The remarkable etymology of sine

(or, How to go from Sanskrit to Latin in four easy steps)

"The ancient Indian mathematician, Aryabhata, had made extensive use of the concept of "sine" (central to modern trigonometry), in the fifth century. He called it jya-ardha, which literally means half-chord in Sanskrit. From there the term moved on in an interesting migratory way, as Howard Eves describes, in his An Introduction to the History of Mathematics (1990, p. 237):
Aryabhata called it ardha-jya ("half-chord") and jya-ardha ("chord-half"), and then abbreviated the term by simply using jya ("chord"). From jya the Arabs phonetically derived jiba, which, following Arabic practice of omitting vowels, was written as jb. Now jiba, aside from its technical significance, is a meaningless word in Arabic. Later writers who came across jb as an abbreviation for the meaningless word jiba substituted jaib instead, which contains the same letters, and is a good Arabic word meaning "cove" or "bay". Still later, Gherardo of Cremona (ca. 1150), when he made his translations from the Arabic, replaced the Arabian jaib by its Latin equivalent, sinus [meaning a cove or a bay], from whence came our present word sine."

[Amartya Sen, Keynote Speech at Beijing Forum 2006]

Murphy's Meta-law

Just a thought (surely unoriginal) regarding the ubiquitous Murphy's Law. I propose this be called "Murphy's Meta-law":


People only think about Murphy's law when something has gone wrong. Thus, Murphy's law is self-validating.


This is related to the notion of hindsight bias.

Bhrigu and Bharadwaja on the caste system

1 [भृगु]
असृजथ बराह्मणान एव पूर्वं बरह्मा परजापतिः
आत्मतेजॊ ऽभिनिर्वृत्तान भास्कराग्निसमप्रभान
2 ततः सत्यं च धर्मं च तपॊ बरह्म च शाश्वतम
आचारं चैव शौचं च सवर्गाय विथधे परभुः
3 थेवथानवगन्धर्वथैत्यासुरमहॊरगाः
यक्षरास्कस नागाश च पिशाचा मनुजास तदा
4 बराह्मणाः कषत्रिया वैश्याः शूथ्राश च थविजसत्तम
ये चान्ये भूतसंघानां संघास तांश चापि निर्ममे
5 बराह्मणानां सितॊ वर्णः कषत्रियाणां तु लॊहितः
वैश्यानां पीतकॊ वर्णः शूथ्राणाम असितस तदा

"Bhrigu said, 'Brahman first created a few Brahmanas who came to be called Prajapatis (lords of creation). Possessed of splendour equal to that of the fire or the Sun, they were created out of the energy of that First-born Being. The puissant Lord then created Truth, Duty, Penance, the eternal Vedas, all kinds of pious acts, and Purity, for enabling creatures to attain to heaven (by practising them). After this, the Deities and the Danavas, the Gandharvas, the Daityas, the Asuras, the great snakes, the Yakshas, the Rakshasas, the Serpents, the Pisachas, and human beings with their four divisions, viz., Brahmanas, Kshatriyas, Vaisyas, and Sudras, O best of regenerate ones, and all the other orders of creatures that exist, were created. The complexion the Brahmanas obtained was white; that which the Kshatriyas obtained was red; that which the Vaisyas got was yellow; and that which was given to the Sudras was black.'"

6 [भरथ्वाज]
चातुर्वर्ण्यस्य वर्णेन यथि वर्णॊ विभज्यते
सर्वेषां खलु वर्णानां थृश्यते वर्णसंकरः
7 कामः करॊधॊ भयं लॊभः शॊकश चिन्ता कषुधा शरमः
सर्वेषां नः परभवति कस्माथ वर्णॊ विभज्यते
8 सवेथमूत्र पुरीषाणि शलेष्मा पित्तं सशॊनितम
तनुः कषरति सर्वेषां कस्माथ वर्णॊ विभज्यते
9 जङ्गमानाम असंख्येयाः सदावराणां च जातयः
तेषां विविधवर्णानां कुतॊ वर्णविनिश्चयः

"Bharadwaja said, 'If the distinction between the four orders (of human beings) be made by means only of colour (attribute), then it seems that all the four orders have been mingled together. Lust, wrath, fear, cupidity, grief, anxiety, hunger, toil, possess and prevail over all men. How can men be distinguished by the possession of attributes? The bodies of all men emit sweat, urine, faeces, phlegm, bile, and blood. How then can men be distributed into classes? Of mobile objects the number is infinite; the species also of immobile objects are innumerable. How, then, can objects of such very great diversity be distributed into classes?'"

