Light on a Society and Culture – through Mathematics

Lockdown has impacted many things, one of which is ability to buy new books. Kindle does not cut in our household, for any age group. And since the 10-year-old was getting cranky, innovative ideas were required. One came from the now defunct library in our society’s club house. When the mother-son duo raided the dusty shelfs, one Marathi book caught the eye of my wife. Her friend reacted who will ever read such a book. And my wife said she knows one person who might!

It’s a different story that I took over eight months to finish a 200+ pager.

The book

The curious book is श्री भास्कराचार्यकृत लीलावती पुनर्दर्शन a treasure trove of mathematics (An Analysis of Leelavati, originally written by Bhaskaracharya II). The writer is Prof. Narayan Hari Phadke, mathematics teacher and writer of many books on mathematics. The book was first published in 1971 by Varada Books, Pune.

History and mathematics made me curious. The book has original Sanskrit verses from Leelavati, lucidly explained theorems and practice sums, and information about Bhaskaracharya II – his life and times. It includes possible stories about the daughter after whom the book is purported to be named. But there’s more to it – the book reflects the society and culture of 1100s – the nature, flora and fauna, lifestyle of people, standardization of units of measure etc. The book demonstrates an advanced understanding of principles of mathematics and physics. If you study the underlying concepts, solve the sums/problems, and adopt the methods given, you will have revised all the math that is taught in the high school and junior college.

Bhaskaracharya II was very close to postulating basic laws of Physics based on his understanding of calculus. However, details of this are missing from the book and are not found in the works of contemporaries or immediate successors. Nevertheless, the expanse of the book cannot be undermined by what is missing.

Leelavati is the first part of Siddhanta-shiromani (सिद्धांत शिरोमणी). Other three parts deal with advanced Algebra (बीजगणित), General Mathematics (गणिताध्याय), Study of spheres, shapes, and other objects – possibly dealing with planetary motions (गोलाध्याय). 

Bhaskaracharya II considered Leelavati as a primer and recommended that students master it before studying other three books. Leelavati remains a popular treatise on mathematics.

The book – a look inside

As with all the Sanskrit books written up to that era, Leelavati is lyrical in style. The book starts by seeking blessings from Ganesh, the deity of knowledge and good beginnings. Prof.  Phadke says that the style and grammar are impeccable. Explaining a dry subject like mathematics in a lyrical style can be done only by a genius. 

Maheshwar, a renowned mathematician and astrologer and Bhaskaracharya’s father was also his teacher. The practical usage of mathematics also covered astrology, an important aspect of those times.

What does the math tell us about the society?

The initial part of the book lists the decimal system and the place value system. In one of the shlokas, names of numbers up to 10 to the power 18 are given, and the student is reminded of the progression by 10 (दशमान पद्धत).

Standardization is seen across measurements of length, area, distance, weight, volume, and time. Even currency was uniformly measured. 

Length

Small lengths were measured in units of flaxseeds (जवस) and phalanges of the fingers (अंगुल). Even though the original Sanskrit word is often translated as a full finger, Prof. Phadke asserts that the length is only a phalange long and not the entire finger. It also appears that there was a difference in the units of measurements when the flaxseeds were placed beside each other breadthwise versus when placed lengthwise. Bigger lengths were measured as equivalent to a hand (full arm) and bamboo shoot. In today’s equivalent, the bamboo shoot measure is approximately 5-and-a-half meters. Conversions from one unit of measure to other was standard. Which means, just as the western system had foot and inches, the Indian system of flaxseeds, fingers/digits and hands was a well-defined one. One hand-measure for example, was same everywhere and did not depend on the length of the hand of a person!

Time

Time measurement was based on the blink of an eye. Today’s German measure Augenblick mirrors this. The larger measures of time were based on the number of breaths. Even larger measures accounted for the movement of the Sun and Moon, e.g., days, months, seasons etc.

