How was Galileo's approach to mathematics different than Descartes's approach?

How was Galileo's approach to mathematics different than Descartes's approach?

According to Galileo "the world is written in the language of mathematics," and a natural philosopher must learn to read it. How did this approach differ from Descartes' notion of a mathematical universe?

Here's what I've come across so far-


  • Invented the telescope, looked at planets, and for the first time, there was hardcore visual proof that not all astronomical bodies orbited the Earth.
  • Got branded a heretic (saved himself by recanting) for it, so Descartes stood down with his ideas since he feared that he'd have to go through the same process.
  • Was slightly accepted after Descartes laid down that God, being a perfect creature, would never try to deceive us, so we can trust our senses. So Galileo trusted his senses and hence trusted what he saw through the telescope.
  • Galileo was more focused on coming up with the math to solve math and physics problems.


  • Invented the Cartesian coordinate system and the analytical geometry that we have right now.
  • Descartes believed that Mathematics was the only certain thing in the universe, hence it could be used to reason things out.
  • Descartes, unlike Galileo, wanted to develop math so that he could reach any truth whatsoever.

You have the difference in the last bullet points of your two lists. Galileo was an experimental scientist, engineer first - math for him was the most comfortable tool to describe the nature's phenomena he studied. From his works, it seems that "why" was less important than "how" for him. Also, note that Galileo's mathematical methods were not very different from the ones used by his peers. Descartes' position, on the other hand, is better captured in his Wax Argument. The experimental results for him are secondary - the thing that captures the nature of the phenomena is the mind. His focus was on philosophy, not on natural science - and success in applying his works in natural sciences only reaffirmed this focus:

Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principal, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.


And there you have it - the classical example of an experimentator versus a theorist, a natural scientist versus a philosopher.

The differences were best summed up in a Wikipedia article on France's Descartes:

Descartes laid the foundation for 17th-century continental rationalism, later advocated by Baruch Spinoza and Gottfried Leibniz, and opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, and Hume. Leibniz, Spinoza[16] and Descartes were all well-versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well.

The article might have included Galileo with the English philosophers.

Followers of Descartes were "rational," and tended to have their actions dictated by a chain of reasoning. They saw mathematics as a "unifier" of science, from which the principles of science could be deduced; they approached science from a "top down" direction; theory first, then applications. The "Cartesian" coordinate system was a major step in this direction.

People like Galileo were more "empirical, that is, more likely to react intuitively to what their senses and data told them. They preferred to discover their science by "trial and error," (which is the approach Galileo took to religious issues as well, and drove the Catholic Church crazy). To people like Galileo, math was a tool with which to understand scientific discoveries, not a "framework" by which science was deduced.

An overview of the history of mathematics

Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.

In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60 . It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.

Geometric problems relating to similar figures, area and volume were also studied and values obtained for π.

The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration.

The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry.

The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11 th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.

Major progress in mathematics in Europe began again at the beginning of the 16 th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe.

The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in France.

The 17 th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.

Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17 th Century.

Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18 th Century.

However we must also mention Leibniz, whose much more rigorous approach to the calculus ( although still unsatisfactory ) was to set the scene for the mathematical work of the 18 th Century rather than that of Newton. Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.

The most important mathematician of the 18 th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.

Toward the end of the 18 th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.

The 19 th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.

The 19 th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20 th Century.

Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.

Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19 th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers

Analysis was driven by the requirements of mathematical physics and astronomy. Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.

The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm's work led to Hilbert and the development of functional analysis.

Notation and communication

There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.

For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.

What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton's notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.

It was not always like this: Harriot used a a a as his unknown as did others at this time. The convention we use ( letters near the end of the alphabet representing unknowns ) was introduced by Descartes in 1637 . Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.

Brilliant discoveries?

It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period.

For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered. Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.

Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at "the right time". This too would be completely unfair ( although it does go some why to explain why two or more people often discovered something independently around the same time ) . There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly.

How we view history

We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.

Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.

A challenge

If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs.

Then the world changed. The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever.

Here is the challenge. What will replace the calculator? You might say that this is an unfair question. However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us.

We can think of faster calculators, smaller calculators, better calculators but I'm asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, . .


