Egyptian astronomical contributions to civilization

Egyptian Science

1. Astronomy – Celestial Sphere 2. Medicine 3. Mathematics 4. Numbers 5. Arithmetic 6. Imhotep 7. Lady Peseshet

151

SCIENCE IN REMET

Introduction

One must abandon the preposterous notion that the ancient Egyptian people only knew “empirical

recipes”, instead of “science.” Indeed, the ancient Egyptians possessed a wealth of knowledge, which

encompassed metaphysics, astronomy, medicine, mathematics, and geometry. It is impossible to build

pyramids with “empirical recipes.”

Astronomy

Although Egyptian astronomical contributions to civilization such as calendar making and astrological

prognostication are widely acknowledged, there still exits a prevalent belief that Egyptians had very little

to say regarding the realm of astronomy. However, in order to engage Egyptian knowledge of astronomy,

The Book of the Dead, or more precisely, The Book of Going Forth Toward the Light in the new edition

translated by Raymond Faulkner (1994) is essential reading material. In reading the introductory hymn

to the sun-god Ra in the Papyrus of Ani, it is clear that the question deals with the celestial sphere.

By focusing on the celestial sphere, we are able to introduce critical aspects of Egyptian astronomy as

articulated by Egyptians themselves. And, in doing so we are able to illuminate a small but very

important part of Egyptian astronomy that will ultimately facilitate more descriptive scientific

explorations into Egyptian knowledge. Before presenting the Egyptian text, let us recall the meaning of

the celestial sphere in modem astronomy. Dr. Mario Beatty, a young African American Egyptologist

completed his Ph.D. on this very acute topic.

153

The Celestial Sphere and its Defining Components

The celestial sphere is the imaginary sphere surrounding the Earth on which all heavenly bodies are

projected. It is, so to speak, the vault of the sky as it appears to be spread above us. This celestial sphere

of stars surrounds the Earth on every side. In the daytime we are unable to see the stars because of the

superior light of the sun. As the Earth rotates, the celestial poles seem to remain stationary, while the rest

of the sky moves around. Now, because the celestial sphere is divided into two equal hemispheres by the

celestial equator, one half of the celestial sphere is constantly visible to us, which is dependent upon the

observer’s position on Earth. But, if the observer is north of the equator, the south celestial pole cannot

be seen, as it always stays below the horizon. Similarly, observers south of the equator cannot see the

north celestial pole.

On the surface of the celestial sphere, which is concave, the three primary circles that are imagined are

crucial to an understanding of the more common problems of astronomy. They are the following:

■ Celestial horizon;

■ Celestial equator;

■ And, the ecliptic.

The celestial horizon is a great circle defined by the intersection of a plane perpendicular to the plumb

line and the celestial sphere. This alone may be thought of as passing through the observer, or through

the center of the Earth. The visible horizon is usually two or three degrees above the true one. This can

be easily determined by timing the rising and setting of heavenly bodies such as the rising sun. The

astronomer usually defines the plane of the horizon with a surface of mercury, which by physical laws

must be a level surface.

154

The Egyptians understood that the orbit of thè sun, which because of its oblique course, moved in an ¡5

opposite direction in relation to the other stars. According to Diodorus Siculus of Agyrion, Oenopides

of Chios, a Pythagorean learned this astronomical discovery from the Egyptians. Indeed, in Egyptian and

Dogon astronomy, the ecliptic is the great circle of the celestial sphere.

Furthermore, it seems that the ancient Egyptian astronomers possessed the knowledge of a full horizon

system dealing with the celestial sphere because of their knowledge of both the celestial and visible

horizon. Implicit in the knowledge of the celestial horizon is the knowledge of the celestial sphere, but to

provide further concrete evidence for the Egyptian astronomical observations of the celestial sphere it is

necessary to show precisely what the Egyptians called the celestial sphere:

“You appear and you shine (on) the back of your mother (the sky).”

From the textual analysis of the introductory hymn to the sun-god Ra, the celestial sphere was called

psdw, psd by the Egyptians. And, psdw, psd means “back” or spine and the determinative ® generally

denotes meanings that deal with the sun or time. Of course, the determinative provides us with the

precise context of how the foregoing sound-signs should be defined and described. If psd (“back”,

“spine”) was a question of anatomy then the determinative would be » (a piece of flesh). The “back” or

“spine” is not to be taken literally in this passage. Indeed, the Egyptian use of two determinatives, one

being the backbone and ribs and the other being, the sun © suggests that they themselves did not

want psd to be taken literally. In this specific passage, psd is used metaphorically by the Egyptians to

reveal an astronomical observation, not anatomy. The heavening bodies are seen shining and the sun

© symbolizes all stars.

155

One half of the celestial sphere (against which the stars are seen) is constantly visible to us. The surface

of the celestial sphere is concave, and the Egyptian astronomers used the image of the “back” of a bent

woman to convey this idea.

Numerous scholars have a built-in skepticism concerning Egyptian scientific innovations, particularly

as it relates to the field of astronomy. In this way, distorted scholarship becomes a consequence of the

ideological positions of many of the chroniclers of Egyptian civilization. Neither of the major

dictionaries of the Egyptian language written by Erman and Grapow (1926), Lesko (1982), and

Faulkner (1991) have the word psd with the determinative of the sun © that represents the term that

Egyptians equated with the celestial sphere. Budge (1898) replicates the word psd correctly with the

determinative of the sun ® in his translation of the Papyrus of Ani. However, he translates the passage

incorrectly as “thou risest up in the sky, illumining thy mother (Nut)”, which does not address the specific

meaning of the celestial sphere.

