapshot of Galileo at work.
The cycloid is the curve traced by a point on a rolling circle, like a piece of chalk attached to a bicycle wheel(Figure 1). 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. Findingareas of shapes like that is what geometers had been doing for thousands of years. Archimedes for instancefound the area of any section of a parabola, and the area of a spiral, and so on. Galileo wanted nothing morethan to join their ranks. The cycloid was a suitable showcase. It was a natural next step following upon theGreek 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 forthe life of him come up with one of those clever geometrical arguments for which the Greek mathematicianswere universally admired. All those brilliant feats of ingenuity that Archimedes and his friends had blessedus 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 theshape out of thick paper and got his scales out to have this instrument do his thinking for him. As best ashe 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. This was no way to audition for the pantheon of geometers. Galileo was left red-faced when mathemat-ically 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 gen-erating circle, even though Galileo had explicitly concluded the contrary on the basis of his little cardboarddiorama.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,” [Drake (1978), 19, 406]. Roberval in 1634. [Struik (1969), 232–238], [Whitman (1943)], [Kline (1972), 350–351]. he pleads, tacitly acknowledging his ownunmistakably inferior mathematical abilities. It is interesting to contrast this with the very different reactionto the same problem by Galileo’s contemporary René Descartes, the famous philosopher who was also a vastlybetter mathematician than Galileo. When Descartes heard of the problem he immediately wrote back to hiscorrespondent that “I do not see why you attribute such importance to something so simple, that anyonewho knows even a little geometry could not fail to observe, were he simply to look.” He then immediatelygoes 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 helacked the ability to tackle the problem as a mathematician. If Galileo could have used a more mathematicalapproach he would unquestionably have done so. It is my contention that what is so glaringly obvious in thiscase holds for Galileo’s science generally. Galileo’s celebrated use of experiments in science is not a brilliantmethodological 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, Galileogot 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,” said Huygens, and all other mathematicians with him. It had alwaysbeen obvious that mathematics and science can be explored using experiment and observation. As Galileosays: “You may be sure that Pythagoras, long before he discovered the proof . . . , had satisfied himself thatthe square on the side opposite the right angle in a right triangle was equal to the squares on the othertwo sides” —presumably by making numerical measurements on various concretely drawn triangles. Butable mathematicians had always known that haphazard trial and error had to be superseded by rigorousdemonstration for a treatise to be worth the parchment it is written on. This and only this is why you don’tsee experimental and numerical data defiling the pages of masterpieces of ancient mathematics and sciencesuch as those of Archimedes. In this case as in so many others, Galileo’s inglorious contribution to the history of thought is to cut offmathematical reasoning at the training-wheels stage, and to air in public what true mathematicians consid-ered unworthy scratch work. He experiments because his intellect comes up short. He cannot reach insightsby reason, so he turns to more simplistic, hands-on methods. In physics this ignominious shortcoming hasbeen 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 isscience being dumbed down. Galileo to Cavalieri, 24 February 1640, [Drake (1978), 406]. Cavalieri did not take up the problem—“I too left itaside” [Freguglia & Giaquinta (2016), 34]—but Torricelli solved it soon thereafter. Descartes to Mersenne, 27 May 1638,AT.II.135, [Jullien (2015), 171]. [Palmerino & Thijssen (2004), 189]. Oeuvres.XI.115. Galileo,
Dialogue , OGG.VII.75,[Wootton (2010), 85]. §8.1. .2 Mathematicians versus philosophers There are mathematicians and then there are other people. Mathematicians have always felt strongly aboutthis. Profound respect for other mathematicians and borderline contempt for anyone else is a persistenttheme in their writings. “I have no doubt that talented and learned mathematicians will agree with me,” says Copernicus with complete confidence when introducing his new sun-centered astronomical system in1543, decades before Galileo was born. Copernicus promises to make everything “clearer than day—at leastfor those who are not ignorant of the art of mathematics.” As for non-mathematicians, who cares whatthose 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 theirjudgements.” These are the kinds of people who “on account of their natural stupidity hold the positionamong philosophers that drones hold among bees.” “The studious need not be surprised if people like thatlaugh at us. Mathematics is written for mathematicians.” Johannes Kepler—the best mathematical astronomer of Galileo’s age—says the same thing. “Let all theskilled mathematicians of Europe come forward,” he implores at one point, confident that mathematicalreason compels them to speak with one voice. Those who are not “skilled mathematicians,” on the otherhand, 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 understandastronomical science, . . . I advise him that, having dismissed astronomical studies and having damnedwhatever philosophical opinions he pleases, he mind his own business and betake himself home to scratch inhis own dirt patch.” This is and always has been the worldview of mathematicians. My thesis in this essay consists in littlemore than taking them at their word. What does the history of science look like if we adopt the worldview ofthe 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 leastfor 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. Forone thing, mathematicians certainly did not need Galileo to tell them that Aristotle was wrong. Decadesbefore Galileo’s first book, Aristotle was already ridiculed by mathematicians. As a contemporary sourcenotes, “these modern mathematicians solemnly declare that Aristotle’s divine mind failed to understand[mathematics], and that as a result he made ridiculous mistakes.” “And they are right in saying so,” notedGalileo in the margin next to this passage, “for he committed many and serious mathematical blunders.” Hence Descartes’s judgement of Galileo’s science: “He is eloquent to refute Aristotle, but that is nothard.” This is essentially the thesis of my essay in so many words. It was also, I claim, the standardopinion of Galileo among mathematicians.Galileo’s supposed greatness can only be maintained by ignoring this unequivocal sentiment in the math-ematical tradition. Galileo’s claim to fame rests on the assumption that everyone but him was a ravingAristotelian. Galileo himself went out of his way to ensure this framing. His two big books are both dia-logues in which Galileo spends hundreds and hundreds of pages in back-and-forth squabbles with a fictiveAristotelian 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 feebleopposition. But that’s fish in a barrel, not a scientific revolution.No other mathematician, ancient or modern, ever engaged with Aristotelian science to anywhere nearthe same extent as Galileo, let alone devoted the entirety of their main works to refuting it. As Kepler putit: “The foolish studies of humans have come to such a pitch of vanity that no one’s work becomes famousunless he . . . either fortifies himself with the authority of Aristotle, or takes a stand in the battle againsthim, seeking to show off.” To the mathematicians this was obviously “foolish vanity” either way, becausethey considered it self-evident that Aristotelians were so ridiculous that one should simply dismiss them as [Copernicus (1995), 6]. [Copernicus (1995), 24]. [Copernicus (1995), 7]. [Copernicus (1995), 4]. [Copernicus (1995), 7]. [Kepler (1984), 157]. Kepler,
Astronomia Nova (1609), 6r, [Kepler (2015), 33]. Colombe, c. 1611, [Galileo (1957), 223]. OGG.III.253–254. Galileo, [Galileo (1957), 223]. OGG.III.253–254. Descartes to Mersenne, 11 October 1638, [Drake (1978), 390]. Kepler,
Astronomiae Pars Optica (1604), 29,[Kepler (2000), 43].
4o many “idiots” in one sentence of the introduction and leave it at that, just as Copernicus and Kepler doin their major books. Descartes did the same, calling followers of Aristotle “less knowledgable than if theyhad abstained from study.” Tycho Brahe—a prominent astronomer in the generation between Copernicusand Galileo—likewise lamented “the oppressive authority of Aristotle”: “Aristotle’s individual words areworshipped as though they were those of the Delphic Oracle.” Kepler once castigated a colleague withthe ultimate insult: “your theory will attract lecturers and philosophers.” Lest anyone mistake this for acompliment, Kepler explained what he meant: “it will offer a way out to the enemies of the physics of thesky, the patrons of ignorance.” Such was always the universal opinion of philosophers among mathematicians. Mathematically competentpeople were united and had nothing but complete contempt for Aristotelian philosophy and the like. By thetime Galileo comes along and belabours this point it has been old news for hundreds of years. As Galileohimself says: “the philosophers of our times . . . philosophize . . . as men of no intellect and little better thanabsolute fools.” Precisely. Which is why there is little value in writing several thick books hammeringhome this point and little else, which is what Galileo did. That’s just beating a dead horse as far as themathematicians are concerned.The standard view among modern scholars is very different: “Galileo’s significance for the formationof modern science lies partly in his discoveries and opinions in physics and astronomy, but much morein his refusal to allow science to be guided any longer by philosophy.” But in this refusal Galileo isjust following the mathematicians, who had always maintained that “only mathematics can provide sure andunshakeable knowledge” and that other “divisions of theoretical philosophy should rather be called guessworkthan knowledge.” The only difference is that, while competent mathematicians took this for granted andgot 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 theslightest interest in Archimedes.” This was the one thing that set Galileo apart from the philosophicalhordes of his day. “What was novel in Galileo’s approach, when contrasted with that of his Aristotelianopponents, was . . . his faith in the relevance of mathematics.” It is true that Galileo got this muchright. He had a crush on the right person. Insofar as Galileo ever accomplished anything, it stems from hisinfatuation with Archimedes. His only regret was that his own mathematical abilities were so vanishinglysmall compared to those of his hero, as he frankly admitted: “Those who read his works realise only tooclearly how inferior are all other minds compared with Archimedes’, and what small hope is left of everdiscovering things similar to those he discovered.” This is Galileo’s unique position in the history of science. He is the worst of the mathematicians andthe best of the philosophers. He is too incompetent to do mathematics but competent enough to admireit. Relative to the mathematicians he is a dud who has to spend his time spelling out obvious philosophicalpoints because he lacks the ability to do real mathematics. Relative to the philosophers he is a revelationand a watershed, articulating for the first time in philosophical prose many central tenets of the modernscientific worldview.Evaluating Galileo thus comes down to understanding the historical relation between mathematics andphilosophy. If you believe that mathematicians from Greek times onwards considered it obvious that mathe-matics encompassed an expansive, daringly creative, empirically informed, interconnected study of all quan-tifiable aspects of the world, and that philosophers talking about mathematics and science are basicallyjust derivative commentators, then Galileo is nobody. If you believe that mathematics before Galileo was atechnocratic enterprise that was intrinsically compartmentalised and limited to traditionally defined specifictasks, and that the empirical-mathematical method of modern science was a great “conceptual” revelationcoming 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. [Descartes (2008), 57]. Tycho Brahe, 1589, [Westman (1980a), 124]. Kepler to David Fabricius, 10 November1608, [Baumgardt (1951), 81]. Galileo, [Santillana (1955), 10]. [Drake (2001), xvii]. Ptolemy,
Almagest , I.1,[Toomer (1998), 36]. [Drake (1970), 10]. [Shea (1972), 11]. Galileo, [Shea (1972), 1]. OGG.I.215–216.
Opere di Galileo Galilei , Bologna, 1656.6igure 3: Galileo’s “geometric and military compass”: a slide-rule-style calculation device.
Galileo was a “Professor of Mathematics” for some twenty years. But we must not let the title fool us. Theposition had nothing to do with the research frontier in the field. In modern terms it is more comparableto that of high school teacher. Galileo taught basic and practical courses, and his lectures were unoriginaland usually cribbed from standard sources. His mathematical lectures went no further than elementaryEuclidean geometry. He also had to teach a basic astronomy course “mainly for medical students, who hadto be able to cast horoscopes.” They “needed it to determine when not to bleed a patient” and the like. Perhaps Galileo didn’t mind, for he seems to have been quite open to astrology judging by the fact thathe cast horoscopes for his own family members and friends without renumeration. Alas, he did not enjoymuch 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.” As soon as Galileo had made a name for himself he eagerly resigned his lowly university post in favourof a court appointment. This freed him from teaching duties and boosted his finances, but Galileo also hadan additional demand:I desire that in addition to the title of “mathematician” His Highness will annex that of “philoso-pher”; for I may claim to have studied more years in philosophy than months in pure mathemat-ics. 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 andin any case included what today is called science (then “natural philosophy”). But I think a more literalreading is in order. Galileo is not only declaring himself a philosopher; he is also confessing his ignorancein mathematics. Taken literally, his statement implies that he could not have spent more than two or threeyears on pure mathematics—which indeed sounds about right considering his documented mediocrity in thisfield. 7 .4 Sector
For his teaching, Galileo developed an instrument—his “geometric and military compass” or sector (Figure3)—that could be used to facilitate certain computations, somewhat like a slide rule. Is this a great testamentto his mathematical genius? Should Galileo be credited with inventing “the first mechanical computingdevice,” as one sensationalistic headline has it? Obviously not, since far more sophisticated mechanicalcomputing 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 instrumentwas original in its time. “It has always been supposed that Galileo took over a calculating device already inservice,” and various at least partial precursors are documented, but it is hard to say anything definitivesince very few instruments from this era have survived. One recent study declines to pronounce on Galileo’sdegree of originality, noting only that, at any rate, “students did not necessarily buy the compass because itwas 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.” In any caseit is undisputed that a “very similar instrument,” developed in England “entirely independently of Galileo’swork,” was extensively described in print eight years before Galileo published anything about his device. Galileo is upfront about the target audience of his sector: it is “suited to . . . first youthful studies” andan aid “for those who cannot easily manage numbers.” The fact that the latter group arguably includesGalileo himself is perhaps not coincidental. Ancient Greek folklore has it that some king found geometricalstudy too onerous and demanded to be brought to its fruits by an easier route, only to be reprimandedthat “there is no royal road to geometry.” Many a mathematician has shared this beautifully egalitariansentiment. But not Galileo, who leapt at any opportunity to kiss the feet of princes. “Not entirely improperwas the request of that royal pupil who sought . . . an easier and more open road,” Galileo declares inhis instrument manual. While Galileo encouraged this shortcut mindset, more committed mathematicianstook 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.” Thedivide remains as current as ever. Anyone familiar with modern mathematics classrooms have observed thisunmistakeable 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 seriousmathematical work that characterises his career as a whole.
Here is one of Galileo’s most quoted passages:This grand book, the universe, . . . stands continually open to our gaze. But the book cannot beunderstood unless one first learns to comprehend the language and read the letters in which it iscomposed. 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 wordof it. Like so much else Galileo gets credit for, this idea had obviously been a truism among mathematicians forthousands of years. Pythagoras had put it succinctly more than two millennia before: “All is number.” Plato agreed, saying “God is always doing geometry.” Mathematicians have lived by these words eversince. It was obvious even to for instance Grosseteste, a medieval bishop, that “the usefulness of considering [Valleriani (2001), 286–290], [Drake (1978), 34–35, 45–46]. [Drake (1978), 12, 23, 34–35, 51]. [Drake (1978),35–36]. [Heilbron (2010), 46]. [Rutkin (2018)], [Drake (1978), 55], [Heilbron (2010), 68], [Kollerstrom (2001)]. [Machamer (1998), 19]. Galileo, negotiating his Tuscan court appointment in 1610, [Galileo (1957), 64]. [Drake (1976), 104]. [Drake (1976), 106–107]. [Drake (1976)], [Galileo (1978)], [Robertson (1775), vi–x]. [Biagioli (2007), 12]. Drake, [Galileo (1978), 10]. Thomas Hood,
The Making and Use of the Geometricall Instru-ment, Called a Sector , London, 1598. [Galileo (1978), 39]. [Galileo (1978), 21]. [Galileo (1978), 41]. WilliamOughtred,
To the English gentrie, and all others studious of the mathematicks (1634), 27–28. Galileo,
Assayer (1623),[Galileo (1957), 237–238]. [Cornelli (2013), Chapter 4]. [Thomas (1993), I.387], slightly paraphrased. §8.1. Galileo’s version of this platitude is better read as a frank confession of his reactionary restriction tosimplistic mathematics. “Triangles and circles” are hardly the cutting edge. Ellipses, for instance, are passedover in silence by Galileo, not just in this quote but throughout all his works, even though Kepler had alreadydemonstrated many years before that planets move in ellipses about the sun. Unfortunately, for reasons that remain unclear or controversial, Galileo never did pay the properattention to Kepler’s writings and so ignored or neglected the elliptical nature of planetaryorbits. There is nothing “unclear” about it. The reason Galileo ignored Kepler’s writings is abundantly obvious: Hedidn’t have anywhere near the mathematical ability to understand them. Kepler had already anticipated asmuch:It is extremely hard these days to write mathematical books. . . . There are very few suitablyprepared readers. . . . How many mathematicians are there who put up with the trouble ofworking through the Conics of Apollonius of Perga? Not many indeed, and certainly not Galileo. Yet Kepler saw no other option than to write mathematics allthe same and “let it await its reader for a hundred years.” Compared to Kepler’s visionary science, Galileo’s book of nature is a children’s book sticking to babymathematics 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: ifyou don’t know math, you’re screwed. But it can also be read as a confession of an upper limit on Galileo’sown understanding: anything beyond “triangles and circles” and I’m lost. This would also explain Galileo’sequally reactionary rejection of “infinities and indivisibles, the former incomprehensible to our understandingby reason of their largeness, and the latter by their smallness.” Virtually all competent mathematiciansin Galileo’s time were eagerly exploring infinitesimal methods, prefiguring what we call calculus today. ButGalileo found all of this “incomprehensible.” When he did try to dabble with such issues his argumentswere “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.” Suchshortcomings undermine core parts of his physics, as we shall see in the next section.
Galileo’s “technique for dealing with problems involving infinitesimals . . . is incompatible with basic featuresof integration . . . and even leads, if explicated systematically, to absurd consequences.” The core issue isillustrated in Figure 4. Galileo tries to prove areas equal by “matching” the lines that make them up one byone: The aggregate of all parallels contained in the quadrilateral [ABFG is] equal to the aggregateof those included in triangle AEB. For those in triangle IEF are matched by those containedin triangle GIA, while those which are in the trapezium AIFB are common. Since each instantand all instants of time AB correspond to each point and all points of time AB, from whichpoints the parallels drawn and included within triangle AEB represent increasing degrees of theincreased speed, while the parallels contained within the parallelogram represent in the same wayjust as many degrees of speed not increased but equable, it appears that there are just as manymomenta of speed consumed in the accelerated motion according to the increasing parallels oftriangle AEB, as in the equable motion according to the parallels of the parallelogram GB. For Grosseteste, first half of 13th century, [Baur (1912), IX.59], [Grant (1974), 385]. Kepler,
Astronomia Nova (1609). [Finocchiaro (2010), 36] Kepler,
Astronomia Nova (1609), 2r, [Kepler (2015), 17]. [Kepler (1995),170]. [Galileo (1989), 34], OGG.VIII.73. [Wallace (1991), VII.76]. See also [Butts & Pitt (1978), 153–154]. [Damerow et al. (2004), 237]. The problem, as shown in Figure 4, 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 pub-lished texts indicate that he noticed this clearly nonsensical implication of the proof technique.” Many ofhis contemporaries were more alert, including Cavalieri, who “criticizes Galileo for not having emphasizedthat the indivisibles have to be taken as equidistant.” Cavalieri’s reaction is the most charitable option:Galileo neglected to “emphasise” (or even mention, in fact) some implicit assumptions that rule out the falseapplications of his method of proof.A better way of thinking about these kinds of problems is to view areas as composed of thin rectanglesrather than lines. The correct conclusion of Figure 4 works because the two areas can be sliced into thinrectangles that correspond to an equal rectangle in the other figure. The false conclusion of Figure 4, on theother hand, doesn’t go through this way, because while each rectangle “fits” in a natural place in the otherarea (just as the lines do, as illustrated), the rectangles positioned this way do not cover the full figure, butrather leave gaps between them. Galileo’s proof ought to have included some clarifications or precautions tospecify in what way his method avoids false conclusions on these kinds of grounds.If we look only at the examples in Figure 4, 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 arenot. Surely, we may be tempted to say, it is clear what Galileo must have meant, even if he may not haveexpressed it with great precision.But it gets worse. Galileo applies his “matching” technique again later, in a way that very much compli-cates the picture. He applies this method of proof when comparing the motion of a falling body to that of aball rolling down an inclined plane (Figure 5). Galileo wants to show that time it takes the ball to completethe motion is proportional to the distance it has to travel: for example, if AC = 2 AB , then the rolling ballneeds twice as long as the falling ball to reach the ground. Galileo tries to derive this result from the factthat 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 therolling 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 momentaor degrees of speed at both ends of each parallel are always matched with each other. Thus thetwo spaces AC and AB are traversed at the same degrees of speed. But . . . if two spaces aretraversed by a moveable which is carried at the same degrees of speed, then whatever ratio thosespaces have, the times of motion have the same [ratio]. Therefore the time of motion throughAC is to the time through AB as the length of plane AC is to the length of vertical AB, whichwas to be demonstrated. But again the rules of this method is unclear. From Galileo’s words, there seems to be nothing stopping usfrom applying the same proof, word for word, to the situation of a non-straight path, as in Figure 6. Butthis clearly produces false results, for in such cases the time it takes to complete the motion is no longerproportional to the distance travelled. So Galileo’s proof technique can be used to prove false results again.The example of Figure 5 makes it much harder to argue that Galileo’s method can be fixed by adding somecommon-sense implicit assumptions or restrictions. Galileo clearly thought his method could be applied evenin cases where the time-velocity diagrams have different shape and different spacing of the matched lines, asin Figure 5. In terms of the area interpretation mentioned above, this reasoning is dependent on matchingrectangles of one area to thinner rectangles of another area, but for Galileo’s reasoning to work the ratioof the thicknesses of these rectangles must remain the same throughout. This is what sets the permissibleconclusion of Figure 5 apart from the impermissible conclusion of Figure 6. 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 [Galileo (1989), 165–166], OGG.VIII.208–209. [Damerow et al. (2004), 250–251]. [Damerow et al. (2004), 251].Bonaventura Cavalieri to Galileo, June 28, 1639. OGG.XVIII.67. [Galileo (1989), 176], OGG.VIII.216–217. e l o c i t y timeuniformly accelerated motion(such as a freely falling object)uniform motion Result known long before Galileo: distance covered by uniformly accelerated motion = distance covered in same time by uniform motion at half the final speed of the uniformly accelerated motion(= this area) (= this area)
Galileo’s proof attempt: the areas are equal becausethe “parallels” of this part“match” those ofthis part
False consequence of Galileo’s logic: the areas are equal in this case toobecause the “parallels” still “match”one to oneA BFEG I
Figure 4: Galileo’s dubious way of comparing two motions.11 e l o c i t y timefalling ball AB C rolling ball
Figure 5: A falling ball compared to a ball rolling down an inclined plane. v e l o c i t y time Figure 6: Variant of Figure 5. The ramp along which the ball is rolling is no longer straight.12f rectangles. This seems to be directly incompatible with Galileo’s text. Galileo quite clearly highlightsonly the local, pairwise matching of velocities and draws his conclusion from there. There is no indicationin Galileo’s text that he considers a constant-ratio requirement essential to the legitimate application of hisproof technique.Hence we see why scholars have argued that “closer inspection of [Galileo’s] proof [of Figure 4] in the lightof [his use of this technique in the case of Figure 5] indeed makes it clear that this proof also works equallywell for nonuniformly accelerated motion,” and therefore leads to false results. Galileo often portrays himself as defeating obstinate philosophers who would rather cling to the words ofAristotle than believe empirical evidence made clear through experimental demonstration. Thus he imagineshis enemies to say things like: “You have made me see this matter so plainly and palpably that if Aristotle’stext were not contrary to it . . . I should be forced to admit it to be true.” It makes life easy for Galileoto pretend that his enemies are fools blind to facts. In reality, Galileo’s “anti-Aristotelian polemics . . . weredirected 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.” Galileo’s ploy was well calculated. It tricks many of his readers to this day into believing the fairytale ofGalileo the valiant knight singlehandedly fighting for truth in world beset by dogmatism. But “excessiveclaims for Galileo the dragon slayer have to be muted,” 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. Legend has itthat Galileo shocked the world when he dropped some balls of differing weights from the tower of Pisa andrevealed them to fall at the same speed. But the notion that it required some kind of radical conceptualinnovation by a scientific genius like Galileo to realise that one could test the matter by experiment isludicrous. Of course one can drop some rocks and see if it works: this much has been obvious to anyonesince time immemorial. Indeed, Philoponus—an unoriginal commentator—had clearly and explicitly rejectedAristotle’s law of fall by precisely such an experiment more than a thousand years before Galileo. So ifyou want to believe that experimental science and empirical verification was a radical new insight then thisputs you in quite a pickle. If that was a revolution, then why is it found for the first time in this mediocrecommentator from the 6th century? If people like Philoponus had the key to science, then why did theysit around and write commentary upon commentary on Aristotle? Their contributions to mathematics andscience is otherwise zero. How likely is it that such elementary scientific principles eluded generations of thebest mathematical minds the world has ever seen, in the age of Archimedes, only to be then discovered byderivative and subservient thinkers in an age where the pinnacle of mathematical expertise extended littlefurther 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, a result many still believe Galileo was the first todiscover. 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 fatherof modern science”? Yet here is his statement of “Galileo’s” great insight, expressed two decades beforeGalileo was even born:The custom of modern philosophers is always to believe and never to test that which they findwritten in good authors, especially Aristotle. . . . [But it would be] both safer and more delightfulto otherwise and sometimes descend to experience in some cases, as for example in the motion of [Damerow et al. (2004), 250]. [Galileo (2001), 125]. [Wallace (1991), I.377]. [Pitt (1991), 142]. E.g.
DeCaelo , I.6: “A given weight moves a given distance in a given time; a weight which is as great and more moves the same distancein a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will takehalf as long over a given movement.” [Clagett (1955), 172–173]. Lucretius,
De rerum natura , II:225–239. E.g.[Gowers (2010), 4]. The fact that run-of-the-mill humanists like Varchi were arguing this way long before Galileo shows what anobvious idea it was.It is not clear whether Galileo did in fact carry out such an experiment from the tower of Pisa when hewas teaching at the university there, as legend would have it. Personally I find the story plausible inasmuchas a field day throwing rocks surely had great appeal to a professor whose strong suit certainly wasn’t puremathematical theory. But be that as it may. It is in any case perfectly clear that many people carried outsuch experiments around that time, independently of Galileo. Simon Stevin, for example, certainly did, andpublished his results years before Galileo made his experiment, if indeed the latter ever took place. Stevindropped 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. Similarly, Mersenne was already dropping weightsout out Parisian chamber windows before he heard of Galileo’s work. “To crown the comedy, it was an Aristotelian, Coresio, who in 1612 claimed that previous experimentshad been carried out from too low an altitude. In a work published in that year he described how he hadimproved on all previous attempts—he had not merely dropped the bodies from a high window, he had goneto 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.” Girolamo Borro,another Aristotelian philosopher, similarly investigated whether lead falls faster than wood. “We took refugein experience, the teacher of all things,” he says, and hence “threw these two pieces of equal weight from arather high window of our house at the same time. . . . The lead descended more slowly. . . . Not only oncebut many times we tried it with the same results.” These philosophers were united in their emphasis onactually testing their theories practically and even in taking care to mitigate possible sources of error in themeasurements. So much for Aristotelians hating experiments. On the contrary, appeal to experiments werecommonplace long before Galileo.But these people got the result wrong, you say. Maybe they didn’t experiment at all, or if they didthey messed it up somehow. Well, so did Galileo. Galileo’s own early experiments also produced the wrongresults. “If an observation is made, the lighter body will, at the beginning of the motion, move ahead of theheavier and will be swifter,” Galileo reports, but if the fall is long enough the heavier body will eventuallyovertake it. Galileo devotes a full chapter to following this up, “in which the cause is given why, at thebeginning of their natural motion, bodies that are less heavy move more swiftly than heavier ones.” Sowhen Galileo started experimenting on this he got the wrong result, and he also believed himself to a have agood theory “explaining” those false results. We now know that the true cause for the errant results was nota theoretical one like Galileo imagined, but more likely the pedestrian circumstance that we are not good atdropping one object from each hand at the same time. Modern experiments show that people tend to dropthe lighter object sooner, “even though the subject ‘feels’ that she has released both simultaneously.” Thisis why Galileo and others ended up thinking that lead balls started out slower than lighter bodies and onlythen picked up speed.With experiments being so inconclusive, it is no wonder that Galileo relied more on a theoretical argumentin his published account. His supporters would have us believe that “Galileo showed that Aristotle’s rulecould be refuted by logic alone.” Let us listen to his argument, which is supposedly a “splendid andincontrovertible” model example of “cast-iron reductio ad absurdum reasoning.” By a short and conclusive demonstration, we can prove clearly that it is not true that a heaviermoveable is moved more swiftly than another, less heavy . . . If we had two moveables whosenatural speeds were unequal, it is evident that were we to connect the slower to the faster, thelatter would be partially retarded by the slower . . . But the two stones joined together make a Benedetto Varchi, 1544, [Settle (1983), 9]. [Drake (1978), 19–20]. [Stevin (1955), 511]. [Mersenne (1933–1988),III.274–275], [Caffarelli (2009), 249]. [Butterfield (1959), 83]. Girolamo Borro,
De motu et levium (1575), [Settle (1983),7]. Just as they did for the cycloid, §2.1. Galileo, [Drabkin & Drake (1960), 38]. Galileo, [Drabkin & Drake (1960),106]. [Settle (1983), 14]. [Drake (1999), 35]. [Frova & Marenzana (1998), 41, 46]. “From this we conclude that both great and small bodies, of the same specific gravity, are moved with likespeeds.” Furthermore, “if one were to remove entirely the resistance of the medium, all materials woulddescend with equal speed.” With this argument Galileo allegedly exposes a fundamental logical inconsistency in the Aristoteliantheory of fall. But he doesn’t. Aristotle is perfectly clear: heavier objects fall faster. So when you putthe heavy and the light together they will fall faster. The inconsistency arises only when one inserts theadditional assumption that when you put two bodies together the lighter will retard the heavier and slowit down. But there is no basis for this latter assumption in Aristotle. It is a fiction that Galileo has madeup. Only by dishonestly misrepresenting the view he is trying to refute in this way is he able to draw histriumphant 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 tiedto them they still would. Shortening the string could not change that. Hence two bricks tiedtogether end to end would fall at the same speed as either brick alone. Now throw away thestring and glue the bricks together; no reason appears why this double brick of double weightshould fall faster than two bricks tied together—or either one alone. So the real crux of the argument is the claim that two bricks held side by side should behave the same wayin terms of fall whether they are glued together or not. This is not a bad argument, but it falls short ofbeing a matter of “logic alone.” For instance, imagine you are taking a basketball free throw. You can choosebetween trying to hit the hoop with either two bricks glued together or two bricks merely held side by sideas you throw them. Would you really say that “no reason appears” why nature should treat the two casesdifferently, so you might as well go with the loose bricks? I don’t think so. Then why should this assumptionbe accepted in the case Galileo describes? Of course these two cases are quite different, but my point is thateven 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 caseof simple fall, yet that we do not have such a priori knowledge of other scenarios like that bricks thrownthrough a hoop. If one case is “logic alone” then why isn’t the other? So, if we are being honest, we are backto having to rely on experiment after all. It is not “logic” that guarantees that bricks behave the same waywhether 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 actualexperiments do not come out in accordance with his law because of air resistance. But, he says, the fit ismuch better than for Aristotle’s law. To make this point Galileo provides specific measurements of howmuch the slower ball lags behind the heavier one:Aristotle says, “a hundred-pound iron ball falling from the height of a hundred braccia hits theground before one of just one pound has descended a single braccio.” I say that they arrive atthe same time. You find, on making the experiment, that the larger anticipates the smaller bytwo 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, andspeaking only of my tiny error, remain silent about his enormous one. 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 [Galileo (1974), 66–67], OGG.VIII.107–108. [Galileo (1974), 68], OGG.VIII.109. [Galileo (1974), 75],OGG.VIII.116. [Drake (1999), 35], 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 [Drake & Drabkin (1969), 206]. [Galileo (1974), 68], OGG.VIII.109. Of course the statement attributedto Aristotle is not a literal quote but rather an inference from his law of fall. Nevertheless it remains true that Galileo’s law doesn’t fare as poorly as that of Aristotle in this exper-iment. So Galileo has indeed managed to improve on a two-thousand-year-old claim, made passingly by anon-mathematician. Aristotle himself obviously did not intend his so-called law as quantitative science. Heonly introduces it very parenthetically as a stepping-stone toward making the philosophical point that therecan be no such thing as an object of infinite weight. Aristotle has no interest in this “law” beyond drawingthis isolated metaphysical conclusion from it.Galileo’s entire case rests on his readers considering Aristotle to be a great authority. No wonder heclings to this framing and uses it as the trope of his dialogues. If we admit the truth—that Aristotle’slaw had been refuted more than a thousand years before, and that the notion of relying on Aristotle forquantitative science would never have entered the mind of any mathematically competent person in Galileo’stime—then what does Galileo have to show for himself? An unproven claim that doesn’t even fit the fakedata Galileo has specifically concocted for the purpose, let alone the many experiments that proved theopposite, including his own before he knew which way he was supposed to fudge the data. Galileo likes toportray 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 areever so ubiquitous, so that he can inflate the importance of his own contributions, the feebleness of whichwould be all too evident if he addressed actual scientists instead of straw men Aristotelians.
Here is “Galileo’s” law of fall, familiar to everyone from high school physics: ignoring air resistance, theacceleration of a freely falling object is constant. Equivalently, the velocity it acquires during its fall isproportional to time, and the distance fallen is proportional to time squared. The equivalence of thesethree formulations of the law of fall is elementary and immediately suggests itself to any mathematicallycompetent person. In modern terms we can express the three version of the law in calculus terms as y (cid:48)(cid:48) = g , y (cid:48) = gt , and y = gt / , respectively, but the straightforward equivalence of these three relationships longpredates the calculus. Galileo set out to prove as much, but since he botched his treatment of calculus-styledeductions, “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.” Motion on an inclined plane (without friction) is closely related to this, because a ball rolling down aplane will acquire the same speed as it would have in free fall through the same vertical distance. All ofthese laws for free fall and for the inclined plane were indeed clearly stated by Galileo. The equivalencewith the inclined plane is important since it makes empirical verification of the law much easier. A ballrolling 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 notbeen recognized long before the 17th century. Measurements sufficient to put the law withinsomeone’s grasp are quite simple. Equipment for making them had not been lacking—a gentlysloping plane, a heavy ball, and the sense of rhythm with which everyone is born. Relying on the sense of rhythm means that no time-keeping device is necessary. One places small groovesor bells along the ramp at intervals that the ball ought to cover in equal times according to the law of fallto be tested. The ball will make a click every time it rolls across one of the markings, so hearing whetherthe law holds is as easy as telling when a musician is off beat. Nevertheless, it is “rather unlikely” thatGalileo is truthful in his exorbitant claim “that he had obtained precise correspondence in hundreds of trialson inclined planes.” [Hill (1984), 132]. Cf. §8.7. §2.6. [Damerow et al. (2004), 281]. Descartes to Huygens, 18 February 1643,AT.III.807–808. As we would say today, in anachronistic terms, since all balls covering the same vertical distance trade inthe same amount of potential energy, they get the same amount of kinetic energy out of it. In various places in both the
Dialogue and the
Discourse , conveniently compiled and quoted in [Caffarelli (2009), 206–207]. OGG.VII: 47, 50–52, 225, 248,252. OGG.VIII: 198, 202–205, 208–209. [Drake (1989), 1]. [Drake (1978), 88–90]. [Drake (1999), 26]. The question of whether velocityis proportional to distance or to time was indeed a tricky and elusive one. Galileo too got it wrong for along time, and at one point still gets himself confused and in effect uses the wrong form even in his mature Dialogue . In any case, Beeckman independently discovered the correct law of fall, as did Harriot, both at aboutthe same time as Galileo and completely independently of him. “In his mathematical analysis of [uniformlyaccelerated] motion, Harriot discovered that this actually includes two distinct possibilities: either the ve-locity 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 con-cluded that velocity increases in proportion to time and thus arrived at the law of time proportionality, whichis correct also according to classical mechanics.” Of course he also immediately derived the equivalent formof the law that distance fallen is proportional to time squared, in keeping with our observation that thisequivalence is trivial and obvious.
All objects fall with the same acceleration, we have now concluded. But how fast is that exactly? Whatis this same universal acceleration that every object shares? The answer is well-known to any student ofhigh school physics: the constant of acceleration is g ≈ . m/sec . But Galileo messed this up. He givesvalues equivalent to less than half of the true value. According to his defenders, “clearly, round figureswere taken here in order to make the ensuing calculation simple.”
In other words, Galileo “used arbitrarydata.”
And that’s what the people trying to defend
Galileo are saying. Isn’t the law of fall supposed tobe one of Galileo’s greatest discoveries? Why did he use fake data? Why not use real data? It was readilydoable. His contemporaries did it. Why not do a little work to get the details right when you are publishingyour supposed key results in your mature treatise? Is that really too much to ask? Instead of reportingmake-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 thinkit 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 theytry to excuse his inexcusable behaviour. Mersenne put it clearly: “I doubt whether Mr Galileo has performedthe experiment on free fall on a plane, since . . . the intervals of time he gives often contradict experiment.”
Being a serious and diligent scientist, Mersenne did the work to find the correct value, unlike Galileo. Thesame goes for Riccioli, another contemporary of Galileo, who likewise did his own detailed experiments andfound that they did not at all agree with the data Galileo had made up.
Once again the historical record fails to conform to Galileo’s narrative of himself as a novel masterexperimenter refuting dogmas everyone believes in. Instead, contemporaries like Mersenne were alreadydoing free fall experiments of their own before hearing of Galileo’s, and immediately recognised Galileo’sdata as fraudulent. Galileo is not addressing actual scientists. To them, much of what he has to say isdisappointingly shallow and lacks serious scientific follow-through. Galileo’s claim to fame relies on us notknowing this, and buying into the fiction that everyone at the time was an Aristotelian who had never heardof such a thing as experiment and quantitative science. [Butts & Pitt (1978), 142], [Damerow et al. (2004), 10]. §6.3. [Damerow et al. (2004), 24–69], [Drake (1978), 100–102],[Butts & Pitt (1978), 12–13]. For example in a letter to Sarpi, 1604, OGG.X.115. In the course of the argu-ment regarding centrifugal whirling discussed in §3.9. [Chalmers & Nicholas (1983), 321] and [Hill (1984), 121, 124] makethe case that Galileo’s error in effect amounts to assuming that speed in free fall is proportional to distance rather thanto time. [Damerow et al. (2004), 10], [Clagett (1959), 417–418]. [Schemmel (2008), 238]. [Galileo (1953),223]. [Drake (1989), 69]. [Drake (1989), 70]. Mersenne,
Harmonie universelle , I.122, [Lewis (2012), 261]. [Graney (2015), 204]. §3.1.
Before praising Galileo for stating the correct law of fall, it is sobering to consider the foolish uses he put itto. One is what has been called his “Pisan Drop” theory of planetary speeds.
The planets orbit the sun atdifferent speeds. Mercury has a small orbit and zips around it quickly. Saturn goes the long way around in abig orbit and it is also moving very slowly. Galileo imagines that these speeds were obtained by the planetsfalling from some faraway point toward the sun, and then being somehow deflected into their circular orbitsat some stage during this fall (Figure 7). That supposedly explains why the planets have the speeds theydo. Galileo describes it as follows in the
Dialogue . “Suppose all the [planets] to have been created in thesame place . . . descending toward the [sun] until they had acquired those degrees of velocity which originallyseemed good to the Divine mind. These velocities being acquired, . . . suppose that the globes were set inrotation [around the sun], each retaining in its orbit its predetermined velocity. Now, at what altitude anddistance from the sun would have been the place where the said globes were first created, and could theyall have been created in the same place? To make this investigation, we must take from the most skilfulastronomers 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 cancompute “at what altitude and distance form the center of their revolutions must have been the place fromwhich they departed.” According to Galileo this shows that indeed all the planets were dropped from a singlepoint and their orbital data “agree so closely with those given by the computations that the matter is trulywonderful.” Galileo was so proud of this erroneous argument that he repeated it in his second major work, the
Discourse , as well:Our Author [Galileo] . . . may at some time have had the curiosity to try whether he could assigna determinate sublimity [i.e., distance from the sun] from which the bodies of the planets left froma state of rest, and were moved through certain distances in straight and naturally accelerated [Heilbron (2010), 116]. [Galileo (1953), 29].
Galileo omits the details though. He has one of the characters in his dialogue say that “making thesecalculations . . . 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.”
Actually there is nothing “difficult” about it, at least not to mathematically competent people. Mersenneimmediately ran the calculations and found that Galileo must have messed his up, because his scheme doesn’twork.
There is no such point from which the planets can fall and obtain their respective speeds. Galileo’sprecious idea is so much nonsense, which evidently must have been based on an elementary mathematicalerror in calculation.
Galileo tried to compute how long it would take for the moon to fall to the earth, if it was robbed of itsorbital speed. “Making the computation exactly,” according to himself, he finds the answer: 3 hours, 22minutes, and 4 seconds.
This is way off the mark because Galileo assumes that his law of fall—thatis, 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’spurpose with this calculation was to refute the claim of another scholar that the fall would take about sixdays, which is a much better value: in fact it would take the moon almost five days to fall to the earth.
That’s Galileo, the great hero of quantitative science, in action for you: bombastically claiming to refuteothers with his “exact calculations,” only to make fundamental mistakes and err orders of magnitude worsethan his opponents did.
As we have seen, Aristotelians were often as inclined to experiment as Galileo—a point obscured by Galileo’spretences to the contrary when it suited his purposes. Elsewhere it suited Galileo better to feign otherstraw men. “In one of the dialogues of Galileo, it is Simplicius, the spokesman of the Aristotelians—thebutt of the whole piece—who defends the experimental method of Aristotle against what is described as themathematical method of Galileo.”
Consider for example the question of the resistance of the medium (such as air or water) on a movingobject. 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”; insteadone must “employ reasoning at all times rather than examples,” “for we seek the causes of effects, and theseare not revealed by experience.”
Alas, Galileo’s own law as to how resistance depends on density of themedium is also “incompatible with classical mechanics” as one study puts it—a polite, scholarly way of sayingit’s wrong.
Employing some more “reasoning” along the same lines, Galileo decided that air resistance doesn’t reallyincrease appreciably with speed: “The impediment received from the air by the same moveable when movedwith great speed is not very much more than that with which the air opposes it in slow motion.”
Asurprising conclusion to modern bicyclists, among others. Yet “experiment gives firm assurance of this,”
Galileo promises. Alas, once again “the statement is false, and the experiment adduced in its support isfictitious.”
More fake data, in other words. This is quickly becoming a pattern. [Galileo (1974), 233], OGG.VIII.284. [Galileo (1953), 30].
Marin Mersenne,
Harmonie Universelle , II.103–107,[Galileo (1974), 233, note 22]. Later Newton made the same observation, [Newton (1999), 144]. [Galileo (1953), 224]. [Galileo (1953), 480]. [Butterfield (1959), 80]. [Shea (1972), 7]. OGG.I.260, 263. [Damerow et al. (2004),270]. [Galileo (1974), 226], OGG.VIII.277. [Galileo (1974), 226], OGG.VIII.277.
