Late on a rainy night some years ago, a few blocks away from home on Broadway, I happened to give a homeless man a dollar or two. In gratitude, he handed me a book. It was very dark, so I had to wait until I got home to see that it was a wet, worn and torn, pale blue copy of *Studies in Mathematics, Volume XI: Mathematical Methods in Science* by George Polya, edited by Leon Bowden. Leaving it to dry on the radiator overnight, I looked at it the next day. It turned out to be a course of lectures that Polya had given in the summer of 1962 at Stanford to high school teachers of mathematics, under the auspices of the National Science Foundation. (Confession: the book has a Hunter College stamp in it, which may mean that I am in posession of a purloined library book.)

As I skimmed through the book, I found much fascinating material, but most of all, I was struck by a short introduction to the work of Simon Stevinus. I had never heard of Stevinus before this, and I would guess that many of you might not have either. It turns out that Stevinus was a brilliant 16th century Dutch mathematician, engineer, and scientist; a contemporary of Decartes, who even anticipated some of Galileo's work. Stevinus was the first to use decimal fractions and showed their usefullness. As an engineer, he constructed dykes which are in use to this day.

What I want to talk about today is this: contrary to popular caricature, science does not always advance by observation and measurement. Often, it is a simple thought experiment which results in new insight. Einstein's musings about what would happen if he sped along with a wave of light at a speed close to its own resulted in a rather famous conclusion, for example. (Of course, even Einstein's thinking was connected to reality by the experimental and observational work of others.) And this serendipitously-discovered book exposed to me a truly marvelous thought experiment I hadn't ever known about. I would like to explain this physically-intuitive proof in some detail to you, by which Stevinus derived the Law of Inclined Planes. In going through this step-by-step, I hope to illustrate the power of the thought experiment in general, and the elegance of Stevinus's imaginative formulation in particular.

Now bear with me here: it is obvious that pushing an object up an inclined plane is easier than lifting it up vertically. (The weight of an object is simply the force required to lift it up vertically.) This is why a brewer will load his wagon by rolling casks of beer up a ramp, and it is pretty obvious that the steeper the ramp, the harder it is to roll the casks. The closer the ramp becomes to vertical, the closer the force required to roll (lift) the cask up becomes to its actual weight. But what *exactly* is the force required to roll a cask up a given inclined plane? This is the question that Stevinus set out to answer.

His first important move was to ask the question in a clear way. He realized that he must simplify the situation so that only the relevant physical quantities come into play, so he decided to ignore friction (something all of us are now used to doing from high school physics!). Deciding what is relevant and what isn't is, in fact, half the job. This is how he put it: given the setup of inclined plane and pulley in Fig. 1, what weight would Y have to be, to balance X and keep it from sliding up or down the plane? Keep in mind that movement of X on the plane is frictionless (or you can imagine that the weight X has little wheels that allow it to roll on the plane), as is the pulley.

Stevinus's next move was to realize that the vertical drop is just a special (extreme) case of another inclined plane, so he was able to generalize his question to this one: given the new setup in Fig. 2, once again, what weight does Y have to be, to balance X and keep it from sliding up or down the plane? (This time, imagine Y as also having little wheels, so it can slide up or down its own inclined plane without friction.)

Stevinus realized that the shape of the weights X and Y is irrelevant, and in an extraordinary leap of imagination, he replaced them with just a uniform rope (or chain). This situation is shown in Fig. 3. Assuming that the rope can still slide without friction on the inclined planes, it is clear that if the downward force is greater on the segment AB, then the rope will slide down that plane. If the downward force is greater on the segment BC, then the rope will slide down that side, and if the forces are in equilibrium, then the rope will stay balanced as it is. So which is it?

