Mathematics, Astronomy and Calendars in Ancient Egypt

From the early pharaonic state, around 3100 BCE, until Alexander’s conquest in 332 BCE, calculation and observation helped Egyptian officials, builders and temple personnel organize resources and time, including the accounting of institutional goods. Surviving evidence varies by period: Middle Kingdom fragments and coffins preserve mathematical procedures and star tables; New Kingdom objects and records show surveying and clocks. These sources illuminate particular tasks rather than a single unchanging system.

On this page4 sections

How scribes calculated

Egyptian numeral signs expressed decimal quantities, while multiplication could be carried out by repeated doubling. To multiply 13 by 7, a calculator could double 7 to make 14, 28 and 56, then combine the entries for 1, 4 and 8: 7 + 28 + 56 = 91. Division could reverse the process by finding which doubled amounts fit into the quantity being shared. For fractions, scribes commonly used unit fractions, each with a numerator of one, such as 1/3 or 1/8, and special forms for some common fractions.

The Rhind Mathematical Papyrus is a long instructional text copied by the scribe Ahmose around 1650 BCE, probably from a model about two centuries older. Its problems apply calculation to grain, bread and beer, divide loaves among people, and work with shapes.

Middle Kingdom mathematical fragments from Lahun add another view of scribal work. Egyptologist Annette Imhausen distinguishes “problem texts,” which record a procedure, from “table texts,” which help carry one out; the surviving Lahun fragments include both. The Moscow Mathematical Papyrus preserves a geometry problem for the volume of a truncated pyramid, a pyramid whose top has been cut off parallel to its base. For a square base four cubits wide, a top two cubits wide and a height of six, the scribe squares four and two (16 and 4), adds their product (8), then multiplies the total, 28, by one-third of the height, or 2. The result is a volume of 56 cubic cubits. This step-by-step method calculates a solid’s capacity rather than merely listing a formula.

Measuring fields and laying out temples

The cubit was a standard measure of length, reconstructed from surviving rods at roughly 52 centimetres. A cubit comprised seven palms, or 28 finger-widths. Wooden rods survive from Middle Kingdom sites including Lahun and Abydos; New Kingdom rods also appear in elite burials.

For a temple foundation, measuring had a visible physical procedure. The Met preserves a wooden stake and mallet from the foundation deposit of Hatshepsut’s temple at Deir el-Bahri, dated about 1473–1458 BCE. The Egyptian ceremony called “stretching the cord” involved driving stakes into the ground and stretching a cord between them to mark the temple’s plan and perimeter. That joins ritual language to a concrete construction task.

Land measurement also served administration. Old Kingdom written sources already name area units: a setjat measured a square 100 cubits on each side. New Kingdom documents use other land-measure terms, showing that the vocabulary changed over time. The Wilbour Papyrus, a register from year 4 of Ramesses V, around 1142 BCE, records assessed plots in Middle Egypt, including land held by temples and other institutions. Its sections group plots by locality and record measures and obligations. The document shows officials organizing information about thousands of parcels.

The civil year and festival dates

The Egyptian civil calendar had twelve months of 30 days, followed by five additional days outside the regular months. The months were grouped into three seasons: Akhet, the inundation; Peret, the season of emergence and growth; and Shemu, the harvest season. A formal date could identify a day, month and season within the king’s regnal year, the year count that began with his accession and restarted with a new reign. This gave officials a way to locate events within a ruler’s time in power.

At 365 days, the civil year fell about a quarter-day short of the solar year. Without a leap day, its dates gradually moved through the seasons: over roughly four years they shifted by about one day. A festival fixed to a civil date could therefore fall at a different point in the agricultural cycle over time. Records also show lunar dates for some observances, so priests and officials could use more than one way to reckon time.

The five extra days, often called epagomenal days, were associated with the birthdays of Osiris, Horus, Seth, Isis and Nephthys. The later Alexandrian civil calendar, introduced under Roman rule in the first century BCE, added a leap day every four years. That correction belongs to a later period; pharaonic civil dates continued to use the 365-day year.

Reading the night sky and measuring hours

Middle Kingdom coffin lids preserve diagonal star tables: grids arranged in twelve rows and 36 columns. They list decans, star groups whose rising times shifted at ten-day intervals. As successive groups appeared, an observer could use their sequence to mark the twelve night hours. The heliacal rising of Sirius, its first visible return at dawn after a period hidden by daylight, helped anchor the annual cycle and was associated with the Nile flood. The tables also served a funerary purpose, placing the dead within a cycle of celestial and seasonal renewal.

Daylight could be measured with shadow clocks, which tracked either the length or direction of a shadow. A water clock used a bowl with a small opening: as water slowly drained, its falling level could be read against hour marks inside. The earliest preserved Egyptian water clocks date to the New Kingdom; the finest known example was made during Amenhotep III’s reign in the fourteenth century BCE and was found in a buried deposit at the Karnak temple.