1

Rhind Mathematical Papyrus How did people count 4.000 years ago along the Nile? (An excerpt from the book Royal Circles by J. Borbola)

Many asked this question, but no satisfying answer was given yet. Several rows

of calculating-steps are missing from the papyruses, thus we can’t learn or follow the

4000 years old mathematical knowledge in all details. We could say that the calculation happens in a black box. We put in the data on the one side and the right answers appear on the other side. We just may have a hunch about what happened

in between. Thus, it is understandable that beside of the traceable calculations several assumptions, assumed calculation-methods, so called solution-methods appeared. According to our special literature, the old Egyptians used well developed

calculation-systems. They could handle in addition to their fingers and calculating-boards, as assumed, integer and fractioned numbers, first and second grade equations, furthermore arithmetical and mathematical progressions. They supposed

to use well refined geometrical structures, they new the circle, the similarities, relations and the basics of trigonometry too. Their astronomical knowledge is proven by their calendars. The Egyptologists base all their above statements mainly on the

two mathematical papyruses of RMP and MMP.1 Well, how much of those statements can we approve? How much could our

ancestors really know about the secrets of the mathematics and the geometry and

how much we, their late successors add to it? What is veracity and what is assumption?

In the following we narrow down the mathematical and geometrical knowledge

traceable in the lessons of the RMP and the Kahun-Fragment. We complete all those by logical assumptions and conclusions. This picture can’t be perfect, or generalized because we base our assumptions only on the six lessons examined by us. We note

here that the further lessons of the papyrus won’t be taken into account. It is not certain that the mathematical knowledge of the whole papyrus is valid for every exercise.

Looking for answers let us start with the calculation-analysis of one of the most complicated, the Lesson 42.

Let’s see first what we accept. All things we experienced during the analysis as

common attributes.

They wrote the numbers from the right to the left, beside each other and following their declining value.

They calculated in decimal system..

We didn’t meet the numeral zero and negative numbers.

The numerator of their fraction was always 1 but that of 2/3.(Its hieroglyph had the form of 1/1,5.)

They used separate hieroglyphs to write five, six, seven, nine, ten, twenty, thirty, fifty, seventy, and ninety. To write four, eighth, forty and eighty they

used the hieratic sign of ‘n’ as previously told. Six and sixty could be written with the help of three. (They can often be seen as ligatures.)

1 See for these statements Kákosy: Az ókori Egyiptom története és kultúrája (History and Kulture of

ancient Egypt), Osiris (Budapest, 1998), 276, and Puskásné Párdy Jolán: Abakusz, page 60.

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There were extra signs for 1/3, 2/3, ½, ¼ and for the fractions of the Horus-eye.2

They could add (no matter how large values).

We did not see subtraction, but they were able to subtract 1+1/9 from ten.

They could not directly multiply, but they created tables by adding identical values. The desired results, called “product” today, were built and read in

this way.

Integer numbers were decimated and divided into halves. (It’s interesting

the value half, received by decimating five.)

Integer numbers were divided into two and three parts.

Summing up fractions they choose always first the one with the highest value.

Let’s see what we didn’t find, what they probably didn’t know, before we start the detailed analysis.

They didn’t use decimal point and it’s uncertain if they know about the

position-value.

We don’t know, how they added.

They could not directly divide or multiply. Thus, they could not use the little

multiplying table. To reach the products of 8 x 8 = 64 or 9 x 9 = 81 they needed to build number-columns. (See the lessons 48, 41 and 50 of RMP)

They never wrote two identical fractions besides each other.

They could not bring different fractions over a common nominator.

Despite of all the above, their results were exact up to a hair’s width.

In order to learn their thoughts and mathematical knowledge, we have to look into

the “black box”. We have to learn to work with their methods.

Based on their real calculation procedures, we have to examine the following possibilities:

1) They arrived at their results by using exclusively theoretical, calculating methods. (we can even accept that they used previously worked out methods mechanically.)

2) They could not count. They were just taking twice or dividing in two or three parts the numbers and the rest was done by measuring and using devices for help.

3) They used both methods to complete one with the other.

We certainly can’t pull sharp lines between the different possible methods and

one can’t make a picture about the ratio of using No 1 or 2. Think about the 200

years old Hungarian word “mérnök” (engineer), meaning someone who “mér” (is measuring) and uses mathematics as well.

Let’s see all three methods. First the original script again:

2 See table XIV of the Appendix.

3

Picture 13. The mathematical columns of the lesson 42, RMP. A copy of Eisenlohr.

Ad.1. Let’s learn the deduction of Rátky by going through a part of lesson 42.

The exercise: (8 + 2/3 + 1/6 + 1/18)2 = ? We assume, they new the followings:

For example: 1/3 + 1/6 = ½ ¼ + 1/12 = 1/3

1/6 + 1/12 = ¼ 1/8 + 1/24 = 1/6 1/9 + 1/18 = 1/6 1/12 + 1/36 = 1/9

They new further the following equation: It will be informative to express the equations I and II in text:

If the doubled denominator of a fraction can be divided into three parts, then this fraction could be resolved - according to equation I – into the form 1/a +

½a.

If the tripled denominator can be divided into four parts (bisected twice), then

this fraction could be resolved into the form 1/a + 1/3a. (see equation II) This procedure is valid the reversed ways as well. Fractions can be reduced too in this way.

