*Updated slightly to add even more cat goodness!*

The more I research, the more it becomes clear that cats caused all sorts of mischief in the scientific community in the late 1800s! The source of this mischief is the feline ability to turn themselves over in freefall and land on their feet, even when released at rest with no rotational motion. As I have noted in a previous post, this ability is, at a glance, seemingly at odds with the conservation of angular momentum — though in reality it is not! In a rigid body, the angular momentum of the object is directly proportional to its rotational speed. In a flexible body such as a cat, however, different sections can rotate in different ways, producing a net overall rotation even if the cat’s total angular momentum remains zero.

The debate, and confusion, was sparked in 1894 when Étienne-Jules Marey presented a sequence of photographs to the Paris Academy showing a cat flipping over at rest. As was later reported in the New York Herald, Marey’s observations were met with hilarious incredulity at the meeting:

When M. Marey laid the results of his investigations before the Academy of Sciences, a lively discussion resulted. The difficulty was to explain how the cat could turn itself round without a fulcrum to assist it in the operation. One member declared that M. Marey had presented them with a scientific paradox in direct contradiction with the most elementary mechanical principles.

Fortunately for the dignity of the scientific community, researchers quickly realized that Marey was correct: non-rigid bodies can flip over, even starting from rest, while conserving angular momentum. This led to a century-long investigation into how, exactly, a cat achieves this feat (you can read about the history in another blog post of mine).

Other researchers, however, found immediate inspiration in the cat’s newly-appreciated ability and its implications for physics. Inspired by Marey’s work, mathematician Giuseppe Peano in fact argued that the cat’s flipping talent provided a lesson and a solution for a problem in the most unlikely of places: geophysics!

Giuseppe Peano (1858-1932) is not well-known to the general public, but he was a formidable voice and researcher in mathematics, publishing over 200 books and papers during his lifetime. He is perhaps best known for the so-called Peano axioms, a set of axioms describing the natural numbers that he formulated in 1889. Peano was also one of the founders, along with Georg Cantor, of mathematical set theory, a subject also discussed extensively on this blog recently [1]. One of Peano’s most intriguing results is related to Cantor’s demonstration that there are the same number of points in a one-dimensional line and a two-dimensional plane; in 1890, Peano introduced what is now called the Peano curve, an iterative process to construct a continuous line that, in a limit of infinite steps, fills the plane entirely.

This curve is quite fascinating, and I will blog more about it in a future post, but for now let’s focus on Peano’s interest [2] in cats! By the 1890s, Peano had become a very distinguished and well-known academic. In 1890, he became a full professor in the University of Turin, and in that same year he founded his own journal, *Rivista di Matematica* (“Journal of Mathematics”), to promote new and exciting results in the field. Only a year later, he started work on the *Formulario Mathematico*, an attempt to compile all of the fundamental theorems of mathematics into a single “encyclopedia.”

I mention all these achievements to emphasize that Peano was extremely active and ambitious, and there seemingly wasn’t any mathematical issue he wasn’t aware of or afraid of investigating. When Marey’s observations, and the controversy surrounding them, came to his attention, Peano devised his own explanation, that he published in the January 1895 issue of *Rivista di Matematica* [3]:

But the explanation of the cat’s motion appears to me quite simple. When the animal is left to itself, it describes with its tail a circle in the plane perpendicular to the axis of its body. Consequently, by the principle of the conservation of angular momentum, the rest of its body must rotate in the sense opposite to the tail. When it has rotated as much as it wishes, it halts its tail and with this simultaneously stops its rotary motion, saving in this way itself and the principle of angular momentum.

In short: Peano suggests that a cat flips itself over by a rotation of its tail! By rotating its tail over multiple circles in, say, a counter-clockwise manner, the cat’s torso should rotate in the clockwise direction, albeit at a slower rate, keeping the total angular momentum at zero. Bringing back my cylindrical cat from an earlier post, we have something like below.

This model is physically possible, but incorrect! As we have discussed in my previous cat post, the bulk of a cat’s rotation comes from a torso twist, not a tail rotation. Also, a cat would need to rotate its tail in many complete circles in order to turn its much heavier body completely over, and this is not borne out by the photographic evidence. Finally, it is known that even bob tail cats cat turn over without a problem! This isn’t to say that cats don’t ever use their tails to aid in rotation, but that it is not the most important, or even essential, part of their flip.

