Egyptian Math History
The history of Egyptian mathematics must be read in modern
base 10 as well as the ancient base 10 number system. This
two-way street situation, which is about to be explained,
provides a 'confirmation' step, one that currently does not
exist in the professional literature.
For example, one-way street translations, discussing an ancient
text in its oldest terminologies, have been offered as 'final' over
the last 120 + years by Peet, Chace and many others. However,
by showing details of remainder arithmetic and Egyptian fractions
as remainders, in both modern and ancient scribal shorthand
(the two way streets) directly proves that the earlier scholarly
translations were incompletely translated into an ancient form of
base 10 (a one way street). Scholars have therefore been shown
to have omitted big chunks of scribal thinking, missing many
important scribal techniques, like remainder arithmetic, points
that may not have been missed had the scholars been required
to fully translate every problem into modern arithmetic beginning
120, 100 or 80 years ago.
These scribal gaps need to be filled. By building two-way streets,
looking at the meaning of the ancient arithmetic, algebra and geometry
in modern terms, and visa versa, following the clues provided by the
scribal mental and explicit shorthand facts, detailing a deeper forms
of Egyptian arithmetic, algebra and geometry.
This is one of the central purposes of this blog, to validate that
two-streets are more than just sometimes possible, they are
almost always a requirement to fairly read scribal mathematics,
as written by Ahmes and other scribes.
One of the clues that has been under valued is the wide use
vulgar fractions, a center piece of scribal mental arithmetic.
Scholars have long suggested that scribal division consisted
of only unit fraction terms, thereby eliminating the need for
modern scholars to consider vulgar fractions in the majority of
ancient text calculations.
For example, RMP # 32 states in modern terms
x + (1/2 + 1/4)x = 2 or
x + 7/12x = 2, or
19/12 x = 2, or
x = 24/19 = 1 + 5/19
Ahmes' mental shorthand notes worked with the 5/19 vulgar fraction.
Noting his 1 + 1/6 + 1/12 + 1/114 + 1/224 answer, it appears that
a generalized, but not optimized, Hultsch-Bruin method did the following:
Find 5/19 by first subtracting 1/12 or
5/19 - 1/12 = (41)/(12*19)
= (38 + 2 + 1)/(12*19)
= 1/6 + 1/114 + 1/224, or
the final answer
1 + 1/6 + 1/12 + 1/114 + 1/224
Ahmes' choice of 1/12 followed the 2/19 2/nth table suggestion,
where:
2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114
as also used in 2/95, where 2/19 x 1/5 = 1/5 (1/12 + 1/76 + 1/114)
as Wilbur Knorr and many others have seen, in 1982, as published
in the journal Historia Mathematica.
Had Ahmes 'thought of another number' like 1/4, a more optional
series 1/4 + 1/76 would have been found (as the 400 AD Akhmim
Papyrus, from the Coptic era includes in its n/19 table).
Gillings and other authors like Robins-Shute avoid discussing Ahmes'
shorthand that reveals this class of vulgar fractions fact - fragmentary
facts for sure, but facts that clearly point ONLY in the remainder
direction, then rewriting vulgar fractions as Egyptian fractions.
This blog details a wide range of specific textual ways that scribal
shorthand has been hard to read. I have been lucky in several areas,
assisted by many over the last 18 years, to have corrected several
scholarly oversights that had only reported what scholars could read
of ancient texts. Sadly many gaps are still left in the ancient texts, here,
there and everywhere. As a conseqence, to date, the oldest number
system has not been fairly read by modern language speakers, as the
scribes and archaeological luck, has left for us to read.
Special care needs to be taken when reading heiratic arithmetic
and its scribal shorthand. One new new method to achieve
this goal is documented on this blog. Choose whatever name
that fits in your mind for this new method. For myself, I call it
building two-way streets.
