Reminiscences of International Congresses of Mathematicians (ICM): Nice-France

Professor Dr. Hashtroudi was requested and he submitted some of his own reminiscences

of international congresses of mathematicians (ICM) that would be in scope of magazine

readers, in order to publish in the magazine.

The organization of the international congresses of mathematicians that holds in various

countries once every four-year and dates back to two centuries ago. In these congresses,

representatives of universities from various countries and mathematicians who are personally

inclined, take part. Typically few number of eminent mathematicians take part according to

the invitation of the host country. During the World Wars, these congresses did not hold, and

before the World War II, the last congress which was held, was ICM-1935 in Moscow. This

year it will be also held in Moscow.

In each congress, a one-hour lecture is considered for prominent mathematicians that give a

lecture about the general progress of mathematics. The other participants could give a quarter-

lecture and present their new theories, however, possibly discussion about some of these

theories takes a number of congress sessions.

I have taken part in all of the congresses after the war and I have been invited in most of them,

just in Stockholm congress, though I was invited, I could not take part in it, since I was in

Moscow on that time.

To speak about the official talks of congresses is wearisome for magazine readers. Some of

memoirs are about circumstance that has happened in the margin of each congress:

In Nice congress, the international congress of Latin mathematicians, Marchoux was the

honorary president, in closing ceremony, a number of participants' relatives have gone to a

night dancing party. There, one of mathematicians' wife and an artist disputed and finally

brother of mentioned lady and the manager of club had quarreled. A journalist of a newspaper

who had sought for such opportunity, had taken several photos of this incident and it was

anticipated that the next day the happening has been reflected with thousand embellishment

and foliage in a newspaper to become a disgrace for congress. Marchaux asked me to interfere

and somehow quiet it; in Nice according to the policy of tourist attraction, they respect

foreigners and in particular journalists abstain to insert and publish something that makes

them resentment. Marchaux and I met the journalist and convinced him with so deceits and

eventually I could take his camera and blacken all photographic films. Journalist who did

not want to be taken a task by manager of newspaper, complained me but just requested the

compensation of a damaged film roll and the incident was finished here.

In this congress, when Arnaud Denjoy who was rather old in this congress and was one of

great professors in Sorbonne University and was also my professor, wanted to lecture, we

all applauded him but just as he wanted to continue his lecture, a number of youth began

again to applaud and wanted to impede his speech. By the way, some of friends and I tried

to conciliate the meeting and he with a high indignant voice that trembled, ended his lecture.

Indeed, Arnodangoux had worked on some subjects in concerning to modern mathematics

in old age and this had not been at all acceptable for youth. They were illiberal youth who

deemed that the way is closed for old, whereas Élie Cartan argued also the theory of isotropic

spaces in the age of eighty years. By the way, in that meeting, I and some others wanted

to soothe him and declared him that they are young, we were also young and sometimes

naughty. He said with the same indignant tone:” you were at all not so , since youth passes,

but maliciousness remains”.

1Arnaud Denjoy(1884-1974)


Translated by Fariba Elliee

A Memory of Moscow Congress, 1935

Reminiscences of International Congresses of Mathematicians



By Dr. Mohsen Hashtroudi

This congress was the last international congress of mathematicians before the World War
II. One night after finishing the session of differential geometry and topology, the late
Cartan (Élie Cartan) and Schouten (who is now the president of Amsterdam mathematics
center) and Hermann Weyl and a group of researchers in tensor calculus and connection
theory negotiated and discussed. Alleys in Moscow in vicinity of Kremlin almost all lead to
Kremlin, indeed they are centralized. The late Weyl propounded this question that: “is there
a geometrical measure that its all geodesics (the shortest distance, here it should be pointed
out that in all measure spaces, geodesics and straight lines don not coincide with each other,
namely, between two points, there exist a straight line and a shortest curve that are distinct)
are centralized?” That night, after separation of this group Schouten overnight found this
connection that at present in his name in Russia it is known as Schouten connection. Strangely
in western countries this connection is called sometimes as Moscow connection.

The author of this lines determined intimate properties of this connection about 15 years ago
and simultaneously André Lichnerowicz, the professor in Collége de France, proved also
that among point spaces (spaces which are introduced just with point not with a point and a
line nor a point and a plane), the only space that are connected to nonholonomic analytical
mechanics, is the same semi-symmetric Schouten connection. This connection is called semi-
symmetric since torsion of space is determined by a vector just by using main tensor (this
space is obviously is nor a usual space, namely, it has curvature and torsion of space is related
to transportation of the origin of coordinates).

Such as this issue has happened in many congresses that a problem has been posed in a
session of a field or even out of that and an overnight it has been solved by one of mighty
mathematician.


Translated by Fariba Elliee

The Foundation of Modern Mathematics

Part 3

Instances of articles, mathematical problems and the other works of Dr. Hashtroudi



1

In each era, mathematics is the mirror of its civilization. Technical, industrial, scientific and

art progress in any century can be realized in worldview of its mathematics.

The history of evolutions in cultures and modernity of thoughts is simultaneous with the

history of progress in mathematics.

In this century that empirical development in natural sciences and biology and evolution in

observational sciences such as astronomy and cosmology with wave knowledge has made far

scientifical research possible, mathematics as an abstract inductiveand discursive knowledge

has led to where abstract mathematics has been introduced without resorting to the study

of classifications in human sciences and on the other hand evolution of mathematical logic

and modern science semantics has made abstract thought search and research regardless of

statement and words quality and has founded the statement of other abstract sciences.

In this brief article, short mention to the quality of mathematical thought evolution will be

accomplished in particular in the 20th century so that readers would get informed more or less

perfectly in this issue.

The basis foundation of abstract mathematics begins with the set theory. This theory that

had been considered more or less in the 18th century, occupied thoughts of mathematicians

at the beginning of the 20th century and made clear coordination and integration of logic and

mathematics. The basic of this theory, later with other generalities that are always required in

mathematics, founded eventually the discursive school in mathematics.

Antinomies that arose in the set theory, was considered by logicians and the coordination

between logic and mathematics (the set theory and categories) led to the building of

semantics.exclude

In order to clarify the issue according to the historical development, we talk about appearance

of the set theory and the existence of antinomies so that we lead eventually to the situation

and establishment of abstract modern mathematics.

***

Category and union of elements in a category is always arbitrarily possible, but the set that is

obtained in this way, is an indeterminate and more or less an ambiguous set. With a particular

definition that includes all members of the set and excludes non members, it can be defined

and determined (Note that here an inclusive and exclusive logical definition is not necessary,

any definition that is containing the result, suffices and such definitions are equivalent).

Clearly, sets can be defined that the own set is apparently also a member of itself. For

instance, if we form the set of all Persian names and we name this set a Persian name, it will

seem so that the set itself is a member of the set. More logical antinomies in the set theory

begins with this point. Resolving such problems that are now so normal, made difficulties at

the beginning for mathematicians that domination them seemed abstained. Later, it became

clear that such antinomies are not confined to the area of mathematics and it occurs in the area

of logic too. For instance, some cases of these antinomies in the area of logic are mentioned

here:

First. The singular sentence antinomy – This antinomy has been known from ancient

and it is quoted from Aristotle. Ghiyath al-Din Jamshid Kashani says that Kashanians are

mendacious. Obviously, this sentence makes a cycle because this quotation has been said by

a Kashanian thus it is included by the sentence itself and that sentence is not true, namely

Kashanians are truthful. And since the narrator is from Kashan (Kashanian), the sentence will

be true and the cycle becomes evident.

Should note that this sentence actually is not a singular sentence and it has been formed

by two sentences: a – Ghiyath al-Din Jamshid Kashani is from Kashan; b – He claims that

Kashanians are mendacious. Hence, the antinomy appears and makes a cycle, since lie

does not form a group, because lie of lie is equivalent to true but not lie, if Ghiyath al-Din

Jamshid Kashani said that Kashanians are truthful, no antinomy arises, because true of true is

equivalent to true and it forms a group.

Second. The Antinomy of Sentences based on Definition – For every number, it can

be mentioned either mathematical definition or another non mathematical definition that

2

determines that number. For instance, the number (a) is defined such that „the smallest

number that its definition requires less than ten words.“. Obviously, the numbers that are

defined by less than ten words, form a set (finite or infinite) that among them, by comparing,

we determine their smallest and the number (a) is specified. For instance, the number π

is defined such that „the ratio of a circle's circumference to its diameter“. This statement

(or sentence) has 9 words, namely, it is a sentence that has less than ten words, thus the

number „Pi“ is such as these numbers. As well, the number 9 (the number of planets in the

solar system), since the sentence „the number of planets in the solar system“ has less than ten

words, it is such as thsese sentences. If it is contented with these two numbers, the number (a)

will be π that is the smallest between two numbers 9 and π. However, if other definitions are

also mentioned for these such numbers, the number (a) can be always determined. Now, we

see that the number (a) itself has been defined by the statement „the smallest number that its

definition requires less than ten words.“ that has eleven words and here antinomy happens,

because the number (a) should be defined by a statement (or a sentence) that has less than

ten words, namely, according to the result of the above definition, the number (a) exists and

according to the sentence form, this number is not contained in the set of numbers which are

determined by this defintion. With a little attention, it can be found that the set of numbers

which definition needs less than ten words, is a sequence of numbers (finite or infinite,

in terms of numbers) and the defintion of the number (a) is indeed such that „the smallest

number of this set“ and in this case, there would be no antinomy, because the last sentence has

less than ten words.

In the set theory, these antinomies had been already seen and their dependence on the logical

area became later clear. The base and the fundamental of the mathematical logic is on the

basis of sentences and theorems and category theory.