156

Medicine

Like astronomy, medicine was well developed in the Old Kingdom. Hesy-Ra, Imhotep, and Lady

Peseshet and Iri are some of the most well known physicians in Kemet. Further, Isocrates of Athens in

his book Busins, wrote that Egypt was the cradle of medicine. He states that, “the [Egyptian] priests,

because they enjoyed such conditions of life, discovered for the body the aid which the medical art

affords.(Van Hook and Larue, Ed., Isocrates in Three Volumes. Cambridge, Massachusetts: Harvard

University Press, 1984).

Indeed, many of the physicians were attached to the royal palace and there were several degrees of

specialization such as, oculists, dentists and gastroenterologists, to name a few. Medical instructions and

precepts were written down as early as Dynasty 5 (2465-2322 B.C.) For instance, in the tomb of Wash-

Ptah at Saqqara, an event is recorded whereby King Nefer-ir-kare Kakai (2446-2436 B.C.) ordered the

chief of physicians to bring books with which to cure an illness his high official was suffering from.

Moreover, the level of medical science revealed in the Ebers Papyrus and the Edwin Smith Papyrus

makes them the most remarkable medical documents to survive in Antiquity. It is in Kemet that the first

anatomical descriptions appear in a systematic manner. For more than two hundred different anatomical

parts have been described in the extant Egyptian medical literature.

The Ebers Papyrus

The first treatise on cardiology, the Ebers Papyrus (as it was named) dates back to the reign of

Amenophis I, contains theoretical reflections on the problems of life and health. Paragraph 854 a

indicates the anatomy and physiology of the heart precisely. The Egyptians had a clear notion of arterial

vascularisation and of the whole circulatory system. The physicians knew how to take the pulse in an

exact manner. Cardiac pathology had also attracted the attention of Egyptian doctors; paragraph 855 of

the Ebers Papyrus also notes the state of cardiac deficiency and its hepatic and pulmonary repercussions.

157

The Edwin Smith Papyrus

The first treatise on general surgery and bone surgery of mankind is listed in the Edwin Smith

Papyrus, which dates back to the beginning of Dynasty 18 (1570 B.C.) This book contains the study of

forty-eight surgical cases of traumatic injuries from head to toe, and follows the most rigorous clinical

method. Consider the following tasks:

■ documenting the formal name or informal name (obtained from the

complaint of the patient);

■ conducting an examination;

■ diagnosis;

■ prognosis;

■ and prescribing a treatment.

The surgeons of Kemet were able to establish anatomical links between the effects of the central

system and the peripheral effects, hence identifying the unity of innervations (between the central and

peripheral nerves). The physicians in Kemet also practiced dissection: the identification of the masseter,

which goes from the external angle of the lower jaw up to the temporal bone, by Egyptian surgeons

themselves. Moreover, of the 200 anatomical terms identified in the Egyptian medical literature, nearly

100 are found in the Edwin Smith Papyrus. However, this was a surgical treatise, and though filled with

anatomical terms, we cannot assume that it or any of the other surviving medical papyri necessarily

exhaust Egyptian anatomical terminology or contain the sum total of Egyptian anatomical knowledge.

In modem times, only a textbook of anatomy could indicate the thousands of anatomical terms used in

modem medicine. However, Homer praised Egypt as the country “where the doctors are the first

scientists in the world.” (The Odyssey, IV, 231). The most skilled doctors in the world, in Antiquity

(Mesopotamia, Phoenicia, Greece, Persia, etc.), were to be found in Egypt where Hippocrates (460-377

158

B.C.) and Galen (131-201 B.C.) had consulted the Egyptian medical works in the temple of Imhotep in

Memphis. In addition, Manetho reports that Athothis, the second pharaoh of Dynasty 1 was a physician

who wrote a textbook on anatomy.

Medicine in Kemet was already at a very advanced state and in fact, it underwent a decline in later

ages, which was not to be reversed until the scientific breakthroughs of the 18th and 19th centuries.

159

Mummification

Mummification played an integral role for the people of Kemet. The entire process of mummification

lasted a period of seventy days. The process of mummification was a tedious procedure, which involved

the following steps:

1. Extraction of the brain. The brain was typically removed through the nasal

area with long needles by means of hooks on the ends.

2. Removal of the intestines. An incision around five inches long was cut into

the side of the corpse in order to remove the liver, lungs, stomach, and

intestines. The heart was not removed.

3. Disposal of the entrails. The entrails were washed and treated with natron, a

natural salt and left to dry. After drying the entrails they were wrapped in

bandages and put inside canopic jars. There were four different canopic jars.

The jars represent the Four Sons of Horus.

4. Dehydration of the body.

5. The body was rubbed with scared oil.

6. The arms were arranged so that they rested across the chest.

7. The body was wrapped. There were three different wrappings; shrouds,

stuffing and bandages.

8. Disposal of waste.

160

Mummy of Ramesses II, Western Thebes, cache of the royal mummies of Deir el-Bahari. Dynasty 19. Ramesses II was over eighty years old. The mummy of Ramesses II is now in the Egyptian Museum, Cairo (CG61078).

161

Detail of the inner lid from Sennedjem’s “Sweet brother” coffin. Provenance, Thebes, Deir el Medina, tomb of Sennedjem (length: 175 cm., width: 44.3 cm) Dynasty 19. The coffin is made of wood. Cairo Museum,Egypt (JE 27308).

162

Lid of the canopic chest from the tomb of Tut-Ankh-Amun (1334-1325), Provenance, Western Thebes, Valley of the Kings. Dynasty 18. The painted lid is made of calcite alabaster (h. 24 cm). The interior of the canopic chest has four cylindrical cavities that were closed individually with lids in the form of the king’s head from the shoulders upwards. The piece is in the Egyptian Museum, Cairo (JE 60687).

163

31. AIR VIEW OF THE PYRAMIDS AT CIZEH Courtesy Mr, J. H. Breasted, Jr.

165

MUMMIFICATION.