Drake, [Galileo (1974), 226]. .7 Inertia Newton’s law of inertia says: “Every body perseveres in its state of being at rest or of moving uniformlystraight forward, except insofar as it is compelled to change its state by forces impressed.”
Is this importantconception, 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” and that “he neglected to stateexplicitly the general inertial principle” that everyone knows today, which was instead correctly “formulatedtwo years after his death by Pierre Gassendi and René Descartes.”
The charitable interpretation of trying to attribute to Galileo some kind of “restricted law of inertia” is amurky business. According to one author who tries to do so, “in my opinion the essential core of the inertialconcept lies in the ideas . . . of a body’s indifference to motion or to rest and its continuance in the stateit is once given. This idea is, to the best of my knowledge, original with Galileo.”
You could very wellargue that that’s not really inertia at all because it doesn’t involve the straightness of the direction of themotion, nor does it explicitly say that the motion keeps going at a perpetual uniform speed. It only focusseson indifference of motion versus rest and preservation of the state of motion. But it is a dangerous gameto 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 andpreservation 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 stophere 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.
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 physicaltreatment of motion comparable to how we use inertia today. Aristotle as usual is focussed on much morephilosophical purposes. So you might say: that’s a one-off quote taken out of context which sounds muchmore modern than it really is.Indeed. But then again the same could be said for Aristotle’s so-called law of fall that Galileo refutedwith so much fanfare.
This too is only mentioned in passing very briefly and plays no systematic rolein Aristotle’s thought. Yet Galileo takes great pride in defeating this incidental remark, and his modernfans 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 ascompletely inconsequential as well. If we argue that statements such as those of Aristotle don’t count asscientific principles unless they are systematically applied to explain various natural phenomena, then wewould have to conclude that there was no Aristotelian science of mechanics at all. This, of course, wouldbe a disastrous concession to make for advocates of Galileo’s greatness, since so much of Galileo’s claim tofame 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 pointgrasped 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 retainsan impetus to continue its motion along the straight line touching the circle described by themotion of the projectile at the point of separation. . . . The projectile would continue to movealong that line if it were not inclined downward by its own weight. . . . The impressed impetus,I say, is undoubtedly in a straight line.
In Newtonian terms this would be interpreted in terms of the law of inertia. But it hardly seems Galileois thinking about it that way. He calls it “impetus.” What is “impetus” and why should we equate it withinertia? Will “impetus” run out? Is the motion caused by the “impetus” perpetual and uniform? In manyother sources at the time, “loss of impetus by projectiles was likened to . . . the diminution of sound in a
Newton,
Principia (1687), Law 1. [Newton (1999), 416]. [Drake (1964), 608]. [Drake (1964), 601]. Descartes,
Principia philosophiae (1644), II.37, 39. On Gassendi, see [Palmerino & Thijssen (2004), 150]. [Drake (1964), 606].
Aristotle,
Physics , IV.8, 215a. §3.1. [Galileo (1953), 191, 193].
This conception is perfectlycompatible, 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.”
On the contrary, he explicitlyrejects it: “Straight motion cannot be naturally perpetual.” “It is impossible that anything should have bynature the principle of moving in a straight line.”
It is easy to understand, then, why Galileo’s defendersare so eager to insist on characterising “the essential core of the inertial concept” in a way that does notinvolve 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 whichthey have neither inclination . . . or repugnance. And therefore, all external impediments beingremoved, a heavy body on a spherical surface concentric with the earth will be indifferent to restor to movement toward any part of the horizon. And it will remain in that state in which ithas once been placed; that is, if placed in a state of rest, it will conserve that; and if placed inmovement toward the west, for example, it will maintain itself in that movement. Thus a ship, forinstance, having once received some impetus through the tranquil sea, would move continuallyaround our globe without ever stopping.
Motion in a horizontal line which is tilted neither up nor down is circular motion about thecenter; . . . once acquired, it will continue perpetually with uniform velocity.
Again, as with the sling and the projectile, one can debate whether this is inertia per se. In Newtonianmechanics too a hockey puck on a spherical ice earth would glide forever in a great circle, even though thisis not inertial motion. But this agreement with Newtonian mechanics only holds if the object is preventedfrom moving downward, as the puck is by the ice, or the ship by the water. Galileo seems to have believedhorizontal inertia to hold also for objects travelling freely through the air, which is not compatible withNewtonian mechanics. For example:I think it very probable that a stone dropped from the top of the tower . . . will move, with amotion composed of the general circular movement and its own straight one.
Once again it is not entirely clear that this is supposed to represent inertia at all. It is conceivable that, inGalileo’s conception, the circular motion itself is not a force-free, default motion, but rather a motion causedor 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. Hespeaks unambiguously of “the horizontal line . . . which the projectile would continue to follow with uniformmotion if its weight did not bend it downward.”
But he does not make the same claim for projectiles firedin 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 hisdiscussion attempting to prove that the the earth’s rotational speed, no matter how much it increased, couldnot cause objects on its surface to be whirled off into space.
Galileo uses some intricate arguments to tryto prove this, which he could have avoided by a simple appeal to circular inertia if he truly believed in thelatter.In conclusion, “there is no general principle of inertia in Galileo’s work.”
He definitely never statedthe correct law of inertia. He often spoke of what appears to be a kind of circular inertia, in particularfor horizontal motion (that it to say, circular motion around the center of the earth). But altogether itis impossible to attribute to him any one consistent view on the matter. Newton and Descartes, like thegood mathematicians that they are, state concisely and explicitly what the exact fundamental assumption oftheir theory of mechanics are. Their laws of inertia are crystal clear and specifically announced to be basicprinciples upon which the rest of the theory is built. Galileo never comes close to anything of this sort. He [Drake (1970), 244]. [Koyré (1978), 175]. [Galileo (1953), 32]. [Galileo (1953), 19]. [Galileo (1957),113–114]. [Galileo (1953), 28]. See also [Coffa (1968), 265–269]. [Galileo (1953), 165]. See also [Coffa (1968), 270]. [Galileo (1953), 199].
As will be discussed further in §3.9. [Chalmers & Nicholas (1983), 329].
When teaching basic astronomy at Padua, Galileo explained to his students that Copernicus was undoubtedlywrong about the earth’s motion. The earth doesn’t move, Galileo explained. Because, if the earth moved, arock dropped from a tower would strike the ground not at its foot but some distance away, since the earthwould have moved during the fall. In support of this claim, “Galileo observed that a rock let go from the topof a mast of a moving ship hits the deck in the stern.”
This had indeed been reported as an experimentalfact by people who actually carried it out.
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 likelihoodhe simply taught the party line because it was the easiest way to pay the bills. But at least the episodedoes show that the simplistic narrative that “the experimental method” forced the transition from ancientto modern physics is certainly wrong. On the contrary, experimental evidence was among the standardarguments 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 willfall the same way relative to the ship regardless of whether the ship is standing still or travelling with aconstant 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 havewith you there some flies, butterflies, and other small flying animals. Have a large bowl of waterwith 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 allsides of the cabin. The fish swim indifferently in all directions; the drops fall into the vesselbeneath; and, in throwing something to your friend, you need throw it no more strongly in onedirection than another, the distances being equal; jumping with your feet together, you pass equalspaces in every direction. When you have observed all these things carefully (though doubtlesswhen the ship is standing still everything must happen in this way), have the ship proceed withany speed you like, so long as the motion is uniform and not fluctuating this way and that. Youwill discover not the least change in all the effects named, nor could you tell from any of themwhether the ship was moving or standing still. In jumping, you will pass on the floor the samespaces as before, nor will you make larger jumps toward the stern than toward the prow eventhough the ship is moving quite rapidly, despite the fact that during the time that you are in theair the floor under you will be going in a direction opposite to your jump. In throwing somethingto your companion, you will need no more force to get it to him whether he is in the directionof the bow or the stern, with yourself situated opposite. The droplets will fall as before into thevessel beneath without dropping toward the stern, although while the drops are in the air theship runs many spans. The fish in their water will swim toward the front of their bowl with nomore effort than toward the back, and will go with equal ease to bait placed anywhere around theedges of the bowl. Finally the butterflies and flies will continue their flights indifferently towardevery side, nor will it ever happen that they are concentrated toward the stern, as if tired outfrom keeping up with the course of the ship, from which they will have been separated duringlong 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 towardone side than the other.
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 [Heilbron (2010), 71]. [Wootton (2010), 277–278]. [Galileo (1953), 186–187].
So Galileo’s relativity argument, like somuch else he says, is old news. The primary contribution of his version is literary ornamentation—addingsome 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 todistinguish between the ship being at rest or moving with constant velocity in a straight line. But Galileoclearly 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 misconceivedidea of horizontal inertia.
This principle of relativity—the actually “Galilean” one—is of course false.
In fact it’s even worse than that. Galileo’s purpose with this passage about the ship is to argue, erro-neously, that his ship cabin experiment “alone shows the nullity of all those [arguments based on the motionsof objects such as bird and clouds and falling objects relative to the earth] adduced against the motion ofthe earth.”
That is to say, the rotation of the earth has no detectible mechanical effects, according toGalileo. But he is wrong. The earth’s rotation is in fact mechanically detectible. The Foucault pendulum isa 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 theearth was fixed or uniformly rotating, just as there is no way to detect the uniform motion of the ship frominside the cabin.So the attribution of the principle of relativity of motion to Galileo in modern textbooks is doublymistaken. First of all, relativity of motion and the idea of an inertial frame had been noted long before andwas 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 inintroducing it is to draw another false conclusion from it, namely that the earth’s rotation is undetectable.
Galileo wished to refute this ancient argument: “The earth does not move, because beasts and men andbuildings” would be thrown off.
Picture an object placed at the equator of the earth, such as a rocklying on the African savanna. Imagine this little rock being “thrown off” by the earth’s rotation. In otherwords, the rock takes the speed it has due to the rotation of the earth, and shoots off with this speed in thedirection tangential to its motion. The spectacle will be rather underwhelming at first: since the earth is solarge, the tangent line is almost parallel to the ground, and since the speed of the rock and of the earth arethe same they will keep moving in tandem. So rather than shooting off into the air like a canon ball, therock will slowly begin to hover above the ground, a few centimeters at a time. For instance, Figure 8 showshow far the rock has gotten ten minutes after being thrown off the earth. As we can see, the deviation ofthe tangential path from the curved surface of the earth is only now becoming noticeable in relation to thesize of the earth.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 theearth, it is a simple task (suitable for a high school physics test) to calculate how much the rock has risen “When a ship floats on over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement whichis an image of their own, and they think on the contrary that they themselves and all the things with them are at rest. Soit can easily happen in the case of the movement of the Earth that the whole world should be believed to be moving in acircle. Then what would we say about the clouds and the other things floating in the air or falling or rising up, except thatnot 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?”[Copernicus (1995), 17]. §3.7. [Dijksterhuis (1961), 355]. [Galileo (1953), 186]. [Galileo (1989), 220]. g ). The answer is about 4.9 meters. This is why the rocknever actually begins to levitate due to being “thrown off”: gravity easily overpowers this slow ascent manytimes over.But of course this conclusion depended on the particular size and speed and mass of the earth. We couldmake the rock fly by spinning the earth fast enough. For example, if we run the above calculations againassuming that the earth rotates 100 times faster, we find that, instead of rising a measly 1.7 centimetersabove the ground in one second, the rock now soars to 170 meters in the same time. The fall of 4.9 metersdue 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. But Galileo, alas, gets all of this horriblywrong. Even though we are supposed to celebrate Galileo as the discoverer of the law of fall, it is apparentlytoo much to ask that he work out this very basic application of it. As noted,
Galileo did not offer aserious estimate for the constant of gravitational acceleration g , unlike his contemporaries. Therefore he didnot have the quantitative foundations to carry out the above analysis, which high school students today cando in five minutes.Worse yet, Galileo maintains that no such analysis is needed in the first place, because he can “prove” thatthe rock will never be thrown off regardless of the rotational velocity. “There is no danger,” Galileo assuresus, “however fast the whirling and however slow the downward motion, that the feather (or even somethinglighter) will begin to rise up. For the tendency downward always exceeds the speed of projection.” Heeven offers us “a geometrical demonstration to prove the impossibility of extrusion by terrestrial whirling.”
Galileo’s so-called “demonstration” is shown in Figure 9.
It is indeed a qualitative argument that rulesout all possible cases of centrifugal projection, regardless of the rotational speed of the earth V , the radiusof 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 ∞ as t goes to zero. But this is obviously comparing apples to oranges, namely a velocitywith a distance. The relevant comparison is between h ( t ) and the distance d ( t ) covered by free fall in thistime. 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 → d ( t ) h ( t ) = lim t → gt / R − √ R − V t = gRV . In other words, d ( t ) does not always overpower h ( t ) , as Galileo mistakenly believes. Rather, whether d ( t ) isgreater or smaller than h ( t ) for small t depends on the specific parameters of the situation in question. Astrong gravitational acceleration g , or a big radius of the rotational path R , makes it easier for the objectto “catch up” with the surface of the earth, while a big rotational speed V makes it harder. Whether theobject catches up with the surface or flies away depends on the relation between these parameters.Galileo’s claim to fame as a “mathematiser of nature” is certainly done no favours by this episode. Hedoesn’t know how to quantify his own law of fall, and doesn’t understand basic implications of it. Hisphysical intuition is categorically wrong on a qualitative level, and worse than that of the ancients he istrying 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. A rock dropped from the top of a tower falls in a straight line to the foot of the tower. But its path of fallis not actually straight if we take into account the earth’s rotation. Seen from this point of view—that isto say, from a vantage point that doesn’t move with the rotation of the earth—what kind of path does the
By Mersenne. [Bertoloni Meli (2006), 113]. §3.3. [Galileo (1953), 197]. [Galileo (1953), 198]. [Galileo (2001), 231–234]. The errors in Galileo’s argument have been analysed by [Chalmers & Nicholas (1983)],[Hill (1984)]. F H KCcenter of earth R r a d i u s o f e a r t h EBDG I L distance travelled inertiallyin time t inertial path of object h(t) = distance between inertial pathand earth’s surface at time tv(t) = velocity aquired due to gravityin time t when falling from rest Figure 9: Galileo’s “proof” that centrifugal projection can never hurl objects off the earth. If gravity stopsacting on an object at A , it would move inertially in the tangential direction AB . Since inertial motionhas uniform speed, it would reach the equally spaced points AF HK in equal time intervals. If the objecthad instead been dropped from rest, it would have acquired a certain downward speed in those same timeintervals. These speeds are represented in the diagram by
F G , HI , KL . Since the velocity acquired in freefall is proportional to time, AGILE is a straight line. The slope of the line depends on the magnitude ofgravitational 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 ofcentrifugal projection follows, according to Galileo, from the fact that as we consider smaller and smallertime intervals (that is to say, as we zoom it at A ), the distance h ( t ) required to catch up with the earthshrinks 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.25 IBC e a r t h ’ s r o t a t i o n stone dropped from top of towerstone hits foot of towerhypothetical continued path of fall Figure 10: 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 straightone [due to gravity].” [Galileo (2001), 192], OGG.VII.191.rock trace? Galileo answers, erroneously, that it will be a semicircle going from the top of the tower to thecenter of the earth (Figure 10):If we consider the matter carefully, the body really moves in nothing other than a simple circularmotion, just as when it rested on the tower it moved with a simple circular motion. . . . Iunderstand the whole thing perfectly, and I cannot think that . . . the falling body follows anyother line but one such as this. . . . I do not believe that there is any other way in which thesethings can happen. I sincerely wish that all proofs by philosophers had half the probability ofthis one. This is “so obviously false (and besides incompatible with his own theory of uniformly accelerated motionof falling bodies) that one may wonder that Galileo did not see it himself.”
Once again Galileo doesn’tunderstand basic implications of his own law. Mersenne readily spotted Galileo’s error, whereupon Fermatobserved that the path should be a spiral (Figure 11), not a semicircle.
This would be the right answergiven Galileo’s assumptions, namely that the path is generated by composing uniform angular motion withuniformly accelerated radial motion toward the center of the earth.
This implies that the path of fallis r = r − aθ in polar coordinates, which is indeed a spiral. This is still not the true path of fall, sinceGalileo’s assumption that his law of fall remains unchanged in the interior of the earth is itself false. But Iam not concerned here with criticising Galileo on such anachronistic grounds. Much worse is the fact thathe got the wrong answer even if we grant his own assumptions.When his embarrassing error was pointed out to him, Galileo replied that “this was said as a jest, asis clearly manifest, since it is called a caprice and a curiosity.” Some defence this is! Far from offeringexonerating testimony, Galileo actually openly pleads guilty to the main charge: namely, that his science is [Galileo (2001), 192–193], OGG.VII.191. [Koyré (1955), 335]. [Koyré (1955), 336, 342],[Engelberg & Gertner (1981)], [Galileo (2001), 556].
As stated in [Galileo (2001), 192] and again later when headmitted Fermat’s correction [Koyré (1955), 343].
Galileo to Pierre Carcavy, 5 June 1637, OGG.XVII.89, [Shea (1972),135].
If Galileo truly meant his argument to be taken merely in jest, then whydid he say that he “considered the matter carefully” and “sincerely wished that all proofs by philosophershad half the probability of this one” and so on? Many of Galileo’s errors come with these kinds of bombasticclaims where Galileo is editorialising about how remarkably convincing his own arguments are. Isn’t thatconvenient? Throw out a bunch of half-baked guesses, and when they turn out right you can claim credit forstating 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 thatwas what you intended all along, even though you published it with all those extremely assertive phrasesright in the middle of your big definitive book on the subject. It’s easy to be “the father of science” if youcan count on posterity to play along with this double standard.
Pick up a rock and throw it in front of you. The path of its motion makes a parabola. That’s Galileo’sgreat discovery, right? Well, not really. Galileo does claim this but he doesn’t prove it. Even Galileo’sown follower Torricelli acknowledged this. The result is “more desired than proven,” as he says, verydiplomatically. And the reason why Galileo doesn’t prove it is a revealing one. It is due to a basic physicalmisunderstanding.The right way to understand the parabolic motion of projectiles like this is to analyse it in terms of twoindependent components: the inertial motion and the gravitational motion. If we disregard gravity, the rockwould keep going along a straight line forever at exactly the same speed. That’s the law of inertia. Butgravity pulls it down in accordance with the law of fall.
The rock therefore drops below the inertial lineby the same distance it would have fallen below its starting point in that amount of time if you had simplylet it fall straight down instead of throwing it. A staple fact of elementary physics is that the resulting pathcomposed 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 projectileis fired horizontally, such as for instance a ball rolling off a table, then Galileo does prove that it makes a [Koyré (1955), 343].
Torricelli, 1644, [Damerow et al. (2004), 275]. §3.2.
But if you throw the rock at some other angle, not horizontally, then Galileo doesn’t dare to give suchan analysis. “Although [Galileo’s]
Discorsi takes it for granted that the trajectory for oblique projection isa parabola, no derivation of this proposition is presented.” “At the point in the systematic treatmentof projectile motion in the
Discorsi where oblique projection is actually dealt with and correctly stated toyield a parabolic trajectory, there is simply a gap in the argumentation, and no derivation is offered for thisclaim.”
Galileo’s failure is quite clearly due to his not daring to believe in uniform inertial motion in any otherdirection than along the horizontal. He seems to fear that the law of inertia is perhaps not true for suchmotions. 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 butrather as gradually slowing down, like a ball struggling to roll up a hill or an inclined plane. In the lattercase the trajectory is still a parabola, though not an “upright” one. See Figure 12.In his final account, Galileo correctly “postulated upright parabolas for all angles of projection. Galileo’sreasoning for this shape is, however, untenable in classical mechanics. What is more, Galileo was unableto derive it from the consideration of two component motions.” “Galileo was . . . confronted with acontradiction between the inclined-plane conception of projectile motion and his claim that the trajectoryis an upright parabola for all angles of projection, a contradiction he was never able to resolve.”
Sincehe only trusted the horizontal case, Galileo tried to analyse other trajectories in terms of this case. To thisend he assumed, without justification, that a parabola traced by an object rolling off a table would alsobe the parabola of an object fired back up again in the same direction.
In other words, “he takes theconverse of his proposition without proving or explaining it,” as Descartes—a mathematically competentreader—immediately pointed out.Instead, “it was Galileo’s disciples who first derived the parabolic trajectory for oblique projection, al-though they present it merely as an explication of Galileo’s
Discorsi ,” which it is not.
Indeed, “even beforeGalileo’s
Discorsi appeared in print, Bonaventura Cavalieri published a derivation of the parabolic trajectorythat is consistent with classical mechanics and is not restricted to horizontal projection.”
Cavalieri wasa good mathematician, unlike Galileo. He was also Galileo’s countryman and in some sense “disciple,” andwas very generous in deferring credit to Galileo.The failures of Galileo’s treatment of projectile motion confirms his misconception that inertia is limited [Galileo (1989), 217, 221–222], OGG.VIII.269, 272–273. [Damerow et al. (2004), 237]. [Damerow et al. (2004),237]. [Schemmel (2008), 234]. [Schemmel (2008), 234]. [Galileo (1989), 245], OGG.VIII.296. [Schemmel (2008),234], [Damerow et al. (2004), 227, 236].
Descartes to Mersenne, 11 October 1638, AT.II.387. [Drake (1978), 391]. [Damerow et al. (2004), 7]. [Damerow et al. (2004), 284].
28o horizontal motion, which, as we have seen, was already independently suggested by other passages.
Some have tried to argue that “if Galileo never stated the law [of inertia] in its general form, it was implicitin his derivation of the parabolic trajectory of a projectile.”
This would have been a very good argumentif 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 ofprojectile motion are very clumsy. Galileo’s “calculations are unnecessarily complicated, and were greatlysimplified by Torricelli in . . . 1644, a complete revision and enlargement . . . which . . . makes Galileo’sdemonstrations and procedures obsolete.”
Once again Galileo’s text bears the marks of an amateurmathematician, in other words. And once again his followers almost immediately cleaned up his mess inmore mathematically able works that were full of deference to Galileo. “While . . . inspired by veneration ofGalileo, Torricelli is more logical in his treatise.”
Hence later mathematicians who used Torricelli’s betterbut reverential account rather than Galileo’s original for the mathematical details could easily be left with amuch more flattering impression of the mathematical quality of “Galileo’s” theory than if they had studiedGalileo’s own treatise in detail. Perhaps it is not so strange, then, that posterity got a bit confused andattributed much more to Galileo than he actually earned.
Galileo made no theoretical use of his theory of projectile motion. In particular, he made no connectionbetween 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 firingcannons. “He seems to have written [this theory] only to explain the force of cannon shots fired at differentelevations,” as Descartes put it. Descartes correctly denied that Galileo’s idealised theory of projectilemotion would correspond to practice. Galileo was less prudent. “In many passages Galileo remarks that thetheory of projectiles is of great importance to gunners. He made little or no distinction between his theoryand useful ballistics; he believed—though without experiment—that he had discovered methods sufficientlyaccurate within the limitations of military weapons to be capable of direct application in the handling ofartillery.”
This belief, however, was completely wrong. “Galilean ballistics were not and could not be thefruit of experimental method; experiment . . . at once reveals that projectiles do not move in a parabola.”
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 theauthority of Galileo, to which I must be partial, did not support me, I should not fail to havesome doubts about the motion of projectiles, [and] whether it is parabolical or not.
In reply, Torricelli “stated outright that his book was written for philosophers, not gunners.” “According toTorricelli [the Galilean theory] had no connection with practical gunnery or with real projectiles.”
Healso attributed this view to Galileo, but “Torricelli’s assertion that neither he nor Galileo ever believed thatthe science of motion had anything to do with practical affairs . . . is contradicted by Galileo’s letters.” “Here Galileo himself must be charged with confusion.”
This is evident for example from the extensivetables that Galileo included in his big book: ballistic range tables based on his theory. These long tablesmake no sense other than as a practical guide for firing cannons.
So clearly Galileo thought his theorywas practically viable, which it is absolutely not.
The shape of a hanging chain (Figure 13) looks deceptively like a parabola. It is not, but Galileo fell for theruse: “Fix two nails in a wall in a horizontal line . . . From these two nails hang a fine chain . . . This chain
See §3.7. [Drake (1964), 602]. [Buchwald & Fox (2013), 53]. [Hall (1952), 91].
Descartes to Mersenne,11 October 1638, [Drake (1978), 391]. [Hall (1952), 91]. [Hall (1952), 96].
Giovan Battista Renieri to Torricelli,2 August 1647, [Hall (1952), 97–98]. [Hall (1952), 98]. [Hall (1952), 100]. [Hall (1952), 100]. [Hall (1952),90].
More competent mathematicians proved him wrong: Huygens demonstratedthat the shape was not in fact parabolic.
Admittedly, his proof is from 1646, which is four years afterGalileo’s death. So one may consider Galileo saved by the bell on this occasion, since he was proved wrongnot 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 oflooking at it is that a prominent claim in Galileo’s supposed masterpiece of physics was debunked by a mereboy 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 ineither case.
Indeed, Galileo made much of this supposed equivalence and “intended to introduce the chainas an instrument by which gunners could determine how to shoot in order to hit a given target.”
Galileo also tried to test experimentally whether the catenary is indeed parabolic. To this end he drewa parabola on a sheet of paper and tried to fit a hanging chain to it. His note sheets are preserved and stillshow the holes where he nailed the endpoints of his chain.
The fit was not perfect, but Galileo did notreject his cherished hypothesis. Instead of questioning his theory, he evidently reasoned that the error wasdue merely to a secondary practical aspect, namely the links of the chain being too large in relation to themeasurements. Therefore he tried it with a longer chain, and found the fit to be better. In this way heevidently convinced himself that he was right after all.
The catenary case thus undermines two of Galileo’s main claims to fame. First it brings his work onprojectile motion into disrepute. The composition of vertical and horizontal motions that we are supposed toadmire in that case looks less penetrating and perceptive when we consider that Galileo erroneously believedit to be equivalent to the vertical and horizontal force components acting on a catenary. Secondly, Galileo’sreputation as an experimental scientists par excellence is not helped by the fact that his experiments in thiscase led him to the wrong conclusion, apparently because his love of his pet hypothesis led him to a biasedinterpretation 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 thatthe catenary is a parabola in the passage quoted above, later in the same work he returns to the topic againand 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 toparabolic. . . . The similarity is so great that if you draw a parabolic line in a vertical planesurface . . . and then hang a little chain from the extremities of the base of the parabola thusdrawn, you will see by slackening the little chain now more and now less, that it curves andadapts itself to the parabola and the agreement will be the closer, the less curved and the more [Galileo (1974), 143], OGG.VIII.186. [Bukowski (2008)]. [Galileo (1989), 256], OGG.VIII.309. [Renn et al. (2001), 118]. [Renn et al. (2001), 39]. [Renn et al. (2001), 92–104].
So the catenary is expressly not a parabola, only close to it, which is the correct view. But then again Galileoimmediately goes on to add that “then with a chain wrought very fine, one might speedily mark out manyparabolic lines on a plane surface.”
So now we are back to the erroneous view again. The insertion ofthe qualifier that the chain be “very fine” might be taken to suggest that while a typical “cord” will not beperfectly 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 thecatenary had in fact turned out to be equal to the parabola, then Galileo would certainly have been widelypraised for having announced this fact. The passages about approximate fit could then be written off ashaving to do with practical limitations only, while the unequivocal statements that the catenary is paraboliccould be taken as Galileo’s core theoretical claim. On the other hand, now that we know that the catenaryis not in fact a parabola, Galileo’s defenders can twist his words to say that that is what he meant all alongtoo: it is the parts about the deviation from a perfect fit that are the prescient and wise ones worthy of apioneer of science, while the clear statements suggesting perfect fit are just simplifications for pedagogicalpurposes. Galileo left his text so ambiguous that he could always be construed as being “right” regardless ofwhether the facts of the matter turned out one way or the other. “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.”
Galileo’sallegedly experimental report on pendulums in the
Discourse is clearly fabricated—or “exaggerated,” touse the diplomatic term.
Mersenne did the experiment and rejected Galileo’s claim.
Galileo’s friendGuidobaldo del Monte did the same, but when he told Galileo of his error Galileo rejected the experiment andinsisted in his claim.
Instead of admitting what experiments made by sympathetic and serious scientistsshowed, Galileo preferred to defend his false theory with “conscious deception” (more commonly knownas lying).
The brachistochrone problem asks for the path along which a ball rolls down the quickest from one givenpoint to another. Galileo believed himself to have proved that the optimal curve was a circular arc: “Fromthe things demonstrated, it appears that one can deduce that the swiftest movement of all from the terminusto the other is not through the shortest line of all, which is the straight line, but through the circular arc.”
Actually the fastest curve is not a circle but a cycloid. This was only proved in the 1690s, using quitesophisticated calculus methods.
We cannot blame Galileo for not possessing advanced mathematical toolsdeveloped only half a century after his death. Nevertheless it is one more addition to his sobering pile oferroneous assertions about various physical problems. We are supposed to celebrate him for being the firstto discover the parabolic path of projectile motion, and conveniently forget that at the same time he waswrong 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 morethan luck. Throw up enough half-baked guesses and something’s bound to stick.
Archimedes studied the science of floating bodies almost two thousand years before Galileo. Archimedes’swork 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 [Galileo (1989), 256–257], OGG.VIII.310. [Galileo (1989), 257], OGG.VIII.310. [Drake (1999), 26].[Galileo (1989), 97], OGG.VIII.139. [Galileo (1953), 451]. [Renn et al. (2001), 139–140].
Mersenne,
Les nou-velles pensées de Galilée (1639), 72–73, [Buchwald & Fox (2013), 209]. [Renn et al. (2001), 86], OGG.X.97–100. [Hill (1984), 131]. [Galileo (1974), 213], OGG.VIII.263. [Galileo (1953), 451]. [Blåsjö (2017), 184]. i n e p a r a b o l a c i r c l e c y c l o i d t h d e g r ee p o l y n o m i a l Figure 14: The brachistochrone problem. The ball rolls down the fastest on a cycloidal ramp, not a circularramp as Galileo thought. Points of the same shade correspond to the same moment in time. Based on[Sanchis (2014)].however much water would fit into the space occupied by the part of the floating body that is below thewater level. On the open sea, an equivalent way of phrasing this is that the weight of a floating body equalsthe weight of the water it displaces. In a small container such as a bucket or a bowl, however, Archimedes’sprinciple cannot be phrased in terms of displaced water in this way. Figure 15 explains why.Galileo, in his early work on floating bodies, got himself confused on precisely this point. He made theelementary 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 isimmersed is equal to the total volume of the solid immersed whereas it is only equal to the volume of thatpart 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.”
Some scholars, however, have raved that “the whole of the Scientific Revolution is encapsulated” in howGalileo handled the matter, for in this very case we can “watch our modern conception of common sense (asapplied to science . . . ) being born.”
On this reading, Galileo didn’t make a dumb mistake, as I claim.Instead he “discovered” an “anomaly”:
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 blockof 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 coolerand then floating a bottle of wine in it.)
As a result of this “discovery”:Boldly, [Galileo] had gone back, reanalysed Archimedes’ principle, and revised it. He had thentested his new theory with a quite different experiment. This back and forth movement betweentheory and evidence, hypothesis and experiment has come to seem so familiar that it is hard for [Shea (1972), 19]. [Wootton (2015), §8.2]. [Wootton (2015), §8.2]. [Wootton (2015), §8.2]. ater weight of floating body=weight of water that fits into submerged volume=weight of volume of water displaced Floating body in open water: water
Floating body in small container of water: weight of floating body=weight of water that fits into submerged volume ≠ water displacedoriginal water levelundisplaced waterdisplaced water
Figure 15: Archimedes’ principle. In open water, such as an ocean, the rise in water level when a body issubmerged is negligible. Therefore the submerged volume of the body is equal to the volume of water dis-placed. In a closed container, however, this equality does not hold, since the water level changes appreciablywhen a body is submerged. 33 arth MoonHigh tideLow tide
Figure 16: The correct theory of tides.us to grasp that Galileo was doing something fundamentally new. Where his predecessors hadbeen doing mathematics or philosophy, Galileo was doing what we call science.
I beg to differ. Galileo may have “discovered” an “anomaly” in “Archimedes’ principle as traditionally under-stood,” but only because people were much too stupid to read Archimedes. The erroneous formulation ofArchimedes’s principle is obviously not due to Archimedes, and the foolish confusion about it would neveroccur to anyone who had actually read Archimedes’s work with understanding. It is true that Galileo usedexperiment to uncover a mistake, but that mistake would not have occurred to a mathematically competentperson in the first place. Galileo’s use of experiment in this case was not a new innovation leading to progressin the theory of hydrostatics; rather it was a way for the mathematically inept to correct a misconceptionthat 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 andsome blocks of wood. This was not a “fundamentally new” idea “born” with Galileo. It is “common sense” notonly to us but to anyone who ever lived. It was just as obvious to Archimedes and the other ancient Greeksas 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 milesbeyond the kinds of baby steps that occupied Galileo. It is a fact that Archimedes’s theory of hydrostaticsis excellent and correct, even when it goes into highly specific and nontrivial claims (see §7.3). It makesno sense to imagine that he somehow achieved this without scientific and experimental methods. It makesperfect sense, however, that he was more interested in the advanced mathematical theory he was developingthan in explaining how childishly simple experiments can be used to avoid blockheaded misunderstandingssuch as that committed by Galileo.
At the end of his famous
Dialogue , Galileo lists what he considers to be his three best arguments for provingthat the earth moves around the sun. One of these is his argument “from the ebbing and flowing of the oceantides,” 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 fromthe action of the sun and the moon,” as Newton proved.
The moon, and to a lesser extent the sun, pullwater towards them, causing our oceans to bulge now in one direction, then the other (Figure 16). Thiswas clearly understood already in Galileo’s time. Kepler explained it perfectly, and many others proposedlunar-attraction theories of the tides.
In fact, the lunisolar theory of tides is found already in ancientsources, including the causal role of the sun and moon, and descriptions of the effects in extensive andaccurate detail.
Galileo, however, got all of this completely wrong. Why should “the tides of the seas follow the movementsof the fireballs in the skies,” 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, . . . listenedand assented to the notion of the Moon’s influence on the water.” [Wootton (2015), §15.7]. [Galileo (2001), 536]. [Newton (1999), 835]. [Palmerino & Thijssen (2004),200], [Shea (1972), 172]. [Russo (2004), 308].
Kepler to Herwart, 9–10 April 1599, [Baumgardt (1951), 51]. [Shea & Davie (2012), 356], [Galileo (1953), 462].
EarthEarth’s motionabout the sun v Fast-movingwater( v + v )Slow-movingwater( v - v ) v Figure 18: Galileo’s theory of tides.There are many who refer the tides to the moon, saying that this has a particular dominion overthe waters . . . [and] that the moon, wandering through the sky, attracts and draws up towarditself a heap of water which goes along following it.
Yes, many indeed believed such things. And they were right. But Galileo would have none of it. This theoryis not “one which we can duplicate for ourselves by means of appropriate devices.” How indeed could weever “make the water contained in a motionless vessel run to and fro, or rise and fall”? Certainly not bymoving some heavy rock located thousands of miles away. “But if, by simply setting the vessel in motion, Ican represent for you without any artifice at all precisely those changes which are perceived in the watersof the sea, why should you reject this cause and take refuge in miracles?”
That’s Galileo’s objection tothe lunar theory of tides: It’s hocus-pocus. It assumes the existence of mysterious forces that we cannototherwise observe or test. Proper science should be based on stuff we can do in a laboratory, like shaking abowl of water.Galileo’s own theory of the tides can be described as follows (Figure 18). Picture a torus (Figure 17),laying flat on the ground, filled halfway with water. This represents the water encircling the globe of theearth. Now spin the torus in place, around its midpoint, like a steering wheel. This represents the rotationof the earth. For symmetry reasons it is clear that the water will remain equally distributed around the fullcircle. Because there is no reason for the surface of the water to become higher or lower in one place of thetube than another. Every part is rotating equally, everything is symmetrical, and therefore no asymmetricaldistribution 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. Andalso at the same time it is still spinning around its own midpoint like before. Now there is asymmetry in [Galileo (1953), 419]. [Galileo (1953), 421].
Thus, in fact, “the flow and ebb of the seas endorse the mobility of the earth.”
Unfortunately, Galileo’s theory is completely out of touch with even the most rudimentary observationalfacts about tidal waters. High and low water occur six hours apart. In the lunisolar theory this is explainedvery 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 twohighs 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 ratherthan six, as he himself says: “there resides in the primary principle no cause of moving the waters exceptfrom one twelve-hour period to another.”
So Galileo immediately find himself on the back foot, havingto somehow talk himself out of this obvious flaw of his theory. To this end he alleges that “the particularevents observed [regarding tides] at different times and places are many and varied; these must depend upondiverse concomitant causes,” such as the size, depth and shape of the sea basin, and the internal forces ofthe water trying to level itself out.
The fact that everyone could observe two high and two low tides perday Galileo thus wrote off as purely coincidental:Six hours . . . is not a more proper or natural period for these reciprocations than any otherinterval of time, though perhaps it has been the one most generally observed because it is thatof our Mediterranean.
Galileo even mistakenly believed that he had data to prove his erroneous point, namely that tides twelvehours apart are “daily observed in Lisbon.”
That is not in fact how tides behave in Lisbon. Otherscorrected Galileo on this point, which is why he did not include this claim in the published
Dialogue . Butthat 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 forGalileo’s theory. The lunar theory of the tides immediately and correctly explains why these two periods areequal, and why they are both six hours specifically. According to Galileo’s theory, both the equality and thesix-hour period are purely coincidental. Any reasonable scientist interested in objectively evaluating thesetwo hypotheses would find these facts significant. And Galileo clearly understood this perfectly well, whichis why he was so keen to seize upon the false Lisbon data. Yet, in the published
Dialogue , Galileo simplysuppresses these inconvenient facts. Ever the opportunist, Galileo is very happy to highlight the importanceof certain data when he believes it proves his point, only to then pretend that that data doesn’t exist whenit 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 morecompetent readers.”
The inclination of the earth’s axis implies that the effects Galileo describe should [Galileo (1953), 427]. [Galileo (1953), 416]. [Galileo (1953), 432]. [Galileo (1953), 428]. [Galileo (1953), 428–432]. [Galileo (1953), 432–433]. [Shea (1972), 177], [Finocchiaro (1989), 128].OGG.V.388–389. [Shea (1972), 182].
36e strongest in summer and winter. Unfortunately for Galileo’s theory, the reverse is the case. Actually thetides are most extreme in spring and fall because they receive the maximum effects of the sun’s gravitationalpull.
Galileo got himself confused on this point because he was again relying on false data. Galileo—theself-declared enemy of relying on textual authority, who often mocked his opponents for believing thingssimply because it said so in some book—was the one in this case who found in some old book the claim thattides are greatest in summer and winter, took this for fact and derived this supposed effect from his owntheory.
The embarrassing mismatch between Galileo’s theory and basic facts is on display in another episode aswell: Galileo . . . attacked [those who] postulated that an attractive force acted from the Moon onthe ocean for failing to realize that water rises and falls only at the extremities and not at thecenter 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.
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’stheory: it is inconsistent with the principle of relativity he himself espoused. Think back to his scenario ofthe scientist locked in a cabin below deck of a ship that may or may not be moving.
Galileo’s conclusionon that occasion was that no physical experiment could detect whether the ship was moving or not. Butthe torus tidal simulation, if it really worked as Galileo claimed, certainly could detect such a motion. Ifwe put the torus on the floor of the cabin and spun it, one part would be spinning with the direction ofmotion of the ship, and another part against it. Hence high and low water should arise, by the same logicas in Galileo’s tidal theory. If the ship stood still, on the other hand, no such effect would be observed, ofcourse. 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 becauseit 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 makethat 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 annualmotion than when they move in the opposite direction. But this acceleration is only relative tothe annual motion; relative to the body of the earth as well as to the water, the parts alwaysmove with the same speed. They say, therefore, that it is hard to understand how the parts ofthe earth, which always move in the same way relative to themselves and the water, can impressvarying motions to the water.
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, letit run for a second or two, and then pause it again. Now, compare the new positions of the two markedspots with their original position. One will have moved further than the other. But that’s in a coordinatesystem that doesn’t move with the earth. That type of inequality of speed is irrelevant. What is needed tocreate tides is a different kind of inequality of speed of the water. Namely, a difference in speed relative tothe earth itself and to the other water. Tides arise when a fast-moving part of the water catches up with aslow-moving part of the water. But that is to say, these waters are fast and slow in their speed of rotationabout the earth. So inequality of speed in a coordinate system centred on the earth. But no inequality of thistype arises from the motion of the earth about the sun. Galileo had no solution to this accurate objection.So, to sum up, Galileo small-mindedly rejected the correct theory of the tides, based on the sun andthe moon, even though this was widely understood by his contemporaries. He then proposed a completely
Drake, [Galileo (1953), 490]. [Drake (1978), 296]. [Shea & Davie (2012), 420]. §3.8.
Jean-JaquesBouchard to Galileo, 1633, [Shea (1972), 176].
But this is a problem only if one assumes that Galileo was science personified.If we accept instead that Galileo was an exceptionally mediocre mind, who constantly got wrong whatmathematically competent people like Kepler got right, then we see that Galileo’s skeletons belong onlyto himself, not to the scientific revolution. It’s not that the scientific revolution was flawed. It’s just thatGalileo was. If we restrict ourselves to mathematically competent people then we don’t have to deal withthis kind of nonsense.
Does the earth move around the sun, or vice versa? Copernicus worked out the right answer long beforeGalileo was even born, as did the best Greek mathematicians thousands of years earlier.
Yet somehowGalileo has ended up with much of the credit:If one wonders why the Copernican theory, with almost no adherents at the beginning of theseventeenth century, had pretty much swept the field by the middle, the answer . . . is above all[Galileo’s]
Dialogue . Galileo wrote the book that won the war [and] made belief in a moving earth intellectuallyrespectable.
This may be right in a limited sense: maybe indeed the ignorant masses needed a book like Galileo’s to dumbit down for them before they could finally come to their senses. But mathematically competent people werealready convinced long before and had no use for Galileo telling them the ABCs.When Copernicus made the earth go around the sun, he confidently declared that “I have no doubt thattalented and learned mathematicians will agree with me.”
