Rhind Mathemathical Papyrus (RMP) The Hungarian reading and solving of the Lesson No. 43 An excerpt out of the book “Royal Circles” by J. Borbola

Picture 4 Gay Robins and Charles Shute, The Rhind Mathematical Papyrus, an ancient

Egyptian text 14, problems 41-6

The pronunciation of the Hungarian alphabet for English-

speaking people as used in this text:

A < a > „flop” Á < aa > arm B < b > boat C < cz > Czar, Cs < ch > church --------------------------- D < d > door E < ε > gender É < e > arraign F < f > friend G < g > go

--------------------------- Gy < dj > George

H < h > have

I < i > ink Í < ii > evening J < y > young

------------------------ K < k > milk L < l > play Ly < y > young M < m > milk N < n > no ------------------------- Ny < nj > new O < o > hold Ó < oo > hole Ö < œ > her

P < p > pair R < r > run

------------------------ S < sh > dish Sz < s > list T < t > time Ty < tj > tube U < u > book

--------------------------- Ú < uu > ooze Ü < ue > Munich V < v > have Z < z > zoo Zs < ž > treasure

A. Describing the Lesson

The lesson No.43 of the Rhind-Papyrus is on the left inside of the broken roll. It is

registered under Catalog No.: BM 10057 (left) recto. The papyrus is not damaged at this part, thus the signs are easy to read. The lesson 43 is in the lower third of the segment, above it are two more exercises dealing with circle, No 41 and 42. The part right from it and the area’s backside are left empty. Left from it are the lessons 44, 47 and aligned with them the lesson No 48. The lesson is divided into two horizontal galleys. We find in the upper part the text of the exercise, the lower one contains the calculation and the hieroglyphs of the introduction. There are some red-colored

hieroglyphs among the uniformly black signs: No. (1-8), (82-83), (149-163), and (218-234) on the next pages.1

Picture 5

Lesson No. 43 of the RMP. August Eisenlohr, Ein mathematisches Handbuch der alten Ägypter

(Papyrus Rhind des British Museum) Taf. XV.

1 The numbers written under the hieroglyphs help to keep track of them and so simplifying their

discussion. Those are not catalogue-numbers and restart with 1 in every lesson.

B. Discussing the lesson First line, first segment: I/1a and I/1b

31-29 28 27 26 25 24 23-21 20 19-18 17 16 15 14 13 12 11 10-9 8-7 6-5 4-3 2 1

The mirror-picture of the transliteration:

1/a

1 2 3-4 5-6 7-8 9-10 11 12 13-14 15

š ȝr ‛d pr kb - n kr - n mḥ- ‛d 9 m Kȝ f

CS üR öD-BőL Kap-Ni KöR-öN MeG-aD 9 aMi KaR-SoK íV

Csűrödből kapni, körön megad 9-et, ami sok karív <chuerœdbœl kapni, kœrœn mεgad 9-εt, ami shok kariiv>

Receiving out of your granary, giving 9 on the circle, this are many arches

1/b 16 17 18-19 20 21-23 24 25 26 27 28 29-30 31

6 m psdn f ptj tr * h ȝr ȝr t-íw k

6 aMi PCSDöN SoK íV PaJTa TáR * Be R aR T-JU –K

6, ami p’csdön sok ívű pajta-tár mondjuk, berakatjuk <6, ami p’chdœn shok iivue payta-taar mondyuk, bεrakatyuk>

6 on your (measuring)rod, we call it an arching-granary, let’s load it

Notes:

The first six signs build a coherent, in numerous lessons recurring, starting-unit.

The No.(1) is Gardiner’s sign M8 šȝ, among others as ID: “ ‘lotus pool’, ‘meadow’.

Hence phon. šȝ, or used for š.“ Its sound-value is the Hungarian ‘cs’ <ch>, but

the scientific literature is looking rather for the sound ‘sh’. There is an other sign as well for the sound ‘sh’. We can read it as “csűr” <chuer> (barn, granary), see CS-A/ŰR, but it could be the first part of our word CS-ÍR-ádból <chiirádbool> (out of your seed). The

reading ‘(mo)CSÁR <mochaar> (marsh) – as a choice – can’t be excluded either. Therefore, the first four signs could be read as ‘csírádból’ <csiiraadbool> ( out of your

seed), ‘csűrödből’ <chuerœdbœl> (out of your granary) or ‘mocsaradból’

<mocharadbool>, (out of your marsh).2 We are going to see later that the most plausible reading is “csűrödből” (out of your

granary).

We already analyzed the sign No (5) – Gardiner’s F46 ḳȝb, ‘intestine’ – while

reading lesson No 14 of MMP. We identified its meaning there with our verb ‘hajt’ (fold), behajt (collect) as the best solution. The sounding of the frame K-AR-B could be karba-n,

körbe-n <kœrbεn> , or karóba-n as well (in choir, in circle or in a pale). According to Gardiner, the middle-sound AR is often missing and then we could read K-B as kap

(receiving). At this place supposed to stand the predicate of the title anyway.

Looking for further possible meanings, we find the var. dbn

reading. Its sounding could be iDő-Be-N or aDó-Ba-N ,<idœbεn or adóban> (in time or in

tax) Are these agglutinated forms plausible? The dilemma is caused by the abundance of reading-possibilities!

Keeping with the original reading of ‘hajtani’ we come to the reading of aDó-Ba-N

(be)HAJTANI (collect in taxes) and support with this our earlier statement, that the so called DETs have their own sound-value and the signs before it (with little exception) only tell something about it, but not being identical with its consonantal frame.

In this case, our reading could be: “csirádból, csűrödből, mocsaradból (adóban) behajtani…”.

The signs (7-8) occur in the lesson 41 as well – see later – but with numeral 9 in the given circle. The first 8 signs of lesson 42 are identical with lesson 43, the change

happens after sign No. 13. The title is followed by the sign of the number ten. There is no circle in the title of lesson 50, it appears first here, later as a graphic, with the numeral 9 inside, in the calculation.

Seeing all these, the intention of our scribe is clear: he rendered the circle perceptible with his graphic and did not write the sign of the Sun, the very similar N5 hieroglyph.

A new sentence begins here: “Körön megad…” <kœrœn mεgad> (indicate on circle).

The scientific literature sees here the measurement “royal cubit” instead of reading ‘megad/kitesz’ (‘give’ or ‘indicate’). We note that our scribe could have drawn his usual circle as in lesson 41 even with the numeral 9 inside, but instead – for going sure – he

wrote it with “letters” for he is going to calculate his circle again the usual way. This was necessary, because in the lesson before (42) he introduced the surface-calculation of a 10 unit-diameter circle with the method of the “royal way”. In lesson 44 however, we are

able to follow the calculation of the cubic capacity of a granary with a square base. Here he drew a square with thick lines at the beginning of the lesson, for not to have any misunderstanding.

We note here that our scribe wrote the first 11 signs of lesson 41, the first 7 signs of lesson 42 and the first 8 signs of the lesson 43 with red color emphasizing them this way.3

2 We analyzed it in greater detail in the chapter of methodic.

3 We tell about this in more details in the next chapters.

Summarizing, our scribe telling us with the signs (11-15) that he wants to approach this circle’s calculation with the method of nine: “9, ami sok kar ív” (9, which are many arches).

Here we should make a detour around Gardiner’s sign A28. In our text the sign (13)

“ ḳȝ ‘be high”. The holding of the arms is very similar to that of the sign D28.: kȝ. Its

transliteration happens with the other ‘k’, but its official reading is still K-A. The third

example is the hieroglyph of the ‘bika’ (bull). Gardiner transliterates his sign E1 with kȝ as well.

There is something wrong. Too many signs have the same sounding. The signs A28 and D28 - with the arms held up – may symbolize the bull’s horn and

even we can meet the second half of our word ‘bika’ the sounds ‘K-A‘ by their transliteration. The word ‘bi-ka’ itself, though its picture remained, changed to ‘ka’ by its

pronunciation. Where did the sounds ‘bi(k)’, the first half of the word disappear?

To answer this question, look at Gardiner’s sign G6 bik, with ‘falcon’

transliteration. The names of the two holy animals were at the beginning probably similar sounding and during the harmonisation procedure following the unification of upper and

lower Egypt, Horus, the ‘bika’ became the “winner”. 4 However, the original ‘bika’ shouldn’t complain either. As to a God, man could direct its soul (Ká) to it, according to

Gardiner’s sign D28 ‘soul’.

