The Stephenson Abacus™, Math History Conjectures.

by Steve Stephenson
Revised April 24, 2008 (copyright)

Base-60 numbers, 360 degrees in a circle, 7 days in a week; all from Sumer around 2500 B.C.?

From The Economist, Dec 20, 2001,

The Sumerians were great innovators in matters of time. It is to them, ultimately, that we owe not only the week but also the 60-minute hour. Such things came easily to people who based their maths not on a decimal system but on a sexagesimal one.

Why were these clever chaps, who went for 60 because it is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, fascinated by stubbornly indivisible seven? ...

The Sumerians had a better reason for their septimalism. They worshiped seven gods whom they could see in the sky. Reverently, they named the days of their week for these seven heavenly bodies.

I don't think the divisibility of 60 was the primary reason the Sumerians developed a sexagesimal number system. Divisibility was a convenient coincidental consequence.

I think the Sumerians developed a sexagesimal number system from the periods of the two slowest moving of their seven sky gods. Jupiter and Saturn take approximately 12 and 30 years, respectively, to track through the Zodiac. The observant Sumerians knew this. And the least common multiple of 12 and 30 is 60!

In 60 years Jupiter would go through 5 cycles and Saturn 2. We have 5 fingers on each of 2 hands. Playing with the numbers, in both cases 5+2=7, the number of sky gods. The gods reflecting themselves in our anatomy?

The product of 12 and 30 is 360, the number of degrees in a circle; did the Sumerians define the 360 degree circle? Dividing the Zodiac into 360 parts means Jupiter would traverse 30 degrees in a year and Saturn 12 degrees. Thereby coupling the periods of the gods Jupiter and Saturn.

The Sun tracks through the Zodiac in one year (obvious to the Sumerians). Jupiter would track 1/12 of the way in that time. Why not divide a year into 12ths, e.g., 12 months; then the Sun tracks the same distance in one month that Jupiter tracks in one year. Thereby coupling the periods of Jupiter and the Sun. And since the Sun would then track 30 degrees along the Zodiac in a month, why not divide the month into about 30 days? (12 months/year and ~30 days/month probably came a lot later than the Sumerians, though).

But why 24 hours in a day? I like this explanation:

7, 12, 30, 60, 30, 12, 5, 2, 7.  Mystical numbers to the Sumerians, yes?

Exponents from Archimedes?

Archimedes' The Sand Reckoner,,
is cited as evidence that Archimedes "invented" exponents.

In it Archimedes defines a large-number system, which in our notation becomes something like this:

Q := 108, or a myriad myriad, or 10000 times 10000 (The Stephenson Abacus™ can hold this number in the main table).
P := QQ = 108Q, or a myriad myriad to the myriad myriad power.
Largest number defined: P ^ Q = Q ^ Q ^ Q = 108 ^ 108 ^ 108 !

The Stephenson Abacus could represent a standard exponent of 2.9999 myriad, but an abacus with 3 tables, each with 11 lines could be used to represent Archimedes largest number if one of the additional two tables was the exponent for the first table, and the third the exponent for the second. Three tables are used for multiplication and division, so using three tables for numerical computations was probably commonplace (triple tables were certainly used in Medieval Europe).

Could it be that Archimedes used the counting board, ancient even for him, as his inspiration?!


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