He was right. In 1600, long before Galileo hadpublished a single word on the matter, there were already over a dozen committed Copernicans. One histo-rian counted ten, besides Copernicus himself: “Thomas Digges and Thomas Harriot in England; GiordanoBruno and Galileo Galilei in Italy; Diego de Zuniga in Spain; Simon Stevin in the Low Countries; and, inGermany, the largest group—Georg Joachim Rheticus, Michael Maestlin, Christopher Rothmann, and Jo-hannes Kepler.”
This list is not complete. Gemma Frisius publicly “endorsed physical Copernicanism,” and John Feild “apparently accepted the views of Copernicus without qualifications.” “[Paolo] Sarpi’scosmology was Copernican and heliocentric” as well: although he did not explicitly declare this publicly, “theway in which Sarpi referred to Copernicus proves his acceptance of the latter’s theory.”
Another early supporter was Achilles Gasser, a contemporary of Copernicus. He praises Copernicus forhaving brought about “the restoration–or rather, the rebirth—of a true system of astronomy.” Much likeCopernicus himself, Gasser considers it a foregone conclusion that competent mathematicians will see thetruth at once, while the ignorant should be dismissed:[Copernicus] has not only demonstratively proven his theory among the mathematicians, . . . buthas also immediately been regarded as having perpetrated a heresy, and indeed—by many othersincapable of understanding this matter—is already being condemned. [Shea (1972), 186]. §7.6.
Swerdlow, [Machamer (1998), 267]. [Gingerich (2016), 68]. [Copernicus (1995), 6]. [Westman (1980a), 136]. [Tredwell & Barker (2004), 146]. [Dobrzycki (1972),192]. cite[98]Kainulainen. . . . in spite of the critical gaze of the plebs.” Ultimately, “there is no doubtthat this new thing [i.e., heliocentrism] will one day be accepted without bitterness by all educated peopleas something both agreeable and useful.”
That makes fifteen in total. Fifteen professed believers in the new astronomy, before Galileo has publishedanything. Such a number already at this early stage is not “almost no adherents.” What do you expect?How many “talented and learned mathematicians” do you think there were in pest-ridden, blood-letting,witch-burning Europe of 1600? And how many of them were interested in the Copernican question andformed an opinion on it, even though that was a philosophical question beyond the scope of the officialcomputational task of the astronomer? And, among those in turn, how many were prepared to declareallegiance to a flagrantly heretical opinion in an age where the religious thought-police routinely employedvicious torture and burnt dissenters alive? Including one person on the very list just mentioned, in fact.
So fifteen avowed Copernicans may well be regarded as quite a crowd considering the circumstances.Galileo’s own assessment of the situation in 1597 is apt indeed:I have preferred not to publish, intimidated by the fortune of our teacher Copernicus, who thoughhe will be of immortal fame to some, is yet by an infinite number (for such is the multitude offools) laughed at and rejected.
This confirms that social pressures to avoid the issue were very real—enough to “intimidate” Galileo andsurely many others. More importantly, Galileo is also making my main point for me: even at this very earlystage—long before Galileo has published a single word on the matter, and long before the invention of thetelescope—every serious astronomer has rejected the old astronomy already. By Galileo’s own reckoning,there were only “fools” left to convince. On this point he is exactly right.There were most likely a seizable number of closet Copernicans who figured “don’t ask, don’t tell” was thebest policy to avoid needless conflicts with the intolerant. “Scriptural and theological concerns, attempts atcensorship, apologies and forms of self-censorship accompanied Copernicus’s ideas from the beginning.” “The wars of religion may well have inhibited outright advocacy of heliocentrism by late sixteenth-centuryFrenchmen,” for example: it is “suggestive of a causal relationship” that initially positive attitudes towardCopernicus gave way to negative ones in step with religious strictures.
Kepler faced theological pushback for the heliocentrism of his
Mysterium cosmographicum (1597). Regard-ing one of his critics, Kepler says: “I really cannot believe that he is opposed to this doctrine [Copernicanism].He pretends [to be so] in order to appease his colleagues.” Indeed, Kepler himself recommended a similarattitude to his friends: “If anybody approaches us privately, let us tell him our opinion frankly. Publicly letus be silent.”
By some indications, William Gilbert was “a probable Copernican,” even though he stopped short ofopenly endorsing it in print.
The same goes for Gemma Frisius’s student Johannes Stadius, who “wasinclined to accept the Copernican system.”
Similarly, Harriot had a number of followers in England whowere enthusiastic believers in heliocentrism, but “the local political and intellectual milieu . . . forced theminto what we can term preventive self-censorship.”
Another example is Mersenne, who does not go onthe official list because “at no time during his life did he find any proof so overwhelming that he felt likechallenging the Church on the matter.”
And this despite the fact that he was one of the most enthusiasticreaders of Galileo’s
Dialogue . He remained uncommitted for political reasons, it seems.Another indication that many were silently receptive to Copernicanism is the fact that most of the leadingastronomers of the 16th century owned Copernicus’s book, and many of them wrote extensive notes in themargins, as was the habit at the time. Books were printed with generous margins because everyone wasexpected to take detailed notes as they read. And indeed they did. Owen Gingerich conducted a thoroughcensus of all surviving copies of Copernicus’s book. He looked at all this marginalia that this large groupof serious, competent readers of Copernicus’s book has written. That’s a group far larger than those fifteenmentioned above. Some of them were probably secretly convinced that Copernicus was right; others studied
Gasser, writing in the 1540s. [Danielson (2004), 461–462, 465].
Giordano Bruno. [Martínez (2018)].
Galileo to Kepler, 4 August 1597, [Drake (1978), 41]. [Omodeo (2014), 271]. [Baumgartner (1986),83, 85]. [Voelkel (2001), 63, 66]. [Tredwell & Barker (2004), 144, 155]. [Omodeo (2014), 156]. [Bucciantini et al. (2015), 146]. [Hine (1973), 20]. [Gingerich (2002), xvi].
But there is no need for surprise. Galileo was a poor mathematician. He did not have the patience or abilityto understand serious mathematical astronomy, let alone make any contribution to it.
Why were people convinced Copernicans before Galileo? Arguably the most compelling reason was that theCopernican system explained complex phenomena in a simple, unified way. Here are some basics observa-tional facts: The planets sometimes move forwards and sometimes backwards. Also, Mercury and Venusalways remain close to the sun. In a geocentric system, with the earth in the center, there is no inherentreason why those things should be like that. Nevertheless these are the most basic and prominent facts ofobservational astronomy. In the Ptolemaic system (Figure 19), the geocentric system of antiquity, these factsare accounted for by introducing complicated secondary effects and coordinations beyond the basic modelof simple circles. So planetary orbits are not just simple circles but combinations of circles in complicatedways that also happen to be coordinated with one another in particular patterns. In the Ptolemaic systemthere is no particular reason why these complicated constructions should be just so and not otherwise. Wehave to accept that it just happens to be that way. So Ptolemy could account for, or accommodate, thephenomena, but he can hardly have been said to have explained them. The basic idea, that planets movein circles around the earth, is on the back foot from the outset. It is inconsistent with the most basic dataand is therefore forced to add individual quick-fixes for these phenomena.In the Copernican system (Figure 20), it’s the opposite. The phenomena are here instead an immediate,natural consequence of the motion of the earth. It becomes obvious and unavoidable that outer planetsappear to stop and go backwards as the earth is speeding past them in its quicker orbit. It becomes obviousand unavoidable that Mercury and Venus are never seen far from the sun, since, like the sun, they are alwayson the “inside” of the earth’s orbit. Ad hoc secondary causes and just-so numerical coincidences in parametervalues are no longer needed to accommodate these facts; instead they follow at once from the most basicassumptions of the system.Galileo makes this point in the
Dialogue :What are we to say of the apparent movement of a planet, so uneven that it not only goes fast atone time and slow at another, but even goes backward a long way after doing so? To save theseappearances, Ptolemy introduces vast epicycles, adapting them one by one to each planet, withcertain rules for incongruous motions—all of which can be done away with by one very simplemotion of the earth.
You see, gentlemen, with what ease and simplicity the annual motion—if made by the earth—lends itself to supplying reasons for the apparent anomalies which are observed in the movementsof the five planets. . . . It removes them all. . . . It was Nicholas Copernicus who first clarified forus the reasons for this marvelous effect. [Gingerich (2004), 142, 200]. [Galileo (1989), 397]. It is not true, of course, that all epicycles can be done away with.Galileo is exaggerating and oversimplifying, as ever. [Galileo (1989), 400]. i x e d s t a r s S a t u r n J u p i t e r M a r s S u n V e n u s M e r c u r y M o o n Figure 19: Ptolemy’s system. All heavenly bodies orbit the stationary earth. S a t u r n J u p i t e r M a r s E a r t h V e n u s M e r c u r y Figure 20: Copernicus’s system. The planets orbit the stationary sun.41This alone ought to be enough to gain assent for the rest of the [Copernican] doctrine from anyone who isneither stubborn nor unteachable.”
This is all good and well. Galileo is absolutely right. But this was a core point of Copernicanism fromthe outset, that was already a hundred years old and common knowledge by the time Galileo repeated it.Copernicus himself made the point clearly. He challenged those who deny heliocentrism to explain whyMercury and Venus always stay close to the sun: “what cause will they allege why these planets do notalso make longitudinal circuits separate and independent of the sun, like the other planets”?
Similarly,retrograde motion of the outer planets arises because “the movement of the Earth is speedier, so that itoutruns the movement of the planet.”
Thus the motion of the earth gives a unified explanation for whatthe deniers of heliocentrism must attribute to incidental properties of individual planetary models: “And soit is once more manifest that all these apparent movements—which the ancients were looking into by meansof the epicycles of the individual planets—occur on account of the movement of the Earth.”
Thus is itstriking “what power and effect the assumption of the revolution of the Earth has in the case of the apparentmovement in longitude of the wandering stars [i.e. planets] and in what sure and necessary order it placesall the appearances. . . . Accordingly by means of the assumption of the mobility of the Earth we shall dowith perhaps greater compactness and more becomingly what the ancient mathematicians thought to havedemonstrated by means of the immobility of the Earth.”
It is instructive, however, to compare this with Galileo’s theory of tides.
The correct, lunisolar theoryof the tides explains the basic phenomena in a simple and natural way as immediate consequences of the firstprinciples of the theory. That’s exactly the same point that we made about the Copernican system. Thecorrect theory of the tides thus has the same kind of credibility as the Copernican system. So, by Galileo’slogic, this “ought to be enough to gain assent.” But in the case of the tides it seems Galileo was the “stubbornand unteachable” one. He insisted on a theory which—like that of Ptolemy—could only account for basicfacts by invoking arbitrary and unnatural secondary causes unrelated to the primary principles of the theory.It’s a sign that your theory has poor foundations if the foundations themselves are good for nothing andall the actual explanatory work is being done by emergency extras duct-taped on later to specifically fixobvious problems with the foundations. Intelligent people realised this, which is why they turned to thesun-centered view of the universe. Galileo paid lip service to the same principle when he wanted to ride onthe coattails of their insights. But, if he had been consistent in his application of this principle, he shouldhave used it to reject his foolish theory of the tides.
If you move from one side of room to another, your view of everything on the walls will change. The wallyou are approaching will appear to “grow,” while the wall behind you will shrink and occupy a smaller partof your field of vision. This is called parallax.A major problem with Copernicus’s theory is the absence of stellar parallax in the course of a year. If theearth moves in an enormous circle around the sun, we should be at one moment close to some constellationof stars, and then half a year later much further away from them. Hence we should see them sometimes bigand “up close,” and sometimes shrunk into a small area, like a faraway wall at the end of a long corridor.But this does not happen. The night sky is immutable. As far as 17th-century astronomers could detect,the constellations all look exactly the same throughout the year, just as if we never move an inch.To maintain the earth’s motion in spite of this, it is therefore necessary to postulate, as Copernicus does,that “the fixed stars . . . are at an immense height away.”
The diameter of the earth’s orbit is so smallin relation to such an astronomical distance that our feeble little motion is all but tantamount to standingstill. That is why no parallax can be detected. This is the correct explanation, as we now know, but in the16th century it didn’t sound too convincing.Tycho Brahe was one of the sceptics. He was the most exacting astronomical observer in the pre-telescopic era, but even he, with his very advanced and precise observations, could find no parallactic effect. [Galileo (1989), 398]. [Copernicus (1995), 21]. [Copernicus (1995), 240]. [Copernicus (1995), 240]. [Copernicus (1995), 311]. The force of this argument was acknowledged by Tycho Brahe, among others; see §4.3. §3.17. [Copernicus (1995), 27].
42e calculated that, if Copernicus was right, the stars would have to be at least 700 times more distantthan Saturn for this to happen. The universe would not have been designed with so much wasted space, hereasoned.
Tycho Brahe therefore devised a system of his own (Figure 21), in which the earth remained the center ofthe universe, while the planets orbit the sun. This removed the problem of parallax. Moreover, the argumentsfor the Copernican system based on explanatory simplicity (§4.2) apply equally well to the Tychonic system.Tycho indeed acknowledged that Copernicus “expertly and completely circumvents all that is superfluousor discordant in the system of Ptolemy.”
As we have seen, Galileo repeated this old argument in his
Dialogue as proof for the Copernican hypothesis, while conveniently neglecting to acknowledge that seriousmathematical astronomers from Tycho onwards had already reconciled the full force of this argument withgeocentrism.In fact, Tycho’s system is equivalent to the Copernican one as far as the relative positions of the heavenlybodies are concerned. Tycho and Copernicus describe the same planetary motions, but they choose a differentreference point in terms of which to describe them. Kepler illustrates the point with an analogy: the samecircle can be traced on a piece of paper by either rotating the pen arm of a compass around the fixed leg,or by keeping the compass fixed while rotating the paper underneath it.
Because of this equivalence, theCopernican and Tychonic systems are by necessity on equal footing as far as the simplicity arguments of§4.2 are concerned.One might feel that the Tychonic system is less physically plausible than those of Ptolemy or Copernicus.Indeed, traditional conceptions had it that planets were enclosed in translucent crystalline spheres, like thelayers of an onion. Both the Ptolemaic and Copernican systems are basically compatible with such an “onion”conception of the cosmos. The Tychonic system clearly is not: planets are crossing each other’s “orbs” allover the place. But Tycho had some good counterarguments to this.
By a careful study of the paths ofcomets, he proved that they evidently passed through the alleged crystalline spheres with ease. Furthermore,he pointed out that these alleged spheres did not refract light, as glass or other materials had been knownto do since antiquity.All in all, Tycho’s system was a serious scientific theory with good arguments to its credit. This is anotherreason why our headcount of Copernicans above is misleading.
The number of people who rejected thePtolemaic system was certainly greater than the number of outright Copernicans, and the middle road ofTycho was by no means blind conservatism but rather a viable system based on the latest mathematicalastronomy.Galileo, however, liked to pretend otherwise. The full title of his famous book reads:
Dialogue Concerningthe Two Chief World Systems: Ptolemaic and Copernican . It certainly made Galileo’s life a lot easier toframe his fictional debate with fictional opponents in those antiquated terms. That way he could battletwo-thousand-year-old ideas instead of having to engage with the latest mathematical astronomy.More serious and mathematically competent people had a very different view of which were “the twochief world systems.” Around 1600, long before Galileo enters the fray, Kepler considers it obvious that thePtolemaic system is obsolete:Today there is practically no one who would doubt what is common to the Copernican andTychonic hypotheses, namely, that the sun is at the centre of motion of the five planets, andthat this is the way things are in the heavens themselves—though in the meantime there is doubtfrom all sides about the motion or stability of the sun.
Later, after the telescope has brought its new evidence, not much has changed. Kepler is a bit more assuredthat “today it is absolutely certain among all astronomers that all the planets revolve around the sun.”
But the battle between Copernicus and Tycho remained far from settled: “. . . the hypothesis not only ofCopernicus but also of Tycho Brahe, whereof either hypotheses are today publicly accepted as most true,and the Ptolemaic as outmoded.” “The theologians may decide which of the two hypotheses . . . —that [Siebert (2005), 253].
Tycho, [Duhem (1969), 96].
Kepler,
Harmonices Mundi (1619), V, [Kepler (1995), 175]. [Kepler (1995), 16]. §4.1. [Kepler (1984), 147].
Kepler,
Harmonices Mundi (1619), V, [Kepler (1995), 175].
Kepler,
Harmonices Mundi (1619), V, [Kepler (1995), 169]. i x e d s t a r s Figure 21: Tycho Brahe’s system. The planets orbit the sun, while the sun orbits the stationary earth.44f Copernicus or that of Brahe—should henceforth be regarded as valid [for] the old Ptolemaic is surelywrong.”
Nor was this merely Kepler’s opinion. Historians who study much more minor figures have also concludedthat, indeed, “the Ptolemaic system already had been set aside, at least among mathematical astronomers,” well before Galileo wrote his
Dialogue . But Galileo, in his great book, like a schoolyard bully secretly tooscared to pick on someone his own size, preferred to pretend that the old Ptolemaic system was still theenemy of the day. To be sure, there were still a “multitude of fools” left to convince, and perhaps indeedGalileo did so more effectively than anyone else. But that proves at most that Galileo should be praised asa populariser, not as a scientist. To mathematically competent astronomers he was beating a dead horse.
Some scholars have made too much of the above arguments in favour of the Tychonic system, however. Theyhave concluded that “it is fair to say that, contrary to [the standard view], science backed geocentrism.”
If you take the works of Galileo to be the extent of “science” then this conclusion is indeed defensible. Butthe conclusion is false if you include genuinely talented scientists like Kepler. Unlike Galileo, Kepler dared totake on the Tychonic system and he gave a long list of compelling arguments against it.
Tycho’s system isequivalent to the Copernican one in terms of relative position of the planets, but “that Copernicus is betterable than Brahe to deal with celestial physics is proven in many ways.”
Let us sample just two of them.One is based on “the magnitude of the moveable bodies”:For just as Saturn, Jupiter, Mars, Venus, and Mercury are all smaller bodies than the solar bodyaround which they revolve; so the moon is smaller than the Earth . . . [and] so the four [moons]of Jupiter are smaller than the body of Jupiter itself, around which they revolve. But if the sunmoves, the sun which is the greatest . . . will revolve around the Earth which is smaller. Thereforeit is more believable that the Earth, a small body, should revolve around the great body of thesun.
This is of course completely correct also from a modern point of view. It is also verified by elementaryexperience. For instance, take a lead ball and a ping-pong ball, and tie them together with a string. If youflick the ping-pong ball it will start spinning around the lead ball. But if you flick the lead ball it will rollstraight ahead without any regard for the ping-pong ball, which will simply be dragged along behind it. Sothe lighter object adapts its motion to the heaver one but not conversely, just as Kepler says. The planetsare not tied to the sun with a string but the point generalises. You can observe the same principle with abig and a small magnet for example: the little one is moved by the bigger, not the other way around.More generally, the Copernican picture is more readily susceptible to a mechanical explanation of plan-etary motion than the Tychonian picture. Such mechanical explanations were eagerly sought in the 17thcentury. Slings and magnets were powerful analogies for trying to conceptualise the kinds of forces involved inheavenly motions.
Another popular conception was that the planets were carried along in their orbits bysome sort of vortex made up of invisible particles.
Whichever mechanical model one favours, Copernicancosmology is much more naturally explained in mechanical terms than a Tychonic cosmos.Another of Kepler’s arguments against Tycho has to do with the motion of the planets in latitude, that is to say “up and down.” Suppose you are in a dark room, and someone has place a glowing object ina sling and is whirling it around you. If you are asked to describe the motion you will notice first of all thatit is a circular motion. But it may also be the case that the orbit of the object is not exactly parallel tothe floor. You will therefore notice that the object is higher is some part of its orbit and lower in anotherpart. This is the motion in latitude. The orbit of the object in the sling lies in a single plane, but it isinclined with respect to your plane of reference, which is a plane parallel to the floor through your eyes. If
Kepler, 1619, [Baumgardt (1951), 136]. [Magruder (2009), 208]. See also [Bucciantini et al. (2015), 210–211]. [Graney (2015), 132].
Astronomia Nova (1609), 3r–3v, [Kepler (2015), 20–23],
Epitome Astronomiae Copernicanae (1620), IV.5, [Kepler (1995), 71–76].
Kepler,
Astronomia Nova (1609), 3r, [Kepler (2015), 21].
Kepler,
EpitomeAstronomiae Copernicanae (1620), IV.5, [Kepler (1995), 73].
Kepler’s
Astronomia Nova (1609) gives a quasi-magnetic ac-count of planetary motion in great detail.
Descartes,
Traité du monde et de la lumière (1633).
Epitome AstronomiaeCopernicanae (1620), IV.5, [Kepler (1995), 72]. inherently belong the motion of theeach planet. This is physically less plausible than the simple Copernican explanation. It is easy to imaginephysical processes that produce circular motions that take place in a single plane. This happens for examplewith an object in a sling or a point on a rotating solid. But it is much more difficult to imagine a physicalprocess that could move an object in a circle and also make it bob up and down in a systematic way at thesame time.This is just two of Kepler’s many arguments against Tycho. We need not discuss them all. My pointhere is only that from the low quality of Galileo’s case for Copernicus one cannot infer that “big surprise:in 1651 the geocentric cosmology had science on its side.”
The mistake in this argument is the implicitassumption that Galileo represents “science.” Galileo was an opportunistic populariser. If you want to knowthe state of serious science at the time you have to read instead mathematicians like Kepler.
The year is 1609. Galileo is well into his middle age; a frail man not infrequently bedridden with rheumaticor arthritic pains. Had he died from his many ailments, in this year, at the age of 45, he would have beenall but forgotten today. He would have been an insignificant footnote in the history of science, no morememorable than a hundred of his contemporaries. It has often been said that mathematics is a young man’sgame. Newton had his annus mirabilis in his early twenties—“the prime of my age for invention,” as helater said. Kepler was the same age when he finished his first masterpiece, the
Mysterium Cosmographicum (1596). Galileo was already nearly twice this age, and he had nothing to show for it but some confused pilesof notes of highly dubious value. In short, as a mathematician the ageing Galileo had proved little excepthis own mediocrity.It is this run-of-the-mill nobody that first hears of a new invention: the telescope. Here was his chance. Heonly had to point this contraption to the skies and record what he saw. No need anymore for mathematicaltalent or painstaking scientific investigations. For twenty years he had tried and failed to gain scientific famethe hard way, but now a bounty of it lay ripe for the plunder. All you needed was eyes and being first.Galileo first heard about the mysterious new “optical tube for seeing things close” in July 1609. A weekor two later a traveller offered one for sale in Padua and Venice at an outrageous price—about twice Galileo’syearly salary.
The enterprising salesman found no takers, but the sense of opportunity remained in theair. And it was an opportunity tailor-made for Galileo: finally a path to scientific fame that required onlyhandiwork and none of that tiresome thinking in which he was so deficient.The design of telescopes was still a trade secret among the Dutch spectacle-makers who had stumbledupon the discovery. But acting fast was of the essence. Making a basic telescope is not rocket science. Soonmany people figured out how to make their own. “It took no special talent or unique inventiveness to comeup with the idea that combining two different lenses . . . would create a device allowing people to see farawayobjects enlarged.”
Reading glasses and magnifying glasses were already in common use: they obviouslymade text appear bigger, so it was not a far-fetched idea to use them to magnify more distant objects aswell. And the external shape of the telescopes people reported seeing suggested that at least two lenses
Owen Gingerich, back cover blurb of [Graney (2015)]. [Drake (1978), 139–140]. [Bucciantini et al. (2015), 22].
It didn’t take a genius, therefore, to soon strike upon the simple recipeGalileo found: take one convex and one concave lens and stick them in a tube, and look through the concaveend. That’s it. No theoretical knowledge of optics played any part in this; it was purely a matter of hands-oncraftsmanship and trial-and-error.
About a month after first hearing of the telescope, Galileo has managed to build his own, with × magnification, and begins a campaign to leverage it into a more lucrative appointment for himself. Hegives demonstrations to various important dignitaries—“to the infinite amazement of all,” according tohimself—and enters multiple negotiations about improved career prospects. Between hands-on optical trialsand lens grinding, showmanship demonstrations, shrewd self-marketing, and juggling potential job offers,Galileo must have had a busy couple of months indeed. On top of this his regular teaching duties at theuniversity were just starting again in the fall.We can easily understand, therefore, why the scientific importance of the new instrument for astronomywas not realised right away. At first neither Galileo nor anyone else thought of the telescope as primarily anastronomical instrument. Galileo instead tried to market it as “a thing of inestimable value in all businessand every undertaking at sea or on land,” such as spotting a ship early on the horizon. But the moonmakes an obvious object of observation, especially at night when there is little else to look at. Perhapsindeed moon observations were part of Galileo’s sales pitch routine more or less from the outset, thoughas a gimmick rather than science.But this was soon to change. In the dark of winter, the black night sky is less bashful with its secretsthan in summer. It monopolises the visible world from dinner to breakfast; it seems so eager to be seen thatit would be rude not to look. In January, Galileo takes up the invitation and spots moons around Jupiter. This changes everything. Suddenly it is clear that the telescope is the key to a revolution in astronomy.Eternal scientific fame is there for the taking for whoever is the first to plant his flag on the shores of thisterra incognita. For the next two months Galileo goes on a frenzied race against the clock. He writes duringthe day and raids of the heavens for one precious secret after another at night. In early March he has cobbledtogether enough to claim the main pearls of the heavens for himself. He rushes his little booklet into printwith the greatest haste: the last observation entry is dated March 2, and only ten days later the book iscoming off the presses —a turnaround time modern academic publishers can only dream of, even thoughthey do not have to work with hand-set metal type and copper engravings for the illustrations.“I thank God from the bottom of my heart that he has pleased to make me the sole initial observer ofso many astounding things, concealed for all the ages.”
So wrote Galileo, and his palpable relief is fullyjustified. Little more than dumb luck—or, as he would have it, the grace of God—separated Galileo from nu-merous other telescopic pioneers who also produced telescopes and made the same discoveries independentlyof Galileo, such as Simon Marius who discovered the moons of Jupiter one single day later than Galileo. “Adelay of only three or four months would have set him behind several of his rivals and undercut his claimto priority regarding several key discoveries with the telescope.”
Perhaps it was not the grace of God,but Galileo’s desperation, born of decades of impotence as a mathematician, that drove him to publish first.Being incapable of making any contribution to the mathematical side of science and astronomy, he perhapsneeded and craved this shortcut to stardom more than anyone else.Accordingly, Galileo greedily sought to milk every last drop of fame he could from the telescope. “I donot wish to show the proper method of making them to anyone,” he admitted; rather “I hope to win somefame.”
His competitors quickly realised that “we must resign ourselves to obtaining the invention withouthis help.”
Still six years after his booklet of discoveries, people who thought science should be a sharedand egalitarian enterprise were rightly upset by Galileo’s selfish quest for personal glory: “How long will youkeep us on the tenterhooks? You promised in your
Sidereal Message to let us know how to make a telescopeso that we could see all the things that are invisible to the naked eye, and you haven’t done it to the present
It is generally believed that Galileo had not seen a telescope before he made his own, but it is possible that he mighthave. [Biagioli (2010)]. If not, he was relying on eye-witness accounts of others.
As Galileo himself in effect says in the
Assayer (1623), [Galileo (1957), 245].
Galileo, letter to his brother-in-law, [Drake (1978), 141].
Galileo to Donà,24 August 1609, [Heilbron (2010), 149]. [Drake (1978), 142]. [Drake (1978), 146]. [Bucciantini et al. (2015),77], [Drake (1978), 157].
Galileo to Vinta, 30 January 1610, [Heilbron (2010), 153]. [Baumgartner (1988), 171]. [Galileo (1957), 61].
Peiresc to Pace, 7 November 1610, [Bucciantini et al. (2015), 156].
Galileo never missed a chance to mock stuffy Aristotelian professors for thinking “that truth is tobe discovered, not in the world or in nature, but by comparing texts (I use their own words).”
But if hegenuinely wanted them to turn to nature he could have shared his techniques for telescope construction. Intruth it served his own interests very well that these people were left with no choice but “comparing texts”while he claimed the novelties of the heavens for himself.
Some believed “the telescope carries spectres to the eyes and deludes the mind with various images . . . be-witched and deformed.”
Perhaps these peculiar “Dutch glasses” were but a cousin of the gypsy soothsayer’scrystal ball? The “transmigration into heaven, even whil’st we remain here upon earth in the flesh,” mayindeed seem like so much black magic. Add to this the very numerous imperfections of early telescopes,which often made it very difficult even for sympathetic friends to confirm observations, not to mention gaveample ammunition to outright sceptics.Indeed, we find Galileo on the defensive right from the outset, just a few pages into his first booklet. Seenthrough the telescope, the moon appeared to have enormous mountains and craters but its boundary wasstill perfectly smooth. “I am told that many have serious reservations on this point”: for if the surface of themoon is “full of chasms, that is countless bumps and depressions,” then “why is the whole periphery of thefull Moon not seen to be uneven, rough and sinuous?”
Galileo replies that this is because the Moon hasan atmosphere, which “stop[s] our sight from penetrating to the actual body of the Moon” at the edge only,since there “our visual rays cut it obliquely.”
So when we look at the edge of the moon our line of sightspends more time passing through the atmosphere of the moon and that’s why it’s blurred. It is therefore“obvious,” says Galileo, that “not only the Earth but the Moon also is surrounded by a vaporous sphere.”
This is of course completely wrong.Another puzzling fact was that the planets were magnified by the telescope, but not the stars: theyremained the same point-sized light spots no matter what the strength the telescope. Some even mistookthis for a “law that the enlargement appears less and less the farther away [the observed objects] are removedfrom the eye.”
Galileo tried to explain the matter, but once again he gets it completely wrong.
Clearly, in light of all these challenges to the reliability and consistency of the telescope, it was importantto understand its basis in theoretical optics. That is why, presumably, Galileo felt obliged to swear at theoutset that “on some other occasion we shall explain the entire theory of this instrument.”
To thoseaware of his mathematical shortcomings, it will come as no surprise that Galileo never delivered on thispromise. Kepler—a competent mathematician—took up the task instead, and in the process came up witha fundamentally new telescope design better than that of Galileo. “Galileo kept silent [on Kepler’s workon optics]. Apparently, this self-styled mathematical philosopher was not interested in the mathematicalproperties of the instrument that had brought him fame.”
Galileo’s excuse for ignoring Kepler’s excellentwork of the optics of lenses was that it was allegedly “so obscure that it would seem that the author did notunderstand it himself.” A modern scholar comments that “this is a curious statement since the
Dioptrice ,unlike other works by Kepler, is remarkably straightforward.”
Still not straightforward enough for Galileo,evidently. Indeed, Galileo’s naive conception of optics was still mired in the old notion that seeing involvedrays of sight spreading outward from the eye rather than conversely.
Malatesta Porta to Galileo, 13 September 1616, [Shea & Davie (2012), 398], OGG.XII.281. [Shea & Davie (2012),419]. OGG.X.422. [Drake & O’Malley (1960), 80].
Robert Hooke,
Micrographia (1665), 234.
Galileo,
SidereusNuncius (1610), [Shea (2009), 64].
Galileo,
Sidereus Nuncius (1610), [Shea (2009), 65].
Galileo,
Sidereus Nuncius (1610), [Shea (2009), 93].
Grassi, [Drake (1978), 268]. [Shea (2009), 110]. A correct explanation was given in 1665(ibid.).
Galileo,
Sidereus Nuncius (1610), [Galileo (1957), 31].
Kepler,
Dioptrice (1611). Kepler’s telescope uses twoconvex lenses instead of Galileo’s pair of one convex and one concave lens. [Dijksterhuis (2004), 35]. [Shea (2009),11].
Galileo made repeated statements to this effect, collected in [Shea & Davie (2012), 415–416]. .7 Mountains on the moon “The moon is not robed in a smooth and polished surface but is in fact rough and uneven, covered everywhere,just like the earth’s surface, with huge prominences, deep valleys, and chasms.” It is all too easy to castthis report by Galileo as a revolutionary discovery. The “Aristotelian” worldview rested on a sharp divisionbetween the sublunar and heavenly realm. Our pedestrian world is one of constant change—a bustling soupof the four elements (earth, water, air, fire) mixing and matching in fleeting configurations. The heavens,by contrast, were a pristine realm of perfection and immutability, governed by its very own fifth elemententirely different from the physical stuff of our everyday world. If we are predisposed to view Galileo as thefather of modern science, a pleasing narrative readily suggests itself: With his revolutionary discovery ofmountains on the moon, Galileo disproved what “everybody” believed.Every educated person in the sixteenth century took as well-established fact . . . that the Moonwas a very different sort of place from the Earth. . . . The lunar surface, according to the commonwisdom, was supposed to be as smooth as the shaven head of a monk.
In those years virtually no one questioned the ontological difference between heaven and earth.. . . The difference between Earth and the heavenly bodies was an absolute truth for astrologersand astronomers, theologians and philosophers of every ilk and school. . . . If the Moon turnedout to be covered with mountains, just like Earth, a millenary representation of the sky wouldbe shattered.
Using data and hard facts to expose its prejudices, this narrative goes, Galileo sent an entire worldviewcrashing down. Furthermore, by revealing the similarity of heaven and earth, Galileo opened the door toa unification of terrestrial and celestial physics —in other words, led us to the brink of Newton’s insightthat a moon and an apple are governed by the very same gravitational force.The problem with this narrative lies in one word: “everybody.” The Aristotelian worldview is not what“everybody” believed; it is what one particular sect of philosophers believed. As ever, Galileo’s claim tofame rests on conflating the two. If we compare Galileo to this Aristotelian sect—as Galileo wants usto do—then indeed he comes out looking pretty good. Members of this sect did indeed try to deny themountainous character of the moon in back-pedalling desperation, for instance by arbitrarily postulatingthat the mountains were not on the surface of the moon at all but rather enclosed in a perfectly round, clearcrystal ball.
If we mistake this for the state of science of the day, then indeed Galileo will appear a greatrevolutionary hero.But to anyone outside of that particular sect blinded by dogma, the idea of a mountainous moon hadbeen perfectly natural for thousands of years. It is obvious to anyone who has ever looked at the moon thatits surface is far from uniform. Clearly it has dark spots and light spots. If one wanted to maintain theAristotelian theory one could try to argue, as many people did, that this is some kind of reflection or marblingeffect in a still perfectly spherical moon.
Whatever one thinks of the plausibility of such arguments, theyare certainly defensive in nature: the Aristotelian theory is on the back foot trying to explain away even themost rudimentary phenomena that any child is familiar with. The idea of an irregular moon is an obviousand natural alternative explanation. Which is why we find for instance in Plutarch, a millennium and a halfbefore Galileo, the suggestion that “the Moon is very uneven and rugged.”
If we look to actual scientists and mathematically competent people instead of dogmatic Aristotelians,we find that “Galileo’s” discovery of mountains on the moon was already accepted as fact long before. Keplerhad already “deduced that the body of the moon is dense . . . and with a rough surface,” or in other words “thekind of body that the earth is, uneven and mountainous.”
This was also the opinion of his teacher Maestlin
Galileo,
Sidereus Nuncius (1610), [Galileo (1957), 28]. [Maran & Marschall (2009), 30]. [Bucciantini et al. (2015), 7].
Galileo was “staking a claim to enemy territory. Celestial physics would nolonger be an Aristotelian preserve, with astronomers confined to the computations of motions. . . . [Galileo’s observationsregarding] the Moon was to mark a new chapter in physics and to subvert not only the traditional departmental division ofacademic labour, but also the world-picture that sustained it.” [Sharratt (1994), 156]. [Frova & Marenzana (1998), 177].OGG.XI.118. [Bucciantini et al. (2015), 115]. [Frova & Marenzana (1998), 176]. OGG.XI.92. [Shea (2009), 4].
Kepler,
Astronomiae Pars Optica (1604), 228, 251, [Kepler (2000), 243, 262].
In a later edition Kepler added the note that “Galileo has at last throughly confirmed thisbelief with the Belgian telescope,” thereby vindicating “the consensus of many philosophers on this pointthroughout the ages, who have dared to be wise above the common herd.”
Indeed, Galileo himself sayshis observations are reason to “revive the old Pythagorean opinion that the moon is like another earth.”
So Galileo’s discovery of mountains on the moon was not a revolutionary refutation of what “everybody”thought they knew, but rather a vindication of what many thinking people had seen for thousands of years.Another example along the same lines is “Earth shine” (like moon shine, but in the reverse direction):Galileo discusses it in the
Sidereus Nuncius as one of the novelties made clear by the telescope, but in realityit had been correctly explained previously.
A similar reality check is in order regarding the ludicrous idea that Galileo’s discoveries regarding themoon instigated celestial physics. Kepler had already written an excellent book on celestial physics beforethe telescope, while Galileo’s bumbling attempts on this subject are very poor.
The telescope revealed the existence of “double stars.” Some stars that had appeared as just a single pointof light to the naked eye turned out, when studied with sufficient magnification, to consist of two separatestars. Galileo’s former student Castelli was excited about such stars, because he hoped they could be usedto prove that the earth moves around the sun. It seemed reasonable to assume that all stars were prettymuch alike—they were all just so many suns. Then the apparent size differences in stars would be due notto actual size differences, but only to differences in how far away they are from the observer. A double starwould therefore correspond to a geometrical configuration like that in Figure 22. If the earth moves aroundthe sun, it should be possible to see the two stars interchange position in the course of a year. This wouldcertainly not happen if the earth was stationary, so we have striking and undeniable evidence for the motionof the earth.This is a parallax effect, and as we have seen above astronomers had failed to detect parallax in thepast.
But the traditional method to look for parallax was based on trying to detect subtle shifts in the
Kepler,
Mysterium Cosmographicum (1596), [Kepler (1981), 165]. [Kepler (1981), 169].
Galileo,
Sidereus Nun-cius (1610), [Galileo (1957), 342]. [Shea (2009), 107], Kepler,
Astronomiae Pars Optica (1604), 6.10.
Kepler,
Astronomia Nova (1609). §§3.4, 3.5, 5.4. §4.3.
But that didn’t happen. They didn’t change position at all. Everything remainedexactly stationary, as if the earth did not move. Since this failed to confirm his preferred conclusion, Galileodid not publish these results.Today we know that all the stars in the night sky are much further away than Galileo estimated, andmuch too far away for any effects of the above sort to be detectable with the telescopes of his time. Galileo’sdistance estimates were way off because of certain optical effects that make it impossible to judge the distancesof stars in the manner outlined above.
It would be anachronistic to blame Galileo for not knowing thesethings, which were only understood much later.But Galileo’s way of discussing the matter in the
Dialogue is not above reproach. He describes the aboveprocedure but frames it hypothetically: “if some tiny star were found by the telescope quite close to some ofthe larger ones,” they would, if the above effect could be observed, “appear in court to give witness to suchmotion in favour of the earth.” “This is the very idea that later won Galileo renown and for which hewas to be remembered by parallax hunters in the centuries that followed. While it is generally thought thatGalileo never tried to detect stellar parallax himself, he is credited with this legacy to future generations.”
In reality he deserves no renown, because the idea was not his own. It had already been explained to himin detail by Ramponi in 1611, and again by Castelli, who discovered the double star Mizar and explainedits importance for parallax to Galileo.
There is no indication that Galileo knew about these things beforehis friends explained them to him.Furthermore, Galileo’s discussion in the
Dialogue is deceitfully framed as a test he has not himselfattempted. That way he can pretend that the absence of parallax is not really a problem, by implying thatif one just carried out this test one would surely find it. If Galileo was an honest scientist concerned withobjectively evaluating the evidence he should have admitted that he had in fact carried out the test, whichhad failed completely to yield the desired results. But that would have forced him to engage seriously withactual current astronomy like the system of Tycho, which he did not want to do. It was much easier for himto suppress his data and disingenuously insinuate that the outcome of the observation would be the oppositeof what he knew it to be.
The moons of Jupiter were probably the most surprising new discovery made when telescopes were firstpointed at the sky. An anecdote related by Kepler conveys some of the excitement: “My friend the BaronWakher von Wachenfels drove up to my door and started shouting excitedly from his carriage: ‘Is it true?Is it really true that he has found stars moving around stars?’ I told him that it was indeed so, and onlythen did he enter the house.”
It seems Galileo was indeed the first to observe the moons of Jupiter, but only by the smallest possiblemargin. Simon Marius observed them the very next day.
In any case one hardly qualifies as the “Fatherof Modern Science” just by looking. Nor does Galileo’s account stand out for its scientific excellence. For [Graney (2008), 263]. See also [Siebert (2005), 260]. [Graney & Grayson (2011)]. [Galileo (1967), 382–383].The method of distance estimates via apparent sizes is mentioned at [Galileo (1967), 359–360]. [Siebert (2005),254]. [Siebert (2005), 254]. [Siebert (2005), 259].
Kepler to Galileo, 1610, [Santillana (1955), 10]. [Gaab & Leich (2018), Chapter 5], [Pasachoff (2015)].
Systema Saturnium (1659), 47.instance, he tries to “correct” Marius on the issue of whether the orbits of the moons are tilted or not withrespect to Jupiter’s orbital plane. Galileo says they are not: “it is not true that the four orbits of the satellites[of Jupiter] incline from the plane of the ecliptic; rather, they are always parallel to it.”
In reality, Mariuswas right and Galileo wrong.Galileo’s mathematical ineptitude is on display in this case as well. “Galileo’s first calculations [of theorbital periods of Jupiter’s moons] were geocentric, not heliocentric. . . . Galileo was treating Jupiter as ifit revolved around the Earth, not the Sun. How he ever came to make such an error . . . is an interestingquestion.”
Kepler and Marius, meanwhile, understood the matter perfectly and realised at once that thiswas another good argument against the Ptolemaic system.
One Galileo supporter offers a very charitableinterpretation: “this throws in doubt the view that by 1611 Galileo was already a Copernican zealot anxiousto find every possible argument for the Earth’s motion.”
A more plausible explanation, in my opinion,is that Galileo was simply not very competent as a mathematical astronomer. It was not lack of desire toprove the earth’s motion that made Galileo miss the point, it was lack of ability.
Saturn is “made of three stars,” says Galileo.
He is trying to describe the rings of Saturn, but telescopesat the time were not good enough to show that they were rings rather than two companions stars. Thisonly became clear some twenty years after Galileo’s death.
We cannot blame Galileo for this. We can,however, blame Galileo for his lack of balance in evaluating the evidence. He does not say, as an honestscientist might, that this is the best guess on the available evidence and that we can’t know for sure untilwe have better telescopes. Instead he boldly proclaims it as certainty that Saturn is “accompanied by twostars on its sides,” “as perfect instruments reveal to perfect eyes.”
In the same vein, Galileo hubristically declared that the appearance of Saturn’s “ears” would never change:I, who have examined [Saturn] a thousand times at different times, with an excellent instrument,can assure you that no change at all is perceived in him: and the same reason, based on theexperience which we have of all the other movements of the stars, can render us certain that,likewise, there will be none.
