The Berlin Papyrus 6619 (1800 BC) is one of the only surviving witness which demonstrates that the Egyptian escribes knew how to solve certain quadratic equations.
The first problem in the papyrus says: You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares? That is:
There also is a second similar problem equivalent to the quadratic system:
The solving method is the rule of false position. The escribe assumed that x = 0,75 and y = 1 so x2 + y2 = 1,5625. But the result should be 100 = 64 · 1,5625! Therefore, our two squares must be 64 times bigger and their sides must be 8 times bigger. So the result is x = 0,75 · 8 = 6 units and y = 1 · 8 = 8 units, and x2 + y2 = 100.
This papyrus becames unnotices in the Neues Museum of Berlin due to its close position to the famous bust of Nefertiti:
I am going to begin my Egyptian visit to the British Museum with the limestone game-board in the form of a coiled snake used for the game called “mehen” (2890-2686 BC). The body of the coiled snake is divided into rectangular spaces but the number of these spaces is not important for the game.
Game-boards in the form of coiled snakes are known from the Early Dynastic period whengames became a regular item of tomb equipment. Several examples were discovered in the excavation of the Second-Dinasty tomb of King Khasekhemwy at Abydos. The game for which the snake-board was used was called mehen and although the exact method of play is not known, later representations show that it involved two players. The game-pieces consisted of spherical stone marbles and small figures of lions and lionesses usually made of bone or ivory.
The other popular game in Ancient Egypt was the ‘senet’ and there is one ivory sene board with a drawer for storing the gaming pieces with the glazed gaming pieces. This second board is in the special exhibition dedicated to the tomb-chapel of Nebamun (1350 BC):
The secnond mathematicl object found in the British Museum is this sandstone stela of the Egyptian Viceroy of Kush, Merymose, who served under Pharaoh Amenhotep III (c.1400 BC). A hieroglyphic text describes his campaign against the Nubians of Ibhet:
The hieroglyphic text is full of numbers and figures:
Other hieroglyphic numbers are found in the limestone relief of Rahotep (c.2600 BC) which was fixed in the offering-chapel of a brick mastaba tomb.
The relief shows Rahotep seated before offerings which are detailed in a formal list on the right of the slab and all these offerings are accompanied of the number of them. We can see the ‘lotus’ for the thousands…
and a lot of examples of units, tens and hundreds:
It’s also interesting this bone identifying label from an item of funerary equipment (3100 BC). The front of the label bears the name of Queen Neithhotep and on the back is the numeral 135:
Finally, the limestone false door stela of Niankhre (2450 BC) from Saqqara which comes from the mastaba-tomb of the superintendant of the hairdressers of the Palace Niankhre. You can see the number 4.000 in the top of the stela:
There also is this statue in the Ashmolean Museum. Khasekhem was the last king of the 2nd Dinasty (2850-2700 BC) and he is wearing the White Crown of Upper Egypt and is wrapped in a long robe. His right fist is drilled to attach a separate object, perhaps a mace handle or sceptre. The king’s name is inscribed in front of his feet written inside a representation of the palace façade topped by the falcon god Horus. The base of the statue records a military campaign against the northern rebels referring to the inhabitants of the Nile Delta, alongside the bodies of slain enemies, numbered on the front as 47.209:
There also are some other Egyptian objects in which it’s possible to read the Hieroglific figures. One example is the limestone cornice from a false door in the mastaba of Sheri, overseer of mortuary priests tending the cults of the 2nd Dynasty Kings Sened and Persibsen (c.2700 BC):
Framed between two facades, Sheri and his wife Kentyetka are seated at a table laden with food offerings and we can see four numbers “1.000” below it:
Finally, there is a set of funerary stelae in which can be noticed some Egyptian figures:
For example, look at this limestone stela (c.1200 BC) of the Lector Priest Dedusobek, official of the pyramid and town of Senwosret II at Lahun. Dedusobek and his father Menkau are each seated before a table of offerings. You can see two set of four number “1.000” over the table:
The King Philip II of Spain decided in 1550’s that he wanted to have a great library near his court in Madrid and he chose the new Monastery of San Lorenzo de El Escorial to place it in spite of other bigger villages. He didn’t want that the new library was a regular room inside a monastery so it had to be a very important place. Therefore, the library was placed on the second floor of the monastery just above his royal chambers but never above the basilica. Between 1565 and 1576, the king bought almost 5.000 books and manuscripts and the library became one of the most important libraries in all Europe.
