### The Significance of 106 and 117

Can anyone help – especially a mathematician and/or a historian?

The Cologne City Museum has a copy of this 1638 document which consists of a table of multiples of all numbers up to 99 – plus 106, 117, 256, 318 and 365.

It is pretty obvious why 365 is included, as well as 256 (2 to the power of 8).

But why were 106, 117 and 318 (3 x 106) included?

Here is a close up of the bottom left hand corner of the tables:

Responses, Thoughts …

Alex Bellos kindly asked the same question on my behalf back in 2011.  There were some interesting responses but nothing that seemed to get near explaining why scholars in the 1600s might want to be able to calculate multiples of 106 and 117.

Richard Harries has pointed out that both 106 and 117 are the sums of two squares – 81+25 and 81+36 respectively.  And 81+49=130 in effect appears as 13.  Now … does that help?  Is there a link to simple geometry via Pythagoras’ sums of squares?  But 81+64=145, which doesn’t appear.

In case it helps, the panels around the multiplication tables feature  Astronomia,  Geometria Ingeniara,  Arithmetica,  Castrame Tatio – this picture shows preparations for a battle,  Mercatura,  and  Mensuration Alt et Profund – this picture shows the height of a building being measured.  There is no suggestion that the tables were for religious use.

Here are three photos of items along the top border of the tables, for those who read Latin and German.

Martin Stanley
68rtsw8@gmail.com  &  @ukcivilservant

Editor  Understanding Government