Bhaskaracharya and his Leelavati

On the 900th birth anniversary of India's foremost medieval mathematician, Dr Chakravarti Madhusudana reflects on the life and achievements of Bhaskaracharya

A:~ne~~~~1djan mathematician/ astronomers, probably the greates t were Aryabhata (5th Century CE), Varahamibira (505 - 587 CE), Brahmagupta (598670 CE), .Malm rirac harya (c 800 - 870 CE) and Bhaskaracharya, also k n own as Bhaskara 11 ( 1114 - 1185 CE).

There were, of course, several other omscancling ma the m aticians such as Aryab hata II, Bhaskara I, Sridharac harya, P rithudakswa1ni who flourished during th is perio d.

This year marks the 900th birth an niversary of Bhaskaracharya. The special event will. be commemorated with seve ral acaderujc conferences across India. Ir is therefore a fitting rime to reflect u p o n t h e Life and remarkable a chievements of this great Indian.

We know that Bhaskara was born in l 114 AD, because he says so him sel f in 'Gola dhyaya' whjch is a part of h is work

Siddhanta Shirovumi:

"l was born in 1036 of the Sh aka era. I wro t e the Siddhanta Shiro111ani whe n I was tlurty- six yea r s o ld".

Since the Shaka era is believed co h ave scarred from 78 A D , we can calculate that h e was b o rn in 1114 AD. Bhaskara's wo r k Siddhanta Shiiwmmi co n sists of four pa rts :

1 Leelavati - a treatise on arithmetic, geometry and the so lution of i n deter minate equations

2 Bijt1grmita - a treatise on algebra

3. Gmhaganita - mathematics of planets or asu·onomy

4 Golad&Jc!)'tl - mathematics of sp heres

Bhas k ara goes o n to say that bis father was the emin ent schola r Maheswara, a Brahmin of Shandilya go tra. He also indkates that he was born in a place called V ij jal avida near the Sahyadri Moun tains

Bhas k ara's Lee!t111ati, comprising most of the ma [hematical techn iques needed for eve r y day transactions and beyo nd, is o n e of the most popular m athematical textbooks eve r written fr was in use over the lengtl1 and b read th of ln d ia for 700 years until the British system of education was int roduced.

Leelavatiwas wr itten in 1150, before tl1e days of printing, w he n the m aterial and eq ui p m ent required for makfog permanent wr itten reco rd s were not abund ant. Therefore, Li ke al m osr a ll of the scientific

INDIAN LINK an d p hilosophical works written in Sanskrit, Lee/avr1ti is also composed in ve r se form so that pupil s could memorise the rules wi thout the need co refer ro writte n texts

The verses were afao nec ess arily co nci se and needed exp la n atory commentaries, such as the 'Bud dh ivilasinj' by Ga n esba o r 'Lee lavti Vivaranam ' by Mah.idhara to understand them

The first fordgn la nguage tra n slatio n is [houghr to have been by A b u] Faizi Qlrother of Akbar's viz ier, Abu! Faz!) who r end e red the Leelavari into P ersian in 1587. The fi r st two EngLish trans lations of the Leelavati appeared in 1816 The authors wer e Henry T ho ma s Colebrooke, based in Ca lc utta , and John Taylor, based in Bombay. Both of them worked fo r [he East India Company - Cole b rooke was a high -ranking civil servant a nd Taylor was a medical doctor withj n the army.

Some of tl1e problems in Leelawtfi are addressed co a young maiden o f that same nam e. T lus fact has led co tl1e fol.lowing fanc iful sto r y.

The sto r y i s tl1at Bhaskara wrote tbe Lee lavari in orde r to conso le h i s distraught daughter. l t may be n(lted that other prob lem ques ti ons in L ee/a11t1ti a re ad dressed m a mathematicia n , a merchant and o tl1ers. Ir may also be noted that n either Bhaskara nor a ny of his conuuentacors mention this epi sod e. 1t onl y appears in the n anslari on by J\b uJ Faizi.

