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Topic: A longer view of RMP 41, 42, 43, 44, 45, 56, 57, 68, 69, 70, 83 & 84
Replies: 6   Last Post: Aug 13, 2012 6:25 PM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
A longer view of RMP 41, 42, 43, 44, 45, 56, 57, 68, 69, 70, 83 & 84
Posted: Oct 5, 2010 2:46 PM

Attesting metrology and conversions is made more difficult by Ahmes' choices of symbols. Spalinger (SAK 17, 1990) discussed a few of these issues. Sometimes Ahmes' symbols were truncated or abbreviated. Within the problems duplation arithmetic hekat (and hekat multiples) were shown as big dots and/or long strokes and wedges for 1/2 and crosses for 1/4 (Horus-eye parts and other unique ligatures).

Ahmes used a range of related symbols for quadruple hekat and hekat in RMP 41, 42, 43, 44, 45, 46, 47, 68, 69, 70, 83 and 84. A double-hekat symbol was recorded in RMP 84, a topic discussed in an appendix.

Ahmes quadruple hekat symbols suggests to scholars that a quadruple hekat only meant exactly 4-hekat.

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

Spalinger (1990) mentioned that Griffith did not know of a quadruple hehat measure, though Ahmes cited it several times in mixed hard-to read ways.

Bruce Friedman, Griffith and Spalinger report quadruple hekat symbols as representing an indefinite four or more hekats. Griffith concluded that scribal values associated with quadruple hekat symbols can best be determined from the context of each scribal problem.

Leon Cooper, a 2009 "Historia Mathematica" MMP-10 author discussed RMP 44 and RMP 45 citing Spalinger's SAK 17 article and discussions of RMP Book II (pp. 320-323) and to Griffith's writings on the RMP (as referenced by Spalinger) in PSBA 13-16, and to Griffith again in PSBA 14. (Griffith categorized the RMP as containing three books: I, II and III, related topics that can be discussed at another time).

Spalinger reported Griffith in the latter (p. 429) in regard to whether single, double or quadruple hekats are intended: "the meaning is implied by context", and he adds in regard to hundreds of quadruple hekats that: "I conclude, therefore, that the scribe uses a rather inaccurate but perfectly intelligible abbreviation of language, in saying that '75 is the number of quadruple hekat' when the truth is that 75 is the number of complete squares (of 100 each) of the quadruple hekat" (p. 430)."

Cooper, Spalinger and Griffith were referencing "cubes" and not "squares". The hekat was a volume unit and not a two-dimensional unit.

SUMMARY: Concerning Ahmes' raw data there are at least 11 RMP problems that mention quadruple hekat. Of the 11 problems, 10 problems appear to have been scaled to 100-hekats. No problem seems to be scaled to a 4-hekat unit. RMP 84 may offer an exception with respect to a double-hekat unit discussion cited in the appendix.

APPENDIX

This hieratic analysis was produced by Bruce Friedman.The analysis covers Ahmes' quadruple hekat and hekat scripts written in specified RMP problems involving equivalences also shown in the Akhmim Wooden Tablets (AWT). The an analytical approach, was suggested by an Egyptologist, revisits hieratic symbols.

The simple single hekat symbol wss a loop with a tail to the right which is taken to represent the glyph of a corn basket showing pouring of grains. (RMP 41, end of line 4) We find the number 48 over the hekat loop. We [Milo and Bruce] think this is a key to reading this information as 48 hundreds of hekats, but not 4800 quadruple hekat.

Just prior to the end of line 4 we find the "quadruple hekat" shown as the same loop with four slashes above and a pen stroke below:

At the red ink at the end of line 4 in RMP 42 we find 59 [+] 1/4 over the loop and strongly suspect that this was 5925 single hekats (i.e. 59.25 hundred of hekats).

At the end of line 5 in red ink in RMP 43 we find 22 before the loop and 1/2 and 1/4 after which seems (also as per Chace and Peet and Clagett) to show 2275 hekats (i.e. 22.75 hundreds of hekats).

