Galileo had balls

Prompted by a post by Gene Callahan's recent post, I have thought more about Galileo's thought experiment concerning the two balls (of different mass) tied to each other.  Galileo attempted to prove the inconsistency of Aristotelian physics by arguing that two balls tied to each other are a composite object of greater mass than either, so they must fall faster than either constituent ball, but at the same time the lighter ball has a natural tendency to fall slowly and so it will try to slow down the heavier ball.  Based on this, Galileo concludes that the model is inconsistent and so the balls must accelerate at a mass-independent rate.

This is, of course, wrong in one very important sense:  In a world with air resistance things do fall at mass-dependent rates!  Yet, at the same time, we know that Galileo is right about purely gravitatinal effects.  So, how did he get that right, and was it a coincidence?

Well, first, he got it right about purely gravitational effects because of an embedded assumption of inertia.  He is assuming that the lighter ball is trying to retard the motion of the heavier ball, which means that he's assuming that it's harder to move if you're pulling something.  While that's not a fully Newtonian model, it has a general idea of inertia in there.  If you assume a mass-dependent resistance to changes in motion then adding mass to an object should slow its acceleration, which can (if the dependencies are the same) cancel the mass-dependent pull of the earth.

Galileo got this right, but it was partly a coincidence because these cancellations require that mass enter into inertial and gravitational effects in the same way, so that you can divide it out.  Still, he got the qualitative idea.  I wouldn't recommend acceptance of his theoretical arguments at a journal today, but I give him full marks for groping toward the right physical idea and making a major leap in 17th century physics.

The other reason Galileo got this right is that he made the Aristotelian assumptions explicit while keeping inertia implicit.  When your suppositions lead to a contradiction it means that at least one supposition is wrong.  If you only make one assumption explicit then you'll reject that assumption, rather than concluding that there's a tension between two assumptions and we need more investigation to figure out which is wrong.  Galileo may not have realized to what extent he was invoking an assumption of inertia.  I'd probably need to read Two New Sciences to see how well he understood inertia.

Galileo: Parting shots

I'm almost done with selections from "Two New Sciences."  This is mostly about falling objects, buoyance, pendulums, and solid mechanics.  It's well-argued, but it doesn't pack the same philosophical punch as "Dialogue Concerning the Two Chief World Systems."  That's fine--the subject matter doesn't lend itself to the same philosophical punch.  I view "Two New Sciences" as a book that physicists should challenge themselves with (and I intend to take up that challenge by getting the whole thing and reading it cover-to-cover at some point).  If you really get Newtonian mechanics, you'll get "Two New Sciences."  On the other hand, I view "Dialogue Concerning the Two Chief World Systems" as a matter of historical and cultural literacy, being a book that sparked an infamous episode in the history of science and religion, and also brought about a fundamental transformation of how the Western world sees the universe.  I intend to plug that gap in my literacy by re-reading it at some point.  When I read it several years ago it was a hard slog, but now that I've digested some excerpts and contemporary documents with annotation, I'm ready to try again.

Next book blogging project to be announced soon.

Everybody must have stones

I'm reading Chapter 10, excerpts from Two New Sciences (published in 1638, while he was under house arrest).  I like his argument for free fall speed being independent of the mass of the object:  If we take a heavy and light stone and tie them together, then the small stone should retard the large stone while the large stone drags the small stone along, so they should go at a pace intermediate between their "natural" rates.  However, they also form one large composite stone, and the larger stone should go even faster.  So we get a contradiction.

Unfortunately, this argument is not entirely correct.  If we drop the stones from a sufficiently tall building or cliff, they will reach terminal velocity (due to air resistance) and then the larger stone will indeed go faster (if they have the same shape).  However, if we tie them together then the resulting speed will depend on both the increased mass (which, all else equal, will increase the speed) and the new shape (which could potentially increase the drag force).

However, Galileo's argument is sufficient to show that a model that only considers mass, to the exclusion of all other variables, is indeed defective.  It just isn't sufficient to show the universality of free fall.

I first came across this argument a very, very long time ago, back when dinosaurs roamed the earth and Pets.com was a good buy and I was in grad school.  I read Jacob Israelachvili's book on intermolecular forces, and he included Galileo's argument in the first chapter, along with an assertion that the argument is wrong, but he left it as an exercise for the reader to figure out why the argument was wrong.  It took me a while to figure out the problem with the argument.

Bullship

If I'm reading pages 228-229 correctly, Galileo never actually dropped a rock from the top of the mast of a ship, and is just asserting what the outcome of the experiment must be.  That is...not what I was expecting from Galileo.

The words of the prophets are written on...?

