At the always excellent Ptak Science Books blog John Ptak has a nice post about Renaissance multiplication 9876 X 6789 = 67048164 in which he was unable to work out the algorithm used to carry out the multiplication. In the following I show the example from John’s blog as it was displayed followed by the correct form with working method.

Multiplication the Diamond

9876

6789

81

484948

4242

54565654

7272

636463

36

67048164

The Diamond done correctly:

9876

6789

36

4242

484948

54565654

636463

7272

81

67048164

From left to right lower row L1,L2,L3,L4

From left to right upper row U1,U2,U3,U4

Method:

Row 1: L1 X U4

Row 2: L1 X U3, L2 X U4

Row 3: L1 X U2, L2 X U3, L3 X U4

Row 4: L1 X U1, L2 X U2, L3 X U3, L4 X U4

Row 5: L2 X U1, L3 X U2, L4 X U3

Row 6: L3 X U1, L4 X U2

Row 7: L4 X U1

Then sum vertically from right to left

Write 4

8+5+3=16 write 6 carry 1

1+2+4+6+6+2=21 write 1 carry 2

2+6+4+9+5+4+7+1=38 write 8 carry 3

3+3+2+4+6+6+2+8=34 write 4 carry 3

3+4+8+5+3+7=30 write 0 carry 3

3+4+4+6=17 write 7 carry 1

1+5=6 write 6

Added 11/11: Also motivated by John Ptak, Ray Girvan at Journal of a Southern Book Reader has a really good poston the subject of these Renaissance multiplication algorithms, definitely recommended reading.

This was often called the Galley method (It looked like a ship?). The method is pretty simple but when it’s done it’s hard to see the “tracks” because they don’t keep products on the same line. I think I told you once I had an old German student Copy book with a drawing illustrating the “Galley” idea for the divisioin. I tried putting it on twitpic, hope this works.. http://local.twitpicproxy.com/web17/img/444860593-a7b229ba9a3724e4c237b666ec71ac22.4ebc5068-scaled.png

World’s largest URL, sorry

The Galley is a division algorithm although similar the Diamond is a multiplication algorithm.

Neat. I also answered John, but your reference to the Diamond led me to a source: Ivor Grattan-Guinness’s “Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Volume 1″ ( http://books.google.co.uk/books?id=2hDvzITtfdAC&pg=PA205#v=onepage&q&f=false ). It’s not that Taliente is using a garbled version of the Diamond, but that several forms existed: “per campana”, “per coppa”, the diamond, and the circle. Despite being framed by a diamond, the example at John’s blog is the correct form for the circular-framed algorithm.