Deducting the calculation:

8 2/3 1/6 1/18 1

16 1+1/3 1/3 1/9 2 1/3+1/3 and (from II) 1/9 = 1/12 + 1/36

17 2/3 1/12 1/36

34 1+1/3 1/6 1/18 4 (based on I) 1/3+ 1/6 = ½

35 ½ 1/18

70 1 1/9 8 +

17 2/3 1/9 2 reduced 2/3 = ½ = ½ + 1/6

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17 ½ 1/6 1/9

5 2/3 1/6 1/18 1/27 2/3 +

8 2/3 1/6 1/18 (because 2/3 can’t be divided by 3)

8 ½ 1/6 1/6 1/18 1/6+1/6 = 1/3

8 ½ 1/3 1/18 1

2 2/3 1/6 1/9 1/54 1/3 1/9 = 1/12 + 1/36 because it has to be halved

2 2/3 1/6 1/12 1/36 1/54

1 1/3 1/12 1/24 1/72 1/108 1/6 One can’t arrive with 24 and 72 to 108,

therefore, 1/24 + 1/72 = 1/18.

They new that 1/18 = 1/36+1/36.

Now it’s reducible: 1/12 + 1/36 = 1/9.

Using the above

1 1/3 1/9 1/36 1/108 1/6 + They counted with this at the totaling

1/3 1/9 1/27 1/108 1/324 1/18 +

“+” means totaling the line.

70 1 1/9

5 2/3 1/6 1/18 1/27

1 1/3 1/9 1/36 1/108

1/3 1/9 1/27 1/108 1/324

77 1 ½ 1/6 1/9 1/9 1/27 1/27 1/36 1/108 1/324

1/9

78 ½ 1/6 1/9 1/9 1/9 1/108 1/324

1/6 1/3

½

78 ½ ½ 1/108 1/324

═════════════════════════════════════════════════════

79 + 1/108 + 1/324

One can see, looking at Rátky’s deduction that the calculation-process of the

lesson 42 can be solved in certain circumstances without any helping-devices.

According to him, with help of the above system, our scribe had only to look for the

triples and doubles of the denominators and he had to divide them into three or four

parts. But we can state even that he could arrive at the result written on the papyrus

using the pure mathematical way (maybe mechanically applying the above formulas).

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The presented method is in reality a simplified form of bringing different fractions over

a common denominator.

This method however, has some drawbacks too.

To start with, these formulas work only, if the denominators result integer

numbers. One can’t use them , if a = 5, 7, 10, 11, 13…and so forth. Thus, it can

practically be used for one third of the numbers. Big part of the fractions with the

numerator of 1 can’t be reduced over a common denominator . Only every second

even number can be divided by four, thus every time we had to bisect twice. Did they

know quartering? Otherwise, they definitely could not multiply by four using mental-

arithmetic. The knowledge of the presented formulas takes a much deeper

knowledge of theoretical math for granted as what the doubling mathematical

columns indicate. A further fundamental question: if they really counted this way,

where did these methods come from?

An other problem is that the procedures done with the fractions do not bring in

every case the result what our scribe wrote on the papyrus. Here we have to suppose

that he new the results of his next lines in advance. For this he changed the, for him

unmanageable fractions for two others with the help of the formulas above. Let’s see

the doubling of 1/9. We can state that this operation happened invisibly in two steps.

The first step: 1/18 + 1/18 = 1/9. This is right and this came first onto the papyrus. But

our clever man saw that he won’t be able to bisect 1/9 further, thus – as a middle

step – he changed it (by using the above methods) for fractions, which are well

usable in the following procedures. He calculated before the next doubling already

with the value 1/9 = 1/12 + 1/36. This however, he didn’t write down. In the following,

he certainly new, that the halved value of 1/6 = the 1/12 will be very welcome to him

building ½ by adding 1/3 to it in the third step. (Only ½ and 1/18 appears on the

papyrus). Preconception. (We can follow this complication later too by dividing 1/6.)

We wonder, if our scribe’s intention was to get to the fraction 1/108 in the case of

1/24 + 1/72 + 1/108, or might he had an other possibility to bring his fractions over a

common denominator. In this case, he had to calculate his in-between steps in head

(?) or in written rough drafts. But where are the written traces of such calculations?

Or did he do mental calculation? Then, he must have seen that 72 : 3 = 24.

Moreover, we should accept that he was able to triple 18 directly in head by the

dividing procedure of 1/18 by three. But his result of 54 is exact. He steels the show

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by ‘tripling’ 108, because it turns out, seemingly without hesitation, that 108x3 = 324,

more exact, if 1/6 leads to 108, then 1/18 leads to 1/324.

We wonder, if he was able to multiply by three. Believing the procedures of

above, then he must have known 3x108 = 324 beside of 3x24= 72. However, in this

case 9x9 = 81 should have been an easy play for him. Staying with the number three:

3x9 = 27 through 3x27 = 81 to the same result. After all, why did he need the

doubling- columns for 9x9 = 81 in lesson 48? Thinking logically, we just se one

explanation: Our scribe could not directly multiply by three. He added three times

instead of multiplying. He added the starting value to the result of his doubling.

The same happening by the deduction of 8 x 8 = 64: 8 1

16 2

32 4

64 8

The same multiplying tables are clearly to follow in the lessons 41, 48 and 50 of

RMP. They could not directly receive their numbers quadrupled value in “higher”

regions. See it in the lesson 43 of RMP: 1 113 2/3 1/9

2 227 ½ 1/18

4 455 1/9,

or in the case of the Kahun-Fragment:

1 256

2 512

4 1024

Thus, we can certainly state that they could not multiply by three nor by higher

numbers, as we understand it today.