All the more embarrassing then when in fact the appearances changed radically soon thereafter:I found him solitary without the assistance of the supporting stars. . . . Now what is to be saidabout such a strange metamorphosis? Perhaps the two smaller stars . . . have vanished and fledsuddenly? Perhaps Saturn has devoured his own children?
So the very thing Galileo said “thousands” of observations meant we could be “certain” would never happenactually took place almost right away. That was bad publicity at a time when many doubted the reliabilityof his telescope.
Galileo,
Assayer (1623), [Drake & O’Malley (1960), 166]. [Drake (1999), 421]. See also [Shea (2009), 35]. Galileo even-tually realised his error himself when his calculations didn’t match observations. [Drake (1999), 422]. [Drake (1999),429].
OGG.V.237. [Deiss & Nebel (1998), 218].
Christiaan Huygens,
Systema Saturnium (1659).
Galileo, firstletter on sunspots (1612), [Reeves & Van Helden (2010), 102].
Galileo, May 1612, OGG.V.110–111, [Van Helden (1974),106].
Galileo, 1612, OGG.V.237, [Van Helden (1974), 107]. “In classical mythology, Saturn devoured his newborn childrento forestall a prophecy that he would be overthrown by one of his sons.” [Shea & Davie (2012), 403].
Indeed, Galileo liked his model so much that he also “took the courage” to lie abouthaving made an observation verifying it. He claims that he “saw Saturn triple-bodied this year [1612], atabout the time of the summer solstice.”
But modern calculations show that the ring of Saturn would havebeen vanishingly thin at this time. “Clearly . . . [Galileo] could not have observed the ring at the summersolstice of 1612. . . . Yet the picture of the Saturnian system that was accepted by Galileo implied that thering should have been visible, so much so that he made a claim to this effect that we know must have beenuntrue.”
Some years later Galileo found that the satellites had undergone another “strange metamorphosis.” Thistime they had ceased to be independent globes at all and instead appeared as half-ellipses attached to theplanet. Galileo made a prescient-looking sketch (Figure 23), which has led some people to triumph that hediscovered the rings of Saturn.
But, alas, for all his supposedly admirable courage and precious “specialway of thinking” he failed to interpret them correctly. Instead he reverted back to the erroneous idea of thesatellites as “collateral globes” in the
Dialogue . When astronomers first turned telescopes to the heavens, one of the most disturbing sights was “filth on thecheeks of the Sun.” “I saw the sonne in this manner:”
So records Thomas Harriot in his observational diary. Such spots on the sun soon drew much attentionacross Europe. While some “neglected to observe them, being afraid, . . . that the image might burn myeye,” others figured God had given them two eyes for a reason and “burned” them alternately in the nameof science. Thus Harriot “saw it twise or thrise, once with the right ey & other time with the left” before “theSonne was to cleare.”
Soon a method was developed for projecting the image of the sun onto a piece ofpaper so that no burning of the eyes was needed.
This was convenient enough, though there would havebeen no shortage of martyrs of science willing to pay with their eyes for wisdom.“Galileo insisted to his dying day that he was the first to have seen” sunspots but in reality he was“probably preceded by the Dutch astronomer Johann Fabricius, who was the first to publish informationabout them.”
To boost his priority case, Galileo later claimed he had seen sunspots already in 1610, rather than in 1611 as documented, but this seems to be a lie since “if he had first observed them . . . in1610 it is extremely probable that he would have mentioned that fact in his
Letters printed in 1613, whenthe priority issue was first hotly debated.”
At that time of the
Letters , Galileo was keen to establish hispriority over Scheiner, who published on sunspots in 1612. At that time Galileo was evidently unaware of
OGG.XVIII.238–239. [Van Helden (1974), 111].
OGG.V.237. [Deiss & Nebel (1998), 215]. [Deiss & Nebel (1998), 218]. [Bianchi (1834)].
OGG.VII.287. [Van Helden (1974), 110].
AlessandroAllegri,
Lettere di Ser Poi Pedante (Bologna: Vittorio Benacci, 1613), 14, [Reeves & Van Helden (2010), 77]. [Reeves & Van Helden (2010), 26]. This is the earliest recorded telescopic observation of sunspots.
Maelcote,[Reeves & Van Helden (2010), 47]. [Reeves & Van Helden (2010), 26].
Galileo learned this method from Castelli.[Shea & Davie (2012), 402]. [Shea & Davie (2012), 402]. [Galileo (2001), 401]. [Drake (1978), 333].
Hence it later became importantfor Galileo to push his discovery back even earlier, whereupon he conveniently asserts that he saw them in1610. For that matter, even pre-telescopic astronomers had noticed the phenomenon of sunspots: it “wascertainly mentioned in the time of Charlemagne, and possibly was referred to by Virgil.”
In any case, the game was now on to explain the nature of the spots. Scheiner was concerned “to liberatethe Sun’s body entirely from the insult of spots,” for “who would dare call the Sun false?”
He founda way of accomplishing this by arguing that the sunspots were “many miniature moons,” rather thanblemishes on the sun itself.Galileo on the other hand eagerly embraced sunspots as an opportunity to stick it to his Aristotelianenemies, “for this novelty appears to be the final judgement of their philosophy.”
Thus Galileo placedthe spots on the sun itself, arguing that “clouds about the Sun” was the most plausible explanation, for one“would not find anything known to us that resembles them more.”
It is true that sunspots are on thesurface of the sun—a conclusion, incidentally, which Kepler had already reached before reading Galileo.
However, sunspots are not clouds above the surface of the sun, as Galileo believed, but rather dark pits ordepressions in the solar surface. “Scheiner . . . entertained the possibility of this hypothesis, while Galileoresolutely discarded it as unworthy of serious consideration.”
Galileo loved claiming new discoveries as his own and using them as ammunition in his philosophicaldisputes. But he soon lost interest when it came to the detailed work of actual science. Again and againhe makes careless errors and jumps to conclusions with premature confidence, while his competitor Scheinerdoes the meticulous observational work that Galileo had no patience for. For instance, Galileo erroneouslyclaimed—supposedly based on “a great number of most diligent observations of this particular” —that allsunspots had the same orbital period, regardless of latitude. In fact, sunspots near the sun’s equator orbitquicker than those near the poles by a few days. Galileo was corrected by Scheiner.
Similarly, “the sun’sdisc, as we normally see it in the sky, appears to be uniformly bright, but this impression is dispelled byeven the most perfunctory telescopic observation which reveals that the brightness decreases from the centretowards the limb. Scheiner made this discovery but Galileo dismissed it.”
Galileo also did not miss the opportunity to make some mathematical errors as usual. He tried to computethe perspective aspect of the sunspots’ motion: how does their apparent speed along the sun’s disc vary,given that their actual direction of motion turns more away from us the closer they get to the edge? Galileo’sattempted demonstration covers three pages and contains at least as many errors.
Sunspots can be used as evidence that the earth moves around the sun. In his
Dialogue , Galileo consideredthis one of his three best arguments in favour of Copernicus. “For the sun has shown itself unwilling tostand alone in evading the confirmation of so important a conclusion [i.e., that the earth orbits the sun], andinstead wants to be the greatest witness of all to this.”
The Copernican argument from sunspots goes as follows.
Imagine a standard globe of the earthstanding on a table. Its axis is a bit tilted, of course—the north pole is not pointing straight up. Have aseat at one side of the table and face the globe. What do you see? Focus on the equator. What kind ofshape is it? If the north pole is facing in your direction, the equator will make a “happy mouth” or U shape.If you move to the opposite side of the table, where you see mostly the southern hemisphere, the equator isinstead a sad mouth shape. From the sides, the equator looks like a diagonal line.Now, suppose the sun had its equator marked on it. And suppose that in the course of a year we wouldsee it as alternately as a happy mouth, straight diagonal, sad mouth, straight diagonal, etc. That would
De maculis in sole observatis narratio , Wittenberg, 1611.
Drake, [Galileo (1957), 82].
Third letter of
Tres Epis-tolae (1611), [Reeves & Van Helden (2010), 73].
Third letter of
Tres Epistolae (1611), [Reeves & Van Helden (2010),67].
Third letter of
Tres Epistolae (1611), [Reeves & Van Helden (2010), 72].
Galileo to Barberini, 2 June1612, OGG 11:311, [Reeves & Van Helden (2010), 83].
First letter on sunspots (1612), [Reeves & Van Helden (2010),99]. [Drake (1978), 213]. [Shea (1972), 67].
Third letter on sunspots (1612), [Reeves & Van Helden (2010),268]. [Shea (1972), 67], [Reeves & Van Helden (2010), 268]. [Shea (1972), 66]. [Reeves & Van Helden (2010),357–359], [Shea (1972), 57]. [Galileo (1953), 462]. [Galileo (1953), 345].
Cf. [Galileo (1953), 348–351] andFigures 24 and 25.
RosaUrsina (1630). Figure 25: Equatorial circle of a sphere viewed from different vantage points.55orrespond exactly to us moving around the table, looking at a stationary globe from different vantage points.In the same way, if the sun’s equator exhibited those appearances, the most natural explanation would bethat the earth is moving around it and we are looking at its equator slightly from above, from the side,slightly from below, etc.The sun does not have the equator conveniently marked on its surface, but not far from it. The sun isspinning rather quickly, making a full turn in less than a month. As it spins, a point on its surface tracesout an equatorial or at least latitude circle. So by tracking the paths of sunspots over the course of a fewweeks, we in effect see equatorial and other latitude circles being marked on the surface of the sun.So the shapes of the paths show that we are looking at the sun from alternating vantage points. But thisdoes not necessarily mean that we are moving around the sun. The same phenomena could be accountedfor from a geostatic or Ptolemaic point of view by saying that the sun is so to speak wobbling, showing usdifferent sides of itself in the course of a year. You can see this with your globe on the table. Instead ofmoving around the table and looking at the globe from different sides, you can have a friend tilt the globe,pointing its axis now this way and now that. If you let the axis spin around in a conical motion, this willproduce the exact same visual impressions for you as if you had moved around the table.In order to use the sunspot paths as evidence for Copernicus, then, Galileo needed to dismiss thisalternative explanation. He did so by attacking it as physically implausible. To account for the sunspotsphenomena from a Ptolemaic point of view, the sun had to orbit the earth, and spin on its own axis, andhave its axis wobble in a conical motion. These diverse motions, says Galileo are “so incongruous with eachother and yet necessarily all attributable to the single body of the sun.”
Surely this is a geometrical fictionthat would never happen in an actual physical body.Actually, such an “incongruous” combination of motions is not only possible but a plain fact. The earth,in fact, has exactly such a combination of motions, as had been known since Copernicus.
The earth hasa conical wobbling motion which means that the north pole is pointing to a slightly different spot amongthe stars from year to year, returning to its original spot after 26,000 years. This is the explanation for theso-called precession of the equinoxes, an important technical aspect of classical astronomy. So if Galileo’sargument about “incongruous” motions disproves the Ptolemaic explanation of sunspots, it also disprovesCopernicus’s correct explanation of the precession of the equinoxes.Galileo conveniently neglects to bring up this rather obvious problem with his argument. Whetherhe did so out of ignorance or dishonesty is hard to say, but either way is none too flattering. Any seriousmathematical astronomer was well acquainted with the precession of the equinoxes and of course considered itan essential requirement that any serious astronomical system account for this phenomenon. Galileo, though,is not a serious mathematical astronomer. He is a simplistic populariser who simply ignores technical aspectslike these. And it is only because of this oversimplification that he is able to maintain his argument againstthe Ptolemaic interpretation of sunspots.In any case, in the 1610s, when he was first studying sunspots, Galileo completely missed all of this. Helacked the disposition to do painstaking scientific research like Scheiner. Instead, with premature hubris, hesoon imagined that he had “looked into and demonstrated everything that human reason could attain to”regarding sunspots.
Many years later he was still convinced that his was the last word on the matter:“writing . . . apropos of recent news that Scheiner would soon publish a thick folio volume on sunspots, heremarked that any such book would surely be filled with irrelevancies, as there was no more to be said onthe subject than he had already published in his Letters on Sunspots.”
When Scheiner’s much better work on sunspots came out, Galileo realised he had to completely reversehis earlier proclamations, made with such arrogant confidence. With unwarranted pomposity, he had claimedto offer “observations and drawings of the solar spots, ones of absolute precision, in their shapes as well as intheir daily changes in position, without a hairsbreadth of error.”
According to Galileo, the sunspots were“describing lines on the face of the sun”: “they travel across the body of the sun . . . in parallel lines.”
In fact, “I do not judge that the revolution of the spots is oblique to the plane of the ecliptic, in which the [Galileo (1953), 355]. [Topper (1999)]. [Galileo (1953), 346], referring to the 1610s. [Drake (1970), 185].
First letter on sunspots (1612), [Reeves & Van Helden (2010), 104]. The observations themselves were sent with the secondletter (1612).
First letter on sunspots (1612), [Reeves & Van Helden (2010), 90].
Second letter on sunspots (1612),[Reeves & Van Helden (2010), 109].
In other words, every sunspot path is straight as an arrow, just as the equator of a globewould be from every side if its axis was perfectly vertical.But Scheiner showed that the sun’s axis has a ◦ (cid:48) inclination and that the paths exhibit exactly thealternating diagonal and U shapes described above. He published this result in his Rosa Ursina (1630), thefolio Galileo had mocked as bound to be superfluous, but from whence he now realised his error.
When Galileo finally realised that inclined sunspot paths spoke in favour of heliocentrism, he immediatelythrew all his old observations “without a hairsbreadth of error” out the window and rushed the pro-Copernicanargument into print without making any new observations, as is clear from the fact that the publishedargument “displays entire ignorance or complete neglect of the observational data,” his vague descriptionsbeing “utterly wrong” and “almost the exact opposite” of the careful data published by Scheiner.
Galileo did not want to admit his debt to Scheiner, however, so he pretended that he had come upon thisdiscovery independently, and lied that he had made “very careful observations for many, many months,and noting with consummate accuracy the paths of various spots at different times of the year, we foundthe results to accord exactly with the predictions.”
In reality, “the evidence is unequivocal: Galileo . . .must have had a copy of Scheiner’s book in front of him as he wrote this section.” By pretending otherwise,“Galileo has deliberately set out to efface Scheiner from the historical record and to deny his debt to him. Itis impossible to find any excuse for this behaviour.”
To the naked eye, Venus is just a dot of light. But telescopic magnification reveals it as a sphere, only halfof which is bright, namely the half facing the sun. Thus the half of Venus facing us exhibits phases likethe moon, being sometimes crescent, sometimes half full, sometimes gibbous, and so on (Figure 26). Theseappearances show that Venus orbits the sun, contrary to Ptolemaic cosmology.Galileo observed the phases of Venus in 1610. Here is a timeline of events:• On December 5, Castelli wrote to Galileo and pointed out that it ought to be possible to confirm theCopernican hypothesis by observing phases of Venus with a telescope. • On December 11, perhaps right after receiving Castelli’s letter,
Galileo announced something “justobserved by me which involves the outcome of the most important issue in astronomy and, in particular,contains in itself a strong argument for the . . . Copernican system,” namely that he has observed Venusexhibiting phases like the moon.
Before this there is no record that Galileo knew anything aboutthe phases of Venus. • On December 30, Galileo gave for the first time an account of his observations of Venus. At this pointhe claims to have observed Venus for about three months, and gives an accurate and fairly detaileddescription of its appearance during this period.
The timing of Galileo’s December 11 letter is certainly very suspicious. Was Galileo ignorant of the phasesof Venus and their importance before it was pointed out to him? Did Galileo steal the idea from Castelli?Very possibly.In fact, in a letter of November 13 of the same year, Galileo seems to state expressly that he had no newplanetary discoveries to report, implying that he did not yet know about the phases of Venus, contrary to hislater assertions. His defenders claim that Galileo’s phrase should instead be read as saying that he has madeno new discoveries “around” the planets, that is, discovered no new moons.
Perhaps so. It would make
Second letter on sunspots (1612), [Reeves & Van Helden (2010), 113]. The same claim is repeated in the third letter(1612), [Reeves & Van Helden (2010), 255]. [Reeves & Van Helden (2010), 315, 327]. [Drake (1970), 186–187].See also [Drake (1978), 335]. [Galileo (1953), 346]. [Galileo (1953), 352]. [Wootton (2010), 208–209]. [Westfall (1985), 11].
It is hard to say when exactly Galileo would have received the letter from Castelli.[Westfall (1985), 24] discusses evidence regarding mail delivery times and finds it “easily possible” that Galileo could havereceived the letter before December 11. [Drake (1984), 203], on the other hand, finds the probability of this “vanishingly small”on the basis of other evidence regarding mail delivery times, as well as arguments from other references in their correspondence. [Westfall (1985), 24]. Galileo’s announcement was initially in the form of an anagram, the meaning of which he revealedonly later. [Westfall (1985), 25]. [Palmieri (2001), 109–110]. [Drake (1984), 200].
But that’s all the more reason for him to miss Venus’ phases. At this time Venus was well over half full, soits shape would not have been very remarkable unless you were specifically paying attention to it. He couldeasily have missed it if he was too busy moon hunting and looking only “around” the planets.But if Galileo had not observed the phases of Venus before Castelli’s letter, then how could he later givean accurate description of their appearance dating back two months before this letter? Easy: by fabricatingdata and passing them off as actual observations, as he did on many other occasions.
After all, he wassurely concerned to get the important pro-Copernican argument from the phases of Venus on the record asquickly as possible and claim it for himself, and for this purpose it would be important to have observedVenus’ fully gibbous appearance in the fall (the next opportunity to observe it in this form would be monthsaway). So Galileo certainly had a strong motive to fabricate this data. Making observations throughoutmost of December, after receiving Castelli’s letter, would also have been enough to give him great confidencethat the heliocentric explanation for the phases of Venus was right. So faking the data was not risky.Galileo’s defenders have a counterargument to this. They claim that Galileo could not have fabricatedthe data in question even if he had wanted to. According to them, the changes in appearance of Venusduring these months were so complex and “non-linear” that Galileo could never have given such an accurateaccount if he had not if fact made these observations. Specifically, Galileo correctly describes the fact thatthe transition from a gibbous to semicircular phase is quite rapid, while a roughly semicircular phase lingersfor a considerable time.Castelli’s letter cannot have been the spark that ignited Galileo’s programme of observation ofVenus. It was simply too late. If he only then had started observing Venus, he would have seenit already nearing the exact semicircular phase, thus completely missing the non-linear patternsof change. And he could not possibly have been able to calculate the duration of one month forthe “lingering” phenomenon. In other words, Galileo cannot have predicted Venus’s non-linear §§4.9, 4.10. §§3.1, 3.3, 3.6, 3.14, 4.10, 4.12.
I say that, on the contrary, Galileo could easily have reconstructed these phenomena. He would not haveneeded any sophisticated mathematics as all. All he would have had to do would have been to simulate itsappearance by looking at a half-painted sphere representing Venus from vantage points corresponding to theEarth’s position relative to it.I carried out such a simulation using very simple means (Figure 27). The results are shown in Figure 28.I used a white spherical lamp as Venus. I covered half of it in black to represent the half not illuminatedby the Sun. I pointed the white half toward a center point (the Sun) 4.34 meters away. I then marked offa circle of radius 6 meters with the same center, representing the orbit of the Earth. I used the fact thatVenus was seen exactly semicircular on December 18 to find where the Earth must have been it its orbitthat day.
I placed a camera at this position and photographed the Venus sphere. I then used a protractorpositioned at the Sun to reconfigure the setup to correspond to other dates, counted forward and backwardsfrom December 18 in 21 day increments using the simplest possible estimation for the motions of theseplanets (I simplistically assumed uniform circular motions for the Earth and Venus, so there is no advancedmathematical astronomy involved in any way, just basic calculations using the radii and orbital times ofthese two planets). I again photographed Venus from these positions. I did all of this in a rough-and-readyway in an empty parking lot using crude measurements. I also recreated the exact same setup using 3Dsoftware (Figure 29), which shows the results of this simulation without the accidental imperfections of myphysical demonstration.Galileo could easily have completed such a simulation from start to finish in just a few hours. And ofcourse the idea of illustrating the phases of the moon by an illuminated or half-painted sphere had beencommonplace since antiquity, so Galileo would not have needed much imagination to come up with thisscheme.The results of this simple simulation are very close to the true appearances.
In particular, the simula-tion is easily sufficient to reproduce the allegedly so unpredictable “non-linear” phenomena that Galileo gotright in his December 30 report. So the claim that it would have been impossible for Galileo to recreate theseappearances after the fact is definitely false. We may note also that one argument that Galileo’s accounthas “the ring of a record of visual impressions rather than an account coloured by calculations” in that it“has a highly visual character.”
Obviously this is consistent with my simulation just as well as actualobservations.The hypothesis that Galileo simulated his Venus observations by using such a model is lent some furthercredibility by its close parallels with his treatment of sunspots.
As we have seen, Galileo realised thatsunspots constituted an important pro-Copernican argument only quite late, based on the input of others,and needed to act fast in writing something about it without having the time for thorough observations. Isuggest that this is an exact parallel of what happened also in the case of the phases of Venus.This parallel undermines the common assumption that Castelli’s idea must already have been obviousto Galileo. One scholar, for example, thinks it “would be to dignify the idea beyond reasonable measure”to view Castelli’s suggestion as a significant insight; rather, “the thought that Venus might have phases was‘in the air’ ” and hence Castelli’s contribution is to be considered quite trifling.
Another historian arguesalong similar lines that Galileo had no need to be spurred to action by Castelli’s letter, only by news ofothers making advanced telescopic observations. Around a day or two before hearing from Castelli, Galileohad received another letter, reporting that Clavius and his assistants at Rome had observed the moons ofJupiter.
So Galileo now had serious competitors in the realm of advanced telescopic observations, or so it [Palmieri (2001), 117]. [Peters (1984), 212].
Figure 26 shows exactly computed actual appearances. Suchmodern reconstructions of the actual appearances are also given in [Palmieri (2001)], [Gingerich (1984)], [Peters (1984)]. [Peters (1984), 213–214]. §4.12. [Ariew (1987), 92]. [Drake (1984), 200–201].
The sunspots case isa counterexample to this claim: if there was no shortage of “reasoning power,” Galileo should have realisedthe potential importance of sunspots much earlier and not let himself be beaten to the punch about theircurved appearance by his arch-rival Scheiner. The fact of the matter is that the sunspots argument forheliocentrism eluded Galileo for twenty years, despite the fact the was passionately committed to provingheliocentrism in novel ways, and despite the fact that he himself had written specifically and in detail aboutthe very phenomenon at stake, and despite the fact that the argument is very simple. By analogy, thissuggests that Castelli’s idea about Venus could very well have been news to Galileo: if he could somehowmiss the sunspots argument for twenty years despite all of this, then he could certainly have failed to thinkof the Venus argument during his one initial frantic year of telescopic observations, when he had a myriadother novelties and issues to deal with all at once.But perhaps the most interesting aspect of the parallel between the two cases is the possibility that theyboth involved the use of physical models to simulate celestial appearances. For, in the
Dialogue , one speakerreports regarding the appearances of the paths traced on the surface of the sun by sunspots as seen from theEarth that Galileo “assisted my understanding by representing the facts for me upon a material instrument,which was nothing but an astronomical sphere, making use of some of its circles—though a different usefrom that which they ordinarily serve.”
The same sentiment is repeated later: the appearances of thesunspot paths “will become better fixed in my mind when I examine them by placing a globe at this tilt andthen looking at it from various angles.”
This is very closely analogous to the Venus simulation I outlinedabove, suggesting that the latter would have been quite natural to Galileo, and in keeping with his style ofreasoning.
The following, then, are generally accepted facts about the sunspots matter: Galileo claimed to haveconducted careful observations when he had not; according to his own account, Galileo simulated observationsby looking at a physical sphere from a variable vantage point corresponding to the position of the Earth;Galileo failed to see an important pro-Copernican argument for a long time, despite it being simple and verynaturally connected to his own work. The fact that these things did happen in the sunspots case suggeststhat they very well could have happened also in the Venus case.In conclusion, if Galileo had wanted to fabricate or reconstruct Venus observations he had not made, he [Drake (1984), 203]. [Galileo (1953), 348]. [Galileo (1953), 352].
For further examples of Galileo preferringto think with physical objects, see [Machamer (1998), 67–71]. “Have you seen the fleeting comet with its terrifying tail?”
This was the question on everyone’s lips in1618, following the appearance of a comet “of such brightness that all eyes and minds were immediatelyturned toward it.” “Suddenly, men had no greater concern than that of observing the sky. . . . Great throngsgathered on mountains and other very high places, with no thought for sleep and no fear of the cold.” “That stellar body with its menacing rays . . . was considered as a monstrous thing,” and, according tomany, surely a cosmic omen foretelling imminent disasters.Some urged a more dispassionate approach, arguing that “the single role of the mathematician” is merelyto “explain the position, motion, and magnitude of those fires.”
Indeed, “the mathematician” had been soengaged for generations. Tycho Brahe, for instance, had worked extensively on comets.
It would be difficult for Galileo to enter this game, since he was such a poor mathematician. Notcoincidentally, he has an argument for why one should ignore the serious mathematical astronomy of comets,namely that such accounts are hopelessly inconsistent:Observations made by Tycho and many other reputable astronomers upon the comet’s parallax. . . vary among themselves. . . . If . . . complete faith . . . be placed in them, one must concludeeither that the comet was simultaneously below the sun and above it, . . . or else that, becauseit was not a fixed and real object but a vague and empty one, it was not subject to the laws offixed and real things.
Kepler is flabbergasted that someone who calls himself a geometer could write such drivel:Certainly so far as Galileo is concerned, he, if anyone, is a skilled contributor of geometricaldemonstrations and he knows . . . what a difference there is between the incredible observationaldiligence of Tycho and the indolence common to many others in this most difficult of all activities.Therefore, it is incredible that he would criticize as false the observations of all mathematiciansin such a way that even those of Tycho would be included.
The paradox disappears if one recognises that Galileo is not a skilled geometer after all.Unlike serious mathematical astronomers (and perhaps precisely in order to avoid having to engage withthem), Galileo maintained that comets were not physical bodies travelling through space at all, but rather achimerical atmospheric phenomena. He believes that “their material is thinner and more tenuous than fog orsmoke.” “In my opinion,” says Galileo, comets have “no other origin than that a part of the vapour-ladenair surrounding the earth is for some reason unusually rarefied, and . . . is struck by the sun, and made toreflect its splendour.”
Galileo’s vapour theory of comets is inconsistent with basic observations, as he himself admits. If cometsare nothing but “rarefied vapour”—that is to say, some kind of pocket of thin gas—then you’d imagine thattheir natural motion would be straight up, like a helium balloon. Indeed Galileo does propose that cometshave such paths. But then he at once admits that this doesn’t fit the facts: “I shall not pretend to ignorethat if the material in which the comets takes form had only a straight motion perpendicular to the surfaceof the earth . . . , the comet should have seemed to be directed precisely toward the zenith, whereas, in fact,it did not appear so. . . . This compels us either to alter what was stated, . . . or else to retain what has beensaid, adding some other cause for this apparent deviation. I cannot do the one, nor should I like to do theother.”
Bummer, it doesn’t work. But Galileo sees no way out, so he just leaves it at that.
Grassi, [Drake & O’Malley (1960), 4].
Grassi, [Drake & O’Malley (1960), 6].
Grassi, [Drake & O’Malley (1960),4, 6].
Grassi, [Drake & O’Malley (1960), 6–7].
As we noted in §4.3.
Galileo,
Assayer (1623),[Drake & O’Malley (1960), 257–258].
Kepler, appendix to
Hyperaspistes (1625), [Drake & O’Malley (1960), 351].
Galileo,
Assayer (1623), [Galileo (1957), 254]. [Shea (1972), 81]. OGG.VI.94. [Shea (1972), 82–83]. OGG.VI.98.
For instance, the speeds of comets do not fit Galileo’s theory. According to Galileo’stheory, the vapours causing the appearance of comets rise uniformly from the surface of the earth straightupwards. Therefore the comet should appear to be moving fast when it is close to the horizon, and thenmuch slower when it is higher in the sky. Just imagine a red helium balloon released by a child at a carnival:it first it shoots off quickly, but soon you can barely tell if it’s rising anymore, even though it keep goingup at more or less the same speed, because your distance and angle of sight is so different. But comets donot behave like that. Detailed observations of the comet of 1618 showed a much more constant speed thanGalileo’s hypothesis requires.Galileo also offered another very poorly considered argument against the correct view of comets as orbitingbodies, namely that their orbits would have to be unrealistically big: “How many times would the world haveto be expanded to make enough room for an entire revolution [of a comet] when one four-hundredth part ofits orbit takes up half of our universe?”
This is a poor argument, because the universe must indeed bevery big and then some according to Copernican theory, in order to explain the absence of stellar parallax.
Since the earth’s motion is observationally undetectable, the orbit of the earth must be minuscule in relationto the distance to the stars. That means there is plenty of room for comets. But Galileo convenientlypretends otherwise in his argument against comets. Evidently, Galileo “was so intent on refusing Tycho thathe failed to notice that he was pleading for a universe in which there would be no room for the heliocentrictheory” either.
The Cambridge Companion to Galileo poses for itself the question: “What did Galileo actually do that madehis image so great and so long-standing?” Its answer is not a list of great scientific accomplishments butrather: “Certainly his was the first main effort that fired the vision of science and the world that went wellbeyond limited intellectual circles.”
Galileo was a populariser, in other words. “It was to the man ofgeneral interests that Galileo originally addressed his works.”
Indeed, Galileo embraced this role, praisinghimself for “a certain natural talent of mine for explaining by means of simple and obvious things otherswhich are more difficult and abstruse.”
I agree with these learned authors that Galileo wrote for the vulgar masses. I must add only onepoint, which they omit, namely that Galileo was driven to turn to popularisation because he was so bad atmathematics . “Galileo scarcely ever got around to writing for physicists.”
Yes, and he was scarcely able to do so either. The two are not unrelated.Take for instance the “new stars” that appeared in Galileo’s lifetime. One appeared in 1572. It wasstudied with great care by Tycho Brahe. Another appeared in 1604, when Galileo was 40 years old and anestablished professor of mathematics. But Galileo didn’t make a contribution based on serious astronomyas Tycho had done. Instead he gave public lectures on the nova to a layman audience totalling more than athousand people.
Galileo’s little science extravaganzas were a hit at bourgeois dinner parties, as contemporary witnessesdescribe:We have here Signor Galileo who, in gatherings of men of curious mind, often bemuses manyconcerning the opinion of Copernicus, which he holds for true. . . . He discourses often amidfifteen or twenty guests who make hot assaults upon him. . . . But he is so well buttressed thathe laughs them off; and although the novelty of his opinion leaves people unpersuaded, yet he [Shea (1972), 84].
Galileo, [Shea (1972), 77]. §4.3. [Shea (1972), 88]. [Machamer (1998), 1–2].
Drake, [Galileo (1957), 3].
Galileo,
Assayer (1623), [Galileo (1957), 265].
Drake, [Galileo (1957), 2].
Novasor supernovas in modern parlance. [Drake (1978), 105].
Again: Galileo’s speciality is burlesque astronomical road shows, not serious science. Galileo’s defendersrefuse to admit the obvious, and instead go out of their way to try to save face on his behalf:The technique . . . is actually a very sound, wise, and proper one; it really amounts to beingconcerned to avoid the straw-man fallacy; that is, before criticising an opponent, it is a sign ofa serious critic to first strengthen the opposing argument as much as possible and interpret itin the most charitable manner; by so doing, one’s criticism will really undermine the argument,rather than destroying one’s own caricature invented to make one’s own task easy.
The notion that Galileo is concerned with being charitable and avoiding straw men and caricature is pre-posterous. The truth is the exact opposite. We have already seen this for example in §3.1. The followingtwo sections provide further illustrations of the same point.
Space is three-dimensional. Why? Because “three is a perfect number”? Galileo alleges that this is Aristotle’sanswer, and goes on to refute it triumphantly: “I do not understand, let alone believe, that with respect tolegs, for example, the number three is more perfect than four or two.”
You can almost hear across thecenturies the laughter erupt among the tipsy dinner-party guests as this clever put-down is delivered. ClassicGalileo! Just picture a three-legged beast hobbling about like a limp dog—could anything be more farcical?And yet Galileo’s opponents have unwittingly committed themselves to calling this ludicrous spectacle thepinnacle of perfection. So ridiculous are the consequences of Aristotle’s metaphysical teachings.Galileo cannot be beat when it comes to soirée entertainment, but does his argument have any genuinescientific merit? No. Refuting the idea that three is a perfect number is not cutting-edge research by anyone’smeasure. And even if it was, Galileo’s argument still wouldn’t be much of contribution to the debates sinceit is quite clearly more concerned with opportunistic showmanship than serious and balanced engagementwith Aristotle’s thought. Let us look at the passage from Aristotle on which his critique is based:The three dimensions are all that there are. . . . As the Pythagoreans say, the world and all thatis in it is determined by the number three, since beginning and middle and end give the numberof an ‘all’. . . . Further, we use the terms in practice in this way. Of two things, or men, we say‘both’, but not ‘all’: three is the first number to which the term ‘all’ has been appropriated. Andin this . . . we do but follow the lead which nature gives. Therefore . . . [three-dimensional] bodyalone among magnitudes can be complete.
When Galileo sets out to attack this point of view, he has Aristotle’s mouthpiece Simplicio argue thatAristotle “has sufficiently proved that there is no passing beyond the three dimensions, . . . and that thereforethe body, or solid, which has them all, is perfect.”
We see that there are some differences between Aristotle’s account and Galileo’s paraphrase. First ofall Aristotle hardly pretends to have “proved” the matter; rather he merely supports it with some charminghistorical and etymological considerations. Furthermore, the term “perfect”—on which Galileo’s refutationhinges—is not used by Aristotle in the sense needed for Galileo’s argument. The term does not occur at allin the translation I quoted, which instead uses “complete.” Regardless of which of these two renditions ismore accurate from the point of view of Greek linguistics, it is obvious in any case that Aristotle doesn’tmean “perfect” in the sense needed for Galileo’s rebuttal, which would be something like “optimal” or “ideal.”Galileo is obviously “playing on this ambiguity in order to weaken Simplicio’s position.”
Aristotle is merelymaking the rather obvious point that three dimensions are the “whole” or “entirety” of what there is. But
Querengo, 1616, [Koestler (1959), 452]. [Finocchiaro (2007), 311–312]. [Galileo (1967), 11].
Aristotle,
DeCaelo
I.1, trans. J. L. Stocks. [Galileo (1967), 10]. [Galileo (1967), 468].
64t would be no fun for Galileo to admit as much since it would spoil his punchline about the three-leggedogre. So we must conclude that Galileo’s argument is not really engaging with Aristotle in an honest way,but rather relies on a cunning equivocation to score an easy point.Still in modern times though there is no shortage of philosophers who fall for Galileo’s argument hook,line, and sinker, and even use it as an exemplary illustration of his argumentative prowess:Galileo is quoting Aristotle accurately. Simplicio is not inventing these reasons and merelyattributing them to Aristotle. . . . The force of Galileo’s rejection . . . is strengthened sinceAristotle’s account is foolish.
It is hard to know how to reply to such hogwash. The exact negation of each of those statements would beright on the money.As for his own explanation of why space has three dimensions, Galileo purports to offer a “geometricaldemonstration of threefold dimensionality.”
It goes like this. Consider three lines meeting perpendicularly,such as one can effect by, for instance, drawing an “L” on a piece of paper, placing it flat on the ground, andletting a plumb line touch the corner of the L. From this configuration Galileo concludes: “And since clearlyno more lines can meet in the said point to make right angles with them . . . then the dimensions are nomore than three.”
The notion that this is a “proof” that there are three dimensions is ridiculous. It’s anassertion that the conclusion is “clear,” not a proof of it. And yet—perhaps because it comes with letteredgeometrical diagrams—it is enough to fool philosophers even today into believing that it is an “elegant littleproof” and even a “rigorous demonstration.”
As long as readers remain this gullible, Galileo’s undeservedreputation as a great geometer is sure to thrive.
Another prime example of Galileo’s rhetorical skills is his debate with Grassi on comets. It is a fact thatGalileo is dead wrong on the scientific issues at stake, and yet he somehow managed to “win” the debate,in the eyes of many. Galileo’s rousing mockery of his opponent is so satisfying that many readers are seducedinto celebrating it as proof of Galileo’s philosophical acumen. You can read Galileo’s triumphant put-downsof his opponent and go “yeah, crush him!” It’s the same kind of pleasure as watching the villain get punchedin the face in an action movie. But a little reflection shows that this hero-versus-villain dynamic that Galileotries to cultivate is a dishonest fiction that has very little to do with reality.One of Galileo’s most celebrated passages concerns eggs. The context is this. Grassi makes the absolutelycorrect point that comets, if they entered the earth’s atmosphere, would quickly heat up to very greattemperatures due to friction of the air. In support of this point, Grassi quotes a 10th-century Byzantineauthor, Suidas, who claimed that “The Babylonians whirl[ed] about eggs placed in slings . . . [and] by thatforce they also cooked the raw eggs.”
Grassi also quotes passages describing similar phenomena in Ovid,Lucan, Lucretius, Virgil, and Seneca. “For who believes that men who were the flower of erudition and speakhere of things which were in daily use in military affairs would wish egregiously and impudently to lie? I amnot one to cast this stone at those learned men.”
Galileo is unable to answer the substantive point. Indeed, he thinks comets entering the atmospherewould cool down because of the wind rather than heat up because of friction. Galileo is wrong and Grassiis right about the actual scientific issue about comets. But that’s nothing Galileo’s trademarked sophistrycan’t work around. Galileo finds a way to “win” the debate without actually offering any correct scientificclaim regarding the actual subject of comets. He does this by gloatingly attacking Grassi for relying onbooks rather than experimental evidence:If [Grassi] wants me to believe with Suidas that the Babylonians cooked their eggs by whirlingthem in slings, . . . I reason as follows: “If we do not achieve an effect which others formerlyachieved, then it must be that in our operations we lack something that produced their success. [Pitt (1992), 29]. [Galileo (1953), 12]. [Galileo (1967), 14]. [Pitt (1992), 32, 30].
Recall Galileo’svery poor work on comets from §4.14.
Grassi, [Drake & O’Malley (1960), 119].
Grassi, [Drake & O’Malley (1960),119–120].
Not a few modern philosophers blindly and uncritically fall for the ruse: “Galileo shot back with . . . ablistering critique in which he pillories [Grassi] and articulates a tough-minded empiricism as an alternativeto the mere citation of venerable authority.”
Galileo would no doubt be very pleased that so many readers still to this day come away with theimpression that “tough-minded empiricism” is what sets him apart from his opponents. That is precisely theintended effect of his ploy. It has very little basis in reality, however. Just a few pages earlier in the sametreatise, Grassi describes extensively various laboratory experiments he has carried out himself with regardto another point. “I decided that no industry or labor ought to be spared in order to prove this by many andvery careful experiments,” says this supposed obstinate enemy of empirical science. So the notion thatGalileo is the only one “tough-minded” enough to reject authority in favour of experiment is very far off themark.Even in the passage criticised, Grassi is clearly not engaged in “the mere citation of venerable authority.”Rather he honestly and openly cites sources purporting to truthfully report empirical information, just likeany scientist today cites previous works without re-checking all the experiments personally. Grassi doesnot believe that these authors are automatically right because they are “venerable authorities.” Rather heexplicitly considers the possibility that they are wrong, but estimates (reasonably but falsely) that they areprobably right. For that matter, Galileo himself was not above believing falsehoods on the basis of “venerableauthorities.” We have seen him make an error of this type in his theory of tides.
He also considers “credible”the ancient myth of Archimedes setting fire to enemy ships by means of mirrors focussing the rays of thesun.
Descartes sensibly took the opposite view.
Altogether, Galileo is scoring easy points with his taunts about the eggs, by dishonestly pretending that asimplistic point about empiricism was the crux of the matter. It is worth keeping the context of the passagein mind. Indeed, the pro-Galileo interpretation I quoted above comes with its own origin story: “In the courseof his career [Galileo] engaged in many controversies and made powerful enemies. One of those enemies wasthe Jesuit Grassi, who published an attack on some of Galileo’s works.”
This framing goes well with thenotion of the “tough” Galileo bravely defending himself against “attacks” from the “powerful” establishment.But the reality is quite different. Grassi was not a “powerful enemy”: he was a middling college professorjust like Galileo. And the conflict did not start with Grassi “attacking” Galileo, but precisely the other wayaround. Grassi published a fine lecture on comets in which he argued, correctly, that the absence of parallaxshows that comets are beyond the moon. Galileo is not mentioned in this work. Galileo read the lecture andfilled the margins, as one scholar has observed, with an entire vocabulary’s worth of savage expletives.
Galileo then published an attack on Grassi which was not much more restrained than these marginal notes.Grassi replied to it. It is this reply that is called “an attack on some of Galileo’s works” in the pro-Galileanquotation above.In sum, Galileo’s celebrated “pillorying” of Grassi was not a “tough” defence against an “attack” on“some of his works” by “powerful enemies.” The “enemy” was not a “powerful” arm of “authority,” but aconscientious scholar who was right about comets based on good scientific arguments that Galileo rejected.And the enemy was not a cruel aggressor going after “some works” by Galileo unprovoked; rather, the “someworks” in question was an aggressive attack initiated by Galileo in the first place. Furthermore, Galileo’s
Galileo,
Assayer (1623), [Galileo (1957), 272]. [McGrew et al. (2009), 135].
Grassi, [Drake & O’Malley (1960),115]. §3.17. [Galileo (1989), 48], OGG.VIII.86.
Descartes to Mersenne, 11 October 1638, [Drake (1978), 389]. [McGrew et al. (2009), 135]. “There is no misplaced gentleness in the marginal jottings . . . The expletives alone wouldmake a vocabulary of good Tuscan abuse: pezzo d’asinaccio, elefantissimo, bufolaccio, villan poltrone, balordone, barattiere,poveraccio, ingratissimo villano, ridicoloso, sfacciato, inurbano.” [Santillana (1955), 152].
The Bible says next to nothing about astronomical matters. It is more concerned with war, and it is in thiscontext only that it has occasion to speak of the motions of heavenly bodies. Thus, in the Book of Joshua,we find our hero with the upper hand in battle, but alas dusk is drawing close. What a pity if the enemies“delivered up . . . before the children of Israel” should be able to get away under the cover of darkness. “Thenspake Joshua to the Lord,” and he said: “Sun, stand thou still.” “And the sun stood still . . . until the peoplehad avenged themselves upon their enemies.”
