INTRO TO ELEVENTHS
The ACHMIM [or AKHMIM or AKHMIN]
Mathematical Wood Tablets [from about 2000 BCE]

Page Created: February 1, 2003.

Presented by Bruce C. Friedman

Last modified: October 10, 2004 as per H. Vymazalova.


These two small tablets are among the very few purely mathematical Ancient Egyptian artifacts.  They are now at the Cairo Museum but are sadly never on display!
Items CG 25.367 and 25.368 were first published within:
“Catalogue General des Antiquites Egyptiennes du Musee du Caire.”
See the 1901 volume with [ostraca] No.s 25001-25385 par [by] M. G. Daressy.


My prior (grossly inaccurate) personal analysis of the image below can still be found at the bottom of Milo Gardner's excellent EMLR paper:
http://www.math.buffalo.edu/mad/Ancient-Africa/gardner.EMLRL.160202.doc
and also at:

http://www.mathorigins.com/B_emlr050602.htm

Please ignore my ACHMIM analysis within these 2 links above and instead refer to the revision and translation below.

 

(from CG 25.368)

Readers please note that the source image found in the above referenced text by G. Daressy, was very poor and very small and that much effort was required to produce the image below. As always, note that all errors of reproduction, analysis or transcription are my responsibility alone.

(These identities are demonstrated in hieratic, in black ink on a plastered wooden plank. Georges Daressy reports this item [C.G. 25.368] to measure .475 m by .260 m. G. Daressy reports that the other tablet [C.G. 25.367] measures .465 m by .250 m.  These tablets are assigned by the style of script used and the Old Kingdom styled names that appear on them to the XIIth Dynasty. They are further known to date from the 28th reign year of an unnamed King.  Daressy assumes the unnamed Pharaoh to be either Usertesen or Amenemhat.

In defense of this dating, note that the double symbol 1/6 is identical to the symbol with a value of 1/6 from the XIIth Dynasty Kahun Mathematical Papyrus. I have not seen this double symbol elsewhere.

Note that (2/3 + 1/22 + 1/66) is shown in this identity, which of itself is exactly 8/11. The scribe, or his teacher, in keeping with the style of calculations in the EMLR and RMP and KMP, must have been adept at such workings.

 

LEARN MORE ABOUT THE ACHMIM MATHEMATICAL ARTIFACTS:
http://www.mathorigins.com/A.htm

Note that many of the hieratic representations of specific complex hekat fractions have been reduced to one symbol. These are standard fractions (generally) only used in ancient Egypt when referring to binary divisions of a hekat [or hkt or hk3t!].
Examples include:
1/2; 1/4; 1/8; 1/16; 1/32; 1/64.
Many of these symbols are similar to, or reminiscent of, Old Kingdom Horus-Eye parts.
In some cases the same symbol has two values!
Refer to the GUIDE page.

 

TABLE OF ELEVENTHS:

LINE #
            1 10 1   1
            10 100 10   2
            20 200 20   3
            2 20 2   4
            4 40 4   5
            8 80 8   6
            1   11   7
        /11 4 ro /64 /16 1   8
    /66 /6 3 ro /64 /32 /8 2   9
    /33 1 ro /64 /32 /16 /4 4   10
/66 /22 //3 2 ro /32 /16 /8 /2 8   11
 


The typical Old Kingdom hkt divisions are simply repeated halvings:
1/2
1/4
1/8
1/16
1/32
1/64

**1/320 = a daily ration = 1/32*1/10 = 1 ro = 1 [r3]


Lines one through seven are simple and correct.

Line Eight:
/11_4 ro _/64_ /16_1
Read this as {(/11*/320)+4/320+1/64+1/16=1 [eleventh of a hkt]

20+5+4+/11=24+ /11 ro

(29 +/11)* 11= 319+1 = 320 ro = 1 hk3t = correct.

 

Line Nine:
/66_/6_3 ro_/64_/32_/8_2
Read this as {(1/66+/6)*/320)+3/320+1/64+1/32+1/8=2 [elevenths of a hkt] = 1 eleventh of 2 hk3t

40+10+5+3+(6+/66) ro = 58+ 12/66 ro = 58 2/11 ro

(58 + 2/11)* 11 = 638 + 2 = 640 ro = 2 hk3t = correct.

*See the left end of line nine on the image which perhaps reads 1/68 due to damage/smudging/age or possibly a scribal error. I have adjusted the similar difficulty at the left end of line eleven and treated both as if they describe 1/66, NOT 1/68. The scribe seems to have known this meant 1/66.

Unfortunately all examples of the determination of four elevenths contain an error of omission.

Line Ten:
/33_1 ro_/64_ /32_ /16_ /4_ 4
Read this as {(1/33*1/320)+1 ro +1/64+1/32+1/16+1/4=4 [elevenths of a hkt] = 1 eleventh of 4 hk3t

80+20+10+5+1+(/33) ro = 116+ /33 ro

(116 + /33)* 11 = 1276 + 11/33 ro, is less than correct by the omission of one-third ro in the identity and subsequently the error cumulatively brings the workings to a shortage of eleven thirds or 3+2/3 ro.

Line Eleven:

1/66 1/22 2/3 2 ro 1/32 1/16 1/8 1/2   8 [checked off]

8/11= 1/2+1/8+1/16+1/32+2 ro +[(1 ro)*(2/3+1/22+1/66)]
So, reading the entry as:
8/11= 1/2+1/8+1/16+1/32+2/320 +[(8/11) ro]
____= 160+40+20+10+2+ 8/11 = 232 8/11 ro = 8/11 hk3t.

(232 + 8/11) * 11 = 2552 + 8 = 2560 ro = 8 hk3t = correct.

Further note that the scribe has checked three lines on the image above and appears to have separately verified that his calculations for 1/11, 2/11, and 8/11 total 11/11 or Unity!
The total is 100% accurate less the error of omitting one third ro in line ten.



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