Let’s ask the serious question: did they really count by the above methods and

could they solve all the analyzed exercises just by mathematical deduction?

Does the “black box” really contain only these deductions?

Ad.2.

It is worth to examine, if a contemporary “mér”-nök (‘measuring-man’ = engineer)

could reach the same results by only measuring? Did they have devices, which were

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able to replace calculation? Can we imagine that people who created stone-buildings

only measured instead of calculating?

Try to answer the most basic question, before looking for devices: How did they

add integer numbers? There aren’t any written traces for their method, but their

results were exact. We could say going by the official standpoint: mentally. Indeed,

not impossible, but as a control, let’s repeat the Kahun Fractal’s 14th column:

256 1 /

512 2

1024 4 /

85 1/3 2/3 /

together 1365 1/3 bravo!

We should try to add the ticked off numbers in head: 256 + 1024. How much is it?

If we let time for our lazy neurons, then we come slowly to the partial-result of 1280

and we can complete it by 85 1/3. But we choose the easier way. Starting at the

lower end, we get even more slowly. Even this little mental arithmetic creates a

problem.

Based on method (1) of above, we supposed to accept that our ancestors – who

were not able directly to multiply 8 by 8 – could do complicated additions, with no

mistakes and no auxiliary-calculations. Is it possible? In every respect, it’s a thought-

provoking assumption.

They wrote the values of ten, hundred or thousand mixed under each other.

Looking at this, we can state that even if they knew about the position-value, they

must have handled it differently from our system.

Is it possible that they reached their results by measuring, like by pacing out a

distance?

Let’s imagine that they put at the beginning the values to be added beside each

other and later, using their fingers, figured out the number of pieces. Later on,

devices were made and with the help of them larger numbers could be handled as

well.

One found archaic calculation-tables, abacuses all over the world, but along the

Nile only measuring rods were unearthed. We think of the cubit-rods, displayed in the

Museum of Cairo and Turin.3 These rods were made of bronze, basalt or wood and

3 Schlüsser, Pyramiden (Köln, 1983, 331. Abb. 15

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are divided by marks into 3, 4, 5, 6, 12, 16 and 24 parts. These, (not yet deciphered)

system of marks made it possible that the rods sliding beside-each other or in angled

constellation or turned by an axis were used to figure out distances, degrees of

angles and as we assume to reduce fractions with different denominators. They were

using them as we used our slide-rules (slip-sticks) just 50 years ago, before the time

of electro-calculators and computers appeared. 4

We assume, their process of addition was linear-based.

They combined two or more measuring rods in a way that the added lengths of

the wanted measures could be read off directly.

They probably simplified, reduced by ten or hundred in case of larger numbers.

Thus, instead of 1200 one could use 12, which already was marked on their rod.

According to this assumption, they changed the numbers to lengths.

We introduce on the next picture a possible device, created with two rods,

Picture 14

Dividing the length OA into three parts

;;;;;;;;;;;;;;;;;;;;;;;;;;;;/4. Ellenstöcke, Kairo Ägypt. Mus. Steinere Elle des Osoroeris, Turin 6348, 52,38 cm; Rivista. Turin Nr.11, Mai 1961. 4 We describe the rods more in details on the table XII of the Appendix.

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One finds beyond the first theses of plane-geometry the adding and dividing of

sectors. The OA sector of the picture 14 can be divided into three parts by parallel

secants based on the OB auxiliary line. Obviously, the same procedure is suitable for

bisection or for any other divisions. The graduation of the measuring-rods in Turin

and Cairo strengthens our theory.

Think it over again: If somebody is forced to add by mental arithmetic, then he will

certainly meet multiplication as well. Adding the same amounts is nothing else, than

multiplying.

This obvious experience could only stay hidden by adding with the help of

mechanical devices.

Summarized: following the above train of thoughts, we assume that our smart

scribes along the Nile, used devices, probably measuring rods, for their adding up.

This assumption would reduce their mathematical procedures mainly to measuring.

The question is now: could they become the builders of the pyramids along this way?

It seems for us impossible to manage one of the largest empire of the earth for

thousands of years with this knowledge.

The contemporary “engineers” could certainly calculate as well, not only measure.

Let’s now look at the two method’s combination as experienced by us:

Ad.3.

We believe, thinking humans along the Nile wrote down their scientific

accomplishments, arranged them in groups and may have created tables. But they

continued to measure, because doing so, it was easier to get the results in many

cases instead of following complicated calculation-methods.

Remember his words in RMP, where he called the procedure of

adding/multiplying as (‘szaporítás’) augmentation and subtraction as ‘vegyétek’ (take

it off).

Let’s see the alloy of calculation and measuring, especially how did our clever

scribe divide the integer number one?

Looking at his signs too, we stated that he was able to bisect and divide the one

in three parts. We met decimation and multiplying by hundred as well. Knowing

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bisection he could split in four or eight by repetitive bisections. He had extra signs for

a half and for a quarter. If he could take the third part, then he arrived at 1/9 by

repeating it. Using both together, he got 1/6 by taking the third part and bisecting it.

The hieroglyph of the smallest integer, the one was and its hieratic equivalent

a dot, which appeared above the denominator. He wrote 9/٭ instead of 1/9. Other

fractions were built the same way.

How did he subtract?