This blog's argument shows that ancient words and numbers
are only snapshots of ancient statements. To fully translate a
snapshot and find the ancient context, the fragmented scribal
shorthand must be read with rigorous methods. Filling in a
scribal gap at one's leisure, skipping ones that are not readily
understood, has commonly left out major segments of the
mathematical texts, one being remainder arithmetic.
To resolve several fragmented aspects, wider translations of the
scribal math shorthand must be developed that covers all the
available data. One technique that is being developed translates
the ancient data into modern base 10 decimals as a bilingual text.
The new method is comparable to the Rosetta Stone's use of
phonetic symbols to link the third language's decoding to the
first two. The new method uses quotients, remainder and common
divisors a central ideas.
Math scholars have often taken only an aspect of a text, and reported
interpretations outside of the original context when mathematics and
metrology were involved. This 'translate what you can' problem has
existed since the 1870's when the RMP was first published. This blog
will focus on clearly up one of the ancient math systems that have been
overlooked, that being the remainder arithmetic cited in the RMP and
other confirmationsal texts.
For example, RMP 65 states that
100/13 = 7 + 2/3 + 1/30
But what does that mean?
The Reisner Papyrus will be shown to have written:
n/10 = Q + R
where Q was a quotient and R was a remainder, first stated as
vulgar fraction, that was easily converted to an Egyptian fraction
series. The Reisner cites:
8/n1 = 1/2 + 1/4 + 1/20 = 8/10
48/n2 = 4 + 1/2 + 1/4 + 1/20 = 4 + 8/10
18/x3 = 1 + 1/2 + 1/10 = 1 + 6/10
64/x4 = 6 + 1/4 + 1/10 + 1/20 = 6 + 4/10
36/x5 = 3 + 1/2 + 1/10 = 3 + 6/10
such that x1 = x2 = x3 = x4 = x5 = 10
amid tables of data, as further explained on
http://reisnerpapyri.blogspot.com
That is,
100/13 = 7 + 8/13 or 7 + 2/3 + 1/39
with 7 being a quotient and 8/13 being a remainder
that was converted to 2/3 + 1/39.
Additional proof is provided by the word ro, a term often read
in metrology as 1/320th of a hekat. The word has a broader
meaning, including rest, remainder and related ideas that connect
more to arithmetic than metrology. Ro will be shown to have taken
on several numerical values other than 1/320 (as research has already
found, with the details being published in the near future).
For now, the word ro will be shown to have most often meant
'common divisor' to ancient scribes. Ro was linked to binary fraction
and Egyptian fraction series, a point that modern scholars are slowly
becoming aware. Ro did take on several numerical values. The
different values of ro were needed to exactly scale the Egyptian
fraction series, remainders of Horus-Eye divisions, as needed,
working with hekats, cubits (per Masse & Gewiche, 1994) and
other measures.
Stated in other terms, scholars have often reported the ancient
math data in fragmentary ways since the 1880's. By 1923 scholars
like Peet began to falsely conclude that his work was complete.
Peet conclued that ro only equaled 1/320th of a hekat and, by
implication, could not be re-valued within another context, such as
partitioning a cubit.
Peet's analysis stressed the idea of Egyptian division, as a limiting
factor on how and why the Akhmim Wooden Tablet scribe
partitioned a hekat by 1/3, 1/7, 1/10, 1/11 and 1/13. Peet
did not go into detail to show how the scribe proved his answers
by computing 3/3, 7/7, 10/10, 11/11 and 13/13, respectively. Peet
also did not seek out cubit partitioning methods, as later reported
by Masse and Gewiche, or any other weights and measures context.
Peet was not the only scholar that introduced misconceptions by
skipping over 'unreadable fragments' within a given text, while
attempting to create sweeping conclusions concerning the scope
and content of Egyptian mathematics.