I'm on chapter 8 of The Essential Galileo now.  Chapter 5 was letters from Galileo to supporters, largely arguing that a figurative interpretation of Biblical verses on natural phenomena need not be a threat to faith.  Countless scientists throughout the ages have been followers of religious faiths and would agree with him.  I find it interestingly that Galileo did not mention the Gospels, which are full of parables, as evidence that the Bible is awash in figurative language.  Also, in two of his letters he says (paraphrase) that if God only wanted us to learn about the natural world from the Bible then He would not have given us the ability to observe and reason.  Countless scientists who are also religious agree whole-heartedly with our illustrious predecessor on this point.

Chapter 6 is largely documents and testimony from Galileo's first trial, in 1615.  These documents are consistent with a distinction that a historian friend of mine drew:  Galileo didn't get in trouble for heliocentric theory so much as for public statements concerning the Church's authority on astronomical questions.  The testimony of the witnesses is less about heliocentrism vs. geocentrism and more about Galileo proclaiming that his heliocentric-compatible interpretation of Scripture is superior to the Church's.  Also, the witnesses refer ominously to "disciples" of Galileo.  They feared his theory less than his ability to persuade people to favor his teachings over those of the Church.

Chapter 6 ends with a 1616 edict adding several books to the Index of forbidden books.  Nothing by Galileo was on the list, but Copernicus' work made the list.  What else got banned in that year?  Well, it will be no surprise that a book on Calvinism made the list (poor Puritans :( ), nor that some books on law and politics got banned.  But it is amusing that a book called Scotanus Redivivus, or Erotic Commentary in Three Parts, was banned alongside Kepler.  A quick Google search mostly returns pages about Galileo and Copernicus, which is rather disappointing.  (Though there is a link to Fark!)  I could get my freshmen more interested in physics if I did a comparative study of Copernicus' writings and those of his contemporaries.

Chapter 7 was boring, so I skipped it.  Something about comets and disputes with contemporaries.

Chapter 8 is a series of excerpts from Dialogue on the Two Chief World Systems.  I haven't gotten very far in that chapter, but I like this quote from his preface:

Many years ago I had occasion to say that the unsolved problem of the tides could receive some light if the earth's motion were granted.  Flying from mouth to mouth, this assertion of mind has found charitable people who adopt it as a child of their own intellect.

Yeah, that burns.

Galileo, Chapter 5

Chapter 5 is mainly documents written by Galileo in 1615, answering his critics at trial.  Much of it concerns interpretation of Scripture, and how a robust, honest, and defensible Catholicism will require a figurative interpretation of passages whose literal meaning would contradict observations.  However, in the midst of that, he also challenges the argument that there's no point in believing Copernicus if even Copernicus regarded his model as only a tool for calculation.  Galileo elides questions of what Copernicus "really" thought (as high as his self-esteem was, Galileo did not consider himself capable of performing necromancy and consulting the dead Copernicus).  Instead, he noted that in many ways it's the geocentric model that is a calculational tool.  The heliocentric model enables fairly simple calculations, but if one wishes to put those results into a form that can be compared with observation, one must switch to a geocentric frame.  This blurs the distinction between steps taken for convenience and steps taken to reflect reality.  (Einstein finally shattered that distinction altogether.)

I like this argument.  I'm not, however, quite so favorable towards another argument.  He also tries the argument that everyone who favors the heliocentric model started off favoring the geocentric model, so they must have had a good reason to change their minds.  Furthermore, only smart and informed people will even think of adopting the heliocentric model, because they will understand the subtle observational and theoretical arguments for it, whereas less informed people (and well-informed but cautious people) will reject it.

I don't like that argument.  I've met smart people who believed all sorts of dumb things for all sorts of dumb reasons, and I've met ignorant people who embraced alternative viewpoints simply because they wanted to be in the opposition, not because they actually understood the merits of the idea.  I get that Galileo was on trial and trying to persuade people rather than trying to carefully strike at deep truths of human psychology.  However, I am nonetheless unimpressed by the argument.  Surely he must have known that fools were, are, and always will be everywhere.

Galileo! Galileo! Galileo Figaro! Magnifico!

Well, I've now read a selection on sunspots, and also his letter to Castelli (a monk and mathematics professor). I'm in the middle of his letter to the Grand Duchess Christina.  The man clearly likes to argue, he clearly knows when he's right, he clearly knows just how wrong his enemies are, and he has an incisive intellect and wit.

Shame he died so long ago.  I'd have enjoyed hanging out with him.  And since he's Tuscan, he probably had awesome culinary taste.

Next book: The Essential Galileo

A while ago I read snippets of Galileo.  I liked his style, but because I just read it without journaling or discussing I retained far too little.  (See, I can admit the  problems of a "wall of text": Those problems come when you don't follow it up with something to sear it into your memory.)  Now I'm reading an anthology of his writings, edited and translated by Maurice Finocchiaro.  More details to follow.