The document provides an overview of ancient Egyptian science, including astronomy, medicine, mathematics, and mummification. It discusses how the Egyptians understood concepts like the celestial sphere in astronomy and describes important medical texts like the Ebers Papyrus that showed a sophisticated understanding of anatomy and disease. The document also outlines the intricate process of mummification and how mathematics in Egypt formed a coherent system with concepts like fractions, squares, and proportions.

1. 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

2. 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

3. 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:

4. ■ 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

5. 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

6. “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

7. 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.

8. 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

9. 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);

10. ■ 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

11. 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

12. 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

13. 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

14. 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

15. 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. ■

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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;

23. 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

24. 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

25. ■ 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

26. ■ 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

27. 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:

28. 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

29. 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.

30. 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.

31. 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,

32. 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)

33. 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

34. 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

35. 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:

36. 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.

37. 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).

38. 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”.

39. 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

40. 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

41. 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

42. 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

43. 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

44. (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.

45. 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

46. 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.

47. 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.),

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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.
206