Carved into the brickwork of a cylindrical tower at Cambridge University’s New Museums Site is a great crocodile. It was commissioned by Pyotr Kapitza, who had moved to Cambridge from Russia expressly to work with Ernest Rutherford, the father of nuclear physics. Kapitza called his mentor “crocodile,” a title that Russians traditionally confer on great men (and also, Kapitza said, because Rutherford’s thunderous voice announced his approach, just as the crocodile in Peter Pan was announced by the ticking watch in its belly).

Eric Gill carved the animal into the side of the Mond Laboratory, which was erected in 1933 with Rutherford’s backing to support Kapitza’s work in low-temperature physics. Unfortunately, after a holiday in Russia the following year, Kapitza was barred from leaving the country, and he never returned to Cambridge.

A few quotations by Rutherford:

  • “Don’t let me catch anyone talking about the Universe in my department.”
  • “An alleged scientific discovery has no merit unless it can be explained to a barmaid.”
  • “We’re like children who always want to take apart watches to see how they work.”
  • “We’ve got no money, so we’ve got to think.”
  • “When we have found how the nucleus of atoms is built up we shall have found the greatest secret of all — except life.”

Paul Langevin and Rutherford served together as research assistants at Cavendish Laboratory. Asked afterward whether they were friendly, Langevin said, “One can hardly speak of being friendly with a force of nature.”

Early Work

Image: Wikimedia Commons

Amazingly, the notion of a black hole was first posited in 1783, by the English natural philosopher John Michell. In a paper read before the Royal Society that November, he wrote:

Let us now suppose the particles of light to be attracted in the same manner as all other bodies with which we are acquainted; that is, by forces bearing the same proportion to their vis inertiae (or mass), of which there can be no reasonable doubt, gravitation being, as far as we know, or having any reason to believe, an universal law of nature. … [I]f the semi-diameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity.

“From these quotations it is clear that Michell in 1783 understood many of the basic principles of black hole physics which are in daily use almost 200 years later,” writes Cambridge physicist Gary Gibbons. Indeed, Michell’s talent doomed him to obscurity: His breakthroughs were lost on his contemporaries and forgotten by the time the world could appreciate them. His notion of a “dark star” was rediscovered only in the 1970s. The American Physical Society says, “[H]e remains virtually unknown today, in part because he did little to develop and promote his own path-breaking ideas.”

(Gary Gibbons, “The Man Who Invented Black Holes,” New Scientist, June 28, 1979.) (Thanks, Alejandro.)

A Stretch


Biologist and mathematician D’Arcy Thompson advanced a strange new idea in his 1917 book On Growth and Form: He found that if you draw the outline of an animal or plant on an ordinary Cartesian grid, and then you put the grid through some mathematical transformation (stretching it, for example, so that its squares become rhombuses), very often the resulting shape is that of a related real creature.

What can that mean? Thompson doesn’t really say. He thought that the biologists of his day overemphasized evolution in explaining the form and structure of living things; he preferred to look for physical and especially mathematical laws. But he didn’t present his ideas as principles that might be tested, so his book has (so far) remained only a notable curiosity.

“This theory cries out for causal explanation, which is something the great man eschewed,” writes zoologist Wallace Arthur. “Perhaps the time is close when comparative developmental genetics will be able to provide such an explanation.”

(Wallace Arthur, “D’Arcy Thompson and the Theory of Transformations,” Nature Reviews Genetics, May 2006, 401-406.)

A New Perspective


In 1880 Charles Hinton (inventor of the baseball gun) turned his attention to the fourth dimension, that unseen world whose behavior seems so baffling to ordinary thinkers.

In his 1888 book A New Era of Thought, he announced a unique way to think about it, a set of 81 colored cubes that correspond to the 81 parts of a 3 × 3 × 3 × 3 hypercube. By creating a set of wooden cubes, painting them according to Hinton’s instructions, and working through the prescribed exercises, the reader could learn to visualize the fourth dimension as intuitively as the third:

The square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube; and the assemblage of cubes thus placed constitutes the tessaract in layers. We suppose the cube to change its colour directly it begins to move. Its colour between 1 and 2 we can easily determine by finding what colours its different parts assume, as they move in the unknown direction.

Hinton’s method drew few adherents, but he was sure that it worked — he had proved it for himself. “The particular problem,” he wrote, “at which I have worked for more than ten years, has been completely solved. It is possible for the mind to acquire a conception of higher space as adequate as that of our three-dimensional space, and to use it in the same manner.”