Considering the original text, and sketchy translations of the fraction
based mathematics reported in several texts was straight forward
and often trivial. Interestingly, it will be shown that the math texts
have not been fairly translated from the combined Horus-Eye and
Egyptian fraction data cited in the RMP and other texts. Over 40
examples of binary and Egyptian fraction data have been ignored,
or grossly misread, since the data did not fit the scholarly additive
and subtractive paradigms of Egyptian arithmetic operations, as
detailed on:
http://mathorigins.com/image20%grid/awta.htm
and,
http://akhmimwoodentablet.blogspot.com
This blog will freshly analyze Egyptian division based on reading
the binary and non-binary data reported in single scribal statements
mentioned above.
As a goal, this blog for the first time will attempt to offer complete
translations for a couple of example problems within the RMP and
one other text to modern base 10 fractions. This blog also offers a
rigorous method that creates a new set of abstract facts that can be
easily double checked by qualified experts.
Continuing to step back, scholars, mostly working alone, had
unfairly thought that a rigorous translation to modern base 10 was
not necessary. Frances I. Griffith in 1891 sensed that ro meant
'greatest common measure' within the RMP. Others scholars working
in other texts, Daressy in 1906 with the Akhmim Wooden Tablet
sensed other interesting arithmetic features. Daressy sensed exactness
in all but the 13/13 case, where scribal typographical errors hindered
his work. Yet, nine Daressy examples showed that divisions and
proofs were exact, a clue that was not followed up by Peet and
others.
Interestingly, by 1923 Peet came along and tried to close off debate
related to the highest form of Egyptian division operations by refuting
the 1906 analysis of Daressy. Gee, didn't I get the gist of the math
discussion correctly, Peet might have retorted? Sadly, the majority
of early scholars had tried to stay within a singular view of Middle
Kingdom math texts, often created by themselves, with few or no
auditors, picking out readable aspects from ancient scribal
shorthand, skipping over 'unreadable aspects' (often without
mentioning their oversights).
Oddly, by 1923 scholars tended to agree on an additive aspect of
the mathematical texts. But were all the Egyptian mathematical texts
only additive or subtractive in scope, as David Silverman (and others)
continue to suggest in 1994?
It will be shown that math historians have not completely, and
therefore not fairly, reported important Egyptian mathematical
facts due to the oversight of not completing a rigorous translation
to modern base 10 fractions. This blog wll show that Egyptologists
and math historians have created no formalized methodology that
fairly and generally links the original scribal base 10 to modern
base 10. A fair 'decoding' system is needed such that modern
fraction views of the ancient texts can take into account all the
ancient text's contents. Such a 'decoding system exists in limited
circumstances. The proposed system can be rigorously double
checked in ways that offers great hope for the expansion of the
method to other poorly read mathematical texts.
Correcting oversights may be difficult. A formal methodology has
not been agreed to. However, one will be made available in this blog
that achieves a rigorous decoding key to ancient scribal thinking in
certain situations, the first being a hekat with a divisor of 64 or smaller.
The results of this methodology clearly opens up scribal remainder
arithmetic (a form of abstract math) Therefore, it will be shows that
non-additve or subtractive scribal mathematics did not dominate
Middle Kingdom mathematics, as has been assumed for over 100
years.
Proof lies in several examples, each showing the trivial side of
the discussion, within the Rhind Mathematical Papyrus. The
RMP lists several beginning type problems that our modern
4-6th graders would recognize, data that appears additive,
but actually is not. One problem is #83, where three classes
of birds were fed three different amounts, 1/6th of a hekat of
grain for three birds, 1/20th for one bird, and 1/40th for each
of three birds. Ahmes, the RMP scribe, then asks: how much
grain did all seven birds eat in one day?
A modern discussion of this trivial problem requires the
addition of 1/6, 1/6, 1/6, 1/20, 1/40, 1/40 and 1/40,
Note that an easy to reach common multiple (1/120) is
required to solve this problem if no modification in procedure
is introduced. Today, kids might improve the ease of working
the ancient problem by first adding the three 1/6 fractions,
obtaining 1/2. Then they could add 1/20 and 3/40. In this
manner only (1/40) is required to be used as a common divisor.