Why Did they do it? * Egyptian culture obsessed with death. * Before the era of mummification, corpses were ritually dismembered • Legend of Osiris, god of the dead- ■■’ Preservation ofthe body to unite with ka in the afterlife.

When Was It Done ♦ Mummification experiments begin during the 4th Dynasty, around 2600 B.C. * Developed process described here not readied until the’18th Dynasty, 1000 years

later. The process fell into decline’500 years’ later, during the 20th Dynasty.

» Burial rites and embalming performed on western bank of the Nile. » Corpses first brought to purification tents (ibu) and treated with purifying waters.

•. Pharaohs erected their own embalming temples. Ordinary people treated in portable’huts, erected where needed- Embalming practitioners regulated by area-

• ”Cutters’and”Salters’’ • -Mostly men, but female embalmers rise after instances of rape of female corpses. . Corpses treated on a heavy wooden table and a funerary bed- Embalmers also

.used several types of tools- and dried natron- • Natron: A mixture of salts., including sodium carbonate, a natural cleanser

believed to purify the body.

166

The Order Of operations “ Removal of the brain. • Removal of the entrails. • Treatment of the orga ns. • dehydration of the corpse. • Washing of the body. • Stuffing of the body to restore shape. ’ Nails, eyes and genitals. • Anointig .of the. body • Final preparations before wrapping. ■

Wrapping of the corpse.

After Mummification * Embalming process lasted 70 days, including ‘15 days for wrapping. * The funerary mask.. ■ * Afterwards, the mummy entombed in the necropolis, along with everything that

came into contact with the flesh during the embalming. * 500,000.000 estimated mummies of Egypt.

167

A Guide through the Underworld l6l

168

Mathematics

Egyptian mathematics forms a homogeneous, harmonious, and complete whole. In spite of its

“pragmatic, empirical, and concrete” applications, “the rigorous procedure of Egyptian calculus

absolutely reveals mathematical thought…” (Léon Brunschvicg, Les étapes de la philosophie

mathématique. University of France Press, Paris, 1947, p. 32). Therefore, there are rules behind the

calculations of the Egyptian papyri, “The operations themselves are similar because there are rules: the

rules are co-ordinated because they are part of an organized and rational arithmetic. Whole numbers,

even, odd, primary numbers, fractions of one, of two, of any number, fractional expressions, and

numbers, squares, powers, square roots, addition, subtraction with a search for the remainder,

multiplication, and division under multiple forms, proportions, progressions, etc… Everything is there

and treated in such a way that modem critics have nothing to add.” (O. Gillain, La science égyptienne.

L’Arithmétique au Moyen Empire, Bruxelles, F.E.R.E., 1927, p. 301).

Egyptian mathematicians could calculate the area of the square, the rectangle, the triangle, the

trapezium, and the circle with a good approximation of pi, the hemisphere. They could also calculate

with precision the volume of the cube, the prism, the circular cylinder, the pyramid, and the truncated

square-based pyramid.

For example, problem forty-one from the Rhind Papyrus deals with the volume (shaa) of a circular

cylinder (deben), diameter 9, height 10. The deben, a revolving cylinder or right circular cylinder is the

solid figure created by the rotation of a rectangle around one of its sides. The two circles created are

equal; these are the bases, and the distance between them is the height of the cylinder. Thus, the volume

of a cylinder is equal to the area of the circle (i.e., of the base), multiplied by the height. This is a formula

applied by the Egyptian scribe around 1650 B.C., that is, ten centuries prior to the birth of the premier

mathematician of the Greek world, Thales (640-546 B.C.).

Around 1850 B.C., Egyptian mathematicians could accurately calculate the volume of the trunk of a

truncated pyramid (problem 14 of the Moscow Papyrus’). The trunk of a pyramid is the portion of the

169

pyramid situated between the base and the section through a plane parallel to the base. The bases of the

trunk, the great and little bases are polygons. And, the lateral faces are trapeziums. The distance between

the bases is the height of the trunk. So, when the trunk of the pyramid is regular, that is, when it is

obtained by cutting a regular pyramid through a plane parallel to its base, we obtain the following:

■ its bases are regular polygons,

■ its lateral faces are equal isosceles trapeziums,

■ the height of one of them is the apothem of the trunk,

■ the volume of a pyramid is equal to a third of the area of the base x the height.

Therefore, the volume of the trunk of a pyramid is equal to the sum of the volumes of three pyramids

having in common the height of the trunk, and respectively the great base, the little base and the

proportional average of the two bases. There is no formula to accurately find the volume of the trunk of a

truncated pyramid. Indeed, the solution of the scribe is faultless. In Kemet, the volume of the pyramid

was calculated mathematically around 1850 B.C., and in Greece this kind of problem only appears with

Euclid in his Elements, which comprises thirteen books (465 terms) that can be traced back to 300 B.C.

(see Book XIII, construction of regular polyhedrons).

Around 1850 B.C. when the people of Kemet were solving difficult mathematical problems (area of a

hemisphere, volume of a circular cylinder, volume of the truncated pyramid, calculation of the co-tangent

of the angle of the pyramid, etc), there was strictly speaking, nothing of this land in Greece regarding

math or the sciences in general. Greek science was not bom out of thin air, not even from Greek myths

but truly from the knowledge gathered millennia earlier by the Babylonians and the Egyptians. This is the

absolute historical truth. Herodotus, traced the origin of Greek geometry back to Egypt. (Histories, II,

109). In the same manner, Aristotle (384-322 B.C.) indicated that Egypt was the cradle of “mathematical

arts.” (Metaphysics, 1, 981b23).

170

The only great historical text that we have inherited from Greek Antiquity concerning the history of

mathematics is the Prologue of Proclus’ commentary on the Elements of Euclid. And in fact, this

Prologue affirms the fact that Thales learned geometry in Kemet and introduced it to Greece later on.