•Bhaskara had a daug hter n amed Lee lavati According to h er horoscope, she woul d remain unmar r ied and chil clJ ess unJess her weddi n g took p lace at a precise time on a particular, ausp ic ious day. In order to ensure th a t this moment was not mi ssed, Bhas k ara construc t ed a device where a cup w ith a s mall hol e was placed i n a vessel of water. The water would slowly enter the cup and, at the precise moment in question, the cup would sink to [he botto m of the vessel (see figure)

Leelavati, out of cur iosi ty and unbeknownst co her fa[her, peered in to the d evice and, as she rud so, a pearl from one of her ornam ents feU i nto the cup, b lockjng the hole.

Con sequently, [he auspicious m oment passed without m e cup sin king and the wedding could not be performed.

The Leel1111ati cons ists of 277 verses of ru les and exam p les The mrun coments are:

• Basic operations of arithmetic including finding square roots and cube roo ts, fractions, and the effect of cip he r

• The rul e of three, rul e of five and so on

• Bartering, b u ying and selling

• Permutations and comb inatio n s

• Prog r essions and series

• Geometric al. operations

• Solutions co indeterminate equatio n s

The qu es tio n s in Leelt111ati are kn own for rneir variety, storylike prob lems and the ch allenge they offer ro students.

In 181 6, when Coleb r oo ke published his transl ation of the works of Bhaskara and Brahmagupta, ir was generally believed tha t progress in Indian mathematics came to a standsriJJ after the rime of Bhaskara O n e reaso n g iven for tl1is was that the 12th century w imessed the stare of subjugation of lndj an s by fore ign invaders and mosr of Inrua wa s in a stare of turmoil

H oweve r, in 1835, C.M. Whish published a paper in the Transactions f the Royal Asiatic Socie91that proved that d evelopments in Mathematics had continued unabated in Kera.la, a region that was the least affected by the invaders. One of the lat e r Kerala m athematicians was Madbava (1340 - 1425 AD) of Sangamagrama (present day Ir inj alaku da) who had developed power series expr essions for 7! and for the sin e function some three hundred years before Leibrutz and Newto n

There were several other Kerala m athematicians of note in the post-medieval period B ue that i s an e n tirel y dj ffe r enr story.

Some Example Problems From The Leelavati

Atraveller, engaged In apilgrimage, gave half (ln) his money at Prayaaga; two-ninths (1/9) of the remainder at Kaashi (Benares);aquarter (1/4) of the residue in payment ofraxeson the road; six-tenths (6/10) d what was left at Gaya; there remained.sixty-three (63) Nishkas (gold coins) with which he returned home. Tell me the amount of his original stock d money, if you have learned the method of reduction offractions of residues.

HERE THE RULE IS: Divide the product oft he denominators minus the numerators by the product dthedenominators;and by the quciient obtained, divide the product dthe known quantity multlpfled by the iliSumed. Thus:

Let 1be the number iliSumed.

Subtract the numerators from the denominators; thus:

2-1 =1; ~2= 7;4-1 =3; 10-6=4

Product rl denominators minus the numerators= 1x7x3x4 = 84, and

Product of the original denominators= 2x9x4x 10= 720.

Dividing the first product by the second, we get84 /720, = 7/60.

Produa d the known quantity by the assumed = 63 x1 =63

Therefore, his original stock d money= 63 / (7/60) =540.

The interestedreader may verify the result by v.orking out the problem from first principles.

) Asnake's hole isat the foot ofa pillar of nine cubits height, and a peacoclt is pe!Ching on itssummit. Seeing a snake, ata distanced thrice the pillar, gliding towards his hole, he pounces obliquely upon him. Say qukkly at how many cubits from the snake's hole do they meet, both proceeding an equal distance7

Ruie:Thesquare[of the height] of the pillar ls divided by the distance dthesnake from his hoie;the quotient is to be subtracted from that distance. The meeting of the snake and the peacock is, from the snake's hole, half the remainder, incubits.

Referring to the figure, a = [e - (c 2/e)]/2 =[27-9 2/27]/2 =12 c 9;u+d e Z.7;b rJ ! Answe r: o 121 d

It's a tough world out there in the desi dating

enough? Good enough? Talented enough? Rel igious enough?

Recently, I searched on the L1rerner for 'single happ y' The search engine remrned over a million hi ts in 0.29 seconds. I am hesitant to add to the impressive array of material already om there, but this article is not only meant co be informative, but also reassuring, especfally from a Bengali/Indian poim of view. Everything I have read so far on this subject seems either very Bridget Jones or very l'vfargarec Atwood, when l am living proof chat you can be eitl1er, or both , or neither, on any given day.

