ancient egypt: history and culture
Ancient Egypt: Counting and measuring
The number system
Counting
Measuring
   Time
      Sundials and shadow clocks
      Water clocks
   Length
      Long distances
      Short distances
   Area
   Capacity
   Inclination
   Weight

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Counting and measuring

The number system

Numerals     The Egyptian number system was decimal, seemingly with a word for every digit and for all the tens from 10 to 90. The hundreds from 200 to 900, the thousands, the ten thousands, the hundred thousands and the millions were counted like any other thing [13]: two thousand, three thousand etc. The numerals for a million and for a hundred thousand fell into disuse during the course of history [14], leaving the finger sign for 10,000 as the largest numeral. The concept of zero as a number was unknown.
    In the later hieratic each of the groupings of ones, tens, hundreds etc. was represented by an abstraction of the hieroglyphic signs making it up: nine was written as 9, which one can still recognize as a cursive version of nine vertical strokes, but fifty, written as 50, has lost any such visual connection. Numbers between 1 and 10,000 needed therefore a maximum of 4 signs, making the text concise.
    There was an additional way of representing large numbers, writing the signs for the multiplier underneath the sign for a multiplicand - 300,000 was thus represented by the tadpole sign above three single strokes. Demotic followed these conventions.
    In hieroglyphics, on the other hand, the seven basic numerals were generally repeated as needed: 9999 would have been written using nine ones, tens, hundreds and thousands, a total of 36 signs, somewhat cumbersome and easily misread. The system of multiplicand above multiplier was also used occasionally.
    Numbers were mostly written with the largest components preceding the smaller ones, but there was little importance to that as the system was not positional, and aesthetic considerations overrode, as was their wont, all others.

Counting

    People have always counted or measured their possessions in order to be able to keep them safe, while their governments have done so to be able to tax them. Cattle were among the most expensive investments a peasant could make. In Egypt they furnished the power for the cultivation of the fields, but otherwise they were little exploited: their milk yield - it was seemingly the docile cows rather than the stronger but less amenable males whose strength was harnessed - was low and dropped even lower during plough time, and the ordinary person rarely ate beef. A peasant, therefore, might own just a single cow, while the great estates counted their beasts by the thousands. In the following tomb scene the count of the horned cattle is given as 835, and of the hornless as 220. The tomb owner had 3209 goats of various kinds and 760 donkeys, the main beast of burden. Tomb 75, Giza, Source: Lepsius
    One of the recurring events organized by the administration were the cattle counts, held yearly or every other year. These numberings as recorded on the Palermo Stone are one of the cornerstones of modern dating of the earliest kings. Under Neterimu of the second dynasty the cattle counts were performed with great regularity every second year, while during the reign of Snefru the eighth occurrence followed the seventh after one year only.
    Population censuses were seemingly also taken occasionally, thus in the 4+x-th year of an unknown first dynasty king there was - - if Breasted's interpretation of the following, badly understood, passage is correct - - a
[Numbering of (?)] all the people of the nomes of the west, north and east.
J.H.Breasted, Ancient Records of Egypt, Part One, § 106
and the coptic tradition held as late as the 7th century CE that there was a census under Snefru when 500,000 men were counted.
    Years were counted from the rulers's accession to the throne [4]. Dates were composed of the season, the number of the month in the season and the day of the month, in official documents generally preceded by the year of the reign, often in conjunction with the name of the ruling pharaoh. In later times the twelve months were at times given individual names.
3rd Shemu 9
New Kingdom, Ostrakon Berlin P 10655 found at Deir el Medine.
Year of the reign 28, 3rd Akhet 9
New Kingdom, Ostrakon Berlin P 1121 found at Deir el Medine.
Year 11, 1st month of Summer of Pharaoh Ptolemy (VI) son of Ptolemy
Ptolemaic Period, Papyrus BM 10591
Nekhtemonth, son of Petemestou, has written (this) in the year 11, on the 18th of Epiphe
    The Egyptian administrators justified their existence at least in part by listing and enumerating anything that went through their system: bunches of flowers, blocks of stone, jars [of the canal (?)] filled with fish, having wooden [lids (?)], or loaves of bread. At times we may wonder what all these lists were for and whether the amounts listed were real or fictitious. And, like much of the handiwork of bureaucrats, reading their records does not necessarily gladden our hearts, even if it did theirs. One of the more gruesome aspects of their efforts were the calculations they performed, totting up limbs and other body parts cut off from, as we hope, enemies already dead. Ramses III's scribes recorded data concerning his first Libyan war:
Total of foreskins, [1]2,535
Total of hands, 12,535
Total of /// (according to the accompanying picture, genitals), 12,759 (+x)
Total of hands, 12,520 (+x)
Total of hands, 12,635 (+x)
J.H.Breasted, Ancient Records of Egypt, Part Four, § 52
    These "official" numbers are mercifully small compared with the boastful claims made by the king himself who had been slaying hundreds of thousands in their place under his horse. In later inscriptions he speaks of despising a million, holding in contempt two millions, firm-hearted charging into hundred thousands, exploiting the effect such huge numbers would have on the beholders of the monuments and, hopefully, on his gods.
    With the notion of statistics unknown all these records were of little practical consequence, and remain a monument to the king's ambitions and a not altogether very reliable source of information for modern scholars.

