Seked

Seked, also spelled seqed, is an ancient Egyptian term denoting the slope or inclination of a pyramid's triangular face relative to its base, expressed as the horizontal distance in palms for every vertical rise of one cubit (where one cubit equals seven palms, approximately 52.3 cm).^[1] This measurement system, akin to the modern cotangent of the slope angle, allowed Egyptian architects to standardize and calculate pyramid profiles with precision during construction.^[2] The seked's practical application is evident in surviving mathematical texts, particularly the Rhind Mathematical Papyrus (composed around 1650 BCE, though copying earlier sources from the Middle Kingdom), which dedicates problems 56 through 60 to its computation using ratios derived from pyramid height and base dimensions.^[3] For instance, problem 56 determines the seked of a pyramid with a height of 250 cubits and a base side of 360 cubits as 5 + 1/25 palms per cubit, demonstrating the Egyptians' use of fractional arithmetic to achieve proportional accuracy.^[1] These calculations reflect a geometry grounded in empirical observation and similar triangles, essential for surveying and building stable monuments.^[2] Notable examples include the Great Pyramid of Giza, built for Pharaoh Khufu around 2580–2560 BCE, which features a seked of 5½ palms (5 palms and 2 digits), yielding a face slope of approximately 51.84 degrees and ensuring structural integrity through a balanced gradient.^[2] Variations in seked values across pyramids, such as steeper slopes with a seked of approximately 5 palms in earlier structures like the Bent Pyramid, highlight evolving architectural preferences from the Third Dynasty onward, underscoring the seked's role in pharaonic engineering and mathematics.^[2]

Definition and Units

Etymology and Terminology

The term seked derives from the ancient Egyptian noun škd (transliterated as s-k-d), meaning "slope" or "to incline," represented in hieroglyphs by the consonantal skeleton s-k-d with a determinative of a papyrus scroll (Gardiner Y1), denoting a conceptual or calculative notion.^[1] The biliteral sign for kd (Gardiner Aa28) likely depicts a builder's tool for measuring inclines, evoking symbols of rising or inclining paths in architectural contexts.^[1] Transliteration of škd varies across Egyptological sources, commonly appearing as "seked" or "seqed," with less frequent forms like "sheked" reflecting evolving phonetic interpretations.^[1]^[4] Seked is distinguished from related terms sharing the s-k-d root, such as the verb škdy meaning "to sail" or "to travel," which employs a barque determinative (Gardiner P1) to convey motion rather than fixed inclination; this clarifies seked as a precise run-over-rise measure for static slopes, unlike broader notions of gradient in other contexts.^[1] In ancient Egyptian architecture, seked specifically applied to the inclination of pyramid faces.^[1]

Measurement System

The seked (sḳd) was an ancient Egyptian unit for quantifying the slope or inclination of a surface, defined as the horizontal distance, measured in palms, corresponding to a vertical rise of one royal cubit, equivalent to 7 palms.^[5] This system standardized slope measurements by fixing the vertical component to the royal cubit, a fundamental linear unit approximately 52.3 cm in length, derived from the distance from the elbow to the tip of the middle finger.^[6] The royal cubit was subdivided into 7 palms, with each palm further divided into 4 fingers (also called digits), yielding 28 fingers per cubit; thus, the seked captured fine gradations in slope by expressing the horizontal run in whole palms supplemented by fractional parts in fingers.^[7] To express a seked, values were typically denoted as a mixed unit, such as 5 palms and 2 fingers, where the integer part represented complete palms and the remainder converted from the proportional finger subdivisions (e.g., half a palm equaling 2 fingers).^[5] This notation allowed precise communication of inclinations without decimal systems, relying instead on the Egyptian preference for unit fractions in calculations.^[6] The measurement process involved determining the ratio of the actual horizontal offset to the vertical height of the slope, then scaling it to the standard of 7 palms vertical rise.^[7] In practice, this was achieved using a plumb-line square to align the surface, dividing the horizontal distance (in cubits) by the vertical height (in cubits) and multiplying the result by 7 to obtain the equivalent palms horizontal per cubit vertical, with any fractional palm converted to fingers by multiplying by 4.^[6] This method ensured consistency across constructions, as documented in mathematical texts like the Rhind Papyrus.^[5]

