出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

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MATHEMATICSLEARNING THROUGHHISTORY OF MATHEMATICS.
FROM ARAB MATHEMATICIAN OMAR-AL-KHAYYAM METHOD.

MASTER PROGRAME OF EDUCATION, UNIVERSITY OFTSUKUBA.

M S T. ZIAUN NAHAR

ABSTRACT :1. Introduction,.

This paper illustrates the significance of the

2. The purpose of study and the aim of study,

geometrical representation, Arabic culture

3. An contents of class,and modern technology Cabri geometry. In

3.1 Explanation of materials,

this regards I posed history of mathematics.

3.2 Surroundings of the class,In this case I used Omar-Al-khayyam’s

3.3 Development of class,method wherein he used to solve cubic

4. Result of questionnaire.

equation considering unknown numbers. I

5. Conclusion.

used known numbers and justify modern
technology and change student’s idea about

Key Word: Solid, cubic equation, parabola,

mathematics.

Hyperbola , square.
1.

INTRODUCTION :Mathematics is a subject which student doesn’t feel interest to solve. The teacher who knows
little of the history of Mathematics is apt to teach techniques in isolation, unrelated to the
problems and ideas which generated them. History exposes interrelations among different
mathematical domains, or, of mathematics with other discipline (for example culture). History
tells us that the tools used for mathematical inquiry are themselves reformulated using
mathematical (especially geometrical) representations, as mathematicians in the scientifichttp://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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revolution tried to select, find or construct convenient representations and instruments fortheir research. This paper also discussed software Cabri Geometry from the view point ofintegrating traditional instruments (compass, ruler etc.) to Modern Technology.
Historically we can find that Arab mathematician Omar-al-Khayyam, Giyat ed-din Abul Fath,was born at Nishabur(IRAN) 1048 A.D. and died 1131 A.D. He could solve algebraic cubic
equation by using geometrical representation. For solving three degree (cubic) equation heused Greek mathematical work, Euclid and Apollonius. Omar made no addition to the theoryof conics, he did apply the principle of intersecting conic sections to solving algebraic problems.
In doing so he not only exhibited his mastery of conic sections, but also showed that he wasaware of the practical applications of what was a highly abstruse area of geometry.

History of Arab Mathematics : The year A.D. 622 is a momentous one in World history. The Prophet Muhammad fled from
Mecca to Medina, about 350 kilometres away and spread the wholly Quran and established Islam.
There is a long history, I would like to overtake that, cause of mathematics history of Arab
especially Baghdad (IRAN). The process was aided by the creative tension between two main
traditions of astronomy and mathematics represented in Baghdad, even from the early years ofIslamic rule. One tradition derived directly from India and Persian sources and is best exemplified
in the astronomical tables and the algebraic approach to mathematics. One of the greatest
exponents of this tradition, who left an indelible mark on the subsequent development of Arabmathematics, was al-Khwarizmi. To him mathematics had to be useful and help with practicalcorners such as determining inheritances, constructing Calenders or religious observances. The
other tradition looked to Hellenistic mathematics, with its strong emphasis on geometry anddeductive methods. They established a school in Baghdad. One of the best-known exponents of thisschool was Thabit ibn Qurra, who was both an outstanding translator of Greek texts and an
original contributor to geometry and algebra. That the two traditions eventually merged is evident
in the work of later Arab mathematicians such as Omar Khayyam and al-Kashi. Abu Jafar
Muhammad ibn Musa-al-Khwarizmi (780-850AD) wrote Hisab al-jabr w’al-muqabala (calculation
by restoration and reduction). His book on algebra contained an analysis of property relations andthe distribution of inheritance according to Islamic law, and rules for drawing up walls.

But hetechnically avoid three degree equation. In this tradition the other scientists who were largely
responsible for lying the foundations of modern science. These included Thabit ibn Qurra (836-901),
al-Razi (865-901), al-Haytham 8965-10399,al-Biruni (973-1051), ibn Sina (980-1037), Omar
al-Khayyam (1048-1131), Sharaf al-din al-Tusi (d1213), Nasir al-Din al Tusi (1201-74), andal-Kashi (d 1429). Among them Ibrahim ibn Sina’s commentary extending Archimedes work on thequadrature of the parabola has been described as one of the most innovative approaches known出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

3before the emergence of the integral calculus.
2.