Measuring

    Defrauding one's neighbour by using false measures was an offence against the divine order. The forty-two declarations of innocence, generally referred to as Negative Confessions, included
I have not diminished from the bushel.
    The consequences for such behaviour in the next life was utter destruction. In this world measures were taken to prevent it [10], and according to Diodorus Siculus the authorities did not take such crimes lightly:
Counterfeiters and falsifiers of measures and weights, or forgers of seals, scribes who made false entries in the public books or deleted something from the register, as well as those who substituted documents, had both hands cut off.
Diodorus Siculus, Historic Library Vol 1, Chap. 78, after a translation by Julius Friedrich Wurm

Time

    Most people had little need for measuring time. Their life was governed by the sun: They generally rose at break of day and went to bed after nightfall. At midday when the sun stood highest, they probably had a rest. In the vein of the Satire of the Trades the life of the peasant is described as tough in the Lansing papyrus:
By day he cuts his farming tools; by night he twists rope. Even his midday hour he spends on farm labor
M. Lichtheim, Ancient Egyptian Literature, Vol. 2, p.170
Clocks     Life in the temples was more closely regulated, and services were held at prescribed hours of the day. At night hour-priests were seemingly performing services in anticipation of the rising of the Sothis. The appropriate time for beginning and ending them was decided upon by the hour-priest in his role of "observer of the hour", the jmj-wnw.t, based on the position of the sun and stars.
    To this purpose both the day and the night were divided into twelve hours. A day and a night hour were not of equal duration; their length changed with the seasons, day-hours being longer in summer than in winter.

Water and sun clocks
Source: Petrie Museum website [5]

    The division of the whole day into 24 equal hours was the idea of Hipparchus of Niceae [6] who worked in Alexandria in 127 BCE. Another scholar active in Alexandria during the second century CE, Claudius Ptolemaeus, divided the hour into sixty minutes [7].
    The first attempts to use implements in the determination of the time of day were apparently made during the second millennium BCE.

Sundials and shadow clocks

    The shadow of a tall, thin object could be evaluated as to its length, which would be shortest at noon, or its direction, which would be due north at midday.

Water clocks

    The oldest surviving water clock dates to Amenhotep I. Water clocks were generally bowls of stone or pottery which had a small hole at the bottom through which the water could run out. The inside of the bowl had markings where the level of the remaining water could be determined. Some clocks had a number of marker sets for the different seasons as the length of these temporal hours changed. The Greeks added an hour dial and a hand connected to a float for easier use.

Length

Long distances

    Long distances were measured by counting the days or months it took to traverse them. They were therefore dependent on the means of transportation and the route used, which meant shipping and the river Nile in the case of Egypt, and not distances "as the crow flies". The iter, a river unit, grew out of these travel estimates. Thutmose III wrote about his eighth campaign into Naharin that he [pursu(?)]ed after them an iter of sailing [2]. In his Semna inscription viceroy Mermose describes the muster of his army in Nubia
He made troops, commanded by commanders, each man with his village; from the fortress of Beki to the fortress of Taroy, making 52 iters of sailing.
J.H.Breasted, Ancient Records of Egypt, Part Two, § 852
While the fortress of Beki lay near Kubban some fifty kilometres above the first cataract, the location of Taroy is unknown, and the length of an iter cannot be defined with the help of this inscription. Breasted accepted the length of an iter to be about 1.4 miles (2.2 km), while nowadays it is thought by some to have been 20,000 cubits, about ten and a half kilometres.
    In the Late Period it was identified by the Greeks with the schoine of 60 stadia [3]. Herodotus who travelled along the Nile gave estimates which agree less well with reality than do similar estimates of distances known from Greece.

