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.