This is a justifiable question. He divided 10 by 9 in the lesson 42 and subtracted

the result 1+1/9 from 10. However, instead of 8+8/9, he wrote down

8+2/3+1/6+1/18.on the papyrus. It’s funny. It is theoretically possible that he knew the

numerators larger then one and calculated with them, but for some reason, he

reorganized the results again into fractions with the numerator of one. In reality, we

should see some traces of that calculation. It’s just not believable that somebody,

who is able to use 800 hieroglyphs, did not have signs to write fractions with higher

numerators. However, we met nowhere fractions with numerators higher then one,

but with 2/3.

Consequently, he calculated differently. Instead of dividing he was cutting,

chopping into pieces. Could he cut up without devices?

Let’s see again the procedure of dividing into 9. Take ten long ‘bagets’ (French

bread) and divide them between 9 persons. Everybody get one whole and the ninth

part of the tenth bread. Are we going to take one package of 1+1/9 bread out of the

ten, then 8+8/9 will remain. But this is only by our calculation. He didn’t divide by

nine; he cut two times into three.

Our scribe divided the tenth bread mentally into three, cut off one third and

divided this again into three parts. After this, he just took away one whole and a little

piece out of the one-third of the tenth bread. He told us: ‘vegyétek!’ (take it!).

This was the procedure of the united division/subtraction: 10 – 1 1/9 = ?

Well, how much did he retain?

Recounting the rest, he found 8 whole breads and the 2/3 of the tenth bread; and

two little pieces of the last third.

Now he put away one of this little pieces (1/9) and bisects the other one (1/18).

He added the half, received by this bisection 1/18), to the one, which was not

bisected (1/9) and realized that both together (1/9 +1/18 = 1/6) make up half of the

1/3 bread.

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He finished the cutting up and could hold an inspection.

Remained with 8 whole breads

He considered the 2/3 again, being put aside by the first cut into three.

After the subtraction (taking) he still remained with two little pieces. He

recognized that the added pieces make half of 1/3 = 1/6 and the rest is

the result of twice cuts into three and an addition bisection = 1/18

His final result was: 8 + 2/3 + 1/6 + 1/18.

He wrote this onto the papyrus in lesson 42. It looks complicated.

The result is fine. However, it’s possible that he was not thinking this way. He

could use his measuring rods as well. He could “move” around the numbers with it

easier. He could cut everything into three using his rods. Was he able to do this by

mental arithmetic? A precise answer to this question is possible only after a more

intensive examination of the known measuring-rods

We recognized during the above analysis that in case of the numbers smaller

then one, he just created the most necessary pieces. He never saw nine identical

fractions beside each other. Therefore he couldn’t get into a situation where he had

to take 1/9 out of 9/9 and build a fraction of 8/9. This would be our method. Our

scribe was not able to divide 1 by 9 as we do it today.

We receive the following picture by projecting it into a linear form:

2/3 1/9 1/9

X::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::/:::::::::::::::/::::::::: ::::/::::::::::::x

/…………………………………………………………./……………./…….../

2/3 1/6 1/18

This was the first procedure of lesson 42.

We call the next step today squaring.

The task of our scribe was as seen previously at the deduction by Rátky:

(8 + 2/3 + 1/16 + 1/18) x (8 + 2/3 + 1/16 + 1/18) = ?

We solve this problem with our fractions differently today:

(8 8/9)2 = 80/9 x 80/9 = 6400/81 = 79 + 1/81

The result of our smart scribe is exact up to a hair’s width, but his result is:

79 + 1/108 + 1/324 (reducing the fractions above a common denominator:

we receive 4/324 = 1/81 again).

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He certainly used a different method for multiplication. We believe again, that he

measured and calculated as well.

He reduced his multiplier to its components, thus he built his mathematical

columns first with 8, then with 2/3, with 1/6 and at last with 1/18. He added the

received results. Let’s follow his deduction again:

8 2/3 + 1/6 + 1/18 x 1

Then he doubled this: 8 + 8 = 16, possibly even by mental arithmetic. The

problem starts by 2/3 + 2/3. We apply again the linear form:

integer

X::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::x

/…………………………………../……………………/…………………./

2/3 + 2/3

from this /…………………../

1/3

The procedure of adding on the above picture can be looked at as the reverse

process, previously done while dividing into three parts. Did he need 1/3 of an

amount, then he cut off that much and the rest 2/3 remained in one piece. If he

added to this 2/3 (the same length), then he realized that one half of it (1/3) hangs

over the length of the one whole. He took out the largest fraction (2/3) from this

“rakottság” (amount), in this case the one whole and cut off the overhanging part, the

half of the second 2/3, the 1/3 and put it aside for the being.

He proceeded probably the following way by adding 1/6 to 1/6:

integer

x:::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::x

/………………………………./ = 1/3

/-------------------/..……………/ 1/6 + 1/6 = 1/3

He arrived at 1/3 playing this “puzzle”.

His procedure could have been the same in the case of 1/18 + 1/18. His building

of 1/18 was probably different as already seen while subtracting. He took twice the

third and bisected afterwards.

As yet, instead of doubling, he set the identical tractions side by side and

compared them with the larger fractions received and put aside after earlier cutting

ups. Doing this, he might have recognized that it gives the same result, if he bisects

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the nominators of the fractions to be paired. We think, he was calculating by doing

this.

Having done these, he set the fractions, which he received as results, side by side

and read now their common amount (length).