Short distances

cubit rod, Middle Kingdom - Source: UCL

Common cubit rod with markings for palm divisions, Middle Kingdom
Source: UCL

    The royal cubit, mH-nswt, almost 52 cm, was the base of the official Egyptian measuring system used in the construction of temples etc., while the shorter common cubit of about 46 cm was used in every day measuring until the Late Period. There were a plethora of subdivisions, generally based on the relative sizes of different body parts, and multiples of these and other basic units, some imported from abroad and some in sectorial use like the 65 cm long 'pole' used by tradesmen. A few of them are listed in the table below:
 
     Royal cubits (mH nsw.t)     Small cubits (mH Srr)
iter (jtr)20,000 ?c. 10.5 km
stick of rope (xt n nw)100c. 52 m
fathom (Hp.t)5c. 230 cm
cubit (mH)1c. 52 cm1c. 46 cm
shoulder (rmn) - ½ of the diagonal of a 1 cubit square20/28c. 37 cm
sacred cubit (mH Dsr) - 4 palms16/28c. 30 cm
fist (Amm.t)6/28c. 11 cm
hand (Dr.t) - 5 fingers5/28c. 9 cm
palm (Szp) - 4 fingers1/7c. 7.5 cm1/6c. 7.5 cm
finger (Dba) - ¼ of a palm1/28c. 1.9 cm1/24c. 1.9 cm

Area

    Being able to establish the size of areas was vital for the determination of taxes peasants had to pay and for the reallocation of farmland in case the inundation had changed the landscape and obliterated familiar landmarks. The Egyptian scribes knew how to calculate the approximate areas of triangular, quadrilateral and even circular fields.[15][16] They may have calculated the areas of fields with irregular features by dividing them into simple forms like triangles and quadrilaterals and adding all their areas, though there is no direct evidence for such a procedure.[17]
    The basic unit was the setat (sTAt), called aroura by the Greeks, which originally may have been of varying size in the different nomes. Later it was univerally based on the royal cubit.
 
1000 land-units (xA-tA)NK10 setat, 100 by 1000 cubits  c. 27,600 m²
setat (sTAt), arouraOK, MK, NK, LP  100 by 100 cubitsc. 2,760 m²
a 'thousand', i.e square cubits, (xA)  OK10 by 100 cubitsc. 276 m²
land-unit, (tA / mH-tA)OK, MK / NK, LP10 by 10 cubitsc. 27 m²
shoulder, (rmn)½ tAc. 13.5 m²
account unit, (Hsb)½ rmnc. 6.8 m²
sA½ Hsbc. 3.4 m²
 

Capacity

Measures, Source: Petrie Museum website     Grain, the most important commodity that required accounting for was measured in barrels, heqat (HqAt), its multiples, khar (XAr), Oipe (jp.t) and subdivisions, hin (hnw). The relationships between the various units were not constant over the millennia. During the Old (OK) and Middle (MK) Kingdoms the system was decimal, but became increasingly binary during the New Kingdom.
 
sack (khar)OK, MK10 heqat48 litres
sack (khar)NK, 3IP16 heqat76.8 litres
oipeNK4 heqat19.2 litres
double barrelOK, MK, NK2 heqat9.6 litres
barrel (heqat)OK, MK, NK4.8 litres [12]
jar (hin)OK, MK1/10 heqatc. 0.5 litre
roMK, NK1/64 heqatc. 15 ccm [12]

Measures
Source: Petrie Museum website [5]
(The scale measures accompanying the illustrations are marked in centimetres)

    Other substances were measured using various units: liquids were measured in jugs, (ds), which contained about half a litre, goose-fat in Aaat-jars of 3 heqat, or wine in hbn.t. The ro (rA) measured 1/32 of a hin, about 15 millilitres, the sA 1/3 of a hin.
    Ramses III supplied his temples with white fat, goose fat, olives and olive oil, honey , shedekh, white incense and other commodities in mn-jars, pomegranates in crates (pdr), blossoms in baskets (Htp), beer in Trf-measures and incense in ka-Hr-ka-jars. But the question remains whether all these containers were standardized and used as measuring units, or just items in bureaucrats' lists having little to do with reality [8].