The Modern Electronic Calculator and the Egyptian Scribe

The Rhind Papyrus begins with a table whereby each fraction of the form 2/a is broken down into a

sum with the fractions of the numerator 1, for the uneven values between 1 and 100. The complex

calculation leads to a special factoring table, transcribed in The Rhind Papyrus. This Egyptian table was

used from 1650 B.C. to the time of Cleopatra of Alexandria (69-30 B.C.), and later up to the Western

Middle Ages.

In 1967, Professor C.L. Hamblin (University of New South Wales) programmed a computer (KDF-9)

at the University of Sydney (Australia) to create a similar calculation. According to Richard Gillings the

computer did not find a decomposition superior to that given by the ancient scribe. (Gillings,

Mathematics in the Time of the Pharaohs. Dover publications, New York, 1982, p. 52. First edition,

MIT Press, 1972).

171

Egyptian Mathematics

Original Sources

There are three rudimentary sources of first hand information dealing with Ancient Egyptian

Mathematics:

* The Rhind Mathematical Papyrus, written in Hieratic script, now in the British Museum,

no. 100057 and 10058. The Rhind Mathematical Papyrus was found at Thebes in the ruins

of a small building near the Ramesseum. It was purchased in 1858 by A. Henry Rhind and

after his death in 1863, the British Museum possessed the piece. It is a copy that was made

by a scribe-mathematician named Aahmose (Ahmes), approximately 1650 B.C. One

fragment, which was missing is now in the possession of the New York Historical Society.

■ The Moscow Mathematical Papyrus, also written in Hieratic script around 1850 B.C.

This papyrus was purchased at Thebes in 1893 by Viadimer Golenischeff, and it is now in

the possession of the Pushkin Museum, no. 4676.

■ The Demotic Mathematical Papyri (in the Egyptian Museum at Cairo, and the British

Museum, London), was written during the Late Period.

Modern Scholars who Studied Egyptian Mathematics in Depth

a) Rhind Mathematical Papyrus: Eric P. Peet, London, 1923; Arnold Buffum Chace, Henry

Parker Manning, and Raymond Clare Archibald, Oberlin, Ohio, in 1927;

b) Moscow Mathematical Papyrus: W.W. Struve, Berlin, in 1930 (in German);

c) Demotic Mathematical Papyri: Richard A. Parker, Brown University Press, 1972.

172

Egyptian Arithmetic

The Egyptians conceived of two kinds of number, an ascending series from 1 to 1,000,000:

what we now call integers, and a corresponding descending series consisting of 2/3 and reciprocal

numbers or unit fractions. For our 1 3/4, they say 1 1/2 for 2 times 1/5, %1/21/15, and so on (without

any sign of addition, just as we write 1 1/2).

The Egyptians understood thoroughly the four arithmetical operations, addition, subtraction,

multiplication, and division. They also had a well-defined decimal notation.

Problem 30 (Rhind Papyrus): find the quantity of 2/3 1/10 that must be multiplied in order to

produce 10. For the Egyptian scribe the answer is …

Problem 67 (Rhind Papyrus): find how many cattle there are in a herd when 1/6 1/18 of them

equals 70, the number due as tribute to the owner.

Problem 79 (Rhind Papyrus): a problem which is calculated in two ways the sum of a

geometrical progression.

Egyptian Measures

Measures of capacity are as follows:

■ Hekat, 292,24 cubic inches

■ Parts and multiples of hekat

■ Khar, that is, 2/3 of a cubed cubit it takes 20 khar to make 100 quadruple

hekat or 400 simple hekat

■ And the hinu, equal to 1/10 of a hekat

173

Measures of area and length

• khet, the linear unit which is 100 royal cubits

■ setat, which is 10,000 square cubits

■ meh, the royal cubit, equal to 20.62 inches

■ shesep, the palm or hand-breath, which is 1/7 of a cubit, and which also

denotes an arc over the number; and 1/4 of a palm is called a finger-breath or

finger, djeba. Digit (djeba) = 1/28 cubit. Digit (a numeral) = 10,000.

Egyptian Geometry

a) Areas and Volumes

Egyptian mathematicians were able to determine the following:

■ the areas of rectangles, triangles, and circles

■ the volumes of cylinders and prisms, of pyramids and truncated pyramids

■ the relation of the lengths of two sides determined one of the angles in a

right triangle

■ the word seked, seqed, is for the relation of the lengths of the two lines

(pyramid slope)

■ the word meret is the length of a line

■ the word for pyramid is mer.

174

b) Trigonometry

At the beginning of the 20 century B.C., when the greater pyramids were build, the

Egyptian mathematicians possessed a full understanding of “right angle”, “pyramid slope”, “isosceles

triangle”, etc. Problems 56-59 (Rhind Papyrus) deal with the question of the cosine of the angle

which the lateral edge makes with the diagonal of the base. In Problem 60 (Rhind Papyrus), the

question at hand deals with the tangent of the angle which the lateral face makes with the base. The

seked also means the cotangent of the latter angle.

Theoretical Interest

“A careful study of the Rhind Papyrus (emphasis mine, Th. Obenga) convinced me several

years ago that this work is not a mere selection of practical problems especially useful to determine

land values”, division of loaves, quantity of grain, etc., “rather I believe that they studied mathematics

and other subjects for their own sakes”. (A.B. Chace et alii, vol. I, The Rhind Mathematical Papyrus,

Mathematical Association of America, Oberlin, Ohio, U.S.A., 1927, p. 42). For instance:

■ the problems of area

■ the problems of volume

■ the dimensions of a pyramid

■ the purely numerical problems

• theoretical problems put in concrete form as pedagogy is concerned

The title of the Rhind Mathematical Papyrus is translated as follows:

“Accurate method (tep-heseb) of investigating {net hrt) into nature (em khet) in order to

know {rekh) all existing things {netet nebet)” (My translation, Th. Obenga).