Let’s see:

integer 2/3

x:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::::::::::::::::::::::::x

out of: 2/3+2/3 = 1+1/3 out of 1/6+1/6 = 1/3 out of 1/18+1/18=1/9

x------------------------------------ /----------------------------------/------------/

1/3 1/3 1/9

=1/3 + 1/3 1/9 = 2/3 + 1/9

We can well follow his thinking-method on this picture. Setting his fractions side by

side, he could examine what their common length was. Recognizing that their sum is

bigger then the 2/3 of the whole, he took again out of the “rakottság” (amount) the

largest (2/3) fraction. It’s a linear method. He took the one whole, received by adding

the 2/3-s and added this (by mental calculation) to the 16, he already had. Thus the

integer numbers grew to 17. Thus, must have been calculating too.

Now we can get back to his multiplying-table and write down the resulting

numbers:

8 + 1/3 + 1/6 3 1/18 x 1

17 + 2/3 + 1/9 x 2

A change happened at the next pairing-procedure.5 When he added to 17 + 2/3 +

1/9 its pair, adding 2/3 to 2/3, he received again 1 + 1/3. Setting the fractions side by

side he took out again the largest fraction, which in this case did not come from

dividing by three:

Integer

x::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::://:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::x

/-----------------------------------/,,,,,,,,,,,,,,,,/,,,,,,,/,,,,,,/ 1/3 + 1/9 + 1/18 + 1/18 = ½ + 1/18

1/3 1/9 1/18 1/18

5 Compare it with things discussed in chapter 1.

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The largest removable fraction with the numerator of one was in this case the ½.

The overreaching piece was the half of the second 1/9 = 1/18. He had to change his

method from dividing to three into bisection. His result: 34 + 1+ ½ + 1/18.

He mostly must have measured here.

35 + ½ + 1/18 x 4

With this he arrived at the last step of his multiplier table, to the pairing of

35+1/2+1/18. It shouldn’t have caused him any problem. 35+1/2+1/18 + 35+1/2+1/18

resulted in 70 + 1 + 1/9. He certainly counted.

71 + 1/9 x 8

He stated that (8 + 2/3 + 1/6 + 1/18) multiplied by 8 resulted in 71 + 1/9.

Considering the procedure from above we can state that he could read his results

directly by using his measuring-rods and doing mental arithmetic. He didn’t need a

collection of tables or additional calculations.

1/3

The procedure of adding on the above picture can be looked at as the reverse

process, previously done while dividing into three parts. When he needed 1/3 of an

amount, then he cut off that much and the rest 2/3 remained in one piece. If he

added to this 2/3 (the same length), then he realized that one half of it (1/3) hangs

over the length of the whole. He took from this “rakottság” (amount) out the largest

fraction, in this case the integer one and cut off the overhanging part, the half of the

second 2/3, the 1/3 and put it aside as for the being.

He proceeded probably the following way by adding 1/6 to 1/6:

integer

x:::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::x

/………………………………./ = 1/3

/-------------------/..……………/ 1/6 + 1/6 = 1/3

He arrived at 1/3 playing this “puzzle”.

His procedure could have been the same in the case of 1/18 + 1/18. His building

of 1/18 was probably different as already seen while subtracting. He took twice the

third and bisected afterwards.

15

As yet, instead of doubling, he set the identical tractions side by side and

compared them with the larger fractions received and put aside after earlier cutting

ups. Doing this, he might have recognized that it gives the same result, if he bisects

the nominators of the fractions to be paired. We think, he was calculating by doing

this.

Having done these, he set the fractions, which he received as results, side by side

and read now their common amount (length).

Let’s see:

integer 2/3

x:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::::::::::::::::::::::::x

out of: 2/3+2/3 = 1+1/3 out of 1/6+1/6 = 1/3 out of 1/18+1/18=1/9

x------------------------------------ /----------------------------------/------------/

1/3 1/3 1/9

=1/3 + 1/3 1/9 = 2/3 + 1/9

We can well follow his thinking-method on this picture. Setting his fractions side by

side, he could examine what their common length was. Recognizing that their sum is

bigger then the 2/3 of the whole, he took again out of the “rakottság” (amount) the

largest (2/3) fraction. It’s a linear method. He took the one whole, received by adding

the 2/3-s and added this (by mental calculation) to the 16, he already had. Thus the

integer numbers grew to 17. Thus, he must have been calculating too.

Now we can get back to his multiplying-table and write down the resulting

numbers:

8 + 1/3 + 1/6 3 1/18 x 1

17 + 2/3 + 1/9 x 2

A change happened at the next pairing-procedure.6 When he added to 17 + 2/3 +

1/9 its pair, adding 2/3 to 2/3, he received again 1 + 1/3. Setting the fractions side by

side he took out again the largest fraction, which in this case did not come from

dividing by three:

Integer

x::::::::::::::::::::::::::::::::::::::::::/::::::::::::::::::::::://:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::x

/-----------------------------------/,,,,,,,,,,,,,,,,/,,,,,,,/,,,,,,/ 1/3 + 1/9 + 1/18 + 1/18 = ½ + 1/18

6 Compare it with things discussed in chapter 1.

16

1/3 1/9 1/18 1/18

The largest removable fraction with the numerator of one was in this case the ½.

The overreaching piece was the half of the second 1/9 = 1/18. He had to change his

method from dividing to three into bisection. His result: 34 + 1+ ½ + 1/18.

He mostly measured here.

