In Memory of Bacon

Not the fatty breakfast-food, although you may remember that if you like. The great Sir Francis Bacon died 384 years ago today. And in memory of Bacon, here is your assignment:

THIS PASSAGE IS A FRANCIS BACON CIPHER WHERIN THE BOLD LETTERS REPRESENT B AND THE PLAIN LETTERS REPRESENT A

Cheers!

1907 Crisis in Mathematical Physics According to Poincaré

With 20-20 hindsight, we all agree that Einstein's discoveries of 1905 revolutionized nearly every area of fundamental physics. But what did scientists think at the time? One telling source is Poincaré's 1907 account of the "new crisis" in physics (available here, on the newly released Popular Science archive). Poincaré identifies five fundamental principles he thought were in danger of being overturned:

1. Carnot's principle of heat transfer. Brownian motion was thought to violate Carnot's principle of heat transfer, since it apparently involved an unlimited source of motion. Poincaré wrote, "to see the world return backward, we no longer have need of the infinitely keen eye of Maxwell's demon; our microscope suffices."
2. The principle of relativity. Although Einstein had recently defended this principle, Poincaré wasn't convinced, and in particular worried about the prohibition on superluminal signaling. Anticipating a coming revolution in gravity, he wrote: "are such signals inconceivable, if we admit with Laplace that universal gravitation is transmitted a million times more rapidly than light?"
3. Newton's third law (of action-reaction). Electrodynamics seemed to be suggesting that not every action corresponds to an equal and opposite reaction. In particular, the action of one electric charge on another doesn't necessarily give rise to a simultaneous reaction.
4. Lavoisier's principle of fixed mass. Alluding to Einstein, Poincaré wrote that electrodynamics suggests a body's mass might increase with velocity, refuting principle of fixed mass: "And now certain persons think that it seems true to us only because in mechanics merely moderate velocities are considered."
5. Mayer's principle of energy conservation. Finally, the recent discovery of radiation by the Curies suggested to Laplace Poincaré that radium might be a limitless source of energy, and hence that energy is not locally conserved.

What I find striking about this list is Poincaré's recognition of the deep and difficult consequences of taking classical electrodynamics seriously -- and in particular, of retaining the principle of relativity. Of course, only 3 and 4 were actually overturned, and a version of 4 may still be salvageable (by replacing "mass" with "rest mass"). And it's somewhat surprising that as late as 1907, Poincaré isn't mentioning Einstein by name.

But then, I suppose it's never clear what the revolution will bring until well after it's over.

Galilean Freefall Confirmed

It wasn't centuries of successful theories of mechanics, planetary dynamics, and interplanetary travel that confirmed Galilean freefall. Oh no. It was this.

Galileo's Manuscripts: Still Learning the Lesson

When I studied history of science, my teachers -- especially Ken Manders and Paolo Palmieri -- hammered two principles like drill sergeants:

1. Always read the original manuscripts;
2. Always pay attention to the diagrams.

Okay, that seems obvious. We've known that since the mid-20th century. Yes, these lessons are easy to under-estimate -- until you realize how many historians still haven't learned them.

Here's an example. Galileo's personal notebooks on mechanics have survived the last four centuries. I recently took interest in one famous page among them, in which Galileo did very early work on acceleration in freefall. It's distinguished title is Folio 152r of Manuscript 72, written sometime between 1604 and 1610. The Max Planck institute has done us the extraordinary service of putting the whole manuscript online.

I first spent a few days assembling my own interpretation of this manuscript page. Then I took a look at how other people interpreted it. Surprisingly, two features of the manuscript that I thought were significant seem to have been completely ignored by scholars. And no wonder -- it turns out that the only available English reproductions of the document omit a signficant portion of the diagram.

Compare the following. Here's Drake's (1973) English reproduction -- click to enlarge it:

Damerow et al. (1992) later spruced things up with a nicer type-face:

Finally, here's the original. Can you see what's missing?

Two important things. First: the little upside-down triangular diagram in the middle. (The circle visible near it is actually printed on the back of the page.) Both triangular diagrams represent distance along the vertical, and velocity along the horizontal. So, if this is small triangle is a scaled-down copy of the larger one, then the scaling rule will actually indicate which rule of freefall Galileo was considering. (It turns out to be the incorrect "double-distance" rule, which says that velocity in freefall is proportional to the square of the distance fallen.)

Second: the languages used. While the top-left, top-middle, and bottom-most paragraphs are all written in Italian, the rest of the text is written in Latin. Galileo wrote comfortably in both, but his train of thought probably only followed one language at time. That suggests a connection between these three paragraphs, which might not have otherwise been apparent, given their physical separation on the page.

So, here's the way a translation of 152r should look:

Ah, yes. Isn't that better with the diagram restored? Over a hundred years reading Galileo's manuscripts, and we're still learning this lesson!

Did All Calorists Believe in Caloric?

There is a lot of literature about how it is that important successes in science apparently stemmed from false belief. Belief in the existence of caloric is a common example: it seems to have led Laplace (1816) to the discovery of the correct speed of sound equation.

However, it seems that many calorists of the time were very cautious about the purported existence of caloric. Here are a few passages worth thinking about.

