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* "Christianity as a Mystical Fact" GA8

The book Christianity as Mystical Fact  (text available from RSArchive.org) is the written elaboration of a series of 24 lectures given by Rudolf Steiner from October 1901 to April 1902 at the Berlin Theosophical Library at the invitation of Cay Lorenz, Count von Brockdorff, and Sophie, Countess von Brockdorff. This led to Rudolf Steiner being invited to be the leader of the German Section of the Theosophical Society, a post he held for 10 years, before the Anthroposophical Society was founded.  

For those who have Amazon Kindle, the text is also available from their library.

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Tania Diedericks Graduation Performance

7pm Sunday 20th September
The Steiner Centre, 401 Whitehead Road.

All are welcome to Tania's graduation in Creative Speech. Come and enjoy a rich evening of poetry, story and music: old ballads; a charming Irish folk tale; Milton's 'Paradise Lost'; through to modern poets exploring themes of inner struggle to enlightenment. Music will enhance the evening.
Enjoy a rich tapestry of the English language brought to life by Tania.
Koha is invited.

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Anthroposophic Meditation Week
Connecting to the Angelic World and to Christ
New Zealand

December 14 to 19, 2026, Mon-Sat
in the Āio Wīra Retreat Centre
8 Aio Wira Road, Waitākere,
www.aiowira.org.nz , 20 km from Auckland
Course leaders: Agnes Hardorp, Thomas Mayer
Times: Arrival day 6 pm - 9.30 pm,
seminar day 9 am - approx. 9.00 pm,
departure day 9 am - 2 pm
Course fee: 850 NZD, reduction possible
Accommodation: Between 350 and 650 NZD.
Meals: 325 NZD

To register: meditation333@protonmail.com
www. anthroposophical-meditation.info  

See POSTER

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Arithmetic Musings 18 – NL36

When I started these musings I pointed out that numbers are pure ideas - they are not physical but of a spiritual nature, thus Arithmetic is a form of spiritual activity.  It has been a key part of education since Pythagoras.

Multiplication as 2 Dimensional pattern

Clock arithmetic

• The digits 0 to 9 are arranged into a clock face.
  0-9
• The final digit of each times table forms a pattern.
• 1, 2, 3, 4, 5, 6, 7, 8, 9, 0
• 2, 4, 6, 8, 0, 2, 4, 6, 8, 0,
• 3, 6, 9, 2, 5, 8, 1, 4, 7, 0
• 4, 8, 2, 6, 0, 4, 8, 2, 6, 0
• 5, 0, 5, 0, 5, 0, 5, 0, 5, 0
• 6, 2, 8, 4, 0, 6, 2, 8, 4, 0
• 7, 4, 1, 8, 5, 2, 9, 6, 3, 0
• 8, 6, 4, 2, 0, 8, 6, 4, 2, 0
• 9, 8, 7, 6, 5, 4,3, 2, 1, 0
•  E.g. for the 7x table:
  0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 or 07418529630 (the reverse of 3x table)

In Table or Lattice format, this pattern looks like this:

Note how the UNIT digit appears in the Lower Right sector of each small square, while the TENS digit appears in the Upper Left sector.

 John Napier (1550–1617) was the 8th Laird of Merchiston (near Edinburgh) and the brilliant Scottish polymath best known for inventing logarithms.

Of local relevance, the city of Napier in Hawkes Bay is named after Sir Charles James Napier (1782–1853)—famous for his 1843 conquest of Sind in India—was a descendent of John Napier.

John Napier used strips of the lattice above to multiply very large numbers.

To see how this works, let us start with a 2-digit number, 87, multiplied by a single digit number, 8:

 87 x 8 = 640 +56 = 6 / (4+5) / 6 = 6/9/6 = 696

 Now a 4-digit number, 4793, multiplied again by a single digit number, 7:

4,793 x 7 = 28,000 + 4,900 + 630 + 21

                   =  2 / (8+4) / (9+6) / (3+2) / 1

                   = 3/3/5/5/1

                   = 33,551

 

Now we will tackle multiplying a 4-digit number by a 3-digit number by stages, using colour to clarify.

2,459 x 367 = 902,453

Firstly create a lattice form with one number along the top and the second down the right side.

Now fill in the central block, using the product of the number above times on the number right, with the ten’s digit in the upper left half of the square.

Now to get the tens and digit part of the answer.  Add the numbers in the diagonal slope.

 3 = 3

4+6+5 = 15  carry 1 to the next slope.

 

Now the next two slopes for the hundreds and thousands.

For the hundreds and thousands. 7+5+0+3+8+1=24; 2+5+3+4+2+4+2=22

Completing the process.

 

How does this compare with the way you were taught to write out long-multiplication?  Did you understand how it worked?  Or just how to mechanically manipulate the numbers?

With Lattice Multiplication, everything is visible, (but still compact!), not hidden or carried mentally, and so fully understandable, I believe.  Once you understand it, the process can be condensed or speeded up by carrying parts mentally, that is by not writing it down.

The standard algorithm usually taught for long multiplication uses vertical columns rather than slopes, but all the digits written in the upper left sector of each square in the Lattice method is carried and added mentally, and is thus invisible.

 

RB