Note that we needed the Hungarian name of this animal, our word bi-ka, in which the accent could slip from the first to the second syllable. We still keep the whole form in our language. The name ‘cow’ in English is probably not random either.

The scientific literature states that the circle and the numeral 9 written in there means that the diameter of that circle is 9 (unit?).

In reality, neither in the lesson 42 nor in 43 is the diameter of the circle to be

calculated 9 (units?). Not even, if the scientific literature wants us strongly to believe it in lesson 43.

Look again Gardiner’s A28 sign: ḳȝ(i) ‘be high’, means according to him “being

tall”. This is the reason, why scientists call the number standing before it as the first characteristic of the circle to be calculated.

Logically, the next number should be the diameter. The transcription of the sign (18,

135) is not uniform in the literature. Eisenlohr transcribed Gardiner’s N9 sign: as psdn,

based on Möller’s collection volume I-sign 303 and volume II-sign 573. But Peet sees

there the sign 492 in Müller’s volume I, which is identical with Gardiner’s W 10 i‛b

meaning a ‘cup’ or ‘basket’. We see in reality a small circle with a short horizontal line in there. Gardiner’s N9 is

still picturing the diameter, thus Eisenlohr’s mistake is understandable. He did not see that our scribe drew only a partial line into the circle. The variant of Peet intended to support his desired reading and he failed to notice the possibility that this sign is not a

4 See further in chapter II, Some Useful knolledge of matter.

hieroglyph and it may picture one attribute of the circle. Anyway, the picture points to this probability.

The psdn reading makes different pronunciations possible. Among those the

variation with the vowel ö (œ) changes the second consonant sounding a cs (ch):’p’csödön’. Accepting a [p <> f] change, the reading will be ‘f’szodon’. Both

expressions are used today in Hungarian for “prick” in some dialects. It could have been

a tool, a rod, a stick, on which one could mark measures of length. It stands in our lesson for the radius, not for the diameter. Fixing one end of a stick in the middle of a circle, then moving the other end around, it will describe a Dom-shaped geometrical form, the cupola-formed top or roof of a granary. The reading in this lesson: ’p’csödön’ sok ív <pchdn shok ív> or ‘f’szodon’ sok ív (many arches on your stick) means the

volume of the cupola of a granary.

The word ‘ív’ means a part of the [word öv <œv> (girdle)] the whole circle. The following signs banish any doubt about the form to be calculated. It is a ‘pajta tár’

<payta taar> (barn, granary, depot). Eisenlohr took Möller’s sign No 270 mistakenly over,

but it is visibly missing its sole. The right transcription is Möller’s 271 and Gardiner’s sign

M6 tr ‚reason’. The sounding is not simple for us, there are too many possibilities: ‘tar, tár, tér, tor, túr, tűr’, but it is not standing alone. The decision is made by the word before

it: ‘pajta’ (barn). However we note that after a [p <> f] change the word ‘fajta’ (variety, type) could be read as well and give us a little different but acceptable meaning.

The signs (26-31) belong together. We can read the (26) ‘h’ sound elsewhere as an ID ‘ház’ (house). Looking at its picture-value, despite the sign ‘pr’, it means ‘be’ (into) as

well, thus we can give a direction to its meaning. The reading is clear “ha (be)rakatjuk..”<ha bεrakatyuk> (if we let stuck into, ….if we let ensilage).

First line, second segment, II/2a and II/2/b.

The mirror-picture of the transliteration:

2/a

32-33 34 35 36 37-38 39-40 41 42 43-44

r - f m mr bd-k ír-t ti j ḫpr

íR-Va Mi Mé-Rő GaboNa SzáM-íT úTi Ja KaP-oL

…írva (hogy) mennyi a mérő gabona. Számítás útja: Kapol…

< iirva (hodj) mεnnji a merœ gabona: saamiitaash uutya: kapol…> (..it is written, how many mérő is the grain. The way of calculation: you get..)

2/b 43-44 45-46 47-48 49 50 51 52-53 54 55 56- 57 58

ḫpr ḫ ḫ-r k l ḫnt t-n 9 ḏȝ t-k 8

KaP-o-L * Ke-Re – K l ** meNe-T 9 veGYÉ-TeK 8

Kapol a kör osztásnál(*) kerek 1-t, kiöntő menet, 9-ből vegyétek, az 8…

<kapol a kœr ostaashnaal kεrεk 1-t, kiœntœ mεnεt, 9-bœl vεdjetεk, az 8>

you get by dividing the circle: round 1, (moulding course) take it from 9, get 8

Notes:

Our word írva/róva5 is clearly readable however, it could mean in this case ‘számolni’ (count) as well. A further sounding could be ‘(be)érve’ (ripened). The following subordinate-clause needs further attention. The transcription of the sign (35) is wrong in

Eisenlohr’s and Peet’s writing as well. They both transcribed Müller’s sign No 520

instead of 509. Using the right sign by Möller’s classification, we got Gardiner’s W 19 “ mhr, mr, mí (old mr) milk jug” The meaning ‘gabona’ (grain) or ‘bödön’ <bœdœn> (tub,

bucket) of the signs (37-38) – see Gardiner’s U9 sign – needs further checking: It’s well possible that the first sign means mérő/méret/merő <merœ/merεt/ mεrœ> (measurer/

measure/ scoop) see (35-36) and the second part (37-38) with the plural sign below it stands for corn/grain.

The sign (41) is very similar to the sign (35), but some differences are present. It

resembles mostly Möller’s 401 sign, which is identical with Gardiner’s U33: phon ti

pestle. Should we accept the identity of the signs (35 and 41) then we will come to the

reading of “számít” (count) or “méri” (he measures). We read the signs (43-44), after Peet’s palaeographic correction as KaP-oL (you

get). A hieroglyph is missing between the signs (45-46). (The here missing ‘b’ sign is

regularly written even in the lesson 42, in the company of others.) We assume that the

correct way of writing could have been here as well. Its reading: KöR-Be <kœrbε>

(in circle) TÖRJÜK <tœryuek> / VÁGJUK / OSZTJUK (we-break/cut/divide). We could read – interestingly – the first measurement-unit on this papyrus by the possibility to supplement sign (45) with the sign of the foot as it is written in the previous two lectures:

KŐ-LÁB (?), or KÖR-LÁB (?) <kœlaab or kœrlaab> (stone-foot or circle-foot). The signs (47-49) build an interesting group. It possibly plays here an adjective /

adverbial roll, contrary to the custom of MMP. The sign (47) will be pronounced here as ‘k’ and the whole group as our word ‘kerek’ <kεrεk> (round).

5 There is only a dialectical difference between the Hungarian ’rov’ and Greek ’graph’, both meant

earlier engraving, scratching. (There the name of the flower rose is coming from, because its thorns are rós=scratchy) The Hungarian variant was used in Egypt.

It is further interesting that reading together the signs (43, 44, 47, 48, 49) by letting out the signs (45-46), we arrive at our known word: KaP-o-L-o-K or KaP-oL É-L-eK (I get or you get, I am alive). It’s better understandable after the reading of ‘szem’ (eye): SzeM-É-L-eK→ számolok (I am counting). (The verb ‘él’, ‘élek’ (live, I am living) stands in the

background. [We still use the expression: ‘élek a lehetőséggel’ <elεk a lεhεtœsheggεl> (I live with the possibility)]. We understand the direct reading of the word ‘kerek’ (round)

better after this explanation. Summing up. The adjective/adverbial roll of the signs (47-49) in this lesson is well

acceptable despite of their earlier experienced verb-attribute. Our reading is therefore:

KaP-oL KöR-(Be)-Osztásnál Ke-Re-K 1-t <kapol kœrbεostaashaanaal kεrεk one> (you get by circle-dividing round one). It is well understandable that we get round 1 by dividing the circle (9/9=1).

