By That Pure, Holy, Four Lettered Name On High | Pythagoras

The Pythagorean Oath mentions the Tetractys:
"By that pure, holy, four lettered name on high,
nature's eternal fountain and supply,
the parent of all souls that living be,
by him, with faith find oath, I swear to thee."
Credits: Michæl Paukner

Albert Mackey (1873) - "The Greek word 'Tetractys' signifies, literally, the number four, and is synonymous with the quaternion; but it has been peculiarly applied to a symbol of the Pythagoreans, which is composed of ten dots arranged in a triangular form of four rows."   

The first four numbers symbolize the harmony of the spheres and the Cosmos as: (1) Unity (Monad); (2) Dyad - Power - Limit/Unlimited; (3) Harmony (Triad); (4) Kosmos (Tetrad) - The four rows add up to ten, which was unity of a higher order (The Dekad).  The Tetractys symbolizes the four elements — fire, air, water, and earth. The Tetractys represented the organization of space: the first row represented zero dimensions (a point). The second row represented one dimension (a line of two points). The third row represented two dimensions (a plane defined by a triangle of three points). The fourth row represented three dimensions (a tetrahedron defined by four points). 

The Pythagorean musical system is based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. The diapason, 2:1 (octave), and the diapason plus diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the tetractys of the decad. The diapason plus diatessaron, 8:3 (compound fourth or perfect eleventh), is not.

Squaring the Circle with the Earth and the Moon

The perimeter of a square around the Earth equals the perimeter of a
circle drawn through the center of the Moon.
Credits: Michæl Paukner

Common Wisdom (2015) - Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.