Egypt Math Glossary

Introduction

This glossary is broken down into two parts. Part I cites standard weights and measures units that scholars have struggled to scale
for over 100 years. The Old Kingdom, 3,000 BCE to 2050 BCE. names were derived from algorithmic-type statements recorded in binary ways. The Eye of Horus numerations system defined

one (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...

with a 1/64 unit thrown away (rounded off in modern terms).

By 2050 BCE the Old Kingdom's round off problem was solved and formalized in six 1900 BCE Akhmim Wooden Tablet examples. An additional 29 examples are found in one 1650 BCE Rhind Mathematical Papyrus problem, RMP 81. The AWT spread sheet reports a Middle Kingdom's finite division method that was prepared by Bruce and myself over a two week period (in 2005 after reading Hana Vymazalova's AWT paper). The AWT and related volume and area weights and measure units were exact.

The Old Kingdom's Eye of Horus binary number metaphor continued in use during the Middle Kingdom. One purpose was to read the BOOK OF THE DEAD and other spiritual documents in the old manner. Secular weights and measures in the Middle Kingdom were written in exact AWT units.

PART I: CONVERSIONS FOR ANCIENT EGYPTIAN UNITS OF MEASURE

Refer to T. E. Peet’s English 1923 RMP and also Tanja Pommerening’s German article in Berichte Zur Wissenschaftsgeschichte #26, (2003) pp.1-16: for further metrological discussions.

Cubit

1 LINEAR royal cubit = 7 palms = 28 fingers
Note that all units are reduced to linear or square or cubic cubits as follows:
(Length as determined by Isaac Newton) ~20.6 inches = ~523.5 mm

See source text:
“Miscellaneous works of Mr. John Greaves [1602-1652]…To which are added: I. Reflections on the Pyramidographia…; II. A dissertation upon the Sacred Cubit of the Jews…; III. Tracts upon various subjects…; IV. A description of the grand seignor’s seraglio. To the whole is prefix’d an historical and critical account of the life and writings of the author, pub. By Thomas Birch [1705-1766].” London, 1737, two volumes. (800 pages)
Including:
I. Pyramidographia: A description of Egypt’s Pyramids.
II.With discourse on Roman foot and Denarius.
III. Tracts; Letters; Poems
IV. Turkish Emporer’s Court
Volume 2 [of 2] Includes Isaac Newton’s Dissertation!
“A Dissertation upon the Sacred Cubit of the Jews and the Cubits of the several Nations ; in which, from the dimensions of the greatest Egyptian Pyramid, as taken by Mr. John Greaves, the antient [ancient] Cubit of Memphis is determined.”/ “Translated from the Latin of Sir Isaac Newton, not yet published.”

Remen

5 palms;
20 fingers;
5/7 LINEAR cubits;
(*sometimes also one-half setat which = 5,000 SQUARE CUBITS)
A special sign was *sometimes used for 1/2 setat also called rmn (remen) and referring to the fact that 5/7 of
a royal cubit squared is roughly 1/2 square cubit. These VERY different uses of remen can be gleaned easily from local context.

Khar

.0956 CUBIC METERS;
6400 ro;
20 hekats;
5 quadruple hekats;
50 quadruple henu;
two-thirds CUBIC cubits

Hekat

.0048 CUBIC METERS;
1/20 khar;
320 ro;
10 henu;
one-thirtieth CUBIC cubits

Double Hekat

.00956 CUBIC METERS;
1/10 khar;
640 ro (or sometimes 320 double-ro?);
20 henu;
5 quadruple henu;
one-fifteenth CUBIC cubits

Quadruple Hekat

.0191 CUBIC METERS;
1/5 khar;
1280 ro (or sometimes 320 quadruple-ro?);
40 henu;
10 quadruple henu;
two-fifteenths CUBIC cubits

Henu

.0005 CUBIC METERS;
32 ro;
1/200 khar;
One-tenth hekat;
1/40 quadruple hekat;
1/300 CUBIC cubits

Quadruple Henu

.0019 CUBIC METERS;
128 ro;
1/50 khar;
two-fifths hekat;
1/10 quadruple hekat;
1/75 CUBIC cubits

Hin

a/p Tanya: 1/10 hekat or 1/10 other units??; same as henu??