Historical Usage

Origins in Egyptian Mathematics

The seked emerged during the Old Kingdom (c. 2686–2181 BCE), a period marked by significant advancements in monumental architecture, particularly the construction of true pyramids. This unit of slope measurement, defined as the horizontal run in palms per vertical rise of one cubit (seven palms), developed as a practical tool to achieve precise inclinations in pyramid faces. Its origins are closely tied to the pyramid-building initiatives of the Fourth Dynasty, exemplified by the reign of Pharaoh Khufu (c. 2589–2566 BCE), whose Great Pyramid at Giza employed a seked of 5½ palms, corresponding to a slope angle of approximately 51°50'.^[8]^[7] Seked calculations were seamlessly integrated into the Egyptian system of arithmetic, which relied heavily on unit fractions to express ratios and proportions. Slopes were thus quantified using these fractions, allowing builders to scale designs accurately from smaller models to full structures. This fractional approach facilitated the manipulation of geometric proportions without decimal notation, reflecting the Egyptians' preference for practical, divisible units in mathematical problem-solving.^[8]^[9] The development of seked built upon earlier surveying and alignment tools, such as the merkhet—a plumb-line instrument used for establishing vertical references and stellar orientations since at least the Fifth Dynasty. While the merkhet primarily ensured horizontal and cardinal alignments, it complemented the precision required for seked-based slope control during pyramid construction. This progression underscores the iterative refinement of mathematical tools in response to the demands of large-scale engineering.^[8] The seked's conceptual framework is later referenced in Middle Kingdom texts like the Rhind Mathematical Papyrus.^[7]

References in Ancient Texts

The Rhind Mathematical Papyrus, dating to circa 1650 BCE and copied by the scribe Ahmose from an earlier Middle Kingdom source, provides the earliest and most detailed references to seked as a measure of slope in pyramid construction. Problem 56 explicitly demonstrates the calculation of seked for a pyramid with a base side of 360 royal cubits and a height of 250 royal cubits, resulting in a seked of 5 palms and 1/25 palm per cubit of height.^[10] This problem illustrates the practical application of seked in determining the inclination of pyramid faces, using the formula involving half the base divided by the height, converted to palms (with one royal cubit equaling 7 palms).^[11] Subsequent problems 57 through 60 in the same papyrus extend seked calculations to variations in pyramid dimensions and even non-pyramidal structures like pillars, emphasizing its versatility in geometric problem-solving.^[3] The Moscow Mathematical Papyrus, composed around 1850–1800 BCE during the Middle Kingdom, includes computations for the volume of structures with inclined sides, such as in problems involving granaries and frustums. Problem 14 addresses the volume of a truncated pyramid (frustum) with a height of 6 cubits, base of 4 cubits, and top of 2 cubits, deriving the geometric form from the differences in base and top dimensions. Such calculations adapted pyramid geometry to everyday engineering tasks in storage structures.^[12] These mathematical papyri functioned as instructional tools in scribal schools, where seked was taught as a fundamental standard for architectural planning and execution by masons and overseers. Training emphasized practical problems like those in the Rhind Papyrus to equip scribes with skills for surveying, building, and documenting slopes in temples, tombs, and utilitarian structures, underscoring the integration of mathematics into professional scribal education.^[13]

Architectural Applications

Slopes in Pyramids

In ancient Egyptian pyramid construction, the seked served as a key metric for establishing the incline of the pyramid's faces, allowing builders to standardize and vary slopes based on structural considerations. Typical seked values ranged from approximately 5 palms per cubit for steeper profiles to higher values around 7 palms for shallower ones, influencing the overall stability and aesthetic of the monument. For instance, the Red Pyramid of Sneferu featured a seked of about 7.3 palms, yielding a gentler slope, while adjustments in other designs pushed toward steeper inclines for visual impact.^[14] The Great Pyramid of Giza, built for Khufu during the 4th Dynasty, exemplifies precise seked application with a value of 5 palms and 2 fingers (equivalent to 5.5 palms per cubit rise), corresponding to a face slope of approximately 51°50'40". This incline was meticulously achieved through the original polished Tura limestone casing stones, which set the angle before much of the casing was removed in later centuries. Measurements from surveys confirm this seked aligned with the pyramid's base-to-apex proportions, ensuring a harmonious taper.^[14] Across dynasties, seked usage evolved to balance ambition with engineering feasibility, with steeper slopes (seked around 5 palms) prevalent in early 4th Dynasty pyramids like the lower section of Sneferu's Bent Pyramid at Dahshur, which aimed for a dramatic 54°27' angle but required mid-construction adjustment to a shallower seked of about 7.3 palms for the upper portion to prevent collapse. This transition to shallower inclines in later periods, such as the Red Pyramid's consistent profile, reflected growing emphasis on stability amid larger-scale builds, as evidenced by surviving structures and ancient mathematical texts like the Rhind Papyrus that document seked computations for pyramid design.^[14]^[11]

