The historian Socrates tells us that the Emperor Tiberius, who was much given to astrology, used to put the masters of that art, whom he thought of consulting, to a severe test. He took them to the top of his house, and if he saw any reason to suspect their skill, threw them down the steep. Thither he took Thrasyllus, and after a long consultation with him, the emperor suddenly asked whether the astrologer had examined his own fate, and what was portended for him in the immediate future. Now the difficulty is this: If Thrasyllus says that nothing important is about to befall him, he will prove his lack of skill and lose his life besides. If, on the other hand, he says that he is soon to die, either the emperor will set him free, in which case the prophecy was false and he ought to have destroyed him; or Tiberius will destroy him, while he ought to have spared him as a true revealer of the future. Of course the solution is easy. Thrasyllus, after some observations and calculations, began to quake and tremble greatly, and said some great calamity seemed to be impending over him, whereupon the emperor embraced him and made him his chief astrologer.

— The Ladies’ Repository, July 1873

In his 1916 book The Science of Musical Sounds, Dayton Clarence Miller uses harmonic analysis to convert the line of a woman’s profile (left) into an equation of 18 terms. Then he uses this equation to reproduce the profile synthetically (right). “If mentality, beauty, and other characteristics can be considered as represented in a profile portrait,” he writes, “then it may be said that they are also expressed in the equation of the profile.”

He repeats the synthesized profile to produce a waveform:

“In this sense beauty of form may be likened to beauty of tone color, that is, to the beauty of a certain harmonious blending of sounds.”

In Noise, Water Meat: A History of Sound in the Arts, Douglas Kahn writes, “The simple beauty of the female expressed in the line thus becomes also the simple beauty of mathematics, graphic representation, and instrumentation, let alone mediation and reproduction, involved in the production of the equation and profile. Thus, we move beyond Lord Kelvin’s fascination with a beauty of mathematics to a fascination with a mathematics of beauty.”

We ask for the probability that a number, integer or fractional, commensurable or incommensurable, randomly chosen between 0 and 100, is greater than 50. The answer seems evident: the number of favourable cases is half the number of possible cases. The probability is 1/2.

Instead of the number, however, we can choose its square. If the number is between 50 and 100, its square will be between 2,500 and 10,000.

The probability that a randomly chosen number between 0 and 10,000 is greater than 2,500 seems evident: the number of favourable cases is three quarters of the number of possible cases. The probability is 3/4.

The two problems are identical. Why are the two answers different?

— Joseph Bertrand, Calcul des probabilités, 1889 (translation by Sorin Bangu)

A triangular number is one that counts the number of objects in an equilateral triangle, as above:

1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21

Some of these numbers are palindromes, numbers that read the same backward and forward. A few examples are 55, 66, 171, 595, and 666. In 1973, Charles Trigg found that of the triangular numbers less than 151340, 27 are palindromes.

But interestingly, every string of 1s:

1
11
111
1111
11111
…

… is a palindromic triangular number in base nine. For example:

11₉ = 9 + 1 = 10
111₉ = 9² + 9 + 1 = 91
1111₉ = 9³ + 9² + 9 + 1 = 820
11111₉ = 9⁴ + 9³ + 9² + 9 + 1 = 7381

The pattern continues — all these numbers are triangular.

02/12/2017 UPDATE: Reader Jacob Bandes-Storch sent a visual proof:

“Given a number n in base 9, if we tack a 1 on the right, the resulting number is 9*n + 1. (By shifting over one place to the left, each digit becomes nine times its original value, and then we add 1 in the ones place.) So given a triangular number, there’s probably a way of sticking together 9 copies of it with a single additional unit to form a new triangle. Sure enough:”

Philosopher and logician Raymond Smullyan passed away on Monday. He was 97.

From my notes, here’s a paradox he offered at a Copenhagen self-reference conference in 2002:

Have you heard of the LAA computing company? Do you know what LAA stands for? It stands for ‘lacking an acronym.’

Actually, the above acronym is not paradoxical; it is simply false. I thought of the following variant which is paradoxical — it is the LACA company. Here LACA stands for ‘lacking a correct acronym.’ Assuming that the company has no other acronym, that acronym is easily seen to be true if and only if it is false.

In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.

If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:

That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.

Bonus: Displace the rows still further and they’ll identify prime numbers.

Dentist William T.G. Morton was the first to use sulfuric ether as an anaesthetic, but he’d learned about this property at the chemistry lectures of Charles T. Jackson.

Which of them deserved a monument? Oliver Wendell Holmes suggested setting up statues of both men on the same pedestal, with the inscription: