Galileo has been called “the father of science” , “the greatest mathematician in Italy and, perhaps, in the world” (of his time), the first thinker who abandoned superstition in favor of science, the person who united the physics and mathematics, the central figure of the Scientific Revolution, a martyr of science… I imagine he would also have gleaming white teeth and a handsome presence, a kind of Captain America of science.
It seems that the person who saw Galileo in this way was… Galileo. If we could psychoanalyze him, we might be surprised to learn that he was a somewhat ambitious and narcissistic character . In his Letter to the Grand Duchess Cristina, Galileo says of those who doubt his theories that: «I should pay no more attention than those who have previously contradicted me — at whom I always laugh, because I know the result for sure — [. ..]». Although Galileo was proven wrong numerous times, his self-esteem was through the roof. But it seems that his mathematics were not up to his self – esteem .
Despite being such a towering figure in the history of science, there is evidence that, at least as a mathematician, Galileo was not stellar. Mathematics historian Viktor Blåsjö has published a podcast and written an informed essay with endless details, but the bottom line would be: Galileo was, at heart, a failed mathematician. It could even be that his problems with the Inquisition stemmed from the fact that he could not maintain a mathematical argument for heliocentrism and turned to philosophy.
This article does not want to be a revisionist treatise to demystify his figure, but rather a look at Galileo’s relationship with mathematics. But alas, his relationship with mathematics was not entirely good, despite having been a professor of mathematics at the universities of Pisa and Padua.
One last anecdote about the “Galileo myth” . Supposedly, Galileo was the first man to leave superstition behind and rely solely on scientific knowledge. The reality is that Galileo participated in the society in which he lived, including its superstitions. Thus, payments from various people for making horoscopes appear in his account books and, probably, the writing of horoscopes would be one of the most important functions of the Padua mathematicus , since part of his job was to teach horoscopes to the Medicine students. As an anecdote, in 1609, Galileo wrote a horoscope for the Grand Duke Ferdinando I de’ Medici, in which he predicted a long and successful life, but unfortunately the Duke died in February of that year.
Galileo was perhaps not a great giant of science. What we can say is that he was a human person, with his mistakes, his weaknesses and his vanity, like all of us. Let’s see some chapters of his mathematical life and the contributions he could have made.
The mathematization of science
Galileo is frequently credited with the mathematization of science, or the bringing together of mathematics and natural phenomena . However, since ancient times, mathematics was the best way to know the universe. Archimedes, the great inventor, whom Galileo admired and wanted to emulate, also made great contributions to mathematics and invented hydrostatics; Pythagoras considered that everything is number, and yet another example courtesy of Viktor Blåsjö: Iamblichus wrote that we attack everything in nature mathematically. Greek science was firmly based on mathematics.
Perhaps he was the first to apply mathematics to refute the Aristotelian heritage? Descartes, a contemporary of Galileo, said of him that he is eloquent in refuting Aristotle. Indeed, Descartes was more brutal in his criticism of Galileo : “I see nothing in his books that makes me envy him, and almost nothing that I wish to confess was mine.” The scientific community of the seventeenth century had long since abandoned Aristotle. Descartes, with his usual amiability, said that the Aristotelians knew less than if they had not studied.
And the new sciences of Galileo? Tartaglia published in 1537 a book called La nova scientia , ( The new science ), with which he began the mathematization of ballistics. A little over a hundred years later, in 1638, Galileo published his famous Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze , which we will call The Two Sciences . The two sciences in question are materials engineering and kinematics.
Galileo necessarily knew Tartaglia’s La nova scientia because the link between Tartaglia and Galileo was Ostilio Ricci, who was Tartaglia’s student and Galileo’s teacher. Apparently, being a student of Ricci was one of the reasons why Galileo abandoned his medical studies and devoted himself to mathematics . Not only that, but Ricci also viewed mathematics as a practical tool for solving physical problems. In fact, his best-known work is called Problemi di Geometria Pratica: L’uso dell’Archimetro .
