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

22
Let it be required to divide 7583 by 5.
Since division is the reverse of multiplication, the process of
division is performed by reversing the operation of multiplication,
with this difference, that the process of division begins with the figure
on the left hand of the dividend and proceeds towards the right, but
the process of multiplication begins with the figure on the right of
the multiplicand and proceeds towards the left.
The dividend 7583 = 7000 + 500 + 80 + 3.
First. 7 thousands divided by 5 units give 1 thousand, the first
partial quotient, and 2 thousands, or 20 hundreds, over.
Secondly. 20 hundreds added to 5 hundreds make 25 hundreds,
and 25 hundreds divided by 5 units give 5 hundreds, the second partial
quotient.
Thirdly. 8 tens divided by 5 units give 1 ten the third partial
quotient, and 3 tens, or 30 units, over.
Fourthly. 30 units added to 3 units make 33 units, and 33 units
divided by 5 units give the fourth partial quotient 6 units, and 3
units remainder. The process is thus exhibited:5)7583
1516 - 3 remainder.
In practice the names of the orders of units are omitted, and the
process is performed naming only the figures. 7 divided by 5 gives
quotient 1 and 2 over. Secondly, 25 by 5 gives 5 and no remainder.
Thirdly, 8 by 5 gives 1 and 3 over. Fourthly, 33 by 5 gives 6, and 3
the last remainder.
9. PROB. To divide any greater number by a less number.
Let it be required to divide 1730808 by 5342.
Since it has been shewn (Art. 4) that the sum of 300 times, 20
times, and 4 times 5342 is equal to 1730808, it follows that 300 times,
20 times, and 4 times 5342 successively subtracted from 1730808 leaves
the remainder 0.
And the method whereby 300, 20, and 4 can be found from
1730808 and 5342 will be simply a reversal of the process by which
1730808 was found from 5342 and 324.
First. The divisor 5342 is contained 300 times in the dividend
1730808.1 When 300 times 5342 is subtracted from the dividend, the
remainder is 128238.
1 Some difficulty may be experienced in the first attempts to find the exact
number of times the divisor is contained in the successive partial dividends. If it
be borne in mind that the product of the divisor by any quotient figure, when subtracted from any dividend, must always leave a remainder less than the divisor, the
difficulty may soon disappear.
By using trial divisors (as in the case above, 5000 and 6000, one less and the
other greater than the given divisor), the limits can be found within which the
correct quotient figure lies. The best and most ready mode of acquiring facility in
the process of division, is by finding the product of two numbers, and then reversing
the operation.