Egypt and Mesopotamia Flashcards
the gift of the Nile
Egypt
Fertile banks of Nile and sea and dessert protect Civillisation
Without the Nile Agriculture and Civilisation couldn’t thrive
flooding allowed for this to happen
Upper and Lower egypt 3100 BC until conquest by Alexander the Great
pyramids
The Great Pyramids were built around 2500 BC the height of the Egyptian culture
people were aware of the culture but not sure on meaning of hieroglyphics
Hieroglyphs
Egyptian script, originally used all-purpose then only in religious contexts and monumental stone carving
The art of reading ancient E scripts was lost:
the key to these was the discovery in 1799 of the Rosetta Stone
read from left to right or right to left depending on the facing of the symbols
Hieratic
script developed after hieroglyphs more fluid and better suited for writing with pen on papyrus
Herodotus
about 450 BCE Herodotus, the inveterate Greek traveler and narrative
historian, visited Egypt
“father of history”
recorded the customs of many peoples
observed the majesty of the Nile and the achievements of those working along its banks.
ROSETTA STONE
Found near Rosetta. Rosetta Stone, a trilingual basalt slab with inscriptions in hieroglyphic, demotic, and Greek writings that had been found by members of Napoleon’s Egyptian expedition in 1799
- Rosetta on the Nile delta
- most viewed exhibit in the british museum
- contains decree of 196 BC three times in hiero,dem and greek
- unable to read hieroglyphs at the top but many read greek and tells it has the same info in three languages
*By 1822, Champollion was able to announce a substantive
portion of his translations
Earning title Father of Egyptology
by using cartouches?
hieroglyphs
French scholar Jean-Franc¸ois Champollion, working with multilingual tablets, was able to slowly translate a number of hieroglyphs
phonetical and alphabetical
*because of variability of hieratic and demotic scripts Egyptologists transcribe all texts into more standardized hieroglyphic they dep on the scribe
papyri and sources
- rhind papyrus
- moscow papyrus
- MLR(hieratic?)
- the royal mace (hieroglyph)
- Rosetta stone (demotic)
rhind papyrus
on exam
RMP
1858, the Scottish antiquary Henry Rhind purchased a papyrus roll in Luxor that is about one foot high and some eighteen feet long. Except for a few fragments in the Brooklyn Museum, this papyrus is now in the British Museum. It is known as the Rhind or the Ahmes Papyrus, in honor of the scribe by whose hand it had been copied in about 1650 BCE. The scribe tells us that the material is derived from a prototype from the Middle Kingdom of about 2000 to 1800 BCE. Written in the hieratic script,
it became the major source of our knowledge of ancient Egyptian mathematics
- 84 hieratic problems
- 18ft x 1 ftm 84 Hieratic problems
- Rhind mathematical papyrus 1675 BC?
Moscow papyrus
MMP
Moscow Papyrus, was purchased in 1893 and is now in the Pushkin Museum of Fine Arts in Moscow. It, too, is about eighteen feet long but is only one-fourth as wide as the Ahmes Papyrus (3 inch) . It was written less carefully than the work of Ahmes was, by an unknown scribe of circa. 1890 BCE. It contains twenty-five examples, mostly from practical life and not differing greatly from those of Ahmes, except for two that will be discussed further
on
18ft x 3 inch 25 problems in hieratic
practical problems
mathematical leather roll
a leather roll containing a list of fractions.
The only leather roll
10 in x 17 in 26 HIERATIC fractions
Royal mace
A museum at Oxford has a royal mace more than 5,000 years old, on which a record of 120,000 prisoners and 1,422,000 captive goats appears.
These figures may have been exaggerated, but from other considerations it is clear that the Egyptians were commendably accurate in counting and measuring.