Currency

Did you know that various words we use in Marathi and Hindi to mean either insignificant or valuable find their roots in the currency measures of the yore? Kavadi (कवडी), Chhadam (छद्म) / Kakini (काकिणी), Daam (दाम), PuN (पण) and Nishka (निष्क) were the measures of currency, increasing in value in that order. The memory of these measures lives in our language without realizing that centuries ago it meant something valuable to people. References to Rupee are missing though.

Similar logic was used for weights and volumes. Special measures were used for gold. Grains were measured based on volume instead of weight.

When humor met math

Unlike the metric system, none of the conversions followed a standard pattern. Each conversion had a different multiplier. On one such occasion, Bhaskaracharya asked his students a convoluted problem of conversion of currency units when a wealthy merchant gave alms to the poor. After much usage of fractions and conversions from one type to another, the result – in a comical style, the merchant gave out only one Kavadi, the lowest amount there could be.

When nature met math

Many problems deal with the flora and fauna. Lotus, Malati (white and pink creeper flowers), Champa (Magnolia) find a mention in solving mundane equations. The equations were made interesting when elephants or swans decided to play in a lake or bumblebees moved from one flower to another or when the peacock decided to snap the snake. Sometimes the zephyr caused the lotus to hide under the water, and Bhaskaracharya made his students curious about the angle at which the flower swayed.

Instead of using unknows x, y, or z, Bhaskaracharya made it interesting by using animals and flowers.

Ratio Proportion

We all know how to find the fourth value if relationship between three is given. For example, 10% of 200. After having taught this basic (त्रैराशिक), Bhaskaracharya moves to more complex relationships and unknowns.

If you have dealt with a home loan, you know the moving factors – principal amount, interest amount, total period, number of payments, rate of interest and the installment. These six have an interplay and the values can be easily determined by pancharashik (पंचराशिक). And then Bhaskaracharya goes on to introduce problems related to eight variables (सप्तराशिक) and 10 variables (नवराशिक).

What is the practical usage of this? Evidently, the mixing of various portions of gold at various levels of purity. Today we use 18, 22 and 24 karats as standards for gold purity. During Bhaskaracharya’s time, it appears there were more than eight to 10 different varieties of gold purity – measured in a unit called kas (कस). In modern Marathi, this word is used to mean ‘proof of the ability or effort’. It is only natural that a valuable commodity had to prove its purity! A measure of 16 kas (कस) was considered purest form of gold. This is at odds with the Marathi phrase बावन कशी सोने – gold that is pure to the measure of 52 kas. However, there is no additional information to derive a relationship between the two.

This showcases an advanced knowledge of metallurgy and ability to distinguish pure forms of metals or other substances.

In one example of ratio-proportion, a traveler is asked to determine how much rice and dal (lentils) he could buy for a specific amount and in a specific proportion. That tells us the good old khichadi is as ancient as our civilization! This also tells us the measures of weight, currency and proportions worked well together and were standard across geographies.

In another example, a borrower is left wondering when the loan will be paid after the lender demanded money. The students were implored to quickly calculate the remaining amount.

This showcases a vibrant economy, travel and trade and interrelations between the peoples of India.

Advanced Mathematics

The book covers other areas such as the arithmetic and geometric progressions, series and induction, factorials, permutation and combinations, differentials and infinitesimals, trigonometry, mensuration, and Diophantine and polynomial equations. 

Some items show that Bhaskaracharya had a good understanding of the curvature of earth and that in case of trigonometry, straight lines were more sensible, and the curvature could be ignored. Problems like Sherlock Holmes’ Musgrave Ritual were presented. However, the context was Indian – a peacock on the tree, jumping onto the snake moving some distance away from the tree or a lamp casting the shadow of a conch shell. 

Bhaskaracharya reminded students that both the peacock and the snake travelled in a straight line! He exhibited an understanding that the light travelled in straight line. He also showed a basic understanding of gravity. Bhaskaracharya could have extrapolated and postulated theorems of instantaneous velocity. Although this extension is missing.

In tune with the style of the day, proving conjectures, hypotheses and theorems was left to the reader.