If among members of a set (finite or infinite), there are relations such that the result of using

these relations for two members of that set gives another member of the set, that set is called

structu For instance, if we consider the set of integers and the addition operation, since the

sum of two integers is also an integer number, the set of integers with addition opertaion is

called a structure. Without mention to the generalities of structure in mathematics, we content

ourselves with this instance and equivalence of this structure to the structure of integers with

multiplication operation, because the set of integers with multiplication operation forms also

a structure. (In addition operation of the first structure, the number zero is as identity element,

because adding it to every number keeps preserved that number, but in multiplication of the

second structure, the identity element is the number one, because multiplying it by every

number keeps preserved that number). It is possible that two relations are defined between

elements of a set (like the set of integers with multiplication and addition operations), in

this case this such structure is called ring. The equivalence between rings and structures

with two multiplication and addition operations is always obvious. The equivalent structure

to the structure with addition is called module and the equivalent structure to the structure

with multiplication is called group. Thus, ring is indeed a structure both a module and a

group. Field of numbers is a set that the result of the four basic arithmetic operations over

its elements keeps preserved that set (except divisionon zero of module which becomes

exception). For instance, the set of rationals forms a field of numbers, because the result of

the four basic operations on rationals is a rational number. The set of irrationals is also a field

of numbers. If an irrational which integer and decimal parts are integers, is called an integer

irrational number, it is observed that the set of integer irrationals is not a field of numbers, but

rather is just a ring. So, the main classification mathematical structures are:

1- Modules that is also called an additive group and as well group that means multiplicative

group which is called in short group. If in such structure: a . b = b . a, the group is called

abelian group. The equivalence between module and group is obvious, namely, these two

structures are indeed equivalent.

2- Ring is a structure that is either module and also group, namely, the three operations:

addition, subtraction and multiplication are possible in it.

3- Field of numbers that the four basic arithmetic operations are possible in it.

Structures are divided into two fundamental parts: algebraic structures (what are so far

mentioned, are algebraic structures) and logical structures. For every logical structure, there

is a particular algebraic computations that is called correspondence structure. For instance,

Boolean algebra is the fundamental of computations of logical Aristotelian structure (i.e.

binary logic). The importance of this structure is so much in modern mathematics and the

fundamental of electronical machines or automatic systems is based on it.

***

In the history of mathematics, a is well-known, geometry has a distinguished situation. The

evolution of scientific ideas in this area is the oldest and the most acquainted history of

mathematics. Axiomatic geometry that argues about shapes with particular methods, is the

oldest geometrical system that has been ever known (Euclid's elements known as Euclid's

axioms). By applying the principles of algebraic computations in geometry, Descartes has

been known as the founder of analytic geometry (Khayyam's studies and inquiries, without

applying system of algebraic equations is indeed the same analytic geometrical method that

Descartes has laters used). And this method has been finally led to affine and projective

geomtries. Laters, with Poncelet's studies and inquiries, these geometries became

independently and anew the method of analytic geometry was founded much more firmly and

logically than before as axoimatic method. Extensive efforts that have been done in this

method, generalized gradually to other mathematical desiplines and made the basic of other

branches in mathematics to the axiomatic method. And this issue is also researched and

inquired now by mathematicians. By founding the axiomatic method in mathematics and

renewed independence of geometry from algebra, algebraists proceeded anew to establish a

branch called algebraic geometry that is now researched and inquired. Applying the principles

of computations in structures and in particular logical structures in definitions and theorems

in geometry helps suficiently to make clear the basic of geometry. For instance, in 3-

dimensional Euclidean geometry, we denote points (the set of points) with capital Latin letters

such that A, B, C, D, … are the different members of the point set. As well, we denote lines

with small Latin letters like a, b, c and surfaces with Greek letters like α, β, γ, … Now, we

define the classification ring of these factors: the addition operation between two factors of

the universal set S (A, B, C, …, a, b, c, ..., α, β, γ, …) is defined as follows: between two

factors of the members, we define a relation that we show with the sign +, such that the sum

of two factors of members is another factors of members with the least dimension. The set S

contains the subset: S (A, B, C, …), S (a, b, c, …), S (α, β, γ, …) such that every subset has a

dimension which is shown by the index of the subset, namely, the dimension of A, B, …. is

zero, the dimension of a, b, … one and of α, β, γ, … two. The sum of two factors of the

members in S is the other factor of the universal set S such that both given factors lie on it and

this factor has a minimal dimension. For instance, if [P] and [Q] are two indeterminant factors

of S (we denote unknown factors with the capital Latin letters), then [P] + [Q] = [R] is

defined so that R is chosen from the subsets of points, lines or surfaces so that P and Q lie on

it and the index of R (the index of correspondence subset that [R] belongs to it) has a minimal

amount. For example, if we sum a factor like a (line) with a factor like B (point), then we

have: a + B = γ in which the index of a, B and γ equal respectively to one, zero and two,

namely, the sum of a line and a point is a surface which passes through them. Obviously, if

the point B lies on a line a, the sum of them is the same line a, because in this case, the point

lies on the line and all of points of the line a lie also on the line a, so the factor with minimal

dimension which contains both of them, is the same line a. In this case, a + A = a (on the

contrary to usual algebraic addition, the sum of two factors is one of those factors without one

of factors is zero). In order to complete these operations, we need also another definition. We

denote the universal set S (in terms of content) with U (i.e. the whole of geometrical space

which contains all of points, lines and surfaces) and the null set which has no member, with N

(which is like zero in arithmetic); Adding these two concepts generalizes additive and

multiplicative operations (which are later defined). In this case, we observe that the sum of

two lines, a + b, if they are not parallel or intersecting, equals to U, namely, a + b = U. And if

these two lines are parallel or intersecting (i.e. they are in the same surface), the sum of them

is the surface which contains them, namely, a + b = α.

The multiplication of two factors in the universal set S is defined as: the multiplication

of two unknown factors [P] and [Q] as the members of the set S is another factor of S

which is common in both factors [P] and [Q] and has maximum dimension. We denote the

multiplicative operation with the sign × and define as [P] × [Q] = [R]. The index of R is the

greatest index which has the mentioned property in the definition of multiplication in the

subsets. For instance, the multiplication of two intersecting lines is their common point: a × b

= C, and the multiplication of two skew lines is the null set N: a × b = N.

Now, we consider the solution of equation [P] × [X] = N in which P is a point and X is a

factor of shapes (point, line or surface) and N is the null factor or set. Obviously, one of

solution is the same null set N, because the multiplication of each factor in the null factor

will be the null set, the other solution is the same point P, because a geometrical shape

which contains the point P and has the minimum dimension, is the same point P, namely,

the solutions of above equation are: X = [P] and X = N. We consider if the factors of the left

side in the above equation belong to P, then we will see that the point P has no other member

except the point P itself and the null factor, namely in other words, the part and whole of a

point are equal to each other and it is the same Euclidean definition of a point.

The above instance is an example from algebraic geometry which is a subject of mathematical

logic and is necessary for comparison, development and extension of geometrical definitions.

***

The fundamental structures of modern mathematics are logical, algebraic and topological

structures that are requirements in the basic of all mathematical issues. Mathematical analysis

is a particular discipline that applies different methods in real and complex functions (it

means functions of real and complex variable). In the late 19th and at the beginning of the 20th

century, some functions were considered that are not differentiable in any points of interval

continuity and furthermore, there are continous functions that their graph covers completely

inside a square. The concepts of derivation and dimension were needed new inquiries and in

topology, the theory of dimension and measure were defined anewly.

The existence of functions with real variables that all rational points are their disconnected

points and all of these points are also limit (accumulation) points, made to review in basic

of mathematical analysis. In other hand, analysing of harmonic functions and integrals, in

one side, and the convergence theory of Fourier series, in other side, opened a new issue in

mathematics that eventually led to axiomatize potential functions.

***

The applied mathematics, in particular mathematical and theoretical physics, made to appear

especial functions that admit just particular values and thus from theory of Dirac function

to distribution theory, there was just one step to go. To apply the calculation axioms of

probability in applied and theoretical problems led to appear information theory and after that

difficulty or complexity theory that is even now researched and inquired. The appearance of

particular functions in physic such as Dirac function that was mentioned, led mathematicians

in other hand to the problem of eigenvalues or proper values that are necessarily applied in

statistical and Quantum physics and eventually ergodic theory to solve differential equations

brought into existence a general and exact discipline in theoretical physic that classified most

problems under a method.

***

In applied problems and difficulties that arise in emperical sciences, empirical formula and

rules are mostly required and until now mentioned empirical issues were frequently inquired

by convergent extension with appropriate series and validity of formula. To apply method and

principles of dificulties and complexity theory, appropriate functions are before determined

for extension of empirical formulas by analytical criteria and thus applied or empirical

formula is obtained with a high validity.

Applied mathematics and calculus, for instance in numerical solutions of algebraic systems

or differential equations, have developed greatly by electronic computing machines or simple

electrical machines and have evolved much more with respect to saving time. In applied

technical problems and in particular in cosmic problems, determination of ranges of orbital

position and stability and movings have developed considerably by using exact computational

methods known as Lyapunov method. The asymptotic (Poincaré) integration method has

attained maturity and encommpassed much more problems.

Geoemtric interpretation of differential equations (or differential systems) predetermines

solutions of equations in long distances by using global geometry and it made possible to

change movings in distant times and far distances by using topology.

Mathematical analysis has attained considerable development under coordination of different

methods that were mentioned and many unsolved problems have now been in the area of

elementary mathematical knowledge.

The concept of finite and infinite groups and its applications in analysis and geometry has

made coordinated interpretation for various problems in one side and in other side, by transfer

data from a close area to wider, it makes solvable many problems that were not verifiable

in analysis area, in geometry. Hence, the various solutions of Schrödinger equation are

determined by using rotation groups of 3- and 4-dimensional spaces.

