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 PharaohDjer the height of the river Nile was given as measuring 6 cubits and 1 palm. This is equivalent to approximately 320 cm (roughly 10 feet 6 inches).
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.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 see also,
Lengths could be measured by ordinary cubit rods, such as the examples of which have been found in the tombs of officials for lengths up to the sizes measured by remen and royal cubits which were used for land measures such as roads and fields utilizing rods, poles and knotted cords. Fourteen such rods, including one double cubit rod, were described and compared by Lepsius in 1865. 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 ca 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 so that for example one foot is given as three hands and fifteen fingers and four palms and sixteen fingers(see the second register of the Turin cubit illustrated below)
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 ties 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.
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.
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).
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.
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.
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.
Years were not numbered but rather named. When named after rulers they are considered regnal years. The Egyptians divided their year (rnpt) into three 120 day seasons of four months of 30 days (hrw) named 3ht or Akhet (innundation), 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 so the Egyptian calendar had a total of 12 months (abd) of 30 days each plus 5 epagomenal days (hryw rnpt) making 365 days rounded down from an actual 365.25 so that over time summer began to fall in winter.
1 day = 24 hours, 12 hours of the day and 12 of the night
1 day = 1/30 month = 24 hours
1 month = 30 days
Akhet = 4 months = 120 days
Peret = 4 months = 120 days
Shemu = 4 months = 120 days
1 year = 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 Table 11.1
Quotients of 1,2,...9 divided by 10 as listed in the RMP
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." 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