Pythagoras 

Bhaskaracharya simply states the Pythagoras theorem and goes on to use the triad values to determine areas of various non-standard shapes. Prof. Phadke says that Indians knew the theorem since the time of Shulbasutras, written circa 800 BCE. 

The case of pi

Like Pythagoras theorem, the Indian mathematicians knew about pi and its significance to a circle. The standard manner of using a polygon with a large number of sides to determine close approximation of pi was abundantly used. Bhaskaracharya used a polygon of 21600 sides. Bhaskaracharya also demonstrated a method of taking infinitesimal or really small pieces of a quadrant of a circle and using series and induction to determine the value of pi. In modern mathematics this method is attributed to Newton. This circumvented the need of using a polygon. Bhaskaracharya demonstrated this about 400 years before Newton.

It is interesting that Bhaskaracharya gave two values of pi – accurate (सूक्ष्म) and approximate (स्थुल). For accurate measure he recommended using 3927/1250 and for approximate measure he recommended 22/7. In modern advanced mathematics, the former number is a more accepted value of pi.

अंकानां वामतो गतिः |

We all use the place-value system today. It originated in India, moved to Arabs and from there to the Europeans. Some call it an Arabic system; some call it a Hindu-Arabic system tentatively acknowledging the origin. Through an active trade between the Indians and the Arabs a lot of knowledge moved westwards. As the numbers moved, they adopted the Arabic style of writing from right to left. 

Therefore, today the right most numeral in a number has the lowest place value. However, in the original Indian style, the left most numeral had the lowest place value. As the numbers moved from left to right, their place-value increased – exactly opposite of how we treat numbers today.

In the lyrical style numbers were often written in words and not numerals. Sanskrit is a very methodical language with an intricate case-system. As a result, the word order is not as important as the case of the words. Which means, to understand a number, it is important to understand the underlying sentence. And to a novice in Sanskrit, this may be difficult.

Number synonyms

Culture plays an important role in how numbers are represented. And many synonyms are used for the numbers in Sanskrit. A few examples are

• A bird = 2

• Vedas = 4

• Moon and all its synonyms = 1. However, the waxing or waning period of moon represent 15

• The sun’s chariot was pulled by seven horses, and therefore, a chariot = 7. The sun represents 12

• Jain Tirthankars (Jinas) represent 24

Therefore, if you find a bird, the sun and the Tirthankars together in a sentence, you must decipher 2 times 12 is equal to 24.

Unless the cultural background is clear, the shlokas can be cryptic or confusing and may seem out of place.

Where did Bhaskaracharya II come from?

Bhaskaracharya II was born in a town called Vijjalaveed. No such town exists today. Taking into consideration the vowel and consonant shift in the language over time, many scholars postulate that Bhaskaracharya was from either Bijapur or Bidar Karnataka, or Beed Maharashtra.

However, in his other books, Bhaskaracharya has left behind enough clues – his town was not too far from the Sahyadri mountains and the river Godavari. His town lay west of Vidarbha and in the middle of Dandak-van. Using these clues, Prof. Phadke theorizes that Bhaskaracharya must have come from either Khandesh area or from the Nasik district.

Bhaskaracharya’s grandson Changdev was given a land grant to start a school in the town of Patan in today’s Jalgaon district. Sanskrit copper plates detail out the existence of such a school. A revenue grant record exists in Ahiri language, a dialect of Marathi spoken in the Khandesh region. Is it likely that Vijjalaveed was close to Patan? No records exist to confirm this.

Wikipedia says Bhaskaracharya II taught at the university in Ujjain. However, Prof. Phadke says there is no proof of Bhaskaracharya having a patron. He was tutored by his father Maheshwar and he in turn was a teacher focusing on mathematics.

Who was Leelavati?

Common understanding is that Leelavati was Bhaskaracharya’s daughter. Many shlokas are addressed to a daughter, a friend, a sharp girl, a curious student. One legend says that according to astrology that Bhaskaracharya understood, unless Leelavati got married at a specific time, she would be widowed. Bhaskaracharya made every effort to get her married off at the date/time. But the destiny was different. The specific time was missed, and as expected, Leelavati lost her husband. To keep his daughter engaged Bhaskaracharya may have taught her mathematics. Other legend says Leelavati was his wife.