The cat’s ability, however, served to inspire Peano in tackling a significant and hot topic in geophysics at the time, related to the overall motion of the Earth itself!

By Peano’s time, astronomers had already long known that the direction of the Earth’s axis of rotation is not fixed. Analogous to the motion of a spinning top or gyroscope, the axis traces out a circular path, known as a precession, with a period of 26,000 years, and wobbles slightly about that circle, or undergoes a nutation, with a period of 18.6 years. This precession and nutation are driven by the interaction of the Earth with the gravitational forces of the Sun and Moon. (A nice video of gyroscope precession and nutation can be seen on YouTube.)

However, another form of nutation was predicted in 1765 by the great mathematician Leonhard Euler, who suggested that the spheroidal (slightly non-spherical) shape of the Earth allows for a free nutation: an additional small wobble of the Earth’s axis with respect to the solid Earth that is self-contained and not induced by external forces. With some serious mathematical gymnastics, Euler predicted that this free nutation should have a period of 305 days.

The amplitude of this small wobble — which is equivalent to a variation in lattitude — is very small, and it was not until 1891 that astronomer Seth Carlo Chandler presented the first definitive measurements of what became known as the Chandler wobble. Chandler observed a variation of axial position of about 30 feet (9 meters) with a period of 433 days.

The discrepancy between Chandler’s measurements and Euler’s prediction was initially somewhat baffling: in fact, numerous astronomers had previously tried and failed to find the Euler nutation simply because they were too focused on Euler’s estimate. In a February 1891 meeting of the Royal Astronomical Society [4], a simple explanation of the difference was given:

Astronomers had hesitated to accept the 427-day period, even in face of the very strong evidence of the 1860-1880 observations, owing to the difficulty in accounting for it theoretically. It had been pointed out by Euler that, treating the Earth as a rigid body, the period of rotation of the pole must be 306 days. Professor Newcomb, however, happily pointed out that a qualified rigidity (either actual viscosity or the composite character due to the ocean) afforded an explanation of this longer period; and after this suggestion Mr. Chandler’s 427-day period was well and even warmly received.

In short: Euler had assumed the Earth to be a perfectly rigid body; however, the flow of material in the planet’s interior, as well as the motion of the crust and the oceans, result in significant differences from Euler’s model [5].

It is one thing to explain an unexpected phenomenon, and another thing entirely to provide a rigorous theory to back up the explanation. When Peano encountered the problem of the falling cat in 1894, he immediately saw in it a kindred spirit to the nutating Earth, and began working on mathematics to explain the latter. Both problems involve an object changing its orientation in space entirely in the absence of external forces, and both problems can be qualitatively explained by internal motions of the object in question.

There is something terribly ironic about Peano’s inspiration: where physicists are normally known for oversimplifying problems — there is the famous joke about approximating a cow by a sphere — Peano went in the other direction, envisioning a sphere as a cat!

In an 1895 paper titled “Sopra lo spostamento del polo sulla terra” (“Concerning the pole shift of the Earth”), Peano presented his own mathematical theory of the phenomenon, giving due acknowledgement to the cat:

At the end of last year at the Academy of Sciences in Paris it was proven by experiment that certain animals, such as cats, can, as they fall, through internal actions, change their orientation. It is immediately explained by mechanics the possibility of this motion. In a short article published in the

Rivista di Matematica(in the beginning of January 1895), I discussed briefly the question, I tried to describe the cyclic motions by which the cat actually rightens, and added other examples.This naturally brings us to the question: Can the globe change its orientation in space, through internal actions alone like every other living being? From the mechanical aspect the question is the same. But Prof. Volterra has the merit of proposing it first. He made it the object of some notes presented in this Academy, and the first of which is on the 1st of February 3.