RMP 44: At the end of line 3 again red ink shows 75 just before the loop and this is takes as 7500 hekats (75 hundreds of hekats)

In RMP 44, Clagett translated a 10 by 10 by 10 cubit box containing 1000 CC and 1500 khar, finding a volume of 7500 hekat. Consistently in RMP 45, Ahmes asked the inverse, what was the size of a container that contained 7500 quadruple hekat? Ahmes found a 10 by 10 x 10 cubit box. At the end of line one in RMP 44 "quadruple hekat" [= QH] is shown as the loop with a dark underscore and a line conects the underscore through the tail of the loop to make a large wedge which points toward the loop. At the end of RMP 44 line 3 we see in black the loop with four slashes above for QH and then in red ink the simple loop for hekat preceded by the quantity 75, from context this appears to mean 7500 hekat or 75 quadruple hekat of 100 hekats each.

In line 1 of RMP 45 we see the loop with one connected slash and an underscore and this is taken as a single hekat (the symbols appear to be inconsistently applied to each case, yet they self-clarify upon analysis- I think) then we see the 4-slash above loop (typical form) for QH.

In RMP 46 the simple loop form of hekat is preceded by the quantity 25 which appears to equal 2500 hekats or 25 QH.

In RMP 47: the "quad hekat" symbol appears in line 3 followed by a long vertical stroke which seems to show a quantity of 10 "quad-hekats" which drives most translators to assume all the divisions are divisions of 100 "quad- hekats" but all the other workings do not show any hekat symbols except as abbreviated old kingdom forms (i.e. dots). See below.

RMP 68 is said to divide 100 great "quad-hekat" among the foremen of 4 work gangs of different sized crews. The workings show on line 2 the "quad-hekat" symbol and this is followed by the 100 quantity shown of the hekat [ = H] symbol and seems to be correctly taken to be 100 hundreds of hekats or 100 "quad hekats." the columns E and F workings primarily show the truncated forms for the "quad-hekat" such as column F line 3 where two vertical strokes apparently represent twice 10 "quad-hekat" which is aligned with the number 20, implying 20 "quad-hekat" has been calculated and proven correct.

RMP 69 and 70 do not show QH but these pesu problems shows mixed format workings with the hekat loop and some change in ro units which is consistent with the attached work on the AWT (H. Vymazalova) and G. Moeller's Palaographie.

RMP 83 and 84 deal with feed for domesticated animals and show mixed format hekat and ro calculations which are consistent as above with Hana and Georg. RMP 84 is the only example of these RMP problems which shows one use of double hekats (which seems to be likely equal to 2H).

One contextual reading 1 khar equaled 5 hekat. A second reading i khar equaled 5 quadruple hekat.

A basic question separates mixed scholarly points of view. What was the RMP hekat value of one quadruple hekat, and related scalings to khar and cubit-cubit units?

The hieratic item named "quadruple hekat" was contextually scaled to one hekat in 10 of 11 RMP problems. But what was the hekat value?

Scholary reviews of RMP 41, 42 and 43 have often scaled one khar to 20 hekats, a context that should hold for RMP 44, RMP 45 and other RMP problems. But does it?

Comparing RMP 41, 42= and 43 with RMP 44 and RMP 45, the hekat units were in harmony within proposed metrology conversions noted at the end of this post.

In terms of Clagett (1999), RMP 41 reported Volume (V) = 640 CC; V= 960 khar; V = 48 quadruple hekat which would be better if called 48 "hundreds of hekats" or 48 "great hekats" and V = 4800 hekat.

Looking at RMP 47 raw data closely (Clagett and Chace translations) lines 1 and 2, were compared to lines 3, 4, 5, 6, 7, 8, 9 and 10 with the X representing a quantity which is not specifically shown but which we are inclined to assume is the same 10,000 single hekats or 100 "quadruple-hekats" [of 100 hekats each - we expect] from lines one and two:

After this Ahmes' abbreviated considerably:

3. X divided by 30 = (3 dots + 1/4 + 1/16 + 1/64) [hekat] + ( 1 + 2/3) [ro]

4. X divided by 40 = (2 dots + 1/2) [hekat]

5. X divided by 50 = 2 dots [hekat]

6. X divided by 60 = (1 dot + 1/2 + 1/8 + 1/32) [hekat]

7. X divided by 70 = (1 dot + 1/4 + 1/8 + 1/32 + 1/64) [hekat] + **(2 + 1/14 + 1/21 + 1/42 ) [ro]

**FYI: Robins-Shute have omitted the actual hieratic workings and showed the equivalent:
(2 + 1/7) ro

8. X divided by 80 = (1 dot + 1/4) [hekat]

9. X divided by 90 = (1 dot + 1/4 + 1/8 + 1/64) [hekat] + (1/2 + 1/18) [ro]

10. X divided by 100 = 1 dot [hekat]

The dots are either abbreviations for hekats or their multiples**** but if these dots refer to quadruple hekats (****which may mean 4 hekats each) or hundreds of hekats or any hekat multiple then the same multiplier must refer to the ro portions of the "answers."

As written, Bruce Friedman believes the dots referenced to single hekats; and Ahmes knew that each "answer" from lines 3-10 was to be multiplied by a factor of 100, and not 400 (the 100-quadruple hekat).

Considering line 1 quadruple hekat likely meant 10 hekat (100-quadruple hekat divided by 10)

line 2, 5 quadruple hekat likely meant 5 hekat (100 quadruple hekat divided by 20)

We can see no application that suggests or confirms that 1 "quadruple hekat" was equal to 4 hekat! We see no 400-hekat scaled value in lines 3, 4, 5, 6, 7, 8, 9 or 10.

In every case the quadruple hekat (an implied value of X) indicated that 100 hekat or more were listed. No 100-quadruple hekat units (glyphs) were directly shown in lines 3, 4, 5, 6, 7, 8, 9 and 10, that scaled an alternate conclusion.

For example, line 3 listed

(3 + 1/4 + 1/16 + 1/64) hekat + ( 1 + 2/3) ro

(likely 1 or 100-quadruple hekat divided by 30)

Lines 3 thru 10 of RMP 47 show answers were written in a binary quotient (Q) and a ro remainder(R) two-part notation.

A scaled ro remainder was published in 2006 based on applying Hana Vymazalova's 2002 view that scribes scaled hekat to (64/64). Vymazalova named (64/64) a hekat unity. Vymazalova made in an analysis of the wooden tablets by introducing divisors n = 3, 7, 10, 11 and 13 of an unknown initial hekat value. Five division created five two-part answers.To add specificity to the unknown initial value Vymazalova showed in five cases that a (64/64) hekat unity was returned when two-part answers were multiplied by the initial divisors, showing the initial problem was (64/64)/n, and

(64/64)/n = Q/64 + (5R/n) ro

was decoded in 2006.

In our case 100-hekat (and 100 quadruple hekat) hieratic symbol cases also used a divisor n = 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 in RMP 47. Rather than dividing (64/64), Ahmes divided 100-quadruple hekat or 100-hekat.

Please note that RMP 83 reported three Akhmim Wooden Tablet [AWT] divisions following (64/64)/n two-part notation.

Within RMP 83 a "set-duck" was valued at 1/20 a hekat recording

a. (1/32 + 1/64) hekat + 1 ro

which meant

3/64 + 20/29(1/320) = (64/64)/20

b. Working the (64/64)/20 problem in reverse order

(64/64)/20 = 3/64 + 4/(64)(1/64) = (2 + 1)/64 + (20/20)(1/320)

= 1/32 + 1/64 + 1 ro

We have taken the liberty to expand RMP 83's scaled values by working the AWT method generally in reverse order such that we can see how and why Ahmes would have shown a table of fowl valuations, given a "set-duck" was valued at 1/20 of a hekat:

Hekat
=====

1/20: (64/64)/20 = 3/64 + 20/20 ro =

(2 + 1)/64 + 1 ro = 3/64 + 1/320 =

16/320 = 1/20 hekats

1/16: (64/64)/16 = 4/64 = 1/16 hekat

1/10: (64/64)/10 = 6/64 + 20/20 =

[(4 + 2)/64 = 1/16 + 1/32] + 1 ro

(the AWT shows the (64/64)/10 two-part result)