The signs (51-53) do not belong directly to the course of calculations. Our scribe interjected a remark here. Its Hungarian reading is: Ki-öN-Tő meN-eT <kiœntœ mεnεt> (moulding-course). We read our word Ö-N-TŐ <œntœ> (pouring, moulding) while

accepting the ó/ő reading of the hieroglyph . Even its picture-value points to this

meaning. We concluded earlier that the “moulding-course” meant a certain form of the surface-calculation of the circle.6

It is worth to deal with this problem in details. The signs W17 and W18 stand in Gardiner’s collection with the following

transliterations: “ │ water-pots in a rack (Dyn. XII-XVIII). Ideo. in 2 ḫntW ‘racks

for water-pots’. Hence phon. ḫnt, ex. var. ḫnt’ in front of’ (§174) and derivates.7 First, it is noticeable that Gardiner supplies the age of this sign. He means

that his transliteration is right from the Dynasty XII on. What could have been the

transliteration of this sign before his given date? Reading the sign as ‚k’ or ‚kh’, then the frame would be K-N-T, when reading it in Hungarian. Pronouncing it with the general

vowel ‘e’ of our scientific literature, the reading would be KeNeT <kεnεt> (ointment) but we see kannákat (cans)! K-NN-T → KaNNáT (can). Even two words could hide in this

notion and we come to the reading of Ki-öN-T <kiœnt> (pours out, moulds). Its picture-

value strengthens our assumption and according to Gardiner, it is placed in a kind of ‘rack’, thus a frame as K-(R)-N-T could be accepted as well. Going this way, the following readings are possible: KeReT – öN-T, KöRé-öN-T <kœreœnt> (pours around

it), KöRüL-úT <kœruelút> (circular way), KeRüLő-úT <kεruelœ-út> (detour, circuitous route). All these readings are possible using Gardiner’s original transliteration. The question is, how good are these transcriptions?

Could it be possible that instead of or beside the sound ‘n’ even the sound ‘r’ was in use before the time of Dynasty XII?8 In this case ‘kerülő út’ <kεruelœ út> (circuitous

route) tells the exact description of this mathematical procedure: a different, a detour-way of the circle’s calculation.9 We ask even today “mibe kerül?” <mibε kεruel?> (how much it cost?) It may have sounded in the Nile-valley: ‘mennyi a körül út’? <mεnnji a

kœruel-út?> (how much is the circuitous route?) We go even one step further, can we

call this mathematical procedure the ‘KeRáL-úT’ → Király-út <kiraay-út> (Royal-way)?.

6 See the lesson No. 10 of MMP.

7 Gardiner, op. Cit. Sign-list, 529.

8 Zonnhoven, Middel-Egyptisch Gramatica. Een practische Inleiding in egyptische taal en het hierogliefenschrift

(Leiden, 1992),9. 9 See table V in the Appendix.

It is possible. As we are going to see it later, our late teachers choose this way to solve the relation of the circle and the quadrate.

Our words ‘menet’ (way, route, procedure) and ‘vegyétek’ (take it!) have been

explained before. The reading ‘vegyétek’ <vεdjetεk> (take it!) could be divided into the signs ‘V-Gy’ and

‘T-K’. The sound ‘k’ is in this case the sign for plural and supposed to be placed after

nouns. The right reading supposed to be ‘VéGY-éTeK’ <vεdjetεk> (take it). Second line, first segment: III/3a and III/3/b.

The mirror-picture of the transliteration:

3/a 59 60 61 62 63 64 65 66 67-68 69 70 71 72-73-74

sk ḥ – tp m 8 ir ḫ r k 1/3 f ḥr r SzűK-Ha-SéG FEJ aMi 8 SzáM –O-L – oK 1/3 íV Arc-L

Szükséges a fej (gondolkozz): ami 8, számolok 1/3 ívet, arcegységű… <sueksegεsh a fεy (gondolkozz): ami 8, saamolok 1/3 ívεt, arczεdjsegue…>

(head needed [think about it!] that is 8, I am calculating (8x)1/3 bow (sheet),

face-unit

3/b

72-74 75 76-77 78-79 f m 10 2/3 84 85 86-88 89 90

…arcegységű ív, kapol ívet, ami 10 egész 2/3. Szükséges a fej: ami 10…

<arczεdjshegue iiv, kapol iivεt, ami 10 εges 2/3. Suekshegεsh a fey: ami 10>

face-unit surface , you get 10 +1/3 (added to 8)

Notes: The group of signs (59-63), which we read as “szükséges a fej” (head needed, think!) build a closed unit. Our scribe rectified them even afterwards in the case of the signs (86-88). Möller’s signs No 399 transliterated as šk are identical with Gardiner’s V29

sign: ‘swab’ or ‘wipe’. FON ‘śk’, in Hungarian it is sounding Z-K or S-K. We choose the vowel ‘ű’ (ue) out of numerous fill-in possibilities for this consonantal frame: SZ-ű-K. We do not have to explain the signs (60) ‘ḥ’, the reading of (61) ‘sok’ (-sek, -ség) <shok,

shεk, sheg> neither the sign of the face with ID.

Reading together: ‘SzüK-Ha-SoK a FEJ’ → ‘ha szükséges a fej’ <ha suekshegεsh a

fεy> (if head needed). Seeing the sign of the head we could think of our word ‘gondolkozz!’ (think!) either.

The hieratic value of the sign (59) is a kind of human-form. The sign (60) is different, it certainly doesn’t picture a human. The difference is smaller at their hieroglyphs. The brioche form of the sign ‘h’ (some people mean that it is the sign of a fish) is well visible

in the sign (59) as well, but man built a doubled thick bow from the first loop. It is possible, considering our reading, that this sign could mean even “ész” (brain) as well. The head-part (skull) of the symbolic body rules the picture. This would better explain the

large difference of the two hieratic signs. Well, if we read this sign as ‘ész’ (brain), than the whole group could be read as ‘eSZüK Ha MAGYAR FEJ’ that is : ha ‘magyarul gondolkozunk’ (if we think Hungarian). We note as interesting that the consonantal

frames of our basic word-roots can be filled in bidirectionally with vowels. This way we can read ‘ész’ and ‘szűk’ (brain and narrow, restricted) out of this group of signs.

Discussing the signs (67-69) in the previous line, we could identify them as our adjective ‘kerek’. The reading is in this case: ‘szem kerek’ → ‘szám kerek’ (eye round – number round) with the meaning: ‘számolok’. Our word ‘szemez’ means ogle and

inoculate as well, but szemel means counting one by one. Szem (eye) and szám

(number) have the same consonantal frame and the two words can often be exchanged in sentences.

We have seen earlier that the ‘ív’ (arch, bow) means a piece, a part of. The ‘öv’ <œv>

(girdle) can mean the whole circle.10 The ‘arc egység’ <arcz εdjsheg> (face-unit) is identical with ‘felszín egység’ <fεlszin

εdjsheg> (surface unit). We see the writing KaP-oL-KéL as a new variant of the reading

of ‘kapol’. We note that the signs (82-83) are painted red in the original papyrus.11 It is to be remarked that the mother-tongue of our scribe probably wasn’t the

language he was writing the lessons in. He most likely only learned his ancestor’s language but did not speak it and this say the Egyptologists as well. This could be an explanation for the often wrong placement of the otherwise well chosen suffixes.

Second line, second segment: IV/4

The mirror-picture of the transliteration:

IV/4

89 90 91 92-93-94 95 96 97-98 99-100 101 102 103 104 105 106 107 108

m 10 2/3 s-p-rȝ 10 2/3 ḫpr-r ḫ-r f m 100 10 3 2/3 1/9 ir

10

See in chapter X/D. 4.about meter 11

See picture 4.

aMi 10 2/3 Sza-Po Ra 10 2/3 KaP-oL Kö-R ív aMi 100 10 3 2/3 1/9 SzáM.

..ami 10 egész 2/3, (azt) szaporít 10 egész 2/3-dal, kapol kör-ívet, ami 113 egész 2/3 és 1/9-es szám

10 + 1/3 multiply by 10 + 1/3 and you get circle-surface of 113 + 2/3 + 1/9 (number)

Notes: The reading of the signs (97-101) could be KaPoL KöR-íV in this case, it deserves attention. Our scribe finished with this the basic-surface-calculation of a form with 6 unit radius.