The mathematician and architect Juan de Herrera (1530-1597) designed a large room (54 m. long x 9 m. wide x 10 m. high) with big windows in both sides under a great barrel vault. This vault had to be decorated by an important painter and Philip II decided that Peregrino Tibaldi (1527–1596) had to be the right artist to do the work. Philip II was advised by Juan de Herrera and other humanists and he decided that the main subject of the paintings of the vault had to be the Liberal Arts. Furthermore, the seven arts would be together with the Philosophy and the Theology on both ends of the room. The Philosophy represented the compendium of the Human knowledge and she is accompanied by Aristotle, Plato, Seneca and Socrates:
The Theology is on the side next to the convent and she represented the Divine knowledge. Therefore the vault represented the way from the Human Philosophy to the Divine knowledge through the seven Liberal Arts: the Arithmetic, the Geometry, the Astronomy (Astrology), the Music, the Rhetoric, the Grammar and the Dialectic. We can see a mathematical detail on the fresco below the Philosophy: it represents the School of Athens and there is a discussion between the Academics leaded by Socrates and the Stoics leaded by Zeno of Elea.
The scholars aren’t listening to the speakers because each of them is “playing” with something different. We can see at the lower left corner a man measuring something with a compass and two books, a sphere and an armilar sphere, a dodecahedron and a compass in the middle of the picture:
Going from the Philosophy to the Theology, we arrive at the Arithmetic after admiring the Grammar, the Rhetoric and the Dialectic. The Arithmetic is a woman turned to a table with simple mathematical operations rounded by muscled young men with tablets with arithmetical operations ans counting with their fingers:
There is also a representation of the Queen of Saba talking with King Solomon According to the Book of the Kings (I,10,1), the Queen of Saba went to meet Solomon to ask some enigmas to him so we can see a ruler, a balance and a tablet with some numbers written on it. In the red tablecloth we can read “Everything has number, weight and measure” in Hebrew:
The other panel next to the Arithmetic represents the school of the Gymnosophists who lived near the Nile and thought their philosophical theories from the numerical computations. In the middle of the picture we can see one of the gymnosophist with a compass looking at a triangle with the word “Anima” and the arithmetic progression 1, 2, 3 and 4 and the geometric 1, 3, 9 and 27 written on it. The other gymnosophists are computing with numbers written on the sand:
Finally, at both sides of the Arithmetic on the roof we find four people related with this subject: Archytas of Tarentum (c.428–c.347 BC) and Boethius (c.480-c.525) in one side and the Platonic Xenocrates (c.396/5 – 314/3 BC) and Jordan in the other. They are writing numbers in their tablets.
There is the Music after the Arithmetic and we find the Geometry after it:
She has a compass in one of her hands and the young men around her have different geometrical instruments. The two scenes which are on the corresponding walls next to her are dedicated to some Egyptian monks drawing geometrical figures on the sand…
and Archimedes’ death:
Notice that Archimedes is drawing the demonstration of the Theorem of Pythagoras made by Euclid!
Finally, the four chosen figures are the Astronomer Aristarchus of Samos (IIIrd c. BC) and the Persian astrologer Abd del Aziz also known as Alcabitius (Xth century) in one side and Archimedes (c.287-212 BC) and Regiomontanus (1436-1476) in the other. Aristarchus is measuring angles and has a dodecahedron at his feet, Alcabitius has a carpenter’s square, Archimedes has a compass and a sphere to measure the Earth and Regiomontanus is pointing at a dodecahedron.
The last Liberal Art is the Astrology. She is backed on a terrestrial globe and her eyes are looking at the sky. She has a compass in one of her hands and the little boys around her have an armilar sphere and some astronomical books:
In one of the two panels on the walls we can see Dionysius the Areopagite observing a solar eclipse the day of Jesuschrist’s death in Athens (Luke, 23,45) We can notice a quadrant and an astrolabe in the hands of the amazed men!