35 + ½ + 1/18 x 4

With this he arrived at the last step of his multiplier table, to the pairing of

35+1/2+1/18. It shouldn’t have caused him any problem. 35+1/2+1/18 + 35+1/2+1/18

resulted in 70 + 1 + 1/9. He certainly counted.

71 + 1/9 x 8

He stated that (8 + 2/3 + 1/6 + 1/18) multiplied by 8 resulted in 71 + 1/9.

Considering the procedure from above we can state that he could read his results

directly by using his measuring-rods and doing mental arithmetic. He didn’t need a

collection of tables or additional calculations.

The question is after all, did our scribe count really the way described above or

chopped his lengths as we demonstrated? See both methods again, or did he have

easier ways to follow? Theoretically, he could have had ready made mathematical

tables and be able to read his results by calculating fractions. However, this

possibility looks very complicated, because he should have included an immense

amount of different mathematical columns into these tables. (We met such series of

numbers dealing with the lessons of the RMP. See the ‘n’ and ‘2/n’ tables.). How did

he count while traveling? Did his many tables always travel with him?

The linear method was much simpler in his case.

Let us return to his calculations again. Up to this moment we just multiplied by 8

doing the procedure (8+2/3+1/6+1/18) x (8+2/3+1/6+1/18). The fractions follow after

this. It stands at the bottom of the first column:

5 + 2/3 + 1/6 + 1/27 x 2/3

But it stands on top of the second column already the division into three parts

necessary for his next step. How did he arrive at the 2/3 of 8+2/3+1/6+1/18? Did he

measured or calculated his result? This question emerged, because the 1/3 result

17

appeared only at the top of the second column. Could he directly divide into 2/3 just

by mental arithmetic, or did he write his result later under the first line? 7

It is most probable that he could not multiply directly by 2/3 on the mathematical

way.8 He did not have to mark the 1/3-results at the top of the second line if he could

do it. He didn’t need these data.

He could divide 8 even mentally by three and doubling the result arriving at 5+1/3.

We don’t see his method, but it is probable that he divided first by three and paired

the result after.

Let’s go this way, because he started immediately dividing 8+2/3+1/6+1/18 into

three in the second line: 8/3 resulted in 2+2/3.

After this he took 1/3 of 2/3. Could he do this even mentally? We can’t exclude

this possibility, but then we should accept that he was able to triple numbers. Tripling

the denominator of (any integer number) With this knowledge he should be able to

divide the volume of a fraction by three. 2/3 x 1/3 = 2/3+3+3 = ? It’s funny that by

doing that he did not see the possibility of 3x3 = 9 (and the further step, the 3x9 = 27

shouldn’t have caused too much problem for him). We would write the result today as

2/9. However our ancient teacher never wrote two identical fractions like 1/9 + 1/9

beside each other. He reduced them after taking the possible largest fraction out of

them.. Thus, we meet the two-steps-procedure again. Is it believable?

It was certainly much easier for him to measure all these.

1/6 1/3

X:::::::::::::::::::::::/:::::::::::::::::::: /

/--------------/------------/ : 1/9 + 1/9

/--------------------/------/ : 1/6 + 1/18

He could take the measurement-reading of his lengths (1/6 + 1/18) directly.

The further dividing by three did not cause him trouble: 1/6 became 1/18 and from

1/18 he received 1/54.

Now he had to add them up: 2+2/3+1/6+1/18+1/18+1/54 = ?

7 It would be possible too mathematically, that he divided first the doubling-result of the second line

into three, Rátky. 8 After all, we find some references fort his, because he wrote the sign of 2/3 with an open mouth and

two strokes with uneven length under it. We could look at this fraction as 1/1,5. Theoretically, he could multiply by 1,5 the denominator of the fraction to be multiplied by 2/3. But this method doesn’t work in the case of 2/3. It seems to be a much easier way the measuring-method, where the value of 2/3 can directly be created.

18

The result was fine, he didn’t have to change it, the largest fraction (2/3) was

taken out, but then he realized the pair of 1/18 fractions. He reduced them again by

taking out the largest (1/12) and defined the size of the rest (1/36).

1/12 1/6

X:::::::::::::::::::::::::::/::::::::::::::::::::::::::/

/----------------/---------------/ : 1/18 + 1/18

/------------------------/-------/ : 1/12 + 1/36

He used most probably his measuring-rods for the procedures above.

On the papyrus appeared the result of (8+2/3+1/6+1/18) x 1/3 = 2 + 2/3 + 1/6 + 1/18

+ 1/54. He had only to double this for the multiplication by 2/3:

2 2/3 + 2 2/3 = 5 1/3

1/6 + 1/6 = 1/3

1/12 + 1/12 = 1/6

1/36 + 1/36 = 1/18

1/54 + 1/54 = 1/27

Our scribe reduced the two 1/3-s, thus he wrote at the bottom of the first column

the result of (8+2/3+1/6+1/18) x 2/3 = 5+2/3+1/6+1/18+1/27.

The procedure (8+2/3+1/6+1/18) x 1/6 came into the second line of column 2.

His job was easy here, he just had to bisect compared to the first line:

See the first line and you will see the bisections:

2 + 2/3 + 1/6 + 1/12 + 1/36 + 1/54 x 1/3

1+ 1/3 + 1/12 + 1/24 + 1/72 + 1/108 x 1/6

Dividing these by three he arrived even to the line with 1/18:

1/3 + 1/9 + 1/36 + 1/72 + 1/216 + 1/324. It would be fine, but he did not write

this on the papyrus!