See: 6 X 6 X 3,1605 = 113,78 = 113+2/3+1/9. From here on, he started to calculate the cubic capacity of a spherical form. By the way, the gist of this calculation is still not understood by the Egyptologists.12

Third line, first segment: V/5a and V/5b

The mirror-picture of the transliteration:

5/a 109 110 111-113 114 115 116 117 118 119 120 121-123 124-125

SzűK-Ha-SoK FEJ aMi 100 10 3 2/3 1/9 r s-p-w r‛ -

Szükséges a fej: ami 113 egész 2/3 és 1/9 rá/el-szaporít… <sueksegεsh a fεy: ami 113 εges 2/3 és 1/9 raa/εl saporit>

Think about it! 113 + 2/3 + 1/9 multiply

5/b 124-125 126 127 128-130 131-132 133 134-136 137 138-139 140

r‛- 4 2/3 p-w-n mḥ‛d 6 n-t – j m psdn - ḫpr

Rá-K 4 2/3 B-U-N MeG-aD 6 Ne T-Je aMi P’CSDN KAP

…szaporít 4-t (ez) 2/3-ban a megad(ott) 6 (me)netje, ami póznádon (ív?), kap… <saporit 4-t (z) 2/3-ban a mεgadott 6 mεnεtye, ami poznaadon (iy), kap>

…multiply by 4 (this is 2/3 of 6 given on your shaft) you get

12

We will say more about this later while discussing the Kahun-fragment.

Notes:

The signs (109-125) did not open a new aspect compared to the previous ones. Reading the sign (120) as an ‘l’, we are going to get our word ‘el-szaporít’ <elsaporiit>

(multiply). The agglutination of the word ‘szaporít’ is wrong in the papyrus.. We find the suffix ‘-rák’ instead of ‘-rít’ at the signs (124-125). The scribe may have used

(unnecessarily) the plural form in connection with the number 4. But it is possible that the signs (124-125) should be read as KÖR-öK <kœrœk> (circles) instead of RÁ-K while looking at the form of the hieroglyph (124).

The signs (128-130) gave us a new variant of the usual reading of B-A-N. The signs (131-132) entitle only to the reading of ‘megad’ (gives), we put the suffix signalling the

completed-participle after it (-ott) only for better understanding. The unit ‘cubit’ being

read here by the scientific literature belongs to the serious mistakes. We dealt with the meaning of the word ‘menet’ (course) before. We see here its form supplied with possessive ending: ‘(m)enet-je’ (his/its course). It is a phonetic writing! It is clearly

understandable in this sentence that the number 4 is in reality the 2/3 of the given 6, which we know as the only given data, the radius. See: “6 (m)enetje, ami póznádon 13 (ív)” (6 course – unit? - on your shaft). We want to point to the fact that the (139) hieratic

sign does not signal the diameter of a circle. Its form with a short line points rather to the radius of the circle and we transcribed it this way. The sign of the ‘ív’ was not written this time.

Line three, second segment: VI/6a and VI/6b

The mirror-picture of the transliteration:

6/a

137 138-139 140 141-142 143 144 145 146 147 148 149

m psdn sk ḫpr ḫ r f m 400 50 5 1/9 ḫ

aMi PózNáDoN SOK KaP-O-L íV aMi 400 50 5 1/9 k

..ami póznádon sok, kapol ívet, ami 455 egész 1/9-es… <ami poznaadon shok, kapol iivet, ami 455 εges 1/9-εs>

big amount on your shaft, you get result 455 “ 1/9

13

Pózna (pole), could have been a rod with measurement-units carved on it.

6/b ?

149 150 151-152 153 154 155-156 157 158 159 160-161 162 163 ḫ r t - - f p w m ḫȝ r w m -

K Ra TaSoKaK íV B –W aMi Csú KaR áR U * K

…(ra-k-ta-ság-ok) eredmény. Ívében, ami karcsú áru udvarok. <rakottságok, εrεdmeny, ami karchu aaru udvarok>

Filling, result. In its size it is a “slender” granary. Notes:

We can read the signs (140-142) here regularly as KaP-O-L. The sign (149) had a side-slip. It should stay under the sign (150) as in the row. According to scientific literature, the three perpendicular strokes mark not only the

plural at this time of history (Hyksos?), but those could have been used as a hyphen. A new part, a new sentence starts after them. Following this rule, our reading could be: R-Kh-T SoK/SáG → Ra-Ka-T-SOK → Ra-Ka-T-SáG or simply RaK-ha-T. We say today: ‘rakat’ or ‘raktár’ (lets store or storage). In reality, we can still read the sign (153) as a ‘k’, the Hungarian suffix of plural. Thus our reading is ‘rakottságok’ meaning ‘quantity’

(amount, load) or ‘result’.

We should stop here for a moment.

Our scribe finished here the calculation of the granary’s dimension. He writes that ‘ami 455 egész 1/9 rakottság’ (455 1/9 is the saturation-point), the result. We offer the following calculation to control his data: If we look at the granary as a regular semi-globe with a radius of 6 units, then calculating by our knowledge of today:

Vsemiglobe = 2/3 x r3π = 2/3 x r d2 (8/9)2 = 2/3 x 6 x 144 x 64/81 = 455,11 = 455 1/9

(We used the π-value of 3,1605 as it was calculated in ancient Egypt). We must state

that the ancient calculation-procedure, even by considering the value of the π-difference, was exact to a hair’s breadth..

However, our scientific literature didn’t recognize all this.

The signs (149-163) were written with red ink on the original papyrus, thus our scribe

laid special emphasis on those among the black-colored signs. They were very important for him.

Indeed, he describes his result more precisely in the following.

We read the signs (154-156) as íVé-B-EN (in its arch, meaning surface, sheet here). [The reading övében <œvebεn> (in its girdle) could be acceptable as well.] The sign (157) is seemingly the sound ‘m’, but it could mean a number too: 1/32.

Man says that the sign (158) supposed to be a short-form of the bird . Möller does not know such a hieratic sign. Reading it together with the transliteration of the fish-sign, it could give words(?) like AR-HAR, HAR-AR (gibberish) or in combination AL-KaR

(forearm), but even HAL-ÁL (death). All these are side-tracking for us.

However, Möller’s sign No. 324 fits into this text (see the Westcar, Sinuhe- columns

of his collection). This sign is identical with Gardiner’s N23-sign and we already

discussed earlier its reading. For us it means now the starting-sound ‘cs’ (ch) of the expression: ‘csatornázott terület’ <chatornaazott tεruelεt> (canal-drained area). Thus, we come to two different readings:

HaR-CSa <harcha> (wells, silure) or KaR-Csú <karchú> (slim, gracile) We can read our word ‘harcsa’ (silure) and its consonantal frame carries its attribute ‘karcsú’ (slender) as well. Reading it backward: Csu-Ka (R) (pike) a slender predatory

fish as well. The general reading of the signs (160-161) is at least doubtful. Our scientific literature reads them instead of W-W as R-W, which supposed to be some measurement-unit and to be read as ‘ró’. We pronounce them in Hungarian as ‘árú’ <aarú> (article, goods).

According to Möller, the hieratic sign (162) ȝ and the sign (182) do have

the same hieroglyphic transcription into: dHr→uD-VaR.14 Both signs appear in the

same hieratic surrounding, they are exchangeable. After all these, we think that the signs (154-163) made the results more precise. The

form of the granary, which was named at the beginning of the calculation, was not identical with the calculated semi-globe. Our scribe corrects here the difference. (Parabolic?). The result, the volume of 455 1/9, is that of a (by him corrected)

“karcsúsított áruudvar”: an “attenuated granary”. A granary formed like a rick-shaped oven, he calls it a slim semi-globe. This was the reason for highlighting those signs.15 Line four, first segment: VII/7a and VII/7b

The mirror-picture of the transcription:

7/a 164 165 166-167 168 169 170 171-72 173-175 176 177

gm m ḫ- r k 1/20 n r ḫ t - - f m

KéM Mi Ke-Re - K 1/20 Nyi Ra–Ka–T-SáGoK íV aMi

Kém(lel), mi (mennyi) kerek 1/20-nyi rakottságok íve, ami…. <kemlεl, mi (mεnnyi) a kεrεk 1/20-nyi rakottsaagok íve, ami>

Spy out, how much is the round 1/20th of the loading of

14

Möller, op.cit. page15, line 166. 15

We present a detailed description of this form in the chapter named ’Kahun fragment’.