Djar or Dja

a/p Tanya: 5 ro; 1/1280 khar; 1/64 hekat

Khet

52.35 LINEAR METERS;
100 LINEAR CUBITS; (sometimes called a reel or cord)

Square Khet

2740.5 SQUARE METERS;
1 setat;
100 "cubits of land";
1/10 of a "thousand of land";
10,000 SQUARE cubits

Setat

2740.5 SQUARE METERS;
1 square khet;
100 "cubits of land";
1/10 of a "thousand of land";
10,000 SQUARE cubits
*Note: example of doubling 7 setat from RMP #53 - result is: 1 "thousand of land" and 4 setat

Cubit of Land

27.405 SQUARE METERS;
a strip 100 LINEAR cubits by 1 LINEAR cubit;
1/100 setat;
100 SQUARE cubits
• AKA "cubit strip"

Thousand of Land

27405 SQUARE METERS;
1000 "cubits of land";
ten setat;
ten square khet;
100,000 SQUARE cubits

Deben

Monetary - Greek term
Egyptian grain and beer used hekat units .

Cubic Cubit

.1434664_ CUBIC METERS;
3/2 khar;
30 hekats;
7 +/2 quadruple hekats;
300 henu;
75 quadruple henu;
9600 ro;
343 CUBIC palms

Square Cubit

.274052_SQUARE METERS;
2.946944_SQUARE FEET;
784 SQUARE fingers;
49 SQUARE palms

Square Remen

.1398226 SQUARE METERS;
25/49 SQUARE CUBITS;
400 SQUARE fingers;
25 SQUARE palms
*possible reference to double remen linear measure of 10/7 cubits.

PART II: MIDDLE KINGDOM CORRECTIONS

Part II of the glossary outlines Middle Kingdom (MK), 2050 BCE to 1500 BCE, corrections to Old Kingdom algorithmic statements. Modern scholars report conflicting weight and measure unit values (Part I). The MK corrections are best read in complete scribal mathematical sentences. Word definitions changed in the MK as scribes applied old concepts to new problem solving situations. For example, ro, 1/320 of a hekat, was written as a remainder, and at other times a quotient and remainder. To determine which ro was being discussed complete sentences, and at times, complete paragraphs need to be read.

MK corrections often mentioned the hekat, a volume unit, first reported in a narrative:

http://en.wikipedia.org/wiki/Hekat_%28volume%29

Scribes used two classes of correction systems.

A. The First Correction System

The first correction system almost always limited rational number divisor n to the range

1/64 < n < 64

such that

(64/64)/n = Q/64 + (5R/n)*1/320

with Q a binary quotients and A a remainder scaled by 5/5 to 1/320

B. Ahmes' Exception

As an exception, Ahmes replaced one hekat within the 64/64 unity idea (decoded by Hana Vymazalova in 2002) by replacing 100 hekat with 6400/64. The 100 hekat case allowed divisor n to exceed the usual 64 limit. Robins-Shute in 1987 muddled RMP #47, problem that allowed 100 hekat to be divided by n = 70 such that:

(6400/64)/70 = 91/64 + 30/(64*70)*(5/5) =

(64 + 16 + 8 + 2 + 1)/64 + (150/70)*(1/320) =

(1 + 1/4 + 1/8 + 1/32 + 1/64) hekat + ( 2 + 1/7)ro

with quotient 91/64 hekat and remainder 30/(70*64) became ro units

written as two part statements.

C. The Second Correction System

The second correction system allowed divisor n to be any positive rational number such that sub-units of the hekat , 1/10 was a henu (hin). 1/64 was a dja and 1/320 was

10/n hin

64/n dja

320/n ro

and so forth.