175

This definition is the most abstract one in the field of mathematics, pure and applied.

Demotic Mathematical Papyri

Some problems correctly resolved in Kemet:

Problem 36: Given a circular figure in which is an equilateral triangle is laid out, with

sides of 12 cubits each, determine the area of the circular figure.

Problem 37: Given a circular figure with an area of 675 square cubit and a diameter of

30, in which is laid out a square, determine the size of the square.

Problem 39: Given a pyramid 300 cubits high, with a square base 500 cubits to a side,

determine the distance from the center of any side to the apex.

Problem 46: Given the fraction 1/5, which is added to 1, determine what fraction of 1

1/15 must be subtracted from it to give 1 again.

Problem 48: Given the fraction 1/7, which is added to 1, determine what fraction of 1

1/7 must be subtracted from it to give 1 again.

Problem 54: A multiplication table for 64 from 1 to 16.

Problem 57: Multiply 1/3 1/15 by 2/3 1/21.

Problem 62: Extract the square root of 10.

Problem 67: Construct a multiplication table for 1/150 from 1 to 10

176

Problem: Given a pyramid with a vertical height of 10 cubits and a square base 10 cubits to a

side, determine the volume.

Richard A. Parker, Demotic Mathematical Papyri, Brown University Press, 1972, pp. 52-53,

problem no. 40, plate 14: S 18-23.

Volume of the pyramid is one third of the product of the surface area of the based by the

height:

V= h/3 a2 OR V = Bxh/3

B = the base

H = the height

The volume of a truncated pyramid from an ancient Egyptian papyrus, about 1850 B.C.

(Homer is not bom yet): W.W. Struve, Mathematischer Papyrus des Staatlichen Museums der

Schonen Kiinste in Moskau, Berlin, 1930, problem 14.

The volume of the truncated pyramid is the sum of the volumes of 3 pyramids, which have in

common the height of the trunk and respectively, the great base (B) and the same base (b); to this it is

added the proportional mean between the two bases:

V = h/3 (a2 + ab + b2)

Height=6

Great base = 4

Small base — 2

OR V = h/3 (B + b + sqrt(Bb))

178

Selected Bibliography

Beckmann, P. A History of n. New York, St. Martin’s Press, 1971,200 pages.

Boyer, C.B. A History of Mathematics. Princeton, Princeton University Press, 1985, SVII – 717 p.

Chapter II: “Egypt”, pp. 9-25.

Finch, C.S., African and the Birth of Science and Technology. A Brief Overview. Decatur, Georgica,

Khenti Publications, 1992, p. 40.

Gillings, R.J., Mathematics in the time of the Pharaohs. New York, Dover Publications, 1982, XI –

286 p.; 1st edition, M.I.T. Press, 1972.

Kamalu, C. Foundations of African Thought: A Worldview Grounded in the African Heritage of

Religion, Philosophy, Science and Art. London, Karnak House, 1990, XI-176 p., 14 fig.

Miller, G.A. A Few Theorems Relating to the Rhind Mathematical Papyrus, “The American

Mathematical Monthly”, vol. 38, no. 4, 1931, pp. 194-197.

179

Egyptian Numbers

180

Origin of Numbers

Numbers first derived from counting, that is, from concrete objects. Even the word calculus,

which designates the combined mathematics of differential and integral method of analysis comes

from the Latin word calculus, which is a small stone used in reckoning, diminutive of calx, a lime,

limestone, from the Greek word khalix/chalix, meaning pebble, or a small stone.

The human body may have been the first object to be numbered. For example, in the Greek

civilization we have the following facts:

of length, 4 palms (palastal) or 6 fingers, about 1/8 of an inch longer than the English foe

(inch is equal to 1/12 of a foot; inch is derived from the Latin unus, meaning “one”; to

inch, to move slowly or by small degrees, small steps, one by one).

Egyptian Measures of Length

The main Egyptian measures of length are as follows:

mh, meh, cubit

1 cubit = about 20.6 inches = 523 millimeters

1 cubit = 7 palms = 28 digits

Ssp, shesep, palm (Greek palaste)

dbr, djeba, digit (Greek daktylos)

nbrw, nebiu, pole

ht, khet, rod (of 100 cubits)

itrw, iteru “river-measure” – 20,000 cubits =10.5 kilometers

For comparison: 1 mile = 5,280 feet, 1,760 yards, or 1,609.34 meters (English mile from Latin

milia, thousand). So, how many miles does 1 iteru contain?

181

Writing of Numbers

A vertical stroke | is used for the units, and special signs for the various power of tens, as

follows:

182

Write the Equivalent in English Numerals

For example:

183

Exercise: Papyrus of Accounts (13th Dynasty)

The glyph , which is read dmd means “total”. Compute and provide the answers in English

184

The Names of the Numbers

185

The numeral follows the noun, which, as a general rule, exhibits the singular form:

2, “two men”

rnpt, 20 “twenty years”

The ordinal numbers:

tpy, tepy “first”

2-nw, “2nd”, “second”

6-nwt, “6th” expedition of the Pharaoh, etc.

186

The Eye of the Falcon-god Horus (Heru)

The Eye of Horus was torn into fragments by the god Seth. Later, the Ibis-god Thoth

completed (mh, meh) the Eye, miraculously. Thus, the Eye became the “the Sound Eye” with total

vision. Human eyes are not complete. Because of this fact, the eye is not able to see itself. In

Paradise, the divine Light (Râ) will open our eyes so that we can see Light, its beauty and its full

power for all eternity.