Primary source and largest number mention in ancient Egypt
HIEROGLYPHIC NUMERALS
demotic
script 7th century BC cursive script, mad doctors handwriting at the end of a bad day
ON EXAM
Q1 part a) Egyptian mathematics, Rhind papyrus (not leather roll)
nothing on pyramids and seqts nor on the 3 classical problems
need dates: Rhind 1675 Egyptian? 1546 beg egypt 3500BC know order they came in aswell as dates
three ancient Egyptian scripts are in chronological order
hieroglyphic 3500BC pictorial carved on monuments
hieratic 2500BC
flowing ink on papyrus, religious
demotic 1850 BC
cursive, everyday
scripts in egypt occur on:
hieroglyphic - Narmer Royal Mace
hieratic- MLeatherR
demotic-
Rosetta stone
Hieroglyph notation fractions
ALL fractions, except 2/3 are reduced to sums of unit fractions ie numerator 1.
They are indicated by placing an elongated oval over its denominator , except 1/2 and 2/3
eg
1/3 = “eye shape”over”111”
1/12= “eye shape” over “11n”
if we write over-lined n as 1/n
we write 2/3 by oval above one and a half vertical strokes, overlineoverline3
The RMP opens with a table expressing fractions of the form 2/n as sums of unit fractions for odd values of n from 5 to 101
eg 2/25 = overline15 overline75
adding= written in juxtaposition
Egyptian multiplication 13 x 11
need to know
they multiplied by doubling and then addition
\1 11
2 22
\4 44
\8 88
add the lines marked with a backslash to deduce that these are the ones you add:
(1+4+8)11= (13)11 = 11+44+88=143
division is performed by reversing the multiplication process.
Chronology
(Egypt was split)
3100 BC
Upper and Lower Egypt united into one country- Egypt
2500 BC Great Pyramids
1850 BC Moscow Papyrus
1650 BC Rhind Papyrus
332 BC Alexander’s conquest invaded Egypt
1799 AD Rosetta Stone 1822 AD translated by Champoliion 1858 RHIND papyrus purchased 1865 AD comes to british museum 1877 AD maths translated by Eisenlohr (he numbered the problems on the papyrus, still used now)
hieroglyphic numerals
1 is |
10 “n”/yoke
100 swirl/coil
1000 looks like moon / lotus flower/balloon with weight
10,000 boomerang/finger
100,000 face? animal tadpole
1,000,000 man with arms up/kneeling
ie number system based on powers of ten
Decimals. Repetitions of symbols for powers of ten
RMP ALGEBRA
There is no algebraic notation but in RMP problem 28 2 legs walking left to right indicate addition
right to left are subtraction
in MMP problem 14 two legs walking left to right indicate squaring
problems 24-27 aha or heap problems are single linear eq in one unknown could be classed as algebraic
They are solved by method of false position, in which an initial guess is made and then adjusted to yield the answer . This is then checked by substitution back into original eq.
aha or heap is the unknown, stated verbally
RMP 25
the heap and it’s half is 16, what is the heap
x+0.5x=16
a) guess x=2. Setting in LHS the eq yields 3, not 16.
b) dividing 16 by 3 gives 5 and 1/3. This is what 2 must be multiplied by to give the answer
c) multiply the initial guess of 2 by 5 and 1/3 to obtain answer 10 and 2/3
d) final check 10 and 2/3 + 0.5(10 and 2/3) = 16
Note initial guess 2 was likely chosen because its half is an integer ,1.
Assume 2 \1 2
\0.5 1
sum 3
as many times as 3 must be multiplied to give 16, so many times 2 must be multiplied
\1 3
2 6
\4 12
2/3 2
\1/3 1
sum 5 1/3
1 5 1/3 \2 10 2/3
Do it so: heap is 10 2/3
0.5 is 5 1/3
sum is 16
PRIMARY SOURCES
ancient egypt
3100 BC Royal Mace, ashmolean Museum , Oxford. High valued, hieroglyphic numerals
NOT***
1850 BC Moscow papyrus, Moscow museum of fine arts. 25 hieratic problems
1650 BC RHIND papyrus British Museum 10057/8 in two parts, 84 hieratic problems
***
1650 BC MLR British museum 26 sums of hieratic unit fractions twice
1799 Rosetta Stone- hieroglyph demotic greek (primary source)
Area of a circle RMP problems
REVISE WELL
RMP problems 48 and 50 both take the area of a circle
(also 41,42,43 concern finding the volumes of cylindrical granaries diameters 9,10,9 and 10,10,6 areas of bases times Heights. Area of circular base of diameter d is always
(d-(d/9))^2 = ((8/9)d)^2)
both problems take the area of a circle of diameter 9 to be that of a square of side 8, ie 64
the equation pi(9/2)^2=64 shows both give an implicit value of Pi of 256/81 . The units of measurement using the problems show their concern circular areas of land.
