Elementary arithmetic, with brief notices of its history... by Robert Potts.

17
The truth of this is evident from the fact, that if three rows of 5
counters be disposed in order under each other; in whichever way
they may be counted, their sum will be same number of counters.
By counting them in one way, they constitute 3 rows of 5
c )unters in each row, or three times 5 counters.
By counting them the other way, they make 5 rows of 3 counters
in each row, or 5 times 3 counters. Hence 3 times 5 counters is the
same as 5 times 3 counters; or 3 times 5 is equal to 5 times 3.
This property is applicable for any two integral abstract numbers.
The following numbers exhibit the products of every two numbers
not exceeding 9:-l
'1 time is 1 2 times 1 is 2 3 times 1 is 3,, 2,, 2,, 2,, 4,, 2,,. 6,, 3,, 3 3,, 6,, 3,, 9,, 4,, 4 4,, 8, 4,, 12,, 5,, 5, 5,,,10,, 5,, 15,, 6,, 6,,, 12,, 6,, 18, 7,, 7,, 7,, 14,, 7,, 21,, 8,, 8,,,16,, 8,, 24,, 9,, 9,, 18,, 9,, 27
4 times 1 is 4 5 times I is 5 6 times I is 6, 2,, 8,, 2,, 10, 2,, 12, 3,,12,, 3,, 15 3,, 18,, 4,,16,, 4,, 20 4,, 24
5,, 20,, 5,, 25,, 5,, 30, 6,,24,,,,30,, 6,, 36,, 7,,28,, 7,, 35,, 7,, 42, 8,, 32,, 8,, 40,, 8,, 48,, 9, 36,, 9,, 45, 9,, 54
7 times 1 is 7 8 times 1 is 8 9 times I is 9,, 2,,14,, 2,, 16,, 2,,18,, 3,,21,, 3,, 2,, 3,, 27, 4,,28,, 4,, 32,, 4,, 36,, 5,, 35, 5,, 40,, 5,, 45,, 6,, 42,, 6 48,, 6,, 54,, 7,, 49,, 7,, 56,, 7,, 63, 8,, 56,, 8,, 64,, 8,, 72,, 9,, 63,, 9,, 72,, 9,, 81
3. PROB. To find the product of two numbers, one of which consists of
one figure only.
The process depends on the following
AxIoM. The product of any two numbers is equal to the sum of
the products obtained by multiplying one of the numbers by each of
the parts into which the other can be divided.

1 If the products of two numbers were extended to 12 times 12, or to 20 times
20, and committed to memory, additional facilities would be acquired in the processes of multiplication and division of numbers.