Thus, we see that Galileo is not the first to devise a mathematization of science, but that he connects with an entire tradition that comes from ancient times of applying mathematics to physical, concrete and tangible problems. He even had examples of recent mathematization.
the first calculator
Galileo is also the name of a global positioning satellite system , which combines two areas in which Galileo Galilei worked: astronomy and calculations. In this sense, that of facilitating calculations, Galileo published in 1606 a book called Le operazioni del compasso geometrico e militare , where he describes an instrument to perform calculations. Some authors call it the «first mechanical calculation instrument» and on occasion I have heard the opinion that without Galileo the path that, after hundreds of years and refinements, ends in the personal computer, would not have been started.
In reality, Galileo’s ‘invention’, the geometric or sector compass, was a device similar to several in circulation at the time. What Galileo did stand out in was what we would call branding today. In 1606, Galileo published 60 copies of Le operazioni del compasso geometrico e militare , each containing a compass. Galileo used a worker who was housed in his house for the manufacture of the instruments. As Drake recounts, the benefits of direct selling may not have been excessive, but where Galileo did benefit a great deal were private lessons to teach students how to use the compass.
As if that were not enough, there were quite a few discussions about the authorship of the compass. Even one of Galileo’s students, Baldassarre Capra, wrote a treatise claiming authorship in 1607 and it was not the first time that Galileo had to defend himself against the accusation of plagiarism in relation to the compass . Galileo prevailed, and the university presidents decreed the destruction of Capra’s books. Incidentally, Galileo gave the matter as much publicity as possible and published the Difesa contro alle calunnie et imposture di Baldessarre Capra .
The concept of infinity
As is well known, the book Dialogo sopra i due massimi sistemi del mondo ptolemaico e coperniciano , from 1632, cost Galileo a dislike with the Inquisition . Not for defending heliocentrism, which many astronomers of that time and later discussed without problem, but for disobeying an order from 1616, and also making fun of the pope, white on black, in the middle of Italy during the Counter-Reformation.
In the dialogue, Galileo approaches the concept of infinity on a couple of occasions. In one case it is an argument about infinitely large quantities and the other about infinitely small quantities, and here we find one of Galileo’s great missed mathematical opportunities.
Salviati’s character demonstrates to Simplicio — an ignorant Aristotelian who always turns out to be ridiculously wrong — the fact that there are the same number of natural numbers (1,2,3,…) as there are square numbers (1,4,9). ,…). The argument is mathematically sound: every square is the square of some number, and every number can be squared. Therefore, there is a one-to-one correspondence between both sets of numbers. Today we would say that both sets are bijective.
However; precisely this is the property that defines infinite sets; which can be put into bijective correspondence with a strict subset. At this point, where Galileo could have made a great contribution to mathematics, he decides not to continue his investigation and simply says that the attributes equal, greater, or less do not apply to infinite quantities. Today we know that there is a whole arithmetic of transfinite numbers , although we would have to wait for Georg Cantor at the end of the 19th century to formalize it.
The other time Galileo approaches the concept of infinity is when discussing whether it is possible for an object to leave the surface of the earth (here Galileo was wrong in arguing that it could not). At one point, he seems to claim that certain infinitesimals are less than others. However, he did not carry this idea to term either. Mathematical historian Carl Boyer claims that Galileo wanted to write a treatise on infinity, but if he did, nothing has survived. In any case, it would take a generation to adequately justify these ideas: in 1642, the year Galileo died, Newton was born, who developed the theory of what we now call infinitesimal calculus in 1669. Newton was also a controversial figure and is his dispute with Leibniz about the priority of the calculus is well known.
The cycloid
Suppose that with a hot glue gun we glue a marker to the inside of a glass and that we roll it with the marker touching a vertical wall. The curve that the pen traces on the wall is called a cycloid.
Galileo was interested in this curve ; he even proposed it as a curve that could be fitted to bridge arches and wanted to study it in some detail in 1599. One of the areas of interest for 17th-century mathematicians was the problem of squaring a curve; that is, to calculate the area under it, so a natural problem for Galileo would have been to calculate the area under an arc of the cycloid. Archimedes, the mathematician admired by Galileo, had calculated the area of all the sections of the parabola, the area of the spiral, etc. Here Galileo had the opportunity to resemble his hero.