***

To apply the concept of connection in infinitesimal geometry and generalize the concept to

existence of general groups and correspond the space of parameters in groups to the space

of geometry of group has solved many problems of differential geometry and mechanics

with a new interpretation so that it was not possible before these concepts. Some particular

differential equations are solved without using quadratic forms in their general solutions.

Combination of eigenvalues with the concepts of finite groups in Quantum and statistical

physics and the problems relating to electrons and the other particles of material to face

with differential equations bases the theory of spinors in Euclidean space that is applied

considerably in relativity theory to solve problems.

***

3

4

The above introductions makes possible to found operator theory in physics and mechanics

and the combination of this concept with the concept of mathematical transformations

introduces the concepts such as homomorphism and isomorphism for general mathematical

and physical problems that can be also applied with a little generalization in problems of

human sciences (such as problems in transportations, sociology, psychology). Gradually,

biology introduces and sets forth its specific problems in mathematics and the abstract

mathematical organizations are applied in this area as well.

***

The history of science has made clear since long that attempt and endeavour for solving some

problems sets forth some other problems that make to introduce some other new sciences,

fields and branches. The 20th century is not also excluded from this general rule. In solving

problems and difficulties of human sciences, though success and evolutions have been

obtained, the other problems have been introduced that no step has been yet taken to solve

them or if a few steps have been gone ahead, it is not yet clear how problem can be solved.

For instance in physics, will gravity like electromagnetics become particle and will quantum

of gravity be discoverd? It is a mystery that

has not been yet expressed any opinion. In spite of all, the 20th century is one of the centuries

that human activities have been efficient and undoubtedly, the continuation of researches that

have been begun in this century, will make generally to change technologies.

As it has been mentioned in introduction that mathematics in an era is the mirror of its

civilization, mathematics in the 20th century is the mirror of development and modernization

of thoughts and ideas. The astonishing result of this development and modernization that are

just now more or less conceivable, is openning gates of unknown skies, whether the sky of

this material universe or whether sky of thought universe.

1The first part of this article, by Dr. Mohsen Hashtroudi, has been firstly published in Yekan magazine, No. 1
(January 19??) and then all of article in Yekan magazine (March 19??).

2Ghiyath al-Din Jamshid Kashani (1380-1429) Persian astronomer and mathematician

3Aleksandr Mikhailovich Lyapunov (1857-1918): Russian mathematician, physicist and mechanician.

4Erwin Schrödinger (1887- 1961) Austrian physicist



Translated by Fariba Elliee

Hashtroudi in words of excellence teacher

Hossein Ghayour

Mr. Hossein Ghayour, associate professor in Teachers Training University, with due attention

to his proficiency and conversancy in geometry, in particular in solving its problems, was

Dr. Mohsen Hashtroudi's full confirmed and cared. The late professor, whether in speech or

lectures and whether in his writings about some of problems, mentioned Hossein Ghayour

as an initiator of the most beautiful solutions of these problems. Hence, Mr. Ghayour was

requested to send memoirs which he had from Prof. Hashtroudi, to publish in this collection.

He wrote his first own memoir.

The first visit with Professor Hashtroudi

In 19?? when I studied in Daneshsara Aali, the National Teacher Training College, in the first

visit with Professor Hashtroudi, the following problem which I myself had propounded and

due to unfamiliarity with advanced mathematics I attempted vainly to solve it, I put up it for

discussion with him.

The problem was this:

Draw a line which creates three chords with equal length in three circles.

The professor, without writing anything on paper, after some minutes walking and thinking,

siad: “the solution of this problem is related to drawing of common tangent to two parabolas

and so it is not solvable by geometrical drawing.”

At the same night of that day, after reading, studying and caring in the problem, I realized

correctness of his words and was surprised his quickness of apprehension and mental acuity.

Laters when I was honoured to become one of his students, I found out frequently that this

answer has been not at all accidental and or precedent.

Most his students who are now professors and teachers, have many memoirs such as this issue

from that great noble man.

For that the statement of above memoir is not attributed to exaggerartion or student's extreme

affection to the late teacher, young students and those who deal with mathematics, think about

this problem and prove how its solution leads to drawing a common tangent to two parabolas.



Translated by Fariba Elliee

Hashtroudi in words of an immemorial friend

In rememberance of a dear friend and an illustrious scientist

By Ahmad Birashk

1

It was 1930. October forreached and a new semester began. A number of youths who had

finished and graduated from high school and were paasionate to become teacher, had joined to

Dar al-Moalemin Aali, the National Teachers' Training College.

Dar al-Moalemin Aali where became later Daneshsara Aali and is now University of

Teachers' Training, began its second academic year in that year. This academic system was a

developed school where had been working to train school teachers since 1918 and Abolhasan

Foroughi undertook to manage it for some years. In 1930, it was set up in one of branching

alley from Amiriyeh Street, in the parish called Sabzikar Takht Zomorod, in a building where

became later the seat of Maarefat high school. There were no trace of gaudiness, glory, luxe,

many employees and servants in it. It had even no head and director. Its headmaster was

undertaken by Hasan Farzani who was from the first alumni of Law College in 1923 and its

registrar was Mohammad Taghi Kerman. A number Iranian teachers such as Gholamhossein

Rahnama, Abbas Eghbal, Dr. Rezazadeh Shafagh, Badia Zaman Forouzanfar and three

French teachers, Dr. Loui Lang, Dr. André and Dr. Arma, taught there.

With all simplicity and perhaps for its simplicity, it was wish Kaaba for those who intended to

become teacher.

***

In that year, 20 people (and apparently exactly 23 people) among who has graduated

with high school diploma in mathematics, had enrolled, among them whom I remember:

Abdolhossein Aghakhani, Ahmad Ahsani, Abolghasem Ahmadvazir, Hosseinali

Akbarnia, Mostafa Bahrami, Mohammad Rokni Qajar, Mohammad Zayandehroud,

Mohammad Taghi Sajadian, Bagher Shahnai, Hossein Gharib, Ali Akbar Gorkani,

Hossein Majzoub, Ali Asghar Mousavi Gharavi, Ahmad Mehran, Taghi Hourfar and

myself.

Among us, Ahmad Mehran was witted, improvisative and he has already been in Europe. He

was sent with despatched students to Europe (to France). I do not know how was happened

that Mehran had returned to Tehran and had been among scholars of mathematics.

***

The semester and courses began.

In the first or second week, I do not remember exactly, a young joined our group, according to

Saadi “spokesman, wise, sweet-spokeman and articulate”.

2

After the graduation from high school, he had studied medicine for some years and then had

joined mathematics and had gone to Europe to study. I do not know why he had also returned

to Iran. He had noble and lovely face, grace words and sweet-spoken, demure and at the

same time magnanimous behavior and above all, predominance which he showed and had in

courses, made him shortly the luminous in circle of friends.

He was Mohsen Hashtroudi.

In that year, three of my friends and I studied and worked with each other and we mostly read,

studied and discussed in our houses and when we got tired, we entertained ourselves with

chess. Someday, I invited Hashtroudi to join us. He said: “how is chess useful? Man must

play with stars in the sky.” It was a sample of his mentality.

***

In 1932, he garduated from Dar al-Moalemin and in order to continue his study, he started

on a journey to Europe. Four or five years later, he returned with PhD degree from Sorbonne

University and imparted in an institute that according to English it was his “almamater” and

as we say, it was his spiritual mother.

I speak nothing about his scientifical works, since others have said.

In 1941 due to my insistence, he and a number of academic teachers, who were Daneshsara

Aali alumni (such as: Dr. Kamal Jenab, Dr. Mohammad Monajemi, Dr. Taghi

Hourfar, ...) accepted my invitation to membership of Daneshsara Aali Alumni Society. On

that time, Daneshsara Aali Alumni Society was adjoint and aid for the Culture Ministry away

from political scuffle and the society was confered in many affairs and contributed in many

projects and was respected by the Culture Ministry.

Again due to my insistence, Dr. Mohsen Hashtroudi who had become the president of the

society, became also the head of high school teaching department. About 1951, he undertook

the chancellorship of Tabriz University. For some time, he became the head of science faculty

in Tehran University. But none of these is discussed by me. I speak about a noble man, about

a person whose vacancy will not soon fill in our society that is unfortunately with this respect

poor. I speak about a person who had always remained demure and submissive despite of his

eminency in science and society (he was university teacher and indeed professor, namely, he

had one of the beloved position in developed countries). He was professor in mathematics,

but he had also profound knowledge in literature, he was philosopher namely he knew well

philosophy, he wrote well and composed also well poems. He was dexterous in other sciences

too, he was a comprehensive man and he supported his friends and indeed he was their

scientific support. If they asked him anything that he could do, he did not stint to do that.

In the last years of his life, the mourning of his older daughter's death made him recluse. A

severe heart rending grief lessened his energy and eventually the light which could remain

still luminous for years and made our needy society light, kept silent.

His rememberance would survive with his friends, his students and in hearts of those who

knew him until they live.

In my opinion, he can be a great pattern for youths who are much more fervent to home. He

did not stop even for an instant to learn. He was one of those who learned until the end of life

and taught. He was active and had dominant over knowledge of his time.

May we learned him.

Peace be to his departed spirit and May he be ever remembered.

1 Ahmad Birashk (1907-2002):Author of a comparative calendar of the Iranian, Muslim lunar, and Christian eras
for three thousand years: 1260 B.H.-2000 A.H./639 B.C.-2621 A.D.

Ahmad Birashk founded Hadaf Educational Group (Goruh Farhangi Hadaf) a pioneering private educational
complex in 1949-50 with a number of well-known high school teachers of mathematics and natural sciences,
including Ahmad Anwari, Taghi Hourfar, Ali Motemadden and Ahmad Reza Gholi-zada. The main objective
of the Group was to offer high quality education from elementary to high school, comparable to that of

American preparatory schools.

2Abu-Mohammad Moslih al-Din bin Abdollah Shirazi, Saadi Shiraz was one of major Persian poets of the
medieval period.