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Measuring motion

Galileo probably gained his introduction to experimental technique while assisting his musician father, who conducted home experiments in the physics of sound. Galileo began his own experimental studies of motion while serving as a young mathematics professor at Pisa, where he is said to have dropped cannonballs from the Leaning Tower to demonstrate how objects of different weights fall with the same speed [see Falling Objects]. He continued his experiments during nearly two decades of teaching at the University of Padua, near Venice, where he measured the swinging of pendulums until he could describe their periods by a mathematical law, and he rolled bronze balls down inclined planes a thousand ways to derive the rate of acceleration in free fall.

Through such pursuits, Galileo discovered and described phenomena that generations of philosophers had not even noticed. For example, the shape of the path traced through space by a hurled or fired missile, Galileo showed, was not just "a line somehow curved," as his predecessors had said, but always precisely a parabola. And when lemons dropped from treetops, or cannonballs from towers, each one picked up speed in the same characteristic pattern, tied to the elapsed time of its fall: Whatever distance the object covered in one instant—measured as a pulse beat, a sung note, the weight of water that dripped from Galileo's timing device—by the end of two such instants it would travel four times as far. After three instants, it wound up at nine times the initial distance of descent after four instants, 16 distance units—and so on, always accelerating, always covering a distance determined by the square of the time passed [see Inclined Plane].

Galileo uncovered this fundamental relationship between distance and time without so much as a reliable unit of measure or an accurate clock. Italy possessed no national standards in the 17th century, leaving distances open to guesstimate gauged by flea's eyes, hairbreadths, lentil or millet seed diameters, hand spans, arm lengths, and the like. Galileo perforce delineated his own arbitrary units along the length of his experimental apparatus. As long as these units matched one another, he reasoned, he could use them to discern mathematical relationships. Lacking any kind of precision timekeeper, Galileo literally weighed the moments of his experiments: He allowed water to drip through a narrow pipe during the interval of interest then he balanced the collected water's weight against grains of sand.

". a large and excellent science"

Aristotelian philosophers of Galileo's day railed at such a mathematical approach to physics, on the grounds that mathematicians pondered immaterial concepts, while Nature consisted entirely of matter. They looked down on mathematicians and denigrated the study of mathematics as inferior—even irrelevant—to natural philosophy. Nature, in their view, could not be expected to follow precise numerical rules.

But Galileo correctly envisioned the experimental, mathematical analysis of Nature as the wave of the future: "There will be opened a gateway and a road to a large and excellent science," he predicted, "into which minds more piercing than mine shall penetrate to recesses still deeper." Among the first to bear out this prophecy was Sir Isaac Newton, born within a year of Galileo's death, who codified mathematical laws of motion and universal gravitation.

Posterity agrees that Galileo's great genius lay in his ability to observe the world at hand, to understand the behavior of its parts, and to describe these in terms of mathematical proportions. For these achievements, Albert Einstein dubbed Galileo "the father of modern physics—indeed of modern science altogether."

How Did Galileo Impact the World?

Galileo's main impact on the world was his improvement upon the telescope and being the first to use it in the science of astronomy. He also supported the Copernican system that stated that planets orbit the sun rather than the Earth as the Catholic Church said at the time. His other contribution was to contradict Aristotle's teachings that heavier objects fall faster than lighter ones.

Galileo Galilei was an Italian astronomer who challenged many of the commonly held ideas of his time. His discoveries of the laws of motion and telescope improvements are still considered the foundations of many scientific beliefs today. Galileo worked extensively with weights to counter and disprove Aristotle's theory about weight. He found that all weights fell at the same speed regardless of their mass.

It is sometimes thought that Galileo actually invented the telescope, but the truth is that he took an invention already in place and improved it. Further, he began using it in the study of astronomy, which was new at the time. His improvements enabled him to magnify things eight to nine times versus three. This is how he confirmed the theory of Copernicus that stated that Earth revolved around the sun.

