Islamic science
(including mathematics and astronomy).
9721201 王重臻 9721204 吳旻駿 9721119 吳仁傑 9720117 亓天毅
Islamic Mathematics
Algebra
About Algebra
• To use notations to represent numbers and operations .
• To solve equations :
Linear equations (ax + b = 0)
Quadratic equations (ax
2+ bx + c = 0)
Cubic equations (x
3+ ax
2+ bx + c = 0)
Quatic equations (x
4+ ax
2+ bx + c = 0)
Al-Khwarizmi
• The father of Algebra
• The book Algebra
Algebra
• Ch I : Squares equals to roots (ax
2= bx)
• Ch II : Squares equals to numbers (ax
2= b)
• Ch III : Roots equals to numbers (ax = b)
• Ch IV : Squares and roots equal to numbers (ax
2+bx=c )
• Ch V : Squares and numbers equal to roots (ax
2+b=cx)
• Ch VI : Roots and numbers equal to squares (ax+b=cx
2) In middle Arabic Mathematic . They have not
accepted “ non-positive” numbers yet . So that
every terms and coefficients should be positive ,
including the solutions .
How to solve x
2+10x=39
Our modern method:
Factorization !!!
x
2+ 10x = 39
x
2+ 10x – 39 = 0
(x-3)(x+13) = 0
x = 3 or x = -13
yes!!!
How to solve x 2 +10x=39
Al-Khwarizmi ‘s
GEOMETRIC FOUNDATION :
x 2
2 1/2． x2 1/2． x
2 1/2 ． x
2 1/2 ． x
+ =39
25/4
25/4
25/4
25/4
The whole square = 39 + 25 = 64 Side of the largest square = 8
x = 8 – 2 ． 2
1/
2= 3
How to solve x
2+21=10x
Our modern method:
I’m too LAZY to calculate ….. XD
X = 3 or 7
How to solve x
2+21=10x
Al-Khwarizmi ‘s
GEOMETRIC FOUNDATION :
x
x 21
10 5
5
5 5-x
5-x x
= 2
= 3
Omar Khayyam
• Omar Khayyam had tried to solve cubic equations by some algebraic method , but failed .
• He construct geometric solutions .
• Omar Khayyam also claimed that
Algebraic sol’n to general cubic equations is impossible
Which turned out to
be possible !!!
NOTE: Cardano(Italian)-Tartaglia(Italian)
Formulathen q
px x
that
Suppose
3 0 ,
3
3 2
3
3 2
3 2
2 3
2
2
q q p q q p
1
x
3
3 2
3 2
2
q q p
3
3 2
3 2
2
q q p
ω +ω
2+ω ω
22
x
3
x
3 2 2 2 3 3
q q p
3
3 2
3 2
2
q q p
How to solve x
3+x=1
2 1 1
x x
= y
［ Sol ］ We want to separate it into two proportions.
The original equ.
x 1 x
2
1
1
2
1 xy
x
y
How to solve x
3-30x
2+500 = 0
= y
30 5 100 0
2 x
x
［ Sol ］ By the same method , we get :
The original equ.
x x
30 5 100
2
5 30 )
( x
y
y x
2 100
(29.422 , 8.657)
(4.421 , 0.195)
(-3.844 , 0.148)
Besides , Omar Khayyam divided all cubic equations into 14 types :
x
3= c ;
x
3+bx=c , x
3+c=bx , x
3=bx+c;
x
3+ax
2=c , x
3+c=ax
2, x
3=ax
2+c;
x
3+ax
2+bx=c , x
3+ax
2+c=bx , x
3+bx+c =ax
2,
x
3=ax
2+bx+c , x
3+ax
2=bx+c ,x
3+bx=ax
2+c , x
3+c
=ax
2+bx .
