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