This is a basic form of modern logic overlay (a bilingual text, to
refer to a cryptanalysis standard) can be restated several ways.
A closely related arithmetic form states, that by adding:
1/2, 1/20 and 3/40 using the common divisor 1/40, or
(20 + 2 + 3)/40 = 5/8th of a hekat of grain, as Ahmes
the requested to know, how much did seven birds eat in one
day?
Ahmes performed this same arithmetical task, but chose another
interesting value for his common divisor, 1/320. Ahmes wrote his
fractional divisors of a hekat, his feeding rates, in terms of a larger
one that he named ro, one that was commonly seen in several
additional RMP problems, #35-38, 47, and 81.
To understand Ahmes' reason for selected 1/320th for a class of
problems, any one of which could have used a smaller common
divisor, let us examine Ahmes used of the word ro.
Stated in terms of ro (common divisor) units, Ahmes
first added up 53 1/3ro, 53 1/3 ro, 53 1/3 ro, reaching
160 ro. Ahmes then added 16 ro, 8 ro, 8 ro and 8 ro,
reaching 200 ro, or 200/320 = 5/8th of hekat. Trivial
right?
At this point in this discussion includes a few non-trivial points.
Why did Ahmes apparently chose only only one common divisor,
1/320th to be specific? One clue is given by, no other common
divisor was needed for any divisor n, where n was not greater
than 64, when that common divisor is multiplied by the Egyptian
fraction series associated with it.
Let me say that again. The trivial side of the modern discussion
shows that several common divisors can come into play to solve
this seven bird eating grain problem. In Ahmes' case all of his
common divisors were large, all multiples of 1/320. This multiple
of 1/320 fact was repeated in several other hekat division
problems, as it will be shown in five Akhmim Wooden Tablet
cases. There were 29 additional cases in the RMP, written in
terms of the hin unit, so there is adequate data to discuss this
issue, in depth.
This blog, and one other, will clearly report that Ahmes was
using remainder arithmetic to exactly partition a hekat, a volume
unit, adding remainder corrections that scribes nameed ro. This
none trivial statement needs to proven in a rigorous manner.
Remainder arithmetic used the Horus-Eye series as its quotient,
with the Egyptian fraction series including the 1/320th common
divisor (multiple) as its remainder term. That is, whenever a hekat
unity, 64/64, was divided by any relatively prime number, 3, 7,
11, 13 and so forth, a remainder term was created, one that
included 1/320th, the Egyptian word ro. Why did ro appear in
every relatively prime division answer, when the divisor was no
greater than 64?
Ahmes and his mentor writing in the Akhmim Wooden Tablets,
or an earlier unreported text, substituted the fraction 5/320 for
a 1/64th fraction that always appeared in the remainder
arithmetic statement:
(64/64)/n = Q/64 + R/(n*64)
such that Ahmes remainder arithmetic statement looked like this
(64/64)/n = Q/64 + (5*R/n)* ro
Further, it will be shown that for n > 64 another form of remainder
common divisor was used, with ro being excluded. Ro certainly
was not used in the medical texts, as the Papyrus Ebers and other
texts, when several divisor greater than 64 were in use, but why?
A definitive discussion will take place on this subject at a later time.
To limit our discussion to the n > 64, as RMP # 47 implies, an
increase in the size of the hekat unit numerator, from 64/64 to
256/64, 320/64 or even 640/640 was made to allow the Q term
to contain a value > 0. Until this addition info is published, this
blog will continue to be updated, reporting bits and pieces of
the search for additional proof to detail the scope and depth of
ancient scriibal solution. The ancient Horus-Eye problem was
solved not larter than 2,000 BC, as reported by the volume
measure system, using large and small unit divisors.
How early was it solved, in a manner that birthed the Egyptian
fraction system itslef, as required to scale the remainder term,
with or without ro?