1/9: (64/64)/9 = 7/64 + 5/9 ro =

(4 + 2 + 1)/64 + [5/9 =

(10/18)ro = (9 + 1)/18)] =

(1/16 + 1/32 + 1/64) hekat + (1/2 + 1/18) ro

1/8: (64/64)/8 = 8/64 = 1/8 hekat

1/7: (64/64)/7 = 9/64 + 5/7 ro =

(8 + 1)/64 + [(10/14) ro =

(7 + 2 + 1) /14 ro] =

(1/8 + 1/64) hekat + (1/2 + 1/7 + 1/14) ro

(the AWT also shows (64/64)/7, a two-part answer)

1/6: (64/64)/6 = 10/64 + 20/6 ro =

(8 + 2)/64 + (3 + 1/3) ro =

(1/8 + 1/32) hekat + ( 3 + 1/3) ro (re-goose*)

1/5: (64/64)/5 = 12/64 + 20/6 ro =

(8 + 4)/64 hekat + (3 + 1/3) ro =

(1/8 + 1/16) hekat + (3 + 1/3) ro

1/4: (64/64)/4 = 16/64 = 1/4 hekat (terp-goose*)

1/3: (64/64)/3 = 21/64 + 5/3 ro =

(16 + 4 + 1)/64 + (1 + 2/3)ro =

(1/4 + 1/16 + 1/64) hekat + (3 + 2/3) ro,

(the AWT recorded this (64/64)/3 two-part answer)

1/2: (64/64)/2 = 32/64 = 1/2 hekat (djendjen*)

1/1. (64/64)/1 = 64/64 = 1 hekat (one set duck*)

* asterisks above refer to work by Clagett (and Chace).

Ahmes implied 100-quadruple hekat was used in line 8 of RMP 84, where we find two strokes over the hekat symbol that appears to be total 200 hekats or 2 times the unit of 100 hekats.

In line 9 of RMP 84 we find this double hekat symbol with the typical loop with tail (for single hekats) and 2 short strokes attached above it which Chace and others translate as a 2-hekat unit.

Citing and modifying Clagett's raw data from RMP 84 we may come to agreement that the context of all RMP quadruple hekat problems only implies hekat values.

Line 1: 4 bulls ate 2402 hekat (or 242 hekat)

Line 2: 2 bulls ate 2206 hekat (or 226 hekat)

Line 3: 2 cattle ate 2002 hekat (or 202 hekat)

Line 4: 1 oxen ate 2000 hekat (or 200 hekat)

Line 5: total animals ate 86010 hekat (or 870 hekat)

Line 6: made 9 + 3/4 hekat of malt for one day

Line 7: 97 + 1/2 hekat of malt for 10 days ( " )

Line 8: 292 + 1/2 hekat of malt for 30 days ( " )

Line 9: seems to convert line 8 into a double hekat by:

1/2 of 100-hekat + (11 + 1/2 + 1/8)hekat +

3 ro + (1/4)hekat + 5 hekat

may mean: 50 + 11 + 1/2 + 1/4 + 1/8 + 5 + 3 ro =

(66 + 7/8)hekat + 3 ro

Where did 3ro come from? Considering line 8

a. 66 (64/64) = 4224/64

b. 7/8(64/64) = 56/64

c. 3/5(1/64)

d. shows that 320/320 was the scaling factor (used in RMP 38) such that:

4290/64 + 3/5(1/64) = 21453/320

Aspects of RMP 84, especially the line nine, are hard-to-read. Can anyone assist reading line nine, double checking the previous eight lines? As a group the nine lines should be read by one scribal context.

9/14 VERSION: Conversion chart: By Bruce Friedman

1 khar = 5 single hekats =

1/20th of a "quadruple hekat" =

2/3rds of a cubic cubit

ALSO:

1 cubic cubit = 3/2 khar =

7 and 1/2 hekats = 3/40ths of a "quadruple hekat"

ALSO:

1 hekat = 1/100th of a "quadruple hekat" =

1/5 khar = 2/15 cubic cubits

ALSO:

1 "quadruple hekat" = 100 hekats =

20 khar = 13 and 1/3 cubic cubits

Date Subject Author
10/5/10 Milo Gardner
10/10/10 Milo Gardner
10/21/10 Milo Gardner
5/16/11 Milo Gardner
5/22/11 Milo Gardner
8/10/12 Milo Gardner
8/13/12 Milo Gardner