D. Overview for the July 21st, 2010 Webinar

Part II of the glossary will exclude monetary units and the PESU (an arithmetic proportion). The PESU was used in scaling bread and beer that also assisted in solving two second degree equations in the Berlin Papyrus. A PESU was an inverse proportion, a topic that will not be presented in next month's Webinar..

The topic of the Webinar will be the MK rational number system that generally converted rational numbers to optimized, by not optional, unit fraction series.

The first set of scribal definitions applied to 1/p and 1/pq conversions. A 1900 BCE scribe wrote the EMLR:

http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll

"The scribal writing consisted of characters recorded from right to left. There were 26 rational numbers listed. Each rational number was followed by an equivalent Egyptian fraction series. There were ten Eye of Horus numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There were seven other even rational numbers converted to Egyptian fractions: 1/6 (twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. Finally, there were nine odd rational numbers converted to Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15, training patterns for scribal students to learn the RMP 2/n table method. "

E. Ahmes' Discussion of EMLR Conversions

Ahmes discussed awkward EMLR conversions of 1/4 and 1/8 by LCM, 72 in RMP 37:

RMP 37: Find 1/90 of a hekat in ro units from:

1. 320 ro*(1/90)=3+1/2+1/18=64/18

2. Ahmes playfully reported four unity sum methods, the first being:

a. 320*(1/180)=64/36
b. 320*(1/360)=64/72
c. 320*(1720)=64/144
d. 320*(1/1440)=64/288
e. 320*(1/2880)=64/576
f. unity sum (b+e)=64/72+64/576=1

Three inverse red number calculations included EMLR-like conversion of

1/4 = 72/288 = (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1)/288,

with additive numerators recorded in red ink. Ahmes aligned red numbers (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1) below a non-optimal(1/32
+ 1/16 + 1/12 + 1/96 + 1/36 + 1/288 + 1/36 + 1/288) series. The paired lines with red integers were inverses of unit fractions.

Ahmes recorded

1/8 = 72/576 = (8 + 36 + 18 + 9 + 1) below (1/72 + 1/16 + 1/32 + 1/64 + 1/576),

ending the playful problem.

F. 2/n tables

This topic has confounded scholars for 120 years. In 1995 Kevin Brown analyzed the patterns with Ahmes' 2/n table, and found that all consistent, reporting:
http://www.mathpages.com/home/kmath340/kmath340.htm

Ahmes generally converted 2/n and n/p discussed by three links:

All show that an LCM m generally scaled rational number n/p to mn/mp such that divisors of mp were best summed to numerator mn, creating finite unit fraction series.

When an m could not be found, ie. 30/53 and 28/97, a second rule was introduced, rewriting n/p to (n - 2)/p + 2/p. In this case two LCM m values scaled (n -2)/p and 2/p separately.

Example (from RMP 36) : 30/53 = 28/53 + 2/53
with,
28/53 x (2/2) = 56/106 = (53 + 2 + 1)/106 = 1/2 + 1/1/53 + 1/106
2/53 x (30/30) = 60/1590 = (53 + 5+ 2)/1590 = 1/30 + 1/318 + 1/795
30/53 = 1/2 + 1/30 + 1/53 + 1/106 + 1/318 + 1/795

Example: (from RMP 31): 28/97 = 26/97 + 2/97
with
28/97 x (4/.4) = 104/388 = (97 + 4 + 2 + 1)/388 = 1/4 + 1/97 + 1/194 + 1/388
2/97 x (56/56) = 112/5432 = (97 + 8 + 7)/5432 = 1/56 + 1/679 + 1/776
28/97 = 1/4 + 1/56 + 1/97 + 1/194 + 1/388 + 1/679 + 1.776

G. Greek Rational Numbers

Greek arithmetic followed 1,500 older Egyptian ciphered numbers, and scaled rational numbers recorded in unit fraction series.

