The Story of Omar Khayyam

Gabrielle Birchak/ August 6, 2024/ Middle Ages, Post Classical

Inter­dis­ci­pli­nary think­ing. Some­times, I can have intel­lec­tu­al rev­e­la­tions by tak­ing my brain into places where it has­n’t been before. And I know there are so many peo­ple out there who do the same. For exam­ple, when I worked at NASA, I dis­cov­ered that there were so many peo­ple on the lab who were musi­cians, part-time actors, res­olute golfers, sports addicts, food­ies and chefs, and more. They enjoyed being cre­ative and immers­ing them­selves in a world that was out­side of their occu­pa­tion. Through these cre­ative out­lets, they could find a new focus and make new dis­cov­er­ies with­in their work in sci­ence. And that was excep­tion­al­ly inspir­ing to be around. Rem­i­nisc­ing about NASA led me to cre­ate this pod­cast about some­one in his­to­ry who was an excep­tion­al inter­dis­ci­pli­nary thinker.

I note Hypatia’s work on con­ics in my book. It’s a fas­ci­nat­ing sub­ject that lends itself to fur­ther under­stand­ing alge­bra and geom­e­try. Its con­tri­bu­tion to alge­bra can be seen through the bril­liant work of the Per­sian math­e­mati­cian Omar Khayyam. But, like Hypa­tia, he was more than just a math­e­mati­cian. Like Hypa­tia, he was also an astronomer, a philoso­pher, and a polit­i­cal advi­sor. Fur­ther­more, he was not just ana­lyt­i­cal; he was also high­ly cre­ative and wrote beau­ti­ful poet­ry.

Omar Khayyam was born on May 18, 1048, in Nisha­pur, a city in north­east­ern Iran. His full name was Ghiy­ath al-Din Abu’l‑Fath Umar ibn Ibrahim al-Nis­aburi al-Khayya­mi. Nisha­pur was a cul­tur­al and intel­lec­tu­al hub dur­ing Khayyam’s time, pro­vid­ing a rich envi­ron­ment for his edu­ca­tion and intel­lec­tu­al growth. Khayyam showed an ear­ly apti­tude for math­e­mat­ics and sci­ence, which set the stage for his lat­er achievements.

Khayyam’s most sig­nif­i­cant con­tri­bu­tions to math­e­mat­ics lie in his work on alge­bra. His piv­otal book, “Trea­tise on Demon­stra­tion of Prob­lems of Alge­bra,” was a land­mark text that sys­tem­at­i­cal­ly pre­sent­ed meth­ods for solv­ing cubic equa­tions. Khayyam employed geo­met­ric meth­ods, unlike his pre­de­ces­sors, who approached cubic equa­tions alge­braical­ly. He used the inter­sec­tion of con­ic sec­tions, includ­ing parabo­las, hyper­bo­las, and cir­cles, to find solu­tions to alge­bra­ic equa­tions. In the eleventh cen­tu­ry, this was an inno­v­a­tive approach ahead of its time. His method­olo­gies laid the ground­work for future devel­op­ments in alge­bra and geometry.

For exam­ple, Khayyam tack­led cubic equa­tions of the form.

x^3+ax=b

He demon­strat­ed that one could find the roots of the cubic equa­tion by inter­sect­ing a parabo­la with a cir­cle. This rev­o­lu­tion­ary method illus­trat­ed Khayyam’s deep under­stand­ing of alge­bra and geometry.

Before Khayyam, solu­tions to cubic equa­tions were large­ly unknown. Ear­ly math­e­mati­cians had devel­oped meth­ods for solv­ing lin­ear equa­tions, qua­drat­ic equa­tions, and equa­tions with expo­nents divis­i­ble by two. How­ev­er, it was still dif­fi­cult to find gen­er­al solu­tions to cubic equa­tions and those equa­tions with expo­nents divis­i­ble by three.