7/b 176 177 178 179 180-81 182-3 184–187 188 189 190 191 192 193 194 195

f m * ḫȝ r w dḥr-ḫpr - - f m 20 2 ½ ¼ 1/40 5 íV aMi CSú KAR áR-Ú uDVaR-oK, KaP- O-L íV aMi 22 egész ½ ¼ 1/45

…íve, ami karcsú áru-udvarok, kapol (kör)ívet ami 22 egész ½ ¼ 1/45.…

<íve, ami karchu aaru udvarok, kapol (kœrívεt), ami εgész ½ ¼ 1/45>

slender granarys. You get 22+1/2+1/4+1/45

Notes:

Sign (164) is Gardiner’s: G28 „black ibis , phon. gm”. We see In Hungarian a GÉM (heron) and read it as ‘gém’. It means officially ‘to find’, in Hungarian it GÉM-LeL

‘kémlel’ <kemlεl> (peers into), what this bird really does before strikes with its beak. If this sign is not an ID, then we can read it as ‘en-gem’ (me in accusative).

We read the sign (165) as ‘m’; mi? mennyi? (what? How much?). It is no DET in this papyrus. We met the signs (166-168) before. It means the ‘egész’ (integer), the round number.

“Mennyi a kerek huszad része…?” (How much is the round twentieth part of the?) The group of the signs (171-175) could be read with the following meanings: rakottság <rakottshaag> (loading), telitettség <tεliitεttsheg> (saturation), tömeg <tœmεg>

(mass), tartalom (contents).

The question according to the lines above is: How much is the twentieth part of the previously calculated volume? Partcularly:..Kém(lelek), mi a kerek 1/20-nyi rakottságok

íve, ami…? (I am peering into, how much is the 1/20 of the contents.) Well, he stated previously that this volume is not really identical with the contents of the calculated semi-globe. He even called it an “attenuated granary”, which looks rather like a hay-stack. It is

interesting that the hieratic picture of the sign (182) in contrary to sign (162) looks very similar to its hieroglyphic transcription. There are no novelties among the signs (184-195), but a mistake in the calculation of

our scribe. Dividing numbers must have been difficult for him. He could not divide 455 1/9 directly by 20. He first decimated the number and bisected the result. All these he wrote

separately in the next segment of the papyrus. Even there he repeated his mistake. (See our later explanation). Based on this, we can state contrary to our scientific literature that this wasn’t a copy-error, because he would not have repeated the same error again. He calculated. He multiplied two numbers instead of dividing them:

There is 1/9 : 10 = 1/90 right, but 1/90 : 2 = 1/180 and not 1/45 as he wrote. We come back to this later on.

Line 4-5, second segment: VIII/8a, VIII/8b and VIII/8c.

217 216 215-14 213-209 208-206 205-201 200-198 197 196 195 194

The mirror-picture of the transcription:

8/a 194 195 196 197 198 199-201 202-203 204-205 206 1/40 5 h ȝr ȝr t-íw-k p w r f m 1/40 5 BE Ar Ar T-JU-K B U íR Va aMi

…1/45, berakatjuk beírva, ennyi…

<1/45, bεrakatyuk bεirva, εnnyi..> 1/45, we let load and register

8/b 206 207-8 209-10 211 212-13 214 215 216 217 m mr – bdt-k hk k t bdt m tr ír

…ennyi mérő gabona a hektó bödön, ami telít. <εnnyi merœ gabona a hεkto bœdœn, ami tεlit>

This many hekto-bödön the mérő grain, which saturates

8/c mr – bdt k 20 2 k ½ ¼ ½ 1/32 1/64 ? ½ ¼ 1/30 6

…Mérő gabona 22 bödön (akó) és ½+1/4 …1/2+1/31+1/4; meg ½+1/4+1/36. <merœ gabona 22 bœdœn (ako) es ½ + ¼ …1/2+1/31+1/4; meg ½+1/4+1/36.>

Mérő-grain 22 bödön (akó) + ½ + ¼ ….1/2+1/32+1/64 and ½+1/431/36

Notes:

The last hieratic line is longer than usual; our scribe ‘added’ something to it. We

assume that it did not belong to the standard procedure of the lesson, because these following calculations are not written among the others in the next segment. However, it was important for him; he highlighted it by using red ink. According to its irregular form,

we discuss it in two steps. The signs (196-201) are the repetition of signs discussed before and gave the reading ‘berakatjuk’ <bεrakatyuk> (we fill it into, we store it into). It is obvious that the

hieroglyph of the house does have the reading of ‘be-’ (into) as well. It’s noteworthy that this way of writing signifies not only the agglutination (conjugation) of a verb but its causativeness as well. The signs (202-205) accentuate our earlier reading: ‘beírva’

<bεiirva> (written into). In this case, the prefix ‘be-‘ (‘the pw-variant of the trinomial

sentences’ according to the scientific literature) puts our verb ‘írni – számolni’ (to write –calculate) into a past-perfect time.

The signs (206-217) further complete this seemingly superfluous procedure. Here our scribe summarized the reason of his previous calculations and told us why the figuring out of the twentieth part was necessary. The ‘m’ sound of the sign (206) means the

relative pronoun, perhaps our pointing- or questioning-word, in this case ‘ennyi’ (this much). It tells: that many are the ‘hekto bödöns’ <hεkto bœdœn> of the mérő <merœ> measurer16. grain, which saturates the granary. This sentence needs further explanation: We can read clearly the measure ‘hekto-bödön’, but a measure with this name is not

known to us. Our word ‘hekto’ means today internationally 100, if our transliteration is right. We don’t know the size of a bödön, but it means certainly a real unit of

measurement. The volume of the “slender granary” in this lesson was divided by 20. The result of this procedure is: 455 ‘mérő’ (measurement-unit) / 20 = 22,75 hekto-bödön (?) of grain.

This means that 20 ‘mérő’ of grain were one ‘bödön’ (1250 litres = 1,25 m3)?. These were 20 sacks of grain, well transportable on a medium size boot seen on contemporary pictures.

Let’s look again at the signs (211-213): . The bishop’s- or shepherd’s-crook means according to Gardiner ‘h-ḳ(ȝ)t and the signs (212-213) ‘K-(AR)_aT’’. We could read them as ‘akarat’ (will) as well. The crook however, points to the royal power, thus

the three signs together could mean very well the ‘king’s will’, the ‘royal measure’.

Hat(almi)-aK-(AR)-aT (will of power). Our scribe finished with these the volume-calculations of the “attenuated granary”.

We see in the next segment mainly the repetition of the calculations, only numbers. The lesson however, is not finished here. Our scribe completed his last line with 16 additionally red signs.17

The signs (218-226) are known from preceding discussions. They mean here as well the calculated ‘mérő’ grain expressed in ‘hekto bödön’: 22 ½ + ¼. The following signs are important, for he recognized his earlier mistake and corrected it.

Remember, he divided 455+1/9 by 20 and his result was 22 ½+1/4+1/45, but the right result should have been 22+1/2+1/4+1/180. His correction starts after ¼ with the sign (227). The correction of the following (now missing) fraction-number seemed to be

16

Mérő is a known, but old and now obsolete Hungarian measure of grain, around 62.5 litres, around one sack. 17

See picture No.4.

problematic for him; he followed therefore a different way: he used the known ‘Horus-eye’ fraction-system. (The sum of the Horus-eye’s fractions: ½+¼+1/8+1/16+1/32+1/64 is only 63/64. The missing 1/64-fraction belongs to god Toht (god of tudás <toodaash>

knowledge), according to a legend.) The signs (227-229) represent ½+1/32+1/64 of the Horus-fractions. Stop here again for a moment. In this case the smallest fraction is almost three-times

larger, than what our scribe wanted to write (1/64 ↔ 1/180). Thus, he changed the form as he is used to and multiplied it by 100 = 100/180 = 10/18 = 0,555.18 Now he could write it with Horus-numbers too: ½+1/32+1/64 = 35/64 = 0,547. But this was only an excellent

approach to 10/18 = 0,555. Thus, he wrote the difference (in 100x size) again: ½+1/4+1/36 = 28/36 = 0,778. If we add the 1/00 part of this (0,00778) to his previous number: 0,547 + 0,00778 = 0,555 and receive the right result (10/18).