Use in Other Structures

In ancient Egyptian architecture, the seked principle extended beyond pyramids to facilitate inclines in utilitarian and ceremonial features, ensuring structural integrity and functional accessibility. Temple ramps and causeways, designed for processional rituals, typically employed higher seked values to create gentle slopes suitable for crowds and sacred processions. Obelisks and mastabas utilized seked for batter—the slight inward slope of walls—to enhance stability against settling and erosion. In mastabas, such as Meidum Mastaba 17, batter slopes of approximately 5.5 were observed on the outer sides, mimicking protective inclines seen in larger monuments.^[15] Obelisks, often capped with pyramidions, applied seked to their tapering forms. Funerary complexes integrated seked in valley temples and serdab roofs to manage drainage and water flow, preventing accumulation during rare rains. Milder inclines were used for these features, directing water away from sacred spaces. In Djedkare's funerary complex, the satellite pyramid featured a steep face slope of approximately 62 degrees.^[16]

Mathematical Interpretation

Relation to Slope Angles

The seked, as employed in ancient Egyptian mathematics, conceptually aligns with the cotangent of the slope's inclination angle θ, where θ is the angle between the sloping face and the horizontal base. In this framework, the seked value in palms equals 7 cot θ, with cot θ defined as the ratio of the horizontal run to the vertical rise. This scaling factor of 7 derives from the Egyptian royal cubit, subdivided into 7 palms for the vertical rise, allowing the horizontal offset to be measured directly in the same unit.^[17]^[1] Unlike the modern slope ratio of rise over run, which corresponds to tan θ and emphasizes vertical change per horizontal distance, the seked inverts this by specifying the horizontal distance per fixed vertical rise of one cubit. This orientation facilitated practical construction techniques, as Egyptian surveyors could establish a standard vertical height and then determine the necessary horizontal extension for the desired inclination.^[17]^[1] The geometric model underlying the seked is a right triangle, with the vertical leg fixed at 7 palms representing the rise, the horizontal leg measuring the seked in palms as the run, and the hypotenuse tracing the slope. Here, the base angle θ satisfies \cot \theta = \frac{\text{seked}}{7}, providing a direct trigonometric bridge between ancient measurement and modern angular analysis.^[17]^[1]

Calculations and Examples

The seked of a pyramid is calculated using the formula \seked = \left( \frac{\base/2}{\height} \right) \times 7 palms, where the base and height are measured in cubits, reflecting the horizontal run per vertical rise of one cubit (equivalent to 7 palms).^[1] This yields the slope in palms, often expressed as a mixed number of whole palms and fractions. For instance, consider a hypothetical pyramid with a base of 440 cubits and height of 280 cubits: half the base is 220 cubits, so \seked = \left( \frac{220}{280} \right) \times 7 = \frac{11}{14} \times 7 = \frac{77}{14} = 5\frac{1}{2} palms.^[14] This calculation verifies the seked of the Great Pyramid of Giza, which has measured dimensions approximating a base of 440 cubits and height of 280 cubits, resulting in an exact seked of $5\frac{1}{2} palms (or 5 palms and 2 fingers).^[14] This value corresponds to a slope angle of approximately 51.84°, as derived from the relation \theta = \arctan\left(\frac{7}{\seked}\right).^[6] An illustrative example appears in Problem 56 of the Rhind Mathematical Papyrus, which computes the seked for a pyramid with a base of 360 cubits and height of 250 cubits. The steps are as follows: first, take half the base to get 180 cubits; divide by the height, \frac{180}{250} = \frac{18}{25} = \frac{1}{2} + \frac{1}{5} + \frac{1}{50} cubits; then multiply by 7 palms per cubit, $7 \times \left( \frac{1}{2} + \frac{1}{5} + \frac{1}{50} \right) = 7 \times \frac{18}{25} = \frac{126}{25} = 5\frac{1}{25} palms.^[1] This demonstrates the Egyptians' use of unit fractions in handling divisions for practical geometry.^[1]