He moved on to other things, but he’s left us one permanent calling card — Hinton coined the word tesseract.

Flying Ash

Images: Wikimedia Commons

Here’s a striking sign of the pervasive influence of the Industrial Revolution: It darkened England’s moths. Before 1811, the peppered moth, Biston betularia, had a white body. But as soot darkened trees, lighter-bodied insects became more visible to birds and other predators. By 1848 the frequency of dark-bodied moths in industrial regions had increased dramatically, one of the first documented instances of Darwin’s principle of natural selection. American geneticist Sewall Wright called it “the clearest case in which a conspicuous evolutionary process has actually been observed.”

Somewhat related: A curious wartime observation by Gertrude Stein, in Alsace, from The Autobiography of Alice B. Toklas:

Another thing that interested us enormously was how different the camouflage of the french looked from the camouflage of the germans, and then once we came across some very neat camouflage and it was american. The idea was the same but as after all it was different nationalities who did it the difference was inevitable. The colour schemes were different, the designs were different, the way of placing them was different, it made plain the whole theory of art and its inevitability.


In 1837 English journalist Albany Fonblanque wrote, “Sir Robert Peel was a smooth round peg, in a sharp-cornered square hole, and Lord Lyndenurst is a rectangular square-cut peg, in a smooth round hole.”

Which of these is the better fit? In other words, which is larger, the ratio of the area of a circle to a circumscribed square, or the area of a square to a circumscribed circle?

In two dimensions, these ratios work out to π/4 and 2/π, respectively, so a round peg fits better into a square hole than a square peg into a round hole.

But, strangely, Berkeley mathematician David Singmaster discovered in 1964 that this is true only in dimensions less than 9. For n ≥ 9 the n-dimensional unit cube fits more closely into the n-dimensional unit sphere than vice versa.

There’s a moral in there, but I don’t know what it is.

(David Singmaster, “On Round Pegs in Square Holes and Square Pegs in Round Holes,” Mathematics Magazine 37:5 [November 1964], 335-337.)

The Floating Water Bridge

When a high voltage difference is applied, a liquid bridge of deionized water can sustain itself between two beakers even when they’re separated by 25 millimeters.

British engineer William Armstrong first reported this in an 1893 lecture. The phenomenon is known to be founded in surface polarization, but it’s still not completely understood.

Just So

Postscript of a letter from Benjamin Franklin to the Abbé André Morellet, July 1779:

P.S. To confirm still more your piety and gratitude to Divine Providence, reflect upon the situation which it has given to the elbow. You see in animals, who are intended to drink the waters that flow upon the earth, that if they have long legs, they have also a long neck, so that they can get at their drink without kneeling down. But man, who was destined to drink wine, is framed in a manner that he may raise the glass to his mouth. If the elbow had been placed nearer the hand, the part in advance would have been too short to bring the glass up to the mouth; and if it had been nearer the shoulder, that part would have been so long that when it attempted to carry the wine to the mouth it would have overshot the mark, and gone beyond the head; thus, either way, we should have been in the case of Tantalus. But from the actual situation of the elbow, we are enabled to drink at our ease, the glass going directly to the mouth.

“Let us, then, with glass in hand, adore this benevolent wisdom; — let us adore and drink!”



This just caught my eye in an old issue of the Mathematical Gazette, a note from P.G. Wood. Suppose we’re designing a cylinder that’s closed at both ends and must encompass a given volume. What relative dimensions should we give it in order to minimize its surface area?

A young student thought, well, if we slice the cylinder with a plane that passes through its axis, the plane’s intersection with the cylinder will form a rectangle. And if we spin that rectangle, it’ll sweep out the surface area of the cylinder. So really we’re just asking: Among all rectangles of the same area, which has the smallest perimeter? A square. So the cylinder’s height should equal its diameter.

It turns out that’s right, but the student had overlooked something. The fact that the volume of the cylinder is fixed doesn’t imply that the area of the rectangle is fixed. We don’t know that.

Wood wrote, “We seem to have arrived at the right answer by rather dubious means.”

(P.G. Wood, “73.5 Interesting Coincidences?”, Mathematical Gazette 73:463 [1989], 33-33.)


Advice in problem solving:

“You must always invert.” — Carl Gustav Jacob Jacobi

“Whenever you can, count.” — Francis Galton

“Each problem that I solved became a rule, which served afterwards to solve other problems.” — Descartes

“By studying the masters, not their pupils.” — Niels Henrik Abel

“Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.” — Isaac Newton