## Rank of quotient in division with monastic abacus

Dear all,
this is my first post on this forum. I am not sure if it is the right place, anyway my question is this.
I am reading the very interesting book by Karl Manninger Number words and number symbols. In it, the author introduces the monastic abacus developed by Pope Sylvester II (Gerbert). To illustrate the use of the abacus, Manninger makes the division 7825/43 with the method of the divisio aurea. The process itself is clear; the problem is to understand where to place the quotient.

To solve this problem, Menninger introduced the concept of ranks: each digit of a number is a rank, so 1, 10 and 100 are of rank 1, 2 and 3 respectively. He also gives the rules derived from the Roman abacus to calculate the ranks during multiplication: given the multiplication

z =
xy

the rank of z (Rz) is

Rz = Rx + Ry  1

Conversely, in the division z = x/y, we have

Rx = Rz  Ry + 1

How to apply these rules? Menninger is not quite clear in this respect.

In the first step of the division we have:

rank 4 3 2 1
divisor (d) 4 3
dividend (D) 7 8 2 5

then:

quotient = 78/43 = 1 plus remainder. Where to place 1?

Rx = Rz  Ry + 1 hence:
R1 = RD  Rd + 1 = 4  2 + 1 = 3

which is correct since the whole quotient is 181 (rank 3). but if I do 7825/86, for instance, I get:

782/8 = 9 with remainder
RD  Rd + 1 = 4  2 + 1 = 3

but the answer is 90 with a remainder, thus 9 should have a rank of 2, not 3. Thus this process is not suitable for all numbers. Maybe I need to take into account that in the last computation I used 782 instead of 78, but how?

Also, when multiplying the quotient by the dividend, what would be the rank of the results? For instance, multiplying 1 × 4 and 1 × 3, where should the result be placed? If 43 is of rank 2 and the quotient is of rank 2 because I calculated it before, then I get for the first digit:

Rx + Ry  1 = Rq + RD  1 = 2 + 2  1 = 3

and for the second:

Rq + RD  1 = 1 + 1  1 = 1

but the values should go on rank 4 and 3, respectively, because 43 needs to be subtracted from 78 before moving the divisor one place rightwards.

Could you please help with this?

Thank you.