(effectively found by taking 8/9 of diameter and square it- they squared by doubling )
*the Egyptian circle area that is not an integer is that of the circle of diameter 10-in problem 42 which by the Egyptian method is 6400/81 = 79 1/81
IMPORTANT
7 marks question on Egypt
need to be able to write a paragraph on the rhind papyrus:
Henry Rhind felt ill, doctor told him to take holidays further and further away. Winter in egypt.
Henry Rhind bought the Papyrus at Luxor in 1858.
Rhind papyrus is in two parts and listed as two numbers in the british museum, found half . Fragments from the break are found by the NY Hisorical society in 1922, now in Brooklyn Museum.
This is a training manual for scribes, copies by Ahmes in 1650 BC comprises
84 worked problems and promises insight into all that exists.
Translated by Eisenlohr, Peet and CHace
Robins and Shute’s Rhinf Mathematical Papyrus has 24 coloured plates of the Papyrus.
red ink for problem and black ink for answer
problems on the RMP
- 24 to 27 aha problems false position, algebra
- 48,50 area of a circle (implicit pi)
- 41,42,43 vlumes of cylindrical granaries
- 56-60 seqts, slopes, pyramids (trigonometry)
*79 geometric progression (recreation)
NO PROBLEM ON THE RMP or MMP concerns finding the circumference of a circle
RMP problem 50
round field diameter 9. Find its area
subtract 1/9 of diameter, 1. Remainder is 8 X 8 to make 64. This is the area.
Do it thus: 1 9
1/9 1
is taken away leaves 8.
1 8
2 16
4 32
\8 64
Area is 64
- passing from the diameter 9 of the circle to the side 8 of the square is affected by subtracting 1/9 of 9 from itself to leave 8. The Egyptians had no concept of the fractions such as 8/9, operating only with unit fractions, like 1/9 here. The diagram shows a circle enclosing a number 9
SEQTS
not on exam?
a cubit is 7 hands
pyramid builders measured vertical distances in cubits and horizontal distances in hands, slopes are measured in seqts.
(the seqt s of a face of a pyramid P is half its base in hands / height in cubits)
the sect of a pyramid is the ratio of the horizontal displacement of a face (half base) enhance to its vertical displacement (height) in qubits, and is an inverse measure of steepness with gentler slopes having higher seqts.
Seqt of a pyramid with height h cubits and base b cubits is 7(0.5b)/h hands per cubit
The angle of slope theta of P is the angle between one of its faces P and its horizontal base, tan theta=2h/b = 7/s and s =7cot theta.
RMP problems 56/59 concerning the seqts of pyramids, lie one below each other at the end of the recto of BM10057. Each is accompanied by a sketch of the pyramid, the numerical values of its base and height written alongside.
RMP problem 57 the seqt of a pyramid is 5 hands and one finger per cubit, the side of its space is 140 cubits. What is it height? A finger is a quarter of a hand.