Galileo’s Copernicanism

Galileo’s increasingly overt Copernicanism began to cause trouble for him. In 1613 he wrote a letter to his student Benedetto Castelli (1577–1644) in Pisa about the problem of squaring the Copernican theory with certain biblical passages. Inaccurate copies of this letter were sent by Galileo’s enemies to the Inquisition in Rome, and he had to retrieve the letter and send an accurate copy. Several Dominican fathers in Florence lodged complaints against Galileo in Rome, and Galileo went to Rome to defend the Copernican cause and his good name. Before leaving, he finished an expanded version of the letter to Castelli, now addressed to the grand duke’s mother and good friend of Galileo, the dowager Christina. In his Letter to the Grand Duchess Christina, Galileo discussed the problem of interpreting biblical passages with regard to scientific discoveries but, except for one example, did not actually interpret the Bible. That task had been reserved for approved theologians in the wake of the Council of Trent (1545–63) and the beginning of the Catholic Counter-Reformation. But the tide in Rome was turning against the Copernican theory, and in 1615, when the cleric Paolo Antonio Foscarini (c. 1565–1616) published a book arguing that the Copernican theory did not conflict with scripture, Inquisition consultants examined the question and pronounced the Copernican theory heretical. Foscarini’s book was banned, as were some more technical and nontheological works, such as Johannes Kepler’s Epitome of Copernican Astronomy. Copernicus’s own 1543 book, De revolutionibus orbium coelestium libri vi (“Six Books Concerning the Revolutions of the Heavenly Orbs”), was suspended until corrected. Galileo was not mentioned directly in the decree, but he was admonished by Robert Cardinal Bellarmine (1542–1621) not to “hold or defend” the Copernican theory. An improperly prepared document placed in the Inquisition files at this time states that Galileo was admonished “not to hold, teach, or defend” the Copernican theory “in any way whatever, either orally or in writing.”

Galileo was thus effectively muzzled on the Copernican issue. Only slowly did he recover from this setback. Through a student, he entered a controversy about the nature of comets occasioned by the appearance of three comets in 1618. After several exchanges, mainly with Orazio Grassi (1583–1654), a professor of mathematics at the Collegio Romano, he finally entered the argument under his own name. Il saggiatore ( The Assayer), published in 1623, was a brilliant polemic on physical reality and an exposition of the new scientific method. Galileo here discussed the method of the newly emerging science, arguing:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it.

He also drew a distinction between the properties of external objects and the sensations they cause in us—i.e., the distinction between primary and secondary qualities. Publication of Il saggiatore came at an auspicious moment, for Maffeo Cardinal Barberini (1568–1644), a friend, admirer, and patron of Galileo for a decade, was named Pope Urban VIII as the book was going to press. Galileo’s friends quickly arranged to have it dedicated to the new pope. In 1624 Galileo went to Rome and had six interviews with Urban VIII. Galileo told the pope about his theory of the tides (developed earlier), which he put forward as proof of the annual and diurnal motions of Earth. The pope gave Galileo permission to write a book about theories of the universe but warned him to treat the Copernican theory only hypothetically. The book, Dialogo sopra i due massimi sistemi del mondo, tolemaico e copernicano ( Dialogue Concerning the Two Chief World Systems, Ptolemaic & Copernican), was finished in 1630, and Galileo sent it to the Roman censor. Because of an outbreak of the plague, communications between Florence and Rome were interrupted, and Galileo asked for the censoring to be done instead in Florence. The Roman censor had a number of serious criticisms of the book and forwarded these to his colleagues in Florence. After writing a preface in which he professed that what followed was written hypothetically, Galileo had little trouble getting the book through the Florentine censors, and it appeared in Florence in 1632.

In the Dialogue’s witty conversation between Salviati (representing Galileo), Sagredo (the intelligent layman), and Simplicio (the dyed-in-the-wool Aristotelian), Galileo gathered together all the arguments (mostly based on his own telescopic discoveries) for the Copernican theory and against the traditional geocentric cosmology. As opposed to Aristotle’s, Galileo’s approach to cosmology is fundamentally spatial and geometric: Earth’s axis retains its orientation in space as Earth circles the Sun, and bodies not under a force retain their velocity (although this inertia is ultimately circular). But in giving Simplicio the final word, that God could have made the universe any way he wanted to and still made it appear to us the way it does, he put Pope Urban VIII’s favourite argument in the mouth of the person who had been ridiculed throughout the dialogue. The reaction against the book was swift. The pope convened a special commission to examine the book and make recommendations the commission found that Galileo had not really treated the Copernican theory hypothetically and recommended that a case be brought against him by the Inquisition. Galileo was summoned to Rome in 1633. During his first appearance before the Inquisition, he was confronted with the 1616 edict recording that he was forbidden to discuss the Copernican theory. In his defense Galileo produced a letter from Cardinal Bellarmine, by then dead, stating that he was admonished only not to hold or defend the theory. The case was at somewhat of an impasse, and, in what can only be called a plea bargain, Galileo confessed to having overstated his case. He was pronounced to be vehemently suspect of heresy and was condemned to life imprisonment and was made to abjure formally. There is no evidence that at this time he whispered, “Eppur si muove” (“And yet it moves”). It should be noted that Galileo was never in a dungeon or tortured during the Inquisition process he stayed mostly at the house of the Tuscan ambassador to the Vatican and for a short time in a comfortable apartment in the Inquisition building. (For a note on actions taken by Galileo’s defenders and by the church in the centuries since the trial, see BTW: Galileo’s condemnation.) After the process he spent six months at the palace of Ascanio Piccolomini (c. 1590–1671), the archbishop of Siena and a friend and patron, and then moved into a villa near Arcetri, in the hills above Florence. He spent the rest of his life there. Galileo’s daughter Sister Maria Celeste, who was in a nearby nunnery, was a great comfort to her father until her untimely death in 1634.