So for the one or two people who haven't heard my mother's huge sigh of re lief reverberacing around the globe, I got married on d1e 22nd of March 2014 After 10 active years of dism issing proposals, arguing with my mother, looking for love, being clesponclent, being angry, being lost and finall y being resigned to being single I gave up. l gave up thinking that my single status is something that I could change.

1 gave up thinking d1ac I was in charge of m y destiny a nd I truly left ic to God.

This is eas ier said cban done. I know because for the past two or three years I had been SAYING that I had 'given up' and that ' God knew best', bur deep clown I didn't uuly believe it. Deep down J was scill scramb ling ro control my relation ships and, whether or not anyone else saw, 1 was scill desperately hoping chat I could do something to change d1e course of rny fate

The start of the beginning

The moment you are born a Bengali/Desi child, your parents begin to worry The y worry as you grow up, d1ey worry as you start school, they worry ,vhen you get to uni versi L-y but none of tbat compares to the worry di.at begins when the fuse person casually mentions co them, 'She's all grown up now! She'll be getting married soon' There goes a peaceful night's sleep for the parents of a Bengali girl. From now uncil the clay she says 'I accept', every waking moment is haunted by: What will happen to my daughter? Who will marry her? ls she educated enough? Pretty

It isn't long before dus worry, and sometimes frustration, finds its way, displaced or cl.ii.·ecdy, tO tbe child in question. And chat, m y dears, is when a girl TRULY becomes a woman. Ir's not when yo ur body changes, it's noc when yo u begin to notice or like boys, but it's when yoll stare realising that you parents cannot sleep because the)' a re so worried about yo u. And that brings with it Guilt. LOTS of guilt.

The beginning

Two things happen at dus point.

1. The very se lect lucky few women on ch is planet find their Prince Charnung (o r meir Prince Charming finds d1em) and they can happil y dispel dleir pa.rents' anxiety and ,vork towards their happily ever after. If that's yo u, pat yo urself on the back. Good job ow go away. Because for the rest of us your life is just as dreamlike and just as elllsive as Jasmine and Aladdin or Elizabeth and Mr Darcy. We don't hate you \Xie ARE genuinely happ y for you Bue seriously don't tell lls your Life is hard ever again because what we go through, chat you a re spared, is much harder.

2. The rest of us are primped, primed and positioned to within an inch of our lives as we are dragged to da1wts. i\ny remotely nice picmre of us is tacked to a bio data, whic h is a restune for marriage purposes, listing your Life history, bobbies tbat s how what a good girl yo u are (gardening and kriirting anyo ne?), your famil y tree and their respective occupacions to show you come from good lineage. And the worst part of all of this is chat every single family friend i s n ow eligible to ask yo u,

'When are you getting married?'

People will delight in pro jecting yo ur life for the next five years, including who you will marry and where yo u will live. More often than not these rununations from an overactive in1agination become firm fast rum o urs doing a marathon around your respective social circle, so che next time you are ou t you have to start every sentence wid1, ' o I am not getting married. Where did yo u hear that?'

If dus is you, in the latter situation, please s it down wim some chocolate and give yourself a hug. You will need it. This is only the beginning of a process which will defi ne your womanhood. It will thrust you into the reality of soc ial circles, it will enlighten you, quite harshly at tin1es, to who your real friends are, and it will reach you how to deal wid1 unwa.nced attention, grief, guilt and depression. Bue know, through all your turbulent emotions, that your parencs are doing chis out of their love for you A Desi parent bas no otbe r avenue of showing you how much mey care. Trying to find you someone who will l ove yo u anywhere near as much as they do is their so le ambicion in d1is elaborate and sometimes painful process. So strap yourse lf in, d1icken your skin and buckle up for the ride. It gees harder before it gets easier.

The process

Typically by the time. the first serious proposal char your parents consider comes around, yo u are around 18. Just beginning uni, jusr getting over chat first high school crush, and there are too many firsts going on in life that you a.re still crying to figure our for you to deal with the idea of marriage and d1e n o tion of 'forever'. But this first proposal will sec the tone for tl1e countless od1ers you will deal with. Because it will be the precedent on wh ich yow· parents and you interact on this subject.