THE PURPOSE OF STUDY AND THE WAY OF STUDY :-The purpose of study : - I would like to change clearly students idea through experiencing byhistory of mathematics.
The way of study : -To achieve the above mentioned aim I established the following :ⅰ The students may be don’t know the meaning of geometricalrepresentation of algebraic equation. Through my lecture they’lllearn that.
ⅱ May be they don’t have any idea about Arab culture, though my
lecture they will introduce a new culture.
ⅲ Through my lecture they could apply old mathematics in Modern
Technology.

For the above purposes I posed questionnaire for students before the lecture and after thelecture.
3.

An outline of the class.
3.1 Explanation of materials : The Algebra of Omar Khayyam ( page 75 to 79 ), Euclid Elements( pages 69-70 ),Euclid Archimedes Apollonius of perga (pages
616-618,627-628,640,669-670,661-662),

Rubaiyatof

Omar

Khayyam (original Persian language pages 3-6, poem 17) Arab
Genius in science and Philosophy ( page 48, original Arabic
language).
3.2 Surroundings of the class : (1)The object of study : Ten undergraduate students, six of them are fourth grade, two ofthem are third grade, two of them are second grade undergraduate
students. Among them four students are related mathematics andsix students are related human science.
(2)Preparation
Computer (Windows), Cabri GeometryⅡ, Microsoft Power Point,Video

Projector,

beforelecturequestionnaire,

after

questionnaire, Text book, Work sheet, Home work sheet.

lecturehttp://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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3.3 The development of the lesson : For studying Omar khayyam as an ancient great mathematician the text-book based on
Historical type was developed. He was the first who did solve algebraic cubic equations by using
geometrical representation. To solve cubic equation he took help from Greek mathematics. He
applied Euclid famous “Two mean proportion” formula and Apollonius conic section for solving
cubic equations. He solved cubic trinomial and tetranomial equations. He always considered asolid (rectangular parallelogram).
First lesson : - Problem1 :- A cube and a sides are equal a number i.e. x3+bx=a.
Before solving the problem I confirmed students knowledge with using historical text suchas Euclidean Elements, definition of Apollonius parabola.
ⅰ.Two mean proportion : To describe a square that shall be equal to a given rectilinear figure, whatever known as
two mean proportion.
Description :- Let A be a rectilinear figure : it is required to describe a square that shallbe equal to A.
How :- The rectangular parallelogram BCDE equal
to the rectilinear figure A. If BE = ED, then it is asquare. Produce BE to F such that EF = ED. On BF
describe a semi-circle, center at G, produce DE to
H. Join GH. Then, because the straight line BF is
divided into two equal parts at the point G, and into
two unequal parts at the point E, the rectangle BE,
EF, together with the square on GE, is equal to thesquare on GF. But DE = EF. So square on EH isequal to BCDE. If EH = x, BE = a and ED = EF = b, then x2 = a 。b.
This Euclidean theorem is known as two mean proportion.
Mathematically :AC=a
BC=b
DC=x
X2 ＝a・b出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

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ⅱ. Definition of a Parabola : In this case I used definition of parabola from
Apollonius, whatever historical.
AB = parameter
BZ = diameter
DZ = ordinate
D Z 2 = AB・BZ
ⅲ. Solid (rectangular parallelogram) : In this case I used a model of solid.
In the figure ABCG is a solid, whose side
AB i.e. ABKH is a square. AB is the given
number of roots b, so AB2 = b.
Volume of the solid is a
i . e . A B 2.BC=aHeight BC=a/b
BC is perpendicular to AB.

http://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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In English.

OMAR-Al-KHAYYAM

could

solve

three

degree equation from third and fourth term byusing geometrical shape, and this is thehighest peak (top) reached by ARAB’S inalgebra and it is higher than what somemathematician havereachednow-a-days,

because still now we can not solve equations
from fifth degree or higher by a common way.
Gregory had praised Omar Khayyam, andsaid, ”solving

cubic equations byusinggeometrical representation, was one of thegreatest achievements of ARAB scientists”

出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

7These above are from original book whatever I used in the lectures.
Now we posed first problem of Omar Khayyam, below.

A cube and a side equal to a number.
i . e . X 3 +bX = a.

http://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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In the first lesson I used a model of a solid anddescribe smoothly and asked the following
questions justifying their understanding.
T : AB is the given number of roots b so AB2 ?
S1 : b.
T : Volume of the solid ABCG is a, AB is theStudent thinking.

side is a square is b, BC is the height of the solidand perpendicular to AB so mathematicallyBC=?
S2 : BC = a/b.
T : DBH, a parabola whose
S3 : Vertex at A.
S4 : AB= Parameter.
S5 : BZ＝ Diameter.
S6 : DZ= ordinate.
S7

Understanding and solving:

Sofrom thedefinition ofparabola

DZ2=AB.BZ.
Draw a semicircle CDB so that the semi circle
and the parabola necessarily meet at the point D.
Now draw perpendicular DZ to ZB and DE to BC.
Then BE=DZ and ED=BZ. Now BEDZ is arectangle. I also told them to follow the
worksheet and try to fill up that. They did that.