Translated by Fariba Elliee

Hashtroudi in words of a contemporary poet

1

Manouchehr Atashi

Quiescence of a enlightened man in dark years and inspiring hope of fecund years

Regret of Dr. Mohsen Hashtroudi's death

We are infants whereas he was magnanimous. We are ignorant whereas he was wise.

We knew a little whereas he knew so much more and when we knew, he was referee,

an experienced referee with profound awareness based on philosophy, art and inherent

intelligence.

Before we got to know Dr. Mohesen Hashtroudi, we had become to acquainted with a similar

name “Zia Hashtroudi”, Dr. Mohsen Hashtroudi's brother, at a time when confirmation

of modern art, literature and poem, music, ... if not as mortal sin, but was ambiguous and

inferential questionable. A collection of Nima's poems and some other poets, more than

twenty years ago, was an evidence of opportune clairvoyance, and furthermore, an evidence

of awareness of the nation's cultural need that was so far from its intellectuals.

And just, it was simultaneous with such relative wonder event that we found out the existence

of much more profound and far-sighted insight and though it was too late for us, we knew a

person who was eqipped with profound and widespread knowledge from different phenomena

and wisdom and artistic events of his time, namely, of our time.

Dr. Mohsen Hashtroudi, PhD in mathematics, was one of the first Iranian students who was

simultaneous with proficiency in mathematics and modern physics, equipped with knowledge

of art, literature and modern painting, entered intellectual Iranian circles and he was known as

a hub to resolve and look after difficult problems and arduous intellectual subjects.

Dr. Mohsen Hashtroudi's attempt was more devoted to find out the relation between art (in

general) and modern sciences and to inform scholars.

2

My first meeting with the wise and intelligent of this time was at the time of beginning of

publication of “Ketab hafteh” (book of the week) in the newspaper Keyhan. In that first

meeting, Dr. Hashtroudi was introduced as the head of editorial board that could naturally

be impressive and encouraging, and lead youth and inquiring and searching thoughts in

ideal direction to effectual working (as it could, and we saw and accepted that the most

brilliant period in the relative short life of “Ketab hafteh” has been indeed the period of Dr.

Hashtroudi's editorship.)

From Prof. Hashtroudi's cultural personality traits was stupendous young attitude, dynamic

thought, uninterrupted query in many intellectual fields and “ism”s. He spoke about modern

art at a time when the most renounced it, and, he succeeded to find out the precise inscrutable

relation between art and sciences and stated it in a brief treatise at a time when others did

not believe in such coordination and unity. What Hashtroudi has done in Iran, certainly in a

smaller criterion, wa similar to what philosopher Russell has done with English literature,

and in a better interpretation, Hashtroudi had thought proximity with many contemporary

intellectuals of the world and he attempted to make to think and to move eminent required

ideas at least in a closer area.

He was a compassionate teacher, a real teacher. And all who have enjoyed to learn from him

however in a short time, know it and testify it. We said that he was extreme young attitude

and it is important that he was aware of truth of this attitude and its naturality and since he

took the most time of his life for it.

From other Hashtroudi's intellectual personality traits was Ishraghi aspects of his thoughts.

The aspects that had origin in his profound knowledge of Iranian theosophy and on the other

hand, with emphasis on it, he attempted to search new bases to justify modern industrial

knowledge and to expose divine relations between art and sciences and from this perspective

to disclose human presence (existence) in all of achievements.

Hashtroudi passed away and we missed one of our wise teacher. We wish, we would make

believes to accept this reality that: he is just asleep, but he has not been dead, because we hold

over his prominent thoughts.

1This article has been quoted from Tamasha magazine No. 278

2Nima Yooshij (1896-1960) contemporary poet, he is known as the father of modern Persian poetry

3

3Ishraghi or illuminationism is a doctorine in theology. Shahab al-Din Sohrevardi (1155-1191), Iranian

philosopher founded the school of Illumination which draws on Avicennism thought and the original idea of

Sohrevardi. In this philosophy, Sohrevardi argued that light operators at all levels and hierarchies of reality.



Translated by Fariba Elliee

Some Excerpts from the book “Knowledge and Art” Compiled by Dr. Mohsen Hashtroudi

The Preface of the Article “Review books: Phoneme, Grape Poetry, Apprehension “

By Mohsen Hashtroudi

The new art schools, for instance from impressionist painting style to surrealist or

expressionist that each has been somehow impressive in literature, have not appeared

uniformly in various arts and in particular the period of their efficacy has been more or less

short on these arts and has not left high durable and persistent impression (just on music,

architecture and statuary).

Music and architecture that are from a view situated in the both sides of series chains of arts,

more than other art fields has progressed and been developed and new styles (whenever has

occurred in them) has made more persistent impression in them. If in other fields in particular

in painting and specially in literature despite these styles have been highly diversified and

often very modern, such impression does not become evident.

This circumstance is for the reason that music and architecture have been more coincident

with academic criteria almost since early old times, whether in music proportion of harmonic

scales and whether in architecture acquaintance with geometrical rules and relation, these

arts have been granted technical aspects. Whereas in painting and literature and the other

artistic fields, constant and certain rules are not yet accessible and maybe the notion of such

legislation is also an unorganized and a crude notion. The meaning of painting in this paper

is the art of coloring and portraiture that is considered free of natural constraint, otherwise

photography (or development that has occurred in colored photography) is more capable and

satisfiable in this purpose.

In music, architecture and statuary, the art is transfigured in matter or its source is the matter.

For instance, in music instrument should be tuned (to tune tar or tonality of violin and piano,

…) and in architecture, geometrical proportions and scales in stones, brick, clay and limestone

should be considered, provided in painting it appears that it is not so and aspect, dimensions,

figures and their relationship and even their colors remain abstract, since they are painted on

paper and there is no incarnation and the instruments that indicate this abstract art (paper,

plate, metal tablet, egg, …) are arbitrary and the quality of presentation depends on joy, tact

and the quality of painter's interpretation. Hence, contemporary painter disclaims perspective

rules and geometrical proportions and cubism and surrealism styles or deformation appear

insofar as this denial even occurs in colors and painter imagines the statement of his own

thoughts and ideas free of green grass, blue sky, …

If we enumerate photography and moving picture also as arts, it seems that the above claim

is confirmed in this case, since camera perceives geometrical proportions and perspective

rules willy nilly (inevitably)much more perfect than painter's eyes and applies with much

more accurate quality. In modernization of arts because of perfection in photography ,

contemporary painting embarks perhaps on somehow revolt that it seems in the first

observation in the form of passion and gimcrack of pixilated and playful infant, provided if

the key of interpretation and statement of artist's sentiment was available, maybe it could be

interpret and state one's artwork.

In literature, the issue is spontaneously clear and it needs no statement, since if architecture

opposite to painting is a constraint art and well grounded, music opposite to literature is taken

also into account as a motile art. Poem and literature neither are transfigured in matter nor

follow like music passing instantaneous process of time, but as the author of these lines has

mentioned to it in a previously discourse, they have had partly a loan of these arts and are

located at the interface among music, architecture and statuary. Contemporary poet (as painter

rises to modernize his own art to deny the previous tradition and classical current style and

according as she or he joins music or plastic art, she or he creates a new style in her or his

own poet that at the first look it seems indeed like stupendous delirium. Whereas if in this

case, the key of realization, sentiment, interpretation and statement the artist is accessible,

interpretation and explanation of his artwork would not be so difficult.

The preface of “scientific review and artistic criticism”

The scientific thought is independent, namely it does not follow thinker's inclination, verve

and mentality and has no relevance to races and species. Reasonable and inferred thought

occurs to mind with uniform and perfect manner and in this respect, the history of science has

been general and indeed belongs to no particular nation. The history of scientists is a part of

the history of science. Every scientist belongs to a particular land, whereas science is free of

these designations.

Artistic thought or sense is not so and not only it differs in various nations and different races,

but it varies in different persons. In artistic sense that is creator of art and is the explanatory of

artist's thought, thinker contemplates with ancestors' thought, her or his ailments, rejoicings,

prosperity and failures are all inherited of instantaneous life in the pasts. Such thought cannot

be independent and free. The history of art has interlaced and incorporated into the history of

artistic. The history of Iranian literature is the history of Persian writers and poets. The history

of religious is the history of races and species and of nations and ethnics. The history of music

or statuary and architecture is the history of the livings' lives, rejoicings and ailments. Iranians

realize Mevlana 's anguish and mood and Hafez 's request and seeking and these such wails

are not acquaintance except for Iranian's ear. Tavern, cupbearer are not translated into other

languages except Persian and each of these words are results and produce of endeavor and

efforts and also results of periods of search and seeking.

Each word, each sentence, each thought for writers and poets in the same time is the

representative of living in the past centuries and epochs, such issues that are sometimes

interpreted as art, literature and in other time according to nationality, the mentioned issues

are interpreted as philosophy, is different and diverse for various nations, ethnics and tribes.

Mevlana's anguishes, Khayyam's apprehensions, Hefez's wails, Zakaria Razi's , Farabi's ,

Avicenna's and Khwaja Nasir's philosophical discussions define better the nationality of

Iran rather than their scientific research. Some of Khayyam's scientific works that are the

continuation of Greek researches, despite of their competence and worth, do not have validity

as much as one of his quatrains with deliberation balance of worth.

Hence, in the result of lifetime search and seeking, every ethnic chooses a custom and in order

to reach to purpose destination selects a way. In bright shade of transient life, in particular

some minutes that pass occult and veiled away from consideration, the result and produce of

searching, seeking and exploration of mysteries and abstruse are stated mostly by familiar

words and sometimes by newfangled words and tavern, cupbearer, fortuity, destiny and

3

7

hundred nights hangover are found. Each of these words such as a man of society includes its

particular own history and such that others are not mentioned in the history of nation except

of particular persons, a clear history for these words is also not accessible, their narrations are

occult and hidden in the darkness of the past times.