10 [भृगु]
न विशेषॊ ऽसति वर्णानां सर्वं बराह्मम इथं जगत
बरह्मणा पूर्वसृष्टं हि कर्मभिर वर्णतां गतम
11 कामभॊग परियास तीक्ष्णाः करॊधनाः परिय साहसाः
तयक्तस्वधर्मा रक्ताङ्गास ते थविजाः कषत्रतां गताः
12 गॊषु वृत्तिथं समाधाय पीताः कृष्युपजीविनः
सवधर्मं नानुतिष्ठन्ति ते थविजा वैश्यतां गताः
13 हिंसानृत परिया लुब्धाः सर्वकर्मॊपजीविनः
कृष्णाः शौचपरिभ्रष्टास ते थविजाः शूथ्रतां गताः
14 इत्य एतैर कर्मभिर वयस्ता थविजा वर्णान्तरं गताः
धर्मॊ यज्ञक्रिया चैषां नित्यं न परतिषिध्यते
15 वर्णाश चत्वार एते हि येषां बराह्मी सरस्वती
विहिता बरह्मणा पूर्वं लॊभात तव अज्ञानतां गताः
16 बराह्मणा धर्मतन्त्रस्दास तपस तेषां न नश्यति
बरह्म धारयतां नित्यं वरतानि नियमांस तदा
17 बरह्म चैतत पुरा सृष्टं ये न जानन्त्य अतथ्विथः
तेषां बहुविधास तव अन्यास तत्र तत्र हि जातयः
18 पिशाचा राक्षसाः परेता बहुधा मलेच्छ जातयः
परनस्त जञानविज्ञानाः सवच्छन्थाचार चेष्टिताः
19 परजा बराह्मण संस्काराः सवधर्मकृतनिश्चयाः
ऋषिभिः सवेन तपसा सृज्यन्ते चापरे परैः
20 आथिथेव समुथ्भूता बरह्म मूलाक्षयाव्यया
सा सृष्टिर मानसी नाम धर्मतन्त्र परायना

"Bhrigu said, 'There is really no distinction between the different orders. The whole world at first consisted of Brahmanas. Created (equal) by Brahman, men have, in consequence of their acts, become distributed into different orders. They that became fond of indulging in desire and enjoying pleasures, possessed of the attributes of severity and wrath, endued with courage, and unmindful of the duties of piety and worship,--these Brahmanas possessing the attribute of Passion,--became Kshatriyas. Those Brahmanas again who, without attending to the duties laid down for them, became possessed of both the attributes of Goodness and Passion, and took to the professions of cattle-rearing and agriculture, became Vaisyas. Those Brahmanas again that became fond of untruth and injuring other creatures, possessed of cupidity,--engaged in all kinds of acts for a living, and fallen away from purity of behaviour, and thus wedded to the attribute of Darkness, became Sudras. Separated by these occupations, Brahmanas, falling away from their own order, became members of the other three orders. All the four orders, therefore, have always the right to the performance of all pious duties and of sacrifices. Even thus were the four orders at first created equal by Brahman who ordained for all of them (the observances disclosed in) the words of Brahma (in the Vedas). Through cupidity alone, many fell away, and became possessed by ignorance. The Brahmanas are always devoted to the scriptures on Brahma; and mindful of vows and restraints, are capable of grasping the conception of Brahma. Their penances therefore, never go for nothing. They amongst them are not Brahmanas that are incapable of understanding that every created thing is Supreme Brahma. These, falling away, became members of diverse (inferior) orders. Losing the light of knowledge, and betaking themselves to an unrestrained course of conduct, they take birth as Pisachas and Rakshasas and Pretas and as individuals of diverse Mleccha species. The great Rishis who at the beginning sprang into life (through Brahman's Will) subsequently created, by means of their penances, men devoted to the duties ordained for them and attached to the rites laid down in the Eternal Vedas. That other Creation, however, which is eternal and undecaying, which is based upon Brahma and has sprung from the Primeval God, and which has its refuge upon yoga, is a mental one.'"

