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 (ax2 + bx + c = 0)

Cubic equations (x3 + ax2 + bx + c = 0) Quatic equations (x4 + ax2 + bx + c = 0)

Al-Khwarizmi• The father of Algebra• The book Algebra

Algebra• Ch I : Squares equals to roots (ax2 = bx)

• Ch II : Squares equals to numbers (ax2 = b)

• Ch III : Roots equals to numbers (ax = b)

• Ch IV : Squares and roots equal to numbers (ax2+bx=c)• Ch V : Squares and numbers equal to roots (ax2+b=cx)

• Ch VI : Roots and numbers equal to squares(ax+b=cx2)

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 x2+10x=39

Our modern method: Factorization !!! x2 + 10x = 39 x2 + 10x – 39 = 0 (x-3)(x+13) = 0 x = 3 or x = -13 yes!!!

How to solve x2+10x=39Al-Khwarizmi ‘s GEOMETRIC FOUNDATION :

x2 2 1/2 ． x 2 1/2 ． x

2 1/2 ． x

2 1/2 ． x + =3925/4

25/4

25/4

25/4

The whole square = 39 + 25 = 64

Side of the largest square = 8

x = 8 – 2 ． 21/2 = 3

How to solve x2+21=10x

Our modern method:

I’m too LAZY to calculate ….. XD

X = 3 or 7

How to solve x2+21=10xAl-Khwarizmi ‘s GEOMETRIC FOUNDATION :

x

x 21

10 55

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) Formula

thenqpxxthatSuppose , 0 3

3

32

3

32

322322

pqqpqq1x

3

32

322

pqq3

32

322

pqqω +ω 2

+ω ω 2

2x

3x 3

32

322

pqq3

32

322

pqq

How to solve x3+x=1

112 xx

= y

［ Sol ］ We want to separate it into two proportions.

The original equ.

xx

112

1

12

xy

xy

How to solve x3-30x2+500 = 0

= y

01005302 xx

［ Sol ］ By the same method , we get :

The original equ.

x

x

30

5

100

2

530 )( xy

yx 1002

(29.422 , 8.657)

(4.421 , 0.195)

(-3.844 , 0.148)

Besides , Omar Khayyam divided all cubic equations into 14 types :

x3 = c ;x3+bx=c , x3+c=bx , x3=bx+c;x3+ax2=c , x3+c=ax2 , x3=ax2+c;x3+ax2+bx=c , x3+ax2+c=bx , x3+bx+c =ax2 ,

x3=ax2+bx+c , x3+ax2=bx+c ,x3+bx=ax2+c , x3+c =ax2+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 =3×2n−1−1, q=3×2n−1, r=9×22n−1−1,

where n>1 is an integer, satisfy that p, q, r are prime.

Then, 2npq and 2nr 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||

||

||

||

||

||

||

||

||

||

||

||

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

xxx

xx

sinsinsin

cossin22sin

sin212cos

cossincossin)sin(2

Engineering─somearchitectures 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&AQ: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 theaniaml and upright shaft.

Lantern pinion :The lantern pinion is two large wooden discs held apart byequally spaced pegs. The vertical gear-wheel carriers the pot-garland wheel.

Potgarland wheel : The potgarland wheel is a vertical gear-wheel carries thechain-of-pot.

Cylindrical pot : Cylindrical pot consists of two continuous loops ofrope 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 across-beam which is supported on plinths. Potgarland wheel is supportedcentrally over the well or other source of water on a wooden axle. On oneside 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 thisrotates the potgarland wheel. The pots dip into a water in continuousone by one and pour water at the top of the wheel into a channel connectedhead 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

• RamadanA 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.

Achievement: Published a star table with more than one thousand suns.

Contribution

• Reformation of Ptolemy’s theory about planetary astronomy.

• Compiling numerous astronomical tables (ex. 《 Toledan Tables 》《 Alfonsine Tables 》 ).

• Translated scientific works from the ancient Greece into Arabic, passing them on to Western Europe through Islamic Spain.

«References»• Boyer, Carl Benjamin, 1906-A history of

mathematics• 傅鍾鵬 , 三次方程風雲記 • http://en.wikipedia.org/wiki/Al-Khwarizmi• Elizabeth Rogers, Islamic Mathematics• http://

en.wikipedia.org/wiki/Thabit_ibn_qurra

• Routledge, Encyclopedia of the history of Arabic science

• Michael Hoskin ,The cambridge illustrated history of astronomy