Milo Gardner milogardner@juno.com
base 10 as well as the ancient base 10 number system. This
two-way street situation, which is about to be explained,
provides a 'confirmation' step, one that currently does not
exist in the professional literature.
For example, one-way street translations, discussing an ancient
text in its oldest terminologies, have been offered as 'final' over
the last 120 + years by Peet, Chace and many others. However,
by showing details of remainder arithmetic and Egyptian fractions
as remainders, in both modern and ancient scribal shorthand
(the two way streets) directly proves that the earlier scholarly
translations were incompletely translated into an ancient form of
base 10 (a one way street). Scholars have therefore been shown
to have omitted big chunks of scribal thinking, missing many
important scribal techniques, like remainder arithmetic, points
that may not have been missed had the scholars been required
to fully translate every problem into modern arithmetic beginning
120, 100 or 80 years ago.
These scribal gaps need to be filled. By building two-way streets,
looking at the meaning of the ancient arithmetic, algebra and geometry
in modern terms, and visa versa, following the clues provided by the
scribal mental and explicit shorthand facts, detailing a deeper forms
of Egyptian arithmetic, algebra and geometry.
This is one of the central purposes of this blog, to validate that
two-streets are more than just sometimes possible, they are
almost always a requirement to fairly read scribal mathematics,
as written by Ahmes and other scribes.
One of the clues that has been under valued is the wide use
vulgar fractions, a center piece of scribal mental arithmetic.
Scholars have long suggested that scribal division consisted
of only unit fraction terms, thereby eliminating the need for
modern scholars to consider vulgar fractions in the majority of
ancient text calculations.
For example, RMP # 32 states in modern terms
x + (1/2 + 1/4)x = 2 or
x + 7/12x = 2, or
19/12 x = 2, or
x = 24/19 = 1 + 5/19
Ahmes' mental shorthand notes worked with the 5/19 vulgar fraction.
Noting his 1 + 1/6 + 1/12 + 1/114 + 1/224 answer, it appears that
a generalized, but not optimized, Hultsch-Bruin method did the following:
Find 5/19 by first subtracting 1/12 or
5/19 - 1/12 = (41)/(12*19)
= (38 + 2 + 1)/(12*19)
= 1/6 + 1/114 + 1/224, or
the final answer
1 + 1/6 + 1/12 + 1/114 + 1/224
Ahmes' choice of 1/12 followed the 2/19 2/nth table suggestion,
where:
2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114
as also used in 2/95, where 2/19 x 1/5 = 1/5 (1/12 + 1/76 + 1/114)
as Wilbur Knorr and many others have seen, in 1982, as published
in the journal Historia Mathematica.
Had Ahmes 'thought of another number' like 1/4, a more optional
series 1/4 + 1/76 would have been found (as the 400 AD Akhmim
Papyrus, from the Coptic era includes in its n/19 table).
Gillings and other authors like Robins-Shute avoid discussing Ahmes'
shorthand that reveals this class of vulgar fractions fact - fragmentary
facts for sure, but facts that clearly point ONLY in the remainder
direction, then rewriting vulgar fractions as Egyptian fractions.
This blog details a wide range of specific textual ways that scribal
shorthand has been hard to read. I have been lucky in several areas,
assisted by many over the last 18 years, to have corrected several
scholarly oversights that had only reported what scholars could read
of ancient texts. Sadly many gaps are still left in the ancient texts, here,
there and everywhere. As a conseqence, to date, the oldest number
system has not been fairly read by modern language speakers, as the
scribes and archaeological luck, has left for us to read.
Special care needs to be taken when reading heiratic arithmetic
and its scribal shorthand. One new new method to achieve
this goal is documented on this blog. Choose whatever name
that fits in your mind for this new method. For myself, I call it
building two-way streets.