[Mahabharata, Book 12, Chapter 181; English translation by Kisari Mohan Ganguli]

If only we better knew and appreciated our great epics...

The pitfalls of political analysis

Did the BJP lose because it was too right-wing, or because it was not right-wing enough?

It never ceases to amaze me how intelligent and well-informed people can have such drastically opposed viewpoints. Not that I'm equivocating between the two; I certainly think the first one is much closer to the truth, and the second one seemingly fails to answer a simple question: why did the people vote for the Congress?

Are we a democracy or a shoethrowcracy?

P. Chidambaram, Naveen Jindal, L. K. Advani and now the Prime Minister himself, all within the space of 20 days, in a country where even pointing one's feet towards someone is considered rude. What is going on?

Only in the IPL, Only in India

I'm not particularly fond of the uber-commercial Indian Premier League, but it does occasionally throw up some incredible cricketing stories. One such came in the recent Rajasthan vs. Kolkata game. At the death, with Ganguly going well and Kolkata apparently cruising, Shane Warne (Rajasthan's captain) brought on to bowl... Kamran Khan. Who is he? The 18-year old son of a woodcutter from Azamgarh, a place that has been in the news for all the wrong reasons of late. Someone who is yet to play a single first-class game, but can apparently bowl at 140 kph. There he was, in distant Cape Town, up against the veteran former Indian captain, only 7 runs needed by Kolkata from the last over with 4 wickets in hand. Who would have given him a chance? Yet at the end of those 6 balls, Ganguly had been dismissed and only 6 runs scored. Kamran finished with figures of 4-0-18-3; Warne, arguably the greatest bowler of all time, managed 4-0-25-2 in comparison.

That wasn't the end though. This being the IPL after all, ties are unacceptable. So we have the spectacle of a "Super Over"; each side gets to bat for one extra over and the one scoring more runs wins. So whom did Rajasthan choose to bowl this all-or-nothing over? The legendary Shane Warne, surely? Nope; Kamran Khan. Up against the fearsome duo of Brendon McCullum and Chris Gayle, two of the hardest hitters around, he managed to restrict them to 'just' 15 runs. Yusuf Pathan then came out and finished things off in 4 balls: 6,2,6,4. Yusuf was man of the match (he'd scored 42 earlier as well), but surely Kamran was a strong contender too. Warne said afterwards: "I thought about Yusuf and myself for that Super Over but I went with Kamran and I thought he bowled a very good over". No small praise, coming from a man with 708 test wickets. Two years ago, who could have even imagined a 39-year old legend from Melbourne playing on the same team as an 18-year old rookie from Azamgarh, let alone all this drama? Now that's what I call awesome; and despite the physical location of the events, this story was truly "made in India".

Are Tescos the new cathedrals?

Recently I was in Blackwater, a small town situated within the London commuter belt, at a point where the counties of Berkshire, Hampshire and Surrey all meet. It is a fairly typical English riverside settlement, except for one thing: just outside the town is a shopping centre called The Meadows, which contains two of the largest retail outlets in the country. One is a Marks and Spencer and the other a Tesco "hypermarket". Strolling through the latter (not a bad place to get some exercise, as it took several minutes to walk end-to-end), I was somewhat stunned by the bewildering variety of products on offer: bread in hundreds of shapes and sizes, myriad juices representing virtually all imaginable fruit combinations, cheeses from a host of different European countries, and so on and so forth. Not to mention plenty of non-grocery products: TVs, DVDs, books, toys, video games, cameras and more.