Now we can state that sign (230), not identified previously, means in our case 1/100 and certainly not ‘2 Ro’ value as imagined in the scientific literature.19 We have to note further that we don’t see the origin of our scribe’s last calculations-

process. We don’t know how he could recognize his inaccuracy by using the Horus-fractions and how could he finally arrive at the right result. We assume that he had tables with ready mathematical calculations.

C. The Hungarian-English reading of lesson No. 43 1. line: Receiving out of your granary. Indicate the procedure 9 on a circle and give 6

on your shaft, which we call a large granary. We load it and write how many mérő-s of grain we can get into it? Process of calculation goes: dividing the circle by 9 (9/9) you get round 1 (this is the circular moulding method). Take it from 9 (subtract), you will get 8.

2. line: Think about it! Take 8 and give 8x1/3 (face-unit) to it and you get 10+2/3. Think again! Multiply 10+2/3 by 10+2/3 and you get a circular surface of 113+1/3+1/9.

3. line Think! You multiply 113+1/3+19 by 4 (which is 2/3 part of 6 which was given on your shaft) and you will get 455+1/9 (mérő) loading (saturation). This we call a

“slender granary”. 4. line: Figure out, how much is the round 1/20 part loading of the slender granary.

You get an amount, which is 22+1/2+1/4+1/45. Recalculated, this is the number of hekto-bödöns out of the (455+1/9) mérő grain, which will fill the granary up.

5. line: (Mérő) grain 22 bödön + ½ + ¼+…1/2 +1/32 +1/64 and (1/100) ½ + ¼ + 1/36. 6-7-8-9. line: Asking scholars about the results.

18

They used only fractions with the numerator one and did not know or did not use 5/9 in calculations. We could write much more about their calculation-method, but here we just have to note that the sign for the numerator unit was that of the mouth. Its ancient reading was ‘l’: ‘éLő’ <elœ> (the living), the wholeness, the size and the one God as in Hungarian. (See the Aramaic Eli, Elu and Ilah/Allah in Arabic). 19

This sign has been used to express 1/1000 in the lesson 69 of RMP.

D. Summing up the calculations

First segment:

First column, 6. line:

235-236

237

238

239

240-241-242

243-244

The hieroglyphs from the left to right, and grouped horizontally:

235 236 237 238 239 240 241 242 243 244

ḳȝȝ I ki - n s m hi t -

K-éR I/J TUTó SOK oN Szá - M - Hí - Tá SOK Kéri tudósokon számítások(at)

<keri tudóshokon saamitaashokat> Asking scholars about the results

Notes:

We can find the text of the vertically written first column (235-244), occasionally only the second half (240-244) of it, in several lessons of RMP. It contains the signs of three words: (235-36), (237-39) and (240-244). The last word stands alone in the lessons 44

and 46, while the whole text stands in lesson (42) as well. We note that the transcription is not unified in the scientific literature. Eisenlohr

transcribed the sign (242) into Gardiner’s D53 sign but Peet preferred Gardiner’s

T31 sign for the transcription. To make a decision, we called Möller’s 96th and 441th

lines for help. We go for Eisenlohr’s decision having compared the signs. The transcription of the other signs is identical at both authors.

The Hungarian reading sounds clearly as ‘szám-hí-tá-sok’ → ‘számítások’

<saamiitaashok> (calculations). Indeed, our scribe repeats in the followings the calculation-procedures of the text-exercise.

Reading the sounds ‘sz’, ‘m’ and ‘t’ is certain. Gardiner’s sign D53 “ phallus with

liquid issuing from it“ has at least 6 different transliterations while it appears in diverse surroundings. For us is the right transliteration ‘hí(m)’ (male) and our otherwise properly written word suspects the sound ‘(h)Í’ .

The sign (238, 243) has the known transliteration of mḏȝt and as we told before, it

means since the time of New Kingdom our word ‘sok’ <shok> (many) or the suffix ‘-ság/-

ség’. Thus, our reading ‘számítások’ (calculations) is right semantically as well as morphologically.

The following (235-36) signs offer some surprises. The sign (235) , Gardiner’s N29

has the transliteration of ‘ḳȝȝ’. Its Hungarian reading is K-AR-AR, pronounced softly K-

AR-AL. According to the scientific literature, the reading should be K-I as a phone. The sign (236) means ‘i/j’’. Read together: K-AR-AL-I → királyi (royal). Both signs would

officially cover the vowels of the following standing mummy. Thus, Gardiner’s A53 sign is transliterated as ‚ki’ based on the signs before it. The signs (235-36) are officially the

phonetic helpers of the DET A35. Otherwise, Gardiner’s A53 should be transliterated differently in other surrounding: ‘wi’, ‘twt’ and ‘ḫprw’. The last two transliterations should mean stature, form and sculpture.

We note here again that it is unlikely to refer several different definitions of sense and

morphology to the same sign. Thus, Gardiner’s phonetic definitions can not be exact in every case. We are convinced that every sign has its own meaning, therefore its own phonetic form. The hieroglyphs around a ‘DET” are not its phonetic determinants. They

are independent words and in most cases adverbs or attributes of the assumed “DET”. With other words, the variants in Gardener’s “Sign-list” mostly don’t carry the articulation of the particular sign. Otherwise, we can often meet the right phonemes as

well. We see as our further job to review and even correct the wrong transliterations, if necessary. Back to Gardiner’s D53, we assume that this has as well its own sense and based on

its picture, it could mean ‘dead’, ‘body’, ‘mummy’ or ‘ancestor’. At the same time it could

be the symbol of the one “scholar pharaoh”. Officially, king Thut was responsible for the introduction of writing, counting and general science along the Nile valley. Several successors, probably out of respect of him, have chosen names like Tutmos → Tudó-

Mása = (Tut’s replica), Gardiner says “..statue, likeness, ex. var. 2 twt’ statue“. We choose this second variant. Therefore, we met our word királyi/akarat/kér (royal/will/asking) and then tudósok

<tudoshok> (scholars). The following ‘n’ sound refines the previous reading: ‘tudósokon’

(Számon)- kéri = calling scholars to account, examine.

Reading together: kéri jó ősökön / tudósokon számítások eredményét. <keri yó œshœkœn / tudóshokon saamiitaashok εrεdmenjet>

(asking good ancestors / scholars to account on the result of the calculations)

first mathematical table, reading from the right to the left:

● ← 8 1 x ←

1/3+5 2/3 x

→ 2/3+2 1/3 x ←

2/3+10 =

The course of calculation: 1(x) 8; 2/3 (x8), that is 5 1/3; 1/3 (x8) 2 2/3; it becomes magyar (true): 10 + 2/3. We should add the first and third line: 8 + 2 2/3 = 10 2/3.20

Second mathematical table:

● 2/3+10 1 x

2/3+6+100 10 x ←

1/9+7 2/3 x ←

1/9+ 2/3+3+10+100 =

20

One of the methods calculating circle’s surface without the π, used by our scribe, was the ‘körbeöntő menet’, the ‘moulding course’. diameter divided by 9, multip lied by 8 and the result squared. The diameter in this lesson is 2x6 = 12 unit, 1/3 larger then 9, thus his formula changed: 8+ 8 times 1/3 = 10+2/3 had to be squared. (See more about it in the next chapter 7-th paragraph.)

The course of calculation: 1(x) 10+2/3; 10 (x) 10+2/3 (=) 106+2/3; 2/3 (10+2/3) (=) 7+1/9; It becomes magyar (true) = 113+2/3+1/9. The result of the second and third line has to

be added (are marked with an aslant line in the hieratic and with an arrow in the transliteration): 106+2/3 and 7+1/9 = 113+2/3+1/9.

Note: We don’t have any dates about the calculation of the partial result. It’s regarded as a big accomplishment for example, to conduct the multiplication of 2/3 x 10+2/3 = 7+1/9

(2/3x10 = 20/3; 2/3x2/3 = 4/9; 20/3+4/9 = 64/9 = 7+1/9). 2. Second segment:

The third mathematical table:

● 1/9+2/3+3+10+100 1x

●● 1/18+1/2+7+ 20+200 2x

1/9+5+50+400 4x

The course of calculation: 1x 100+10+3+2/3+1/9; 2x (113+2/3+1/9) = 227+1/2+1/18;

4x (113+2/3+1/9) = 455+1/9. The fourth mathematical table:

→

● 1/9+5+50+400 1 x

1/90+1/2+5+40 1/10 x

x

1/45+1/4+1/2+2+20 1/20 x

The course of calculation: 1 x (455+1/9); 1/10 x (455+1/9) = 40+5+1/2+1/9; 1/20 x (455+1/9) = 22+1/2+1/4+1/45. We repeat again that our scribe made a mistake here at the last division. The right result would be1/180 instead of 1/45.