Galileo was then 70 years old. Yet he kept working. In Siena he had begun a new book on the sciences of motion and strength of materials. There he wrote up his unpublished studies that had been interrupted by his interest in the telescope in 1609 and pursued intermittently since. The book was spirited out of Italy and published in Leiden, the Netherlands, in 1638 under the title Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla meccanica ( Dialogues Concerning Two New Sciences). Galileo here treated for the first time the bending and breaking of beams and summarized his mathematical and experimental investigations of motion, including the law of falling bodies and the parabolic path of projectiles as a result of the mixing of two motions, constant speed and uniform acceleration. By then Galileo had become blind, and he spent his time working with a young student, Vincenzo Viviani, who was with him when he died on January 8, 1642.

Galileo helped prove that the Earth revolved around the sun

In 1610, Galileo published his new findings in the book Sidereus Nuncius, or Starry Messenger, which was an instant success. The Medicis helped secure him an appointment as a mathematician and philosopher in his native Tuscany.

He became close with a number of other leading scientists, including Johannes Kepler. A German astronomer and mathematician, Kepler’s work helped lay the foundations for the later discoveries of Isaac Newton and others.

Kepler’s experiments had led him to support the idea that the planets, Earth included, revolved around the sun. This heliocentric theory, as well as the idea of Earth’s daily rotational turning, had been developed by Polish astronomer Nicolaus Copernicus half a century earlier. Galileo and Kepler exchanged correspondence around Kepler’s ideas of planetary motion, and their detailed studies and observations helped spur the Scientific Revolution.

Their belief that the Sun, and not the Earth, was the gravitational center of the universe, upended almost 2,000 years of scientific thinking, dating back to theories about the fixed, unchanging universe put forth by the Greek philosopher and scientist Aristotle. Galileo had been testing Aristotle’s theories for years, including an experiment in the late 16th century in which he dropped two items of different masses from the Leaning Tower of Pisa, disproving Aristotle’s belief that objects would fall at differing speeds based on their weight (Newton later improved upon this work).

Lesson to All

A final lesson and warning applies to the Church, Science, and the modern Creationist movement today. Beware of holding steadfastly to a particular interpretation of Scripture and/or a scientific model, which may be in error. For instance, there are various scientific challenges to the Young-Earth Creationist position. We should hold many of our scientific views and their corresponding Biblical interpretations loosely. For we will never have all the right answers this side of heaven.

27 thoughts on &ldquo The Case Against Galileo &rdquo

This all may be true, but it still makes me a little sad. It’s a little like I felt when I learned that Thomas Jefferson may not have truly believed that “All men are created equal.”

Yeah, I feel that – though moral credibility is a very different thing than scientific credibility! It’s Jefferson’s soul on the line, whereas Galileo has only his C.V. to lose.

On the brighter side, this revisionism lets us shift the focus to other scientific leaders, less famous than Galileo and perhaps worthier of discussion: Mersenne, Kepler, Cavalieri…

The notion of “the Humanities” as opposed to “the Sciences” is a modern one, and would have baffled people at the time of Galileo (as would the distinction between philosophers and mathematicians). Galileo’s position as a populariser rather than an innovator has long been known — this seems simply to stir in modern Social-Media-style anger and bluster, and a lack of understanding of intellectual history. Better to read an actual book, peer reviewed, by someone who knows what she’s writing about — there are plenty out there. Podcasts… now rinse and spit.