Understanding and solving
Discussion about the class : This cubic equation is possible only for the positive unknown numbers. If the semicircle CDB
and the parabola DBH does not meet then the solution is impossible. Omar’s geometrical methods
were more comprehensive than the algebraic methods developed by the Italian algebraists, notably
Girolano Cardano and Niccolo Tartaglia
For the next class I divide the number of students in two groups. I distribute two problems as their
homework.

出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

9These two problem are as follows:-

http://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

10From the beginning of the second lecture I divide the all number of students into two groups fortwo problems and made leader who present the problem in the blackboard. They discussed
themselves.
For their understanding I made clear about the definition of hyperbola.

I helped her to understand the problem.

A cube and a number are equal to sides i.e.
X3 + a = bx.

I helped to understand individually.

A cube is equal to sides plus a number
A group thinking how to solve.

One group is describing.

I.e., X3 = bX +a .

Other group leader is describing.

For the third lecture I posed some known numbers instead of the volume of the solid a, and the出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

11side b is 6 and 2 respectively.
So from last two lecture height = volume / sides.
BC = 6 /2.
= 3.
And root 2 is 1.41.
Using previous formula students drew x3 + 2x = 6 (original equation is x3 + bx = a).
The following are the Cabri Geometry whatever students draw. We also helped them.

I helped to draw figure in computer.

S t u d e n t i s v e r y a c t i ve in computer.
At the end of the third lecture I compared Omar and Cardano’s method for cubic equation.
After then we conclude that Omar is the father of cubic equation.
4.

RESULT OF THE QUESTIONNAIRE : Before the lecture I posed MCQ (multiple choice question) type question. Which are below:Q : - Do you have any idea about Arab and their culture.
A : - Most of them write no, some of them write Arabia language.
Q : - Do you have any idea about geometric representation.

http://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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A : - No.
Q : - In general whatever you use to draw figure.
A : - Pencil, Compass, Ruler.
Since my students are University undergraduate. So after my three lectures I asked thefollowing question and justifying their understanding.
For modern technology I posed the following question to students.
Q : - If you are teacher how can you teach Omar Khayyam’s mathematics.
A : - If I’ll be a teacher I use Cabri Geometry, because there is no difficulty in handling.
A : - It is possible to draw Parabola by the software of CABRI GEOMETRY. So, I’ll teachstudents by Personal Computer.
A : - It is easy to draw figure by using Cabri Geometry. Using this Software we can also justify
and confirm about our figure.
For the culture I posed the following question.
Q : - What are the time and cultural difference between Omar mathematics and themathematics you have learned.
A : - Comparison now and long time ago, We learn mathematics whatever mathematician
made、but Omar made mathematics himself.
A : - We and mathematician are different standpoint. We learn only pattern mathematics. As aresult different time culture idea is different.
A : - By turning my attention to regional mathematics differ from place to place. I have learned
different through the three lectures. I feel diversity (variety).
About the geometrical representation I posed the following question.
Q : - Through three lecture what you have learned new.
A : - Before I don’t know the idea how to solve the equation by figure. Aim is same but theprocess is different.
A : - Not only calculation, we can solve equation by figure too.
A : - I came to know that there are many way of thinking about solving mathematics, One ofthem is equation with figure.
Generally equation expressing with figure is called geometric representation. In the previous
lectures we find that algebraic equation could represent with geometrical figure. In this world
everything has own culture. As for example if we catch a bird and hanged it, after sometimes it
must die. That means it is apt with its own culture. Similarly Arab has own culture.
Technology means the development of science. We are studying mathematics so for thedevelopment of mathematics Cabri Geometry is the software whatever used to draw geometricfigure.

出典：中学校・高等学校数学科教育課程開発に関する研究(9)発行：筑波大学数学教育学研究室、pp.122-135、2002年3月

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Discussing the above questionnaire, conclusion is that students gathered knowledge on
Modern Technology, Arab culture and Geometrical representation whatever my aim.
.
5.