The Eye is not sound (udft, udjat): 1/2 + 1/4 + 1/8 + 1/32 + 1/64 = 63/64. The missing 1/64

was supplied magically by Thoth (Djehuty). Mathematics, mythology, philosophy, and spirituality

are usually interwoven in Ancient Egypt. It is difficult to have a complete vision when dealing with a

question or a problem. Even despite this lack, we still want to know. Human knowledge is increased

by ignorance guided by sound method. The most genuine acquisition of human civilization is

scientific method.

187

Fractions

A) Terminology

The Egyptian word for “fraction” is hsb, heseb. Hence:

hsb, heseb, means “count, reckon”, i.e. to break

hsbw, hesebu, “reckoning”, “account”

tp-hsb, tep-heseb, “correct method” (Rhind Papyrus, title)

B) Writing Fractions

The use of the word r, er, “part”, was written to describe what we designate in English as

the denominator. Thus, r-5 stands for “part 5”, which is equivalent to our modem expression 1/5;

r-276 “part 276” for 1/276.

The word gs, ges, means “side” and stands for 1/2. (Refer to the Rhind Papyrus, problem

no. 34).

For 5 +1/2+1/7+1/14 = 5 5/7.

The hieroglyphic sign in hieratic script is written as » , which equals 1/4. (Refer to

Rhind Papyrus, problem no. 41).

For 2 + 1/2 + 1/4 + 1/14 + 1/28 = 2 6/7

The fraction rwy, rui, “the two parts” stands for 2/3, and r-3, (hmt rw, khemet ru)

stands for 3/4. (Refer to Rhind Papyrus, problem no. 42).

For 2 + 2/3 + 1/6 + 1/12 + 1/36+ 1/54 = 2 26/27.

188

Schist palette of King Narmer, i.e., “The Striking Catfish” (3150-3050), carved on both sides in low

raised relief was found in the temple of Hierakonpolis. Naqada III. King Narmer wears the white crown of

Egypt. Behind the king is an attendant carrying the King’s sandals. The palette is made of greywacke (height

63.5 cm, width: 42 cm). This palette is the finest monumental palette with relief ornamentation. The scene

depicts the king’s victory over an enemy. Egyptian Museum, Cairo (JE 32169).

189

Egyptian Arithmetic

I. Words for “counting” and “number”

As mentioned previously, the verb hsb, (heseb) means “to count, reckon” and, as a noun means

“counting”. The expression wr hrt m hsb tnw means, “Greatly skilled in the counting of numbers”

(Urk, IV, 968,4).

II. Addition

The word hr (her) as a preposition means “added to”. For example, from the Rhind Papyrus,

gs.f hr.f means, “its half is added to it”.

The verb h3w (hau) means “increase”, and conveys the idea of addition, which is “more than”.

HI, Subtraction/Deduction

The verb hbi (khebi) means “to subtract”. From the Rhind Papyrus, problem no. 46,2.

w3h tp 500 ir 1 /lO.f qf 500= 50. This expression means “Take 1/10 of 500 and it equals

50”.

190

IV. Multiplication

According to the context the verb iri means “to multiply”. We also have the same with the

following verb in its idiomatic use: w3h tp, “bow the head”. The verb iri means “to do, make”.

Consider the following:

ir 6 sp 10, which means “multiply by 6 (times) 10”. The word sp means “times” and “by”.

w3h tp m 10 r sp 2 “multiply 10 by 2”, literally “bow the head of 10 to 2 times”

V. Division

The preposition hnt (Khent) means “out of’ is used in the following way:

nis 10 hnt 2 “divide 10 by 2”, literally “summon 10 out of 2”.

The verb nis means “evoke” or “summon”. For example, nis tnwt, “evoke a number”, that is,

make a reckoning. Also, w3h tp m20 r gmt 4, “divide 20 by 4”, literally “bow the head in 20, 4

being able to do so”.

Conclusion

Egyptian arithmetic made use of idiomatic expressions. Words were clear and precise in

describing every single reckoning. For example, the verb iri “to do, make” means “multiply”,

“divide”, and “square”. Thus, ir 10 m nbtt “square 10”, literally make 10 in movements”.

191

Architecture

The Egyptian architects called kedu in Egyptian language built many forms of buildings and

monuments such as:

temples (hut netjer),

palaces (per-nesu)

pyramids (mer)

obelisks (tekhen, pl. tekhenu),

and tombs (per-djet “house of eternity”).

The first people in world history to build large monuments with hewn stones were Egyptians around

2300 B.C. Imhotep was the inventor of this technique. In order to calculate land area and to build

temples, tombs, and state buildings the Egyptians used the royal cubit, a standard unit of measure (the

length of royal cubit was 20.6 inches).

The Step Pyramid built, thanks to Imhotep’s technique still stands today. Besides Imhotep we also

have other architects such as: Hemiunu, Sennemut, Kha, Amenhotep (son of Hapu), Bakenamon,

May, etc.

All of the buildings were linked to spirituality and the divine world. Ptah, the god of Memphis was

patron of craftsmen and architects. The temple was the replica of the cosmos on this earth. The pyramids

and temples were built according to the orientation of the four cardinal points and the position of some

stars, for example, the constellation of Orion. The obelisks were the symbol of the sunrays, to illuminate

Upper and Lower Egypt. The Egyptians developed a very organized and complex system of architecture,

which has yet to be surpassed. Modem scientists, engineers, navigational experts, and archeologists have

yet to figure out how the Egyptians produced and moved these magnificent pieces of art. The temples,

pyramids, and obelisks in Egypt still attract tourists from around the world. Some sites are internationally

known, for example: Saqqara, Giza, Memphis, Luxor, Karnak, Abydos, Edfu, Dendara, Kom Ombo,

192

Deir el-Bahari, Aswan, and Abu Simbel. Indeed, architecture is one of the greatest achievements of

Egyptian civilization and Plato believed that Egyptian art was divine.