This is the extent of astronomy in the holy book. Nowheredoes it say that the earth is in the center of the universe, for example. Just this one passage about how “thesun stood still” that one time to ensure that all the infidels could be killed.Obviously scientists have little reason to engage with such a tangential allusion to cosmology. ButGalileo’s philosophical enemies saw an opportunity. By persistently and prominently accusing Galileo ofproposing theories contrary to scripture they forced him into a dilemma: either let the argument standunopposed, and hence let his enemies have the last word, or else get involved with the dangerous matter ofscriptural interpretation.
Galileo foolishly took the bait. Now all the Aristotelians had to do was to sitback and watch Galileo march to his own ruin in this minefield.In this context, Galileo offered some common-sense platitudes on the relation between science and religion.Many have found it appealing to see in these writings modern conceptions being born. Was it Galileo whoshowed how faith and science can coexist? How they need not undermine or conflict with one another sinceone is about the spiritual and the other about the physical? Galileo indeed makes such a case:Far from pretending to teach us the constitution and motions of the heavens and the stars, . . .the authors of the Bible intentionally forbore to speak of these things, though all were quite wellknown to them. . . . The Holy Spirit has purposely neglected to teach us propositions of this sortas [they are] irrelevant to the highest goal (that is, to our salvation). . . . The intention of theHoly Ghost is to teach us how one goes to heaven, not how heaven goes.
Even a recent Pope praised Galileo for his supposed insight on this subject: “Galileo, a sincere believer, showedhimself to be more perceptive [in regard to the criteria of scriptural interpretation] than the theologians whoopposed him.”
I disagree with this papal statement on two grounds. First of all, Galileo was not pioneering a new visionfor the roles of science and religion more perceptively than anyone else. Rather, he was merely recapitulatingelementary ideas that were virtually as old as organised Christianity itself. “[Galileo’s] exegetical principleswere not in any sense novel, as he himself went out of his way to stress. They were all to be found in varyingdegrees of explicitness in Augustine’s
De Genesi ad Litteram ”—written twelve centuries before Galileo—“and,separately, they could call on the support of other [even] earlier theologians.”
Galileo indeed quotes atgreat length from Augustine and the church fathers. Not that Galileo knew anything about the history ofbiblical interpretation: “He had no expertise whatever in that area, so he evidently asked his Benedictinefriend, Castelli, to seek out references that would support the exegetical principles he had outlined.”
Furthermore, it is highly doubtful whether Galileo was truly “a sincere believer,” as he purported tobe. Recent scholarship has made a compelling case for “two Galileos, the public Catholic and the privatesceptic.”
Joshua, X.12–13. [Blackwell (1992), Ch. 3].
Galileo,
Letter to Duchess Christina (1615), [Galileo (1957),184–186].
Pope John Paul II, “Lessons of the Galileo case,”
Origins: Catholic News Service , November 12, 1992, 22,372, [McMullin (1998), 291]. [McMullin (1998), 314]. [McMullin (1998), 287]. [Wootton (2010), 249].
It is indeed striking that God plays an essential role in the scientific systems of so many other 17th-centuryscientists—such as Kepler, Descartes, Leibniz, and Newton—but no part whatsoever in Galileo’s. Whilemany of Galileo’s contemporaries made great efforts to synthesise scientific and transcendental knowledgethat can be nothing but sincere, Galileo brings up religion only when it conveniently serves his purposes,such as when professing in eloquent terms to be more committed to finding the true meaning of scripturalpassages than his opponents who used those texts against him.This sense of opportunism is reinforced by how poorly Galileo’s patchwork of trite ideas hold together.“He says first that saving the appearances is not enough to demonstrate the truth of a hypothesis and endsby remarking that saving the appearances is the most that can be demanded of an hypothesis.”
He arguesthat “officials and experts in theology should not arrogate to themselves the authority to issue decrees inprofessions they neither exercise nor study,” but apparently considered himself entitled to issue decreesin a profession that he neither exercised nor studied, namely that of scriptural interpretation. He also saysthat one should not rely on the Bible in scientific matters, then does precisely this himself, “violating his ownprohibition against using Scripture to support a philosophical thesis about the natural world.”
There islittle consistency in his views, except of course that he consistently chooses to espouse whatever principlesserve his rhetorical purposes at any given moment.Galileo’s interpretation of the Joshua passage is a prime example of his shameless drive to score rhetoricalpoints at any cost. It is perfectly reasonable to argue that the phrase about the sun “standing still” shouldnot be taken too literally. It is commonly accepted, as Galileo observes, that various things in the Bible “wereset down in that manner by the sacred scribes in order to accommodate them to the capacities of the commonpeople, who are rude and unlearned.”
Indeed, if the Bible is read literally, “it would be necessary to assignto God feet, hands, and eyes,” as Galileo says, but those passages are only figures of speech, according toorthodox Christian understanding. When the Old Testament says that the commandments handed to Moseswere “written with the finger of God,” the intended point is of course not that God has an actual physicalfinger and that he needs it to write. It doesn’t make a whole lot of sense that he could create the entireuniverse in under a week, or flood the entire earth at will, yet if he has to write something down he has topainstakingly trace it out in clay with his finger. Perhaps it is the same with the sun “standing still.” It’sjust a phrase adapted to everyday speech, not a scientific account. In fact, even Copernicus himself speaksof “sunrise” and “sunset,” as Galileo points out, even though the sun doesn’t move in his system. So itis hardly unreasonable to think that “the sacred scribes” used this kind of common parlance as well, even ifthey knew that the sun is always stationary.That’s all fine and well. But Galileo does not stop with this balanced point. He did not become a salonsensation by sensibly looking for middle ground and mutual reconciliation. His audience expects a moretriumphant and extravagant finishing blow. Seemingly to this end, Galileo makes the outlandish claim thatthe Joshua passage in fact literally agrees best with heliocentrism rather than geocentrism: [Wootton (2010), 247–248]. [McMullin (1998), 285–286]. [Finocchiaro (1989), 100]. [McMullin (1998),280–281].
Galileo,
Letter to Duchess Christina (1615), [Galileo (1957), 181].
Galileo,
Letter to Duchess Christina (1615), [Galileo (1957), 202].
68f we consider the nobility of the sun . . . I believe that it will not be entirely unphilosophicalto say that the sun, as the chief minister of Nature and in a certain sense the heart and soul ofthe universe, infuses by its own rotation not only light but also motion into other bodies whichsurround it. . . . So if the rotation of the sun were to stop, the rotations of all the planets wouldstop too. . . . [Therefore,] when God willed that at Joshua’s command the whole system of theworld should rest and should remain for many hours in the same state, it sufficed to make thesun stand still. . . . In this manner, by the stopping of the sun, . . . the day could be lengthenedon earth—which agrees exquisitely with the literal sense of the sacred text.
This is a terrible argument. It is so unscrupulous that its absurdity can be exposed simply by quoting thewords of Galileo himself, written in another context:If the terrestrial globe should encounter an obstacle such as to resist completely all its whirlingmotion and stop it, I believe that at such a time not only beasts, buildings, and cities would beupset, but mountains, lakes, and seas, if indeed the globe itself did not fall apart. . . . This agreeswith the effect which is seen every day in a boat travelling briskly which runs aground or strikessome obstacle; everyone aboard, being caught unawares, tumbles and falls suddenly toward thefront of the boat.
So in this manner “Joshua would have destroyed not only the Philistines, but the whole earth.”
Not tomention that the idea that the sun’s rotation on its axis is the only thing moving the planets is completelyunsubstantiated in the first place. It seems that Galileo pretended to believe in it on this occasion solely forthe sake of being able to make this scriptural argument. Once again the hypocrisy and unbridled opportunismof Galileo’s forays into biblical interpretation are plain to see. It is very difficult, if not impossible, to see thisinterpretation of the Joshua passage as a scientific argument that Galileo genuinely believed. It does makesense, however, as a crowd-pleasing bit of sophistry. If you are an Italian aristocrat who enjoys seeing thelearned establishment lose face but don’t want to rock the boat yourself, then you can live vicariously throughGalileo’s snappy comebacks and provocations. To this end it matters little whether they are scientificallysound or not.
Galileo’s famous conflict with the church was entirely unnecessary. It arose precisely because Galileo wasa lampooning populariser rather than a mathematical astronomer and scientist. “[Galileo] was far fromstanding in the role of a technician of science; had he done so, he would have escaped all trouble.”
Copernicus’ book had long been permitted, and Galileo’s own
Letters on Sunspots of 1613 had been censoredonly where it referred to scripture, not where it asserted heliocentrism. The church establishment had nointerest in prosecuting geometers and astronomers.Today many take for granted that a fundamental rift between science and religion was unavoidable. Somehave imagined for instance that Galileo defied the worldview of the church by demoting the earth from itssupposedly “privileged” position. 20th-century playwright Bertolt Brecht appreciated the dramatic flare offraming the conflict in such terms when he wrote a play about Galileo. He has one of the characters arguethe privilege point passionately:I am informed that Signor Galilei transfers mankind from the center of the universe to somewhereon the outskirts. Signor Galilei is therefore an enemy of mankind and must be dealt with as such.Is it conceivable that God would trust this most precious fruit of his labor to a minor frolickingstar? Would He have sent His Son to such a place? . . . The earth is the center of all things, andI am the center of the earth, and the eye of the Creator is upon me.
But historically this is nonsense, to be sure. Nobody was concerned about this at the time. In fact, classicalcosmology clearly stipulated that the Earth was not at all in a privileged position but rather condemned to
Galileo,
Letter to Duchess Christina (1615), [Galileo (1957), 212–214]. [Galileo (1953), 212]. [Koestler (1959),439]. [Santillana (1955), vii].
Bertolt Brecht,
Galileo , [Brecht (1966), 72–73], [McMullin (1998), 271].
Even Galileo himself added to the pile, writing in an Aristotelian mode that “after themarvellous construction of the vast celestial sphere, the divine Creator pushed the refuse that remained intothe center of that very sphere and hid it there lest it be offensive to the sight of the immortal and blessedspirits.”
Contemporaries reasoned alike: considering “the Vileness of our Earth,” it “must be situatedat the center, which is the worst place, and at the greatest distance from those Purer and incorruptibleBodies, the Heavens,” wrote John Wilkins, an Anglican bishop.
This is the very opposite of the argumentretrospectively imagined by Brecht and other modern minds.In reality, “a major part of the Church intellectuals were on the side of Galileo, while the clearest oppositionto him came from secular ideas” and philosophical opponents.
It was only because Galileo got involved withbiblical interpretation that he ended up in the crosshairs of the Inquisition. Nobody minded mathematicalastronomy, but the question of who has the right to interpret the Bible was the stuff that wars were made of.Luther challenged church authority and emphasised personal understanding of the Bible—“sola scriptura,”as the motto went. This was the core belief of protestantism, and eradicating protestantism was top of theagenda for the catholic church. Once they baited him into commenting on the Bible, it was all too easyfor Galileo’s enemies to connect Galileo’s otherwise harmless dabbling to this heresy du jour—a matter onwhich the church could not afford to show any weakness.There is only one mystery: Why did Galileo walk straight into such an obvious trap? The answer lies,as ever, in his mathematical ineptitude. Galileo was told by church authorities that “if Galileo spoke onlyas a mathematician he would have nothing to worry about.”
Galileo would presumably have followedthis advice if he could. The problem, of course, was that he did not have anything to contribute “as amathematician.” Since a mathematical defence of heliocentrism was beyond his abilities, Galileo was leftwith no other recourse than to roll the dice and try his luck in the dangerous and unscientific game ofscriptural interpretation.The church was thus reluctantly drawn into these astronomical squabbles and had to do something.The Inquisition settled for a slap on the wrist: in the future, Galileo must not “hold, teach or defend [theCopernican system] in any way whatever,” they decided.
They also ordered mild censoring of Copernicus’book, namely the removal of a brief passage concerning the conflict with the Bible and a handful expressionswhich insinuated the physical truth of the theory.
Galileo did indeed keep quiet for a number of years after being ordered to do so by the Inquisition. But timeschanged. After waiting for over a decade, Galileo felt it was safe to try the waters again. A new Pope wasin power, Urban VIII, who was quite liberal. even said of the 1616 censoring of Copernicus that “if it hadbeen up to me that decree would never have been issued.”
Galileo had good personal relation with thisnew open-minded Pope. So Galileo sensed an opening and obtained a permission to publish the Dialogue in1632. Or rather, as the Inquisition would later put it, he “artfully and cunningly extorted” this permissionto publish.
For when the permission was granted the Pope did not know about the private injunction of1616 for Galileo to keep off the subject. When this came to light the Pope was outraged and felt, with goodcause, that Galileo had been deliberately deceitful and reportedly stated that “this alone was sufficient toruin [Galileo] now.”
A special commission was thus appointed. It found many inappropriate things in the
Dialogue , but thiswas not a major issue, they noted, for such things “could be emended if the book were judged to have [Finocchiaro (2019), 377].
OGG.I.344, [Shea (2009), 109].
John Wilkins, [Shea (2009), 110]. [Santillana (1955), xii]. The same point is made by [Drake (1978), 288]. [Drake (1978), 249], referring to astatement of 1615. [Finocchiaro (1989), 147]. [Finocchiaro (1989), 149, 200–202]. [Drake (1978), 312]. [Finocchiaro (1989), 290]. [Drake (1978), 340–341].
The problem was instead that Galileo “overstepped hisinstructions” not to treat heliocentricism.
The same report also points out that Galileo had disrespected the Pope on another point as well. ThePope had asked Galileo to include the argument that since God is omnipotent he could have created anyuniverse, including a heliocentric one. So even though the church does not agree with Copernicus, their ownlogic, namely belief in God’s omnipotence, can be used to legitimate at least considering the possibility ofthis hypothesis. So that’s a useful argument that Galileo could have used to try to find at least a little bit ofcommon ground with his opponents. But instead of using it for such purposes of reconciliation as intended,Galileo used it to fuel the fires of conflict even more. He made had placed the Pope’s favourite argument“in the mouth of a fool,” the commission observed.
Galileo made Simplicio, the dumb character in the
Dialogue who constantly expresses the wrong ideas and is proven wrong at every turn, be the one who spokethe Pope’s words. He hardly did himself any favours with this disrespectful move.Following these findings, the second Inquisition proceedings took place in 1633: 17 years after the firstInquisition where Galileo had gotten off easy, and one year after the publication of his inflammatory
Dialogue in defence of Copernicanism. The outcome was a forgone conclusion. Galileo’s defence was transparentlydishonest. He pretended that, in the
Dialogue , “I show the contrary of Copernicus’s opinion, and that Coper-nicus’s reasons are invalid and inconclusive.”
This is of course pure nonsense. In private correspondenceshortly before, Galileo had spoken more honestly, and stated that the book was “a most ample confirma-tion of the Copernican system by showing the nullity of all that had been brought by Tycho and others tothe contrary.”
But now before the Inquisition he had to pretend otherwise. In light of the accusations,Galileo continued, “it dawned on me to reread my printed
Dialogue ,” and “I found it almost a new book byanother author.”
These transparent lies did little to save him. He was forced to abjure. The
Dialogue was prohibited, but not for its contents but rather, in the words of the Inquisition’s sentence, “so that thisserious and pernicious error and transgression of yours does not remain completely unpunished” and as “anexample for others to abstain from similar crimes.”
There is a popular myth that Galileo muttered “eppur si muove”—“yet it moves” (the earth moves, thatis)—as he rose from his knees after abjuring before the Inquisition. But this is certainly false.
Obviouslythe Inquisition would not have tolerated such insubordination, especially since the whole point the trial inthe first place was to punish Galileo for his defiance. Galileo had been shown the instruments of torture,and such a rebellious exclamation would have been the surest way to have them dusted off for the occasion.Today no historian believes the myth that Galileo mumbled these words before the Inquisition. Yet it remainsinstructive in warning us of the lengths many Galilean idol worshippers are willing to go to, who do notwant to admit the many ignominious historical facts about their hero. The sheer multitude of such mythsnow universally regarded as busted should leave us open to the distinct possibility that we have not gottento the end of them yet.A similar myth, appealing to anti-religion ideologues, is that “the great Galileo . . . groaned away hisdays in the dungeons of the Inquisition, because he had demonstrated . . . the motion of the earth.”
Butin reality Galileo was sentenced more for his provocateurism than for his science, and furthermore he wasnever imprisoned in any “dungeon.” He was sentenced to house arrest. A visitor “reported that [Galileo] waslodged in rooms elegantly decorated with damask and silk tapestries.”
Soon thereafter he retired to “thislittle villa a mile from Florence,” where “nearby . . . I had two daughters whom I much loved” and wherehe also received many friends and guests. Many today would pay dearly for such a retirement. Galileo gotit as a “punishment.” [Finocchiaro (1989), 222]. [Finocchiaro (1989), 219]. [Finocchiaro (1989), 221]. [Finocchiaro (1989), 262].
Galileo to Diodati, October 1629, [Drake (1978), 310]. [Finocchiaro (1989), 277–278]. [Finocchiaro (1989), 291]. [Drake (1978), 356–357]. [Voltaire (1762), 120]. [Finocchiaro (2010), 166].
Galileo to Diodati, 25 July1634, [Heilbron (2010), 328]. Galileo evaluated
Galileo’s catalogue of errors is extensive. It makes for quite a list even if we restrict ourselves only to centralclaims and only to matters his mathematically superior contemporaries did better than him: wrong value forcycloid area (§2.1); erroneous infinitesimal geometry (§§2.6, 3.1); wrong value of gravitational acceleration g (§3.1); erroneous theory of planetary speeds (§3.4); erroneous claim that path of fall is semicircular inabsolute space (§3.10); lack of proof that projectile motion is parabolic (§3.11); erroneous theory that tidesare caused by the motion of earth (§3.17); erroneous claim that the simple pendulum is isochronic (§3.14);erroneous theory that comets are an atmospheric phenomena (§4.14). Within a few years of his death hewas also corrected on inertia (§3.7) and the shape of a hanging chain (§3.13).To be sure, other people made mistakes too. Suppose I concede that everyone has an equal comedy oferrors to their name. Even so, this would still prove my point that Galileo was a dime a dozen scientist andnot at all a singular “father of modern science.” But I do not need to concede this much. Galileo’s sum oferrors are not just par for the course. They are exceptionally poor, and in matters of mathematics altogetherastonishing.The persistent myth of “Galileo’s mathematical genius” must certainly die. Historians will never seeGalileo’s true colours as long as they keep taking it for granted that Galileo was “the greatest mathematicianin Italy, and perhaps the world” in his time. In reality, tell-tale signs of mathematical mediocrity permeateall his works. Like all too many modern mathematics students, he reaches for a calculator or instrumentinstead of thinking (§§2.1; cf. §§2.4, 4.13), doesn’t do the reading (§§4.1, 4.6), makes computational mistakes(§§3.4, 4.11), methodological mistakes (§4.9), fundamental mistakes in the entire structure of his theory(§3.11), presents technical material in a clumsy way (§§3.1, 3.11), doesn’t seem to know the differencebetween proof and assertion (§5.2), and ignores the latest research which is much too advanced for him(§§2.5, 4.3–4.4, 4.14). Some pages of Galileo would not be out of place somewhere in the middle of the pilesof slipshod student homework that some of us grade for a living.Furthermore, Galileo’s errors concern some of his core achievements. I have discussed all of his notablescientific contributions and found much to object in every single case. Notably, Galileo uses “his” law of fallerroneously on a number of occasions (§§3.4, 3.5, 3.9, 3.10). He also not infrequently presents arguments thatare demonstrably inconsistent with his core beliefs, such as his tidal theory contradicting his own principleof relativity (§3.17), his Joshua argument contradicting his own principle of inertia (§5.4), and his objectionto the geocentric explanation of sunspots being inconsistent with his own heliocentrism (§4.12).Apollo 15 astronauts performed an experiment on the moon. They dropped a hammer and a feather andfound that they fell with the same speed. “Galileo was correct,” they concluded in a famous video recordingstill often shown in science classrooms today. Actually, Lucretius was correct, because he is the one who saidthis would happen in the absence of air (§3.1). Galileo was wrong because he considered it “obvious” thatthe moon had an atmosphere (§4.6). If the astronauts wanted to test Galileo’s theory they should not havedropped a hammer and a feather. They should have taken off their helmets and suits and tried to breathe.That would have showed you how “right” Galileo really was.Posterity have chosen to remember only Galileo’s successes while forgetting his numerous errors. It iseasy to be a hero of science if you can count on such selective amnesia to always put you in the most flatteringlight. If there had been air on the moon, the astronauts would have hailed Galileo for this “discovery” instead.The point generalises. Galileo made many claims that would have earned him not a little credit if theyhad been correct. This includes a number of the errors mentioned above, plus his erroneous claims thatthe motion of the earth is undetectable by experiment (§3.8), that meteors entering the earth’s atmospherewould cool down due to wind (§5.3), that centrifugal force is independent of radius (§3.9), and that the curveof fastest descent is a circular arc (§3.15). [Costabel & Lerner (1973), I.41]. [Heilbron (2010), 303].
Erroneous theory of planetary speeds (§3.4); erroneoustheory of tides (§3.17); erroneous claim that the simple pendulum is isochronic (§3.14); erroneous theory that comets are anatmospheric phenomena (§4.14); erroneous notion of horizontal inertia (§3.7); erroneous claim the shape of a hanging chain isparabolic (§3.13).
Actually these discoveries are not identical with those of Galileo but rather go beyond them, because Galileonever “calculated the orbits of heavenly bodies using methods and data of Kepler,” as Harriot did, who wasa better mathematician.In the history of science, virtually any given scientific discovery has a complicated genealogy of precursorsand independent discoveries and near-discoveries if one only looks closely enough. We must not get carriedaway and let this kind of thing take away all the honour of discovery, for then no one would ever be creditedwith anything. It may seem unfair, therefore, to catalogue so many instances in which Galileo was preemptedby others, since hardly any scientist would come out of such an examination without having to share theirclaims to fame with others.Nevertheless, the list of independent discoveries by others does tell us something. First of all, it showsthat we must reject the simplistic idea of Galileo as “the father of modern science,” as if his achievementwas somehow singular and different in kind from his contemporaries. On the contrary, Galileo’s science wasbusiness as usual at the time, which is why so many of his discoveries were made independently by others.Furthermore, telescopic astronomy is somewhat of a unique case, as we have noted. If Galileo had nevertouched a telescope, the march of science would not have been set back more than a matter of months. Manywere doing this kind of work, and the fact that Galileo managed to monopolise almost all the fame for it hadvery little to do with science and much to do with shrewd jostling. But for authors of story-book history ofscience it is all too tempting to ignore this and focus the narrative on one convenient protagonist.It is also instructive to compare Galileo to Kepler in these kinds of terms. We can find independentcontemporary discoveries for almost everything Galileo did, but not so for Kepler’s achievements, eventhough many of them are still central in modern science. Harriot was a “second Galileo” and you could goon to a third or a fourth stand-in without much loss. It would be much harder to find a “second Kepler.” In [Büttner et al. (2001), 184].
73y view it is not hard to see why: Kepler was an excellent mathematician who worked on difficult things,while Galileo didn’t know much mathematics and therefore focussed on much easier tasks. The standardstory has it that Galileo’s insights were more “conceptual,” yet at least as deep as technical mathematics.
On this account it is imagined that basic conceptions of science that we consider commonsensical today wereonce far from obvious: we greatly underestimate the magnitude of the conceptual breakthroughs requiredfor these developments because we are biased our modern education and anachronistic perspective. But ifthis is true, how come that Galileo’s ideas—for all their alleged “conceptual” avant-gardism—spontaneouslysprung up like mushrooms all over Europe? And how come all of those ideas can easily be explained to anyhigh school student today, if they are supposedly so profound and advanced? The same cannot be said forKepler’s ideas. They were neither simultaneously developed by dozens of scientists, nor can they be taughtto a modern student without years of specialised training. Perhaps this contrast between Galileo and Keplersays something about what genuine depth in the mathematical sciences looks like.
Already in his own day, “Galileo was a celebrity and a hero, receiving endless praise during his lifetime fromadmirers, friends, and opponents alike.”
Or so
The Cambridge Companion to Galileo tells us. There issome truth to this. Galileo was, after all, a spirited populariser and polemicist, a lightning rod in the conflictwith the church, and a successful self-promotor in connection with the telescope. All of which contributedto make him a household name for reasons not primarily based on scientific achievement.But I challenge the notion that Galileo received “endless praise” from all quarters, especially if we focuson the judgements of mathematically competent people. I believe that, as Figure 30 suggests, Galileo’sfame has snowballed over time, while contemporary scientists, who understood Galileo’s work in the contextof his day, were much less impressed. Indeed, it is my contention that Galileo erroneously gets credit forinsights that were already well established in the Greek mathematical tradition and the works of peoplelike Archimedes. This squares well with the statistical trend that Galileo’s prominence stands in inverseproportion to that of Archimedes. Archimedes loomed large in scientific consciousness in the 17th century,but as his works moved from scientists’ desks to dusty shelves of ancient history, people began to forgetwhat had been obvious to those who had studied him with care. Meanwhile, Galileo’s lively prose was a lotmore accessible than abstruse Archimedes. And as knowledge of Archimedes and the true state of science inGalileo’s time faded from living memory, readers more readily bought Galileo’s self-aggrandising narrativethat everyone but him was a foolish Aristotelian.
I have already quoted Descartes’s opinion that Galileo “is eloquent to refute Aristotle, but that is nothard.”
Descartes continued in the same vein: “I see nothing in his books to make me envious, andhardly anything I should wish to avow mine.”
Galileo’s mathematical demonstrations in particular didnot impress Descartes: “he did not need to be a great geometer to discover them.”
These were not emptywords. Descartes backed up his assessment with an extensive laundry list of specific critiques, a number ofwhich we have had occasion to mention above.
Defenders of the standard view of the great Galileo have been puzzled by this and often tried to write it offon the grounds that Descartes was “perhaps too much [Galileo’s] rival to judge his merits quite impartially.”
But “can Descartes’ critique of Galileo’s last work on motion really be discarded as the envious comment ofa stubborn philosopher, unable or unwilling to acknowledge that this work inaugurated a new physics?”
More recent scholarship says: no. “Descartes was not only right in asserting that the theory of motion
Examples of such allegedly deep conceptual developments are for instance the relation between mathematics and thephysical world ([Gorham et al. (2016), Ch. 1] and below), and the conception of velocity as an instantaneous quantity([Gorham et al. (2016), Ch. 3]). [Machamer (1998), 389].
Descartes to Mersenne, 11 October 1638, [Drake (1978),390]. §2.2.
Descartes to Mersenne, 11 October 1638, [Drake (1978), 392].
Descartes to Mersenne, 11 October 1638,[Drake (1978), 391]. §§2.5, 3.11, 3.12, 5.3. [Hall (1983), 114]. [Damerow et al. (2004), 264]. Galileo,Kepler,Copernicus,Tycho,Archimedes,Huygens (click on line/label for focus)
GalileoKeplerCopernicusTychoArchimedesHuygens
Search in Google Books
Books Ngram Viewer !" % English (2019) % Case-Insensitive Smoothing of 50 % Figure 30: Mentions of Galileo compared to other scientists in the Google Books corpus.presented in the
Discorsi did not cohere but also in claiming that some of its foundational concepts werequestionable.”
So Descartes—one of Galileo’s most mathematically competent contemporaries—thought his work wasquite useless. And he supported this assessment with compelling and perceptive arguments. “Descartes’judgement of Galileo’s mechanics deserves more sympathy than it usually receives.”
Kepler was the best mathematical astronomer in Galileo’s day. What was his relation to Galileo? Certainlynot as substantive as one might expect.One wonders why these two great men, who were both present and actual participants at thevery birth of some of the most world-shaking scientific events, and who apparently were verymuch in accord in their astronomical views, did not engage in a more on-going correspondenceover these years.
This is a puzzle and a paradox if one accepts the standard view of Galileo. But of course it becomes perfectlyunderstandable as soon as one realises that Kepler, who was a brilliant mathematician, had very little tolearn from a dilettante such as Galileo.Their correspondence began when, “in 1597, as a lowly high-school teacher of mathematics and a fledglingauthor, Kepler . . . vainly implored Galileo, the established university professor, to give him the benefit ofa judgment of his first major work.”
Galileo replied briefly, declaring himself in agreement with theCopernican standpoint of Kepler’s book, although “I have preferred not to publish, intimidated by thefortune of our teacher Copernicus, who though he will be of immortal fame to some, is yet by an infinitenumber (for such is the multitude of fools) laughed at and rejected.”
Kepler is happy to hear that Galileo,“like so many learned mathematicians,” has joined in supporting “the Copernican heresy.”
Galileo shouldgrow a spine though, “for it is not only you Italians who do not believe that they move unless they feel it,but we in Germany, too, in no way make ourselves popular with this idea.”
Kepler urges Galileo to focus [Damerow et al. (2004), 267]. [Moody (1966), 25]. [Postl (1977), 329]. [Rosen (1966), 263]. The work inquestion is Kepler’s
Mysterium Cosmographicum (1596).
Galileo to Kepler, 4 August 1597, [Drake (1978), 41].
Kepler,[Baumgardt (1951), 40–41].
Kepler to Galileo, 13 October 1597, [Baumgardt (1951), 41].
75n compelling mathematics instead of on the number of fools: “Not many good mathematicians in Europewill want to differ from us; such is the power of truth.”
At this point, Kepler naively mistook Galileo for a serious scientific interlocutor. In connection withtheir discussion of Copernicanism, Kepler noted the importance of parallax and asked Galileo if he couldhelp him with observations for this, adding detailed instructions regarding the exact nature and timing ofthe requisite measurements.
Kepler also sent additional copies of his book, as Galileo had requested, and“asked only for a long letter of response as payment—which was, however, never forthcoming.”
Galileostopped replying, presumably since this kind of actual, substantive mathematical astronomy was beyond hisabilities.Kepler’s was not the only scientific correspondence Galileo shrunk from. He also neglected to reply toall three letters he received from Mersenne, offering only “the rather limp excuse that he found Mersenne’shandwriting too hard to read.”
It seems he had a point, for others complained similarly of Mersenne’sletters that “his hande is an Arabicke character to me.”
Nevertheless these are further instances of Galileofailing to reply to a serious scientific interlocutor.The tables were turned in 1610. While Galileo had not seen the greatness in Kepler’s book, moremathematically competent people had, and consequently Kepler had succeeded Tycho Brahe as the ImperialMathematician of the Holy Roman Emperor. “In that capacity Kepler’s help was sorely needed by Galileoin 1610, when his momentous telescopic discoveries were being received on all sides with skepticism andhostility.” “To Kepler’s credit . . . he manfully swallowed his justifiable resentment” and “ungrudginglygave Galileo the authoritative support he could find nowhere else.” “In spite of Galileo’s earlier silenceafter his own request in 1597, Kepler quickly and enthusiastically responded to Galileo’s findings, within 11days.”
Galileo surely had this in mind when, in reply, he praised Kepler for “your uprightness and loftinessof mind”—“you were the first one, and practically the only one, to have complete faith in my assertions”regarding the telescopic discoveries.
Kepler’s support was indeed crucial, and Galileo keenly flaunted itto his advantage.
Galileo did not take the occasion to revive their scientific discussion or comment on Kepler’s brilliantnew book, the
Astronomia Nova (1609). Instead he only wanted to make fun of dumb philosophers:Oh, my dear Kepler, how I wish that we could have one hearty laugh together! Here at Paduais the principal professor of philosophy, whom I have repeatedly and urgently requested to lookat the moon and planets thorough my glass, which he pertinaciously refuses to do. Why are younot here? What shouts of laughter we should have at this glorious folly!
Kepler wasn’t there because he was busy doing real science. He ignored idiotic philosophers, as all math-ematically competent people had done for thousands of years. Galileo, however, had nothing better to dothan to sit around and laugh at idiots. To him, it seems, the most desirable application of science is a cleverput-down and the last laugh.Kepler eventually grew weary of Galileo’s dilettantism. When, in later years, he found himself having tocorrect errors in Galileo’s superficial writings, he fully justifiably took a patronising tone:Galileo rejects Tycho’s argument that there are no celestial orbs with definite surfaces becausethere are no refractions of the stars. . . . Rays reach the earth perpendicular to the spheres,says Galileo, and perpendicular rays are not refracted. But oh, Galileo, if there are orbs, it isnecessary that they be eccentric. Therefore, no rays perpendicular to the spheres reach to theearth except at apogee and perigee. Hence, Tycho’s argument is a strong one, if you are willingto listen.
Galileo denies that the Ptolemaic hypothesis could be refuted by Tycho, Copernicus, or others,and says that it was refuted only by Galileo through the use of the telescope for observation of
Kepler to Galileo, 13 October 1597, [Heilbron (2010), 113].
Kepler to Galileo, 13 October 1597, [Baumgardt (1951),42]. [Postl (1977), 326]. OGG.X.71. [Lewis (2012), 744], “scrivere in carattere intelligibile,” OGG.17.370.
Charles Cavendish, [Halliwell (1841), 72]. [Rosen (1966), 263]. [Rosen (1966), 263, 264]. [Postl (1977),327].
OGG.X.421. [Rosen (1966), 264]. [Galileo (1957), 60].
Galileo to Kepler, 1610. [Burtt (1932), 66–67].
Kepler, appendix to
Hyperaspistes (1625), [Drake & O’Malley (1960), 350]. orbis magnus in Copernicus.
In both cases, Kepler exposes Galileo’s true colours. Galileo doesn’t treat the matter as a serious mathe-matical astronomer, but rather as a superficial and unscrupulous rhetorician. Kepler is right to scold him ashe does.
Huygens was perhaps the greatest physicist of the generation between Galileo and Newton. He is oftenportrayed as continuing the scientific program of Galileo. But, in the 22 thick volumes of his collectedworks, one searches in vain for any strong praise of Galileo, let alone anything remotely like calling hima “father of science.” The closest Huygens ever gets to mentioning Galileo favourably is in the context ofa critique of Cartesianism.
In the late 17th century, the teachings of Descartes had attracted a strongfollowing. In the eyes of many mathematicians, the way Cartesianism had become an entrenched beliefsystem was uncomfortably similar to how Aristotelianism had been an all too dominant dogma a centurybefore. Huygens makes the parallel explicit:Descartes . . . had a great desire to be regarded as the author of a new philosophy . . . [and] itappears . . . that he wished to have it taught in the academies in place of Aristotle. . . . [Descartes]should have proposed his system of physics as an essay on what can be said with probability.. . . That would have been admirable. But in wishing to be thought to have found the truth, . . .he has done something which is a great detriment to the progress of philosophy. For those whobelieve him and who have become his disciples imagine themselves to possess an understanding ofthe causes of everything that it is possible to know; in this way, they often lose time in supportingthe teaching of their master and not studying enough to fathom the true reasons of this greatnumber of phenomena of which Des Cartes has only spread idle fancies.
It is in direct contrast with this that Huygens slips in a few kind words for Galileo:[Galileo] had neither the audacity nor the vanity to wish to be the head of a sect. He was modestand loved the truth too much.
Historians have observed that Huygens in all likelihood quite consciously intended this passage to apply tohimself as much as to Galileo.
Perhaps this is why Huygens is surely too generous in praising Galileo’salleged “modesty.”In any case, it is very interesting to see what Huygens says about Galileo’s actual science in this passage.Let us read it, and keep in mind that this is as close as Huygens ever gets to praising Galileo, and thatthe context of the passage—a scathing condemnation of Cartesianism—gives Huygens a notable incentiveto put Galileo’s scientific achievements in the most positive terms for the sake of contrast. In light of this,Huygens’s ostensible praise for Galileo is most remarkable, I think, for how qualified and restrained it is.Here is how I read it:
Kepler, appendix to
Hyperaspistes (1625), [Drake & O’Malley (1960), 344–345]. The passage criticised is Galileo,
As-sayer (1623), [Drake & O’Malley (1960), 257–258].
Huygens, 1693,
Oeuvres
X.403–406.
Ibid. [Westman (1980b),98].
Ibid. [Westman (1980b), 97]. [Dugas (1954), 32], [Westman (1980b), 97], [Dijksterhuis (2004), 247–248],[Andriesse (2005), xi]. . . .
Meaning: He said all the right things about about mathematics and scientific method, but he didn’t actuallycarry through on it. Given his rhetoric, he ought to have been able to do it, but be didn’t. . . . and one has to admit that he was the first to make very beautiful discoveries concerning thenature of motion . . .
He wasn’t the first, as we now know, but even though Huygens is overly generous his formulation is stillvery restrained: “one has to admit” ( il faut avouer )—a phrase that suggests reluctance to concede the point.Who speaks of their greatest hero in such terms? One “has to admit” that he made some discoveries? Thatseems more like the kind of phrasing you use to describe the work of someone who is overrated, not someoneyou esteem as the founder of science. . . . although he left very considerable things to be done.
Exactly. What is most striking and remarkable about the work of Galileo is not the few discoveries he“admittedly” made, but how very little he actually accomplished despite all his posturing about mathematicsand scientific method. It seems to me that Huygens and I agree on this.
Newton famously said that “if I have seen further it is by standing on the shoulders of giants.”
Many haveerroneously assumed that Galileo was one of these “giants.”
One scholar even proposes to explain that“when Newton credits Galileo with being one of the giants on whose shoulders he stood, he means . . . ”
Wedo not need to listen to what this philosopher thinks Newton meant, because the first part of the sentenceis false already. The assumption that Galileo was one of the scientific giants in question has no basis in fact.The closest Newton gets to praising Galileo is in the
Principia . After introducing his laws of motion,Newton adds some notes on their history.The principles I have set forth are accepted by mathematicians and confirmed by experiments ofmany kinds. . . . By means of the first two laws and the first two corollaries Galileo found thatthe descent of heavy bodies is in the squared ratio of the time and that the motion of projectilesoccurs in a parabola.
The laws and corollaries in question are: the law of inertia, which Galileo did not know;
Newton’s force law F = ma , which Galileo also did not know; and the composition of forces and motions, which was establishedin antiquity. Of course, once you are looking at the world though Newtonian mechanics it is natural to think thatsurely Galileo must have had these laws, because that is so obviously the right way to think about parabolicmotion. Hence, “[according to] the myth in which he appears as the founder of classical dynamics, . . .[Galileo] must surely have known the proportionality of force and acceleration. . . . But to those who havebecome acquainted with Galileo through his own works, not at second hand, there can be no doubt thathe never possessed this insight.”
Indeed, “Newton almost certainly did not read [Galileo’s]
Discorsi —if,indeed, he ever did—until some considerable time after he had published the
Principia .” “Hence Newton (rather too generously, for once!) allowed to Galileo the discovery of the first two laws ofmotion.” The reason for Newton’s excessive charity is not hard to divine. Newton’s
Principia is marked byan obvious and vehement “anti-Cartesian bias.” “Because of his strongly anti-Cartesian position, Newtonmight . . . have preferred to think of Galileo rather than Descartes as the originator of the First Law.”
Ibid.
Newton to Hooke, 5 February 1676,
The Correspondence of Isaac Newton
I.416. [Brake (2009), 106]. [Pitt (1992), 5]. [Newton (1999), 424]. Note that Newton doesn’t say Galileo was the discoverer of these laws. Indeed,“Newton’s Latin contains some ambiguity” for it “can have two very different meanings: that the two laws were completelyaccepted by Galileo before he found that projectiles follow a parabolic path, or that these two laws were already generallyaccepted by scientists at the time that Galileo made his discovery of the parabolic path.” [Cohen (1967), xxxviii]. §3.7.
The Pseudo-Aristotelian
Mechanics has a law for the composition of motions that “has all the important features of theParallelogram Rule” of Newtonian physics [Miller (2017), 162]. [Dijksterhuis (1961), 344]. [Cohen (1967), xxvi]. [Hall (1983), 108]. [Newton (1999), 46]. [Cohen (1967), xli].
Prima Lex [i.e., the law of inertia] of Newton’s
Principia was derived directlyfrom the
Prima Lex of Descartes’s
Principia .” Clearly, then, Newton’s attribution of these laws to Galileo means next to nothing. In fact, there isfurther evidence that it was not meant as high praise in any case. For when Newton continues his historicaldiscussion he says on the very same page: “Sir Christopher Wren, Dr. John Wallis, and Mr. ChristiaanHuygens, easily the foremost geometers of the previous generation , independently found the rules of thecollisions and reflections of hard bodies.”
So evidently Newton was of a mind to point out who “theforemost geometers” of the past were, yet he had no such words for Galileo—a telling omission. Altogetherthere is no evidence that Newton regarded Galileo particularly highly, let alone considered him anywherenear a “father of modern science.”
My polemic against Galileo is over. I now wish to consider the broader implications of the story I havetold. There are much more important matters at stake than deciding whether Galileo was smart or dumb.The traditional picture of Galileo is the linchpin of an entire historical worldview. When we pull the rugunderneath him, a cascade of misconceptions come crashing down.Galileo’s status stands and falls with our willingness to accept radical relativism. His discoveries areso basic and obvious that the only way to consider them profound is to maintain that they were once not basic and obvious. In other words, that they are fundamentally different in character from anything theGreeks were doing, for example. Believing in Galileo’s greatness means believing that the history of scienceis a story of “conceptual” revolutions that made previously unimaginable things suddenly obvious. It meansbelieving that basic principles of scientific method that seem so obvious today were in fact once completelyoutside the cognitive universe of even extremely sophisticated mathematical scientists like Archimedes. Itmeans believing that first-rate mathematicians wasted enormous efforts on specialised technical work whenvastly greater advances were to be had by simply postulating a few basic philosophical ideas.The idea that modern science was born in a Galilean revolution is thus based on seeing history as soakedin cultural relativism and replete with dramatic Gestalt shifts. This is the worldview and historiographicalapproach of many who are far removed from mathematics. Mathematicians find this hard to stomach. Theyare more ready to say: There is a spiritual unity of scientific thought from ancient to modern times. Greatminds think alike. What is obvious to us was obvious to the Greeks. It is ludicrous to think that generationsof Greek mathematical geniuses of the first order, with their extensively documented interest science, allsomehow failed to conceive basic principles of scientific method. Such is the historiographical outlook ofmathematicians.How did modern science grow out of mathematical and philosophical tradition? The humanistic perspec-tive is that science needed both: it was born through the unification of the technical but insular know-howof the mathematicians with the conceptual depth and holistic vision of the philosophers. The mathematicalperspective is that science is what the mathematicians were doing all along. Science did not need philosophyto be its eye-opener and better half; it merely needed the philosophers to step out of the way and let themathematicians do their thing.This is why Galileo is the idol of the humanists and the bane of the mathematicians. The philosopherssay he invented modern science; the mathematicians that he’s a poor man’s Archimedes. The issue cutsmuch deeper than merely allotting credit to one century rather than another. Much more than a question ofthe detailed chronology of obscure scientific facts, it is a question of worldview and how one should approachand understand history.If we think there is only one common sense, and that mathematical truth and thought is the same foreveryone, then we are strongly inclined to see Galileo’s achievements as trifling. On the other hand, if wereject the very notion of a universal scientific common sense, then we are primed to think that Galileoopened up an entirely new world with his style of science, and that the Greeks couldn’t even think suchthings, because the way they approached the world was just inherently and profoundly different from ours. [Cohen (1967), xxvii]. [Newton (1999), 424]. Emphasis added.
