Count Like An Egyptian

The Bizarre Fractions of Ancient Egypt
Wrath of Math From Wikipedia:

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics.

It is one of two well-known mathematical papyri, along with the Moscow Mathematical Papyrus. The Rhind Papyrus is the larger, but younger, of the two.

In the papyrus' opening paragraphs Ahmes presents the papyrus as giving "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries ... all secrets". He continues:

This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre. The scribe Ahmose writes this copy.

Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these stand out. The Rhind Papyrus was published in 1923 by the English Egyptologist T. Eric Peet and contains a discussion of the text that followed Francis Llewellyn Griffith's Book I, II and III outline. Chace published a compendium in 1927–29 which included photographs of the text. A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute.

Farther down the Wikipedia page is this image:

Ancient Egyption Units of Measurement in the Rhind Mathematical Papyrus

The image on Wikipedia contains some text, but the lines are very long which makes it difficult to read. Printing the image large enough to make the text legible requires two sheets of paper. I used Windows Paint. I had completely forgotten about this program, but it is around, at least on my Windows computer.

Anyway, I thought the text in this image ought to be converted to text, so I used OCR Online to extract the text and then edited it to correct the errors. The only words OCR had trouble with were the  highlighted words in color. Other than that it was perfect. Then I went ahead and chopped it up to make it more readable, at least in my mind. Here it is:

The present table is a concordance of ancient Egyptian units of measurement which are used throughout the Rhind Mathematical Papyrus, an ancient Egyptian document which is a record of elementary mathematics. The papyrus consists of four sections:

1. a title page with historical information,
2. a table of fractional calculations for 2/3 - 2/101, or the "2/n table" (where n is always odd),
3. a much smaller table of fractional calculations for the nine fractions 1/10 - 9/10, or the "1-9/10" table, and finally
4. a series of 91 "problems" or numbers, which are numbered from 1-87 (a modern convention imposed on the document to differentiate the problems) and include four additional items designated by moderns as 7B, 59B, 61B, and 82B.

Many of the latter problems make use of certain ancient Egyptian units of measurement, whether of length, volume, time, or otherwise, and this is what the present table summarizes. It happens that none of the 2/n table, the 1-9/10 table, or even most of the first several problems 7, 7B, 8-34 make any mention of units, and they are therefore not included in this table.
Almost all usage of units of measure is confined to the later problems in the papyrus (with some early exceptions, also listed), being the title page, problems 1-6, and problems 35-87, which include three additional items designated as 59B, 61B, and 82B.
For the title page and numbers 86-87, the context of the usage of units is not mathematical, but historical.

The units are grouped by unit type, and color-coded. From top to bottom, the unit types are:

• length,
• area,
• volume (deben),
• monetary (sha'ty),
• trigonometric (seked),
• food/manufacturing (pefsu), and finally
• "foodstuff" (loaves, des-measure).

The seked is not strictly speaking a trigonometric item, but its context is so close to our modern understanding of trigonometry that we identify it with that word.
The pefsu is also a kind of derived unit of measure relating to food preparation and manufacturing.
Finally, the "loaf' is not really a standard unit of measure within the papyrus as such, but it is mentioned so frequently in related contexts that it merits its own entry. In any one problem, it can be interpreted as a measure of solid food.
Likewise, the "des-measure", mentioned in only a few problems, can be interpreted as a volume unit of liquid measure (especially in the context of food and drink) which is not immediately related to the other volume units.

Entries in black indicate that a given unit of measure is explicitly named or entailed in the problem, in the original document.
Entries in gray indicate that although the unit's word is not expressly stated in that problem, the context of the document makes clear that the unit is implicitly being used in the course of the problem's calculation, statement, etc.
Note that neighboring groups of problems often tend to entail similar units, although in some cases the units are merely strongly implied by context, as opposed to being explicitly stated in the original document.

In the Rhind Papyrus, units of a given type have precise, exact, simple relationships to one another, which conversion factors we can and therefore do easily express in modern terms, using only integers.
Although there is historical evidence that certain units of measure can be "concretely" compared with modern units, these exact specifications are not essential to reading the Rhind Papyrus as a mathematical document.
Indeed, one can read the papyrus has having an internally consistent system of units of measure, which is all that is presented in this table and explanation.
The reader is cautioned that ancient Egypt had a very long history, and that the present information about units pertains directly to the Rhind Papyrus only.
At other periods of ancient Egyptian history, certain units were taken to have different conversion factors relative to other units.

Among units of length, 1 khet = 100 cubits = 700 palms = 2800 fingers.
Among units of area, one square khet is called one setat, and 1 setat = 100 cubit strips, where one cubit strip is a rectangular strip of area being 1 cubit by 100 cubits (or any other sector of equal area).

Among units of volume, the following holds:

2 cubic cubits = 3 khar = 60 heqats = 600 hinu = 19200 ro.