The other fresco represents King Ezekiel resting in bed and looking how time is delayed 15 years by God because of the repentance of his sins:
The four famous men are Euclid, Ptolemy, Alfonso X and Johannes of Sacrobosco. Euclid is represented here meaning the relationship between Astrology and Geometry. He has drawn three geometrical schemes. One is a triangle and a square inscribed in a circle and another square. Another scheme seems to be two overlaid squares partially hidden by Euclid’s name. In the middle of both pictures there is a man measuring the stars. Johannes of Sacrobosco has a quadrant in his right hand.
King Alfonso X of Castile (XIIIth. c) is the author of the Libros del Saber de Astronomía (“Books of the Astronomical knowledge”) and on the tablet which he has in his hands we notice a compass and the Ursa Maior (the compass is anachronistic!). His left hand has an open book with a horoscope
So you can see that this wonderful vault is an open mathematical book designed by Tibaldi and Juan de Herrera. I’ve been twice in the library and now I am waiting for the next time that I could enjoy this artistic part of the monastery of San Lorenzo de El Escorial.
Location: San Lorenzo de El Escorial (map)
One of the most important objects kept in the Pushkin Museum of Fine Arts is the Moscow Mathematical Papyrus. This papyrus was purchased by Vladimir Golenishchev in 1893 from one of the brothers who found the Pharaoh’s tomb in Deir al-Bahri. The text is written in hieratic and it’s supposed to be compiled around 1850 BC. It measures 5 meters long and 8 cm wide and containing 25 mathematical problems. Unfortunately, not all the problems are readable so we only have information about some of its sections. The 25 problems are:
- Damaged and unreadable.
- Damaged and unreadable.
- A stick measures 1/3 and 1/5 [=1/3 + 1/5] of 30 = 16.
- Area of a triangle: 1/2 of 4 x 10 = 20.
- Pesus of loaves and bread.
- Rectangle of area = 12 and base = 1/2 and 1/4 of the height. Find the base and the height.
- Triangle of area = 20 and height = 2 and 1/2 of base. Find the base and the height.
- Pesus of loaves and bread.
- Pesus of loaves and bread.
- Area of a curved surface of a cylinder or a hemisphere.
- Loaves and basket. Unclear.
- Pesus of beer.
- Pesus of loaves and beer.
- Volume of a truncated pyramid.
- Pesu of beer.
- Pesu of beer.
- Triangle of area = 20 and base = 1/3 and 1/15 of height. Find the base and the height.
- Measuring cloth in cubits and palms.
- Solve the equation (1 and 1/2)x + 4 = 10.
- Pesu of 1.000 loaves. Oudjat.
- Mixing of sacrificial bread.
- Pesus of loaves and beer.
- Computing the work of a cobbler.
- Exchange of loaves and beer.
- Solve the equation 2x + x = 9.
The section of the papyrus which is exhibited in the Pushkin Museum is problem number 14. We can see clearly a figure similar to a trapezium that represents a truncated pyramid and the scribe computed its volume through the formula: V = (h/3) · (a2 + ab + b2). The exact text is:
It is said to thee, a truncated pyramid of 6 ellen in height.
Of 4 ellen of the base, by 2 of the top.
Reckon thou with this 4, squaring. Result 16.
Double thou this 4. Result 8.
Reckon thou with this 2, squaring. Result 4.
Add together this 16, with this 8, and with this 4. Result 28.
Calculate thou 1/3 of 6. Result 2.
Calculate thou with 28 twice. Result 56.
Lo! It is 56! Thou has found rightly.
Location: Pushkin Museum of Fine Arts in Moscow (map)
One of the most famous Egyptian circular zodiacs is located in the Parisian Louvre Museum since 1828. This archaeological piece was found in the temple of Dendera and it’s dated around year 50 BC. The circle of sky is supported by the four goddesses corresponding to the four cardinal points and eight falcon-headed gods more. Rounding the periphery of the circle we can see the 36 decans and the twelve zodiacal constellations and the planets in the middle. It is possible to distinguish some different constellations:
- The Hippopotamus Isis-Daymet in the center of the circle.