In the opinion of philosophers, scientific method for inquiry of artistic and philosophical

issues is authentic and it can be brought up these such issues like a scientific issue and can be

studied and inquired them by the same scientific inquiry.

Though this allegation seems true, it is just sophistry. Inquiry in artistic or philosophical

issues, if the purpose would be to realize the time of issue and its progress and evolutions

during the time, it would be taken into scientific account and such that the historical issues

are brought up and inquired, they could be studied. But if the purpose is just to bring up a

philosophical or artistic issue, the scientific method and even scientific statement is infirm in

these issues and justification and explaining causes in scientific method is also unorganized...

The extent of the universe – the dimension and geometry of the cosmos

The new theories and experiences in sciences and in particular in celestial physic and

cosmology have made our view and cognition to the universe totally metamorphose and have

laid on new horizons in proscenium of human's view.

Human's thought accepted difficultly the imagination of an unlimited and infinite universe

since early times and the controversy of finite dimensions or discrepancy of opinion about

inseparable part is famed in the history of philosophy by philosophers.

Now, the controversy in finite and infinite between mathematicians is evident. Thothe ugh

this discrepancy of opinion does not relate to the universe issue and its dimensions and it is

brought up in fundamental mathematics and its basic, in spite of all that it is an evocation and

inherit that has come down by inheritance since early times.

In the opinion of Aristotle such as is famed, space has no sense as in the current meaning

and situation of objects was just taken into authentic by him. Indeed, as if objects were

obliterated, their situations are also obliterated and since theoretical geometry, in his opinion

and before that, spoke about didactic objects that the properties of natural objects in them was

considered.

Progressively, since the late of Medieval later on, the theoretical sciences got succeed

eventually to develop and progress the foundation and establishment of the modern sciences

that more significant and older than the others was geometry. But this geometry was yet

discussed in Euclidean area and geometrical space was homological, isotrope and three

dimensional space. With progress of mathematics and scrutiny and inquiry that was realized

in basic geometry in particular in parallel postulates and the realization that the famed Euclid

axiom can not be proved for these lines, the denial of validity of this axiom seemed absolutely

necessary and eventually various mathematicians got succeed to found and establish new

geometries that known as non-Euclidean geometry. In these geometries, the most sentences

in Euclidean geometry do not satisfy. For instance, the sum of three angles in a triangle is not

constant and changes but never equals to 180°. Namely, if the sum of angles in a triangle is

more than 180°, this property satisfies in all of triangles in that space and if the sum is less

than 180°, it satisfies in all of other triangles. Here, it should be mentioned that if a triangle

becomes small or big, this property does not change, just the amount of mentioned summation

becomes more or less. In these non-Euclidean geometries, there exists no similar shapes and

can not for instance enlarge an image or picture, however, if difference between angles of

two triangles is closer to each other and their two sides are held proportional, the third side

remains more or less proportional and two triangles seem almost similar but the definite

and certain similarity such that exists in Euclidean geommetry, does not exists in these

geometries.

This introduction is hence considered as we will see later, in order to determine the kind of

physical space, we will require this introductions.

The development of geometry, at the beginning of the 20th century exceeded also more

than this step and with inquiry in base and fundamental in one hand and definitions and

introductions in other hand, it appears that the coincidence of straight line that conjoins

two points, with the shortest distance between those two points, is not necessary, namely,

geometries can be founded that in those, a line (curve) could be drawn between two given

points such that would be the shortest way between those points and the other way could

be the same straight line. Namely, if a mobile moves from a point to another point, its

prollongation remains stable and does not change.

The history of science begins freom the history of mathematics and the other human's

sciences have developed and progressed under the shade of progress in mathematics. The

foundations and establishment of various geometries that was briefly mentioned, becomes the

introduction of human's development and progress.

Since the mid-19th century, physicists' and astronomers' inquiries and investigations have

confronted dificulties and problems that the rules and principles of classic physics were not

able to explain, interpret, comment and state them and it made inevitably necessary to revise

the basic of these principals and that was all to make to establish and found the relativity

principle and eventually the concepts of time and physical space missed its primary meanings.

The most significant problem was to be constant the speed of light and in general the speed

of transfer and diffusion of electromagnetic waves. And briefly it can be taken affidavit for

instance by well known Michelson 's and Morely 's experiences. The base of this experience

is on the principle that is called in Newtonian classical mechanics as the principle of speeds

combinations. For instance, if a boat is abandoned in water flow in a river, it will move with

the same speed of river flow and an observer that has stood on the beach, can determine the

speed of boat and thus the speed of water.

We suppose such that the speed of river flow is 10 meter in hour, so for observer the boat

will move with the same speed 10 meter in hour. Now we suppose that oarsmen have sat

in boat and by roaing, they drive boat in stationary water with 70 meter in hour. If these

oarsmen make boat to move with the same speed in mentioned river, according to Newtonian

mechanics principle if they oar in the same direct of river, for observer on beach, the moving

of boat seems 80 meter in hour, though boat is driven by oarsmen with speed 70 meter in

hour and river flows in the same direct of boat with speed 10 meter in hour and consequently

the speed of boat will be the sum of two speeds for observer. Whereas if oarsmen unlikely

oar boat in opposite direct of water flow, for observer on beach the speed of boat will seem

60 meter in hour. The experience in the case of light and measurement its speed does not

corroborate this principle, however, the tensional speed of light transferor environment is

considerable large amount. Indeed, the electromagnetic waves do not follow Newtonian

mechanics principle.

The other problem that was not interpreted and commented with basic principles of classical

physic, was the problem of being postponed Mercury perigee and tendency of star spectrum

toward the infrared.

In revision that in basics of natural sciences was fulfilled, th famed scientists have attempted,

but eventually, the solution of this problem and resolving of difficulties be accomplished by

Einstein's genius. The statement of firm and scientifical Einstein's relativity principles is not

possible except by mathematical language and it is not contained here in the comprehension

of these words and I try to mention briefly for readers its more or less perceptible results in

particular from the aspect of the universe organization.

Before the explanation of the issue, I have to mention two points:

1. To confine the dimensions to three dimensions is not the result of nature of space or

objects, but it is just the human's experimental possible in the three dimensions. In

other words, we can say that man has created three dimensional space. For instance,

human's maintaining balance system such as tactile sense, sense of sight and in

particular half crescent channels in ears that balance maintaining is accomplished by

them (in darkness the possibility of balance maintaining is the result of these channels)

have been created as three dimensional and indeed it is our experiences that build a

three dimensional space. The imagination of spaces with more dimensions is possible

and the lack of contradiction that is observed in the multidimensional geometries,

makes indisputable this possibility.

2. In non-Euclidean geometries that have been mentioned synoptically at the first of

these words, it can be founded spaces that are closed, surrounded and however infinite

or unlimited. For instance, it can be drawn a closed curve in the plane such that its

surrounded surface is closed but the perimeter of this curve is unlimited.

From these two introductions, it can be easily realized that perhaps problems and questions

such as:

1. what is the beyond of the universe?

2. Where does space end?

3. Are the universe and dimensions finite or infinite? That seems apparently reasonable

and meaningful but indeed is meaningless and absurd.

Now, we can state synoptically the results of Einstein's relativity principle about

physical universe.

In the opinion of Newton, the space and bodies were realized and determined whereas

time was interpreted as a subjective and conceivable issue. Furthermore, the space and

time that one was realized and the other was conceivable, both of them were absolute

and independent from each other. Namely, their scale and measuring did not depend

on observer's quiescince or moving situation. Time or local interval were considered

as two uniform issues for everyone in everywhere.

Einstein's principle denies all of these elements. Firstly, each of space and time alone has no

authenticity and realization but rather the fact of nature is a mixed both of them. However,

it can be imagined or perhaps described each of these by abstraction and abruption, but the

possibility of experience, sclae and measuring the mixed part of them that in terms of Einstein

is called space-time, is not possible. Secondly, according to this introduction, abstract and

absolute measuring is abstained from these two factors, therefore time or local interval of two

supposed issues will change according to the observer's situation and moving.

Einstein's space-time seems thus similar to 4-dimensional mathematical space. In the first

model of the universe, the time dimension of the universe seems infinite and direct. Whereas

the local dimensions are closed and surrounded but infinite. Light that emits from a point

in the universe, will come back eventually again to this point and this time dimension is

representative of diameter in Einstein's universe and is farther up miliard miliard years.

The other model of the universe has been planed by Danish astronomer De Sitter in that the

universe is hyperbolic, time and local dimensions are all infinite but none of them are direct.

In these two models, there exist areas in the universe in that time is stationary and in french

term , it is called “Les Zones du temps stationnaire”. In these areas, there is no motion and no

moving. Indestructible supremacy of time has no sovereignty in these areas.

The progress of physics and in particular astrophysics showed that indeed all stars keep away

from each other and at the beginning such a thing is immutable. For instance, if some points

move on the surface a sphere, they all cannot keep away from each other and some of them

have to approach the others. This evident issue made revise in the mentioned models of the

universe and to bring up the theory of expanded universe. In the instance that was mentioned,

if some points have been drawn on the surface of a sphere (or a closed convex surface) like

a balloon and a child blows in it, since the surface of sphere expands, it seems so that these

points keep away from each other. The model of expanded universe has been founded by

Eddington and his student, the famed Belgian priest mathematician George Lemaitre . In this

model, the local dimensions of the universe is every moment changeable. In Einstein's and De

Sitter's models, the local dimensions (i.e. the abstarct three dimensional space that is obtained

in the universe after abstraction by eliminating timedimension) was immutable but in George

Lemaitre's model, this local dimension is a function of time in terms of mathematicians.

