## Principle

Suppose we have to perform a large number of divisions by 36525, which could be the case if we do calendar calculations. Then we can simplify the task by creating a specialized division table for this divisor. Following what is stated in the chapter: Guide to traditional division, we will start by calculating the following three Euclidean divisions:

Creating a special division table for 36525
100000÷36525 200000÷36525 300000÷36525
Quotient Remainder Quotient Remainder Quotient Remainder
2 26950 5 17375 8 07800

Which can be summarized in the following specialized division table:

 1/36525>2+26950 2/36525>5+17375 3/36525>8+07800

And now we can use this table to do divisions without touching the multiplication table. For example, how many Julian centuries of 36525 days can fit in 1 000 000 days?

1000000/36525
Abacus Comment
ABCDEFGHIJKLM
36525 1000000 Use rule: 1/36525>2+26950 on column G
36525 2000000 change 1 in G into 2
36525 2269500 Use rule: 2/36525>5+17375 on column H
36525 2569500 change 2 in H into 5
36525 2586875 revise up
+1
-36525
36525 2650350 revise up
+1
-36525
36525 2713825 Done! 1000000÷36525=27, remainder 13825

And we have done a division by a five-digit divisor without using the multiplication table!

## Two-digit division tables

In the past, special division tables were used for divisors between 11 and 99[1].

11 12 13 14 15 16 17 18 19 1 9+01 8+04 7+09 7+02 6+10 6+04 5+15 5+10 5+05 4+16 4+12 4+08 4+04 4+00 3+22 3+19 3+16 3+13 9+11 9+02 8+16 8+08 8+00 7+18 7+11 7+04 6+26 3+07 3+04 3+01 2+32 2+30 2+28 2+26 2+24 2+22 6+14 6+08 6+02 5+30 5+25 5+20 5+15 5+10 5+05 9+21 9+12 9+03 8+28 8+20 8+12 8+04 7+34 7+27 2+18 2+16 2+14 2+12 2+10 2+08 2+06 2+04 2+02 4+36 4+32 4+28 4+24 4+20 4+16 4+12 4+08 4+04 7+13 7+06 6+42 6+36 6+30 6+24 6+18 6+12 6+06 9+31 9+22 9+13 9+04 8+40 8+32 8+24 8+16 8+08 1 1+49 1+48 1+47 1+46 1+45 1+44 1+43 1+42 1+41 3+47 3+44 3+41 3+38 3+35 3+32 3+29 3+26 3+23 5+45 5+40 5+35 5+30 5+25 5+20 5+15 5+10 5+05 7+43 7+36 7+29 7+22 7+15 7+08 7+01 6+52 6+46 9+41 9+32 9+23 9+14 9+05 8+52 8+44 8+36 8+28 1+39 1+38 1+37 1+36 1+35 1+34 1+33 1+32 1+31 3+17 3+14 3+11 3+08 3+05 3+02 2+66 2+64 2+62 4+56 4+52 4+48 4+44 4+40 4+36 4+32 4+28 4+24 6+34 6+28 6+22 6+16 6+10 6+04 5+65 5+60 5+55 8+12 8+04 7+59 7+52 7+45 7+38 7+31 7+24 7+17 9+51 9+42 9+33 9+24 9+15 9+06 8+64 8+56 8+48 1+29 1+28 1+27 1+26 1+25 1+24 1+23 1+22 1+21 2+58 2+56 2+54 2+52 2+50 2+48 2+46 2+44 2+42 4+16 4+12 4+08 4+04 4+00 3+72 3+69 3+66 3+63 5+45 5+40 5+35 5+30 5+25 5+20 5+15 5+10 5+05 7+03 6+68 6+62 6+56 6+50 6+44 6+38 6+32 6+26 8+32 8+24 8+16 8+08 8+00 7+68 7+61 7+54 7+47 9+61 9+52 9+43 9+34 9+25 9+16 9+07 8+76 8+68 1+19 1+18 1+17 1+16 1+15 1+14 1+13 1+12 1+11 2+38 2+36 2+34 2+32 2+30 2+28 2+26 2+24 2+22 3+57 3+54 3+51 3+48 3+45 3+42 3+39 3+36 3+33 4+76 4+72 4+68 4+64 4+60 4+56 4+52 4+48 4+44 6+14 6+08 6+02 5+80 5+75 5+70 5+65 5+60 5+55 7+33 7+26 7+19 7+12 7+05 6+84 6+78 6+72 6+66 8+52 8+44 8+36 8+28 8+20 8+12 8+04 7+84 7+77 9+71 9+62 9+53 9+44 9+35 9+26 9+17 9+08 8+88 1+09 1+08 1+07 1+06 1+05 1+04 1+03 1+02 1+01 2+18 2+16 2+14 2+12 2+10 2+08 2+06 2+04 2+02 3+27 3+24 3+21 3+18 3+15 3+12 3+09 3+06 3+03 4+36 4+32 4+28 4+24 4+20 4+16 4+12 4+08 4+04 5+45 5+40 5+35 5+30 5+25 5+20 5+15 5+10 5+05 6+54 6+48 6+42 6+36 6+30 6+24 6+18 6+12 6+06 7+63 7+56 7+49 7+42 7+35 7+28 7+21 7+14 7+07 8+72 8+64 8+56 8+48 8+40 8+32 8+24 8+16 8+08 9+81 9+72 9+63 9+54 9+45 9+36 9+27 9+18 9+09

## Some examples

Dividing by numbers that start with 1 is awkward, the following table may be used to divide by 19[2].

 19 1 5+05
 99 1 1+01 2 2+02 3 3+03 4 4+04 5 5+05 6 6+06 7 7+07 8 8+08 9 9+09
9801÷99
Abacus Comment
ABCDEFGHI
9801   99 Dividend AD, divisor HI
9891   99 A: Rule 9/99>9+09
9899   99 B: Rule 8/99>8+08
+1 revising up
-99
99     99 Done! No remainder, quotient: 99

Dividing by 𝝅 is common in applications, here are the tables for two approximations of this irrational number.

 314 31416 1 3+058 1 3+05752 2 6+116 2 6+11504 3 9+174 3 9+17256

Finally, the division by 666 table.

 666 1 1+334 2 3+002 3 4+336 4 6+004 5 7+338 6 9+006

However, It is not advisable to divide by this number; results can be unpredictable… and uncontrollable! In any case, remember the advice:

I say to you againe, doe not call up Any that you can not put downe; by the Which I meane, Any that can in Turne call up somewhat against you, whereby your Powerfullest Devices may not be of use.

:)👿

• Murakami, Masaaki (2020). "Specially Crafted Division Tables" (PDF). 算盤 Abacus: Mystery of the Bead. Archived from the original (PDF) on August 1, 2021. `{{cite web}}`: Unknown parameter `|accesdate=` ignored (`|access-date=` suggested) (help)