- A thigh representing the Ursa Major.
- A jackal with a bring between them.
- Cancer under the thigh.
- Leo under the Cancer.
- Taurus over the thigh.
- Gemini between Leo and Taurus.
- A god with two feathers next to Gemini.
- Virgo behind Leo.
- A little god sitting on a snake just over Leo.
- Libra behind Virgo.
- A set of three figures over Libra: a falcon-headed god with the solar disk on his head sitting on a throne, a god with his finger in his mouth sitting on a lotus flower and a jackal.
- Scorpio under this set of three figures.
- Sagittarius behind Scorpius.
- A man and a duck between Sagittarius and Capricorn.
- Capricorn behind Sagittarius.
- An headless animal over Capricorn.
- Pisces behind Capricorn.
- Aries over Pisces.
- A man holding the horns of an animal next to Pisces.
- An oryx and a baboon in Aries’ back.
- An oudjat between Aries and Pisces.
- Two goddesses under Aries.
- A goddess holding the legs of a pig inside a circle next to Pisces.
- Orion over Taurus and a bird following him.
- A cow representing Sirius on a ship.
- A falcon between Sirius and Orion.
- Satis with a bow and an arrow behind Sirius.
- God Anukis behind Satis.
- A lion next to Libra.
- A bull-headed god with a bring next to the lion.
Seti I was pharaoh of Egypt around the year 1290 BC and he was the second king of the Nineteenth Dynasty to which his son Ramses II belonged. His tomb has the reference KV17 in the King’s Valley and it’s another extant example of astronomical ceiling divided in two panels. One of them have 35 columns and in the first 23 columns we find the decans with their numbers of stars and their representative gods and goddesses. For example, column 22 describes Orion and column 18 describes the Egg. There also are Jupiter, Saturn and a retrograding Mars behind Isis in columns 24-26 and Mercury and Venus are described in columns 34 and 35:
In the other panel we have the circumpolar constellations in the central part between two series of gods and goddesses walking to the center of the picture:
Font: Wikimedia Commons
At the right we can see the Hippopotamus with the crocodile holding a cane where there are two ropes tied to the bull’s tail. This bull represents the Ursa Major and there is the goddess Serqet in front of it who has a falcon next to her feet. Below this falcon there is the Lion and the crocodile and the man fighting as we have also seen in Senenmut’s tomb. A man with a solar disk on his head is holding the ropes and Anu is standing on the cane.
Location: Seti I’s tomb in the Valley of the Kings (map)
Font: Wikimedia Commons
The Egyptians divided the stars of the night sky in different constellations (called “decans”) in a similar way that we do today. All these constellations traveled through the sky on ships. We have some examples of representations of these decans in the painted ceilings of different Egyptian tombs and we can establish that the number of them from one tomb to another. However it seems that there were 36 decans which always were painted.Egyptians divided the whole year in 5 days plus 36 periods of 10 days corresponding to the 10 days of consecutive risings of the decans:
- tpc-knmwt. A set of three stars represented by a man with a solar disk on his head. It is associated to Geb.
- knmwt. A boat with a head of a snake in the ship’s bow with a standing boy with one finger in his mouth. It is associated to Isis or Ba.
- hr-hpt-knmwt. A set of three stars represented by a tailed man with a bird’s head with a double crown. It is associated to Isis or Horus’ sons.
- h3t-d3t. A set of four stars represented by a falcon-headed man with two big feathers. It is associated to Isis or Hours’ sons.
- phwy-d3t. A set of three stars represented by a man with a white crown on his head and a scepter in his hand. It is associated to Horus’ sons.
- tm3-hrt. A set of three stars represented by a man with two big feathers on his head and a scepter in his hand. It is associated to Duamutef or Mesti.
- tm3-hrt. A set of three stars represented by a falcon-headed man with horns and a solar disk over them. He also holds a scepter in his hand. It is associated to Duamutef or Qeehsenuef.