Following a tradition established by Eudoxus. Archimedes used the Eudoxian 1/4 geometric series, one phase of the Eye of Horus, to find the area of a parabola. The infinite series side of Archimedes' calculus stressed the modern 'math of exhaustion' in a PBS "Nova" program, as recommended by a small team of Stanford U. researchers.

4A/3 = A + A/4 + A/16 + A/64 + ...

The finite side of the Western Tradition's first calculus reported

4A/3 = A + A/4 + A/12

in "Archimedes", by E.J, Dijksterhus, Princeton Princeton Press, 1987, ISBN 0-691-084321-1. On page 129, Dijksterhus cites a review of Heiberg's 1906 infinite series and finite series analysis of the raw data, a point a view that is adopted in the pending Webinar.

H. Arab and Medieval Changes of Numeration

Arab and Medieval arithmetic changed 2,800 years of ciphered numeration systems. Arab and medieval scribes, however, did followGreek arithmetic conventions by scaling rational numbers, much as Greeks built arithmetic definitions and operations on 1,500 year older Egyptian scaling methods.

Math historians fairly report that Hindu-Arabic numerals replaced ciphered numeration in 800 AD. Math historians report that Fibonacci learned unit fraction arithmetic from Arab texts, but have not reported an Arab and medieval rational number conversion rule that defined LCM m within a subtraction context:

(n/p - 1/m) = (mn -p)/mp

with remainder' numerator (mn-p) set to one, unity. Fibonnaci dedicated 125 pages of the 500 page Liber Abaci (Sigler's 2002 translation) to demonstrate seven classes of rules.

http://www.ebook3000.com/Fibonacci-s-Liber-Abaci_60105.html

Four pages of Sigler's 2002 translations of the Liber Abaci summarize the seven conversion rules as distinctions:

http://liberabaci.blogspot.com/

Distinction number seven reported the 4/13 case, in which an LCM m could not be found. that equaled unity and a 2-term series. Fibonacci selected two LCMs, first converting 4/13 by LCM 4 to

(4/13 - 1/4) = (16 - 13)/52

with

(3/52 - 1/18) = (54 - 52)/936 = 2/936 = 1/468

meaning

4/13 = 1/4 + 1/18 + 1/468

reported a 3-term series.

I. Summary

Old Kingdom Egyptian mathematics recorded rational numbers and mathematical solutions in algorithmic statements. By 2050 BC Egyptian scribes corrected rounded off infinite series within exact finite methods. Middle Kingdom proto.number theory allowed scribes, like Ahmes in 1650 BCE, to record 50 member 2/n tables by converting 2/3, 2/5, ... 2/101 to optimized, but not optimal unit fraction series. Ahmes scaled each rational number by selecting a LCM m that scaled n/p to mn/mp, and considering the divisors of mp that summed to numerator mn, to record unit fraction series.

Greeks, Arabs and medieval scribes modified Ahmes rational number system in minor ways. One modification selected an LCM m in a subtraction context, converting n/p by often setting (n/p - 1/m) = (mn - p) = 1 within seven distinctions.

A 1637 AD Arab text is the last know Egyptian fraction text, recorded in pre-modern Arabic script. In modern Arabic script uses a base 10 decimal algorithm, applying the binomial theorem, ironically replacing an exact numeration system that had lasted for 3,700 years.

The history of Egyptian fraction numeration systems can be compared to our modern computers' development of binary numbers that used zero and one, and later other available numbers. Old Kingdom numeration can be compared to early machine language written in a 6-term FIXED point system; Middle Kingdom's scaled finite arithmetic to DOS; Greek arithmetic notations to Windows 95; Arab un-ciphered numeration systems to Windows 98, and medieval numeration systems to Windows 2000.

Directly stated, Egyptian, Greek, Arab and medieval scribe began with infinite series (algorithmic) statements, and corrected round off errors by writing finite series (non-algorithmic) statements, recorded in unit fractions.