Typ ically this co n versa ti on is rnotber-claughter domain, and it is a conversation so e lectricall y charged it has very real potential co quickly disintegrate into a shouting match. Your mod1er will see th.is 'boy' co b e a 'great match' and 'exactly what we are looking for for you'. She will be astounded by cbe fact that you ,vill say, 'J am not ready' and '1 don't even know !urn' and you might ger the ' in my day' story. My advice to you : shut up

Your mum or dad is not interested in what yo u have co say, so save yo ur breath. They are here. to advocate and advertise their IDEA of what d1ey dunk is best for you. So just let them. This is important. Because i t shows them chat you respect them, under stand d1em and value d1eir opinion. In doing chi s you set the tone for fun1re co nversations so that when YOU wane to be heard, respected and understood - they will do so (hopefully). Wid1 d1is first conversation out o f the way, jus t keep quiet and wait. Because the first proposal will come and go and so will a fair few otl1ers before THAT ONE D UDE comes along and you need to be ready for d1is one.

That one dude

Every s ing le person that has gone th.rough d1e arranged marriage process has encountered what J call tl1e 'That one dude' phenomena He is that one person that makes you question every single argument yo u ever put to your parents. He is d1e one person that makes you want co give up because i t's just too hard to keep fighting This may have n othing to do wid1 the guy himself or the proposal, and everything to do with the time at which chis phenomena occt1rs in your life

Nevertheless 'tl1at one dude' will come along. He is tbe 'perfect' potential on paper. He is tall (or so his bio-data says) , fair and handsome (.refer to the 'best of I 00' picture d1a t his mum sent with the bio data), and has a great job (doccor, lawyer, accoumant, engineer). So yo u agree to meet him in person and d1e meeting might happen at a dt1w11t, gad1ering or over coffee by yo urselves or in a group.

It won't take you long (30 seconds) before you reali se you have ZERO chemistry with this guy and he is BORING you co death. I need to clarify here that this has NOTHING to do with the guy (if you are THAT guy reading this you are REALLY nice. And sweet but just not the right fit) It's just something is missing and you sense ir right away. So you go home and when your mum asks you how it went you say 'it was ok' because it was. He wasn't rude or smel.l.y or stupid He was a perfectl y nice guy. And so tl1e internal struggle begins. Because by now your about 23- 25 and the pressure is really o n. Your friends are getting married, you're going to weddi.ngs every other weekend and there is always weddi ng talk goin g on in the house, and your parents have already gone through a cycle of potential hope, frustra tion and tl1en despair. This is when you are at your most vulnerab le.

It is when yo u open Facebook and t he plethora of wedding pictures that affront you is assaulting. Ir is when you have conversations wim your mum th at sotmd like Simo n Cowell judging a contestant: 'You gained a fair bit of weight. You have crooked teeth. I really d t ink this is the best you can do. l am sayi n g this because I love you ' It is when this whole process coi:nes ver y close to breaking you. When you lie awa ke. at night and mink, is this re.ally the best I ca n do? Maybe it is. Maybe those butterflies aren't meant to Hutter in m y sto mach after all, I mean h e's a nice. guy, sure he bores me. but h e can provide for m e. So what if we have n othing in common? He's got a good job. So what if he has no interest in asking me about myself? He comes from a good family. So what if he never asks me for my opinion? Marrying him will make my parents h appy. And all mar grult tl1ar's been building up inside you th roughout this process of tug of war wi tli your parents will come to a head and you will be very, very tempted ro g ive up. Don't. I r epeat. DO NOT GIVE IN.

Yo11 are so 11111cb 1110r-e thatJ'0lf perceive J'O!lrself to be. !'011 are ivorth SO much 11Jore than)'Ott think and)'OIi deseme so 11111ch fllore tha11J'0II cot!ld possib!J imagine. DO NOT SELL YOURSELF SHORT. If you want chocolate cake tlieo bold out for chocolate cake dammit! No m atter h ow good vanilla cake. i s it will n ever be chocolate cake and yo u will never be satisfied. You might be ab le to fake i t for a day, a week, a month, a year, but eventually your parents will. see tl1at you are not satisfied and tl1ey wiU n ot be h appy. If you feel like giving in for yo ur pa.rems - it won't work. Listen to your parents when they talk ro you about him. But be polite and Ii.rm in your refusal. Use that thick skin you built up earlier, and th e guidelines of open conversation yon established earlier, to maintain your position of th anks, bur no thanks. I want chocolate cake and I w ill wait. For my Chocolate Cake