79o studying Galileo is a mirror to much larger questions. Either you are a cultural relativist and you thinkGalileo was a revolutionary, or you think mathematical thought is the same for you, me and everybody whoever lived, and then you think Galileo was just doing common-sense things. Those are the two possibilities.You have to pick sides. You can’t mix and match. You can’t have both mathematical universalism and
Galileo being a revolutionary. The two contradict one another.
Plato is sometimes seen as the mathematician’s philosopher. According to legend, an inscription above theentrance to his Academy admonished: “Let no one enter here who is ignorant of geometry.” Indeed, Platofounded his entire epistemology on the example of geometry, and speculated at some length about themathematical design of the universe.
But Plato was no mathematician. Many misconceptions about Greek mathematics and science stemfrom mistakenly assuming that he speaks on behalf of these fields. The image of mathematics conveyedin Plato’s works has aptly been called “Mathematics: The Movie” —that is, a dumbed-down, vulgarisedpage-turner, rich in grandiose sentimentality and sanctimonious moralising that strikes anyone familiar withthe real thing as irresponsibly simplistic.In Plato, mathematics is purer than snow. To apply it to the physical world is to defile it. Many modernscholars assume this was the view of Greek geometers generally, but in fact even Plato’s own words showthat this was clearly not the case:No one with even a little experience of geometry will dispute that this science is entirely theopposite of what is said about it in the accounts of its practitioners. . . . They give ridiculousaccounts of it, . . . for they speak like practical men, and all their accounts refer to doing things.They talk of “squaring,” “applying,” “adding,” and the like, whereas the entire subject is pursuedfor the sake of knowledge . . . [and] for the sake of knowing what always is, not what comes intobeing and passes away.
So Plato’s view of mathematics is in fact, by his own admission, in direct and explicit opposition to the viewof actual mathematicians.It is evident that Plato’s role [in the development of mathematics] has been widely exaggerated.His own direct contributions to mathematical knowledge were obviously nil. . . . The exceedinglyelementary character of the examples of mathematical procedures quoted by Plato and Aristotlegive no support to the hypothesis that [mathematicians] had anything to learn from [them].
Mathematicians do not pay any attention to philosophers trying to tell them about their own subject usingone or two basic examples—not then, not in the 17th century, and not today.
In keeping with his praise for the abstraction of mathematics, Plato looked down on empirical scienceand satirised it with no little scorn: “Birds . . . descended from . . . simpleminded men, men who studied theheavenly bodies but in their naiveté believed that the most reliable proofs concerning them could be basedupon visual observation,” leading them to grow wings in order to be able to look at the heavens moreclosely. Focussing on empirical data is for unphilosophical beasts, in other words. Such an attitude is quitean obstacle to science. If you glorify pure and abstract thought as the only worthwhile pursuit of rationalbeings, and deride empiricism as fit only for brutes, then you’re not going to get a whole lot of science done.But again there is very little evidence that mathematically competent people and working scientists evershared this view, and plenty of evidence to the contrary. This would be evident to all “if modern scholarshad devoted as much attention to Galen or Ptolemy as they did to Plato and his followers.”
Plato isadvancing his personal ideology, not describing Greek science. Indeed, “the retarding effect that Platonismcould thus exert on science . . . is encountered particularly during periods in which disparagement of theempirical study of nature on philosophical grounds was supported by a contempt for the material world from
Meno , 82–85.
Timaeus , 31–32, 55–61. [Netz (2002), 215].
Plato,
Republic , VII, 527, [Plato (1997), 1143]. [Neugebauer (1969), 152]. §2.2.
Plato,
Timaeus , 91d, [Plato (1997), 1290]. [Neugebauer (1969), 152]. Seealso [Russo (2004), 194].
In other words, it is a view that has often appealed to philosophers and ideologues forreasons external to science itself.
Dante called Aristotle ‘the Master of those who know’. Aristotle was so regarded by learned menfrom the time of Aquinas to that of Galileo. If one wished to know, the way to go about it wasto read the texts of Aristotle with care, to study commentaries on Aristotle in order to grasp hismeaning in difficult passages, and to explore questions that had been raised and debated arisingfrom Aristotle’s books.
Yes, but note well the crucial qualifier: from the time of Aquinas , not from the time of Aristotle himself.The intellectual quality of the European Middle Ages was indeed so low that subservience to Aristotle waswhat passed for erudition. But in the far more advanced intellectual culture of ancient Greece people werenot so foolish. Theophrastus—who “was head of the Lyceum for some 36 years after Aristotle’s death”—was“highly critical of Aristotle, both of his specific physical theories and of his general doctrine of causation.”
Theophrastus was succeeded by Strato, whose “position is [even] further from Aristotle”: he “rejected many ofAristotle’s ideas” and “broke new ground in his attempts to investigate problems in dynamics and pneumaticsexperimentally.”
And these were Aristotle’s immediate successors at his own Lyceum. Not even they wereparticularly attached to his ideas. Mathematicians—who were used to progress in their field, and who wereused to accepting propositions based on proof rather than philosophical authority—would have had no reasonto adhere to Aristotle’s teachings on mechanics, and indeed there is virtually no evidence that they ever did.It is therefore a mistake to ask the question: “How is it that the scientific enterprise undertaken by theGreeks, with their unique interest in the rational interpretation of nature, nevertheless culminated in theradically wrong natural philosophy of Aristotle?”
It didn’t. Aristotle was not the “culmination” of Greekthought. He was one particular philosopher who lived well before the true flourishing of Greek science andwho didn’t know any mathematics to speak of. The notion of taking Aristotle as their master in mathematicalsciences such as mechanics and astronomy was laughable to mathematically competent people in Hellenisticantiquity.Aristotle, for whom the brain had a cooling function, could not possibly have enjoyed excessiveauthority in the eyes of Herophilus and his disciples, the founders of neurophysiology, nor in thoseof Archimedes and Ctesibius, who had designed machines that could perform operations whoseimpossibility Aristotle had “demonstrated”. [There is] an analogous supersession of Aristotle inAristarchus’ heliocentric theory [and many other domains.] . . . It is clear that the “excessiveauthority of Aristotle” applies only to later ages and is often backdated.
Altogether the spectres of Plato and Aristotle have ruined historical understanding of Greek science—theformer by falsely portraying it as adverse to empiricism, the latter by giving the false impression that ittreated motion and mechanics only dogmatically and through a qualitative, philosophical lens. “Negativeassessments . . . of ancient Greek mechanics need to be reconsidered. . . . The common view . . . is basedpartly on a misguided Platonizing tendency . . . , and partly on a mistaken view that the tradition Galileo isrejecting goes back to ancient Greek thought.”
Plato and Aristotle were not the pinnacle of Greek thought, as people who read too much philosophy andnot enough mathematics are inclined to believe. In the century or two after their death, the Greeks madeenormous strides in mathematics and science. This is the age that gave us excellent mathematicians andscientists like Euclid, Aristarchus, Archimedes, Eratosthenes, Apollonius, and Hipparchus, all of whom were [Dijksterhuis (1961), 15]. [Drake (2001), 1]. [Lloyd (1973), 8–9]. [Lloyd (1973), 19]. [Cohen (1994),243]. [Russo (2004), 233]. [Berryman (2009), 176].
PrincipiaMathematica as a work in which basic physical laws are both formulated and accompanied by superb appli-cations,” namely a detailed investigation of the floatation behaviour of paraboloids that was “the standardstarting point for scientists and naval architects examining the stability of ships” still thousands of yearslater, and that can also be used to explain phenomena such as “the sudden tumbling of a melting icebergor the toppling of a tall structure due to liquefaction of the ground beneath it.”
What more could oneask for? It is quite simply an outstanding masterpiece of science by the standards of any age. Only themathematically illiterate could fail to grasp its immense significance—as indeed they have.Now, poor Archimedes, he is often misunderstood. Many people who don’t care for mathematics hardlyeven know this work exists. But if they do look at it they say: What’s this? It’s just a bunch of technicalgeometry about parabolas and such. Archimedes says not a word about any experiment, not a word aboutany empirical data, nothing about testing his theory. It seems to have very little to do with the real world.It’s not science; it’s an exercise in pure geometry. The Greeks may have been excellent geometers but theydidn’t really do science, you see. They were speculative thinkers, philosophers. They were great with abstractstuff but they didn’t have the sense to ground their fanciful theories in reality.That attitude is completely wrong, in my opinion. Ask yourself: What are the odds that Archimedes gothis detailed, quantitative theory of floating bodies absolutely spot-on right if he was ignorant of empiricismand experiment and scientific method? Was he just sitting around doings speculative armchair geometryand, whoops, it just happened to come out exactly equal to empirical facts in a range of far from obviousways? Are we supposed to believe that was just dumb luck? It doesn’t make any sense.I much more plausible interpretation is that of course Archimedes knew about the scientific method. Ofcourse Archimedes tested his theory by experiment. That is obvious from the accuracy of his results. Histext doesn’t say that because he was too good of a mathematician to think that kind of kid’s stuff countedfor much of anything. He only published the actual theory, not the obvious tests that any simpleton coulddo for themselves. Galileo was precisely that simpleton. He spent his whole life spelling out those partsthat Archimedes thought were too trivial to mention. People ignorant of Archimedes are readily tricked intothinking that this was somehow profound. But mathematicians know better. [Rorres (2004), 32–33]. These applications were not discussed by Archimedes in his treatise. .4 Lost and ignored In the historiography of the Scientific Revolution there has been a notorious difficulty with . . .the idea of “the Archimedean origins of early modern science” . . . : If the 16th-century impactof Archimedes’ work in mathematical physics was so revolutionary, why, then, was its originalimpact in the 2nd century B.C. so negligible?
A simple rebuttal is: it wasn’t. When we study Greek science and mathematics, we must remember that onlya fraction of even the very best works of this era have survived. 17th-century mathematicians had boundlessrespect and admiration for the ancient Greeks, but they never doubted that there were many more “longburied monuments of geometry in which so many great findings of the Ancients lie with the roaches andworms.”
And they were right.In the 20th century a few such masterpieces were recovered. In 1906, a work of Archimedes that had beenlost since antiquity was rediscovered in a dusty Constantinople library. The valuable parchment on whichit was written had been scrubbed and reused for some religious text, but the original could still just aboutbe made out underneath it. “Our admiration of the genius of the greatest mathematician of antiquity mustsurely be increased, if that were possible,” by this “astounding” work, which draws creative inspirationfrom the mechanical law of the lever to solve advanced geometrical problems. If this brilliant work byantiquity’s greatest geometer only survived by the skin of its teeth and dumb luck, just imagine how manymore works are lost forever.Also in the 20th century, divers chanced upon an ancient shipwreck, which turned out to contain acomplex machine (the so-called Antikythera mechanism). “From all we know of science and technology inthe Hellenistic age we should have felt that such a device could not exist.” “This singular artifact is nowidentified as an astronomical or calendrical calculating device involving a very sophisticated arrangementof more than thirty gear-wheels. It transcends all that we had previously known from textual and literarysources and may involve a completely new appraisal of the scientific technology of the Hellenistic period.”
Another area in which the Greeks appear to have been much further ahead than conventional sourceswould lead one to believe is combinatorics. Of this entire mathematical field little more survives than onestray remark mentioned parenthetically in a non-mathematical work:Chrysippus said that the number of intertwinings obtainable from ten simple statements is overone million. Hipparchus contradicted him, showing that affirmatively there are 103,049 inter-twinings. “This passage stumped commentators until 1994,” when a mathematician realised that it corresponds to thecorrect solution of a complex combinatorial problem worked out in modern Europe in 1870, thereby forcing“a reevaluation of our notions of what was known about combinatorics in Antiquity.”
It is undeniablefrom this evidence that this entire field of mathematics must have reached an advanced stage, yet not onesingle treatise on it survives.These are just a few striking examples illustrating an indisputable point: the Hellenistic age was extremelysophisticated mathematically and scientifically, and we don’t even know the half of it. When “human learning. . . suffered shipwreck,” “the systems of Aristotle and Plato, like planks of lighter and less solid material,floated on the waves of time and were preserved,” while treasure troves of much more mathematicallyadvanced works were lost forever.With so many key works being lost, we are forced to rely on later commentators and compilers for accountsof the works of Hellenistic authors. This is not unlike forming an image of modern science and mathematicsfrom popularisations in the Sunday paper. Such coverage is invariably oversimplified and dumbed-down,reducing the matter to one or two simplistic ideas while conveying nothing whatsoever of the often massivetechnical groundwork underlying it. Actually this is a misleading analogy. The situation is much worse thanthis. [Cohen (1994), 277].
Fermat, [Mahoney (1973), 119]. [Heath (1912), 10]. [Solla Price (1959), 60]. [Solla Price (1974), 5].
Plutarch, [Russo (2004), 281]. [Russo (2004), 281]. See also [Acerbi (2018), 282].
Bacon,
Novum Organum (1620), I.77, [Bacon (1999), 115].
Sadly, the lack of appreciation for science among these third-rate commentators continues among scholarstoday.More of Greek “science” survives than does any other category of Greek literature, and yet muchof that is obscure even to classicists.
The state of editions and translations of ancient scientific works as a whole remains scandalousby comparison with the torrent of modern works on anything unscientific – about 100 papers peryear on Homer, for example. And an embarrassingly large number of classicists are . . . ignorantof Greek scientific works.
Classicists include many who have chosen Latin and Greek precisely to escape from science at thevery early stage of specialisation that our schools’ curricula permit: and often a very successfulescape it is, to judge from the depth of ignorance of science ancient and modern that it oftensecures.
Modern scholars, persistently regarding this era as somehow inferior to the Athens of Perikles andDemosthenes, have often disregarded Hellenistic science and technology in favor of later Romanachievements or earlier Greek work.
Little wonder then that Greek science is systematically misunderstood and undervalued, and that simplisticideas of philosophical authors and commentators are substituted for the real thing. “The study of mechanics . . . is eagerly pursued by all those interested in mathematics,”
Greek sourcescredibly report. Perhaps even Aristotle himself wrote on technical mechanics. At any rate a work on thissubject survived under his name, which Galileo and his contemporaries still regarded as a work by Aristotle.In more recent times, “the attribution to Aristotle has been questioned mainly on the ground that thetreatise’s attention to practical problems is ‘quite un-Aristotelian’, which is doubtful reasoning at best.” Inany case, it is “agreed by those who would question its attribution to Aristotle that the treatise was composedby an Aristotelian shortly after Aristotle’s time.”
The authors of the treatise, whoever they are, “discussthe lever, the pulley, and the balance, and expound with considerable success some of the main principlesof statics—the law of virtual velocities, the parallelogram of forces, and the law of inertia.”
Quite serious [Burnet (1929), 253–254]. For further references to the same effect, see [Lloyd (1991), 75]. [Irby-Massie & Keyser (2002), xxi]. [Rihll (1999), 137]. [Lloyd (1991), 354]. [Bugh (2006), 243].
Pappus,
Collection , VIII.1, [Lloyd (1973), 91]. [Clagett (1959), 4]. [Ross (1923), 12].
It is squarely within the realm of possibility that thesemay have contained most or all of “Galileo’s” discoveries.Strato had “an apparently deserved reputation as an experimenter.”
To prove that falling objects speedup, he reasoned as follows.
Pour water slowly from a vessel. At first it flows in a continuous stream, butthen further down its fall it breaks up into drops and trickles. This is because the water is speeding up. Sothe water spreads out, like cars let loose on a highway after a congested area. Another experiment provingthe same point is stones dropped into a sand bed from various heights. The stone makes craters of differentdepths depending on height fallen. These are the kinds of basics preserved in the superficial commentaryliterature. Quite possibly the original treatise backed these things up with a more mathematical treatment.Hipparchus was the greatest mathematical astronomer of the Hellenistic era and certainly more thancapable of giving a mathematically sophisticated treatment of falling bodies. Later commentators tell usthat “Hipparchus contradicts Aristotle regarding weight, as he says that the further something is, the heavierit is.” “The only way to make [this statement] comprehensible is to suppose that Hipparchus meant theweight of bodies inside the earth, recognizing that it decreases as the body nears the center,” which isin accordance with modern gravitational theory. Another commentator evidently had this in mind whendescribing the question posed by “the folks who introduced the thrust toward the center” as to whether“boulders thrust through [a tunnel into] the depths of the earth, upon reaching the center, should stay stillwith nothing touching or supporting them; [or whether] if thrust down with impetus they should overshootthe center and turn back again and keep bobbing back and forth.”
These things are very much in line with 17th-century physics. Dropping stones into sand, thinking abouthow gravity varies on a super-terrestrial scale and inside the earth: scientists spent a lot of serious effort onexactly that in the 17th century. All in all, it is certainly possible that Hipparchus and his contemporarieswere familiar with principles of “Galilean” mechanics. As one modern scholar has observed: “The ease ofstumbling upon this discovery renders it highly improbable that natural philosophers had ever searchedfor the law of fall” before Galileo.
Perhaps a more natural conclusion from this logic is that Hellenisticscientists in fact did search for it and in fact did stumble upon it in those lost works on falling bodies thatwe know they wrote.
Ptolemy is the canonical source for Greek astronomy, and the target of Galileo’s attacks. But it is quiteplausible that Ptolemy, who lived hundreds of years after the golden age of Greek science, was not the pinnacleof Greek astronomy, but should rather be seen as a regressive later author. Ptolemy’s big book possibly did“more damage to astronomy than any other work ever written” by displacing much better Hellenistic Greekastronomy, so that “astronomy would be better off if it had never existed.”
It is in any case certain thatPtolemy was merely the last in a very long tradition of mathematical astronomy.
Sadly, virtually none ofthe technical works on astronomy before Ptolemy have survived.We know for a fact that heliocentrism was pursued very seriously in the Hellenistic era, long beforePtolemy. Aristarchus, who was a good mathematician, wrote a now lost treatise arguing that the earthrevolves around the sun. Archimedes cites this work with tacit approval.
There are some indications thatothers kept pursuing heliocentrism.
Indeed, “several technical elements of Ptolemaic astronomy can onlybe explained as derivatives of an earlier heliocentric model.”
There are clear indications that Greek heliocentrists supported their theory with physical arguments. [Clagett (1955), 70–71]. [Clagett (1955), 69]. [Lloyd (1973), 16].
Simplicius, [Russo (2004), 291]. [Russo (2004), 293].
Plutarch, [Russo (2004), 293]. [Drake (1989), 2]. [Newton (1977), 379].
Theproblem of creating geometrical models for planetary motions goes back at least to Aristotle’s time, as is evident from e.g.
Meta-physics
XII.8. [Heath (2002), 222]. [Russo (2004), 80–82, 285–286, 294–297]. [Russo (2004), 317], referring to[Rawlins (1987)]. See also [Thurston (2002)], [Rawlins (2003)].
But of course other heavenly bodies are round too, such as the moon for example. So that very naturallysuggests that they have their own gravity just like the earth. This conclusion too is explicitly spelled outin ancient sources. Thus Plutarch says: “The downward tendency of falling bodies is evidence not of theearth’s centrality but of the affinity and cohesion to earth of those bodies which when thrust away fall backagain. . . . The way in which things here [fall] upon the earth suggests how in all probability things [on themoon] fall . . . upon the moon and remain there.”
Now, from this way of thinking, it is a short step to the idea that the heavenly bodies pull not only onnearby objects but also on each other. This is again explicit in ancient sources. This is why Seneca, forexample, says that “if ever [these bodies] stop, they will fall upon one another.”
That is correct, of course.The planets would “fall upon one another” if it wasn’t for their orbital speed. The Greeks were well awarethat tides can be explained in terms of such attractions, as we have seen.
This point of view explains the motions of the planets in terms of physical forces. It’s not that the planetshave circularity of motion as an inherent attribute imbedded in their essence, as Aristotle would have it.Rather, circularity is a secondary effect, the result of the interaction of two primary forces: a tangentialforce from motion and a radial force from gravity. There are clear indications that ancient astronomersworked out such a theory, including a mathematical treatment. Thus Vitruvius says: “the sun’s powerfulforce attracts to itself the planets by means of rays projected in the shape of triangles; as if braking theirforward movement or holding them back, the sun does not allow them to go forth but [forces them] to returnto it.”
Pliny says the same thing: planets are “prevented by a triangular solar ray from following a straightpath.”
All this talk of triangles, in both of these authors, certainly suggests an underlying mathematical treat-ment. Indeed, the Greeks knew very well the parallelogram law for the composition of forces or displace-ments, and in fact explicitly used this to explain circular motion as the net result of a tangential and a radialmotion.
It is beautiful how coherently all of that fits together and how naturally we were led from one idea tothe other. Just like the water of the oceans naturally seeks a spherical shape, so the spherical shape of theearth has been formed by the same forces. And just as gravity explains why the earth is round, so it mustexplain why other planets are round. Hence they have gravity. But just as they attract nearby objects, sothey attract each other. So the heavens have a perpetual tendency to lump itself up, except this tendencyis counterbalanced by the tendency of speeding objects to shoot off in a straight line.Hence top mathematicians in the Hellenistic era advocated heliocentrism, and evidently integrated thisview with mechanical considerations comparable in spirit to the works of people like Kepler in the 17thcentury. Many people refuse to believe this. It has recently been claimed, for example, that “pre-Copernicanheliocentrisms (that of Aristarchus, for example) have all the disadvantages and none of the advantages ofCopernican heliocentrism,” because they postulated only that the earth revolves around the sun, not, as hascommonly been assumed, that all the other planets do so as well. This supposedly “explains why Copernicus’sheliocentrism was accepted . . . , while pre-Copernican heliocentrism” was not.
This is completely wrong,in my opinion. And for an obvious reason. Namely: Why would Aristarchus have affirmed and written atreatise on heliocentrism if it had nothing but disadvantages? What possible reason could he have had donefor doing so? None, in fact. Yet this is exactly what this recent article proposes.It is a fact that Aristarchus asserted the physical reality of his hypothesis. And it is a fact that herecognised the parallax argument against it (discussed in §4.3), as Archimedes implies when he mentionsAristarchus.
Even the recent article I cited admits this. So why, then, would Aristarchus write a treatiseproposing this bold hypothesis, discuss a major argument against it (namely the parallax argument) and noarguments in favour of it, and then nevertheless conclude that the hypothesis is true? And, furthermore, [Russo (2004), 303].
Plutarch, [Irby-Massie & Keyser (2002), 72].
Seneca, [Russo (2004), 294]. §3.17.
Vitruvius, [Russo (2004), 297].
Pliny, [Russo (2004), 298]. [Russo (2004), 301–302]. [Carman (2018), 16]. [Heath (2002), 222].
Strikingly, Aristarchus wrote atreatise calculating the relative distances and sizes of the sun, the earth, and the moon.
This treatiseshows that Aristarchus was at any rate a highly competent mathematician. But I think it shows much morethan that. I think it feeds directly into his heliocentrism. Or are we supposed to believe that Aristarchuscalculated the sizes of heavenly bodies just for kicks in one treatise and did not see any connection with theheliocentrism he advanced in another treatise even though the obvious connection was right under his nose?What is the probability that he suffered from such schizophrenia? Virtually zero, in my opinion.In fact there are certain aspects Aristarchus’s treatise that make much more sense when you read it thisway. On its own it is a weird treatise. On the one hand it calculates the sizes and distances of the sun,moon, and earth in a mathematically sophisticated manner. Very detailed, technical stuff, including thecompletely pointless complication that the sun does not quite illuminate half the moon but ever so slightlymore than half, since the sun is larger than the moon. This is “pure mathematical pedantry.”
It makesthe geometrical calculations ten times more intricate while having only the most minuscule and completelyinsignificant impact on the final results.On the other hand, the observational data that Aristarchus uses for his calculations are extremely crude.He says that the angular distance between the sun and the moon at half moon is 87 degrees: a pretty terriblevalue. The real value is more like 89.9 degrees.
Because of this his results are way off. For instance, hiscalculated distance to the sun is off by a factor of 20 or so. So what’s going on? Why do such intricatemathematics with such worthless data? Did he just care about the mathematical ideas and not about theactual numbers? I think it would be a mistake to jump to that simplistic conclusion, even though manypeople have done so.In fact, it is easy to see how Aristarchus had a purpose in underestimating the angle. His purpose withthe treatise, I propose, is to support his heliocentric cosmology based on the principle that smaller bodiesorbit bigger ones. This hypothesis fits very well with the structure of Aristarchus’s treatise. The treatisehas 18 propositions. Proposition 16 says that the sun has a volume about 300 times greater than the earth,and Proposition 18, the very last proposition, says that the earth has a volume about 20 times greater thanthe moon. These are exactly the propositions you need to explain which body should orbit which. And thatis exactly where Aristarchus chooses to end his treatise.Many commentators have been puzzled by why Aristarchus ends in that strange place.
In particular,many have been baffled by why he does not give distances and sizes in terms of earth radii. This seems likethe natural and obvious thing to do, and doing so would have been easily within his reach. Many moderncommentators add the small extra steps along the same lines needed to fill this obvious “gap.”
Except it’snot a gap at all and there is no need to be puzzled by Aristarchus’s choices. If we accept my hypothesis,everything he does makes perfect sense all of a sudden. He carries his calculations precisely as far as heneeds for this purpose, and no further.My hypothesis also explains why he chose such a poor value for the angular measurement. He has everyreason to purposefully use a value that is much too small. Underestimating this angle means that the sizeof the sun will be underestimated. And his goal is to show that the sun is much bigger than the earth. Sohe has shown that even if we grossly underestimate the angle, the sun is still much bigger than the earth.So he has considered the worst case scenario for his desired conclusion, and he still comes out on top. Thatjust makes his case all the stronger, of course.Clearly my interpretation requires that Aristarchus knew that 87 degrees was an underestimate. The §4.4.
His only surviving work,
On the Sizes and Distances of the Sun and Moon . [Neugebauer (1975),643]. [Neugebauer (1975), 642]. [Neugebauer (1975), 636], [Berggren & Sidoli (2007), 221–222]. [Berggren & Sidoli (2007), 222–223], [Van Helden (1985), 8–9]. The argument for this is as follows. You are trying to measure the angle betweenthe sun and the moon at half moon. But to do that you need to pinpoint the moment of half moon, whichcan only be done with an accuracy of maybe half a day. But in half a day the moon has moved six degrees,and therefore radically changed the angle you are trying to measure. So your observational value is going tohave a margin of error of 6 degrees, which is enormous and makes the whole thing completely pointless.But I’m not convinced that it’s as hopeless as all that. One way to work around the problem would beto use not one single observation, but the average of many observations. There is little evidence that theGreeks ever made use of averaging that way, but the idea is simple enough. I did a bit of statistics tosee if this would be viable. Let’s assume that our angular measurements are normally distributed about thetrue value. And let’s accept the assumption in the critique of Aristarchus that one would be lucky to getthe moment of half moon correct to the day.
So we can tell it’s today rather than yesterday, but we can’ttell at what exact hour the moon is exactly half full. Let us roughly translate this into statistical terms bysaying that the observations have a standard deviation of 12 hours, or six degrees.Now, an astronomer active for, let’s say, two decades would have occasion to observe about 500 halfmoons. So say he makes 500 angular measurements and then average them. This would produce an estimateof the true value with a 95% confidence interval of ± about half a degree. A margin of error of half a degreeis easily enough to support my interpretation that 87 is a conscious underestimate.Naturally, Aristarchus would not have reasoned in such terms exactly, but it is not necessary to knowany statistical theory to get an intuitive sense of the order of magnitude of the error in such an estimate.As you keep adding observations, and keep averaging them, you will see the average stabilising over time,of course. So certainly it will become clear after a while that, whatever the true value is, it must be greaterthan 87 degrees at any rate.It is certainly extremely speculative to imagine that Aristarchus might have had something like thisin mind. But in any case my argument shows that it cannot be ruled out as out of the question thatAristarchus could in principle have had solid empirical evidence that his value of 87 degrees was certainlyan underestimate.So, in conclusion: Aristarchus was a good mathematician. He proposed a heliocentric theory that wastaken very seriously by Archimedes. There was a long tradition in Greek thought of trying to account for themotions of the planets in terms of everyday physics. This is naturally connected to heliocentrism becauseof the natural idea that smaller bodies orbit bigger ones. Aristarchus in fact wrote a major treatise devotedspecifically to comparing the sizes of the sun and the earth, and the earth and the moon. Several otherwisepeculiar aspects of the treatise fit like a glove the idea that it was written precisely to lend credibility toheliocentrism.The specific details of this speculative reconstruction are not important as far as our Galilean story isconcerned. But it does indicate the scope of uncertainty that the ancient sources leave us with. For thepurposes of a contextualised assessment of Galileo, the important question is whether we should think of theemergence of modern science as a drastic breakthrough and discontinuity. Was Galileo’s mode of scientificthought fundamentally different in kind from that of even the best minds of previous eras? Some will haveyou believe that it is. I disagree. What I have said about the possible contents of the lost scientific works ofantiquity is tentative and speculative. I have certainly not proved that the ancients made those discoveries.But I have proved something, namely that the assumption that ancient science was fundamentally differentin kind from the “Scientific Revolution” of Galileo’s age is by no means an established fact. My speculativereconstructions do not show that this assumption is false, but they do show that it is possible that it is false.This in itself is enough to call into question the routine assumptions often made about how “everyone” beforeGalileo allegedly lacked this or that scientific insight.To sum up my discussion of Hellenistic science more broadly, it is my contention that there is a strong [Neugebauer (1975), 642–643]. [Babu & Feigelson (1996), 1–2]. [Neugebauer (1975), 642]. Galileo “may properly be regarded as the ‘father of modern science’.”
This is still the accepted viewamong modern historians, as in this quote from the recent
Oxford Companion to the History of ModernScience . This view is considered so unassailable that even the very Pope once conceded that Galileo “isjustly entitled the founder of modern physics.”
There is less agreement on what warrants this epithet.“No one indeed is prepared to challenge [Galileo’s] scientific greatness or to deny that he was perhaps theman who made the greatest contribution to the growth of classical science. But on the question of whatprecisely his contribution was and wherein his greatness essentially lay there seems to be no unanimity atall.”
I shall go though all major attempts at capturing Galileo’s alleged greatness, and offer a consistentrebuttal case against them.
It is a common view that Galileo was the first to bring together mathematics and the study of the naturalworld.The momentous change that Galileo . . . introduced into scientific ontology was to identify thesubstance of the real world with . . . mathematical entities. The important practical result of thisidentification was to open the physical world to an unrestricted use of mathematics.
Galileo is one of the Founders of Modern Science [primarily on account of] his formulation anddefence of the idea of a mathematical method as the most appropriate means for uncovering thesecrets of nature. [This was a great achievement because] explaining how mathematics couldshow anything about the properties of matter [posed] an enormous conceptual difficulty, [since]geometry was recognized to be an abstract discipline, not one appropriate for dealing with physicalmatters.
Galileo was the ‘first inventor’ of mathematical physics: I mean, of truly modern physics. . . . Itwas Galileo who, by consistently applying mathematics to physics and physics to astronomy, firstbrought mathematics, physics, and astronomy together in a truly significant and fruitful way.The three disciplines had always been looked upon as essentially separate: Galileo revealed theirtriply-paired relationships. [Heilbron (2003), 323].
Pope John Paul II, 1979, [Poupard (1987), 198]. [Dijksterhuis (1961), 333]. [Crombie (1953), 310]. [Pitt (1992), xii, 5, 6]. [Drake (1999), xxi, 64–65].
What was fundamentally original and revolutionary in the conception of Galileo . . . was the assur-ance that, in principle, the potentialities of mathematical reasoning went far beyond the narrowlimits allowed by traditional philosophy. . . . [Thus Galileo was] the founder of the mathematicalphilosophy of nature.
This standard view obviously rests on the assumption that, before Galileo, mathematics and natural sci-ence were fundamentally disjoint. This assumption is plainly and unequivocally false. In Greek works bymathematically competent authors, there is zero evidence for this assumption and a mountain of evidenceto the contrary. “We attack mathematically everything in nature” said Iamblichus of Greek science, andhe was right. Far from being unable to conceive the unity of mathematics and science, “Hellenistic naturalphilosophers often took mathematics as the paradigm of science and sought to mathematize their study, thatis, to ground all its claims in mathematical theorems and procedures, a goal shared by modern scientists.”
How can so many historians get it exactly backwards? By ignoring the entire corpus of Greek mathematicsand instead relying exclusively on philosophical authors. Thus we are told that, following “the classificationof philosophical knowledge deriving from Aristotle,” a sharp division prevailed among “the Greeks” between“natural science (or ‘physics’), which studied the causes of change in material things,” and “mathematics,which was the science of abstract quantity.”
This was perhaps a problem for philosophers who spent theirtime trying to classify scientific knowledge instead of contributing to it. But I challenge you to produce onesingle piece of evidence that this division had any impact whatsoever on any mathematically creative personin antiquity.The alleged divide doesn’t exist in Aristotle’s own works either, for that matter. Aristotle lived well beforethe glory days of Greek science, and he was clearly no mathematician. But even Aristotle lists mechanics,optics, harmonics, and astronomy as fields based on mathematical demonstrations, and even explicitlycalls them “branches of mathematics.”
How can anyone infer from this that Aristotle saw the very notionof mathematical science as a conceptual impossibility? Remarkably, historians in fact do so, by insistingthat these fields are mere exceptions:Previous assumptions [before Galileo], encouraged by Aristotle and scholastic philosophers, heldthat mathematics was only relevant to our understanding of very specific aspects of the naturalworld, such as astronomy, and the behaviour of light rays (optics), both of which could be reducedto exercises in geometry. Otherwise, mathematics was just too abstract to have any relevance tothe physical world.
The implausibility of this view is obvious. If, as Aristotle himself clearly states, mechanics, optics, harmonics,and astronomy are four entire fields of knowledge that successfully use mathematics to understand the naturalworld, who in their right mind would then categorically insist that, nevertheless, other than that mathematicssurely has nothing to contribute to science. It makes no sense. If mathematics has already given you fourentire branches of science, why close your mind to the possibility of any further success along similar lines?It is hard to think of any reason for taking such a stance, except perhaps for someone who themselves lackmathematical ability and want to justify their neglect of this field.The strange habit of writing off the numerous branches of mathematical science in antiquity as so manyexceptions is necessary to maintain triumphalist narratives of the great Galilean revolution. For example, weare told that “it was Galileo who first subjected other natural phenomena to mathematical treatment than the. . . Alexandrian ones.”
In other words, except mechanics, astronomy, optics, statics, and hydrostatics,
Galileo was the very first to take this step. That is to say, if you ignore all previous mathematicians who didthis exact thing in great detail, Galileo’s step was revolutionary. [Koyré (1978), 201]. [Hall (1983), 113, 285]. [Lloyd (1973), 156]. [Bugh (2006), 245]. [Crombie (1953),1–2].
Aristotle,
Posterior Analytics , 1.9: 76a, 1.13: 78b,
Metaphysics , 13.3: 1078a.
Aristotle,
Physics , II, 194a8,[Aristotle (BW), 239]. [Henry (2012), 109]. [Cohen (2015), 114]. §7.3.
But many Greek achievements do not fit that characterisation. ThePythagorean insight that numbers govern musical harmonies can precisely not be interpreted as a man-madeapplication of mathematics; rather it is clearly and unequivocally an indication that the world is funda-mentally, inherently mathematical somehow, in a deep way that is not immediately obvious. The obviousconclusion is the one the Pythagoreans explicitly drew: that “all is number,” not that mathematics sometimeshappens to be applicable to some limited aspects of nature. In astronomy, the Greeks insisted on explainingeverything in terms of uniform circular motions. Aristotle and Plato are both very explicit about this.
What is this but an explicit commitment to the universe being inherently mathematical in its very meta-physical essence? In optics, light chooses the shortest path, whence light rays are straight and are reflectedwith the outgoing angle equal to the incoming angle. Again, not really an application of mathematics butrather an indication that nature herself is doing mathematics to determine the outcome of fundamental pro-cesses. Plato’s
Timaeus is explicit about nature being mathematically designed, down to its very elements,which are regular polyhedra. Others too saw mathematical design everywhere they looked. Honeycombsare hexagonal because it optimises the amount of area per perimeter among regular polygons, as Pappussays.
Ptolemy infers planetary distances by assuming that the universe was designed to fit the epicyclicmodels with no waste in between.
And so on. The Greek universe is through and through the work ofmathematical demiurges.Another strategy for explaining away the obvious fact of extensive mathematical sciences in antiquityis to discount them as genuine science on the grounds that they were abstractions. Thus some claim that,despite ostensible applications of mathematics in numerous fields, “mathematical theory and natural realityremain almost entirely separate entities” due to the “high level of abstraction” of the mathematical theories,which mean that they were “barely connected with the real world.”
Supposedly, Galileo broke this spell—an absurd claim since this critique is all the more true for his science: even Galileo’s supposedly “best”discoveries are often way out of touch with reality, not to mention his many erroneous theories. Meanwhile,Greek scientific laws of statics, optics, hydrostatics, and harmonics concern everyday phenomena that canbe verified by anyone in their own back yard using common household items. Indeed, they are still partof modern physics textbooks—and high school laboratory demonstrations—to this day. Take optics, forexample. Heron of Alexandria proved the law of reflection, which anyone with a mirror can readily check,using the distance-minimisation argument still found in every textbook today.
Diocles demonstrated thereflective property of the parabola and used it to “cause burning” by concentrating the rays of the sun witha paraboloid mirror; a principle still widely applied today, for example in satellite dishes and flashlights.Ptolemy demonstrated the magnifying property of concave mirrors, such as modern makeup mirrors.
These kinds of results, which are not atypical, are clearly not disconnected from reality by any means.The false notion of a divide between mathematics and science also rests on a conception of mathematicsitself as a purely abstract field.Traditionally, geometry was taken to be an abstract inquiry into the properties of magnitudes thatare not to be found in nature. Dimensionless points, breadthless lines, and depthless surfaces ofEuclidean geometry were not traditionally taken to be the sort of thing one might encounter whilewalking down the street. Whether such items were characterized as Platonic objects inhabitinga separate realm of geometric forms, or as abstractions arising from experience, it was generallyagreed that the objects of geometry and the space in which they are located could not be identifiedwith material objects or the space of everyday experience.
This is again a view expressed by philosophers only. Nothing of the sort is ever stated by any mathematicallycompetent author in antiquity. On the contrary, mathematicians routinely take the exact opposite forgranted. Allegedly “abstract” geometry is constantly applied to physical objects in Greek mathematicalworks without ado. [Alexander (2019)].
Plato
Timaeus
De caelo
II.6. [Blåsjö (2005), 526–527]. §8.5. [Cohen (2015), 19]. §§3.1, 3.12. [Clagett (1955), 80], [Cohen & Drabkin (1966), 264–265],[Irby-Massie & Keyser (2002), 195]. [Irby-Massie & Keyser (2002), 189–192]. [Irby-Massie & Keyser (2002),199–200]. [Jesseph (2015), 205].
De Pictura (1435)—an example of the geometryinvolved in perspective painting. The construction is based on the elementary geometrical fact that theprojection of a straight line onto the canvas is a straight line. From this fact it follows that the diagonal ofthe first tile is also the diagonal of successive tiles, whence the image of a tiled floor can be drawn accordingto the steps indicated.It is obvious that the long list of Greek mathematicians who studied the natural world always took forgranted the identification of geometry with the space and material objects around us. And why shouldn’tthey? For thousands of years geometry had been used to delineate fields, draw up buildings, measurevolumes of produce, and a thousand other practical purposes—exactly “the sort of thing one might encounterwhile walking down the street.” Every single theorem of Euclid’s geometry can be verified by concretemeasurements and constructions with physical tools and materials. So why would mathematicians suddenlyinsist that their field is completely divorced from reality? What could possibly be their motivation for doingso? It accomplishes nothing and creates tons of obvious problems when one wants to apply mathematics farand wide in numerous areas, as mathematicians always did. The only people with any motive to take suchan extremist stance are philosophers with an axe to grind.Only those ignorant of the vast tradition of Greek mathematical science can maintain that the unity ofmathematics and science in the 17th century was in any way revolutionary. However, even if one acceptsthis completely wrongheaded view, credit still should not go to Galileo. Some recent historians have begunto stress that “the mathematization of the sublunary world begins not with Galileo but with Alberti,” who wrote on the geometrical principles of perspective in painting in the 15th century (Figure 32).The invention of perspective by the Renaissance artists, . . . by demonstrating that mathematicscould be usefully applied to physical space itself, [constituted] a momentous step . . . toward thegeneral representation of physical phenomena in mathematical terms.
These historians correctly challenge the narrative of Galileo as the heroic visionary who united mathemat-ics and the physical world, but they retain the erroneous underlying assumption that this unification wasrevolutionary to begin with. Perspective painting is fine mathematics, but it wasn’t a “momentous step”“demonstrating” that mathematics could be applied to the world, because that had already been demon-strated over and over again thousands of years before. Vitruvius, to take just one example, had pointed outthe obvious: “an architect should be . . . instructed in geometry,” which “is of much assistance in architec-ture.”
Certainly a strange thing to say if the “momentous” insight that geometry is relevant to “the spaceof everyday experience” is still more than a thousand years in the future! No, the absurd notion that theapplication of geometry to physical space was somehow a Renaissance revolution can only occur to those whospend too much time reading philosophical authors pontificating about the divisions of knowledge instead ofreading authors actually active in those fields.The restriction to “the sublunary world” in the above quotation is also telling. The allegedly profoundconceptual divide between heaven and earth in this period is a standard trope among historians, as we haveseen.
Of course, the Greeks mathematised the sublunary world too, but you have to read specialised worksto find out much about that. Astronomy, on the other hand, is such an obvious example of an extremelysuccessful and detailed mathematisation of one aspect of reality that even philosophers and historians cannot [Wootton (2015), §5.8]. [Conner (2005), 270].