Inclination

Plummets     The human eye is very good at spotting even small deviations from the vertical, when looking on from a distance. Builders, standing close to the walls they were erecting, must often have preferred to work with a plummet to get their surfaces to look straight and upright. And the fact that a plummet is always true made it proverbial:
Is it not wrong, a balance that tilts,
A plummet that strays,
The straight becoming crooked ?
The Eloquent Peasant
M. Lichtheim, Ancient Egyptian Literature, Vol.1. p.173
    A slope (sqd), such as that of a pyramid, was quantified as the vertical rise per royal cubit, measured in palms [11]. Khufu's pyramid with a slope of close to 52° had a quotient of about 5½.

Weight

Width, Source: Petrie Museum website     Bulk was mostly measured in volume, metal, the volume of which cannot be easily ascertained unless it is in the form of dust or a bar, was generally weighed. Few names of weights are reported. There was the deben (dbn), as copper deben weighing probably about 27 grammes, and half that as gold deben [1]; from the New Kingdom onwards the deben was 91 to 93 grammes. The kite (qdt) was one tenth of this, the ring, translit. Sna.tj. one twelfth. The ro (rA), an amount frequently used by physicians, was one fifth of a kite.

Weights made of various substances
Source: Petrie Museum website [5]

    Deben, kite and seniu, the last an Old Kingdom unit which fell into disuse during the New Kingdom, were used to state value, signifying the equivalent weight of metal a commodity was worth. Five deben worth of grain would not mean 450 grammes of wheat, but rather how much wheat you could buy for 5 deben of metal, which - in a case like this - would have been copper.
    The forms weights took on were manyfold. The original deben seems to have been a ring. There were weights in the form of cow statuettes, but most of them had simple geometric shapes like conuses, spheroids and the like.
The balances, accurate and true, of Thoth, which the king of Upper and Lower Egypt, [Make]re (Hatshepsut), made for her father, Amun , lord of Thebes, in order to weigh the silver, gold, lapis lazuli, malachite and every splendid costly stone, for the sake of the life, property and health of her majesty
J. H. Breasted Ancient Records of Egypt, Part Two, § 280
    Errors in weighing were commonplace. The scales, generally hand held, were imprecise, and the weights had a tolerance of 10 to 20 percent [9]. Moreover there was cheating which was not just an offense in this world, but counted heavily against the deceased in the next. The negative confessions include these statements
I have not added to the weights of the scales.
I have not misread the pointer of the scales.
and the deceased waits anxiously for the following words of Thoth, clearing him of any wrongdoing
Hear ye this judgment. The heart of Osiris hath in very truth been weighed, and his Heart-soul hath borne testimony on his behalf; his heart hath been found right by the trial in the Great Balance. There hath not been found any wickedness in him; he hath not wasted the offerings which have been made in the temples; he hath not committed any evil act; and he hath not set his mouth in motion with words of evil whilst he was upon earth.
The Papyrus of Ani translated by Wallis E. Budge

[2] J.H.Breasted, Ancient Records of Egypt, Part Two, § 479
[3] One stadium measured about 185 metres. A schoine was therefore, according to Herodotus, ca. 11 km. The length of the schoine according to Strabo varied from 30 to 120 stadia.
[4] A reasonable enough solution for every day purposes, used all over the ancient world, but which causes a number of problems for historians, e.g. the name of the ruler may not be mentioned in the document or be unknown otherwise, the date of accession may be unclear because of a co-regency etc.
[8] J.H.Breasted, Ancient Records of Egypt, Part Four, §§ 286ff.
[9] cf. Petruso, 1981,   Petrie, Kahun, Gurob and Hawara, pp.42ff,   Petrie, Illahun, Kahun and Gurob, pp.14ff.
[10]
Year 17, second month of winter, day 29, that day the chiefs of the king's tomb said to (the) scribe Ahpet (aHpt) that the oipe measure with which one measures the grain ration is too small, and he asked: "It belongs to whom, this oipe measure?" (who is responsible for it?) and they answered: "It was the scribe Paser (pA-sr) who acquired it and then one acquired a new oipe and one measured it up with the hin (the most accurate hin) and it contained only 38 hins.
Sarah Israelit-Groll A sense-equivalence translation of ostracon Gardiner 5
from Studies in Egyptology, Jerusalem 1990, p.406
An oipe contained 40 hins.
[11] The Rhind Papyrus gives an example of how inclination, the cotangens of the inclination angle, was calculated (The Egyptian notation for fractions is peculiar. Except for 2/3, Egyptians used only fractions the numerator of which was 1. To them ¾ was ½ + ¼):
If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked ?
 