He could not reduce further 1/3 and 1/9 however, when he arrived at

1/36+1/72+1/216, then he already wrote the 1/27 + 1/108 fractions. How did he

receive his results?

Let’s set side by side the fractions waiting to be reduced:

//---------------------------------------------------------------/-------------/------------------------------//

1/36 1/216 ↑↑ 1/72

19

The sequence of the addition is not important, because as our scribe examined

this length, he immediately found on his measuring-rod 1/27, the largest possible

fraction (the arrows point to its size) and he saw the rest, which turned out as the

fraction 1/108. The 1/27 results from the dividing of 1/9 by three and 1/54 and 1/108

are the results of further bisections. He had to measure again.

His result:

1/3 + 1/9 + 1/27 + 1/108 + 1/324 is exact to a hair’s width.

After these, even for him complicated, procedures he had to add his partial

results. This too wasn’t easy. He marked on the papyrus the numbers to be added:

71 + 1/9

5 + 2/3 + 1/6 + 1/18 + 1/27

1 + 1/3 + 1/12 + 1/24 + 1/72 + 1/108

1/3 + 1/9 + 1/27 + 1/108 + 1/324

We can add the integer numbers directly: 71 + 5 + 1 + = 77.

1/3 and 2/3 resulted in one more integer, now together 78.

See the rest:

1/3 + 1/6 + 1/9 + 1/12 + 1/18 + 1/24 + 1/27 + 1/27 + 1/72 + 1/108 + 1/108 + 1/324.

Out of these, the fractions 1/6 + 1/18 + 1/9 build 1/3.and together with the other 1/3

they result in 2/3. When we take out the two smallest fractions of the group (1/108

and 1/324), the remaining fractions are 1/9 + 1/12 + 1/24 + 1/27 + 1/27 + 1/72 and

1/108. It is hardly believable, but all these rest fractions resulted in 1/3. Calculating

by our method of today we need a common denominator of 216 and all nominators

become higher than 1. Our scribe could not deal with this.9

He certainly took out his measuring rods and pushed the lengths together, until he

found the third 1/3 and he probably realized that 1/108 + 1/324 = 1/81. However, he

didn’t finish this last addition. Why?

Adding up everything he could state that the surface of the circle with 10 units

diameter is: 79 + 1/108 + 1/324. (Lesson 42 of RMP. Using their π of 3,1605, his

result of 5x5x3,1605 = 79,0125 was exact.)

There happened one more interesting thing in this lesson.

As a following step, our clever scribe multiplied by ten and later reduced too by

decimation.

Let’s see how somebody, who didn’t know 8x8 = 64, multiplied by ten.

9 It is possible that by grouping differently some more fractions could be reduced.

20

79 + 1/108 + 1/324 1

790 + 1/18 + 1/27 + 1/54 + <1/81> 10

395 + 1/36 + 1/54 + 1/108 + <1/162> ½

1185 <1/6 + 1/54>

118 + ½ + <1/54> 1/10

59 + ¼ + <1/108> 1/20

The numbers between the „angled brackets“ are not on the papyrus.

There are no hints or explanations for the next steps. Thus, we can’t control the

validity of our assumption.

It looks like that he really multiplied by ten. 79 x 10 = 790 is well to follow. The

problem came with (1/108 + 1/324) x 10. The two fractions cannot be multiplied by

ten without increasing the numerators. These fractions had to be changed into forms,

which can be written by our method today using an enumerator ending on zero. Our

scribe needed a method to determine these unknown fractions. He wouldn’t have

received integer numbers by dividing the enumerators by ten. Instead of this, we

think, she really “decimated” by ten. Attention! He did not chopped, or cut off. We

come to this after the deduction. We have to imagine these procedures happened

always in a very simple form.

The method as follows: He multiplied the enumerator by ten, bisected the result,

(he practically multiplied by 5) which he divided again by three and two. It sounds

complicated, but he just did as before: divide by three and bisect after having

multiplied by ten.

108 x 10 = 1080; 1080 : 2 = 540; 540 : 3 = 180; 540: 2 = 270.

This were fractions, thus the results were 1/108 = 1/180 + 1/270. These are

enumerators, which can be divided by ten and the results are integer numbers.

Unfortunately, he wrote only the already decimated results (1/18+ /27) on the

papyrus. Following the above method, he could have proceeded in the case of 1/324

as well.: 324 x 10 = 3240; 3240 : 2 = 1620; 1620 : 3 = 540; 1620 : 2 = 810. He

wrote on the papyrus only the decimated numbers 1/54, but the fraction 1/81 is

missing. Why?

Let us write down the general form of the above procedures, before we try to

answer the last question:

21

We want to note here that this equation is the counter-pole of the Rátky equation

presented earlier, because here we do not have to bring two fractions above a

common enumerator but to divide one fraction into two. Unfortunately, this procedure

too works only if the division of the enumerator results integer numbers, which can be

divided by 3 and by 2 as well. For better lucidity, we did the division by five in

advance, but he should have done this in two steps as presented above.

After all, his decimation process stays invisible. We think, he did not measure

here, he just multiplied his enumerators one by one. It is most probable that he could

multiply or divide by ten. He just stepped into the next or the lower order of

magnitude.

It follows from this that he could not multiply by 5. He received this by bisecting

the result of the multiplication by 10.

Could he get there using only the linear/ (measuring-stacking) method.? Certainly.

