Ancient Egyptian units of measurement

Ancient Egyptian units of measure include units for length, area and volume.

Length

Units of length date back to at least the early dynastic period. In the Palermo stone, for instance, the level of the Nile river is recorded. During the reign of Pharaoh Djer the height of the river Nile was given as measuring 6 cubits and 1 palm. This is equivalent to approximately 3.2 m (roughly 10 feet 6 inches).^[1]

A third dynasty diagram shows how to construct an elliptical vault using simple measures along an arc. The ostracon depicting this diagram was found in the area of the Step Pyramid in Saqqara. A curve is divided into five sections and the height of the curve is given in cubits, palms and fingers in each of the sections.^{[2] [3]} Egyptian Circle

Hiero in tables refers to Gardiner Numbers from the Sign List starting p. 438, values of measurements refer to section 266 p. 199-200;^[3] see also ^[4][5]

Lengths could be measured by ordinary cubit rods, examples of which have been found in the tombs of officials for lengths up to the sizes measured by remen (see list of units below) and royal cubits which were used for land measures such as roads and fields, using rods, poles and knotted cords. Fourteen such rods, including one double cubit rod, were described and compared by Lepsius in 1865.^[6] Two examples are known from the tomb of Maya – the treasurer of Tutankhamun – in Saqqara. Another was found in the tomb of Kha (TT8) in Thebes. These cubits are about 52.5 cm long and are divided into palms and hands: each palm is divided into four fingers from left to right and the fingers are further subdivided into ro from right to left. The rules are also divided into hands^[7] so that for example one foot is given as three hands and fifteen fingers and also as four palms and sixteen fingers(see the second register of the Turin cubit illustrated below)^{[1][3][3][4][5][5] [7]}

Cubit rod from the Turin Museum.

For longer distances, such as land measurements, the Ancient Egyptians used rope. A scene in the tomb of Menna in Thebes shows surveyors measuring a plot of land using rope with knots tied at regular intervals. Similar scenes can be found in the tombs of Amenhotep-Sesi, Khaemhat and Djeserkareseneb. The balls of rope are also shown in New Kingdom statues of officials such as Senenmut, Amenemhet-Surer and Penanhor.^[2]

Units of Length^[1][2]

Name	Egyptian name	Equivalent Egyptian values	Approx Metric Equivalent
Finger	djeba	1 finger = 1/4 palm	1.875 cm
Palm	shesep	1 palm = 4 fingers	7.5 cm
Hand	drt	1 hand = 5 fingers	9.38 cm
Fist	3mm	1 fist = 6 fingers	11.25 cm
Span (small)	pedj-sheser	1 small span = 3 palms = 12 fingers	22.5 cm
Span (large)	pedj-aa	1 large span = 3.5 palms = 14 fingers	26 cm
Djeser	djeser	1 djeser = 4 palms = 16 fingers = 1 ft	30 cm
Remen	remen	1 remen = 5 palms = 20 fingers	37.5 cm
Greek cubit	meh nedjes	1 short cubit = 6 palms = 24 fingers	45 cm
Royal cubit	meh niswt	1 royal cubit = 7 palms = 28 fingers	52.4 cm
Pole	nbiw	1 nbiw =6 hands = 8 palms = 32 fingers	60 cm
Rod of cord	khet	1 rod of cord = 100 cubits	52.5 m[1]
River measure	iteru	1 iteru = 20,000 cubits	10.5 km

Area

The records of areas of land date back to the early dynastic period. Gifts of land recorded in the Palermo stone are expressed in terms of kha, setat, etc. Further examples of units of area come from the mathematical papyri. Several problems in the Moscow Mathematical Papyrus for instance give the area of a rectangular plot of land (measured in setjats) and given a ratio for the lengths of the sides of the rectangles one is asked to compute the lengths of the sides.^[1]

The setat was equal to one square khet, where a khet measured 100 cubits. The setat could be divided into strips one khet long and ten cubit wide (a Kha).^[2][8][9]

Units of Area^[1][2]