Solution in modern Let pyramid have height hqbutts. Then the sect is 7(140/2)/h = 21/4 whence h=(7x140x4)/(2 x 21)=93 1/3
RMP 48
most important of the problems
48:ONLY PROBLEM WITH NO RED INK, NO STATEMENT OF PROBLEM
compare the area of a circle and that of its circumscribing Square
circle diameter 9 1 8 2 16 4 32 \8 64
Square side 9
\1 9 2 18 4 36 \8 72 sum 81
Area circle ~octagon = 63~64
taking the area of the circle to be 64
- only RMP problem with no statement - no red ink
Diagram is square circumscribing irregular polygon , octagon? with a nine drawn inside. Beneath, 2 multiplications are set out : 9x9 =81 and 8x8=64 by doubling and edition with check marks
Geometric series
RMP problem 79:concerns the geometric series. It is unclear what the problem is (as there is no red ink??? only 48??) describe sets out in a column: 7 houses, 49 cats, 343 mice, 2401 ears of Grain, 16807 hekats of grain, total 19607. another column shows 19607 was found as 7(1+7+49+343+2401), so the Egyptians knew something of geometric progressions. the problem may just be a diversion, perhaps a precursor of the nursery rhyme: as I was going to St Ives, I met a man with seven wives; each wife had seven sacks, each sack had seven cats, each cat had 7 kits; kits, cats, sacks and wives, how many we’re going to St Ives?
MMP problem 14
MOSCOW
coined the greatest Egyptian pyramid by ET Bell. This problem calculates the exact volume V = 56 of the frustum of a square pyramid of height h = 6, side of an upper face a = 2, and side of lower face b = 4, using the equivalent of the correct formula:
V= (h/3)(a^2+ab+b^2)
a diagram is given along with the full details of all the sub calculations performed, written both in and around it. On two occasions, a pair of legs walking from left to right indicates operation of squaring.
*the frustum can be seen to have been cut from a pyramid of height 12 and base 4. This had seqt (7x2)/12 = 7/6 hands per cubit.
RMP problems 56-59
concern pyramids as shown by triangles illustrating them. Pyramid builders used seqts to maintain constant slopes in their constructions.
let pyramid have height h cubits and angle of slope theta. Then seqt = 5 1/4 = (7x70)/h and h= (280/3) = 93 1/3 cubits. Now tan theta= height/ 0.5base = (280/3)/70 = 4/3 and theta = tan -1 (4/3) = 53 to the nearest degree
notes on what questions may come up
problem 15 of the rhind papyrus, implicit values of Pi, finding the area of a circle
diagram of problem 48 on the Rhind papyrus finding areas of the circle: only one that does not have the red ink, all it has is a diagram of the square circumscribing an irregular polygon which is possibly an octagon with a number 9 script inside of it. underneath are the multiplications of 9 x 9 and 8 by 8. Approximates area.
fraction notation in egyptian
- *7 marks is on Egyptian mathematics
- *16 marks on Mesopotamia?
MESOPOTAMIA and BABYLONIA
The land between the rivers
3000 BC Sumerian civilization flourished the fertile crescent between the Tigris and Euphrates rivers, now in Iraq. Babylonian refers to any people occupying the region up to 539 BC, then Babylon fell to Persia. Babylonian maths continued through persia and greek rule up to Christianity.
3000 BC SUmerians 2500BC cuneiform script sexagesimal, place-value number system 1800-1600BC old babylonian 539 BC Persia defeats Babylon 330 BC Alxander the Greats conquest 312 BC+ Seleucid period:0 symbol 1849 AD Rawlinson deciphers cuneiform 1935 AD+ Neugebauers pioneering work
*history of Mesopotamia is a story of Invaders who attracted by the richness of the Land conquered their predecessors only to be overcome by new Intruders
cuneiform
the Babylonians inscribe their numbers in clay using a triangular stylus, producing a wedge-shaped script known as cuneiform.
Babylonic cuneiform (wedge-shaped) script made by impressing triangular stylus into moist clay, then baking.
inscribed with the clay is still moist and baked hard in an oven or by The Heat Of The Sun
babylonian references
our knowledge of that mathematics rests on 400 also clay tablets either TABLE TEXTS OR PROBLEM TEXTS from the old Babylonian period (1800 -1600) BC
*published in 20th-century by assyriologists such as:
Thureau-Dangin, Sachs, Neugebauer (last century and a half)
*Exhibits of Babylonian Cuneiform Tablets
National Museums: Berlin, London, Paris
US Universities: Columbia, Pennsylvania, Yale
inscribed on bottom, sides etc
