In 1960 Jane Goodall watched a chimpanzee repeatedly poking pieces of grass into a termite mound in order to “fish” for insects, the first observation of tool use among animals. When she notified anthropologist Louis Leakey of her discovery, he responded with a telegram:

NOW WE MUST REDEFINE TOOL, REDEFINE MAN, OR ACCEPT CHIMPANZEES AS HUMAN.

A magic square by Lee Sallows. The 16 pieces progress in area from 1 to 16, and those in each row, column, and long diagonal can be assembled to form the same target shape with area 34.

Every road in this little town is a one-way street, and each street is colored either red or blue. This has a helpful effect: If you start at any house in town and follow the sequence blue-red-red three times in a row, you’ll always arrive at the yellow house.

If you follow blue-blue-red three times, you’ll always arrive at the green one.

In 1970 Roy Adler and Benjamin Weiss asked whether it’s always possible to create such a coloring in a given network; in 2009 Avraham Trahtman proved that, within certain constraints, it is.

The sum of the squares of the reciprocals of the positive integers is π²/6.

The sum of their fourth powers is π⁴/90.

The sum of their sixth powers is π⁶/945.

The area of the region under the Gaussian curve y = e^–x2 is the square root of π.

The probability that two integers chosen at random will have no prime factor in common is 6/π².

The integer 8 can be written as the sum of two squares of integers, m² + n², in four ways, when (m, n) is (2, 2), (2, -2), (-2, 2), or (-2, -2). The integer 7 can’t be written at all as the sum of such squares. Over a very large collection of integers from 1 to n, the average number of ways an integer can be written as the sum of two squares approaches π. Why?

James Watt perfects the steam engine, 1765:

I had gone to take a walk on a fine Sunday afternoon. I had entered the Green and had passed the old washing house. I was thinking up on the engine at the time and had got as far as the herd’s house, when the idea came into my mind that as steam was an elastic body it would rush into a vacuum, and that if a communication were made between the cylinder and an exhausted vessel it would rush into it and might there be condensed without cooling the cylinder. I had not walked farther than the golf house when the whole thing was arranged clearly in my mind.

Charles Darwin realizes why species diverge, 1840s:

I can remember the very spot in the road, whilst in my carriage, when to my joy the solution occurred to me; and this was long after I had come to Down. The solution, as I believe, is that the modified offspring of all dominant and increasing forms tend to become adapted to many and highly diversified places in the economy of nature.

Henri Poincaré discovers the relation between automorphic functions and non-Euclidean geometries, 1881:

Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure.

Walter Cannon recognizes the fight-or-flight response, 1911:

As a matter of routine I have long trusted unconscious processes to serve me. … [One] example I may cite was the interpretation of the significance of bodily changes which occur in great emotional excitement, such as fear and rage. These changes — the more rapid pulse, the deeper breathing, the increase in sugar in the blood, the secretion from the adrenal glands — were very diverse and seemed unrelated. Then, one wakeful night, after a considerable collection of these changes had been disclosed, the idea flashed through my mind that they could be nicely integrated if conceived as bodily preparations for supreme effort in flight or in fighting.

William Rowan Hamilton conceives the fundamental formula for quaternions, 1843:

But on the 16th day of the same month — which happened to be a Monday, and a Council day of the Royal Irish Academy — I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery.

Hamilton adds: “Nor could I resist the impulse — unphilosophical as it may have been — to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,

i² = j² = k² = ijk = -1

which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.” The bridge now bears a permanent plaque marking Hamilton’s achievement (below), and mathematicians undertake an annual walk from Dunsink Observatory to commemorate it.

In the Minnesota Multiphasic Personality Inventory, the subject is asked whether he agrees with a series of statements, such as “There seems to be a lump in my throat much of the time,” “I am not afraid to handle money,” “I feel uneasy indoors,” and “My sleep is fitful and disturbed.” His responses give insights into his personality and psychopathology.

In 2006, poet Katie Degentesh fed these statements into Google and combined the results into a series of poems, which she published as The Anger Scale. Here’s an excerpt from “As a youngster I was suspended from school one or more times for cutting up”:

Everyone knows about Dallas
and its acts of terrifying gorgeousness

a chef in a tall hat piping meringue
discussing the “brain drain”

dropped a slab of concrete on his left foot
before being lured to the guitar

doesn’t recall details of cutting up friend
to create fake masterpiece

Poets Craig Dworkin and Kenneth Goldsmith call this “a ‘pataphysical nosography, evaluating and diagnosing the mental stability of the Internet itself.” But how do we evaluate the results?

Pick any three points on a line and use each pair of them to define a semicircle, as shown.

Now draw a perpendicular between the two smaller semicircles.

Circles c₁ and c₂ will always have the same area.