And gave each type a geometric sol’n .The same as other mathematician ,
POSITIVE SOLUTIONS ONLY
Geometry
&
Number Theory
Early Islamic Geometry &
Number Theory
Thâbit (Thâbit ibn Qurra) (826-901)
Contributions:
• He translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.
• He generalized the Pythagorean Theorem.
• He found a method for discovering amicable
numbers, known as the Thâbit ibn Qurra rule (or
simply Thabit’s rule) nowadays.
Theorem. ( Generalization of Pythagorean Theorem .)
Given an arbitrary triangle
△ABC, construct B’
and C’ such that AB’B= AC’C= A (as ∠ ∠ ∠ shown below)
Then, |AB|
2+|AC|
2= |BC|(|BB’| + |CC’|)
( Here, |XY| means the length between X and Y.)
Proof of this theorem ：
∵△ABC ～△ B’BA |AB|/|BC| = |B’B|/|BA| ∴ , which implies |AB|
2= |BC|×|BB’|.
∵ △ ABC ～△ C’AC |AC|/|BC| = |C’C|/|AC| ∴ , which implies |AC|
2= |BC|×|CC’|.
Thus, |AB|
2+|AC|
2= |BC|×(|BB’|+|CC’|)
.□Special Case of this theorem (α=90°)
becomes
Applying the theorem, we obtain
|AB|
2+|AC|
2= |BC|×(|BB’|+|CC’|) = |BC|
2, which is the Pythagorean theorem, which we
are familiar with.
Definition ( amicable numbers )
Amicable numbers are a pair of two different positive integers p and q such that the sum of proper divisors of p is q, and vice versa.
(Note: A proper divisor of a positive integer is a positive divisor other than the number
itself.
Ex: 1, 2, 3 are the proper divisors of 6.)
Example:
(220, 284) is a pair of amicable numbers.
(Actually, this is the smallest pair of amicable numbers)
The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110.
1+2+4+5+10+11+20+22+44+55+110=284 The proper divisors of 284 are 1, 2, 4, 71, and 142.
1+2+4+71+142=220
Thabit’s rule:
If p =3×2
n−1−1, q=3×2
n−1, r=9×2
2n−1−1, where n>1 is an integer, satisfy that p, q, r are prime.
Then, 2
npq and 2
nr is a pair of amicable
numbers.
Proof of Thabit’s Rule:
∵p, q, r are prime
∴
The sum of positive divisors of 2npq except for 2npq itself is (1+2+22+…+2n)(1+p)(1+q)- 2npq=[(2n+1-1)/(2-1)]×3×2n−1×3×2n-2n(3×2n−1−1)(3×2n−1)
=9×23n-1-2n=2nr
and the sum of positive divisors of 2nr except for 2nr is (1+2+22+…+2n)(1+r)-2nr
=[(2n+1-1)/(2-1)]×9×22n−1-2n×(9×22n−1-1)
=2n(3×2n−1−1)(9×22n−1−1)
=2npq
Trigonometry
Yusuf ibn Ahmad al-Mu'taman ibn Hud
The Triangle Theorem of Yusuf ibn Ahmad al-Mu'taman ibn Hud
(known as Ceva’s Theorem nowadays)
Consider ΔABC as below.
Then, we have the following property:
Proof of Ceva’s Theorem
| 1
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BOA BOC AOC
AOB BOC
AOC EA
CE DC
BD FB
AF
BOA BOC EA
CE
AOC AOB DC
BD
BOC AOC FB
AF
Abul Wafa Buzjani
The Six Trigonometric Functions
After the work of Abul Wafa Buzjani, mathematicians use six
trigonometric functions:
Sine, Cosine, Tangent, Cotangent,
Secant, Cosecant.
Some Relations of Trigonometric
Functions Discovered by Abul Wafa Buzjani :
c C b
B a
A
x x
x
x x
sin sin
sin
cos sin
2 2
sin
sin 2
1 2
cos
cos sin
cos sin
) sin(
2
Engineering─some
architectures and machines
1.Dam(Kanats ； Karez)
2.Water-raising machine
1.Dam
(Kanats ； Karez)
Glossary
• Mother Well :The first-builded well
• Shaft : It is also a well and convenient to repair dam and remove dirt .