Prof. Phadke disagrees with these theories. He thinks that there was no person called Leelavati. It was just a name that Bhaskaracharya gave to his book and made it interesting for his students.

Conclusion

The genius of Bhaskaracharya was unmatched. And Prof. Phadke’s book is an interesting read. An acknowledgement of Prof. Phadke’s genius is due. His command over mathematics and Sanskrit is unmatched. And explanations are easy to understand.

A student of mathematics, history, Sanskrit, or culture – all will find the book exciting. Through lucid examples and lyrical style, Bhaskaracharya inadvertently throws open an unconventional piece of history and society. The original Leelavati has been translated over 20 times – in Sanskrit, Hindi, Marathi, Farsi, and English.

Bhaskaracharya (born 1114 AD) wrote Leelavati at age 36. He lived up to 79. He must have touched many minds and enlightened the society. His books were regularly used as teaching material in the medieval period. It is not for nothing that he was known as गणकचक्रचुडामणी - a great mathematician.

Know your history – Our geography problems find a root there

(reading time 6 min)

Look at the map from 1948. The border situation was very different then. China never had a border with India. However, the history is not linear. Tibet as a region is culturally influenced by India but historically a part of Qing Dynasty many a times. So was the case with East Turkestan or Xinjiang.

 

Although mired in internal conflicts and turmoil, the Middle Kingdom maintained an expansionist and hegemonistic view of its remote border areas. This was true not only for the Ming and the Qing Dynasties but also for the Republic (1912 to 1949) and now the People’s Republic. For this reason, a pacifist Tibet either enjoyed the status of being a sovereign or a protectorate or part of a hegemonistic rule.

 

Taking advantage of this confusion, the British decided to demarcate the boundary between British-India and Tibet. At that time, Tibet enjoyed de facto sovereignty.  However, when treaties are forced on weak states, the politics gets murky. Perceiving threat to its hegemonistic control, China intervened and the British and the Russians acknowledged that neither nation will engage with Tibet except through the Chinese.

 

Why was Russia a part of this engagement? Their search for a warm water port led them to Afghanistan and further south – a situation that made the British uncomfortable. What ensued was a great game of political confrontation between the British and the Russians over Afghanistan and Central Asia.

 

That also brings us to the nominally independent part of Central Asia and today’s restive province of Xinjiang – known to history variously as East Turkestan, the Dzungar (people), Tamir Basin (a part of it) or Sinkiang.

 

At the time of drawing of the McMahon Line, which China refuses to accept, and which Tibet only accepted under duress, the British maintained a Political Agent in both Lhasa (Tibet) and Kashgar (Xinjiang). In fact, Kashgar was later upgraded to a Consulate-General and was managed through the Indian Political Department of the British India. This indicates that both these regions were perceived as independent by the British.

 

What’s interesting is that Kashmir, Tibet, East Turkestan, and other small principalities, regions or valleys maintained an active trade and cultural exchange amongst themselves and traveled freely between the regions. The borders were porous and ill-defined. The concepts of sovereignty were unknown. Daulat Beg of the Daulat Beg Oldi region of Ladakh today bears the name of a certain Yakub Beg, a Tajik who conquered and ruled the East Turkestan and adjoining areas.

 

The Qing dynasty often controlled these regions either by means of a direct rule or by means of a confederacy.

 

Geography played an important role. The Himalayas made it difficult for the Indians and the British to venture into these areas. That meant, the Chinese and the Russians exercised most of the control.

 

From 1948 onwards Mao’s expansionist view was violent and backed by modern aggressive military. They walked over Tibet and Sinkiang. The resistance was brutally crushed. India, having won the freedom recently, was trying to put its own house in order. And grasp of strategic interests was absent.