Name	Egyptian name	Equivalent Egyptian values	Approx Metric Equivalent
S3	s3	^1⁄8 st3t (Greek Aroura of 1 sq khet) 1250 sq cubits	345 m^2
hsb	hsb	^1⁄4 st3t (Greek Aroura of 1 sq. khet) 2500 sq cubits	689 m^2
rmn	rmn	^1⁄2 st3t (Greek Aroura of 1 sq. khet) 5000 sq cubits	1378 m^2
Khet	khet	100 sq cubits, (sq side of setat)	52.5 m^2 (Gillings)
Setat (setjat)	setat	1 sq khet = 10,000 sq cubits	0.276 ha
h3-t3	kha	1000 of land 10 arouras, 100,000 sq cubits	2.76 ha
Ta	ta	100 sq cubits = 1/100 setat	27.6 m^2[8]
Shoulder (Remen)	remen	^1⁄2 ta = 50 square cubits	13.7 m^2[8]
Heseb	heseb	^1⁄2 remen = 25 sq cubits	6.9 m^2[8]

Volume, capacity and weight

Several problems in the mathematical papyri deal with volume questions. For example in RMP 42 the volume of a circular granary is computed as part of the problem and units of cubic cubits, khar, quadruple heqats and heqats are used.^[1][5]

Problem 80 on the Rhind Mathematical Papyrus recorded how to divide grain (measured in heqats), a topic included in RMP 42 into smaller units called henu:

Problem 80 on the Rhind Mathematical Papyrus

The text states: As for vessels (debeh) used in measuring grain by the functionaries of the granary, done into henu : 1 hekat makes 10 [henu]; 1/2 makes 5 [henu]; 1/4 makes 2½ etc.^[1][5]

Units of volume and capacity^[1][2]

Name	Egyptian name	Equivalent Egyptian values	Approx Metric Equivalent
Deny	deny	1 cubic royal cubit = 30 hekat = 300 hinu = 480 dja = 9600 ro	144 liters
Khar (sack)	khar	20 heqat (Middle Kingdom) 16 heqat (New Kingdom)[10]	96.5 liters (Middle Kingdom) 76.8 liters (New Kingdom)[10]
quadruple heqat	hekat-fedw	4 heqat = 40 hinu = 64 dja = 1280 ro	19.2 liters
double heqat	hekaty	2 heqat = 20 hinu = 32 dja = 640 ro	9.6 liters
Heqat (barrel)	hekat	10 hinu = 1/30 deny = 320 ro	4.8 liters
Hinu (jar)	hnw	1/10 heqat = 32 ro = 1/300 deny	0.48 liters
Dja	dja	5/8 hinu = 20 ro[11]	0.30 liters
Ro	r	1/320 heqat = 1/9600 deny = 1/32 hinu	0.015 liters

Weights were measured in terms of deben. This unit would have been equivalent to 13.6 grams in the Old Kingdom and Middle Kingdom. During the New Kingdom however it was equivalent to 91 grams. For smaller amounts the kite (1/10 of a deben) and the shematy (1/12 of a deben) were used.^[2][5]

Units of weight^[2]

Name	Egyptian name	Equivalent Egyptian values	Metric Equivalent
Deben	dbn		13.6 grams in the Old Kingdom and Middle Kingdom. 91 grams during the New Kingdom
Kite	qd.t	0.1 deben
Shematy	shȝts	^1⁄12 deben

Time

Years were not numbered but rather named. When named after rulers they are thus regnal years. The Egyptians divided their year (rnpt) into three 120-day seasons of four months of 30 days (hrw) named 3ht or Akhet (inundation), prt or Peret, (emergence) and shmu or Shemu (summer). Akhet was the season of inundation. Peret was the season which saw the emergence of life after the inundation. The season of Shemu was named after the low water and included harvest time. Thus the Egyptian calendar had a total of 12 months (abd) of 30 days each plus 5 epagomenal days (hryw rnpt) making 365 days; as this is less than the actual 365.25 (approx), the seasons shifted in the calendar over time.^[1][3][12]

Units of Time^[1][2]

Name	Egyptian name	Values
hour	wnwt
day	hrw	1 day=24 hours
month	abd	30 days
Inundation season	akhet	4 months = 120 days
Emergence season	peret	4 months = 120 days
Harvest season	shemu	4 months = 120 days
year	renpet	365 days

The introduction of equal length hours occurred in 127 BC. The Alexandrian scholar Claudius Ptolemaeus introduced the division of the hour into 60 minutes in the second century AD.