“Social-Media-style bluster” is certainly a fair charge! There’s plenty in the original, although my excerpts no doubt skew the bluster-to-history ratio further.

Is “populariser rather than innovator” the consensus on Galileo among historians and philosophers of science? That view doesn’t seem to filter out into the popular literature (where he’s painted as Bacon’s peer in creating the foundations of empiricism, and Kepler’s in presaging Newtonian physics, plus more). But that would hardly be the first time that the popular literature missed the scholarly consensus!

I’m a bit behind on the history-of-science literature, to be honest — the notion of Galileo as a populariser rather than an innovator was something I learnt as an undergraduate back in the early 1980s, gleaned from books such as Thomas Kuhn’s 1962(!) book “The Structure of Scientific Revolutions”. A more recent book, which is well worth reading, is Michael Sharratt’s “Galileo: Decisive Innovator” (Cambridge Science Biographies) — it presents him as a populariser of the new science, though is a lot more generous to him with regard to his own work. Incidentally, the notion of Galileo dropping rocks (or a cannon ball and a feather, or whatever) from a tower has long been exploded the actual experiment involved an inclined plane, and Galileo wasn’t the first to perform it.

Thank you – I’m reading a little of Kuhn’s discussion now! And I’ll seek out Sharratt’s writing before I pretend to have any notion of what words should complete the sentence “Galileo was a popularizer ___ an innovator.” (Is it “and”? “More than”? “Rather than”?)

I do hope Viktor writes something on Galileo, too! I love podcasts, but they’ve got their limitations, of course.

Personally, I’d rather read a paper on the subject than listen to a podcast, where in the latter, one could inadvertently (or not) be a little looser with sources and accuracy.

I do love me some thonyc, and that is the format that I prefer, but I was more interested in the aspects of Galileo’s relationship with mathematics. Though I guess this does go to show that Galileo is supremely overrated.

This is my first experience with thonyc! I’m impressed.

I was just reading a 2010 post (can’t find the link now) in which thonyc gets more into the weeds on the “geocentrism vs. heliocentrism,” making a case that Galileo was pretty irrelevant to the scientific debate, in large part because he just wasn’t engaging at the appropriate level of mathematical technicality.

Tried to find that piece by searching, but thonyc has a *lot* of Galileo-skeptical writing!

I tend to celebrate Galileo in math history as the guy who mathematized science. He was a math fan boy, and his efforts to describe motion and position vs time were the start of something great. The reasons math became fashionable are connected to this need to describe nature with it.

Yeah, this is more or less the history I’ve always heard, and never had any reason to question. Certainly the intellectual steps attributed to Galileo in this account seem very important!

What this podcast brought to my attention (and what I hadn’t known before) was that there’s a serious strand of scholarship arguing that the steps attributed to him may not really be his. Like, maybe he brought these ideas (heliocentrism, principles of empiricism, the idea that math is the language of physical motion) to a wider literate audience, but didn’t really change the trajectory of science’s internal development.

One thing I’m curious about: Blasjo makes a lot of the church’s censorship of heliocentrism, as a reason why other Copernicans were shy about coming forward. But I’ve seen others making exactly the opposite argument – that the church wasn’t super invested in geocentrism, and that Galileo’s alienation from the church was driven by other factors.

This series seems akin to challenging the notion that Columbus discovered the Americas. Heresy! Heresy, I say!

You point out that ancients had already got to some of the ideas attributed to Galileo: pause to remember, though, that Galileo’s contemporaries were likely unaware of those ancients. They knew a select few of the ancients and regarded only some of them as trustworthy sources. An ancient they’d never heard of, or that wasn’t taken seriously, isn’t relevant to Galileo’s significance: what matters is that he got more folk to take some ideas seriously. Whether what he was saying was original is far less important than the fact that he managed to get it listened to.

Our culture likes myths of heroes and genius, so tends to paint a few people as such, exaggerating their achievements while ignoring all the other folk who made their (actual) achievements possible. Inevitably, the popular myth of Galileo thus grew beyond the reality, ignoring his deficiencies along the way. Scholarship usually has a rather toned-down view of figures that culture has magnified in this way the reality is usually that plenty of their contemporaries were having thoughts along similar lines, some of them taking them further and closer to what we’ve later settled on, but the ones who are remembered managed to get public attention, for one reason or another, so they’re who gets the credit. Occasionally two have to share, as Newton and Leibniz with the calculus, but even then the popular account makes it sound like it came out of the blue – ignoring the well-established work that surely contributed to the idea behind it.