 

India inherited the consulate in Kashgar. However, when the consulate was closed by the Chinese, there was no curiosity to find out what was happening under the hood. In fact, Captain Ram Sathe, who was the Consul General in Kashgar and later on the Ambassador to China, unceremoniously walked back to India via Lhasa facing great difficulties along the way. The construction of the road passing through Aksai Chin may have begun as early as 1948, a reason to close the consulate. This road and the Chinese annexation of the region was dismissed as irrelevant by the then government.

 

Thrust on Tibet and Sinkiang should have alarmed us. But either our mandarins were woefully unprepared or did not care. We tirelessly worked on the Panchsheel Agreement. We believed in the ‘peaceful rise of China’ when this was an example of oxymoron. We never understood the duplicity of the Chinese. And largely don’t understand it well even today. Sun Tzu has proved wiser than Kautilya.

 

We may have partially learnt our lessons from the 1962 war, but the psychological defeat is still alive. The incident of 1967 is unknown to many. And the Doklam standoff presided over a bickering that brought out the worst of Indian politicians.

 

The needling at the Pangong Tso lake or Daulat Beg Oldi or Tawang could have been avoided with buffer states. Today, we seem to have lost Nepal, opening up one more theater for the conflict.

 

Our strategy to counter China does not seem cohesive. Our muscle power and soft power has definitely increased. But it is wise to acknowledge that there is still a long way to go.

 

A case for two time zones

As we drove from Guwahati to Shillong, the road started to turn, and turn again and turn again. The winding mountainous road kept climbing up. The rain was watching us travel and the fog and mist was making it difficult to watch the natural beauty outside. By the time we reached Shillong, it was cold, misty, rainy and dark – pitch dark. I thought, now would be the time to eat some dinner, and go to sleep. However, the tour manager said, after check-in, come down for a hot cup of tea, and some meet and greet.

Tea? At this hour? I had clearly lost track of time through the winding roads and steep valleys. It was only 5:20 PM IST! 

An equal but opposite shock awaited in the morning. As I lay awake in the bed, the small sliver in the curtain suggested something bright outside. Well rested, as I stretched and walked towards the window to draw the curtains aside, I was shocked to see bright sunlight piercing through. And it was only 6:25 AM.

The daylight hours had shifted as we travelled eastwards. The Indian Standard Time did not keep up.

An eastern quip – give us a separate time zone

Many people – scholars, leaders, industrialists etc. from the Northeastern region have demanded a separate time zone for years together. However, it has not materialized into anything concrete. Energy saving, health and well-being of people and optimum utilization of resources are primary concerns. I am sure many back-of-the-envelope and official calculations must have shown the benefits of the two different time zones. However, it has fallen on deaf ears so far. Assam follows the concept of ‘Chai Bagan time’, but that is a tweak only to the start and end of working hours. It does not affect the time zone setting itself. Therefore, I will not pontificate the benefits. Instead, I will just put forward a solution/suggestion that I think will benefit our country.

Divide that Country – into time zones!

Let’s create a Northeast Indian Time Zone, which is aligned to UTC+6:30, which is a full hour ahead of the current IST. One look at the time zone map of current UTC+6:30 will easily convince you that this aligns better with the geography. Andaman and Nicobar Islands can also use this time zone.

Let’s continue to use UTC+5:30, which is the Indian Standard Time, for rest of the country.

However, I can think of another radical alternative. Align the UTC+5:30 only with the remaining northern, eastern and central Indian states. UP, Chhattisgarh, West Bengal, Bihar, Odisha, NCR, Punjab, Haryana, J&K, Ladakh, HP can use this time zone.

And let’s create another time zone, which is aligned to UTC+5:00. This will be used by the remaining western and southern states of India.

If you study the longitudes carefully:

1.     UTC+6:30 would roughly correspond to 104 degrees East. This passes through the middle of the Northeastern region.

2.     UTC+5:30, which corresponds to 82.5 degrees East, passes through major central Indian states, but is aligned more eastern than the western half of the country. And interestingly, it does not pass through any of the south Indian states.