This blog's argument shows that ancient words and numbers
are only snapshots of ancient statements. To fully translate a
snapshot and find the ancient context, the fragmented scribal
shorthand must be read with rigorous methods. Filling in a
scribal gap at one's leisure, skipping ones that are not readily
understood, has commonly left out major segments of the
mathematical texts, one being remainder arithmetic.
To resolve several fragmented aspects, wider translations of the
scribal math shorthand must be developed that covers all the
available data. One technique that is being developed translates
the ancient data into modern base 10 decimals as a bilingual text.
The new method is comparable to the Rosetta Stone's use of
phonetic symbols to link the third language's decoding to the
first two. The new method uses quotients, remainder and common
divisors a central ideas.
Math scholars have often taken only an aspect of a text, and reported
interpretations outside of the original context when mathematics and
metrology were involved. This 'translate what you can' problem has
existed since the 1870's when the RMP was first published. This blog
will focus on clearly up one of the ancient math systems that have been
overlooked, that being the remainder arithmetic cited in the RMP and
other confirmationsal texts.
For example, RMP 65 states that
100/13 = 7 + 2/3 + 1/30
But what does that mean?
The Reisner Papyrus will be shown to have written:
n/10 = Q + R
where Q was a quotient and R was a remainder, first stated as
vulgar fraction, that was easily converted to an Egyptian fraction
series. The Reisner cites:
8/n1 = 1/2 + 1/4 + 1/20 = 8/10
48/n2 = 4 + 1/2 + 1/4 + 1/20 = 4 + 8/10
18/x3 = 1 + 1/2 + 1/10 = 1 + 6/10
64/x4 = 6 + 1/4 + 1/10 + 1/20 = 6 + 4/10
36/x5 = 3 + 1/2 + 1/10 = 3 + 6/10
such that x1 = x2 = x3 = x4 = x5 = 10
amid tables of data, as further explained on
http://reisnerpapyri.blogspot.com
That is,
100/13 = 7 + 8/13 or 7 + 2/3 + 1/39
with 7 being a quotient and 8/13 being a remainder
that was converted to 2/3 + 1/39.
Additional proof is provided by the word ro, a term often read
in metrology as 1/320th of a hekat. The word has a broader
meaning, including rest, remainder and related ideas that connect
more to arithmetic than metrology. Ro will be shown to have taken
on several numerical values other than 1/320 (as research has already
found, with the details being published in the near future).
For now, the word ro will be shown to have most often meant
'common divisor' to ancient scribes. Ro was linked to binary fraction
and Egyptian fraction series, a point that modern scholars are slowly
becoming aware. Ro did take on several numerical values. The
different values of ro were needed to exactly scale the Egyptian
fraction series, remainders of Horus-Eye divisions, as needed,
working with hekats, cubits (per Masse & Gewiche, 1994) and
other measures.
Stated in other terms, scholars have often reported the ancient
math data in fragmentary ways since the 1880's. By 1923 scholars
like Peet began to falsely conclude that his work was complete.
Peet conclued that ro only equaled 1/320th of a hekat and, by
implication, could not be re-valued within another context, such as
partitioning a cubit.
Peet's analysis stressed the idea of Egyptian division, as a limiting
factor on how and why the Akhmim Wooden Tablet scribe
partitioned a hekat by 1/3, 1/7, 1/10, 1/11 and 1/13. Peet
did not go into detail to show how the scribe proved his answers
by computing 3/3, 7/7, 10/10, 11/11 and 13/13, respectively. Peet
also did not seek out cubit partitioning methods, as later reported
by Masse and Gewiche, or any other weights and measures context.
Peet was not the only scholar that introduced misconceptions by
skipping over 'unreadable fragments' within a given text, while
attempting to create sweeping conclusions concerning the scope
and content of Egyptian mathematics.