Expanded universe cannot always expand more and there exists border for this expansion that

at that time the universe will collapse and will repeat the primary chaos. According to George

Lemaitre's computations, the universe is now close to this border of desolation and perhaps

miliards years after this will fall down and collapse.

Here, it is not irrelevant that after mentioning an introduction, it is pointed out to recent

astronomical observations. Matter possesses the property of expansion, diluted and as well

contraction and density. Namely it can be put an object under pressing and reduced its

volume and consequently made it more condensed and denser or as the result of reduction of

pressure and some other physical factors, its volume can be increased and made the object

more expanded and diluted. But anyway, these two have border possibility, namely, as if an

object expands so much it will consequently explode, also as if an object is compressed more

than permitted, i.e, in the both cases of unlimited expansion or pressing, the object becomes

unstable and maybe easily collapse.

Recent astronomical observations, in particular the modern science radio astronomy has

observed dead extinct suns in dark areas of the universe that due to extinct, visible waves are

not emitted by them. These suns are so compressed and heavy that some of them are smaller

than the moon of the earth and thousands thousand times are more heavy than the sun, for

instance, each cubic centimeter of these suns has about 700 kg mass. These suns are in border

of their own unstable explosion and eventually they will explode and perhaps will become in

the form of spiral galaxies. In George Lemaitre's system, these compressions of extinct suns

are the result of the expansion universe and the statement of quality is out of comprehension

of these words.

12

The universe has boarder in other aspect and that is the primary elements that form matter in

one side and enormous galaxies and nebulars on the other side and inscrutable relationship

of compass matter in suns that emits electrons and the other particles in the space and some

fameless stars attract and compress them and eventually they abandon unstable compressed

matter in cycle of physical transformation into its own cosmological predestination. This is

another story and with honorable readers' permission, I finish word here.

1This book is a collection of eighteen essays that has been published by “Dehkhoda Publication House“ in 19??.
The essays of the book are either lectures that were given by Professor Hashtroudi in various assemblies or
articles that had been already inserted and published in press.

2Iranian musical instrument

3Jalal ad-Din Mohammad Balkhi also known as Rumi or Mevlana (1207-1273)

4Khawja Shamsu Din Mohammad Hafez Shirazi(1325/6-1389/90)

5Mohammad ibn Zakariya Razi (865-925)

6Abu Nasr Mohammad ibn Mohammad Farabi(872-950)

7Abu Ali al-Hossein ibn Abdollah ibn Sina known as ibn Sina or Avicenna (980-1037)

8Khwaja Mohammad ibn Mohammad ibn Hasan Tusi (1201-1274)

9Albert Abraham Michelson(1852-1931)

10Edward Morely(1838-1923)

11Willem De Sitter (1872-1934)

12Arthur Eddington (1882-1944)

13George Lemaitre (1894-1966)



Translated by Fariba Elliee

Hashtroudi in words of his colleague

Dr. Mohsen Hashtroudi

A Scientist – A Professor – An Eminent Man

by Dr. Ali Afzalipour

*

It is nearly 60 years that I know the deceaced Dr. Mohsen Hashtroudi. Our familiarity and

friendship began when we were both 10 years old children and spent primary school in

Aghdasiyeh school. Later, this familiarity that led to a very profoud friendship and close

sincere cooperation whether at the time of studying in Tehran and Paris and whether at the

time of teaching in faculty of science in Tehran University and continued until the end of his

honourable life.

Perhaps it seems so since I have had very close sincere friendship and collaboration with

Hashtroudi, it is easy for me to speak about this great man. But his prominent and exceptional

personality and missing this so honourable friend and dear colleague that has made me

profound sorrow, is so hard for me. Nevertheless, I will attempt so far as I am able to write

a short of what I myself remember about this venerable scientist, sublime professor, pious

nobleman and of what I have heard from students, other colleagues, friends and relatives

about him.

As the title of this short essay shows, I want briefly to reminisce about Dr. Mohsen

Hashtroudi at first as a scientist and then as a professor and at the end as an eminent man.

Hashtroudi: Scientist

The deceased Dr. Mohsen Hashtroudi studied a lot and read ver carefully and interestingly

any books in various fields of mathematics, physics and other scientific subjects and also

philosophical, art and literary books. It is interesting that he could read a thick book in two to

three days and came to know its contents in whatever subjects.

Meanwhile, like all scientists, his scientifical curiosity was so that he could not overlook

profound attention to any new problems in mathematics or physics or other subjects and even

their circumferential problems. The other interesting point is that he could have assimilated

their contents and prepared new problems for discussion and talk to academic colleagues or

students or others. Hence, Hashtroudi was familiar with many scientifical, philosophical,

art and litrary fields in particular various problems in mathematics and had sufficiently

proficiency in them.

With due attention to what was mentioned, it is not at all wonderful that Hashtroudi

propounded a scientifical, philosophical, art or litrary subject in the course about pure

mathemetics with his own specific delicacy, acuity and deep thoughts and talked and

discussed about it with students. Even when he conversed with his academic colleagues,

sometimes the continuation of discussion deviated from main route and interestingly

conclusions which were obtained from these talking, were more profound and worthful than it

was expected fron beginning.

The deceased Hashtroudi greatly cared always about new scientifical, philosophical, art and

litrary problems from youth and even in the last years of his worthful life and passing time

had not at all detracted from his own curiosity and inquisitiveness. Hence he was always

ready to talk and discuss about various fields of sciences and culture and had sufficient

conversance and proficiency in each of them.

Though (and since) the deceased Hashtroudi was himself one of the most worthwhile Iranian

scientists, he honored highly to great men and women in knowledge and scinece who are our

Iranian pride and honor all the world and reminisced thoroughly nicely about them. However,

just those who have sufficient provisions from knowledge, are well aware of dignity of former

scientists and esteem duly them.

In the end of this part should be added that Hashtroudi was a very mighty lecturer and in

addition to his proficiency to discussed subject he spoke with such ability that auditor was

attracted and fascinated from the beginning by the words of this great man.

Hashtroudi: Professor

Dr. Mohsen Hashtroudi was an active strenuous professor. During his youth time and until

his physical ability allowed him, he always attended on time in the faculty and sometimes his

courses became lengthy three to four hours without no fatigue appeared in his lovely face.

In order that he persuaded students much more to acquire what he taught them and to oblige

them to find personally the solutions of problems, he went seldom to write on the blackboard

to solve problems and he made over the writing and following mathematical reasoning to one

of good students in the class.

His public lecture session was also so that his speech and words were utilizable for all

audiences even for those who had no acquaintance with mathematical and scientifical

arguments.

Hashtroudi had so expressive profound recitation and at the same time simple, unadorned,

comprehensible and apprehensible such that he could teach the most difficult scientific

subjects easily to students (or other audience). Hence, from the beginning students (or

audience) relied completely on him.

Hashtroudi almost never obligated and tied himself up completely to determine before a plan

in his courses and deviated by no means from it. Subjects and issues which he taught and

discussed, were function of time circumstance and in particular function of questions which

students asked probably at the beginning of courses.

In order that students learned and realized difficult scientifical parts better and more profound,

Hashtroudi used to ask a scientifical question in courses and asked students for its solutions. If

student did not know duly the answer of his question or even if student's answer was partially

satisfactory, Hashtroudi himself answered and explained correct and complete solutions at the

end of class.

Hashtroudi allowed students and even encouraged them to ask questions in any fields which

were compatible to academical curricula and dignity, and to discuss about them. Then he

himself followed that question and resolved difficulties which existed probably in that

question with his special own proficiency and professionality and supervised and guided

students.

Thus, students became acquainted in addition to mathematics with other fields of human

culture and knowledge and realized though the highest dignity in sciences which human

achieved, undoubtedly

appertains to mathematics, other fields also exist that have worth to know them and to discuss

about them and they are so useful and instructive. Hence,it is not at all surprising that more

than a half of Hashtroudi's courses spent to discussion about various scientific, philosophical,

art and litrary issues and subjects.

From what was said, it is evident that in Hashtroudi's courses students became sophisticated

scientific and cultural facts in our world in where we are living. The words that must be said

is what students learned from this great professor and noble man, heard from a few other

professors and teachers.

Hashtroudi: An Eminent Man

It can be right justified Dr. Mohsen Hashtroudi as a real noble man. He was not only inwardly

a very noble and freeman but outwardly also a very decent and tidy professor. Certainly, this

factor impressed on others and in particular on students.

Since Dr. Hashtroudi had so delicate and profound sentiment and more significant than all he

was a man who behaved always extreme kindly all whether academical colleagues and friends

and whether students and others. Hence, though he was not always on complete welfare so far

as he could help everyone in particular students who were needy.

Professor Hashtroudi was aestheticist. With due attention to that mathematics is one of the

most beautiful (perhaps the most beautiful) manifestations of human culture, he could not be

non-mathematician. He realized completely extraordinary beauty of theorems, reasonings,

proofs and mathematical problems and hence indeed he made love mathematics.

Hashtroudi was a great mathematician. Since a man is a mathematician who realizes

wholeheartedly particular beauty of mathematics. He not only realized in the best manner this

interesting and complete extraordinary aspects of mathematics but also sensed it compeletly

and he enjoyed mathematics beauty and correlation and coordination among its various fields.

At the end of these words, I should add that Professor Dr. Mohsen Hashtroudi, in addition

to his scientific and cultural eminent, was a chaste, noble and honorable man who wore

his heart on his sleeves, in other words, he was honest and had this attractive and precious

characteristic (which is nowadays so infrequent and even unobtainable).

I enjoyed also like the other colleagues and friends greatly to speak and discuss with him and

I made myself the most of his companionship.

Alas that this honorable scientist, chaste and noble man does not live more. Peace be to his

departed spirit and May he be ever remembered.

