- The operation of the law of small numbers in

http://www.naturalsciences.be/expo/old_ishango/en/ishango/riddle.html

reminds me of some messages on this list.

David - On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:

> > > published by Eratosthenes some 2200 years ago,

Sorry for the late response to this thread but I've been rather tied up

> > > and was certainly known some 1000

> > > (maybe 25 000) years earlier.

> >

> > I'm intrigued by the "certainly";

> > I would have said "probably" for 1k BCE.

>

> I'd have said "definitely" for >3k BCE. Base 60 just screams

> knowledge of divisibility properties.

with Real Life(tm) recently.

There is a persuasive suggestion that the divisibility properties of

radix-60 arithmetic is a consequence of its choice, not a reason for its

choice. The argument goes as follows.

A number of cultures have independently invented quinary arithmetic, for

reasons which should be obvious. There are still relics of this in

modern culture --- the five-bar-gate tallying method, for instance.

Bi-quinary has also been widely used throughout history. This uses four

different symbols for the digits 1-4 (the symbols are frequently 1 to 4

identical lines or dots) and another symbol for 5. Digits 6 through 9

are then represented by the juxtaposition of the 5-symbol and the

appropriate symbol for 1 through 4.

A number of cultures have independently invented duodecimal arithmetic.

Many relics of this exist: 12 ounces to the Troy pound; 12 inches to the

foot; 12 pennies to the shilling and so on. The most convincing

survivors to my mind are the survival of the English words "dozen" and

"gross".

Some time around 4000 to 3500 BCE the Sumerians moved into Mesopotamia

and merged with a pre-existing culture. One culture used quinary or

bi-quinary and the other duodecimal. Neither culture supplanted the

other, rather their notations merged. Indeed, the symbols of early

Mesopotamian arithmetic and accounting documents show strong evidence

for a bi-quinary (later decimal) sub-structure in the sexagesimal

notation.

If need be, I'll try and dig up the references from my catastrophically

disorganized library.

Paul - --- In primenumbers@yahoogroups.com,

Paul Leyland <paul@...> wrote:

> Digits 6 through 9 are then represented by the juxtaposition

I learnt to count like that in Chi-Nyanja:

> of the 5-symbol and the appropriate symbol for 1 through 4.

modzi : one

wiri : two

tatu : three

nai : four

sanu : hand

sanu ndi modzi : hand-and-one

sanu ndi wiri : hand-and-two

sanu ndi tatu : hand-and-three

sanu ndi nai : hand and-four

khumi : all-together

It seemed much more sensible than counting in French :-)

David - --- On Wed, 12/23/09, Paul Leyland <paul@...> wrote:
> On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:

That looks like the choice of 60 precisely because of its divisibility properties. They didn't take the LCM and later make a shock discovery that it had all the factors of the two original numbers, shall we say.

> > > > published by Eratosthenes some 2200 years ago,

> > > > and was certainly known some 1000

> > > > (maybe 25 000) years earlier.

> > >

> > > I'm intrigued by the "certainly";

> > > I would have said "probably" for 1k BCE.

> >

> > I'd have said "definitely" for >3k BCE. Base 60 just screams

> > knowledge of divisibility properties.

>

> Sorry for the late response to this thread but I've been

> rather tied up

> with Real Life(tm) recently.

>

> There is a persuasive suggestion that the divisibility

> properties of

> radix-60 arithmetic is a consequence of its choice, not a

> reason for its

> choice. The argument goes as follows.

>

> A number of cultures have independently invented quinary

> arithmetic, for

> reasons which should be obvious. There are still

> relics of this in

> modern culture --- the five-bar-gate tallying method, for

> instance.

> Bi-quinary has also been widely used throughout

> history. This uses four

> different symbols for the digits 1-4 (the symbols are

> frequently 1 to 4

> identical lines or dots) and another symbol for 5.

> Digits 6 through 9

> are then represented by the juxtaposition of the 5-symbol

> and the

> appropriate symbol for 1 through 4.

>

> A number of cultures have independently invented duodecimal

> arithmetic.

> Many relics of this exist: 12 ounces to the Troy pound; 12

> inches to the

> foot; 12 pennies to the shilling and so on. The most

> convincing

> survivors to my mind are the survival of the English words

> "dozen" and

> "gross".

>

> Some time around 4000 to 3500 BCE the Sumerians moved into

> Mesopotamia

> and merged with a pre-existing culture. One culture

> used quinary or

> bi-quinary and the other

> duodecimal. Neither culture supplanted the

> other, rather their notations

> merged. Indeed, the symbols of early

> Mesopotamian arithmetic and accounting documents show

> strong evidence

> for a bi-quinary (later decimal) sub-structure in the

> sexagesimal

> notation.

Phil