“Prof. Volterra” referred to in the passage quoted is Vita Volterra (1860-1940), another Italian mathematician and physicist of significant influence. Superficially, it probably seems quite decent of Peano to have acknowledged Volterra’s work and its primacy in the scientific literature. In reality, however, Volterra took Peano’s entire presentation as a huge scientific slap in the face. First of all: Volterra and Peano were both working together at the University of Turin at the time, and would literally see each other every day, but Peano never mentioned his own work on the Earth’s wobble, though he well knew that Volterra was working on it as well. This snub would have seemed at best uncollegial, and at worst outright deceptive. Second: reading between the lines of Peano’s description above, it strongly implies that Volterra’s own work was inspired by Peano’s cat physics discussions! Note that Peano very clearly emphasizes that his cat article appeared in *January*, and that Volterra’s first article on the Earth appeared in *February*. Volterra had in fact been working on the problem for a significant period of time, and it so happened that the publication only appeared in February.

Reading accounts of the May 1895 meeting of Turin’s Royal Academy of Sciences [6], one can almost picture the smoke that must have been pouring from Volterra’s ears as he listened to Peano speak. In fact, as soon as Peano finished, Volterra leapt to his feet and argued not only that his work had come first but that Peano’s paper was based on flawed incomplete data. Volterra then asked for permission to go home and get an additional paper to present to the Academy to back up his claims; he was given this permission.

What followed was a year-long, very public feud between the two scientist-mathematicians. At the heart of it, there was more than arguments of authorship: the two men had very different, even conflicting, approaches to research, and the wobble of the Earth became the perfect battleground for their views. Peano was very much a pure mathematician, attempting to develop cutting-edge mathematical formalisms for solving problems. In the case of the wobble, he championed the use of “geometric calculus” and argued its superiority over the classical method of solution by Volterra, who was more of a physicist and was employing conventional calculus.

Over the course of the next year, the two mathematical giants would fire shots at each other through various papers, using subtle and personal jabs at each other as well as criticisms of each other’s work. In a paper on 23 June, for instance, Peano fails to acknowledge Volterra’s work on the Earth’s wobble at all! Later, Peano implicitly responded to Volterra with a paper of exactly the same title as Volterra’s. Volterra was much more direct, and criticized Peano in print of “forgetting” to cite his work, and condemning him outright for misrepresentations. Peano, in turn, continued to mention the damn cat in his papers, continually hinting that Volterra had drawn inspiration from it!

Finally, in early 1896, Volterra seems to have tired of the dispute. He wrote a final letter to the President of the Accademia dei Lincei (“Italian Academy of Science”), arguing yet again that Peano had no standing in their quarrel [6]:

Having thus shown to be empty and unfounded any of the points of criticism made of me by Peano, and that his assertions are neither original nor exact, he himself having recognized them as such, for my part I hold this polemic definitively closed.

As I have already noted, both mathematicians would, over the course of their careers, make far more significant contributions to mathematics and physics than the wobble, so I consider their duel a “draw.”

But what is the source of the Chandler wobble? Though it was appreciated that the Earth can wobble, it was less clear what keeps the wobble going or, in fact, occasionally changes the nature of it dramatically. It was only in the year 2000, in fact, that a paper appeared in Geophysical Research Letters that used computer simulations to suggest that 2/3 of the wobble comes from fluctuating pressure on the ocean bottom, and another 1/3 comes from atmospheric fluctuations.

It is remarkable, though, that some of the earliest discussion of this wobble was sparked by a falling cat — and that the cat led to an angry argument among scientists!

*Postscript: I almost wish I could say that this is the last cat physics blog post, but there still may be yet one more coming down the line in the near future!*

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[1] My four-part series on the weirdness of infinity in set theory: part 1, part 2, part 3, part 4.

[2] This discussion is based partly on the book by Hubert C. Kennedy, *Peano: Life and Works of Giuseppe Peano* (D. Reidel, Holland, 1980), and partly on Peano’s original papers.

[3] Here I use Kennedy’s translation of the paper, which is much better than my Google translate version.

[4] Monthly Notices of the Royal Astronomical Society, Vol. LIII (1893), p. 295.

[5] An aside: astronomer Simon Newcomb was also an early author of science fiction, as I have discussed on this blog before.

[6] The feud is discussed in Judith R. Goodstein, *The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician 1860-1940* (American Mathematical Society, 2007), Chapter 9.