Dr. Ali Afzalipour,

The Professor in mathematics Department in Tehran University,

the Permanent Member of Persian Language Academy

Tehran – October 1976


Translated by Fariba Elliee

Mathematical articles based on Dr. Hashtroudi's works

To deceased Professor: Mohsen Hashtroudi

Four Dimensional Space of Circles

Alireza Amir Moez1

Deceased Professor Hashtroudi generalized the inner product of two vectors to the inner

product of two circles and described its applications in geometry of circles and lines 2. This

interesting and useful subject was also translated into English3. Now, we briefly survey some

invariant properties of this product. Then we show that the set of circles and lines in a plane

form a 4-dimensional space in that the inner product of circles is its same vector product.

1. The scalar product of two circles: we suppose that

a ( x ² + y ²) − 2bx − 2cy + d = 0

(1)

is the equation of a circle with radius r. In this equation, if a = 0, the a direct line is

obtained. Hence, the equation (1) shows a circle or a line. Now, we consider two circles

C1 and C 2 with these equations respectively:

C1 : a1 ( x ² + y ²) − 2b1 x − 2c1 y + d 1 = 0
C 2 : a 2 ( x ² + y ²) − 2b2 x − 2c2 y + d 2 = 0

assume that the radii of C1 and C 2 are respectively r1 and r2 . Then we define the scalar or

inner product of C1 and C 2 as following:
(C1 , C 2 ) = ( a1r1 )( a 2 r2 ) cos α

in that α is the angle between two circles. We leave details to readers who refer to [2].
is directed angle. If we obtain the value of ( C1 , C 2 ) in terms of coefficents,

(C1 , C 2 ) = b1b2 + c1c2 −

a1d 2 + a 2 d 1
2

From this the inner product is obtained that the norm of the circle (1) is equal to:

1

|| C ||= (C , C ) 2 = b 2 + c 2 − ad = ar

indeed, the above value is pseudo norm because isotrop circles are created, namely acircle

which is not zero, its norm may be equal to zero; for instance:

x 2 + y 2 − 2 x − 4 y + 20 = 0

However, this circle is virtual. Reader can easily show that if a goes to zero, the value of r

will go to infinity and the value of ||C|| is again meaningful, namely:

|| C ||= b 2 + c 2

however, in this case the scalar product of two circles (two lines) changes to the scalar

product of two vestors which are prependicular to both lines.

2. Invariability of the product of two circles under transfer: the deceased Professor

Mc Duffee proved that the product of ( C1 , C 2 ) in part one remains constant under

transfer and he asked which other transformations hold constantly this product.
Proof: We assume that C1 and C 2 are the circles of part one and we consider the transfer

equations:

x=X+h

and  y=Y+k

we see that after transfer, (1) in part one changes to:

C ' : a ( X ² + Y ²) − 2(b − ah ) X − 2( a − ak )Y − 2bk − 2ck + ah ² + ak ² + d = 0

If we consider the value of C1 and C 2 and we compute ( C1 ´, C 2 ´) , it is again equal to:

b1b2 + c1c2 −

a1d 2 + a 2 d 1
2

3. Invariability of the product of circles under rotation: we consider again the circles

C1 and C 2 in part one. The equations of rotation are generally in the plane:

 x = hX − kY
, h² + k ² = 1

 y = kX + hY

Easily we can see that after rotation, the equation C in part one is:

C ' : a ( X 2 + Y 2 ) − 2(bh + ck ) X − 2( ch − bk )Y + d = 0

We assume again that C1 ´ and C 2 ´ are transformer of C1 and C 2 . We see that

(C1 ' , C 2 ' ) = (b1h + c1k )(b2 h − c2 k ) + ( c1h − b1k )( c2 h − b2 k ) −

= b1b2 + c1c2 −

4. Reflection: we consider the circle C in part one and the following reflection:

a1d 2 + a 2 d 1
2

a1d 2 + a 2 d 1
2

kX

x= 2


X +Y2

 y = kY


X² +Y²

( X , Y ) ≠ (0,0)

and

We see that the transformer of C becomes:

C ' : d ( X ² + Y ²) − 2bkX − 2ckY + ak ² = 0

(C1 ' , C 2 ' ) = b1b2 k ² + c1c2 k 2 −

( a1d 2 + a 2 d 1 )k 2
2

±

1, then ( C1 , C 2 ) will remain constant.

if k =

Still this question remains: in general under which transformations does the value of ( C1 ,

C 2 ) remain constant? Are these transformations confined to the parts 2, 3, 4?

5. 4-dimensional space of circles: we consider again the circle C in part one, we see that

the C can be written as:

a+d
a−d
[ x ² + y ² + 1] − 2bx − 2cy +
2
2

C:

we consider the following elements as base:

B1 = x ² + y ² + 1 = 0 , B2 = −2 x = 0 , B3 = −2 y = 0 , B4 = x ² + y ² − 1 = 0

we see that C is a linear combination of B1 , B2 , B3 , B4 and B1 is a complex circle. Reader

can easily prove that the set of circles forms a vector space on real numbers. If we

consider C in the following form:

C: pB1 + bB2 + cB3 + qB4

The product ( C1 , C 2 ) will be equal to:

(C1 , C 2 ) = − p1 p2 + b1b2 + c1c2 + q1q2

that its difference with common vector product is just the sign (-) in the first sentence. We

leave it to readers.

6. Transformations on the space of circles: among transformations in a vector space,

transfer and reflection are not linear. But if we apply the space of circles, these

transformations will become linear. At first, we survey transfer. We assume that:

F(x, y) = (x + h, y + k) = (X, Y), x = X – h , y = Y - k

Therefore, we consider the circle C in part one:

a+d
a−d
B1 + bB2 + cB3 +
B4 = 0
2
C: 2

Since C is a linear combination of B1 , B2 , B3 , B4 , its transfer will be a linear combination

h² + k ² + 2
h² + k ²
B1 + hB2 + kB3 +
2
2
F ( B2 ) = −hB1 + B2 + hB4

F ( B1 ) =

F ( B3 ) = −kB1 + B3 + kB4

h2 + k 2
2 − h² − k ²
B1 + hB2 + kB3 +
2
2

F ( B4 ) =

So, the transformation has a matrix which is as follows:

 h² + k ² + 2

2

−h

−k

 h² + k ²

2


h k

1 0
0 1

h k

h² + k ² 

2

h

k

2 − h² − k ² 

2


Now, we survey the reflection:

kX

x = X ² + Y ²


 y = kY

X² +Y²
G: 

1+ k²
1− k²
B1 +
B4
2
2
G ( B2 ) = B2

G ( B1 ) =

G ( B3 ) = B3

G ( B4 ) =

1− k²
1+ k²
B1 +
B4
2
2

so the reflection matrix will be as follows:

1 + k ²
 2
 0

 0
1 − k ²
 2


1− k²
2 
1 0
0 

0 1
0 
1+ k²
0 0
2 


0 0

We leave the other transformations to readers.

Khayyam

Poet Mathematician

Mohsen Hashtroudi

Khayyam resembling anyone opened his eyes one day to the world and other day he concealed countenance in veil of tomb. Annalist has recorded the date of his birth and death and possibly occurrences and vicissitudes of time has confiscated him in book of remembrances.

From purposes of what is our discussed, knowing these points are not so significant. Khayyam's prosperity and failures, his all joys and griefs, mirth and sorrow proceeded with him to the transient world. From him, a name, a number of quatrains and some scholarly works in mathematics and natural sciences have been left. If the criteria of people's esteems and worth are their renown, in this regard too, Khayyam is progenitor of scholar dynasty and pioneer of celebrities' convoy. Since by translation of his quatrains to almost all of languages in the world, after Fitzgerald¹ work, his name is colloquialism of people of all classes.

Khayyam's renown as a poet was so that Khayyam as mathematician and scientist is eclipsed by Khayyam as poet and in this regard, by permission of honorable reader, some words would be propounded about Khayyam's quatrains, philosophical thoughts and artistic paradigms so that his literary worth is also mentioned. Though utterance of such duty is so arduous.

Khayyam does not see optimistically this passing transient world at all. He knows end of anything as no return and unrelenting. He wishes, from behind of thousand years “like verdure of hoping, you would grow” and anxious eye would open a way again to the world.

He knows the extensive arena of the world as a field of the living's vain effort and search. In hallucination world he sees paradox that have given hand to hand affectionately and friendly and in brick of crenation of veranda, Caesar's cranium on the bone knee of Anoushirvan has relied and quieted down.

Wine is the mystery of the universe and since it makes us unaware of us ourselves, it would give any predicate, regardless of us and so it is the greatest reality. From transient instants of life, every instant that passes away with negligence, for him is venerable and preferential. No doubt, negligence that he purposes, is to take no negligence of the end, namely, negligence of death. The garden that Khayyam sees, is sepulcher of deceased darling ones. Narcissus eyes of friend, violet hair of mistress and cedar stature of beloved have been missed.

Workroom of potter is warning scenery of deep-minded sage and he sees father's ash that is a gimcrack in potter's hands.

I saw if any unaware did not see

Ash of my father in hand of any potter

Conundrum of the universe and mystery of living is not conquerable and anyone in turning of chalice by cupbearer in turn of oneself own, gets intoxicated and unaware out of the ring.

Friends take seats in nocturnal agora but this night is not followed by a day and in turn, they evacuate their seats. Whatever it is also acquired much more erudition and insight, eventually should pass unaware away.

“Those who went in pursuit of knowledge
Soared up so high, stretched the edge
Were still encaged by the same dark hedge
Brought us some tales ere life to death pledge.”

He sees workroom of existence without no purpose and anything as a gimcrack of this workroom.

Potter of workroom makes any pitcher though subtle and handsome, with the intention of its return he makes and furbishes it.

Goblet was made by the Wise Lord
“With love & care to the heights soared
This potter who shaped with such accord
To make and break the same clay, can also afford.”