He worked with a few method­olo­gies, some of which include the reduc­tion to a stan­dard form. Khayyam reduced the gen­er­al cubic equa­tion to sim­pler forms he could man­age with geo­met­ric tech­niques. For instance, he would change the vari­ables of equa­tions to elim­i­nate the qua­drat­ic term if it was present.

Anoth­er method­ol­o­gy that he used includ­ed trans­lat­ing alge­bra­ic prob­lems into geo­met­ric ones. By look­ing at the geom­e­try of the equa­tions, like parabo­las and cir­cles, he could bet­ter under­stand the alge­bra behind the geom­e­try. This leads us to his third method­ol­o­gy which was eval­u­at­ing alge­bra through the inter­sec­tion of con­ics. He real­ized that the solu­tions to cubic equa­tions could be found by con­sid­er­ing the points of inter­sec­tion between cer­tain con­ic sec­tions, such as the inter­sec­tion between a cir­cle and a parabola.

For exam­ple, let’s look at the cubic equation

x^3+6x=20.

Khayyam would rewrite this equa­tion to read 

x^3=20-6x

which became a form suit­able for geo­met­ric interpretation.

He would then look at the inter­sec­tion of the parabo­la of equation

y=\frac{x^2}{6}

And the inter­sec­tion of the cir­cle of equation

y=20-6x.

By find­ing the points where these two curves inter­sect, he obtained the solu­tions to the cubic equa­tion. This was con­sid­ered ground­break­ing for his time.

He was a math­e­mati­cian whose work exceed­ed the bril­liance of his peers. In addi­tion to his work with cubic equa­tions, HE also con­tributed to the under­stand­ing of the bino­mi­al the­o­rem and the con­cept of com­bi­na­torics, which are foun­da­tion­al to mod­ern alge­bra. His meth­ods and ideas influ­enced lat­er math­e­mati­cians in both the Islam­ic world and Europe, cement­ing his place in the his­to­ry of mathematics.

CALENDAR REFORM

In addi­tion to his work with math­e­mat­ics, he did sig­nif­i­cant work on cal­en­dar reform. Over the past 2000 years, if not more, there have been var­i­ous cal­en­dar reforms. I dis­cussed this in my sea­son one pod­cast about Leap Year. In 46 BCE, Julius Cae­sar and his bril­liant side­kick Sosi­genes real­ized a three-month sea­son­al dis­crep­an­cy in their cal­en­dar. And so, he com­mis­sioned Sosi­genes to devel­op the Julian cal­en­dar. In the same pod­cast, I also men­tion the Gre­go­ri­an cal­en­dar and how that was imple­ment­ed by Pope Gre­go­ry in Octo­ber 1582.

How­ev­er, in that pod­cast, I failed to men­tion the devel­op­ments made in Per­sia and Iran on the cal­en­dar reform imple­ment­ed by Sul­tan Malik Shah I of the Seljuk Empire. Five hun­dred years ear­li­er than the Gre­go­ri­an cal­en­dar, in 1074 the Sul­tan invit­ed Khayyam to Isfa­han to reform the cal­en­dar. The goal was to cre­ate a more accu­rate cal­en­dar to bet­ter pre­dict the solar year and improve agri­cul­tur­al planning.

Khayyam, along with a team of astronomers, devel­oped the Jalali cal­en­dar. The Jalali Cal­en­dar, intro­duced in 1079, was remark­ably pre­cise. It cal­cu­lat­ed the length of the year as 365.24219858156 days, which is only slight­ly dif­fer­ent from the mod­ern val­ue of 365.242190 days. This lev­el of accu­ra­cy was unmatched at the time and remained so until the intro­duc­tion of the Gre­go­ri­an cal­en­dar in 1582.