Unfortunately, Galileo was unable to solve this problem . In the absence of an exact solution, he did an experiment: he cut some pieces of metal in the shape of the cycloid and weighed them, arriving at the conclusion that the area under the cycloid was somewhat less than three times the area of the circumference that the cycloid had. generate; This is how he tells it in a letter to Cavalieri in 1640. The distinction is important, because Galileo explicitly says that he rejects the idea that the area under the cycloid is exactly three times that of the generating circle, which is the correct answer.
To add insult to injury, it was not a matter of a lack of mathematical tools, as was the case with infinities. Descartes, Torricelli and Fermat, all Galileo’s contemporaries, did calculate the area correctly. Descartes in particular, never moderate if he could be offensive, wrote that “I don’t see why you attach so much importance to something so simple, which anyone who knows even a little about geometry cannot fail to observe.”
catenary
Another interesting curve is the catenary. This curve describes a chain (or a rope) suspended between two points. It is shaped like the letter U, more or less open depending on where the chain is suspended. In fact, it takes its name from the Latin word catena, which means, precisely, “chain” . In Spain, an application of the catenary curve is found in the works of the magnificent Gaudí, who used the catenary — inverted — in his designs, especially in the Sagrada Familia. Well, Galileo also studied the catenary; though he mistakenly thought it was a parable . Galileo’s notes survive in which he attempts to fit a parabola to a catenary, in which you can still see the holes in the paper where he suspended a string for the fit. Although the two curves are very similar, the fit is not perfect. Despite this, by 1606, Galileo thought that both were identical.
Depending on how a paragraph of The Two Sciences is interpreted, it is possible to think that by 1638, Galileo had discarded this idea and thought — now yes, correctly — that the catenary and the parabola were different. The passage in question is unclear: it’s talking about a way for gunners to calculate a trajectory, but it’s not clear if it’s approximate or exact. Incorrect statements are found in the same passage, so the matter is difficult to decide.
The parabolic trajectory
Currently, all high school science students solve projectile trajectory problems by breaking down the motion into its horizontal component, with uniform motion, and the vertical, with uniformly accelerated motion, resulting in a curve called a parabola (for this resolution “simple” you have to ignore the friction with the air, which requires a much more advanced treatment).
We have already mentioned The Two Sciences . In it, Galileo correctly states that the motion of a projectile is parabolic . The question of whether he proves it adequately, that is, of the mathematical argumentation of the result, is another. And as if this were not enough, Galileo only discusses the case of a projectile with an initial horizontal trajectory. Evangelista Torricelli, one of his students and who would become a famous physicist, did give a complete mathematical formulation of the parabolic shot. Of his teacher Galileo’s writing, Torricelli would say that the result is “more desired than proven.”
However, the first to correctly publish the proof of the parabolic trajectory of projectiles would be Bonaventura Cavalieri, six years before Galileo. Drake and MacLahan recount in a 1975 article that Galileo was outraged to learn of Cavalieri’s publication. This gave rise to another controversy: Cavalieri was a student of Castelli, who was a student of… Galileo. And Galileo, enraged, upon learning of Cavalieri’s publication, demanded the preferential right of publication. It must be said that Cavalieri was a better mathematician than Galileo and also had more of a left hand: he wrote to Galileo saying that he had recognized the work of Galileo and Castelli in his book, that everyone knew that the discovery was Galileo’s, and that he believed Galileo had already published it. Thus calm, Galileo would have praise for Cavalieri in The Two Sciences .
conclusion
Galileo’s work can be examined, in the Old Fund of the University of Seville, among other places, and digitized and available online. It is impossible not to admire his Sidereus nuncius , in which he describes the positions of the satellites of Jupiter and his engravings showing the mountains and valleys of the Moon.
Perhaps as a mathematician, physicist or scientist his contributions were not as great as his fame; but Galileo did make a contribution that left a mark on his society. Without a doubt, Galileo was a great popularizer of science . Starting from the fact that he wrote in Italian, when most of his colleagues still did in Latin, and he did it exceptionally well, Galileo was a great popularizer of the science of his time. He was also a great popularizer of himself and his achievements, perhaps somewhat exaggerated; but it undoubtedly contributed to making science take root as part of Renaissance society.
His writings use irony, humor, rhetoric, and provocation to make available to many an understanding of scientific work that was then still incipient. Perhaps, if he had had a little more prudence along with his literary abilities, he would not have had his famous disagreement with the Inquisition.