Statue of the scribe Amenhotep, son of Hapu. From the temple of Amun-Ra, tenth pylon (1360 B.C.) Provenance: Karnak. Dynasty 18. Amenhotep is in a seated position and holds a written papyrus roll in his hand. He was the personal advisor to King Amenophis III and the éminence grise at court. He was the “Director of all the royal works” and “Leader of the Festival” at Karnak. This means that he represented the king at the great celebrations in the temple. The statue is made of granodiorite (h. 128 cm). Egyptian Museum, Cairo.

Portico facade of the tombs of Amenemhat II (Nub-ka-Ra). Provenance: Beni Hasan (3,4, and 5). Dynasty 12. The tomb facades were cut vertically into the sloping cliffs. The form of the columns is defined as protodoric, and it anticipates the flues and sharp edges of later Doric (Greek) columns.

194

Tomb of Amenemhat II Provenance: Beni Hasan (BH2). Dynasty 12. Four 16-sided columns stand in the square hall. Two massive longitudinal architraves, that is, the lowest part of an entabulature, a beam resting directly on the tops of the columns connect them in pairs and divide the room into three barrel-vaulted naves. The ceiling vault decorated with a pattern runs parallel to the east and west axis.

195

The “White Chapel” of Sesotris I (Kheper-ka-Ra). Provenance: Karnak, Dynasty 12. The Chapel is made of limestone and has a surface area of 6.54 by 6.45 m. Two ramps lead up to a pillared structure raised on a platform.

196

A colossal seated statue of Ramesses II. Provenance, Luxor Temple. First court at the entrance to the colonnade. The statue is made of black granite (h: 7 m). The king wears the double crown and he is also wearing a royal nemes headdress. The right arm of the King bears his throne name, User-maat- Ra “Powerful is the Justice of Ra, Chosen by Ra.” Ramesses II, known as “Ramesses the Great” acceded to the throne at a very young age and ruled for 67 years. No other pharaoh constructed so many temples or erected so many colossal statues and obelisks. He completed his father’s mortuary temple at Gouma (Thebes) and built his own temple nearby at Abydos. On the west bank of Thebes he constructed a giant mortuary temple, called the Ramesseum. Inscriptions in the sandstone quarries at Gebel el-Silsila indicate that at least 3000 workmen were employed there.

197

Ramesses II greatest achievement was the carving out of the mountainside of the two temples at Abu Simbel in Nubia, where four colossal seated figures 60-ft (18 m) of the king flank the entrance to the temples. This amazing temple was rediscovered in 1813 by the Swiss explorer Jean Louis Burckhardt, and Giovanni Belzoni was the first European to enter the temple on August 1,1817. The orientation was so exact that on February 22 and October 22 the rising sun at the equinox flooded directly through the great entrance to illuminate three of the four gods carved seated in the sanctuary over 200 ft (60 m) inside the mountain.

198

Imhotep: A Man of Multiple Genius

Described as a polymath, Imhotep was bom in Ankh-tawy, Memphis on the 16th day of Epipi, the

Egyptian name for the third month of shemu or the harvest season. This translates to May 31st. The exact

year of his birth is uncertain but is roughly estimated at 3000 B.C. He descended from a distinguished

architect named Ka-nefer and his mother’s name was Kheredu-ankh. His wife’s was Renpet-neferet

and they had one son named Ra-hotep. He became known for his vast learning and striking

achievements. In world history, Imhotep is known as vizier, architect, chief liturgist, sage, scribe,

astronomer, and physician.

As vizier, Imhotep is credited with the legend of the Seven Year’s Famine. It is inscribed on a granite

rock on the island of Sehel near Aswan. The legend describes a period of seven years of starvation due to

the failure of the Nile to flood and irrigate the land for cultivation. The famine was attributed to King

Djoser’s neglect to pay respect to the god Khmim of the First Cataract, one of the deities controlling the

source of the Nile. Djoser went to Imhotep for assistance. On his advice, King Djoser wrote the viceroy

of Nubia requesting the name of the god who resided over the Nile so that the deity could be visited.

Djoser paid a visit to the temple of the god to make prayers and supplications before him. In response,

Khnum appeared in a dream and promised that the Nile would never fail to flood again.

As an architect he designed the well-known Step Pyramid of Saqqara near Memphis for King

Djoser. This pyramid is a transition between the mastaba (tomb) and the true pyramid form. Imhotep

completed the stone facing of the walls of the funerary temple, the chapels, and the walls enclosing the

entire area. One of the chapels symbolized the Sed Festival and the other two were expanded versions of

earlier shrines found in Lower and Upper Egypt. Indeed, Imhotep built the earliest group of stone hewn

structures in the history of humanity.

As chief liturgist, Imhotep served as the high priest of Ra at Heliopolis. The essential duty of the

chief priest was to attend to the daily cult of the god, this included recitation of prayers from sacred texts.

199

Imhotep’s knowledge and performance of this rite led the common people to believe that he had magical

powers.

The reputation of Imhotep as scribe and sage has survived many centuries. For example, the “Song

of the Harper,” written during the New Kingdom (1570-1070 B.C.) recorded the wisdom of Imhotep. It

also indicates that no one had surpassed him in knowledge. Let us remember that Imhotep lived during

Dynasty 3 in the Old Kingdom. Hence, his thoughts were highly regarded for more than fifteen centuries.

As an astronomer, Imhotep was associated with the cult of Ra, that is, the creator of the universe.

Maat, the cosmic harmony, was the daughter of Ra. In this way, Imhotep was linked with the

observation of celestial phenomena and other members of the clergy of Heliopolis. Thus, the

astronomical calendar, which is based on the solar cycle was invented by Imhotep. Our calendar, the

Gregorian calendar comes from him.