• Water Channel : Just water channel .
• Aquifer : A layer which contains water.
• Impermeable layer : A layer which doesn’t contain water.
• Canal :Just canal.
• P.S. The difference between Water Channel and Canal : Water Channel is undergroune ,Canal is on the ground.
Q&A
Q:Why muslim require dams?
A:Water is very precious for muslim.
Dam is a hydraulic system for them.
Q:What advantages do dams have?
A: In wadi irrigation, they are used to trap the floodwaters that result from heavy but infrequent downpours, so that the water-level is raised above that of surrounding fields, to which it can be conducted under gravity.
It is also used to divert water from streams or river into canal network.
The impounding of river behind dams gives more control over the supply.
It also allows the water in the reservoir to be gravity-fed into irrigation and town to supply systems.
2 .Water-raising machine
Glossary
Drawbar : The drawbar is such as the shaft of a pen which connects the aniaml and upright shaft.
Lantern pinion :The lantern pinion is two large wooden discs held apart by equally spaced pegs. The vertical gear-wheel carriers the pot-garland wheel.
Potgarland wheel : The potgarland wheel is a vertical gear-wheel carries the chain-of-pot.
Cylindrical pot : Cylindrical pot consists of two continuous loops of
rope between which the earthenware pots are attached-sometimes chain and metal containers are used.
Pawl : A structure which acts on the cogs of the potgarland wheel
How does the machine work?
The machine is a chain-of-pots driven through a pair of gear-wheels by
one or two animals ,such as donkeys ,mules or oxen, harnessed to a draw- bar and walking around a circular track. The shaft rotates in a thrust bearing at ground level and another bearing above the the gear-wheel located in a cross-beam which is supported on plinths. Potgarland wheel is supported centrally over the well or other source of water on a wooden axle. On one side of it are the pegs that enter the spaces between the pegs if the lantern- pinion and these pegs pass through to the other side of the wheel ,where they carry the chain-of-pots.
As the animal walks in a circular path, the lantern-pinion is turned and this rotates the potgarland wheel. The pots dip into a water in continuous
one by one and pour water at the top of the wheel into a channel connected head tank.
Pawl is important?
In order to prevent the wheel from going into reverse, the machine is provided with a pawl mechanism. This mechanism is essential, because the draught animals is subjected to a constant pull both when moving and when standing still. The pawl actives when the animals is to be unharnessed and in the event of the harness or traces breaking. Without the pawl the machine would turn backwards at great speed and, after one revolution, the drawbar would hit the animal on the head. At the same time, many of the pins of the latern-pinion would break and the pots smash.
Islamic Astronomy
Some Problems
• Ramadan
• Time for prayer
• Positional Astronomy
• Ramadan
A month starts when people “see”
the crescent.
Leap month
• Time for prayer
• Positional Astronomy
• Ramadan
• Time for prayer
Five times a day (Dawn, sunset, the third, the sixth, the ninth “hour”)
al-Khwarizmi (900 AD)created a timetable correspond to the latitude of Baghdad (by using spherical
trigonometry).
• Positional Astronomy
• Ramadan
• Time for prayer
• Positional Astronomy
The mosques must face to the direction of Mecca, the sacred city.
Qibla
The Observatories
• Maragha, the North of Iran (1260 AD)
Built by Hulagu, for Nasir al-Din al-Tusi.
10 feet wide armillary sphere, 28 feet wide mural quadrant
Achievement : 《 Zij 》 ,an astronomical table
based on Ptolemy’s 《 Handy Tables 》
• Ulugh Beg Observatory (1420 AD)
Built in Samarkand, Uzbekistan, using a huge sextant to observe the solar system.