Problems of Equitable Distribution and Accurate Measurement

Not all measurements were units. Ro for example were unit fractions which Egyptians used instead of decimals or other fractions. Tables of unit fraction were used in the RMP. "Table 11.1 is a translation of the one made by the scribe in preparation for the first six problems of the RMP. In these problems which immediately follow the original table, 1,2,6,7,8,and 9 loaves are to be divided equally among 10 men." The table is in effect a ruler which allows a computation. Division of the numbers 1 to 9 by 10^[5] Table 11.1

Quotients of 1,2,...9 divided by 10 as listed in the RMP

Number	1st Quotient	Second Quotient	Third Quotient
1	'10
2	'5
3	'5	'10
4	'3	'15
5	'2
6	'2	'10
7	'3	'30
8	'3	'10	'30
9	'3	'5	'30

A similar computation can use a remen as the diagonal of a square with side a cubit or give a value for pi. In the RMP the scribes method of finding the area of a circle is "subtract from the diameter its '9 part and square the remainder."^[5] At Saqarra an architect used numerical analysis to state a formula in the form of 3 '8 '64...where each added term from the formula would arrive at a better approximation. The formulas from the RMP include finding areas and volumes; the area of a rectangle, the area of a triangle, the area of a circle, the volume of a cylindrical granary, equations of the first and second degree, geometric and arithmetric progressions, the volume of a truncated pyramid, the area of a semicylinder and the area of a hemisphere

References

1. Clagett, Marshall (1999). Ancient Egyptian science, a Source Book. Volume Three: Ancient Egyptian Mathematics. Philadelphia: American Philosophical Society. ISBN 978-0-87169-232-0.
2. Corinna Rossi, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2007
3. Englebach, Clarke (1990). Ancient Egyptian Construction and Architecture. New York: Dover. ISBN 0486264858.
4. Faulkner, Raymond (1991). A Concise Dictionary of Middle Egyptian. Griffith Institute Asmolean Museum, Oxford. ISBN 0900416327.
5. Gillings, Richard (1972). Mathematics in the Time of the Pharaohs. MIT. ISBN 0262070456.
6. Lepsius, Richard (1865). Die altaegyptische Elle und ihre Eintheilung (in German). Berlin: Dümmler.
7. Loprieno, Antonio (1996). Ancient Egyptian. New York: CUP. ISBN 0521448492.
8. Clagett, Marshall (1999). Ancient Egyptian Science A Source Book Vol III Ancient Egyptian Mathematics. Philadelphia: American Philosophical Society. ISBN 0871692325.
9. Digital Egypt: Measuring area in Ancient Egypt
10. Katz, Victor J. (editor),Imhausen, Annette et al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007, p 17, ISBN 978-0-691-11485-9
11. T. Pommerening, Altagyptische Rezepturen metrologisch neu onterpretiert, Berichte zur Wissenschaftgeschichte 26 (2003) p. 1 - 16
12. Marshall Clagett, Ancient Egyptian Science: Calendars, clocks, and astronomy, 1989

External links

• Egyptian ceremonial ruler showing fingers, palms, hands, fists, feet, remen
• Cubit divided into fingers and hands
• Modern replica of Egyptian ruler
• Measuring length in Ancient Egypt Page by Digitalegypt (University College London).
• Irrational numbers and pyramids Article by Gay Robins and C.C.D. Shute
• Introduction to Egyptian mathematics Page contains photographs of Maya's cubit rod from the Louvre and land surveying scenes from the tomb of Menna.