Corresponding with Leibniz about his method of infinite series in 1677, Isaac Newton wanted to advert to his “fluxional method,” the calculus, without actually revealing it. So he used an unusual expedient — after describing his methods of tangents and handling maxima and minima, he added:

The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.

That peculiar string is an inventory of the letters in the phrase that Newton wanted to conceal, Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa, which means “Given an equation involving any number of fluent quantities to find the fluxions, and vice versa.” So “6a” indicates that the Latin phrase contains six instances of the letter A, “cc” means that there are two Cs, and so on. In this way Newton could register his discovery without actually revealing it — the fact that he could present an accurate letter inventory of the fundamental theorem of the calculus proved that he’d established the theorem by that date. (More details here.)

Robert Hooke had used the same resource in 1660 to establish priority for his eponymous law before he was ready to publish it. And Galileo first published his discovery of the phases of Venus as an anagram. The technique today is known as trusted timestamping.

(Thanks, Andy.)

From Lee Sallows, a geomagic square with a staircase theme:

(Thanks, Lee!)

Swiss chemist Hans Heinrich Landolt discovered this startling reaction in 1886: Two colorless solutions are mixed and, after a distinct delay, the liquid turns dark blue due to the formation of a triiodide–starch complex. It’s called the iodine clock reaction.

See A Chemical Traffic Light.

In the 13th century, Cistercian monks worked out a system of numerals in which a single glyph can represent any integer from 1 to 9,999:

Once you’ve mastered the digits in the top row, you can represent tens by flipping them (second row), hundreds by inverting them (third row), and thousands by doing both (fourth row). And now you can combine these symbols to produce any number under 10,000:

The monks eventually dropped the system in favor of Arabic numerals, which reached northwestern Europe at about the same time, but it was being used informally elsewhere as recently as the early 20th century.

A logic exercise by Lewis Carroll: What conclusion can be drawn from these premises?

 
1. No birds, except ostriches, are 9 feet high.
2. There are no birds in this aviary that belong to anyone but me.
3. No ostrich lives on mince pies.
4. I have no birds less than 9 feet high.

From Lee Sallows:

(Thanks, Lee!)

The ancient Chinese philosopher Gongsun Long appeared to claim that a white horse is not a horse:

Is ‘a white horse is not horse’ assertible?

Advocate: It is.

Objector: How?

Advocate: ‘Horse’ is that by means of which one names the shape. ‘White’ is that by means of which one names the color. What names the color is not what names the shape. Hence, one may say ‘white horse is not horse.’

Objector: If there are white horses, one cannot say that there are no horses. If one cannot say that there are no horses, doesn’t that mean that there are horses? For there to be white horses is for there to be horses. How could it be that the white ones are not horses?

Advocate: If one wants horses, that extends to yellow or black horses. But if one wants white horses, that does not extend to yellow or black horses. Suppose that white horses were horses. Then what one wants [in the two cases] would be the same. If what one wants were the same, then ‘white’ would not differ from ‘horse.’ If what one wants does not differ, then how is it that yellow or black horses are acceptable in one case and unacceptable in the other case? It is clear that acceptable and unacceptable are mutually contrary. Hence, yellow and black horses are the same, one can respond that there are horses, but one cannot respond that there are white horses. Thus, it is evident that white horses are not horses.

Interpretations vary; one explanation is that the conundrum blurs the distinction between identity and class, exploiting an ambiguity in the Chinese language — certainly the expressions “white horse” and “horse” do not have identical meanings, but one can refer to a subset of the other.

Whether the philosopher was serious isn’t clear. His other paradoxes include “When no thing is not the pointed-out, to point out is not to point out” and “There is no 1 in 2.”

More trouble with horse color.

03/08/2024 UPDATE: A Swedish Facebook meme of 2012: Horses are a fruit that does not exist. (Thanks, Mikael.)

By Lee Sallows: If the letters BJFGSDNRMLATPHOCIYVEU are assigned to the integers -10 to 10, then:

J+A+N+U+A+R+Y     = -9+0-4+10+0-3+7     =  1
F+E+B+R+U+A+R+Y   = -8+9-10-3+10+0-3+7  =  2
M+A+R+C+H         = -2+0-3+5+3          =  3
A+P+R+I+L         = 0+2-3+6-1           =  4
M+A+Y             = -2+0+7              =  5
J+U+N+E           = -9+10-4+9           =  6
J+U+L+Y           = -9+10-1+7           =  7
A+U+G+U+S+T       = 0+10-7+10-6+1       =  8
S+E+P+T+E+M+B+E+R = -6+9+2+1+9-2-10+9-3 =  9
O+C+T+O+B+ER      = 4+5+1+4-10+9-3      = 10
N+O+V+E+M+B+E+R   = -4+4+8+9-2-10+9-3   = 11
D+E+C+E+M+B+E+R   = -5+9+5+9-2-10+9-3   = 12

Similarly, if -7 to 7 are assigned SROEMUNFIDYHTAW, then SUNDAY to SATURDAY take on ordinal values. See Alignment.

(David Morice, “Kickshaws,” Word Ways 24:2 [May 1991], 105-116.)

Hungarian mathematician József Kürschák offered this “proof without words” that a regular dodecagon inscribed in a unit circle has area 3. If the circle is inscribed in a square, the resulting figure can be tiled by triangles of two families — 16 equilateral triangles whose sides are equal to those of the dodecagon and 32 isosceles triangles with angles 15°-15°-150° and longest side 1. The area of the large square is 4, and the triangles that make up the dodecagon can be rearranged to fill 3 of its quadrants (see the video below). So the area of the dodecagon is 3/4 of 4, or 3.

(To see that the square and the dodecagon can be tiled as claimed, see Alexander Bogomolny’s discussion here.)

(Gerald L. Alexanderson and Kenneth Seydel, “Kürschak’s Tile,” Mathematical Gazette 62:421 [October 1978], 192-196.)