Why do I mention all this? Aren't such stores fairly common in the west, particularly in America? Perhaps they are, but they still amaze me, given that I grew up in a mid-sized Indian city where our notion of a big shop was one we could actually step into, rather than just standing outside at the counter and asking the shopkeeper to get us what we wanted. The thing that interests me most about these megastores, however, is how they seem to have become in a sense focal points of modern industrial society. Even the much smaller local supermarket here in Oxford is always teeming with shoppers whenever I go there; and there are constant streams of people entering and exiting. That one single building is surely by far the most visited in the entire city. Of course busy marketplaces have long existed in towns, both in the west and east, but they are more spread out and generally have a more leisurely feel. The concentration of such large numbers of people and products into a single confined space; the milling queues and constant rhythm of beeping scanners at countless checkout tills; employees rushing around trying to re-stock fast emptying shelves - these are all phenomena unique to the modern superstore. In general, they seem to be the biggest and busiest buildings in our modern urban centres, in the west and increasingly also in India and other developing countries.

Some of the most wonderful buildings in Britain are the cathedrals. You can find one in every prominent town: typically an ancient, massive and awe-inspiring stone edifice. Once upon a time, these were the great centres of urban communities, the places where people in their thousands would gather to pray, to interact and to learn. Today, however, in an increasingly secular society, they have become somewhat irrelevant and derelict, even a burden due to huge maintenance costs, and are often kept going largely by means of tourists paying for exorbitantly priced entry tickets. People once went to cathedrals for evening or weekend outings; now, they are much more likely to go the shopping mall or supermarket.

So cathedrals were once focal points of towns and cities, the biggest, busiest and most central buildings, somewhat like the local Tesco might be today. Of course, there are huge differences between the two, not least that most cathedrals were (and still are) incredibly beautiful, whereas most shopping centres are incredibly ugly. In fact, I think this is a reflection of one major way in which society has changed. It is hard to imagine any cathedrals being built today: they would probably be deemed too expensive, too time-consuming to build, of too little practical utility. And yet in an earlier era, people were willing to invest massive time and resources in them. Those who conceived of and funded a cathedral, generally royals or noblemen, would often not even live to see it completed. They were happy, however, to think that when finished it would be truly impressive, and their children and grandchildren would one day benefit from it. So they were willing to take a long-term view, and even display a sort of altruism. Today, on the other hand, we have much less patience, and our thinking is more utilitarian and predominantly on shorter time scales. The most important considerations for any new construction project are likely to be duration, cost-effectiveness and functionality, with aesthetics lower down the priority list. Our much faster lifestyles are also reflected in the actual usage of the buildings: a visit to a cathedral would be a relaxed social affair, and might involve participation in a service, listening to music and sermons, and talking to other people. Most shopping trips tend to be much more rushed, with the objective being to pick up everything one needs as quickly as possible.

Are these changes in our collective way of thinking, our increased focus on practicality and short-term outcomes, good or bad? I guess they have certain positive aspects, but it seems to me the pros are outweighed by the cons. So many of the great contemporary issues we face, whether it be environmental degradation, food and water shortages, growing social inequality and unrest, or climate change, ultimately stem from our failure to consider the long-term consequences of our actions. Don't get me wrong: I am not saying we should go back to the middle ages, and I certainly believe there have been very many positive social changes since then. My liking for cathedrals also does not stem from any religious beliefs. However, sometimes I do wish we would slow down our lives a bit; perhaps spend a little less time earning and shopping and consuming, and a little more enjoying stained-glass windows and towering spires.

200 years on

Let me start with a piece of trivia in keeping with the theme of the blog's title: two of the greatest men in human history, Charles Darwin and Abraham Lincoln, were born on the same day, February 12, 1809. What are the odds on that? The coincidence doesn't end there, however. We normally think of Darwin as an apolitical scientist, but as a recent Nature article notes, one of the key motivating factors behind his ideas on evolution was his revulsion for slavery and his desire to debunk any purported scientific argument for racism. Just two years after the publication of Darwin's seminal The Origin of Species in 1859, Lincoln became the American President, and the abolition of slavery in the US followed soon after. It is unlikely that Lincoln was aware of Darwin's ideas; seemingly the two men on different continents were destined to be linked by their shared, visionary ideals. It is indeed fitting that the 200th anniversary of their birth has been marked by Barack Obama's inauguration.