Ancient Egypt: Counting and measuring
The number system
Sundials and shadow clocks
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Counting and measuring13]: two thousand, three thousand etc. The numerals for a million and for a hundred thousand fell into disuse during the course of history , 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 , which one can still recognize as a cursive version of nine vertical strokes, but fifty, written as , 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. 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.
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.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 . 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 9The 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, 2,535These "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.
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 , 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.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 laborLife 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
The first attempts to use implements in the determination of the time of day were apparently made during the second millennium BCE. 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.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 . Herodotus who travelled along the Nile gave estimates which agree less well with reality than do similar estimates of distances known from Greece.
Common cubit rod with markings for palm divisions, Middle Kingdom
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.
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 .
Is it not wrong, a balance that tilts,A slope (sqd), such as that of a pyramid, was quantified as the vertical rise per royal cubit, measured in palms . Khufu's pyramid with a slope of close to 52° had a quotient of about 5½. 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
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 majestyErrors in weighing were commonplace. The scales, generally hand held, were imprecise, and the weights had a tolerance of 10 to 20 percent . 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.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.
 J.H.Breasted, Ancient Records of Egypt, Part Two, § 479
 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.
 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.
 J.H.Breasted, Ancient Records of Egypt, Part Four, §§ 286ff.
 cf. Petruso, 1981, Petrie, Kahun, Gurob and Hawara, pp.42ff, Petrie, Illahun, Kahun and Gurob, pp.14ff.
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.An oipe contained 40 hins.
 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 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)
 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 (?)]. 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.The tadpole sign Hfn for 100,000 was at least partially retained in the hieratic but completely dropped in the demotic script.
 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
 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)
 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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| Measuring weight in Ancient Egypt|
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|Deir el-Medina ostraca: a selection of economic records|
|K. Sethe, Von Zahlen und Zahlworten bei den alten Ägyptern|
|The Story of Numbers|
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