Only this way could have sense the 4 and the 6 units-arrangements of the

measurement-rods in Turin and Cairo. Our clever scribe wanted to know the results

of the multiplication by ¼ and 1/6 after the multiplication by 10. In this case he did

not had to bisect the tenfold number, because the fractions 5/3 = 10/6 and 5/2 =

10/4. Thus, he could read directly the enumerators of his Fractions.10

Did everything happen as we deducted in the reality too? We don’t have any

proof of it. In the first line, we are missing the fraction 1/81 in this line. The sum of the

fractions written this way is 1/9, which points to its size of 1/90 before the decimation,

but the common value of 1/108 and 1/324 is 1/81. Thus, his result is not exact; it is

only a close approximation.

Why? Again why? Did our smart scribe forget to write the last fraction? Or might

he not calculated this way? Everything is possible. However we feel that something

different was standing behind it. He just didn’t need these fractioned values in his

lesson. He even marked only the integer numbers in the addition procedure at the

end of the line.

The following decimation and bisection shouldn’t have caused him any more

problem.

10

This recognition led us to the elaboration of the above calculation-method.

22

Summary

As a reminder, we repeat again that we based our above calculations exclusively

on procedures, numbers and numerals found in the lessons of the RMP being

examined by us. We did not deal with knowledge presented somewhere else,

because it is not approvable that our scribe knew about it.11 We examined the above

material exclusively by the methods, signs and terminology experienced by us.

We can state that our scribes calculated using the decimal system in those

lessons. Compared to our contemporary mathematical knowledge, we can state that

they could not do basic procedures. Párosítás <paaroshiitaash>, elvevés <εlvεvesh>

szaporitás <saporiitaash> és dölés <dœlesh> (Pairing, taking, increasing and

inclination12 stood for addition, subtraction, multiplication and division.

They could not divide the number one as we do; they only cut it, chopped into two

or three parts. In order to accomplish their procedures –not for the addition – they

always had to write number-columns, which were built by pairing off. They know how

to take integer numbers and probably fractions tenfold, or to decimate them.

They went several ways by cutting off the number one and enlarged the numbers

of parts by their need. They wrote their fractions (except the 2/3) always by using the

numerator 1. Thus, we had to realize that their mathematical procedures were

different from ours. They used different method by adding the fractions and did not

use a common enumerator. We could demonstrate with the help of the equations of

Rátky that the partial results of the calculation with fractions, written on the papyrus,

could be reached even based purely on mathematics. But at the same time, we

pointed to the limitations of this method. We supposed that our scribe set their

fractions into a linear system, examined the length of the (“rakottság”) amounts. In

this case, they always chose as result the possible largest fraction with the nominator

of 1 and the rest of the next largest fractions also with the nominator of one.

We pointed to the probability, during the lesson 42’s analysis, that they had some

simple portable devices and we named them ‘measuring rods’. They probably

needed several of them and the diverse graduation of the rods made the setting of

11

It is even possible that the scribe of the RMP had a much higher mathematical knowledge as found in the papyrus. It is imaginable, according to the introduction, that he just copied completed lessons from earlier papyruses and we evaluated not his, but the work of an earlier mathematics--teacher. 12

The word inclination (dőlés) is the right name of the procedure. Think of their method of dividing by three using two measuring-rods.

23

the different fractions (lengths) easier for them. The common amounts (rakottság), of

the added lengths could directly be read on these measuring-rods. The measuring-

rods in the museums of Turin and Cairo indicate this kind of procedures. The real use

of these rods is still obscure. A detailed investigation of them could bring us some

new knowledge.

We assumed further that they probably had tables containing ready-made

number-columns with the results of the procedures with fractions too. The linear–

system could be avoided or supplemented with these. However, the number of the

necessary tables would have been enormous, or they had to use general solving

equations. For the use of those, one have to grant them though much higher mental

preparedness. Using these general equations they shouldn’t needed the pairing

multiplication-tablets and should have known the numerators larger then one.

We assume that they enlarged by doing bisections or dividing by three, but the

real details of their method of tripling and dividing into three could not be explored by

us. 13 We could demonstrate that our scribe couldn’t triple by mental arithmetic and

we never met multiplication-columns written for this. (like: 1x9; 3x27; 9x81). At the

same time, using their measuring-rods, they could do all four basic procedures and

even measure the grade of the ‘dőlés’ (inclination).

The question remained open: from where did our clever scribe know the solving-

equation for calculating the semi-globe’s volume?

V = 2/3 r3π = 2/3r x (d 8/9)2 ? Lesson 42 of RMP.

We were even thinking of the possibility that the RMP contains only the high-

school-knowledge of that time and the contemporary “university-mathematics” is still

somewhere under the sand to be found. Unfortunately, this can’t be proved either.

We think, if somebody is able to multiply than he will teach that to his children too and

will avoid the outdated complicated method to use. The writing-signs, they used, are

not bearing any witness from a higher mathematical knowledge either.

We can state summarizing, that neither presented method is perfect. We reached

only partial results. We feel that we could only open the cover of the mentioned

black-box and we have to continue the further investigation of its content under

different conditions.

13

One could believe that they already knew the dividing of the circle’s circumference by 6 using the radius. This way they could even construct equal sided triangles. The added up sides of such a triangle resulted already the 1/3 of the total circumference of the given circle.

24

We could not hope to receive an answer for every details of our question asked in

the title of this chapter. However, we could look into the w itches’ kitchen of the

archaic calculating-methods by examining these lessons.

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