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September 23rd, 2018, 11:44 PM   #1
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Rank of quotient in division with monastic abacus

Dear all,
this is my first post on this forum. I am not sure whether it is in the right place. Aanyway, 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*’. 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
*(D)* 7 8 2 5


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*
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.

Last edited by skipjack; September 24th, 2018 at 03:00 AM.
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September 25th, 2018, 02:22 PM   #2
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Pope Sylvester II was one of the most learned men of his time. Well versed in mathematics, astronomy, and mechanics; he is credited with inventing the hydraulic organ, pendulum clock, and introducing Arabic numerals to Western Europe. He also wrote books on mathematics, natural science, music, theology and philosophy. Pope Sylvester II was the first French Pope and certainly the most significant in the 10th century. Upon his death, rumors began to fly that his great intelligence – and, consequently his inventive genius – ****was the result of a pact with the devil.****
This was most likely due to his regular contact with great scientific minds in the Arab world and his brave attempts to root out simony in the Church
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abacus, division, division or remainder, monastic, quotient, quotient rule, rank

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