He is consternation passing of time and the caravan of life is wandering about deviated paths. In such position, he supposes worrying about tomorrow injudiciously and recommends enjoying transient moment.

“The caravan of life shall always pass
Beware that is fresh as sweet young grass
Let’s not worry about what tomorrow will amass
Fill my goblet again, this night will pass, alas.”

The triple principles of Khayyam's thoughts is abstracted in “transience and shortness of life”, “to make the most of transient life” and “to be free of thought and to conquer mystery of the universe” and this last point is so significant, since it is mentioned to the same point in studying his scientific works and revising his thoughts.

The choice of quatrain form to state thought by Khayyam is a fundamental point. Since transient and short thought in a short form is stated more truthfully and sincerely. Furthermore, quatrain (couplet) in the frame of its own structure is not anomalous to the form of a logic theorem as indeed the philosopher sage sets forth circumstance of deduction of a verdict from other verdicts, Khayyam's literary arts is abstracted in this concise and its comment and explanation requires more time and more leisure, by permission of honorable readers, we leave artist Khayyam in this position.

Erudite Khayyam

To introduce erudite Khayyam, we require some preliminary. At the first we should investigate and study the progress of mathematics and its coordination with other sciences in Khayyam's era and then we would view what Khayyam has fulfilled. Here is necessary to be mentioned that there exists an age in the history of sciences that is well known as Islamic age but it has not been researched sufficiently and in particular in Iran if we relinquish a few people who have proceeded for indigenous verve and interest in rehabilitating and improving the history of erudite in this country, an important work has not been fulfilled. (Meanwhile it could be called Mr. Dr. Mostafavi and his “the world of science” magazine, Mr. Dr. Mosahab and his research about Khayyam, Mr. Abolghasem Ghorbani and his papers about a number of erudite and Mr. Daneshpazhouh and Danaseresht and their study about Abu Rayhan² and Khawaja Tusi³ and some others insofar as I know.)

In mathematics from the Greek era thereafter brief progress has been realized in Medieval and Islamic age. Indeed endeavor of scientists has been more directed to translations of scientific works from Greek and Syrian to Arabic, furthermore it has been mostly contented with description and comment of Euclid and same other Greek scientists. The most significant mathematical work, in the first stage, the regularization of algebra has been fulfilled by al-Khwarizmi⁴ so that in European languages, now, the deductive principles of Algebra have been ascribed to al-Khwarizmi and called algorithm (this word is anagram of the name al-Khwarizmi).

In the second stage, it includes Khayyam's works in geometry about parallel postulate and in Algebra about classification and the solution of cubic equations.

The significance of Khayyam's scientific work is so apparent among experts that demonstrates him relative to his own era four centuries closer to contemporary era, namely, Khayyam's works in Algebra belong to Descartes', Pascal's and Newton's age instead of Khayyam's own time.

Khayyam's study in Euclid's axiom or parallel postulate

In geometry book known as Euclid's Elements that the base of geometry is ascribed to him, an axiom is accepted that is mentioned as Euclid's axiom or parallel postulate and it is such that from a point out of strict line it can be drawn just one parallel line with that line so that they are all in the same plane.

Euclid accepts this axiom definitively and apparently and since clarity and clearness of this postulate like the other Euclid's elements is not so manifest from that time, in order to analysing and leading this axiom to other Euclid's axioms it has been taken efforts that eventually by Lobachevsky⁵'s researches it has been led to the foundation and deduction of new geometry.

Khayyam, in his own contribution to prove this postulate or to lead it to a simpler one, has also proceeded a treatise as “Explanations of the difficulties in the postulates in Euclid's Elements”.

Khayyam's view and analysing method in this treatise is more or less similar to the mathematicians' works early the 19^th century and Khayyam's conclusion is so summarized:

“Maybe the system that I apply to explain and reason for this postulate, be more logic and clear than Euclid's method”. During Khayyam's research in this work, it is observed that Khayyam seems uncertain to accept this postulate as a unique indisputable one. Indeed, as if he sees no impediment logically to deny this postulate. He accepts inevitably it just empirically.

Consideration of two points here is so important: the first is that according to Khayyam's view between logic and mathematics, there exists somehow a close dependency so that introduces parallel postulate in other form which is in his view more logical than Euclid's method. The second is that geometry in Khayyam's view is a science of abstract shapes that are absorbed in abstract space and this point is so important. Since for Greek the space was not authentic and the position of objects according to Archimedes' opinion was considered as the position of objects and shapes and now we know that the imagination of abstract space has worthy contributed to progress of mathematics and physics.

The connection between logic and mathematics in Khayyam's view leads to a postulate that now is considered as a basic foundation of science in philosophy and it is causality in the concept of scientific. The leisure is short and the discussion about this problem is not in scope of these words, it is just sufficient to mention to it what is discussed in sciences as cause and effect and the causality relationship between them, is a kind of coordination and uniformity in measurement and the result of comparisons that has remained constant and does not change and the point that as “to be free of thought and to conquer mystery of the universe” is already mentioned is the same problem that

relationship causality between evident objects whatever is, the manner of appearing of these objects is constant and Khayyam takes care of this issue exactly and in his own quatrains he has mentioned it frequently. Always the sign of beings has been.

Algebra and cubic equations

In the research that Khayyam has carried out, he has required the expansion of various powers of a “binomial” and has found out the formation of the coefficients of this expansion as a rule and regulation that nowadays is known as Pascal's triangle.

The algebraic binomial expansion is nowadays known as Newton's binomial theorem, since for the first time apparently Newton has codified these computations. Considering that Khayyam has used this expansion and the regulation of the formation of its coefficients in his works, it is clarified that Newton's binomial theorem and Pascal's triangle have been discovered and innovated almost four centuries before these scientist by Khayyam. For the first time, this point was mentioned by Mr. Abolghasem Ghorbani, school teacher in the Culture Ministry, in one of magazine in Tehran and he published some papers about it. Sometime after in one of international congress of the history of sciences that held in Rom, foreign scientists mentioned it as well and Rozenfeld, professor in Moscow University, submitted a suggestion concerning to change of the name Newton's binomial theorem and Pascal's triangle to the name Khayyam's binomial theorem and Khayyam's triangle to congress.

About the third degree equations, Khayyam is the first that classified them and mentioned some regulations to solve each of them by using conic sections. If it is considered that this method is indeed analytical and geometrical method, then it can be said that Khayyam is the first that has used analytical geometry to solve algebraic equations and in this regard, he has innovated analytical geometry almost four centuries before Descartes.

If is noticed that notations, form of algebraic equations in the present forms and signs in Khayyam's era has not at all exist, the significance and worthy of Khayyam's mathematical works would be more appreciated and better perceptible.

Khayyam has compiled a brief treatise about determination of carat gold, silver and ingot that has been combined by these two metals, actually is the explanation of know as Archimedes' method and his famous experience.

In this case, Khayyam also uses analytical and argumentative method to explain the famous Archimedes' principle that is not unlike to the present theoretical method.

The corrections of Zij Malekshahi and Jalali have enjoyed also Khayyam's endeavors. And here it is

contented with this brief mention.

In the imputation that it has been attributed to Khayyam contrary to equity and generosity, it should be careful that this erudite impossibly withholds to teach and this is contrary to ethic and knowledge to abstain to teach knowledge to apprentices. No doubt, those who did not know even elementary terminology, asked Khayyam to explain and the sage was not able to answer of necessity due to inadequate understanding of questioner and he has silenced and maybe has avoided to answer. And he has been attributed to withhold and been oppressed.

Khayyam's scientific position is at least in mathematics so much honorable and it is believed that he is the greatest mathematician in his own era and possibly can be said the greatest mathematician in Islamic age.

In comparison with al-Khwarizmi, Abu al-Rayhan and Ghiyath al-Din Jamshid al-Kashi⁶ , all are the top stars in the first place and justly Khayyam is the most eminent of this group.

The spiritual and scientific worth of Khayyam with regard to this that this scientist has not proceeded to found philosophical school, is more appreciated. Since philosophical issue if it has been even precisely presented and asked, does not have confidential and certain answer.

The mind of a scientist like Khayyam with clear and reasonable introductory could not be scientifically and principally like philosophical issues that are pessimistic and unstable with a view of sentiment and impressible existence. Furthermore, hypotheses and sentences of his previous or contemporary philosophers do not convince Khayyam. Hence he is weary of philosophical issues and elusive of philosophers. Similarity, though it would be little, between Khayyam and Buddha can be found that a derisive smile to deceitful manifestations of life has appeared on both their lips and has summoned both of them to silence and amnesia. Both of them have perceived affliction and both have attained maturity through containment, renunciation, connivance and forgiveness for high afflictions and their therapy, but none has argued and was aetiologist and has proceeded to found a new philosophical manner.

Both have seen the life of human as commiseration and compassion and have salved kindly human's trauma whether individually or socially. Tolerance and longanimity of heterogeneous thoughts that is well known as characteristic of Iranian race, is perfectly evident in Khayyam and can be said that Khayyam is the distillate and extract of centuries Iranian thoughts, contemplations, search and acquisition and is the result of a long endeavor, efforts, eminence and perfection of humanity. His doubt is closer to reality rather than of unawares' certainty who are claimant of wisdom and his perplexity is more ordered and regulated than of tranquility of those who reposed. Here is a position that words are unable and incapable to explain.

1Edward Fitzgerald

2Abu al-Rayhan Muhammad ibn Ahmad al-Biruni (973-1048)

3Khawaja Muhammad ibn Muhammad ibn Hasan Tusi(1201-1274)

4Abu Abdallah Muhammad ibn Musa al-Khwarizmi (780-850)

5Nicolai Ivanovich Lobachevsky (1792-1856)

6Ghiyath al-Din Jamshid al-Kashi or al-Kashani (1380-1429)

Translator: Fariba Elliee