Furthermore as one might expect, the somewhat redundant "hundreds/multiples" versions of the heqat and the ro can be equated with those same "base units" in the following wise:

1 hundred quadruple heqat = 2 hundred double heqats = 4 hundred heqats = 100 quadruple heqats = 200 double heqats = 400 heqats = 128000 ro = 64000 double ro = 32000 quadruple ro.

Combining these two chains of equalities, rearranging their terms, and expressing all of the units' conversion factors in the simplest terms of integers yields the following comparison among all of the standardized units of volume measurement in the Rhind Papyrus:

3 hundred quadruple heqats = 6 hundred double heqats = 12 hundred heqats = 300 quadruple heqats = 600 double heqats = 1200 heqats = 40 cubic cubits = 60 khar = 12000 hinu = 384000 ro = 192000 double ro = 96000 quadruple ro.

Among time units, since the Egyptian month is always exactly 30 days, then 6 years = 73 months = 2190 days. The remaining units, being of different types and not having simple or relevant conversion factors into other units of like type (at least as far as the papyrus itself is concerned) are therefore not expanded upon here.

More posts about fractions.

Another post about ancient mathematics: the Babylonian Plimpton 322 clay tablet. It dates from 1800 BC, so roughly the same era the Rhind Papyrus.

Today's Lesson

Stolen entire from JMSmith.

The Secret Badges that we Wear

“She bound the scarlet line in the window.” Joshua 2:21

A shibboleth is a special kind of password, which is to say a key or badge that opens a social door and grants admission to a social group.  As everyone versed in scripture knowledge knows, shibboleth was at first a word that the lisping Ephramites could not pronounce, and that the Sons of Giliad therefore used to identify the survivors of a shattered Ephramite army.  When a bloodied and bedraggled warrior staggered down to the ford of the Jordan, he was challenged to pronounce the word “shibboleth,” and thereby show his secret badge. Those who pronounced it “sibboleth” were immediately slain.

As I explained some years ago, shibboleths often take the form of sacrilege.  The password that grants admission to one social group is in such cases a violation of the norms of that group’s enemy.  Thus anti-Christian secret societies used to require an aspiring member to trample on a crucifix, spit on a Bible, or otherwise treat the sacred objects and words Christianity as profane.  In on-line culture, such sacrilegious shibboleths are called “shill tests,” and in these tests an aspiring member is required to type words that an undercover agent would find it very hard to type.  As I explained in that long-ago post, to get past the sentries of one group, you must often slay another group’s sacred cow.

A shibboleth is a password that prevents infiltration by spies.  A “scarlet thread” is a secret badge with which a traitor makes himself known to his new compatriots, and is thereby passed over when his new compatriots descend like wolves on the people whose compatriot the traitor only pretends to be.  Like shibboleth, the term “scarlet thread” comes from the Old Testament, and more particularly from the curious tale of a traitor known as Rahab the Harlot.

Rahab the Harlot dwells in the city of Jericho, and when Joshua sends two spies into that city, the two young men somehow fall in with Rahab, who we must never forget is a harlot, and she hides these two handsome young spies in her house.  This house was built on, or rather up against, the city wall; and on its city-wall side had a high window that looked out over the countryside.  It was from this high window that Rahab lowered Joshua’s spies on a “scarlet thread,” after telling them that her people were ripe for conquest because they were rotten with fear.

“Your terror is fallen upon us . . . the inhabitants of the land faint because of you.”

Rahab’s reward for betraying her people is that she and her family will be spared when Jericho is sacked and its citizens are slaughtered.  But to assure their deliverance, Rahab must keep her family in her house (of ill repute), and must display the “scarlet thread” in the front window. of that house.

“Behold, when we come into the land, thou shall bind this thread in the window which thou didst let us down by; and thou shalt bring thy father, and thy mother, and thy brethren, and all thy father’s household, home unto thee” Joshua 2: 18.

Thus a “scarlet thread” is a means whereby a traitor escapes the destruction brought down by his or her treason.  One wonders how many who “miraculously” survive some near-universal destruction owe their deliverance to display of a secret badge or “scarlet thread.”  Rahab was a harlot, so she knew how treachery works.

“She bound the scarlet line in the window.”

Not long after, as every Sunday-school scholar knows, Joshua’s army compasses the city of Jericho and “the walls came a tumbling down.”  Then, as many Sunday-school scholars do not know, before the sack and slaughter begins, a house displaying a “scarlet thread” is sought and a traitorous harlot is saved.   Which was very fortunate for that traitorous harlot and her family, because of Joshua’s army we are told:

“They utterly destroyed all that was in the city, both man and woman, young and old, and ox, and sheep, and ass, with the edge of the sword.”  Joshua 6: 21

Like a shibboleth, a “scarlet thread” is a special kind of password, although it grants its possessor the privilege of escape and not admission.  It is a secret badge with which traitors and spies show their true colors to their true friends, and by which traitors and spies are exempted when their true friends utterly destroy their pretended friends with the edge of the sword.