Vitruvius,
De Architectura (1st century BC), I.3–4,[Vitruvius (1914), 5–6]. Indeed, Vitruvius is perfectly well aquatinted with perspective painting, which he describes in VII.11,[Vitruvius (1914), 198]. §4.7. sublunary world.” Aristotle did indeed make much of the difference betweenthe earthly, sublunary world and the world of heavenly motions. But this is one particular dogma ofone particular school of philosophy. There is no reason for any mathematician to accept it, nor is thereany evidence that any mathematically competent person in the golden age of Greek science did so. TheAristotelian dichotomy is far from natural or necessary: “Aristotle argues, against his predecessors , thatthe celestial world is radically different from the sublunary world,” nota bene. For that matter, evenif Aristotle’s dogmatic and arbitrary dichotomy is accepted, it would still be madness to acknowledge theundeniable success of mathematics on one side of the divide, yet consider its application on the other sideof the divide a conceptual impossibility.Ptolemy, who was by all appearances much more philosophically conservative than the astronomers of theHellenistic age, speaks in Aristotelian terms when he contrasts astronomy with physics. The subject matterof astronomy is “eternal and unchanging,” while physics “investigates material and ever-moving nature . . .situated (for the most part) amongst corruptible bodies and below the lunar sphere.”
This is arguablymore of a fact than a philosophical commitment: planetary motions are regular and periodic, whereas fallingbodies, projectile motion, and other phenomena of terrestrial physics are inherently fleeting and limitedto a short time span. But it is conceivable that someone might seize on this dichotomy to “explain” whymathematics is suitable for the heavens only, and not for the sublunary world. This, however, is definitelynot Ptolemy’s stance. He unequivocally expresses the exact opposite view: “as for physics, mathematics canmake a significant contribution” there too.
In sum, the Aristotelian dichotomy was never an obstacle to mathematicians. And this with good reason.The whole business of emphasising the dichotomy in the context of the mathematisation of the world is afigment of the imagination of historians, who find themselves having to somehow explain away astronomy asirrelevant when they want to claim that there was a mathematical revolution in early modern science. Wedo not need to resort to such fictions if we instead accept that the unity of mathematics and science hadbeen obvious since time immemorial.Another argument for Galileo as the unifier of physics and mathematics consists in stressing that othermathematicians of his day were often more concerned with pure geometry than with projectile motion andthe like. For instance, in France there were highly capable “new Archimedeans” like Descartes, Roberval,and Fermat, but their focus differed from that of Galileo.They were indeed good mathematicians, but they did not consider mathematics as a methodfor understanding physical things. Mathematical constructions were only abstractions to them,with which it was fun to play, but which were not to be confused with what really happenedin nature. Moreover, . . . They were not interested in the ways in which motion intervened innatural processes.
In my view, Galileo would have loved to have been this kind of “new Archimedean” too if only he had beencapable of it.
And it is not true that these Frenchmen ignored motion and the mathematisation of nature.We have already noted that Descartes studied the law of fall, and that Fermat corrected Galileo on thepath of a falling object in absolute space.
Both Descartes and Fermat also wrote on the law of refractionof optics, deriving it from physical considerations regarding the speed of light in different media. Also,Descartes explained the motion of the planets, and the fact that they all revolve in the same direction aboutthe sun, by postulating that they were carried along by a vortex. So these mathematicians were clearly notignorant of or averse to studying how “motion intervened in natural processes.”So it is not attention to motion per se, but the study of projectile motion specifically, that sets Galileoapart from these mathematical contemporaries. Does Galileo deserve great credit in this regard? I think not.Why is projectile motion important? With Newton, projectile motion took on a fundamental importancebecause he saw that planetary motion was governed by the same principles. Galileo had no inkling of thisinsight. With Newton, projectile motion is also fundamental as a paradigm illustration of the principles—such as inertia and Newton’s force law—that govern all other mechanics. In Newtonian mechanics this is [Falcon (2005), ix]. Emphasis added.
Ptolemy,
Almagest , I.1, [Toomer (1998), 36].
Ptolemy,
Almagest , I.1,[Toomer (1998), 36]. [Palmerino & Thijssen (2004), 66]. §2.1. §3.2. §3.10. sohe cannot be celebrated for this insight either. Thus we see that praising Galileo for studying projectilemotion is anachronistic. Galileo got lucky: the topic he studied later turned out to be very important forreasons he did not perceive, so that in retrospect his work seems much more prescient and groundbreakingthan it really was. He himself in fact motivates the theory of projectile motion almost exclusively in termsof practical ballistics—a nonsensical application of zero practical value, which one cannot blame othermathematicians for ignoring. “Galileo became (and still is) the model for the empiricist scientist who, unlike the natural philosophers ofhis day, sought to answer questions not by reading philosophical works, but rather through direct contactwith nature.”
This is an image Galileo eagerly (but dishonestly) sought to promote, as we have seen.
Praise for Galileo in this regard goes hand in hand with “the verdict that Greek science suffered from anoverdose of rash generalizations at the expense of a careful scrutiny, whether experimental or observational,of the relevant facts.”
In other words, “Greek thinkers generally . . . overrated the power of unchecked,speculative thought in the natural sciences.”
In reality, an empirical approach to the study of nature is not a newfangled invention by Galileo butjust common sense. It was obviously adopted by the Greeks, especially the mathematicians. In particular,“ancient mechanics never lost [its] empirical, experiential thread.”
Even Aristotle, who practiced “specu-lative thought in the natural sciences” to a much greater extent than mathematicians, was a keen empiricist,and his followers insisted on this as one of the key principles of his philosophy. Aristotle’s zoology largelyfollows a laudable empirical method quite modern in spirit. The same approach was applied by his imme-diate followers in botany and petrology, including for example cataloging extensive empirical data on how awide variety of minerals react to heating.
This was far from forgotten in Galileo’s day, where one oftenencounters passages like these from committed Aristotelians:We made use of a material instrument to establish by means of our senses what the demonstrationhad disclosed to our intellect. Such an experimental verification is very important according to[Aristotelian] doctrine.
Not infrequently, Galileo’s Aristotelian opponents attacked him for being too speculative while they sawthemselves as representing the empirical approach. For example, one critic writes to Galileo:At the beginning of your work, you often proclaim that you wish to follow the way of the sensesso closely that Aristotle (who promised to follow this method and taught it to others) wouldhave changed his opinion, having seen what you have observed. Nonetheless, in the progress ofthe book you have always been so much a stranger to this way of proceeding that . . . all yourcontroversial conclusions go against our sense knowledge, as anyone can see by himself, and asyou expressly say yourself, . . . speaking of the theory of Copernicus, which was rendered plausibleand admirable to many by abstract reasoning although it was against all sensory experience.
It is true that there were also many spineless “Aristotelians” in Galileo’s day who preferred hiding behindtextual studies rather than engaging with actual science. But this was one perverse sect of scholasticism,not the overall state of human knowledge before Galileo. A contemporary colleague of Galileo put is well:The Science of Nature has been already too long made only a work of the Brain and the Fancy: Itis now high time that it should return to the plainness and soundness of Observations on materialand obvious things. §§3.7, 3.14, 6.6. §3.12. [Machamer (1998), 398]. §§3.1, 5.3. [Cohen (1994), 245]. [Dijksterhuis (1961), I.92]. [Irby (2016), 43]. [Lloyd (1973), 11].
Piccolomini, 16th century, [Duhem (1991),147].
Antonio Rocco, [Shea (1972), 183–184]. OGG.VII.712.
Robert Hooke,
Micrographia (1665), preface. Emphasisadded.
The mathematics of Euclid and the physics of Archimedes were . . . necessary, but not suffi-cient, for Galileo’s science. . . . They leave unexplained Galileo’s repeated appeals to sensateexperience.
On a superficial reading this may indeed appear so. Open, say, Archimedes’s treatise on floating bodies andyou will find no mention of any measurement or experiment or data of any kind, only theorems and proofs.It may seem natural to infer from this that Archimedes was doing speculative mathematics divorced fromreality, and that he had no understanding of the importance of empirical tests. This is what it looks like tohistorians who insist on an overly literal reading of the text and lack a sympathetic understanding of how themathematical mind works. The fact of the matter is that Archimedes’s theorems are empirically excellent.It makes no sense to imagine that Archimedes was reasoning about abstractions as an intellectual game, andthat his extremely elaborate and detailed claims about the floatation behaviour of various bodies given theirshapes and densities just happened to align exactly with reality by pure chance. Archimedes doesn’t haveto point out that he made very careful empirical investigations, because it is obvious from the accuracy ofhis results that he did.
This is a better way of putting the relation between mathematics and empirical data:Mixed mathematics were often presented in axiomatic fashion, following the Archimedean tra-dition . . . In this tradition, . . . experiments were often conceived of as inherently uncertain andtherefore they could not be placed at the foundation of a science, lest that science too be taintedwith that same degree of uncertainty. To be sure, experiments were still used as heuristic tools,for example, but their role often remained private, concealed from public presentations.
So the point is not that empirical data is neglected, but that it is a mere preliminary step. Anyone can makemeasurements and collect data. Self-respecting mathematicians do not publish such trivialities. Instead theygo on to the really challenging step of synthesising it into a coherent mathematical theory. Galileo did nothave the ability to do the latter, so he had to stick with the basics, and pretend, nonsensically, that this wassomehow an important innovation. Then as now, there were enough non-mathematicians in the world forhis cheap charade to be successful.
Some draw a distinction between passive observation and active experiment, and take the latter to be thekey Galilean innovation.When Galileo caused balls . . . to roll down an inclined plane . . . a light broke upon all studentsof nature. . . . Reason . . . must approach nature in order to be taught by it. It must not,however, do so in the character of a pupil who listens to everything that the teacher choosesto say, but of an appointed judge who compels the witnesses to answer questions which he hashimself formulated.
The originality of Galileo’s method lay precisely in his effective combination of mathematicswith experiment. . . . The distinctive feature of scientific method in the seventeenth century,
Drake, [Galileo (1989), xvii–xviii]. [Drake (1999), 281–282]. §7.3. [Buchwald & Fox (2013), 200].
Kant,
Critique of Pure Reason , B.xii–xiii, [Kant (1929), 20].
In reality, the use of experiment in Greek science is abundantly documented to anyone who bothers to readmathematical authors.Greek scientists knew perfectly well that “it is not possible for everything to be grasped by reasoning. . . , many things are also discovered through experience.”
This quote refers to the precise numericalproportions needed for the spring in a stone-throwing engine. The same author also offered an experimentaldemonstration that air is corporeal.
Ptolemy experimented with balloons (“inflated skins”) to investigatewhether air or water has weight in their own medium—indeed he “performed the experiment with the greatestpossible care.”
Heron of Alexandria gives a detailed description of an experimental setup to prove theexistence of a vacuum. He explicitly states that “referring to the appearances and to what is accessible tosensation” trumps abstract arguments that there can be no vacuum.
Such arguments had been given byAristotle. In optics, Ptolemy explicitly verified the law of reflection by experiment, and studied refractionexperimentally, giving tables for the angle of refraction of a light ray for various incoming angles in incrementsof 10 degrees for passages between air, water, and glass.
Archimedes performed a scientific experiment (Figure 33) that pleased him so much that he ran nakedthrough the streets yelling “eureka” in excitement.
Such was his love of empirical, experimental science—yet many scholars keep insisting that, like a second Plato, all Archimedes really cared about was abstractgeometry. Evidently, even running naked through the streets and screaming at the top of one’s lungs is notenough for some people to open their eyes. It is hard to imagine what else one can do to draw their attentionto the obvious, namely that Greek mathematicians embraced experimental method through and through.
A prevalent view has it that Galileo was the first to bring together abstract mathematics and science withconcrete technology and practical know-how of craftsmen and workers in mechanical fields. [Crombie (1953), 303, 1].
Philon, c. 200 B.C.,
Belopoeica
Philon,[Irby-Massie & Keyser (2002), 217].
As reported by Simplicius. [Irby-Massie & Keyser (2002), 225],[Cohen & Drabkin (1966), 248]. [Lloyd (1973), 17]. [Clagett (1955), 80–82]. [Irby-Massie & Keyser (2002),200–201]. [Cohen & Drabkin (1966), 271–281]. [Lloyd (1973), 133–135]. [Cohen & Drabkin (1966), 238–239]. Vitruvius,
De Architectura , IX. [Rorres (2019)].
Notizie istoriche e critiche intornoalla vita, alle invenzioni, ed agli scritti di Archimede siracusano , Brescia, 1737.Real science is born when, with the progress of technology, the experimental method of thecraftsmen overcomes the prejudice against manual work and is adopted by rationally traineduniversity-scholars. This is accomplished with Galileo. [Galileo was able] to bring together two once separate worlds that from his time on were destinedto remain forever closely linked—the world of scientific research and that of technology.
Galileo may fruitfully be seen as the culmination point of a tradition in Archimedean thoughtwhich, by itself, had run into a dead end. What enabled Galileo to overcome its limitations . . .seems easily explicable upon considering Galileo’s background in the arts and crafts.
The separation between theory and practice, imposed by university professors of natural philos-ophy, was repeatedly exposed as untenable. Of course the greatest figure in this movement isGalileo.
Galileo himself eagerly cultivated this image. The very first words of his big book on mechanics are devotedto extolling the importance for science of observing “every sort of instrument and machine” in action at the“famous arsenal” of Venice, praising the experiential knowledge of the “truly expert” workmen there.
It is true that universities were filled with many blockheads who foolishly insisted on keeping intellectualwork aloof from such connections to the real world. For example, when Wallis went to Oxford in 1632there was no one at the university who could teach him mathematics. “For Mathematicks, (at that time,with us) were scarce looked upon as Accademical studies, but rather Mechanical; as the business of Traders,Merchants, Seamen, Carpenters, Surveyors of Lands, or the like.”
But it would be mistake to infer from this that Galileo’s step was an innovation. The stupidity ofthe university professors was the doing of one particular clique of mathematically ignorant people. Theirattitude is not natural or representative of the state of human knowledge. Galileo is not a brilliant maverickthinking outside the box. Rather, he is merely doing what had, among mathematically competent people, [Zilsel (2000), 5].
Drake, [Galileo (1957), 78]. [Cohen (1994), 349]. [Henry (1997), 16]. [Galileo (1989),11], OGG.VIII.49.
Wallis, in his autobiography. [Scriba (1970), 27].
The Greek hand worker was considered inferior to the brain worker or contemplative thinker.. . . So, despite the fact that the philosophers derived some of their conclusions as to how naturebehaved from the work of the craftsmen, they rarely had experience of that work. What is more,they were seldom inclined to improve it, and so were powerless to pry apart its potential treasureof knowledge that was to lead to the scientific revolution in the Renaissance.
The fundamental brake upon the further progress of science in antiquity was slave labour [whichprecluded any] meaningful combination of theory and practice.
More specialised scholarship knows better. The recent
Oxford Handbook of Engineering and Technology inthe Classical World is perfectly clear on the matter:Many twentieth-century scholars hit upon . . . banausic prejudice [i.e., a snobbish contempt formanual labour] as an ‘explanation’ for a perceived blockage of technological innovation in theGreco-Roman world. The presence of slave labor was felt to be a related, concomitant factor.. . . [But] this now discredited interpretation [should be rejected and we should] . . . put an endto the myth of a ‘technological blockage’ in the classical cultures.
This is the view of experts on the matter, while the false narrative is promulgated by scholars who focus onGalileo, take it for granted that he is “the Father of Modern Science,” and postulate such nonsense aboutthe Greeks because that’s the only way to craft a narrative that fits with this false assumption.Promulgators of the nonsense about practice-adverse Greeks have evidently not bothered to read math-ematical authors. Pappus, for example, explains clearly that mathematicians enthusiastically embrace prac-tical and manual skills:The science of mechanics . . . has many important uses in practical life, . . . and is zealouslystudied by mathematicians. . . . Mechanics can be divided into a theoretical and a manual part;the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual ofworking in metals, architecture, carpentering and painting and anything involving skill with thehands.
He praises the interaction of geometry with practical fields or “arts” as beneficial to both:Geometry is in no way injured, but is capable of giving content to many arts by being associ-ated with them, and, so far from being injured, it is obvious, while itself advancing those arts,appropriately honoured and adorned by them.
These were no empty words. The Greeks had an extensive tradition of studying “machines,” meaning devicesbased on components such as the lever (Figure 35), pulley (Figure 36), wheel and axle, winch, wedge, screw,and gear wheel (Figure 37). The primary purpose of these machines was that of “multiplying an effort to [Lindberg (1978), 2]. [Brake (2009), 59]. [Cohen (1994), 248], summarising the view of Farrington. A similarview is expressed in [Dijksterhuis (1961), 74]. [Oleson (2008), 5–6]. [Thomas (1993), II.615]. [Thomas (1993),II.619–621]. AC is n times longer than CB , then only one n th of the force is required to lift the stone with the lever compared to lifting it directly. Archimedes wrotean excellent mathematical treatise on the lever. Figure from [Comstock (1850), 69].Figure 36: A compound pulley. Each pulley beyond the first doubles the mechanical advantage. In thiscase, since there are 4 pulleys beyond the first, a weight P of balances a weight W of = 16 . Thus theheavy weight W can be lifted by applying a small force at P . Greek mechanicians wrote extensively on themathematical relations involved in this and other “machines” of this sort. Figure from [Comstock (1850),87].exert greater force than can human or animal muscle power alone.” Such machines were “used primarilyin construction, water-lifting, mining, the processing of agricultural produce, and warfare.” The Greeksalso undertook advanced engineering projects, such as digging a tunnel of more than a kilometer through amountain, the planning of which involved quite sophisticated geometry to enable the tunnel to be dug fromboth ends, with the diggers meeting in the middle.
In short, “while it is crucial to distinguish betweentheoretical mechanics and practitioners’ knowledge, there is substantial evidence of a two-way interactionbetween them in Antiquity.”
Mathematicians were very much involved with such things. Indeed, “the modern distinction betweenphysical and mathematical sciences was alien to Hellenistic science, which was unitary.”
There are manytestimonies attributing to Archimedes various accomplishments in engineering, such as moving a ship sin-glehandedly by means of pulleys, destroying enemy ships using machines, building a screw for lifting water,and so on.
Apollonius wrote a very advanced and thorough treatise on conic sections, which is studiouslyabstract and undoubtedly l’art pour l’art pure mathematics if there ever was such a thing. Yet the sameApollonius “besides writing on conic sections produced a now lost work on a flute-player driven by com- [Oleson (2008), 337]. [Oleson (2008), 324–325]. [Laird & Roux (2008), 15]. [Russo (2004), 189].
Plutarch,
Marcellus , 14.8. [Proclus (1970), 51]. [Dijksterhuis (1987), 14–29].
Collection , VIII.X, [Pappus (1871), 333]). In the case illustrated, a force of 4 units applied at thecrank arm ∆ lifts a weight that it would have taken 160 units of force to lift directly.pressed air released by valves controlled by the operation of a water wheel.” The title page of the Arabicmanuscript that has preserved this work for us reads: “by Apollonius, the carpenter, the geometer.”
Thecliche of Greek geometry as nothing but abstruse abstractions divorced from reality is a modern fiction. Thesources tell a different story. It is not for nothing that one of the most refined mathematicians of antiquitywent by the moniker “the carpenter.”Unfortunately,Renaissance intellectuals were not in a position to understand Hellenistic scientific theories, but,like bright children whose lively curiosity is set astir by a first visit to the library, they found inthe manuscripts many captivating topics, especially those that came with illustrations. . . . Themost famous intellectual attracted by all these ‘novelties’ was Leonardo da Vinci. . . . Leonardo’s‘futuristic’ technical drawings . . . was not a science-fiction voyage into the future so much as aplunge into a distant past. Leonardo’s drawings often show objects that could not have beenbuilt in his time because the relevant technology did not exist. This is not due to a special geniusfor divining the future, but to the mundane fact that behind those drawings . . . there were olderdrawings from a time when technology was far more advanced.
The false narrative of the mechanically ignorant, anti-practical Greeks has obscured this fact, and led to anexaggerated evaluation of Renaissance technology, such as instruments for navigation, surveying, drawing,timekeeping, and so on.Renaissance developments in practical mathematics predated the intellectual shifts in naturalphilosophy. . . . Historians of the early modern reform of natural philosophy have failed to appre-ciate the significance of the prior success of the practical mathematical programme, [which] mustfigure in an explanation of why the new dogma of the seventeenth century embraced mathematics,mechanism, experiment and instrumentation.
This author proves at length that the practical mathematical tradition had much to commend it, which I donot dispute. But then he casually asserts with hardly any justification that there was nothing comparable in [Oleson (2008), 338]. [Hill (1998), X.334]. [Russo (2004), 335–336]. [Bennett (1991), 176, 189].
This is typical of much scholarship of this period. The deeply entrenched standard view ofthe Galilean revolution is basically taken for granted and subsequent work is presented as emendations to it.For instance, if you want to prove the importance of a Renaissance pre-revolution in practical mathematics,you need to prove two things: first that it was relevant to the scientific revolution, and second that it wasnot present long before. It is a typical pattern to see historians put all their efforts toward proving the firstpoint, and glossing over the second point in sentence or two. They can get away with this since the allegedshortcomings of the Greeks is supposedly common knowledge, while the first point is the one that departsfrom the standard narrative. Hence, if the standard narrative is misconceived in the first place, so is all thismore specialised research, which, although it ostensibly departs from the standard view, actually retains itsmost fundamental errors in the very framing of its argument.It is right to emphasise that the practical mathematical tradition stood for a much more fruitful andprogressive approach to nature than that dominant among the philosophy professors of the time. But it isa mistake to believe that these professors represented the considered opinion of the best minds, while themathematical practitioners were oddball underdogs whose pioneering success eventually proved undeniableto the surprise of everyone. The mathematical practitioners stood for simple common sense, not renegadeiconoclasm. They practiced the same common sense that their peers had in antiquity, with much the sameresults. The university professors, meanwhile, should not be mistaken for a neutral representation of the stateof human knowledge at the time. Rather, they formed one particular philosophical sect which retained itsdomination of the universities not because of the preeminence of its teachings but because of the incestuousappointment practices and obsequiousness of academics.
A standard view is that “the Scientific Revolution saw the replacement of a predominantly instrumentalistattitude to mathematical analysis with a more realist outlook.”
Instrumentalism means the following:An explanation which conforms to the facts does not imply that the hypotheses are real and exist.. . . [Astronomers] have been unable to establish in what sense, exactly, the consequences entailedby these arrangements are merely fictive and not real at all. So they are satisfied to assert thatit is possible, by means of circular and uniform movements, always in the same direction, to savethe apparent movements of the wandering stars.
Instrumentalism, as opposed to realism, was supposedly the accepted philosophy of science among “theGreeks”:[Ancient Greek astronomers] balked at the idea that the eccentrics and epicycles are bodies,really up there on the vaults of the heavens. For the Greeks they were simply geometrical fictionsrequisite to the subjection of celestial phenomena to calculation. If these calculations are inaccord with the results of observation, if the ‘hypotheses’ succeed in ‘saving the phenomena’, theastronomer’s problem is solved.
An astronomer who understands the true purpose of science, as defined by men like Posidonius,Ptolemy, Proclus, and Simplicius, . . . would not require the hypotheses supporting his system tobe true , that is, in conformity with things. For him it will be enough if the results of calculationagree with the results of observation— if appearances are saved . The Greek geometer in formulating his astronomical theories does not make any statements aboutphysical nature at all. His theories are purely geometrical fictions. That means that to save theappearances became a purely mathematical task, it was an exercise in geometry, no more, but,of course, also no less.
Galileo, by contrast, brought “a radically new mode of realist-mathematical nature knowledge.” [Bennett (1991), 182]. [Henry (1997), 8].
Simplicius, [Duhem (1969), 23]. [Duhem (1969), 25]. [Duhem (1969), 31]. [Wasserstein (1962), 57]. [Cohen (2015), 146].
Hence the Scientific Revolution owes much to “the novel quality of realism that the abstract-mathematicalmode of nature-knowledge acquired in . . . Galileo’s hands.”
In reality, no mathematically competent Greek author ever advocated instrumentalism. The notion that“the Greeks” were instrumentalists relies exclusively on passages by philosophical commentators. The notionthat Ptolemy believed his planetary models were “fictional . . . combinations of circles which could neverexist in celestial reality” is demonstrably false. First of all Ptolemy opens his big book with physicalarguments for why the earth is in the center of the universe—a blatantly realist justification for this aspectof his astronomical models. Furthermore, Ptolemy has a detailed discussion of the order and distances ofthe planets that obviously assumes that the planetary models, epicycles and all, are physically real. “Thedistances of the . . . planets may be determined without difficulty from the nesting of the spheres, wherethe least distance of a sphere is considered equal to the greatest distance of a sphere below it.”
Thatis to say, according to Ptolemy’s epicyclic planetary models, each planet sways back and forth between anearest and a furthest distance from the earth. The “sphere” of each planet must be just thick enough tocontain these motions. Ptolemy assumes that “there is no space between the greatest and least distances[of adjacent spheres],” which “is most plausible, for it is not conceivable that there be in Nature a vacuum,or any meaningless and useless thing.”
Clearly this is based on taking planetary models to be very realindeed, and not at all mathematical fictions invented for calculation.Nor was Ptolemy an exception in his realism. His colleague Geminos “was a thoughtful realist” as well.
Hipparchus too evidently chose models for planetary motion on realist grounds. His works are lost, butwe know that he proved the mathematical equivalence of epicyclic and equant motion. In other words, heshowed that two different geometrical models of planetary motion are observationally equivalent; they leadto the exact same visual impressions seen from earth, but they are brought about by different mechanisms.How should one choose between the two models in such a case? If Hipparchus was an instrumentalist, hewouldn’t care one way or the other, or he would just pick whichever was more mathematically convenient.But if he was a realist he would be interested in which model could more plausibly correspond to actualphysical reality. So what did he do? Here is what Theon says: “Hipparchus, convinced that this is how thephenomena are brought about, adopted the epicyclic hypothesis as his own and says that it is likely that allthe heavenly bodies are uniformly placed with respect to the center of the world and that they are unitedto it in a similar way.”
So Hipparchus decided between equivalent models based on physical plausibility.This is quite clearly a realist argument.Historians have brought up other “evidence” that “the Greeks” were instrumentalists. One thing theypoint to is the alleged compartmentalisation of Greek science.Phenomena [such as] consonance, light, planetary trajectories and the two states of equilibrium[i.e., statics and hydrostatics] are investigated separately. There is no search for interconnections,let alone for an overarching unity.
This would indeed make sense if mathematical science was just instrumental computation tools with nogenuine anchoring in reality. The only problem is that the claim is false. Greek science is in fact full ofinterconnections, just as one would expect if they were committed realists. Ptolemy uses mechanics to justifygeocentrism; Archimedean hydrostatics explains shapes of planets and “casts light on the earth’s geologicalpast”;
Archimedes used statical principles to compute areas in geometry. Ptolemy applies “consonance”(that is, musical theory) to “the human soul, the ecliptic, zodiac, fixed stars, and planets.”
He also [Hall (1983), 11–12]. [Cohen (2015), 120]. [Cohen (2015), 17].
Ptolemy,
Planetary Hypotheses , I,[Goldstein (1967), 7].
Ptolemy,
Planetary Hypotheses , I, [Goldstein (1967), 8]. [Evans & Berggren (2006), 58].
Theon,
Astronomia
34, [Duhem (1969), 9]. [Cohen (2015), 19–20]. [Russo (2004), 303]. See §7.6.
Ptolemy,
Harmonics , [Solomon (2000), xxxiv].
In Galileo’s time, the same pattern as among the Greeks prevails: mathematically competent people areunabashed realists, while philosophers and theologians often find instrumentalism more appealing for reasonsthat have nothing to do with science. Copernicus’s book, for example, is unequivocally realist. Spinelessphilosophers and theologians could not accept this. One even resorted to the ugly trick of inserting anunsigned foreword in the book without Copernicus’s authorisation, in which they espoused instrumentalism:It is the job of the astronomer to use painstaking and skilled observation in gathering togetherthe history of the celestial movements, and then—since he cannot by any line of reasoning reachthe true causes of these movements—to think up or construct whatever causes or hypotheses hepleases such that, by the assumption of these causes, those same movements can be calculatedfrom the principles of geometry for the past and for the future too. . . . It is not necessarythat these hypotheses should be true. . . . It is enough if they provide a calculus which fits theobservations.
This surely fooled no one who actually read the book, with all its blatant realism. Giordano Bruno, forone, thought “there can be no question that Copernicus believed in this motion [of the earth],” and henceconcluded that the timid foreword must have been written “by I know not what ignorant and presumptuousass.”
But then again the mathematically incompetent people whom this foreword was designed to appeasecould not read the book anyway.In medieval and renaissance philosophical texts it is not hard to find many assertions to the effect that“real astronomy is nonexistent” and what passes for astronomy “is merely something suitable for computingthe entries in astronomical almanacs.”
There were many instrumentalists at the time, to be sure,but the challenge is to find a single serious mathematical astronomer among them. They were exclusivelytheologians and philosophers.All historians nowadays recognise that “Copernicus clearly believed in the physical reality of his astro-nomical system,” but their inference that he “thus broke down the traditional disciplinary boundary betweenastronomy (a branch of mixed mathematics) and physics (or natural philosophy)” is dubious. This was“the traditional” view only in a very limited sense. It was traditional among the particular sect of Aris-totelians that occupied the universities, but outside this narrow clique it had no credibility or standingwhatsoever. Among mathematicians, Copernicus’s view was exactly the traditional one.All mathematically competent people continued in the same vein, long before Galileo entered the scene.Already in the 16th century, “Tycho and Rothman, Maestlin, and even Ursus . . . openly deploy a wide rangeof physical arguments in debating the issue between the rival world-systems.”
Kepler puts the mattervery clearly:One who predicts as accurately as possible the movements and positions of the stars performs thetask of the astronomers well. But one who, in addition to this, also employs true opinions aboutthe form of the universe performs it better and is held worthy of greater praise. The former,indeed, draws conclusions that are true as far as what is observed is concerned; the latter notonly does justice in his conclusions to what is seen, but also . . . in drawing conclusions embracingthe inmost form of nature.
As Kepler notes, this was all obviously well-known and accepted since antiquity, for “to predict the motionsof the planets Ptolemy did not have to consider the order of the planetary spheres, and yet he certainly didso diligently.” [Cohen & Drabkin (1966), 281–283].
Osiander, foreword to Copernicus’s book, [Copernicus (1995), 3]. For theattribution to Osiander, see [Kepler (1984), 152].
Bruno, [Duhem (1969), 99].
Achillini (16th century),[Duhem (1969), 47]. [Jardine (1984), 231–237]. [Osler (2010), 52]. [Jardine (1984), 244]. [Kepler (1984),145]. [Kepler (1984), 145]. .6 Mechanical philosophy
Some say that “the mechanization of the world-picture” was the defining ingredient of “the transition fromancient to classical science.”
A paradigm conception at the heart of the new science was that of theworld as a machine: a “clockwork universe” in which everything is caused by bodies pushing one anotheraccording to basic mechanical laws, as opposed to a world governed by teleological purpose, divine will andintervention, anthropomorphised desires and sympathies ascribed to physical objects, or other supernaturalforces. Galileo was supposedly a pioneer in how he always stuck to the right side in this divide.Galileo possessed in a high degree one special faculty. . . . That is the faculty of thinking cor-rectly about physical problems as such, and not confusing them with either mathematical orphilosophical problems. It is a faculty rare enough still, but much more frequently encounteredtoday than it was in Galileo’s time, if only because nowadays we all cope with mechanical devicesfrom childhood on.
Of course, this “special faculty” is precisely what led Galileo to reject as occult the correct explanation of thetides and propose his own embarrassing nonstarter of a tidal theory based on an analogy with “mechanicaldevices.”
But let us put that aside.There is nothing modern about the mechanical philosophy. “We all cope with mechanical devices fromchildhood on,” but so did the Greeks, who built automata such as entirely mechanical puppet-theatres, self-opening temple doors, a coin-operated holy water dispenser, and so on.
Pappus notes that “the scienceof mechanics” has many applications “of practical utility,” including machines for lifting weights, warfaremachines such as catapults, water-lifting machines, and “marvellous devices” using “ropes and cables tosimulate the motions of living things.”
Clearly, then, “Ancient Greek mechanics offered working artifactscomplex enough to suggest that . . . organisms, the cosmos as a whole, or we ourselves, might ‘work likethat’.”
Thus we read in ancient sources that “the universe is like a single mechanism” governed by simpleand deterministic laws that ultimately lead to “all the varieties of tragic and comedic and other interactionsof human affairs.”
This line of reasoning soon lead to a secularisation of science. “Bit by bit, Zeus wasrelieved of thunderbolt duty, Poseidon of earthquakes, Apollo of epidemic disease, Hera of births, and therest of the pantheon of gods were pensioned off” in the same manner.
Mechanical explanations are widespread in Greek science. The Aristotelian
Mechanics uses the law oflever to explain “why rowers who are . . . in the middle of the ship . . . move the ship the most,” and “how itis that dentists extract teeth more easily by . . . a tooth-extractor [forceps] than with the bare hand only.”
Greek scientists explained perfectly clearly that sound is a “wave of air in motion,” comparable to the ringsforming on a pond when when one throws in a stone.
Atomism—a widely espoused conception of theworld in Greek antiquity—is of course in effect a plan to “make material principles the basis of all reality.”
Greek astronomy went hand in hand with mechanical planetaria that directly reproduced a scale modelof planetary motion. And not just basic toy models, but “complex and scientifically ambitious instruments”that could generate all heavenly motions mechanically from a single generating motion (the turn of a crank,as it were).
The possibility that even biological phenomena worked on the same principle immediately suggested itselfand was eagerly pursued.Just as people who imitate the revolutions of the wandering stars by means of certain instrumentsinstill a principle of motion in them and then go away, while [the devices] operate just as if thecraftsman was there and overseeing them in everything, I think in the same way each of the partsin the body operates by some succession and reception of motion from the first principle to everypart, needing no overseer. [Dijksterhuis (1961), 501]. [Drake (1964), 603]. §3.17. [Humphrey et al. (2003)], [Cohen & Drabkin (1966),224–234].
Pappus,
Collection , VIII.1, [Cohen & Drabkin (1966), 183–184]. [Berryman (2009), 229–230].
Theodorus, as paraphrased by Proclus,
On providence
2, [Jones (2017), 243]. [Rihll (1999), 17]. (Pseudo-)Aristotle,
Mechanics , [Cohen & Drabkin (1966), 194]. [Clagett (1955), 74]. [Irby (2016), 41]. [Jones (2017), 239–242].
Galen,
On the Use of the Parts who also tested his ideas experimentally.
In sum, the world did not need Galileo to tell them about the mechanical philosophy, since it had beenwidely regarded as common sense already thousands of years before.
Many have tried to stress commonalities between Galileo and the Aristotelian philosophers who precededhim. That is to say, they argue for the “continuity thesis” which says that the so-called “Scientific Revolution”was not a radical or revolutionary break with previous thought.Galileo essentially pursued a progressive Aristotelianism [during the first half of his life—theperiod of] positive growth that laid the foundation . . . for the new sciences.
A particular school of Renaissance Aristotelians, located at the University of Padua, constructed avery sophisticated methodology for experimental science; . . . Galileo knew this school of thoughtand built upon its results; . . . this goes a long way toward explaining the birth of early modernscience.
The mechanical and physical science of which the present day is so proud comes to us through anuninterrupted sequence of almost imperceptible refinements from the doctrines professed withinthe Schools of the Middle Ages.
Galileo was clearly the heir of the medieval kinematicists.
I agree with these authors that “those great truths for which Galileo . . . received credit” are not his.
Butthe notion that they were first conceived in Aristotelian schools of philosophy is wrongheaded.The argument of these historians is based on a simple logic. First they show that various concepts of“Galilean” science are prefigured in earlier sources. Then they want to infer from this that these sourcesmarked the true beginning of the scientific revolution. But in order to draw this inference they need twoassumptions: first, that Galileo was the father of modern science; and second, that the Greeks were nowherenear the same accomplishments. These two assumptions are simply taken for granted by these authors, as amatter of common knowledge. But in reality both assumptions are dead wrong, and therefore the inferenceto the significance of the Aristotelian sources is unwarranted.The continuity thesis, then, devalues the contributions of Galileo, yet at the same time desperately needsto reassert the traditional view that “Galileo has a clear and undisputed title as the ‘father of modernscience’,” since this is what gives them the one point of connection they are able to establish betweenmedieval and modern science. Historians have been able to match up some aspects of Renaissance Aristoteliantradition with the scientific methodology and practice of Galileo. From this they are inclined to infer thatGalileo was influenced by this tradition. But there is little direct evidence to this effect. Galileo is saidto have found Aristotelian teachings "boring."
Insofar as there are similarities between him and earlierthinkers who tried to tackle similar questions, this can very well be due largely to common sense rather thandirect philosophical influence.It would be much harder to find substantive links between medieval authors and mathematically com-petent people such as Kepler or Newton. Since Galileo mostly writes philosophical prose and rarely if eversubstantive mathematics, it is much easier to try to construe his works as related to the preceding philo-sophical tradition. The strength of those alleged continuities in philosophical thought between Galileo andpreceding Aristotelian tradition is highly debatable. But it is a moot point in any case unless Galileo isaccepted as a founder of modern science. The entire argument stands and falls with this false premiss. [Lloyd (1973), 80]. [Lloyd (1973), 85]. [Wallace (1991), I.350]. [Cohen (1994), 279], summarising[Randall (1940)]. [Duhem (1991), 9]. [Clagett (1959), 666]. [Duhem (1991), 17]. [Wallace (1987), 57].
Vincenzo Viviani,
Racconto istorico della vita del sig.r Galileo Galileo (1654), OGG.XIX.602, [Gattei (2019), 5].
For example, “although we are left with few monuments from the profound research ofthe Ancients into the laws of equilibrium, those few are worthy of eternal admiration.”
Obviously, “mas-terpieces of Greek science . . . [such as the works of] Pappus, and especially Archimedes, . . . are proof thatthe deductive method can be applied with as much rigor to the field of mechanics as to the demonstrationsof geometry.”
Galileo himself embraced Archimedes as his role model in no uncertain terms. “As far asgenius is concerned, [Galileo] claimed Archimedes had exceeded everybody else, and called him his master.”“Galileo claimed it was possible to walk safely, without stumbling, on earth as well as in heaven, as long aswe remain in Archimedes’s footsteps.”
How, then, can continuity thesis advocates acknowledge these “masterpieces” “worthy of eternal admi-ration” from antiquity, yet at the same time attribute the scientific revolution to medieval or renaissancephilosophers? By writing off those ancient works as minor technical footnotes to an otherwise thoroughlyAristotelian paradigm. Only if this picture is accepted can any kind of greatness be ascribed to the pre-Galileans, as is evident from passages such as these:Some philosophers in medieval universities were teaching ideas about motion and mechanicsthat were totally non-Aristotelian [and] were consciously based on criticisms of Aristotle’s ownpronouncements.
Admittedly, most of these . . . significant medieval mechanical doctrines . . . were formed withinthe Aristotelian framework of mechanics. But these medieval doctrines contained within themthe seeds of a critical refutation of that mechanics.
The medieval mechanics occupied an important middle position between . . . Aristotelian andNewtonian mechanics. . . . [Hence it was] an important link in man’s efforts to represent the lawsthat concern bodies at rest and in movement.
The impressive set of departures from Aristotelianism achieved by medieval science neverthelessfailed to produce genuine efforts to reconstruct, or replace, the Aristotelian world picture.
If Aristotle is taken as the baseline, this looks quite impressive indeed. But why should Aristotle be acceptedas the default opinion? Aristotle was one particular philosopher who was a nobody in mathematics andlived well before the golden age of Greek science. Medieval and renaissance thinkers indeed mustered up thecourage to challenge isolated claims of his teachings almost two thousand years later, while mostly retaininghis overall outlook. This does not constitute great open-mindedness and progress. Rather it is a sign ofsmall-mindedness that these people paid so much attention to Aristotle at all in the first place. In my view,it is not so much impressive that they deviated a bit from Aristotle as it is deplorable that they framedso much of what they did relative to Aristotle, even when they disagreed with him. This is very differentfrom post-Aristotelian thought in Greek times, where there is no evidence that any mathematician paid anyattention to Aristotle’s mechanics.In any case, “extravagant claims for the modernity of medieval concepts” suffer from “serious defects.”
There was no such thing as a fourteenth-century science of mechanics in the sense of a generaltheory of local motion applicable throughout nature, and based on a few unified principles.By searching the literature of late medieval physics for just those ideas and those pieces of [Duhem (1991), 75]. [Duhem (1991), 11]. [Duhem (1991), 149].
Niccolo Gherardini,
Vita di Galileo Galilei (1654), OGG.XIX.645, 637, [Gattei (2019), 159, 145]. [Hall (1983), 30]. [Clagett (1959), 682]. [Clagett (1959),670–671]. [Cohen (1994), 266]. [Clagett (1959), xxi]. appears to form a coherent whole and to be built on new foundations replacing those ofAristotle’s physics. But this is an illusion, and an anachronistic fiction, which we are able toconstruct only because Galileo and Newton gave us the pattern by which to select the rightpieces and put them together.
The main piece of such precursorism is the so-called “mean speed theorem.” This trivial theorem states that,in terms of distance covered, a uniformly accelerated motion is equivalent to a constant-speed motion withthe same average speed (Figure 4). Some people praise this as an “impressive” achievement —“probablythe most outstanding single medieval contribution to the history of physics,” derived by “admirable andingenious” reasoning —even though these authors did nothing with this trivial theorem and only deducedit to illustrate the notion of uniform change abstractly within Aristotelian philosophy. Later the theorembecame central in “Galilean” mechanics since free fall is uniformly accelerated.
But it “was, in fact, neverapplied to motion in fall from rest during the 14th, or even in the 15th century” (only in the mid-16thcentury there is a passing remark to this effect within the Aristotelian tradition, “without any accompanyingevidence”).
Let us not radically inflate our esteem for the Middle Ages by anachronistically praising themfor pointing out a trivial thing that centuries later took on a significance of which they had no inkling.
Let us instead recognise the theorem for the trifle that it is. Then we shall also not have any need to besurprised when it turns out that Babylonian astronomers assumed it without fanfare thousands of yearsearlier still.
In a similar vein we are told that there are “unmistakeable Jesuit influences in Galileo’s work” : “Aboveall Galileo was intent in following out Clavius’s program of applying mathematics to the study of nature andto generating a mathematical physics.”
The preposterous notion that this was “Clavius’s” program canonly enter one’s mind if one only reads philosophy. It was obviously Archimedes’s program, except, unlikeClavius, he proved his point by actually carrying it out instead of sermonising about what one ought to do inphilosophical prose. Philosophers (ancient and modern alike) have a tendency to place disproportionate valueon explaining something conceptually as opposed to actually doing it. After all, that is virtually the definitionof philosophy. Hence they praise certain Aristotelians for explaining some supposedly profound principlesof scientific method even when “it is quite clear that [none of them] ever applied his advocated methods toactual scientific problems.”