Since the construction of pyramids required an extensive organization of a large work force Imhotep

became a physician out of necessity. For accidents and diseases of all kinds could occur during the

construction of a pyramid. His fame for healing led people in the later period to deify him. The Greco-

Egyptians called Imhotep Asklepios and assimilated him with their deity of medicine. As a matter of

fact, medicine was already practiced in Kemet in the earliest prehistoric days, many millennia before

Christ. And more importantly, Imhotep was the impetus for the development of medicine in Egypt and

in the Mediterranean world.

And because of this, many medical texts produced by Egyptian physicians have been found. Some of

them have survived up until the present time, such as, the Kaliun, the Gardiner fragments (2000 B.C.),

the Ebers Papyrus and the Edwin Smith Papyrus. The two earliest papyri deal with diseases of women,

children, and cattle. And as mentioned before the Smith and Ebers papyri have been traced back to the

17th and 16th centuries B.C., and they are the most outstanding medical texts that have come to us.

Indeed, the information within these texts provide us with the most groundbreaking medical achievements

in world history.

200

Summary

The period known as the Old Kingdom in Ancient Egyptian history lasted over 500 years, comprising

the Third through the Eighteenth Dynasties. The pyramids of the Old Kingdom represent the most

spectacular Egyptian achievements in science, technology, and organization to survive up to

contemporary times. And, Imhotep greatly contributed to the development of monumental architecture

in Egypt as the chancellor and chief architect under King Djoser (the second ruler of Dynasty 3). He was

worshipped as a patron of science and medicine.

201

Bronze statue of Imhotep (“Come in Peace!”), the chief architect of King Djoser, scribe, physician, and high priest. Dynasty 3. He has a written papyrus (book) upon his knees. Cairo Museum, Egypt.

202

The Step Pyramid of Saqqara 21

9. The Step Pyramid of Saqqara.

excavate the interior of the Step Pyramid, and none left a description of any lasting value.1

The earliest record of excavations is that of the Prussian Consul- General von Minutoli, in 1821. He penetrated the subterranean galleries and made several discoveries there, the most important of which was one of the blue-tiled chambers. Von Minutoli described his work in Reise zum Tempel des Jupiter Ammon, with several excellent drawings made by the Italian artist Valeriani. He in his turn left a record of his work in Atlante del Basso ad Alto Egitto Illustrate. Von Minutoli found inside the Step Pyramid parts of a mummy and a few inscriptions with the name of Zoser. He sent these to Europe, but they were lost at sea during a “storm. The mummy is often discussed, and is generally at­ tributed to the Third Dynasty. Many years later, archeologists found part of the same mummy in the same place. Dr. Derry, who examined

1 The Arabs knew Saqqara under the name Abusir. For references to this site in the manuscripts of the early travelers and explorers, see Porter and Moss, Topographical Bibliography, III, 85-89.

203

The south face of the Step Pyramid of King Djoser from his mortuary complex. Provenance: Saqqara, Dynasty 3. During the Old Kingdom, Egyptian art flourished tremendously, particularly in the field of architecture. In the construction of the Step Pyramid, large granite blocks were used and Egyptian architects undertook the building method of using a series of layers, with the outer layers inclined approximately 18-20° toward the center that automatically produced a gradient of 70-72° in the outer layer. The pyramid is about 60 m. and has six steps. By doing this, the architects were able to reduce the weight of the building. In 1821, H.M. von Minutoli led the first excavation of the pyramid.

204

Lady Peseshet: First Woman Medical Doctor in World History

In 1930, Dr. Selim Hassan published the stela of Lady Peseshet (Excavations at Giza 1, 1929-1930)

that was found within a tomb from the Old Kingdom. Dr. Hassan translated Peseshet’s title as the

following, “Overseer of the doctors.” In fact, the word imyt-r, “overseer”, is of feminine gender because

it ends in -t. Further, the word swnu, “doctor”, is written in the text with the grammatical ending -t, which

stands for a feminine noun. It is then clear, that Peseshet was a woman doctor (swnwt) and the director

(imyt-r) of women doctors (swnwwt). The word swnw, “physician”, is exclusively used in the field of

medicine.

Lady Peseshet had another title, which is read as the following, imyt-r hm (wf)-k3, that is, “woman

director of the soul-priestesses.” The soul-priests or priestesses were appointed to attend to the funerary

cults of private persons. As we know, women in Egyptian society enjoyed social and professional status

equivalent to that of men. All professions were open to educated women and men, which included the

clergy, administration, business and medicine, among other fields as well.

There was a body of female physicians in Kemet during the Old Kingdom and Lady Peseshet was

their director. The contemporary dilemma of excluding women in “special vocations” was absent in

Kemet. In fact, there were more than a hundred prominent female physicians in Kemet. In contrast,

current scholarship has not identified any female personalities who were physicians in Mesopotamian or

Greek history. It is clear that medical historiography must include the fact that Lady Peseshet was indeed

the first female physician in Africa and in world history. This truth is an essential component to the

consciousness of humanity.

205

Summary

These are but a few facts in metaphysics, astronomy, medicine, and mathematics that are part of an

immense scientific, cultural, and spiritual treasure chest accumulated by the Egyptians for almost thirty

centuries of living history. In the matter of civilization the ancient Egyptians did not imitate any other

people in Antiquity. They themselves were pioneers. They invented their own writing, their own unique

character of monumental architecture (pyramids, temples, etc.), their literature, their mathematics, their

astronomy, their medicine, their religion, their conception of the world, their arts (painting, sculpture,

drawing, carving, music, dance, games, leisure pursuits) etc.

The intellectual experiences were used for the progress of other people in Antiquity, in the glorious

path of human civilization. One can examine the influence of Egyptian thought upon the birth of Greek

thought, particularly since Kernet, in the eyes of many of the Greeks was considered the most civilized

country in the world at that time.