Again, in a brilliant move, Stevinus imagines the two ends of the rope connected by an additional loose length of rope. So now we have a closed loop of rope draped over the inclined planes. The situation now looks as shown in Fig. 4. We can see that although the situation is asymmetrical above the line segment AC, it is the same on both sides (the A side and the C side) below it, where the rope simply hangs in a symmetrical U-ish shape (called a catenary, and while we are on the subject see also this). Whatever forces the rope below AC exerts on the part of the rope above AC, must therefore be the same at A and at C. (The part of the rope below pulls equally on both sides.) So now the startling conclusion: if the part of the rope above AC, on the inclined planes, were to tend to slide down to one side, this would result in perpetual motion in that direction! (Because as it slides down a little bit, an equal part of the rope which had been hanging below would go up the inclined plane on the other side, and the situation would be identical to what we started with, therefore more of the rope would slide down, and it would just keep going like that forever.) We will have constructed a pertpetual motion machine. Since this cannot be right, Stevinus concluded that the parts of the rope above AC on the inclined planes must also be in equilibrium. Since X and Y are in equilibrium, *and* we also know that the weight of the rope is proportional to its length, this means that at equilibrium, the ratio of the weights X/Y justs equals the ratio of the lengths AB/BC. This finally answers our initial question from Fig. 2: *What weight does Y have to be, to balance X and keep it from sliding up or down the plane? *Simple algebraic manipulation shows that since X/Y = AB/BC,

Y = X * BC / AB -- and this is the Law of Inclined Planes

And there you have it! The weight needed at Y will always be as much less than X, as the length of the side it is resting on is less than the length of the other side. If BC is only half the length that AB is, then only half the weight of X will be needed at Y to balance it. And this conclusion holds no matter what the actual inclinations are, because we have (or Stevinus has) derived this result generally, without specifying any particular angles of inclination. In other words, the law will hold even for the vertical case of Fig. 1. (If you still don't get it, you could try the explanation here.) I find this a very beautiful result, especially as it relies on extraordinary imagination guided by good intuition at each step. In addition, the proof exploits considerations of symmetry, which were to become of paramount importance in 20th century physics, through the connection of symmetries with conservation laws.

For all his work with inclined planes, even Galileo's reputation as an experimenter is probably exaggerated. For example, it is unlikely that Galileo bothered to drop objects of different weights from the Tower of Pisa to show that they fall at the same rate. He was too smart to have needed to do this, and had his own thought experiment to show that objects of different weights must fall at the same rate: imagine that you have two objects, say iron balls, one of which weighs 20 pounds and the other 5 pounds. Now, it was thought that the 20 pound ball falls faster (say at some rate F) than the 5 pound one (which falls at a slower rate S). Imagine connecting the two balls with a chain, then dropping them. What will happen? Well, presumably the 20 pound ball should pull the lighter object into a faster rate than S, while the lighter ball should slow down the 20 pound ball from its fast rate of F. In other words, joined together, the balls should drop at some intermediate rate between S and F. But now consider that the two balls joined by a chain can also be construed as *one* object with a weight of 25 pounds, which should fall even faster than the heavier ball alone, or faster than F! Here we have a contradiction, so they must fall at the same rate. Such is the beauty of the thought experiment!

Stevinus is even supposed to have proven that objects of different weights fall at the same rate before Galileo did. He did work in hydrostatics, noting that the pressure exerted by a liquid depends only on its height and is independent of the shape of the vessel containing it. He also invented a sail-powered carriage which could outrun horse-drawn vehicles of the time, shown here in the picture. He was quite a guy.

Thanks to Margit Oberrauch for doing all the inclined plane illustrations.

Have a good week!

Abbas, I am a terrible math student but even I enjoyed this post. I might even have understood some of it, although I still wouldn't pass the quiz. Maybe you should reconsider your profession and become a teacher.

Posted by: J. M. Tyree | Monday, May 09, 2005 at 12:57 PM

Great description of Stevinus' thought-experiment, it's a very elegant one I hadn't seen before. As for Galileo's thought-experiment, it always seemed to me it doesn't quite justify his conclusion--you could apply exactly the same logic to conclude that all objects must fall through a fluid at the same rate, but since the buoyancy force upwards is only proportional to volume, not mass, two objects of the same shape and volume but different masses can fall through a fluid at different rates.