(Aside: polynomial f(x) = sum a_i x^i chord gradient (f(u) -f(v)) / (u – v) for each power i, (u^i -v^i)/(u -v) is the usual sum of i terms, u^ +u^.v +… +u.v^ +v^ multiply by a_i and sum over i to get the gradient of the chord (f(u) -f(v))/(u -v). Since the numerator did have the denominator as a factor, we’re rid of that pesky denominator and can now let u and v get arbitrarily close without having to think too hard about what the gradient’s going to be when they coincide, it’ll be sum i a_i u^, and its value for v close enough to u will necessarily (as the polynomial is continuous) be close to this. Although you might have qualms about using the value at u=v as a gradient of anything, we get a polynomial in two free variables that does give exact chord gradients and does make it entirely natural to interpolate the u=v value as the gradient of a tangent. Newton and Leibniz managed to formalise this without relying on f being a polynomial which *is* an important leap it just doesn’t come out of nowhere.)

Myths of heroes and genius give us a simple story to anchor new ideas to, that helps culture assimilate the idea during the course of doing so, it distorts the truth of whence the idea came because that’s less important than getting the lesson assimilated. Later we can go back and fix up the reality of those historical figures who don’t quite match the myths that got attached to them. The same goes for demons, for that matter – The Spanish Inquisition was grossly misrepresented by protestants (particularly in Holland and the North American colonies), building up a myth that barely resembles the historical reality. This is how cultures mangle their past fortunately, we’ve had writing for a few millennia now, so we often have contemporary sources historians can consult to rediscover the original. Which is worth doing, so I hope you enjoy the podcasts.

… and I neglected to say: so, rather than “the case against Galileo”, think in terms of “the case against the mythology that has accreted around Galileo” and, putting all that mythology aside, take some time to learn what Galileo actually did and to appreciate him for who he actually was. Far more mortal and flawed than the myth, but an interesting and worthy chap, none the less. Try Dava Sobel’s “Galileo’s Daughters” for a sympathetic-ish picture of him.

Worry I don’t have the intellectual rigor of some of the posters, and can’t cite sources. However, I’d heard/read that Arabs already had much more significantly evolved mathematics (and astronomy) long before the Greeks. I probably read it in one of the more scholarly and less sensational articles on the Antykethera device.

Well, the ancient Greek civilisation’s heyday came long before that of the Arabic civilisation, so I suspect there’s some garbling there but Europe got its Greek learning from the Arabs, who’d improved it and enhanced it in various ways along the way, including merging it with some important learning from India (which was probably a source for the Greeks also), notably the better system for representing numbers. An Arab (after whom algorithms are named) was responsible for the invention of algebra and the reinterpretation of lots of geometry in terms of it. (Arabs also worked out how to square Greek philosophy (coming from a polytheistic culture) with a monotheistic religion the resulting synthesis was then taken over wholesale, with minor adaptations, by Thomas Aquinas and others to give Christianity a philosophical rationale. Meanwhile Arab alchemists invented the alembic with which to purify an essence that came to be known as alcohol. You’ll notice a lot of words starting with “al” here it means “the” in Arabic, IIRC.) Christians appropriating all this learning weren’t always eager to credit the Arabs with it, but crediting the Greek precursor sources (where there were any) was totally cool. All of which is roughly why the Renaissance happened.

So *after* the Greeks (and with input also from India and possibly elsewhere), Arabs did indeed significantly advance mathematics (and astronomy) and it’s on the result of the Arabs’ work (and that of some intervening Europeans) that Galileo was building.

The Rolling Ball Experiments: Galileo’s Terrestrial Mechanics

Galileo Galilei was not just an astronomer, but also a scientist who performed many mechanical experiments. (Image: Justus Sustermans/Public domain)

Disproving Aristotle’s Ideas about Falling Objects

In an age when cannons had just been developed (and gunpowder and explosives), people needed to be able to fire objects accurately from one place to another. They needed to know how objects moved on Earth. They needed to know what sort of curving paths objects adopted when they were fired in the Earth’s gravity. Aristotle had said that heavier objects fall faster than light objects, and this is a claim that Galileo demonstrated to be quite false.