The emotions

This who le process can last from anywhere from a ye ar co a decade. It can happen to a girl or a g uy It can happen to a Bengali or nonBengali. lt can happen whether you are rich o r poor. And so there i s no one way of handling your emotions excep t me 011e catdi.nal rule you must never forget: Love )'Ourself. 1 knO\\, As lam e, corny and cliched as it sound s this process, m ore than any other process in life (so far anyway), will make you feel like crap So many times alon g me way family and friend s inten tionally or o tl1erwise wiU make you feel lll1 loved, unwanted, and undervalue d Passing comments from p eople you do not know wi ll sting N ights will be spent crying and sleep will evade you. Be strong. Be the one voic e in your head that is cons istently l oving to deflect die barrage of n egativiry from otl1ers. Because you ARE amaz ing and you ARE gorgeou s and you ARE wonhy and \7 ery soo n a person WILL come a.long that will see that and be wordiy of you.

There is a reason that it doesn't happen automatically. The process of choos ing your life partner is hard BECAUSE it is so inextricabl y linked to the direction your forever will take. You need to ger tl1rough a stage of angst and despair and pain and hur t to come om stronger and witl1 more self-belief so you can choose with confidence your life partner and life path

Find friends that love you a n d understand yo u for who you are and not who diey want yo u to be. And laug h with mem. Laugh the kind of laughte r that wi.11 mal{e you realise tl1ar it doesn't matter if you are single and who knows it.

W hat matters in life i s having die right peo ple arou nd you to keep you on a path that takes you to your destination.

Thedesi way

W hen I told a friend at \vork about the arranged marriage process through which 1 met rn y husband, I expected some sort of remark. \~hat I didn't expect was her tone of wistfulness She actually wished that she had parents and an extended social network that could facilitate ar ranged meet- ups because 'rneeting a good man in rown i s im possible' Through that conversation 1 realised that no matter how frustrating the network of ' aunties ' can get, they actually provide a n invaluable service of vetting men and women and forming connections char would be o therwise impossible. For h er, the process to m eet so meone was at work, the gym, a parry, a bar and go o n at least three to five dates (think e n dless excruciating decisions on c lothes, hair make up, venue, shoes, nor to mention tl1e expense!) before she knew even half of what was provided in a bioda. ta.

So, as m uch as 1 made fun of the aunties in m y spiel above, I am grateful for the vetting and 'Aow of in fo rmati on' mey provide. Just can we stick ro the facts ple ase?

\Vhile the arranged marriage p rocess definitely works, there a.re defin ite!)' a few improvements tha t can be m ade:

1. Aw1ties, reduce the cattiness. You were all young once, you wem mrnugh a similar phase. Have some compassio n. Don't kick a girl w h en she's d own. D on't keep asking her when she's going to get married. TrusL me. \Xlhen she does you will know. You are p art of a net:wo r k much stronger than Facebook.

2 Parents, pl ease be nicer. You ar e the people who love us the m ost in this world Your every wo rd is taken very seriously whether we show i t or not. \When yo u say things like we a.re not good enough even mough you may not have meant i t - it hurts Be our s upport system instead of our critic.

3 Everyone e lse, tone d own the gossip. This process is ha.rd enough as it is witl1out having to contend with the endless ' he said, she said'. If something is happening, let it happen. If someone wants to know what yon think tl1ey will ask you. Ot herwise. Jus t wait and watch.

What it all boils down to

Have faith not only in God, but in YOU who God created. Have faith in your ability Yourself.

Your belie( Your feelings Have confidence to act on them. Don't compare yourself wit h the g irl who married her high schoo l sweemeart. Don't see her as havi ng more tlian you Instead see w h at you can share witl1 her. Deve lop the sk.i.U of sharing because marriage is about sh aring p arrs of you to make a who le.

Don't look for perfection L o one is perfect. Look for someone who is a good person. So meone who respects you. Someone who has ambitions and has a purpose in life.

This process doesn't al ways end in mar r iage. Or even a happ y marriage. Nothing is guaranteed No one knows what is in store fo r us But if you learn the lessons of patie n ce, respect and love, it will put you in good s tead ro tackle any situation life tlirows at youno matter what your marital status.