Intro to the table of thirds
February 1, 2003

Presented by Bruce C. Friedman

Last modified: October 10, 2004.

There are numerous tables of hkt divisions on the two tablets from Cairo.  Some appear as many as five times.

The following table of one-third of various hkt quantities is found twice on these tablets.

Both sets of calculations of thirds are without error.

*Note: the more recent changes to this page stem from the superior translation and source images offered by this text:

Journal: Archiv Orientalni - The Quarterly Journal of Asian and African Studies, Praha, Czech Republic, 2002

Article: The Wooden Tablets from Cairo: The use of the Grain Unit hk3t in Ancient Egypt.

Author: Hana Vymazalova.

ANALYSIS OF 1/3 TABLE:

Modified and corrected 10/9/04 as per H. Vymazalova.

The scribe first determines the value of one-third of five by operating with duplications as follows.
*The results of this effort are applied further as one-third of five is used for one-third of five ro [r3].

Five ro [r3] is equivalent to one-sixty-fourth hkts.
Accordingly, one ro [r3] is 1/320th hkts.

Line 1:
one-third one

To be read as 1/3 of one is = a third


Line 2:
two-thirds two

To be read as 1/3 of two is = two-thirds


Line 3:
one-third one four

To be read as 1/3 of four is = one and one-third

Here, the scribe combined the result of one-third of one, and one-third of four, to find that five-thirds is equal to one-third of five.


Line 4:
//3 ro 1 ro /64

To be read as 1/3 of one/64 hkt is = [(//3* /320)+(/320)] = 5/960 = /192 = correct!
One third of /64 is /192.


Line 5:
/3 ro 3 ro /32

To be read as 1/3 of one/32 hkt is = [(/3* /320)+(3*/320)] = 10/960 = /96 = correct!
One third of /32 is /96.


Line 6:
//3 ro 1 ro /64 /16

To be read as 1/3 of one/16 hkt is = [(//3* /320)+(/320) + (/64)] = 6 2/3 ro = 20/960 = /48 = correct!
One third of /16 is /48.


Line 7:
/3 ro 3 ro /32 /8

To be read as 1/3 of one/8 hkt is = [(/3* /320)+(3*/320) + (/32)] = 13 1/3 ro = 40/960 = /24 = correct!
One third of /8 is /24.


Line 8:
//3 ro 1 ro /64 /16 /4

To be read as 1/3 of one/4 hkt is = [(//3* /320)+(/320 + /64 + /16)] = 26 2/3 ro = 80/960 = /12 = correct!

One third of /4 is /12.


Line 9:
/3 ro 3 ro /32 /8 /2

To be read as 1/3 of one/2 hkt is = [(/3* /320)+(3*/320) + /32 + /8)] = 53 1/3 ro = 160/960 = /6 = correct!

One third of /2 is /6.


Line 10:
//3 ro 1 ro /64 /16 /4 1

To be read as 1/3 of one hkt is = [(//3* /320)+(/320 + /64 + /16 + /4)] = 106 2/3 ro = 320/960 = /3 =correct!
One third of 1 is /3.


Line 11:
/3 ro 3 ro /32 /8 /2 2

To be read as 1/3 of two hkts is = [(/3* /320)+(3*/320) + (/32 + /8 + /2)] = 213 1/3 ro = 640/960 = //3 = correct!
One third of 3 is //3.


Note that the scribed appears here to have tallied line numbers ten and eleven [see the check marks on the image] and found in each case that the result was accurate. Compare this to the similar tallying procedure on the table of elevenths.


I am studying each and every transcribable entry for errors which may more clearly suggest the method(s) that generated these identities.

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