Considering the original text, and sketchy translations of the fraction
based mathematics reported in several texts was straight forward
and often trivial. Interestingly, it will be shown that the math texts
have not been fairly translated from the combined Horus-Eye and
Egyptian fraction data cited in the RMP and other texts. Over 40
examples of binary and Egyptian fraction data have been ignored,
or grossly misread, since the data did not fit the scholarly additive
and subtractive paradigms of Egyptian arithmetic operations, as
detailed on:
http://mathorigins.com/image20%grid/awta.htm
and,
http://akhmimwoodentablet.blogspot.com
This blog will freshly analyze Egyptian division based on reading
the binary and non-binary data reported in single scribal statements
mentioned above.
As a goal, this blog for the first time will attempt to offer complete
translations for a couple of example problems within the RMP and
one other text to modern base 10 fractions. This blog also offers a
rigorous method that creates a new set of abstract facts that can be
easily double checked by qualified experts.
Continuing to step back, scholars, mostly working alone, had
unfairly thought that a rigorous translation to modern base 10 was
not necessary. Frances I. Griffith in 1891 sensed that ro meant
'greatest common measure' within the RMP. Others scholars working
in other texts, Daressy in 1906 with the Akhmim Wooden Tablet
sensed other interesting arithmetic features. Daressy sensed exactness
in all but the 13/13 case, where scribal typographical errors hindered
his work. Yet, nine Daressy examples showed that divisions and
proofs were exact, a clue that was not followed up by Peet and
others.
Interestingly, by 1923 Peet came along and tried to close off debate
related to the highest form of Egyptian division operations by refuting
the 1906 analysis of Daressy. Gee, didn't I get the gist of the math
discussion correctly, Peet might have retorted? Sadly, the majority
of early scholars had tried to stay within a singular view of Middle
Kingdom math texts, often created by themselves, with few or no
auditors, picking out readable aspects from ancient scribal
shorthand, skipping over 'unreadable aspects' (often without
mentioning their oversights).
Oddly, by 1923 scholars tended to agree on an additive aspect of
the mathematical texts. But were all the Egyptian mathematical texts
only additive or subtractive in scope, as David Silverman (and others)
continue to suggest in 1994?
It will be shown that math historians have not completely, and
therefore not fairly, reported important Egyptian mathematical
facts due to the oversight of not completing a rigorous translation
to modern base 10 fractions. This blog wll show that Egyptologists
and math historians have created no formalized methodology that
fairly and generally links the original scribal base 10 to modern
base 10. A fair 'decoding' system is needed such that modern
fraction views of the ancient texts can take into account all the
ancient text's contents. Such a 'decoding system exists in limited
circumstances. The proposed system can be rigorously double
checked in ways that offers great hope for the expansion of the
method to other poorly read mathematical texts.
Correcting oversights may be difficult. A formal methodology has
not been agreed to. However, one will be made available in this blog
that achieves a rigorous decoding key to ancient scribal thinking in
certain situations, the first being a hekat with a divisor of 64 or smaller.
The results of this methodology clearly opens up scribal remainder
arithmetic (a form of abstract math) Therefore, it will be shows that
non-additve or subtractive scribal mathematics did not dominate
Middle Kingdom mathematics, as has been assumed for over 100
years.
Proof lies in several examples, each showing the trivial side of
the discussion, within the Rhind Mathematical Papyrus. The
RMP lists several beginning type problems that our modern
4-6th graders would recognize, data that appears additive,
but actually is not. One problem is #83, where three classes
of birds were fed three different amounts, 1/6th of a hekat of
grain for three birds, 1/20th for one bird, and 1/40th for each
of three birds. Ahmes, the RMP scribe, then asks: how much
grain did all seven birds eat in one day?
A modern discussion of this trivial problem requires the
addition of 1/6, 1/6, 1/6, 1/20, 1/40, 1/40 and 1/40,
Note that an easy to reach common multiple (1/120) is
required to solve this problem if no modification in procedure
is introduced. Today, kids might improve the ease of working
the ancient problem by first adding the three 1/6 fractions,
obtaining 1/2. Then they could add 1/20 and 3/40. In this
manner only (1/40) is required to be used as a common divisor.
