Gelosia mutliplication is the method for multiplying that was
popular in India in the early days when our number notation was first
used. This was quite a long time ago, before the fall of Rome, and they
didn't have cheap paper, so they did it using small blackboards with
white removable paint or a board covered with sand or flour. Here is
how you would multiply 26 times 83 using the gelosia method.
You make a rectangle with a grating in it and the numbers on the
edges like this.
Make sure to write the numbers starting at the bottom left corner and
then just follow around clockwise.
Then you multiply all possible combinations and put the answers into
the square like in a times table or a mileage chart and with the tens
place to the left and the ones place to the right of the diagonal line
like this.
Now add along the diagonals like this.
Then you can read the answer from the bottom around counter clockwise
as 2158. The little ones are the carried numbers for the adding. The
interesting advantage to this method over the method you probably
learned in school is that you only have to carry when doing the
addition at the end, but the slight disadvantage seems to be that it is
hard to find a place to put the numbers that you do have to carry. It
is possible to do that small amount of carrying in your head, but I
find it safer to write it down.
Here's another example of multiplication by the gelosia method, this
time with larger numbers. Here is how to multiply 543 by 7695.
In the first row we have 3 times 7, 21, then 3 times 6, 18, then 3
times 9, 27, and then 3 times 5, 15. In the second row it's 4 times 7,
28, 4 times 6, 24, 4 times 9, 36, and 4 times 5, 20. And in the last
row it's 5 times 7, 35, 5 times 6, 30, 5 times 9, 45, and 5 times 5,
25. Then adding along the diagonal first there is a lonely 5. Then
7+1+0=8. Then 8+2+6+2+5=23, write down the 3, carry the 2. Then
2+1+1+4+3+5+2=18, write down the 8 carry the 1. Then 1+2+5+3=11, write
down the 1 carry the 1. Then 1+3=4. Then reading from the bottom
around, the answer is 4,178,385.