- wsti. A man with the solar disk rounded by a royal snake. It is associated to Duamutef.
- bk3ti. A man with the double crown and a scepter in his hand. It is associated to Duamutef.
- tp-c-hntt. Two stars represented by a falcon-headed man with a solar disk on his head and a scepter in his hand. It is associated to Duamutef or Hapy.
- hntt-hrt. A set of two stars represented by a tailed man with a stick and a scourge in his hands. It is associated to Horus.
- hntt-hrt. A set of four aligned stars represented by the four aligned stars, a head and an arm on a ship. It is associated to Seth.
- tms-n-hntt. Three aligned stars represented by a baboon on a ship in front of a lunar disk. It is associated to Horus.
- s3pt-hnnw. A set of two stars represented by a falcon-headed man with a scepter in his hand. It is associated to Isis or Nephtis.
- hr-ib-wi3. A set of four stars represented by a falcon-headed man with a solar disk on his head and a scepter in his hand. It is associated to Seth or Ur.
- shsmw. Two stars represented by a man with a snake and a scepter in his hand. It is associated to Seth or Ur.
- knmw. A headless man with a solar disk on his shoulders. It is associated to Hapy or Duamutef.
- smdt. Three stars represented by a tailed man with a double crown. It is associated to Horus.
- tpc-smdt. A tailed man with a double crown. It is associated to Hapy.
- srt. A set of four stars represented by a falcon-headed man with the ateph crown and an animal tail in his skirt. It is associated to Isis.
- s3wy-srt. Two stars represented by a woman with a scepter in her hand. It is associated to Duamutef.
- hr-hpt-srt. A tailed man with a scepter in his hand. It is associated to Qebehsenuef.
- 3hwy. A tailed jackal headed man with a scepter in his hand. It is associated to Duamutef.
- b3wy. A set of three stars represented by a tailed ibis headed man with the ateph crown and a scepter in his hands. It is associated to Duamutef.
- hnt-hrw. A tailed falcon-headed man with a tadpole. It is associated to Hapy or Mesti.
- hr-ib-hntw. A tailed falcon-headed man. It is associated to Horus.
- hnt-hwr. A boy with one of his fingers in his mouth sitting on a lotus flower. It is associated to Horus.
- kd. Three stars represented by a boy with one of his fingers in his mouth sitting on a lotus flower and a solar disk on his head. It is associated to Horus.
- s3wy-kd. Two stars represented by a tailed man with the ateph crown and a scepter in his hand. It is associated to Horus.
- crt. A tailed man with a papyrus plant. It is associated to Qebehsenuef.
- h3w. Four stars represented by a tailed falcon-headed man with the double crown. It is associated to Hapy.
- rmn-hrw-inw-s3h. 18 stars represented by a tailed man with the white crown and a scepter in his hand. It is associated to Mesti.
- msdr-s3h. A lioness headed woman with a royal snake and a crosier in her hand. It is associated to Duamutef.
- rmn-hr-s3h. Three stars represented by a man with the white crown with a feather sitting on a throne. It is associated to Horus.
- c-s3h. An ibis headed man with the ateph crown and a scepter in his hand. It is associated to Horus.
- s3h. Its corresponding is Orion and it’s represented by two ships. On the first one there are three people: a boy with one of his fingers in his mouth and a double crown on his head, a horned woman with a solar disk on her head and another woman with two feathers and the solar disk between them. On the other ship there is a snake in a lotus flower. It is associated to Maat-Horus.
- spdt. Its corresponding is Sirius and it’s represented by a tailed falcon-headed man with a solar disk on his head and a scepter in his hand. It is associated to isis.
The first three decans (knmwt) are associated with Canis Major and Cancer, d3t with Leo, tm3, wsti and bk3ti with Virgo, hntt, s3pt-hnnw and hr-ib-wi3 with Scorpio, shsmw and knmw with Serpens, smdt and srt with Capricorn and Aquila, b3wy with Pisces, hnt-hwr with Aries and kd and h3w with Taurus.
