### Adam Adamandy Kochański

October 4, 2021 I was an early adopter of desktop computing, and I had an S-100 bus computer running CP/M on my desktop long before the introduction of the first IBM Personal Computer. Before that, I made good use of an HP-33C Scientific Programmable Calculator for converting xray diffraction data to unit cell dimensions of my epitaxial crystals. This calculator cost about a hundred dollars when I bought it in 1980. This is $325 in today's money, but an equivalent calculator can now be purchased for less than ten dollars. The integrated circuits and liquid crystal displays of modern calculators operate with much lower power, and they might derive their power from photovoltaic cells which were too inefficient and too expensive in 1980.*I've retained my HP-33C Scientific Programmable Calculator, no longer working, as a keepsake.This calculator had an display with an eight digit mantissa and two digit exponent formed from red light-emitting diodes.Nobel Physics Laureate, Kenneth G.Wilson (1936-2013), about whom I wrote in an earlier article (Kenneth G. Wilson, June 21, 2013), remarked about his purchase of an HP pocket calculator."I buy this thing and I can't take my eyes off it, and I have to figure out something that I can actually do that would somehow enable me to have fun with this calculator."[1](Photo by author.)*

This calculator used reverse Polish notation as a method for data entry. Instead of keying

**1 + 1**to get

**2**, you would key

**1 1 +**. While such stack-oriented calculation made things much easier to implement in circuitry, I'm certain that the engineers had a hard time convincing the marketing people that customers would adapt. In my case, adaptation was easy, since I had been programming in assembly language and Forth. In the early days of computing, Forth was a good language for laboratory automation. Polish notation is named after Polish mathematician, Jan Łukasiewicz (1878-1956). Poland has produced many important scientists and mathematicians. The prime example among the scientists is Nicolaus Copernicus (1473-1543), who developed the heliocentric model of the Solar System. Also among the early Polish scientists was astronomer, Johannes Hevelius (1611-1687), the "founder of lunar topography," known for his detailed maps of the lunar surface that he published in his

*Selenographia, sive Lunae descriptio (Selenography, or a Description of the Moon)*. Hevelius also discovered four comets and proposed that comets move in a parabolic path while near the Sun. The list of Polish mathematicians is voluminous, but I'll mention a few that I recognize through their work.

Adam Adamandy Kochański (1631-1700). Kochański discovered a close approximation of the mathematical constant, pi, that's now called Kochański's approximation. I'll present this later in this article. Mark Kac (1914-1984). Kac collaborated with Paul Erdős (1913-1996) on the Erdős-Kac theorem. This is an extension of the Hardy-Ramanujan theorem, which is known also as the fundamental theorem of probabilistic number theory. I'll just mention that it involves a term, log(log(n)), and that brings to mind a number theory joke that a drowning number theorist's last words are not, "glug, glug," but "log, log." Marian Rejewski (1905-1980). Rejewski and colleagues deciphered the German military Enigma cipher machine based on limited documentation from French military intelligence. Over the course of seven years, they continued their decryption, just one step behind modifications to the equipment and the encryption procedures. Jerzy Różycki (1909-1942). Różycki was a cryptologist who worked with Rejewski on the Enigma cipher. Wacław Sierpiński (1882-1969).Polish mathematician, Adam Adamandy Kochański, was also a physicist, a clockmaker, a Jesuit, and the Chaplain and Court Mathematician of John III Sobieski, the King of Poland. Alas, the United States has a Surgeon General, but no Court Mathematician. His best known work is his 1685

Sierpiński is known for the Sierpiński triangle (a.k.a., the Sierpiński gasket).

I wrote about this is an earlier article (Patterns from Randomness, May 18, 2017).

(Wikimedia Commons image. Click for larger image.)

Alfred Tarski (1901-1983). Tarski and Stefan Banach (1892-1945) created the Banach-Tarski paradox that states that a solid sphere in three-dimensional space can be decomposed and reconstructed into two identical copies of the original sphere. This is as strange as anything in quantum mechanics.

It's all clear now - Humpty Dumpty was just trying to clone himself! This is an artistic representation of the Banach-Tarski paradox in which a solid sphere can be broken apart and reconstructed into two identical copies of itself.

(Wikimedia Commons image by Benjamin D. Esham.)

Stanisław Ulam (1909-1984). In the late 1940s, Ulam invented the Monte Carlo method that I've used often. His inspiration came from his playing the solitaire card game while recovering from surgery. He realized that computers could be used to play many "games" to statistically estimate the outcome of physical processes. Józef Hoëné-Wroński (1776-1853). The Wrońskian is a determinant that's used to show linear independence in a set of solutions.

*Observationes Cyclometricae ad facilitandam Praxin accommodatae (Cyclometric Observations to Facilitate the Process of Estimation)*, which considers the ancient problem of squaring the circle. This was published in Acta Eruditorum, the leading scientific journal of his time. Squaring the circle will give a value for pi, and his method leads to a very close value called Kochański's approximation; viz,[2]

√((40/3)-2(√3)) = 3.14153333870509461863...Which is accurate to four decimal places (pi is 3.1415926535...). While pi is known today to 50 trillion decimal places (5 x 10

^{13}), this was a rare achievement in his time.

*Kochański's geometric construction to get a close approximation of the mathematical constant, pi. In the above figure, let the circle radius be one. The circular arc drawn with radius 1 at C intersects the circle at point (-√(3)/2,1/2). At that point, we draw another circular arc of radius 1, and the two arcs intersect at (-√(3)/2,-1/2). We construct the line through that intersection and the circle origin, and this intersects the perpendicular line drawn at C to produce the point at (-√(3)/3,0). Going three radii along this line gives point B at (3-√(3)/3,0), and the length of AB is √(2 ^{2}+(3-(1/3)√(3))^{2})=√((40/3)-2√(3)).[2] (Highly modified version of a Wikimedia Commons image. Click for larger image.)*

### References:

- Martin Weil, "Kenneth Wilson, Nobel winner who explained nature's sudden shifts, dies in Maine at 77," Bangor Daily News, June 19, 2013.
- Eric W. Weisstein, "Kochanski's Approximation." From MathWorld--A Wolfram Web Resource.

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