E. Analysis of the lesson’s contents

This is the most outstanding among all mathematical lessons in this papyrus. Our

scribe, (the smart, born as an image of Árpád), alias Ahmes, demonstrated for us the calculation of the semi-globe’s volume at the time of the pharaohs. He used for this purpose a practical frame-story.

In the lesson 10 of the MMP, highly regarded by the scientific literature and by

mathematicians and physicians21 as well for calculating the surface of a semi-globe. However, the scribe calculated in reality (according to our reading) the surface of a simple basket-form cylinder.22

This time, the topic is really the volume of a semi-globe. Our scribe – by the scientific literature degraded to a simple copier – dazzles us by the use of the radius as a novelty of his time and by calculating the circle’s surface without the π, using the royal-

procedure, the moulding –method. The title: “Csűrödből kapni” <csuerœdbœl kapni> ([the amount] getting out of your

granary) tells everything about the lesson. This is one of the best examples of the compactness of the Egyptian, but especially of his language.

As a good help, he wrote with red ink his most important messages. He defines, just

after the title, the form of the granary. He is not talkative, while telling that the granary’s base is formed as a circle: “körön megad 9-t, ami sok körív” (gives 9 on a circle, which means many sheets). He determined with this few signs his calculation-method: the

circle-shaped granary’s base should be calculated by the method of 9. Be aware of, this number, it is not the value of the diameter or the height. This is simply the name of the procedure supposed to be followed. He tells us the size of the granary by giving the next

data. He writes that the radius is 6 units long: “…6, ami póznádon sokívű pajta-tár” <6 ami pooznaadon sokiivue paytataar> ( 6 (units) on your pole, which is a granary with many sheets). If you take a 6-units large shaft and draw many arches from one spot to

all directions, you get in reality a semi-globe with the radius of 6. Having said these, he puts up the question, how much grain could we load into this

space? Of course, we have to count and register the number of loads during the

procedure. “Berakatjuk megszámolva, mennyi a mérő gabona” <bεrakatyuk mεgsaamolva, mεnyi a merœ gabona?> (we let load it and count, how many mérő is the

grain?)

Until there, the aim of the exercise has been determined. Now he turns directly to the calculation. He introduces the ‘köré-öntő’ <kœre œntœ> (moulding around) or ‘király

menetet’ <kiraay mεnεtεt> (royal-procedure). We explained it in details during the

discussions previously. One subtraction and a multiplication are sufficient to calculate a circle’s surface instead of several multiplications and divisions. However, this lesson put before him a bigger problem. He did not have to calculate with the ‘basic-circle’ of 9 unit.

The diameter of the granary’s base was 2x 6 = 12 units. This again was 1/3, by 3 units (fejméret = face-unit) larger as the usual ‘basic circle of 9’. Thus, with his second step, he increased by 1/3 the number 8 (nyúló-szám= number to be streched) waiting to be

squared. He highlighted his result with red ink: 10+2/3. This was the number to be

21

Sain Márton, Simonyi Károly 22

Borbola, op. cit.

squared. He calculated precisely, the result of the ‘szaporitás’ (multiplication, square) is

perfect: 113+2/3+1/9. With this he already finished the surface-calculation of the given circle. As a control,

let us see the calculation as we do it today. Acircle = r2π. The radius in this lesson is 6 and

the contemporary value of the Egyptian π was: πe = 3,1605. Using these numbers the

result is: Acircle = 36 x 3,1605 = 113,777. Until now, the scientific literature did not recognize this simple fact and our scholars didn’t learn the ancient, but exact calculation

of the circle’s surface either.

This lesson has more surprise in store, because our scribe’s goal wasn’t the circle’s surface, but the volume of a semi-globe built with a radius of 6 units. Surprisingly, he knew the solving-formula for this task as well. We don’t know exactly the deepness of his

knowledge, we see only his calculations and the added text. They probably arrived by observing the solving-procedure of the problems with globe/semi-globe. The question, if they knew the concept of globe or semi-globe, is best answered with this lesson. Back to

the procedure: our scribe multiplied during the next step the surface of the circle by 2/3 of the radius: 113+2/3+1/9 x (6x2/3) = 113+2/3+1/9 x 4 = 455+1/9. He took a deep breath for this step and told that “head is necessary” (you have to think!).

At the next multiplication, our scribe created a new expression using the verbal prefix ‘-rá’ (onto) and distinctly determined the direction of the procedure: ‘rászaporák 4-et’ (go on multiplying 4x), while multiplying the base-surface with the height (455+1/9).

He may have not known his solving-formula in general (he would have written it down for us), but he used it well. Let see again, how we are doing this calculation today:

Vglobe = 4/3 x r3 x π = 4/3 x r x r

2π = 4/3 x r x (d.8/9)2. From this, d2 x (8/9)2 is the surface

of the circle. In our case: Acircle = 12 x 12 x 64/81 = 144 x 64/81 = 113,777. Thus, he did the calculation of the globe in two steps. First, he figured out the surface of the circle and then he multiplied it …..with how much?

More specific: he just wanted to know the volume of a semi-globe, thus, the general formula had to be changed. Vsemi-globe = 2/3 x r x d

2 x (8/9)2 = 2/3 x r x 113+2/3+1/9 = 2/3 X 6 x 113+2/3+1/9 = 4 x 113+2/3+1/9. There we arrived to the expression “rászaporít 4-et”

in the hieratic text. We can state here that our scribe calculated the volume of the semi-globe perfectly.

He finished the calculation with an original expression: ‘..ami 455+1/9 rakottság’.

“Rakottság” = saturation, volume = our result.

The following signs (155-163) help to specify the result and increase the reality-value of the calculations. The form of the granary, named in the title, was possibly not totally a semi-globe, it may have been rather oblong. “Karcsú árúudvarok”. (slender granaries).

However, the semi-globe was his best approach to this “irregular” form. He admits in this sentence that the real figure is more slender built, then the calculated one. The word ‘árúudvarok’ (yards of merchandise) tells the same as our word “Csűr” (granary, barn)

today. Our “smart” accomplished with these the task, which he formulated in the title: he

figured out the volume of the granary.

However, beside of calculating, he had to spy out the 1/20 part of the volume. (We wonder who these pupils may have been. Tax-collectors?)

He says that the granary is somewhat slender and then divides the volume by 20.

The result is almost perfect: 455+1/9 : 20 = 22+1/2+1/4+1/45. We repeat again: dividing 1/9 happened in two steps. First, dividing by 10 = 1/90 and after this he multiplied by 2 instead of dividing. The same mistake happened in the mathematical tablets of the next

segment.

Besides his mistake, he almost perfectly figured out the 20-th part of the volume, the number of the “bödön” <bœdœn>. The finishing thought of the lesson is: “…ennyi mérő

gabona a hekto-bödön, ami telít” <εnnji merœ gabona a hekto-bœdœn, ami tεliit> (these

many “bödön” of the mérő grain are, which overstock /the granary/). The reading of the word ‘hekto’ is not clear. The international transliteration HeK(A)T has been derived from the

signs K(A)T, which are treated as phonetic complements. The remaining sound ‘h’ is coming from the crock-formed sign that however presents power, taxes or wages. We discussed this previously. The ‘bödön’ means a newly recognized unit of volume. Its

form and size have been possibly painted on the wall of Menna’s tomb (See chapter XIV, VI) as a container on a cargo-ship.

Following the calculations we could see that our scribe did a fine job. He figured out almost exactly the volume of the granary as he announced it in the title and additionally calculated the twentieth of the volume. He further worked out the calculations in details .