[i]n our ignorance of the nature of heat, we are left to carefully observe its effects, which principally consist in the dilation of bodies, the rendering of fluids, and the conversion into vapor (Lavoisier and Laplace 1783, 153-154).

Lavoisier and Laplace then go on to suggest a way of translating between caloric and dynamical theories of heat:

• Free caloric :: Force vive


• Combining of caloric :: Loss of force vive


• Disengaging of heat :: Augmentation of force vive

(Lavoisier and Laplace, 154). Lavoisier cautions a few years later:

“we are not even obligated to suppose that caloric is a real substance; it is sufficient... that it be any kind of repulsive cause that separates the molecules of matter, allowing us to imagine its effects in an abstract and mathematical way” (Lavoisier, 1789, 19).

It seems that we must at least tread with great care before inferring what role a particular ontology had in any given scientific discovery.

------------
Laplace. 1816. Sur la vitesse du son dans l'air et dans l'eau. Annales de Chimie et de Physique, 3:328-343.

Lavoisier & Laplace. 1783. Memoire sur la chaleur. Memoires de la Academie des Sciences, 355-408.

Lavoisier. 1789. Traite elementaire de chimie. In Oeuvres de Laplace, premiere tome (1864).

Group Theory Of Your Spine

In more examples of the unreasonable effectiveness of mathematics, there now exists a robust application of group theory to the human spinal cord.

Modeling the Human Spine. A few years ago, Australian DOD scientist Dr. Vladimir Ivancevic unveiled a new model of the human spine. The model, called the Human Biodynamics Engine, was designed to study how various stresses (like those on a soldier carrying a load) can result in spinal injury. However, while older models were built up from the physics of compression, bending and shear, Ivancevic built his model using group theory. He published his result last week on the arXiv.

The 'Spinal Cord' Group. The central idea is to describe the degrees of freedom of each vertebrae using a group. The group that does the trick turns out to be the Special Euclidean Group SE(3). SE(3) is equivalent to the direct product group SO(3) ⊗ R³, which is just the group of rotations around a fixed point:


together with the group of spatial translations:


(both in three dimensions).

Makes sense, right? Those are the degrees of freedom of a piece of your spine. From there, it's easy to model the spinal cord as one big direct product group, made up of as many copies of SE(3) as there are vertebrae.



Of course, to make this idea useful, Ivancevic must argue that most spinal injuries are due to what he calls an 'SE(3)-jolt.' This is, effectively, a certain kind of local perturbation of the forces acting on the 'spinal cord' group. However, if the data continues to support Ivancevic's hypothesis, then he may have discovered a very useful new way to think about these injuries.

History: Applications of Groups. Mathematical groups began their history in the early 19th century, as a play-thing for pure mathematicians. Abel and Galois famously used them to study the roots of polynomials (especially the quintic).

Since then, applications of group theory have turned up in unexpected places. Group theory became inseparable from the study of both quantum theory and relativity in the mid-20th century, leading Wigner to famously call mathematics in general "unreasonably effective." (I recommend Octavio Bueno's very informative analysis of this history.)

In the last two decades, applications of group theory have surfaced in biology and in medicine as well. For example, groups have been applied to population genetics, epidemiology, neurobiology, and even anesthesiology. However, few of these applications have been very robust. Most deal only with combinatorial facts about the symmetric group S_n, the group of permutations of n objects.

Could You Have Defended Galileo?

The Problem. When Galileo died in 1642, there were still two competing schools of free fall. The Galilean School upheld Galileo's law of free fall, which may be posed either of two forms:

1. d ∝ t², or


2. v ∝ t,

for any freely falling body. The first claim is the famous time-squared law. The second claim says that velocity in free fall is proportional to time fallen. It's easy to prove that these two claims are equivalent.

On the other hand, the Jesuit School upheld that

3. v ∝ d.

That is, they believed that velocity in free fall is proportional to distance fallen, which was the traditional view at the time. I call this the 'Jesuit law' of free fall.

There was no agreed upon experimental evidence in the 1640's that could verify one law and falsify the other. (Each side claimed to have experiments that vindicated their law, and disproved their competitor's law.) However, a clever theoretician might still try to use purely theoretical means to prove one side false.

The Challenge. Prove that the Jesuit law is false, without assuming Galileo's law.

Some Remarks Before You Get Going. These aren't really hints, just a few comments about what you should and shouldn't take for granted.

• You may assume that for uniformly moving objects, d = vt. This law is as old as Aristotle, and was agreed upon by both schools of free fall.


• You may assume any mathematics known to Euclid or Archimedes.


• The modern calculus will lead you astray. (It's too easy to accidently assume something that implies Galileo's law when you start using these tools. If you assume Galileo's law, you haven't answered the challenge. And anyway, nobody knew what a derivative was in the 1640's, so that's kind of cheating.)


• I'll sketch a solution to this challenge next week.

A Hint. (Warning! Stop reading if you don't want any hints!) Remember, you go into this knowing almost nothing about the way bodies fall. But you can still try to put upper and lower bounds on motion in free fall, in order to get your result. How can the law of uniform velocity provide bounds on the time it takes for a body to fall?

On Monday, I'll elaborate on that last hint. (It's actually an interesting problem all by itself.) Later, I'll sketch a little bit of the first answer to this challenge, which was given by Pierre de Fermat in an obscure letter to Gassendi, around 1646.

Good luck!