Sesenmut was an important official who worked very close to the queen Hatshepsut (c. 1450 BC). He decided to be buried near her queen’s tomb and the ceiling of his burial monument (see the first picture) is an example of the decans explained above. The sky representation consists in two panels separated by five horizontal lines with prayers for Senenmut. In one of the panels (see picture below) we can distinguish perfectly the constellations Ship and Ram at the bottom right part. The Ship is described in top lines 7-12 from the right to the left and the Ram in top lines 14-16; lines 21-23 describe the Egg represented in the middle of the panel; lines 24-28 describe Orion with its three aligned stars and line 29 shows Isis with the crown with two feathers representing Sirius.
Therefore, lines 3o and 31 represent Horus as a star crossing the sky and they are associated with Jupiter and Saturn respectively. After them the constellation Turtle is described in line 32 and lines 33-37 describe five decans more. Finally Mercury and Venus are in the last lines with a representation of a bird called benu with a star in its head for Venus.
The second panel contains twelve circles divided in 24 circular sectors which correspond with the 12 months of the lunar year. In the middle of the picture the circumpolar constellations are represented. First of all we distinguish the top bull which is the Ursa Major with the stars delta, epsilon and zeta painted in its tail. At its back there is a woman with a solar disk and a scorpion on her head who represents the constellation Serqet (which has stars in Draco and Ursa Minor). We also observe a little crocodile over the scorpion. The bull is looking at falcon-headed god called Anu who is identified with the Swan:
At the bottom we see a crocodile and a man fighting which represent respectively some stars of Ursa Minor and some stars of Draco. There is a lion with crocodile tail over them and another crocodile over the lion. In the other side of the two rays which point to the bull’s tail there is the goddess Hippopotamus (Isis-Dyamut) who has a vertical crocodile in her left hand and a cane in her right hand. There is also a crocodile in her back and all the set could be the representation of some stars of Bootes, Lyra, Hercules and Draco.
Finally there are two series of gods walking to the center. From the left side to the right, the gods and goddess are Iremaua, Tekenu, Shedjeru, Nehes, Aaner, Imysehne-cher and Horhekenu. Behind the Hippotamus there are Isis with Horus’ four sons: Imseti, Hapy (baboon), Duamutef (jackal) and Qebehsenuef (falcon). Behind them there are Maintef (without hands), Irendyetef (without arms), Irrenefdyesef and Haqu.
Location: Senenmut’s tomb (map)
The Pharaonic Egypt was located between latitudes 24º and 32º in the northern hemisphere just over the Tropic of Cancer so all the Egyptians could observe the circumpolar stars in the night sky. It’s well known that the life after death was very important for Egyptians and they believed that their souls went to the region of the night sky where the stars were always visible. Therefore, Egyptian began to improve their astronomical knowledge so they could orient their burial monuments as the great pyramids. Khufu lived around the XXVth century BC and he decided to be buried in a majestic monument which could be the most wonderful building that was ever built. Some years later the pharaohs Khafre and Menkaure (also in the Fourth Dinasty) decided to be buried next to Khufu and ordered to build pyramids for their respectively sarcophagus. The result was the Giza Necropolis and if we observe it carefully we will be able to find some astronomical details which are important to know. For example, the four sides of the Great Pyramid are oriented according to the four cardinal points (the north side has a deviation of 2’28” on the terrestrial parallel and the deviation of the south side is only 1’57”; the east side has a deviation of 5’30” on the terrestrial meridian and the deviation of the west side is 2’30”).
Furthermore, the pyramid has two corridors from the Pharaoh’s room and two more corridors from the Queen’s room. As the pyramid is perfectly oriented, these four corridors drawn to some specific stars of the meridian. The northern corridor from the Pharaoh’s room (right corridor in the picture) points to Alpha Draconis which was next to the pole point in 2500 BC. This star was called “The Immortal” and it was the exact place where the Pharaoh’s soul was supposed to go after the royal death. The meridian corridor from the Pharaoh’s room points to Orion’s Bell and the northern and meridian corridors from the Queen’s room point to the stars Beta Ursa Minor and Sirius, respectively.
Therefore the Great Pyramid is like a very old astronomical observatory.
Location: Pyramid of Khufu (map)