The Jalali calendar’s pre­ci­sion stemmed from Khayyam’s deep under­stand­ing of astron­o­my and his metic­u­lous obser­va­tion­al meth­ods. He and his team exten­sive­ly used obser­va­tion­al data to refine their cal­cu­la­tions, ensur­ing the calendar’s accu­ra­cy over extend­ed peri­ods. The Jalali cal­en­dar is still used in Iran today, a tes­ta­ment to Khayyam’s endur­ing legacy.

POLITICS

Khayyam’s con­tri­bu­tions were not lim­it­ed to math­e­mat­ics and astron­o­my. He was also deeply involved in the polit­i­cal life of his time. He served as an advi­sor to Sul­tan Malik Shah I and lat­er to his suc­ces­sor, Sul­tan San­jar. His role as a court advi­sor allowed him to influ­ence deci­sions on var­i­ous mat­ters, includ­ing sci­en­tif­ic and edu­ca­tion­al policies.

Despite his involve­ment in pol­i­tics, Khayyam remained ded­i­cat­ed to his intel­lec­tu­al pur­suits. He nav­i­gat­ed the com­plex­i­ties of court life while con­tin­u­ing his research and writ­ing. This dual role as a schol­ar and polit­i­cal advi­sor high­lights Khayyam’s ver­sa­til­i­ty and abil­i­ty to thrive in dif­fer­ent spheres.

In addi­tion to his math­e­mat­i­cal and polit­i­cal con­tri­bu­tions, Khayyam is also known in the West for his poet­ry. His col­lec­tion of qua­trains, known as the Rubaiy­at, has been trans­lat­ed into many lan­guages and con­tin­ues to cap­ti­vate read­ers with its themes of love, mor­tal­i­ty, and the pur­suit of knowledge.

Here’s an excerpt from one of his most famous qua­trains, trans­lat­ed by Edward FitzGerald:

“The Mov­ing Fin­ger writes; and, hav­ing writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to can­cel half a Line,
Nor all thy Tears wash out a Word of it.”

Omar Khayyam

These lines reflect Khayyam’s philo­soph­i­cal out­look on life and the inevitable pas­sage of time. His poet­ry often con­tem­plates the tran­sient nature of exis­tence and the impor­tance of liv­ing in the moment. The Rubaiyat’s themes res­onate with read­ers across cul­tures and eras, high­light­ing Khayyam’s uni­ver­sal appeal.

Khayyam’s work in math­e­mat­ics, astron­o­my, and poet­ry inter­sect­ed in fas­ci­nat­ing ways. His math­e­mat­i­cal pre­ci­sion informed his astro­nom­i­cal obser­va­tions, which, in turn, influ­enced his philo­soph­i­cal mus­ings in poet­ry. This inter­play of dis­ci­plines is a hall­mark of Khayyam’s genius and under­scores his com­pre­hen­sive approach to knowledge.

Khayyam’s con­tri­bu­tions to math­e­mat­ics and astron­o­my were ground­break­ing, yet he remained hum­ble and ded­i­cat­ed to the pur­suit of truth. His abil­i­ty to bridge dif­fer­ent fields of study and his pro­found insights into the human con­di­tion make him a time­less fig­ure in the his­to­ry of sci­ence and literature.

Omar Khayyam was a poly­math whose work tran­scend­ed the bound­aries of math­e­mat­ics, astron­o­my, and poet­ry. His solu­tions to cubic equa­tions and his cal­en­dar reform were pio­neer­ing achieve­ments that left an indeli­ble mark on the his­to­ry of sci­ence. His poet­ry, with its philo­soph­i­cal depth and lyri­cal beau­ty, con­tin­ues to inspire read­ers worldwide.

Khayyam’s lega­cy is a tes­ta­ment to the pow­er of inter­dis­ci­pli­nary think­ing and the endur­ing quest for knowl­edge. He reminds us that the pur­suit of truth through math­e­mat­ics, astron­o­my, or poet­ry is a uni­ver­sal endeav­or that tran­scends time and place.

Until next time, carpe diem! — Gabrielle