Posted by: Jesse M. | Tuesday, May 10, 2005 at 03:39 AM

Dear Jesse,

Yes, you are right about the buoyancy and friction effects of fluids, which is why the Aristotelian notion that heavier objects fall faster seems so reasonable. (A feather, an air-filled baloon, and an anvil, dropped together from the Tower of Pisa, certainly won't hit the ground together!)

It was part of Galileo's genius that he was able to ignore such effects, and his results apply perfectly to objects falling in a vacuum.

Posted by: Abbas Raza | Tuesday, May 10, 2005 at 04:49 PM

His results are correct of course, but what I'm saying is that his thought-experiment on its own did not really make a watertight case that they were, before the experiments had been done it was logically possible that the force of gravity would involve something analogous to the buoyancy term, even in the absence of any physical fluid.

Posted by: Jesse M. | Tuesday, May 10, 2005 at 09:51 PM

Once again, Jesse, you are right. In fact, nothing in science is EVER watertight. It is, for that matter,

stilllogically possible, as you say, that Galileo is wrong. In fact there is some reason to think he actually might be, and experiments continue to test this. (See this, for example.)Let me try to explain once more what Galileo was saying: if heavier objects fall faster than lighter ones, then we can imagine two identical objects connected by a thick branch, so it's really just one object. Now make the connecting "branch" flimsier and flimsier so it eventually becomes a microscopic thread, then disappers altogether... When does it cease to be one object, and suddenly start falling much more slowly (because its weight has suddenly been halved)? Why would there be this sudden discontinuity in its rate of fall? It just seems a little unintuitive. But if it doesn't make sense to you, I respect that. I myself find it quite a compellingly intuitive argument.

And actually, Jesse, when you say that "you could apply exactly the same logic to conclude that all objects must fall through a fluid at the same rate" you are not correct. Here's why: imagine two objects of different densities. If they are the same volume and shape, the more dense object will fall faster. If you connect them with a very thin branch, then they will fall at an intermediate rate. But if our assumption is that heavier objects fall faster, then the whole thing, taken as one object should fall even faster than the denser object alone (as it is heavier than either of the two objects). Why doesn't it? This is what Galileo pointed out. You might be confusing the Aristotelian idea that heavier objects fall faster with the idea that denser objects fall faster.

Posted by: Abbas Raza | Tuesday, May 10, 2005 at 11:31 PM

And actually, Jesse, when you say that "you could apply exactly the same logic to conclude that all objects must fall through a fluid at the same rate" you are not correct. Here's why: imagine two objects of different densities. If they are the same volume and shape, the more dense object will fall faster. If you connect them with a very thin branch, then they will fall at an intermediate rate. But if our assumption is that heavier objects fall faster, then the whole thing, taken as one object should fall even faster than the denser object alone (as it is heavier than either of the two objects). Why doesn't it? This is what Galileo pointed out. You might be confusing the Aristotelian idea that heavier objects fall faster with the idea that denser objects fall faster.OK, I see what you're saying. It's true that Galileo's thought-experiment shows that it wouldn't make sense for the rate of falling to be proportional to mass alone, as Aristotle imagined; the fact that objects fall through a fluid at a different rate doesn't contradict this conclusion, because as you say, the rate that an object falls through a fluid is

notproportional to mass alone. I had been thinking of Galileo's thought-experiment a bit differently--I imagined he was trying to show that objects must fall at the same rate, period, regardless of the details of what properties the rate of falling was dependent on. This conclusion would be wrong, because if the rate of falling was dependent on both mass and volume as it is in a fluid, objects would not fall at the same rate, and this would not lead to any self-contradictions when you connect two objects to each other. But if he was just trying to make the more limited argument that however gravity works, it can't be the way Aristotle imagined, and then concluding that the simplest alternative would just be for everything to fall at the same rate, then the thought-experiment makes more sense to me.Posted by: Jesse M. | Wednesday, May 11, 2005 at 08:29 PM

The set of books "A Biographical Encyclopedia of Scientists" by John Daintith, Sarah Mitchell and Elizabeth Tootill, Facts on File, N.Y., 1981, contains entries for both Galileo Galilei (1564-1642) and Simon Stevin, aka Simon Stevinus (1548-1620). Galileo was from Italy while Stevinus was from Netherlands. Since Stevinus predates Galileo by about 16 years, it would be interesting to know if Galileo had been aware of any of his work.