We can follow and control his procedures done with his mathematical tablets. However, there still remained some open thoughts. We did not find in this lesson any

known measurement-units, but the supposed sign of the cubit. To every form in the nature belongs some kind of physical unit and this is seemingly missing here. We are missing a unit of length, radius and a unit of volume. This on the other hand, did not

influence the mathematical procedures. Our scribe calculated in the same unit during the whole process. In which unit? Sorry to say, but we can’t give an exact answer to this question (yet). The ‘bödön’ could give us some help. It could have been the size of a

truly used container 23 (see picture of “container” boat, table VI) of those times and “mérő” <meroe> (a basket in which people carry grain on their shoulder up to the granary as seen on the pictures).

23

Bödön is called in Hungarian a container of different size and material: out of metal (can), wood (tub) or a kibble. Mérő is an old, now obsolete unit of grain in Hungary, around 62 liters of size.

F. An analysis of some translations in the scientific literature. A selection.

August Eisenlohr wrote the first serious elaboration about RMP’s lesson 43. 24 He

wrote more then usual-long and detailed notes to this exercise, but at the end he left

open the question: which form or even forms were calculated by the scribe. He tried his fantasy and calculations on all different figures, even on a semi-globe, which he called a dead end, because he couldn’t connect it with the number 9. He introduced the units of

‘khar’ and ‘ro’. He changed further the given data (6 and 9) and he tried to explain on pages 106-107 of his book, why the change of the data (diameter against height) was right.

T. Eric Peet delivered the “classic” elaboration of the RMP and his views are still kept as valid today. 25 He starts describing lesson 43 on page 82 of his book with the following words: “This is one of the most difficult problems in the papyrus. It professes to

give a method of finding the content of a regular figure in khar without first working out the volume in cubic cubits and multiplying it by 1 ½ as in Nos. 41 and 42” Further down he tells again the same: “Clearly in the attempt to find the result directly in khar without

finding the volume first in cubic cubits…..” He changes the data which are presumably wrong due to mistakes of the scribe. “It will be seen later that the statement of the problem is incorrect, the 9 cubits are really the diameter and the 6 cubits the height of the cylinder.“

We can state here that the two first scholars, who elaborated the problem of lesson 43, were thinking to deal with an open cylinder-formed container or silo for grain. They corrected the data and determined the diameter as 9 and the height as 6. Peet

compared it with the lesson of the Kahun papyrus. According to him, the circle’s diameter is 9, its surface 64 (see previous lessons). Working with a height of 6, the volume supposed to be: 64 x 6 =384 cubic cubit. He took 1,5 as the rate of exchange

between khar and cubit therefore, his result became: 384 x 1,5 = 576 and wrong. The right result in the papyrus is 455 1/9. Peet tried to explain the reason for this difference: “The most probable explanation seems to be that in the original papyrus the statement of

the problem was correct, but the scribe made a mistake in the first line of working as we have seen. A later scribe seeing in the first line of the working the subtraction of a ninth from 9 just as in Nos 41 and 42, very naturally concluded that this nine must be the

diameter and not the height, and so he transposed the two dimensions in the statement of the problem, his mathematical knowledge not taking him far enough to test the result with the statement in its new form”. It is an interesting supposition that scribes

exchanged those data during several copier-work and based on this mistake the calculations went in wrong direction. Peet states that the mathematical capacity of our scribe was not enough to control

the calculation-procedure. He did not bring up the possibility that the scribe was right, but

24

Eisenlohr, op. cit. 106-7. 25

Peet, op. cit. 82.

the modern-age Egyptologist was not able to follow his calculations. We may be sorry, but the haughty, overconfident opinion is still the accepted ruling explanation. As next, please read Arnold Buffum Chace’s26 relating remarks: “This solution for a

long time baffled the ingenuity of Egyptologists, but the correct interpretation was finally discovered by Schack-Schackenburg (1899, see Peet, page 83). 27 He is further convinced: “In the first place, the papyrus states that the height of the granary is 9 and

the breadth (diameter) 6, and in the solution, when we find 4 as 2/3 of 6, the author again calls 6 the breadth; but the solution is for a cylinder in which 9 is the diameter and 6 the height.”

He basically agrees with the statements of his predecessors. On page 87 of his book, just before the translation, he writes the title of this lesson already as: “Problem 43, A cylindrical granary of diameter 9 and height 6.”

The probe is very simple. Please follow the supposed calculation from above. Going by Peet and take a diameter for 9, the calculation is: 9+(1/3x9) = 12; 12x12 = 144; multiplying by 2/3 of 6 = 4; 144 x 4 = 576. Taking the diameter for 6, then: 6+(1/3x6) = 8;

8x8=64; 64x(2/3x9)= 64x6 = 384. Did we calculate in both cases the result in the papyrus (455+1/9), even really in ‘khar’ ?? Forget it. This inaccuracy is not going well with a mathematic professor.

We note again that the unit ‘khar’ derived by the wrong translation of the signs (158-159). The signs (160-160) suppose to mean the unit ‘ro’ and together with (230) sign meaning ‘2ro’. We read it for ‘karcsú’ (slender) instead of ‘khar’ and we supposed ‘árú’

(goods) instead of ‘ro’. The most probable meaning of the sign (230) is 1/100 or 1/000. A further proof of this are the three fractions standing behind it. Our scribe completed with them the inaccuracy of the value expressed by the “Horus-numbers” he used.

Finally, we note again that the scientific literature based on the elaborations above is not aware of the circle’s attributes used along the Nile and that the surface-calculation was finished after the procedure of the squaring. They did not recognize that the next

step belonged already to the third dimension. They did not learn that the number 9 meant the kind of procedure and not the diameter and didn’t see either the use of the radius. They accused the scribe with incorrectness and lack of knowledge, while they

themselves put a mathematical gibberish onto paper.

G. Summary

We just can agree with T.E.Peet as he tells that the lesson 43 offers the most

complex and difficult problem among the exercises of the RMP. For us, it is the most beautiful. It offers almost every dept of the procedures with circles while it amazes us with volume-calculation of the semi-globe. He offers the “kiöntő” (moulding) method

beside of the usual procedure [Acircle = d2 x (8/9)2]. (See MMP lesson 10.) He even connects both methods while he uses a different diameter. After naming the “procedure 9” he gives as a novelty the size of the radius. Our scribe was a precise man. His writing

and calculations are well to be followed. He recognized his mistake and corrected it. 28 The signs, which are highlighted with red ink, tell important messages, novelties.

The language of the text, as we previously stated, differs somewhat from the

classical language of the Middle-Kingdom. This papyrus-roll seems to be even younger,

26

Chace, op. cit. 27

Schack-Schackenburg was the translator of the Kahun papyrus. 28 The fact that he did not wash or scratch off the simple wrong number from the papyrus, but

corrected at the end accurately as a demonstration, keeps even open the possibility that his blunder was done on teaching-purpose to demonstrate pupils how to correct mistakes. (???)

then the famous MMP in Moscow. There are some differences in agglutination and word-order. It looks like, the scribe learned the old language, but it wasn’t his mother-tongue. It seems, he did not copy it blindly, but put the inherited mathematical procedures into his

new language-surrounding (like a “pidgin-Egyptian”). His language is still sparing, resolute and compact, using many of the archaic simple, but vivid expressions. We are convinced that mathematics is figuratively a common language of humanity.

It is independent from politics or from the influence of diverse syndicates. Mathematics is timeless. Its rules were valid 4000 years ago and are today. The only question is, how much did people recognize from them in different ages. The analyzers of archaic scripts

are looking for texts written in multiple languages, to be able to verify their reading. W e are lucky, these mathematical papyruses are bilingual texts. We have got the purest bilingual objective proof through the language of mathematics. We can’t wish to have a better companion, or a better judge to demonstrate the Hungarian reading of the ancient Egyptian hieroglyphs.

We have to repeat our viewpoint about mathematical text-exercises. The knowledge of the language, the text was written in, is absolute necessary. Only then is it possible to interpret the given problem and follow the calculations in detail. This lesson remained

unsolvable until today for Egyptologists missing the knowledge of the Hungarian language.

Egyptologists or anybody else interested in reading ancient Egyptian texts in Hungarian, please contact J.F.T. Borbola, janoslo@kabelfoon.nl.

the author of the book “Royal Circles”, (2001).Reading RMP (Rhind Mathematical Papyrus) in Hungarian.

Let’s read it together in Hungarian (MMP: Moscow Mathematical Papyrus), (2000).

Csillagszoba (Star Chamber), Hungarian reading of the pyramid scripts in Sakkara,

(2004)