Fourth illustration. 1093757 X 33456 
Having found in the middle column of 
the abbreviated Tables 109 3750 
„ complete „ 3757 
1093757 
we thereupon write down the Products obtained 
from the relative columns, in the following manner: 
1,093,757 X 20000 = 21875140000 
3000 = 3281371000 
400 = 487502800 
50 = 54687850 
6 = 6562542 
1,093,757 X 23456 = 25655164192 
Of the separate Products the three first figures 
(Millions, Hundred, and Ten-thousands) are taken 
from the abbreviated Tables, the four last, however, 
(Thousands, Hundreds, Tens, and Units) from the 
complete Tables. 
Dividing the Product by the Multiplicand 
1093757 : 25655164192|23456 
2187514 
” 3780024" 
3281271 
4987531 
4375028 
6125039 
5468785 
6562542 
6562542 
we receive as Quotient the number of the Mul 
tiplicator. 
From this illustration it appears that the same 
Products which in the Multiplication were to be 
cast up, are in the Division to be subtracted; a 
rule which has been set forth already in the Ex 
planation contained in the complete Tables. The 
whole process proves that these present Tables 
are a continuation of the former, and because being 
based on them can be used only along with 
them. 
As to the shortened procedure applied to the 
figures of the middle column which represent 
in Multiplication — the Multiplicand, 
in Divisions — the Divisor, 
the following remarks must be made. 
At the beginning of the Table the number 
10000 is immediately followed by 11112, omitting 
the numbers from 10001, 10002, etc. up to 11,111. 
The omission of these numbers is based on the 
fact, that the ( figures representing the Ten- 
thousands and upwards in the Products resulting 
from multiplying the numbers from 10000 to 11111 
by 2, 3, 4, 5, 6, 7, 8, 9 — remain unchanged, 
an alteration taking place only in the figures 
representing the Thousands, Hundreds, Tens, and 
Units. 
For instance: 10000X 9 = 90000 
11111 X 9 = 99999 
11112 X 9 = 100008 
From this instance it appears that the Pro 
ducts change their figures in the Ten - thousands 
for the first time by the Multiplicand 11112, and 
this being the case the number has been taken 
into the abbreviated Tables. 
Looking further on in the Tables the number 
11113 will be seen followed by 11250. 
Some examples for illustration: 
11112 X 9 = 100,008 
11250 X 9 = 101,250 
11112 X 8 = 88,896 
11249 X 8 = 89,992 
11250 X 8 = 90,000 
All numbers, from 11112 to 11250, being 
multiplied by 9, do not change in their Products 
the figure of the Ten-thousands. But in Multi 
plying them by 8 an alteration in the Ten- 
thousands occurs by the number 11250, for which 
reason the number has been taken into the Tables. 
It further appears that the Ten-thousands in the 
Products change, 
in multiplication by 7 — from the Multiplicand 11429 upwards 
„ „ „ G— „ „ „ 11667 „ 
„ „ „ 5— „ „ „ 12000 „ 
„ „ „ 9— „ „ „ 12223 „ 
„ „ „ 4— „ „ „ 12500 „ 
etc. etc. 
Consequently such numbers of the Multiplicand 
(in Divisions — of the Divisor) only are printed 
in the abbreviated Tables, as suffer an alteration 
in the Ten-thousands of their Products. Such 
numbers as do not undergo this change in the 
Ten-thousands, but only in the Thousands, Hundreds, 
Tens and Units, are omitted here, and must be 
looked for in the complete Tables from 1 to 9999, 
which are intended for that purpose.