Descartes—a mathematically creative person—knew better: “we ought not tobelieve an alchemist who boasts he has the technique of making gold, unless he is extremely wealthy; and bythe same token we should not believe the learned writer who promises new sciences, unless he demonstratesthat he has discovered many things that have been unknown up till now.”
Unfortunately, such basiccommon sense is often lacking among historians and philosophers assigning credit for basic principles of thescientific method.There is a contradiction in the way modern historians try to trace many aspects of the scientific revolutionto roots in the middle ages. On the one hand these historians like to claim that the traditional view of thescientific revolution is ahistorical and based on an anachronistic mindset, whereas their own account thatsees continuity with the middle ages is more sensitive to how people actually thought at the time itself.Ironically, however, their view, which is supposed to be more true to the historical actors’ way of thinking,is actually all the more blatantly at odds with how virtually all leaders of the scientific revolution thoughtof the middle ages. “The scientific achievement of the Middle Ages was held in unanimous contempt fromGalileo’s time onward by those who adhered to the new science. Leibniz’ scathing verdict ‘barbarismusphysicus’ neatly encapsulates the reigning sentiment.”
This was not for nothing. Leibniz was an eruditescholar well versed in the philosophy of the schools. But he was also an excellent mathematician. The latterenabled him to pass a sound judgement on medieval science. [Moody (1966), 42–43]. [Crowe (2007), 25]. [Grant (1978), 56]. §3.2. [Drake (1989), 17].
Forexample, it played no significant role in the thought of the first person to prove it, Oresme. [Damerow et al. (2004), 19]. [Ossendrijver (2016)]. [Wallace (1987), 55]. [Wallace (1987), 57]. [Cohen (1994), 283].
Descartes toVan Hogelande, 1639/1640, [Van de Ven & Bos (2004), 384]. [Cohen (1994), 260]. .8 Epilogue
Galileo could not have asked for better co-conspirators than those modern academia have provided for him.He desperately needs his audience to view Aristotle as the default baseline against which all his works shouldbe evaluated. He desperately needs his audience to be ignorant of mathematical authors. And he’s in luck.The default training of historians of science is not higher mathematics and physics, but seminars based onnon-mathematical authors such as Aristotle. So the people tasked with being Galileo experts are by designthe people most inclined to accept Galileo’s deceit. Pretending that Archimedes doesn’t exist serves boththeir purposes and Galileo’s, since they share Galileo’s aversion to proper mathematics. For example, theannual bibliography of works on the history of science published by the flagship journal of the History ofScience Society consistently lists well over ten times as many works on Aristotle as on Archimedes, andalmost as many more on “Aristotelianism.”
Such are the inclinations and predispositions of the philosophy-trained humanists who dominate the fieldtoday. But people steeped in mathematics see the world differently—a fundamental schism in the outlooksof modern historians of science and the historical figures they are trying to understand. If we read authorslike Copernicus, Kepler, Galileo, Descartes, and virtually everyone else who made a contribution to themathematical sciences, we find endless praise for Archimedes and bottomless contempt for Aristotle. Withthis as our baseline our view of Galileo is radically transformed.Following Galileo’s sentencing by the church, an author of a book to be printed at Florence was toldby the Inquisition to change the phrase “most distinguished Galileo” into “Galileo, man of noted name.”
Though I am not generally on the side of the Inquisition, I have come to the conclusion that this particulardecree is sound. Instead of “Galileo, father of modern science,” we would do better to make it “Galileo, manof noted name.”
References [OGG]
Le Opere di Galileo Galilei , Antonio Favaro (ed.), 20 volumes, Florence, 1890–1909, later reprinted withadditions.[Acerbi (2018)] Fabio Acerbi, Hellenistic Mathematics, in P. T. Keyser & J. Scarborough (eds.),
The Oxford Handbookof Science and Medicine in the Classical World , Oxford University Press, 2018, Chapter C3.[Andriesse (2005)] C. D. Andriesse,
Huygens: The Man Behind the Principle , Cambridge University Press, 2005.[Alexander (2019)] Amir Alexander,
Proof! How the World Became Geometrical , Scientific American / Farrar, Strausand Giroux, 2019.[Ariew (1987)] Roger Ariew, The phases of Venus before 1610,
Studies in History and Philosophy of Science Part A ,18(1), 1987, 81–92.[Aristotle (BW)] Aristotle,
The Basic Works of Aristotle , New York: Modern Library, 2001.[Babu & Feigelson (1996)] Gutti Jogesh Babu & E.D. Feigelson,
Astrostatistics , CRC Press, 1996.[Bacon (1999)] Francis Bacon,
Selected Philosophical Works , edited by Rose-Mary Sargent, Hackett, 1999.[Baumgardt (1951)] Carola Baumgardt,
Johannes Kepler: Life and Letters , Philosophical Library, New York, 1951.[Baumgartner (1986)] Frederic J. Baumgartner, Scepticism and French Interest in Copernicanism to 1630,
Journalfor the History of Astronomy , 17, 1986, 77–88.[Baumgartner (1988)] Frederic J. Baumgartner, Galileo’s French Correspondents,
Annals of Science , 45, 1988, 169–182.[Baur (1912)] Ludwig Baur (ed.),
Die Philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln , Münster:Aschendorff Verlag, 1912.[Bennett (1991)] J. A. Bennett, The challenge of practical mathematics, in S. Pumfrey, P. L. Rossi, & M. Slawinski(eds.),
Science, Culture and Popular Belief in Renaissance Europe , Manchester University Press, 1991, 176–190.
Number of entries in the subject index of the annual
Isis comprehensive bibliography for the years 2003–2017: Archimedes42, Aristotle 482, Aristotelianism 339. There has never been an entry on “Archimedeanism,” though I for one would welcomeit. [Drake (1978), 414].
Berggren & Sidoli (2007)] J. L. Berggren & Nathan Sidoli, Aristarchus’s On the Sizes and Distances of the Sun andthe Moon: Greek and Arabic Texts,
Archive for History of Exact Sciences , 61(3), 2007, 213–254.[Berryman (2009)] Sylvia Berryman,
The Mechanical Hypothesis in Ancient Greek Natural Philosophy , CambridgeUniversity Press, 2009.[Bertoloni Meli (2006)] Domenico Bertoloni Meli,
Thinking with Objects: The Transformation of Mechanics in theSeventeenth Century , Johns Hopkins University Press, 2006.[Biagioli (2007)] Mario Biagioli,
Galileo’s Instruments of Credit: Telescopes, Images, Secrecy , University of ChicagoPress, 2007.[Biagioli (2010)] Mario Biagioli. Did Galileo Copy the Telescope? A “New” Letter by Paolo Sarpi. In A. van Helden,S. Dupré, R. van Gent and H. Zuidervaart (eds.),
The Origins of the Telescope , KNAW Press, Amsterdam, 2010,203–30.[Bianchi (1834)] G. Bianchi, Schreiben des Herrn Bianchi an den Herausgeber,
Astronomische Nachrichten , 11(15),1834, 197–202.[Blackwell (1992)] Richard J. Blackwell,
Galileo, Bellarmine, and the Bible , University of Notre Dame Press, 1992.[Blåsjö (2005)] Viktor Blåsjö, The Isoperimetric Problem,
American Mathematical Monthly , 112(6), 2005, 526–566.[Blåsjö (2017)] Viktor Blåsjö,
Transcendental curves in the Leibnizian calculus , Elsevier, 2017.[Brake (2009)] Mark L. Brake,
Revolution in Science: How Galileo and Darwin Changed Our World , Palgrave Macmil-lan, 2009.[Brecht (1966)] Bertolt Brecht,
Galileo , adaptation by Charles Laughton, New York: Grove Weidenfeld, 1966.[Brophy & Paolucci (2001)] James Brophy & Henry Paolucci (eds.),
The Achievement of Galileo , Bagehot Council,New Edition, 2001.[Bucciantini et al. (2015)] Massimo Bucciantini, Michele Camerota, & Franco Giudice,
Galileo’s Telescope: A Euro-pean Story , Harvard University Press, 2015.[Buchwald & Fox (2013)] Jed Z. Buchwald & Robert Fox (eds.),
The Oxford Handbook of the History of Physics ,Oxford University Press, 2013.[Bugh (2006)] Glenn R. Bugh,
The Cambridge Companion to the Hellenistic World , Cambridge University Press,2006.[Bukowski (2008)] John Bukowski, Christiaan Huygens and the Problem of the Hanging Chain,
College MathematicsJournal , 39(1), 2008, 2–11.[Burke (1984)] John G. Burke (ed.),
The Uses of Science in the Age of Newton , University of California Press, 1984.[Burnet (1929)] John Burnet,
Essays and Adresses , London: Chatto & Windus, 1929.[Burtt (1932)] Edwin Arthur Burtt,
The Metaphysical Foundations of Modern Physical Science , Routledge, 1932.[Butterfield (1959)] Herbert Butterfield,
Origins of Modern Science 1300-1800 , Macmillan, 1959.[Büttner et al. (2001)] Jochen Büttner, Peter Damerow, Jürgen Renn, Traces of an Invisible Giant: Shared Knowl-edge in Galileo’s Unpublished Treatises. In [Montesinos & Solis (2001)], chapter 2.2, 183–202.[Butts & Pitt (1978)] Roberts E. Butts & Joseph C. Pitt,
New Perspectives on Galileo , The University of WesternOntario Series in Philosophy of Science 14, Reidel, 1978.[Caffarelli (2009)] Roberto Vergara Caffarelli,
Galileo Galilei and Motion: A Reconstruction of 50 Years of Experi-ments and Discoveries , Springer, 2009.[Carman (2018)] Christián C. Carman, The first Copernican was Copernicus: the difference between Pre-Copernicanand Copernican heliocentrism,
Archive for History of Exact Sciences , 72(1), 2018, 1–20.[Chalmers & Nicholas (1983)] Alan Chalmers & Richard Nicholas, Galileo on the dissipative effect of a rotating earth,
Studies in History and Philosophy of Science , Part A, 14(4), 1983, 315–340.[Clagett (1955)] Marshall Clagett,
Greek Science in Antiquity , Books for Libraries Press, Plainview, New York, 1955.[Clagett (1959)] Marshall Clagett,
The Science of Mechanics in the Middle Ages , University of Wisconsin Press,1959.
Coffa (1968)] José Alberto Coffa, Galileo’s concept of inertia,
Physis: rivista internazionale di storia della scienzia ,X(4), 261–281, 1968.[Cohen & Drabkin (1966)] Morris R. Cohen & I. E. Drabkin (eds.),
A Source Book in Greek Science , Harvard Uni-versity Press, 1966.[Cohen (1967)] I. B. Cohen, Newton’s attribution of the first two laws of motion to Galileo, in:
Atti del SymposiumInternazionale di Storia, Methodologia, Logica e Philosophia della Scienza ‘Galileo nella Storia e enella Filosofiadella Scienza’ , Gruppo Italiano di Storia delle Scienze, Florence, 1967, xxv–xliv.[Cohen (1994)] H. Floris Cohen,
The Scientific Revolution: A Historiographical Inquiry , University of Chicago Press,1994.[Cohen (2015)] H. Floris Cohen,
The Rise of Modern Science Explained: A Comparative History , Cambridge Univer-sity Press, 2015.[Comstock (1850)] J. L. Comstock,
A System of Natural Philosophy: Principles of Mechanics , Pratt, Woodford, andCompany, 1850.[Conner (2005)] Clifford D. Conner,
A People’s History of Science: Miners, Midwives, and Low Mechanicks , NationBooks, 2005.[Copernicus (1995)] Nicolaus Copernicus,
On the Revolutions of the Heavenly Spheres , translated by Charles GlennWallis, Prometheus Books, Great Minds Series, 1995.[Cornelli (2013)] Gabriele Cornelli,
In Search of Pythagoreanism: Pythagoreanism as an Historiographical Category ,De Gruyter, 2013.[Costabel & Lerner (1973)] Pierre Costabel & Michel Pierre Lerner (eds.). Introduction in
Les nouvelles pensées deGalilée , 2 vols., Paris, J. Vrin, 1973.[Crombie (1953)] A. C. Crombie,
Robert Grosseteste and the Origins of Experimental Science 1100–1700 , OxfordUniversity Press, 1953.[Crowe (2007)] Michael J. Crowe,
Mechanics from Aristotle to Einstein , Green Lion Press, 2007.[Damerow et al. (2004)] Peter Damerow, Gideon Freudenthal, Peter McLaughlin, Jürgen Renn.
Exploring the Limitsof Preclassical Mechanics: A Study of Conceptual Development in Early Modern Science: Free Fall and Com-pounded Motion in the Work of Descartes, Galileo, and Beeckman . Second edition. Springer, Sources and Studiesin the History of Mathematics and Physical Sciences. 2004.[Danielson (2004)] Dennis Danielson, Achilles Gasser and the birth of Copernicanism,
Journal for the History ofAstronomy , Vol. 35, Part 4, No. 121, 2004, 457–474.[Deiss & Nebel (1998)] Bruno M. Deiss & Volker Nebel, On a pretend observation of Saturn by Galileo,
Journal forthe History of Astronomy , 29, 1998, 215–220.[Descartes (2008)] René Descartes,
A Discourse on the Method: Of Correctly Conducting One’s Reason and SeekingTruth in the Sciences , Oxford University Press, 2008.[Dijksterhuis (1961)] E. J. Dijksterhuis,
The Mechanization of the World Picture , Oxford University Press, 1961.[Dijksterhuis (1987)] E. J. Dijksterhuis,
Archimedes , Princeton University Press, 1987.[Dijksterhuis (2004)] Fokko Jan Dijksterhuis,
Lenses and Waves: Christiaan Huygens and the Mathematical Scienceof Optics in the Seventeenth Century , Kluwer, 2004.[Dobrzycki (1972)] Jerzy Dobrzycki (ed.),
The Reception of Copernicus’ Heliocentric Theory , Springer, 1972.[Dugas (1954)] R. Dugas, Sur le cartésianisme de Huygens,
Revue d’histoire des sciences , 7(1), 1954, 22–33.[Duhem (1913)] Pierre Duhem,
Études sur Léonard de Vinci, 3e Série: Les Précurseurs Parisiens de Galilée , Paris,1913.[Duhem (1969)] Pierre Duhem,
To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato toGalileo , University of Chicago Press, 1969.[Duhem (1991)] Pierre Duhem,
The Origins of Statics: The Sources of Physical Theory , Kluwer, Boston Studies inthe Philosophy of Science 123, 1991. Translation of
Les origines de la statique , Hermann, Paris, 1905–1906.[Drabkin & Drake (1960)] I. E. Drabkin & Stillman Drake,
Galileo Galileo on Motion and on Mechanics , Universityof Wisconsin Press, 1960.
Drake & O’Malley (1960)] Stillman Drake & C. D. O’Malley,
The Controversy on the Comets of 1618 , Universityof Pennsylvania Press, 1960.[Drake (1964)] Stillman Drake, Galileo and the Law of Inertia, American Journal of Physics, 32(8), 1964, 601–608.[Drake & Drabkin (1969)] Stillman Drake & I.E. Drabkin,
Mechanics in Sixteenth-Century Italy: Selections fromTartaglia, Benedetti, Guido Ubaldo, & Galileo , University of Wisconsin Press, 1969.[Drake (1970)] Stillman Drake,
Galileo Studies: Personality, Tradition, and Revolution , University of Michigan Press,1970.[Drake (1976)] Stillman Drake, Galileo and the First Mechanical Computing Device,
Scientific American , April 1976,104–113.[Drake (1978)] Stillman Drake,
Galileo at work , University of Chicago Press, 1978.[Drake (1984)] Stillman Drake, Galileo, Kepler, and Phases of Venus,
Journal for the History of Astronomy , 15(3),October 1984, 198–208.[Drake (1989)] Stillman Drake,
History of Free Fall: Aristotle to Galileo , Wall & Emerson, Toronto, 1989. Alsoincluded as an appendix to [Galileo (1989)].[Drake (1999)] Stillman Drake,
Essays on Galileo and the History and Philosophy of Science , Volume 1. Selected andintroduced by N. M. Swerdlow and T. H. Levere. University of Toronto Press, 1999.[Drake (2001)] Stillman Drake,
Galileo: A Very Short Introduction , Oxford University Press, 2001.[Dreyer (1906)] J. L. E. Dreyer,
History of the Planetary Systems from Thales to Kepler , Cambridge University Press,1906.[Engelberg & Gertner (1981)] Don Engelberg & Michael Gertner, A marginal note of Mersenne concerning theGalileian spiral,
Historia Mathematica , 8(1), 1981, 1–14.[Evans & Berggren (2006)] James Evans & J. Lennart Berggren,
Geminos’s Introduction to the Phenomena: A Trans-lation and Study of a Hellenistic Survey of Astronomy , Princeton University Press, 2006.[Falcon (2005)] Andrea Falcon,
Aristotle and the Science of Nature: Unity without Uniformity , Cambridge UniversityPress, 2005.[Finocchiaro (1989)] Maurice A. Finocchiaro (ed.),
The Galileo Affair: A Documentary History , University of Cali-fornia Press, 1989.[Finocchiaro (2007)] Maurice Finocchiaro,
Retrying Galileo, 1633–1992 , University of California Press, 2007.[Finocchiaro (2010)] Maurice Finocchiaro,
Defending Copernicus and Galileo: Critical reasoning in the two affairs ,Springer, Boston Studies in the Philosophy of Science, Volume 280, 2010.[Finocchiaro (2014)] Maurice Finocchiaro,
The Routledge Guidebook to Galileo’s Dialogue , Routledge, 2014.[Finocchiaro (2019)] Maurice Finocchiaro, review of Natacha Fabbri & Federica Favino,
Copernicus Banned: TheEntangled Matter of the Anti-Copernican Decree of 1616 , 2018,
Journal for the History of Astronomy , 50(3),2019, 376–378.[Fox (2012)] Robert Fox (ed.),
Thomas Harriot and His World: Mathematics, Exploration, and Natural Philosophyon Early Modern England , Ashgate, Surrey, 2012.[Freguglia & Giaquinta (2016)] Paolo Freguglia & Mariano Giaquinta,
The Early Period of the Calculus of Variations ,Birkhäuser, 2016.[Frova & Marenzana (1998)] Andrea Frova & Mariapiera Marenzana,
Thus Spoke Galileo: The Great Scientist’s Ideasand Their Relevance to the Present Day , Oxford University Press, 1998.[Gaab & Leich (2018)] Hans Gaab & Pierre Leich (eds.),
Simon Marius and His Research , Springer, 2018.[Gal & Chen-Morris (2005)] Ofer Gal & Raz Chen-Morris, The Archaeology of the Inverse Square Law: (1) Meta-physical Images and Mathematical Practices,
History of Science , 43, 2005, 391–414.[Gal & Chen-Morris (2006)] Ofer Gal & Raz Chen-Morris, The Archaeology of the Inverse Square Law: (2) The Useand Non-use of Mathematics,
History of Science , 44, 2006, 49–67.[Galileo (1953)] Galileo Galilei,
Dialogue concerning the two chief world systems , translated by Stillman Drake,University of California Press, 1953.
Galileo (1957)] Galileo Galilei,
Discoveries and Opinions of Galileo , translated with an introduction and notes byStillman Drake, Anchor Books, 1957.[Galileo (1967)] Galileo Galilei,
Dialogue concerning the two chief world systems , translated by Stillman Drake, 2ndrevised edition, University of California Press, 1967.[Galileo (1974)] Galileo Galilei,
Two New Sciences , translated by Stillman Drake, University of Wisconsin Press,1974.[Galileo (1978)] Galileo Galilei,
Operations of the geometric and military compass (1606), translated with an intro-duction by Stillman Drake, Smithsonian Institution Press, Washington, D.C., 1978.[Galileo (1989)] Galileo Galilei,
Two New Sciences , translated by Stillman Drake, second edition, Wall & Emerson,Toronto, 1989.[Galileo (2001)] Galileo Galilei,
Dialogue concerning the two chief world systems , translated by Stillman Drake,Modern Library Paperback Edition, 2001.[Gattei (2019)] Stefano Gattei (ed.),
On the Life of Galileo: Viviani’s Historical Account & other early biographies ,Princeton University Press, 2019.[Geymonat (1965)] Ludovico Geymonat,
Galileo Galilei , McGraw-Hill, 1965.[Gingerich (1984)] Owen Gingerich, Phases of Venus in 1610,
Journal for the History of Astronomy , 15(3), Oct. 1984,209–210.[Gingerich (2002)] Owen Gingerich,
An annotated census of Copernicus’ De revolutionibus , Brill, Leiden, 2002.[Gingerich (2004)] Owen Gingerich,
The book nobody read: chasing the Revolutions of Nicolaus Copernicus , Heine-mann, London, 2004.[Gingerich (2016)] Owen Gingerich,
Copernicus: A Very Short Introduction , Oxford University Press, 2016.[Goldstein (1967)] Bernard R. Goldstein, The Arabic version of Ptolemy’s Planetary Hypotheses,
Transactions ofthe American Philosophical Society , new series, 57(4), 1967.[Gorham et al. (2016)] Geoffrey Gorham, Benjamin Hill, Edward Slowik, & C. Kenneth Waters (eds.),
The languageof nature: reassessing the mathematization of natural philosophy in the 17th century , Minnesota Studies in thePhilosophy of Science 20, University of Minnesota Press, 2016.[Gowers (2010)] Timothy Gowers,
Mathematics , Sterling Publishing Company, 2010.[Graney (2008)] Christopher M. Graney, But Still, It Moves: Tides, Stellar Parallax, and Galileo’s Commitment tothe Copernican Theory,
Physics in Perspective , 10(3), 2008, 258–268.[Graney & Grayson (2011)] Christopher M. Graney & Timothy P. Grayson, On the Telescopic Disks of Stars: A Re-view and Analysis of Stellar Observations from the Early Seventeenth through the Middle Nineteenth Centuries,
Annals of Science , 68(3), 2011, 351–373.[Graney (2015)] Christopher M. Graney,
Setting Aside All Authority: Giovanni Battista Riccioli and the Scienceagainst Copernicus in the Age of Galileo , University of Notre Dame Press, 2015.[Grant (1974)] Edward Grant (ed.),
A Source Book in Medieval Science , Harvard University Press, 1974.[Grant (1978)] Edward Grant,
Physical Science in the Middle Ages , Cambridge University Press, 1978.[Hall (1952)] A. Rupert Hall,
Ballistics in the Seventeenth Century , Cambridge University Press, 1952.[Hall (1983)] A. Rupert Hall,
The Revolution in Science 1500–1750 , Longman, 1983.[Halliwell (1841)] James Orchard Halliwell (ed.),
A Collection of Letters Illustrative of the Progress of Science inEngland from the Reign of Queen Elizabeth to that of Charles the Second , London, Taylor, 1841.[Heath (1912)] Thomas Little Heath (ed.), The
Method of Archimedes, 1912. Reprinted as appendix in [Heath (2002)].[Heath (2002)] Thomas Little Heath (ed.),
The Works of Archimedes , Dover Publications, 2002.[Heilbron (2003)] J. L. Heilbron (ed.),
The Oxford Companion to the History of Modern Science , Oxford UniversityPress, 2003.[Heilbron (2010)] J. L. Heilbron, Galileo, Oxford University Press, 2010.[Henry (1997)] John Henry,
The Scientific Revolution and the Origins of Modern Science , Macmillan, 1997.[Henry (2012)] John Henry,
A Short History of Scientific Thought , Palgrave Macmillan, 2012.
Hill (1984)] David K. Hill, The projection argument in Galileo and Copernicus: Rhetorical strategy in the defenceof the new system,
Annals of Science , 41(2), 1984, 109–133.[Hill (1998)] Donald R. Hill,
Studies in Medieval Islamic Technology , edited by David King, Routledge, 1998.[Hine (1973)] William L. Hine, Mersenne and Copernicanism,
Isis , 64(1), 1973, 18–32.[Humphrey et al. (2003)] John William Humphrey, John Peter Oleson, Andrew N. Sherwood,
Greek and RomanTechnology: A Sourcebook , Routledge, 2003.[Irby-Massie & Keyser (2002)] Georgia L. Irby-Massie & Paul T. Keyser (eds.),
Greek Science of the Hellenistic Era:A Sourcebook , Routledge, 2002.[Irby (2016)] Georgia L. Irby (ed.),
A Companion to Science, Technology, and Medicine in Ancient Greece and Rome ,John Wiley & Sons, Chichester, UK, 2016.[Jardine (1984)] Nicholas Jardine,
The Birth of History and Philosophy of Science: Kepler’s A defence of Tychoagainst Ursus with essays on its provenance and significance , Cambridge University Press, 1984.[Jesseph (2015)] Douglas Jesseph, Hobbes’s Theory of Space. In V. De Risi (ed.),
Mathematizing Space , Springer,2015.[Jones (2017)] Alexander Jones,
A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of theAncient World , Oxford University Press, 2017.[Jullien (2015)] Vincent Jullien (ed.),
Seventeenth-Century Indivisibles Revisited , Birkhäuser, 2015.[Kainulainen (2014)] Jaska Kainulainen,
Paolo Sarpi: A Servant of God and State , Brill, 2014.[Kant (1929)] Immanuel Kant,
Critique of Pure Reason , translated by N. K. Smith, Macmillan, London, 1929.[Kepler (1981)] Johannes Kepler,
Mysterium Cosmographium: The Secret of the Universe , translated by A. M. Dun-can, Abaris Books, New York, 1981.[Kepler (1984)] Johannes Kepler,
Apologia pro Tychone contra Ursum , c. 1600, translated in [Jardine (1984), 134–207].[Kepler (1995)] Johannes Kepler,
Epitome of Copernican Astronomy & Harmonies of the World , translated byCharles Glenn Wallis, Prometheus Books, 1995.[Kepler (2000)] Johannes Kepler,
Optics , translated by William H. Donahue, Green Lion Press, 2000.[Kepler (2015)] Johannes Kepler,
Astronomia Nova , translated by William H. Donahue, Green Lion Press, newrevised edition, 2015.[Kline (1972)] Morris Kline,
Mathematical Thought from Ancient to Modern Times , Oxford University Press, 1972.[Koestler (1959)] Arthur Koestler,
The Sleepwalkers: A History of Mans Changing Vision of the Universe , Macmillan,New York, 1959.[Kollerstrom (2001)] Nicholas Kollerstrom, Galileo’s astrology. In [Montesinos & Solis (2001)], chapter 3.7, 421–432.[Koyré (1955)] Alexandre Koyré, A Documentary History of the Problem of Fall from Kepler to Newton,
Transactionsof the American Philosophical Society , 45(4), 1955, 329–395.[Koyré (1978)] Alexandre Koyré,
Galileo Studies , translated by John Mepham, Harvester Press, 1978. First publishedas
Etudes Galiléennes , 1939.[Laird & Roux (2008)] Walter Roy Laird & Sophie Roux (eds.),
Mechanics and Natural Philosophy before the Scien-tific Revolution , Springer, Boston Studies in the Philosophy of Science 254, 2008.[Lewis (2012)] John Lewis, Mersenne as Translator and Interpreter of the Works of Galileo,
MLN , 127(4), 2012,754–782.[Lindberg (1978)] David C. Lindberg (ed.),
Science in the Middle Ages , University of Chicago Press, 1978.[Lindberg (1992)] David C. Lindberg,
The Beginnings of Western Science: The European Scientific Tradition inPhilosophical, Religious, and Institutional context, 6000 B.C. to A.D. 1450 , University of Chicago Press, 1992.[Lloyd (1973)] G. E. R. Lloyd,
Greek Science After Aristotle , Chatto & Windus, London, 1973.[Lloyd (1991)] G. E. R. Lloyd,
Methods and Problems in Greek Science: Selected Papers , Cambridge University Press,1991.[Machamer (1998)] Peter Machamer (ed.),
The Cambridge Companion to Galileo , Cambridge University Press, 1998.
Magruder (2009)] Kerry V. Magruder, Jesuit Science After Galileo: The Cosmology of Gabriele Beati,
Centaurus ,51(3), 2009, 189–212.[Mahoney (1973)] Michael Sean Mahoney,
The Mathematical Career of Pierre de Fermat , Princeton University Press,1973.[Maran & Marschall (2009)] Stephen P. Maran & Laurence A. Marschall,
Galileo’s New Universe: The Revolutionin Our Understanding of the Cosmos , BenBella Books, 2009.[Martínez (2018)] Alberto A. Martínez,
Burned Alive: Giordano Bruno, Galileo and the Inquisition , Reaktion Books,2018.[McGrew et al. (2009)] by Timothy McGrew, Marc Alspector-Kelly, Fritz Allhoff (eds.),
Philosophy of Science: AnHistorical Anthology , Wiley-Blackwell, 2009.[McMullin (1998)] Ernan McMullin, Galileo on science and Scripture. Chapter 8 in Peter Machamer (ed.),
TheCambridge Companion to Galileo , Cambridge University Press, 1998.[Mersenne (1933–1988)] Marin Mersenne,
Correspondance du P. Marin Mersenne , 17 volumes, Paris, 1933–1988.[Miller (2017)] David Marshall Miller, The Parallelogram Rule from Pseudo-Aristotle to Newton,
Archive for Historyof Exact Sciences , March 2017, 71(2), 157–191.[Montesinos & Solis (2001)] José Montesinos & Carlos Solis (eds.),
Largo campo di filosofare: Eurosymposium Galileo2001 , Fundación Canaria Orotava de Historia de la Ciencia, 2001.[Moody (1966)] Ernest A. Moody. Galileo and his precursors. In Carlo L. Golino (ed.),
Galileo Reappraised , Universityof California Press, 1966, 23–43.[Naylor (2003)] Ron Naylor, Galileo, Copernicanism and the origins of the new science of motion,
British Journalfor the History of Science , 36(2), 2003, 151–181.[Netz (2002)] Reviel Netz, Greek Mathematicians: A Group Picture, in C. J. Tulpin & T. E. Rihll (eds.),
Scienceand Mathematics in Ancient Greek Culture , Chapter 11, 196–216, Oxford University Press, 2002.[Netz (2004)] Reviel Netz,
The Works of Archimedes Volume 1: The Two Books On the Sphere and the Cylinder ,Cambridge University Press, 2004.[Neugebauer (1969)] Otto Neugebauer,
The Exact Sciences in Antiquity , second edition, Dover Publications, 1969.[Neugebauer (1975)] Otto Neugebauer,
A History of Ancient Mathematical Astronomy , Springer, 1975.[Newton (1977)] Robert R. Newton,
The Crime of Claudius Ptolemy , Johns Hopkins University Press, 1977.[Newton (1999)] Isaac Newton,
The Principia: Mathematical Principles of Natural Philosophy , translated byI. Bernard Cohen and Anne Whitman, preceded by “A guide to Newton’s ’Principia’ ” by I. Bernard Cohen.University of California Press, 1999.[Oleson (2008)] John Peter Oleson,
The Oxford Handbook of Engineering and Technology in the Classical World ,Oxford University Press, 2008.[Omodeo (2014)] Pietro Daniel Omodeo,
Copernicus in the Cultural Debates of the Renaissance: Reception, Legacy,Transformation , Brill, 2014.[Osler (2010)] Margaret J. Osler,
Reconfiguring the World: Nature, God, and Human Understanding from the MiddleAges to Early Modern Europe , Johns Hopkins University Press, 2010.[Ossendrijver (2016)] Mathieu Ossendrijver, Ancient Babylonian astronomers calculated Jupiter’s position from thearea under a time-velocity graph,
Science , 351(6272), 2016, 482–484.[Palmerino & Thijssen (2004)] Carla Rita Palmerino & J.M.M.H. Thijssen (eds.),
The Reception of the GalileanScience of Motion in Seventeenth-Century Europe , Boston Studies in the Philosophy and History of Science 239,Springer, 2004.[Palmieri (2001)] Paolo Palmieri, Galileo and the discovery of the phases of Venus,
Journal for the History of As-tronomy , 32(2), 2001, 109–129.[Pasachoff (2015)] Jay M. Pasachoff, Simon Marius’s Mundus Iovialis: 400th Anniversary in Galileo’s Shadow,
Jour-nal for the History of Astronomy , 46(2), 2015, 218–234.[Pappus (1871)] Pappus of Alexandria,
Der Sammlung des Pappus von Alexandrien: siebentes und achtes Buch , ed.Carl Immanuel Gerhardt, Halle, H.W. Schmidt, 1871.
Peters (1984)] W. T. Peters, The Appearance of Venus and Mars in 1610,
Journal for the History of Astronomy ,15(3), Oct. 1984, 211–214.[Pitt (1991)] Joseph Pitt, The Heavens and Earth: Bellarmine and Galileo, in Peter Barker & Roger Ariew (eds.),
Revolution and Continuity: Essays in the History and Philosophy of Early Modern Science , Catholic Universityof America Press, 1991, 131–142.[Pitt (1992)] Joseph C. Pitt,
Galileo, Human Knowledge, and the Book of Nature , Kluwer, The University of WesternOntario Series in Philosophy of Science, Volume 50, 1992.[Plato (1997)] Plato,
Complete Works , ed. John M. Cooper, Hackett, 1997.[Postl (1977)] Anton Postl, Correspondence between Kepler and Galileo,
Vistas in Astronomy , 21, 1977, 325–330.[Poupard (1987)] Paul Cardinal Poupard (ed.),
Galileo Galilei: Toward a Resolution of 350 Years of Debate ,Duquesne University Press, Pittsburg, PA, 1987.[Proclus (1970)] Proclus,
A commentary of the first book of Euclid’s Elements , translated by Glenn R. Morrow,Princeton University Press, 1970.[Randall (1940)] John Herman Randall, Jr., The Development of Scientific Method in the School of Padua,
Journalof the History of Ideas , 1(2), 1940, 177–206.[Rawlins (1987)] Dennis Rawlins, Ancient heliocentrists, Ptolemy, and the equant,
American Journal of Physics , 55,1987, 235–239.[Rawlins (2003)] Dennis Rawlins, Letter to the editor,
Isis , 94(3), 2003, 500–502.[Reeves & Van Helden (2010)] Galileo Galilei & Christoph Scheiner,
On Sunspots , translated and introduced byEileen Reeves and Albert Van Helden, University of Chicago Press, 2010.[Renn et al. (2001)] Jürgen Renn, Peter Damerow, Simone Rieger. Hunting the White Elephant: When and How didGalileo discover the law of fall? In Jürgen Renn (ed.),
Galileo in Context , Cambridge University Press, 2001,29–149.[Rihll (1999)] T. E. Rihill,
Greek Science , Greece & Rome: New Surveys in the Classics No. 29, Oxford UniversityPress, 1999.[Robertson (1775)] John Robertson,
A Treatise of such Mathematical Instruments as are usually put into a portablecase , third edition, London, J. Nourse, 1775.[Rorres (2004)] Chris Rorres, Completing Book II of Archimedes’s On Floating Bodies,
Mathematical Intelligencer ,26(3), 2004, 32–42.[Rorres (2019)] Chris Rorres,
Archimedes
Isis , 57(2), 1966, 262–264.[Ross (1923)] David Ross,
Aristotle , Methuen, London, 1923.[Russo (2004)] Lucio Russo,
The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to BeReborn , Springer, 2004.[Rutkin (2018)] H. Darrel Rutkin, Galileo as Practising Astrologer,
Journal for the History of Astronomy , 49(3),2018, 388–391.[Sanchis (2014)] Gabriela R. Sanchis, Historical Activities for Calculus,
Convergence , July 2014.[Santillana (1955)] Giorgio de Santillana,
The Crime of Galileo , University of Chicago Press, 1955.[Schemmel (2001)] Matthias Schemmel, England’s forgotten Galileo: a view on Thomas Harriot’s ballistic parabolas.In [Montesinos & Solis (2001)], chapter 2.7, 269–280.[Schemmel (2008)] Matthias Schemmel,
The English Galileo: Thomas Harriot’s Work on Motion as an Example ofPreclassical Mechanics , Volume 1: Interpretation. Springer, Boston Studies in the Philosophy of Science, Volume268. 2008.[Schemmel (2012)] Matthias Schemmel, Thomas Harriot as an English Galileo: The Force of Shared Knowledge inEarly Modern Mechanics. In [Fox (2012)], chapter 5, 89–112.[Schiefsky (2015)] Mark J. Schiefsky, Techne and Method in Ancient Artillery Construction: The Belopoeica of Philoof Byzantium, in Holmes & Fischer (eds.),
The Frontiers of Ancient Science , De Gruyter, Berlin, 2015. 613–651.
Scriba (1970)] Christoph J. Scriba, The Autobiography of John Wallis, F.R.S.,
Notes and Records of the RoyalSociety of London , 25(1), 1970, 17–46.[Settle (1961)] Thomas B. Settle, An experiment in the history of science,
Science , volume 133, Issue 3445, 19–23,January 1961.[Settle (1983)] Thomas B. Settle, Galileo and early experimentation, in R. Artis, H. T. Davis, & R. H. Steuwer (eds.),
Springs of scientific Creativity , University of Minnesota Press, Minneapolis, 1983.[Sharratt (1994)] Michael Sharratt,
Galileo: decisive innovator , Blackwell, 1994.[Shea (1972)] William R. Shea,
Galileo’s Intellectual Revolution , Macmillan, 1972.[Shea (2009)] William R. Shea,
Galileo’s Sidereus Nuncius or A Sidereal Message , Science History Publications,Sagmore Beach MA, 2009. Translated from the Latin by William R. Shea. Introduction and notes by WilliamR. Shea and Tiziana Bascelli.[Shea & Davie (2012)] William R. Shea & Mark Davie (eds.),
Galileo: Selected Writings , Oxford World’s Classics,Oxford University Press, 2012.[Siebert (2005)] Harald Siebert, The early search for stellar parallax: Galileo, Castelli, and Ramponi,
Journal for theHistory of Astronomy , 36(3), 2005, 251–271.[Solla Price (1959)] Derek de Solla Price, An Ancient Greek Computer,
Scientific American , 200(6), 1959, 60–70.[Solla Price (1974)] Derek de Solla Price, Gears from the Greeks. The Antikythera Mechanism: A Calendar Computerfrom ca. 80 B.C.
Transactions of the American Philosophical Society , 64(7), 1974, 1–70.[Solomon (2000)] Jon Solomon,
Ptolemy Harmonics: translation and commentary , Brill, 2000.[Stevin (1955)] Simon Stevin,
The Principal Works of Simon Stevin , Volume 1, edited by E. J. Dijksterhuis, Ams-terdam, C.V. Swets & Zeitlinger, 1955.[Struik (1969)] Dirk Jan Struik,
A Source Book in Mathematics, 1200-1800 , Harvard University Press, 1969.[Thomas (1993)] Ivor Thomas (ed.),
Selections Illustrating the History of Greek Mathematics , Loeb Classical Library,Harvard University Press, 1993 (first published 1941).[Thurston (2002)] Hugh Thurston, Greek Mathematical Astronomy Reconsidered,
Isis , 93(1), 2002, 58–69.[Toomer (1998)] G. J. Toomer (ed. and transl.),
Ptolemy’s Almagest , Princeton University Press, 1998.[Topper (1999)] David Topper, Galileo, Sunspots, and the Motions of the Earth: Redux.
Isis , 90(4), 1999, 756–767.[Tredwell & Barker (2004)] Katherine A. Tredwell & Peter Barker, Copernicus’ First Friends: Physical Copernican-ism from 1543 to 1610,
Filozofski Vestnik , XXV(2), 2004, 143–166.[Valleriani (2001)] Matteo Valleriani, A view on Galileo’s
Ricordi Autografi : Galileo practitioner in Padua. In[Montesinos & Solis (2001)], chapter 2.8, 281–291.[Van de Ven & Bos (2004)] Jeroen van de Ven & Erik-Jan Bos, Se Nihil Daturum — Descartes’s Unpublished Judge-ment of Comenius’s Pansophiae Prodromus (1639),
British Journal for the History of Philosophy , 12(3), 2004,369–386.[Van Helden (1974)] Albert Van Helden, Saturn and his anses,
Journal for the History of Astronomy , 5, 1974, 105–121.[Van Helden (1985)] Albert Van Helden,
Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley ,University of Chicago Press, 1985.[Vitruvius (1914)] Vitruvius,
The Ten Books on Architecture , translated by Morris Hicky Morgan, Harvard UniversityPress, 1914.[Voelkel (2001)] James R. Voelkel,
The Composition of Kepler’s Astronomia nova , Princeton University Press, 2001.[Voltaire (1762)] The Works of M. de Voltaire, Volume 13, translated by T. Smollett and T. Francklin, London.[Wallace (1984)] William A. Wallace.
Galileo and his Sources: The Heritage of the Collegio Romano in Galileo’sScience , Princeton University Press, 1984.[Wallace (1987)] William A. Wallace, Galileo and the Professors of the Collegio Romano at the End of the SixteenthCentury. Chapter 2 in [Poupard (1987)].[Wallace (1991)] William A. Wallace,
Galileo, the Jesuits and the Medieval Aristotle , Variorum, 1991.
Wasserstein (1962)] Abraham Wasserstein, Greek scientific thought,
Proceedings of the Cambridge Philological So-ciety , N.S. 8, 1962, 51–63.[Westfall (1966)] Richard S. Westfall. Th problem of force in Galileo’s physics. In Carlo L. Golino (ed.),
GalileoReappraised , University of California Press, 1966, 67–95.[Westfall (1985)] Richard S. Westfall, Science and Patronage: Galileo and the Telescope,
Isis , 76(1), March 1985),11–30.[Westman (1980a)] Robert S. Westman, The Astronomer’s Role in the Sixteenth Century: A Preliminary Study,
History of Science , 18(2), 1980.[Westman (1980b)] Robert S. Westman, Huygens and the problem of Cartesianism, in H.J.M. Bos et al. (eds.),
Studies on Christian Huygens , Swets & Zettlinger, Lisse, 1980, 83–103.[Whitman (1943)] E. A. Whitman, Some Historical Notes on the Cycloid,
American Mathematical Monthly , 50(5),May 1943, 309–315.[Wootton (2010)] David Wootton,
Galileo: Watcher of the Skies , Yale University Press, 2010.[Wootton (2015)] David Wootton,
The Invention of Science: A New History of the Scientific Revolution , Allen Lane,Penguin Random House, 2015.[Zilsel (2000)] Edgar Zilsel,
The Social Origins of Modern Science , edited by D. Raven, W. Krohn, & R. S. Cohen,Boston Studies in the Philosophy of Science 200, Kluwer, 2000., edited by D. Raven, W. Krohn, & R. S. Cohen,Boston Studies in the Philosophy of Science 200, Kluwer, 2000.