The solution given was:
 
Take 1/2 of 360; it makes 180. Multiply 250 so as to get 180; it makes 1/2 1/5 1/50 of a cubit. A cubit is 7 palms. Multiply 7 by 1/2 1/5 1/50:
1            7
1/2         3 1/2
1/5         1 1/3 1/15
1/50          1/10 1/25
The
seked is 5 1/25 palms [that is, (3+1/2)+(1+1/3+1/15)+(1/10+1/25)=5 1/25]
Eli Maor, Trigonometric Delights, Princeton University Press, 1998
[12] The heqat, like the ro, could have different values, all multiples of the basic unit, even though the same name was used at times. In the records amounts were quite often given without mentioning the unit. The heqat (bushel or barrel) was 4.8 litres, 9.6 litres (common bushel or double barrel), 14.4 litres and 19.2 litres (large bushel or oipe). The ro varied from 15 cm³ to 60 cm³ in steps of 15 cm³.
(cf. Pommerening, Tanja: Die altägyptischen Hohlmaße, Hamburg, Buske 2005)
[13] A letter from Elephantine begins with these well-wishes:
[May your state of health be like life] millions of times, hundreds of thousands, tens of thousands, thousands, hundreds (and) ones [of times (?)].
Letter from the Elephantine archive, pStraßburg A, 6th dynasty
After the transliteration and German translation on the Thesaurus Linguae Aegyptiae web site
[14] HH as a numeral appears to have gone out of use by the Middle Kingdom. It was still used in monumental inscriptions with their preference for the traditional writing style. In everyday usage HH came to mean multitude, as did occasionally smaller numerals such as xA, thousand or the plural xA.w, thousands.
Thousand of clothes, thousand of alabaster vessels, thousand of (loaves) of bread, thousand of (heads) of cattle, thousand of antilopes.
Mastaba of Iunu at Giza (G 4150), offering table inscription
After the transliteration and German translation on the Thesaurus Linguae Aegyptiae web site
The tadpole sign Hfn for 100,000 was at least partially retained in the hieratic but completely dropped in the demotic script.
[15] Marshall Clagett, Ancient Egyptian Science: A Source Book, American Philosophical Society, 1935, Volume 3, pp.68ff. Arnold Buffum Chace et al., The Rhind mathematical papyrus, British Museum 10057 and 10058 , Mathematical Association of America, Volume 1, 1927
[16] It has been suggested, that the Egyptian formula for calculating the area of a circle (d-d/9)² was derived from cutting off of the four corners of a square and turning it into an octagon, which gave a reasonable approximation for the area of a circle inscribed in that square. (cf. Jöran Friberg, Unexpected Links Between Egyptian and Ba, World Scientific, 2005 p.42)
[17] In an exercise in P. Rhind (which is not very well understood) a triangle is cut into three pieces: two trapezes and a small triangle. Some of the data (areas and lengths of sides) are known and the calculation of other data is required, (cf. Jöran Friberg, Unexpected Links Between Egyptian and Ba, World Scientific, 2005 pp.45ff.) a fairly complex problem. Cutting irregular areas into calculable shapes would have been simple in comparison.

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These are just suggestions for further reading. I do not assume any responsibility for the content of these sites
 
Measuring weight in Ancient Egypt[1] Measuring weight in Ancient Egypt
Petrie Museum collection[5] Petrie Museum collection
Hipparchus of Nicaea[6] Hipparchus of Nicaea
Claudius Ptolemy[7] Claudius Ptolemy
Deir el-Medina ostracaDeir el-Medina ostraca: a selection of economic records
Egyptian numeralsEgyptian numerals
Von Zahlen und ZahlwortenK. Sethe, Von Zahlen und Zahlworten bei den alten Ägyptern
NumbersNumbers
The Story of NumbersThe Story of Numbers
Ancient Creation StoriesAncient Creation Stories Told by the Numbers by H. Peter Aleff

 

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