This is a basic form of modern logic overlay (a bilingual text, to
refer to a cryptanalysis standard) can be restated several ways.
A closely related arithmetic form states, that by adding:
1/2, 1/20 and 3/40 using the common divisor 1/40, or
(20 + 2 + 3)/40 = 5/8th of a hekat of grain, as Ahmes
the requested to know, how much did seven birds eat in one
day?
Ahmes performed this same arithmetical task, but chose another
interesting value for his common divisor, 1/320. Ahmes wrote his
fractional divisors of a hekat, his feeding rates, in terms of a larger
one that he named ro, one that was commonly seen in several
additional RMP problems, #35-38, 47, and 81.
To understand Ahmes' reason for selected 1/320th for a class of
problems, any one of which could have used a smaller common
divisor, let us examine Ahmes used of the word ro.
Stated in terms of ro (common divisor) units, Ahmes
first added up 53 1/3ro, 53 1/3 ro, 53 1/3 ro, reaching
160 ro. Ahmes then added 16 ro, 8 ro, 8 ro and 8 ro,
reaching 200 ro, or 200/320 = 5/8th of hekat. Trivial
right?
At this point in this discussion includes a few non-trivial points.
Why did Ahmes apparently chose only only one common divisor,
1/320th to be specific? One clue is given by, no other common
divisor was needed for any divisor n, where n was not greater
than 64, when that common divisor is multiplied by the Egyptian
fraction series associated with it.
Let me say that again. The trivial side of the modern discussion
shows that several common divisors can come into play to solve
this seven bird eating grain problem. In Ahmes' case all of his
common divisors were large, all multiples of 1/320. This multiple
of 1/320 fact was repeated in several other hekat division
problems, as it will be shown in five Akhmim Wooden Tablet
cases. There were 29 additional cases in the RMP, written in
terms of the hin unit, so there is adequate data to discuss this
issue, in depth.
This blog, and one other, will clearly report that Ahmes was
using remainder arithmetic to exactly partition a hekat, a volume
unit, adding remainder corrections that scribes nameed ro. This
none trivial statement needs to proven in a rigorous manner.
Remainder arithmetic used the Horus-Eye series as its quotient,
with the Egyptian fraction series including the 1/320th common
divisor (multiple) as its remainder term. That is, whenever a hekat
unity, 64/64, was divided by any relatively prime number, 3, 7,
11, 13 and so forth, a remainder term was created, one that
included 1/320th, the Egyptian word ro. Why did ro appear in
every relatively prime division answer, when the divisor was no
greater than 64?
Ahmes and his mentor writing in the Akhmim Wooden Tablets,
or an earlier unreported text, substituted the fraction 5/320 for
a 1/64th fraction that always appeared in the remainder
arithmetic statement:
(64/64)/n = Q/64 + R/(n*64)
such that Ahmes remainder arithmetic statement looked like this
(64/64)/n = Q/64 + (5*R/n)* ro
Further, it will be shown that for n > 64 another form of remainder
common divisor was used, with ro being excluded. Ro certainly
was not used in the medical texts, as the Papyrus Ebers and other
texts, when several divisor greater than 64 were in use, but why?
A definitive discussion will take place on this subject at a later time.
To limit our discussion to the n > 64, as RMP # 47 implies, an
increase in the size of the hekat unit numerator, from 64/64 to
256/64, 320/64 or even 640/640 was made to allow the Q term
to contain a value > 0. Until this addition info is published, this
blog will continue to be updated, reporting bits and pieces of
the search for additional proof to detail the scope and depth of
ancient scriibal solution. The ancient Horus-Eye problem was
solved not larter than 2,000 BC, as reported by the volume
measure system, using large and small unit divisors.
How early was it solved, in a manner that birthed the Egyptian
fraction system itslef, as required to scale the remainder term,
with or without ro?
Milo Gardner milogardner@juno.com
