Newsletter 26: Friday 26 June 2026

For the four MONDAYS 6th, 13th, 20th and 27th in July there will be a
conversation on the FOUNDATION STONE at the centre.
Conversations from 7 pm
The Foundation Stone as laid at the Christmas Conference 1923-24 is
such a fundamental event in the life of Anthroposophy that it is not easy to
comprehend all its ramifications. I would like to consider one aspect:
what it cost RUDOLF STEINER to take on his shoulders, the destiny of
the Anthroposophical Society.

Christopher Bacchus

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Arithmetic Musings #13

Odds and Evens

Over the next two weeks, I would like to look at the simple idea of odds and evens that fascinated the famous Greek philosopher Pythagoras, who related them to space and time.

The name Pythagoras originates from Ancient Greek, combining Pythios (a name for the god Apollo) and agora (meaning "marketplace" or "assembly"). Together, the name translates roughly to "spoken by the Pythian [Apollo]" or "oracle of the marketplace," honouring the god associated with wisdom and prophecy.

Apollo and Dionysus are complementary half-brothers in Greek mythology.  

Apollo, god of the sun, prophecy, and music, represents structure, self-control, the conscious mind, reason, logic, and harmony,
while:

Dionysus, god of wine, ritual madness, and theatre, represents raw emotion, instinct, and the unconscious mind, embodying passion, ecstasy, and chaos.

Rather than rivals, the ancient Greeks viewed them as essential counterparts whose energies balanced one another, particularly in the arts and at the oracle of Delphi.

 

First, let us look at ODD numbers.

1                                                                      1                                         1+0

3                                                       1    +      1    +     1                          1+2

5                                                       2    +      1    +     2                          1+4

7                                                       3    +      1    +     3                          1+6

9                                                       4    +      1    +     4                          1+8

11                                                     5    +      1    +     5            and so on.

 

In the diagram below, first lay pebbles in a straight line with the unpaired one in the middle.

Next bend them with the unpaired one in the middle, at right-angles, and they form a series of squares.

A line has length, but a square has area.

One of the great geometrical insights of Pythagoras was about triangles.

Triangles, as the name indicates, have 3 angles between adjacent sides; but they have 3 points (or vertices) and 3 lines (or sides). 3x3 = 3² = 9.

Angles can be measured as a portion of a full turn (= 360° or degrees).

The 360° came from Astronomy when the year was seen as the return of the sun to the same place in the zodiac – it had completed a full circle/cycle in a year of 365 days.

365 – that is awkward when a nearby number, 360, was a very abundant number,  so in due course the calendar became 12 months of 30 days  = 360 days [another small problem: the time from one full moon to the next is only 29.5 days!!)  The 5 days at the end, before the new year could begin, were used to create a holy New Year festival.  So they came to be known as Holidays.  Later, this arrangement was changed and the holidays were distributed through the year, so the odd-numbered months January, March, May, July, etc. got an extra day, 31 in total – they needed 6 but only had 5, so 1 was taken from February so it had only 29!  Perhaps they were trying to make winter shorter?

Anyway, we still use 360° for a full turn, so 180° for a half turn, and 90° for a quarter turn, the corner of a regular square, and 60° for the corner of a regular triangle.

So what was it that Pythagoras noticed?

That the 3 internal angles of a triangled always summed to 180°, the same as only 2 angles of a square.

That is: if one angle of a triangle was a right angle (90°) then the other two angles together must have the same value of 90°.

Now the remarkable discovery was that at the same time if squares were constructed on the sides of this triangle, then the area of the square on the longest side – opposite the right-angle in the triangle – was equal to the areas of the smaller 2 squares combined.

In geometrical diagrams it is a custom to label points or vertices using capital letters A, B, C; and to label lines or sides using small letters a, b, c.  In a triangle point A is opposite line a.  We can refer to the internal angle at A as ⦞A.

 

So, the situation in a triangle as underlined above can be expressed as:

IF, and only IF, ⦞A <=> ⦞B + ⦞C = 90°, THEN a² <=> b² + c².

The most famous example (shown) involves the sequence of three Integers, 3, 4, and 5.

5² = 4² + 3².  [that is: 25 = 16 + 9]

{This relationship is true even if the numbers for the length of the sides are rational or even irrational.}

More next week.  RB