On page 296, the entry for Galileo states: "...While in Pisa cathedral he noticed that the lamps swinging in the wind took the same time for their swing whatever its amplitude. He timed the swing against his pulse..."

Galileo Galilei was, of course, a master intellectual, a master experimenter and a master physicist. He was nearly burned at the stake for advocating the Copernican view, that the Earth revolves around a fixed Sun, the heliocentric system of planetary motion, which the Catholic Church opposed.

The late Pope Paul II must be acknowledged as the prime mover in providing a long overdue apology to the world's greatest physicist after over 350 years, publicly admitting that Galileo Galilei, on whose shoulders so many other great physicists have stood, was right and the Catholic Church was wrong after all.

Posted by: Winfield J. Abbe | Thursday, May 19, 2005 at 08:29 PM

Since you are the expert on this, I am trying to assist my son in an experiment involving four balls of different sizes and weights. His experiment involoved timing each ball in order to determine how fast it hit the ground (note: the experiement was conducted four times for each ball in order for an average rate to be determined). However, his results show the heaviest ball hit the ground faster than the lightest ball. Are these the results he would expect to see or have we not correctly analzed the data?

Please help!!!!

Posted by: Carissa Caswell | Monday, October 31, 2005 at 01:07 PM

Quite clear and easy to understand

but did Galileo really drop balls from the Tower of Pisa?I have mo idea.Can u help me?Thanks a lot

Posted by: Jack | Saturday, January 20, 2007 at 01:19 AM

I was reading through the 1st Volume from Feynman's lecture, and he briefly mentioned Stevinus' experiment. Here you explained it very well! Thank you. And what a great story on how it came into your hands.

(Seems like it's taken awhile for me to comment on this post).

Posted by: Harold | Saturday, December 15, 2007 at 08:21 PM

I got to this post late too. Muse more about this kinda thing, OK?

Posted by: Elatia Harris | Saturday, December 15, 2007 at 09:00 PM

See how Apollo 15 astronaut Dave Scott drops a hammer and a feather on the moon to demonstrate what Galileo predicted:

http://video.google.com/videoplay?docid=6926891572259784994

qed

Posted by: aguy109 | Sunday, December 16, 2007 at 02:39 AM

I think that what Stevinus probably constructed were dikes, with an I, but maybe he was more advanced than we all know.

Posted by: Chris Schoen | Sunday, December 16, 2007 at 02:41 AM

Chris, my trusty Cambridge Dictionary says "dyke" is a perfectly acceptable variant of "dike," and you can do a Google search on "dykes" to find an article on the "Polders and Dykes of the Netherlands." Maybe "Polder" means "gay" and this is a who's who of Dutch homosexuals? I didn't have time to check. :-)

Posted by: Abbas Raza | Sunday, December 16, 2007 at 02:54 AM

Great text, thank you! I cite it in my paper "State, Statistics and Quantization in Einstein's 1907 Paper 'Planck's Theory of Radiation and the Theory of Specific Heat of Solids'" (to be submitted soon).

This type of thinking has also been praised by Feynman (Six Easy Pieces, Reading (MA): Helix Books / Perseus Books, 3rd ed. 1963, Fig. 4.3).

It's that type of thinking what distinguishes Newton and Euler, too. Unfortunately, it has been abandoned as being scholastical at the end of 18th century (v. Weizsäcker, Aufbau der Physik, München: dtv, 4th ed. 2002). The price for this error is a restricted, incomplete understanding of the notion of state, ie, of one of the central (!) notions of physics. As a result, Gibbs' paradox appeared, and it became extremely difficult to found quantum theory...

Looking forward to read more such contributions!

Peter

Posted by: Peter | Friday, January 16, 2009 at 02:16 PM