CONCLUSION : The aim of this paper is to illustrate the significance of using modern technology, Arab culture,
and Geometrical representation of algebraic equation, for this purposes it is used regional mapof that country. Through three lecture there are using technology in the case of teachingmathematics history; specifically, mathematics as a human enterprise through giving students
the chance to experience the hermeneutics in mathematics history. Overall we say that,teaching should n o t b e separated from teaching with tradition. Both historical and newtechnologies are useful tools for knowing the imbedded socio-historical-cultural perspective inmathematics. Teaching history with both traditional and new technologies is effective for thechange of belief in mathematics. We should expect near future mathematics teacher both
historical and modern technology ability.
ACKNOWLEDGEMENTS :The author gratefully thanks to Professor Dr. Mitsuhiro ITOH to revised the solution.
The author would like to thanks with gratefully to Professor Isoda Masami to influence Omar
Khayyam mathematics and also thanks to Sakamoto sensei with all of classmates for theirvaluable suggestions and also thanks all of students for their cooperation。The author also
thanksto

Egyptian Mohammad Ahmed san and Iranian Vahid Nejati san to helped

translate the original Arabic and Persian language.
Remarks : 1. This project is under Professor MASAMI ISODA , University of Tsukuba, Japan.
2. This project is attached with a web page as follows http://www.mathedu-jp.org .
References : (1) The Algebra of Omar Khayyam by Daoud S. Kasir(pages 75 to 79).
(2) Palgrave’s The Golden Treasury to which is appended THE RUBAIYAT OF OMARKHAYYAM with an introduction by LOUIS UNTERMEYER, U.S.A. 1944 pages (393-397,
402).
(3) Introduction to the history of science; vol. 1 From Homer to Omar Khayyam, by George
Sarton, Washington 1927, and reprint 1950 pages (738-741, 759-761).
(4) Miscellanies of Edward Fitzgerald , London: G. Routledge, 1904 pages ( 1-5).
(5) Rubaiyat of Omar Khayyam The Astronomer-poet of Persia, translated by Edwardhttp://math-info.criced.tsukuba.ac.jp/Forall/project/history/2001/Arab-fl/Arabr-index.htm

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Fitzgerald,.London : Noel Douglas, 1927 pages (83-10, 18).
(6) The Rubaiyat of Omar Khayyam translated into English verse by Edward Fitzgeraldpublished in New York by Dodge Publishing Company, page (17-2).
(7) Rubaiyat of Omar Khayyam and The Salaman and Absal of Jami, renderd into English
verse by Edward Fitzgerald , New York, Thomas Y. Crowell & Company, pages (37-47).
(8) Rubaiyat of Omar Khayyam rendered into verse by Edward Fitzgerald , Copyright Thomas
B. Mosher, 1900, pages (1-8).
(9) Rubaiyat of Omar Khayyam translated by Edward Fitzgerald ; edited,with introductory &notes, by Reynold Alleyne Nicholson. Litt.d, Lecturer in Persian in the University ofCambridge, London, 1909 pages (5-7, 13-24,72-73).
(10) Rubaiyat of Omar Khayyam 、Eqbal Publisher, Tehran、Iran (original Persian Language)
pages (3-6, poem 17).
(11) Arab Genius in science and Philosophy by Dr. Phill. Omar A. Farrukh, (original Arabic
language) publishers AL-ILMIEH LIBRARY, Beirut, Second Edition, Beirut 1952, page 48.
(12) A history of mathematics, From al-Khawarizmi to Emmy Noether by B. L. Van derWaerden pages (24-31).
(13) The Crest of the Peacock, Non European roots of Mathematics, by George Gheverghese
Joseph published by the Penguin group, 1994 (pages 301-310, 319-324, 330-331).
(14) Euclid’s Elements edited by ISSAC TODHUNTER introduction by Sir T. L. Heath, K. C B.,
F.R.S., London 1933 (page 69-70).
(15) Euclid Archimedes Apollonius of perga, by Nichomachus of Gerasa, William Benton,
publisher, 1952, GENEVA, pages ( 616-618, 627-628, 640, 669-670, 661-662 ).
(16) A source book in mathematics by David Eugene Smith, volume one, New York, 1929,
pages (203-206).
(17) Inquiring mathematics with history and software, the ICMI study, by Isoda Masami、
2001 pages352.
(18) Integrating history of mathematics in the classroom: an analytic survey, the ICMI study
by Isoda Masami, 2001 pages 205.
(19) www.math.tamu.edu .
(20) www.ummah.net/history/scholars .
(21) www-groups.dcs.st-and.ac.uk .
(22) ハイベルグ編著.中村幸四郎他訳(1998).ユークリッド原論.page(59‐60).
(23) Mathematics History Is Another Door for Using Technology in Education, The ICMI study,
2000, by ISODA MASAMI page 133.