The story that Galileo dropped two balls from the Leaning Tower of Pisa is probably apocryphal, but he did do a similar experiment. He took two objects of different masses and different sizes and dropped them from a high place, and found that they landed at exactly the same time. So, through an empirical experimental approach, he showed that the reasoning, the rationale, of Aristotle was wrong.

This is a transcript from the video series The Joy of Science. Watch it now, on Wondrium.

Galileo’s Rolling Ball Experiment

Realizing that free-falling objects move too fast to measure with any sort of conventional techniques of the day—the watches and clocks that were available at that point—Galileo devised an ingenious, adjustable ramp to dilute the effects of gravity.

What he would do was measure a distance along the inclined plane, and then time the fall. This is called the rolling ball experiment.

Accurate Time Measurement

But the main problem with the ‘rolling ball’ experiment is that accurate time measurements are needed. In Galileo’s day, there weren’t really any accurate timepieces. At first, Galileo used his pulse, but that wasn’t very accurate. Then he invented an ingenious way to measure time.

We employed a large vessel of water and placed it in an elevated position. To the bottom of this vessel was soldered a pipe of small diameter, giving a thin jet of water. We collected this water in a small glass during the time of each descent. The water thus collected was weighed after each descent on a very accurate balance. The differences and ratios of these weights gave us the differences and ratios of the times.

Now that he had the means to measure time, Galileo and his assistants conducted numerous repetitions—another aspect of experimental science.

In such experiments repeated a full 100 times we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the channel along which we rolled the ball.

The Conclusions of the Rolling Ball Experiment

Galileo adapted a water clock so that he could measure time in terms of water collected. (Image: Alexander_P/Shutterstock)

If, for example, it took an object six units of time to go an entire length, then a guess might be that it would only take three units to go half as far as the marked length.

But Galileo found that it takes four units to travel half the distance three units to travel one-quarter of the distance, and six units to travel the entire distance.

So, the distance traveled is proportional to the square of time. That was Galileo’s great discovery with the rolling ball experiment.

A fascinating aspect of this experiment is that Galileo did not conduct the rolling ball experiment to discover a mathematical relationship between time and distance. Rather, he used the apparatus to confirm his conviction that velocity and time bear the simplest kind of relationship to each other.

That is, the velocity of a falling object is proportional to the time of its fall. He called this steadily increasing velocity, uniform acceleration. Galileo also demonstrated mathematically that this result was equivalent to saying that the distance traveled by a falling object is equal to the square of the time of its fall.

Other Experiments in Terrestrial Mechanics

Galileo devised lots of other experiments in his study of terrestrial mechanics. One of his most famous is his study of pendulums where he found that longer pendulums swing more slowly than shorter ones and the rate of speed is independent of the mass of the pendulum.

Galileo also discovered a key principle regarding ballistics, that is, the way objects fly through the air. He found that the horizontal motion of a falling object is completely independent of the vertical fall. For example, he cited the example of a heavy object dropped from the mast of a moving ship.

Aristotelian philosophers held that an object would land some distance behind the mast of the moving ship—if it was dropped from high up the object would fall backward, and the ship would move out from under it. But Galileo said no, the object is moving along with the ship and is going to fall right at the base of the mast.

He tested a lot of other ideas experimentally. For example, he fired cannonballs horizontally off a cliff and observed the curving path of the fall. And what he found is that when you do that, you always find the same type of curved path, called a parabola. He found that all falling objects will follow the same kind of path.

So, Galileo ought to be remembered not just as a great astronomer, but also as the scientist who first discovered the basic rules of terrestrial mechanics: the rules of how objects moved on the Earth.

Common Questions about Galileo’s Rolling Ball Experiments

Realizing that free-falling objects move too fast to measure with any sort of conventional techniques of his age, Galileo devised an ingenious, adjustable ramp to dilute the effects of gravity and slow objects down to observable speeds.

The main problem with the rolling ball experiment was that Galileo needed accurate time measurements . He invented an ingenious way to measure time, which involved weighing the water expelled from a pipe in the time the ball rolled down the incline.

The conclusion of the rolling ball experiments was that the velocity of a falling object is proportional to the time of its fall. Galileo proved that this means that the distance traveled by the ball was proportional to the square of time.


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Watch the video: Galileo - and his big idea