3.     UTC+5:00, which corresponds to 75 degrees East, passes through the middle of the western region.

4.     I am combining the four major south Indian states and UTs with the UTC+5:00 because they are closer to the equator, and length of the day will not change too much with seasonal variations. Even if this is misalignment, it would adjust itself.

Challenges in implementation?

As far as I can see, only political will. While researching on this topic, I came across a poignant observation – this does not need anyone to spend crores of Rupees, hence the political class has no interest in taking this forward.

Many countries operate with multiple time zones. Broadcasters, Telecom Service Providers, Airlines, Train services are used to dealing with this. Public education campaigns will also help. It may create confusion, but that is temporary. If we could adopt a metric system, change our coinage and in recent times, execute demonetization, this task is not complex at all. Is there any political will to do this?

Who’s inconvenienced because of demonetization? Here are three probable candidates

(601 words, less than 5 minutes reading time)

(1) Anyone who has renewed a driver’s license has gotten a receipt of Rs. 250 in return of paying Rs. 1000, which not only includes the agent’s service fees but also the bribe component that he pays. And for the service the agent gives, he’s not even submitting the service tax to the government. I am told the bribe money makes deep inroads. Everyone is part of the chain.

Extent of inconvenience – minor. The new Rs. 2000.00 notes will in fact make it simpler for these people to hoard money.

(2) Anyone who has registered a house purchase or a rental agreement knows the difference between the ‘registration charges’ and the ‘cash’ the builder’s representative asks to carry along. The cash just vanishes in thin air! I am told these chains are rather long and deep.

Extent of inconvenience – minor. Once again, the new currency notes will make it easy to hoard money.

(3) What about the gatekeepers? I think they are the worst offenders. An example. A very close friend of mine sold his house for a new one. He took full cheque payment – meaning entire transaction was in legitimate monies. He reinvested in another house, which means he was exempt from the capital gains as per the appropriate rules. All of this was a part of his income tax return.

By his bad luck, around the time he executed the sale of the house, the so-called ready-reckoner prices were drastically increased. Now, it is a different matter to put the price on the ready-reckoner and expect the market to pay for it (unlike gold, real estate is not controlled by any index). The market did not fetch him what the ready-reckoner suggested. But he took a decision on what he felt was a fair price for his old house.

And because of this he got a notice from the I-T Department. The I-T officer felt that because there was much difference between the two prices, the rest of the money must have come from ‘black’ sources, and this god-fearing, income-tax-paying-salaried person must be sitting on a pile of cash. The scrutiny was detailed and exhausting. He was asked to produce all kind of documents, bank pass books, postal savings etc.

After the entire scrutiny, and the so-called ‘hearings’ between the I-T officer and the CA, he was given a clean-chit – simply because there was nothing ‘black’ in the entire transaction. What transpired after that is interesting to know!

The CA slapped him with a bill of Rs. 84,000! For what? Having produced all the documents required, having tallied all the amounts, and not able to show any unaccounted money?

When he asked for the breakup of the bill, he got to know:

• CA Fees – Rs. 25,000.00
• Bribe money – conveniently shown as consulting fees by the CA – Rs. 50,000.00
• Service Tax – Rs. 9,270.00 (@ 12.36%)

The CA was reluctant to bargain the bribe money on behalf of his client, simply because he has to work with that officer, and couldn’t afford to take a risk.

And the funny (or serious) part was the CA was ready to take all this by a cheque. Which means in his books he will have to cook-up the expense of Rs. 50,000.00. Isn’t it evident that the CA was in cahoots with the I-T officer?

By which logic should my friend have to pay a service tax on the bribe money also?

Extent of inconvenience – there is nothing inconvenient to the unashamed.

It is unlikely that the cash transactions – either legitimate or illegitimate – will stop because of the demonetization. But at least it will make the larger-than-life government bureaucrat and politician think about how the supposed surgical strike has hurt. It is very evident that these are the kind of people, sitting on large piles of cash. They are now in trouble.

What will transpire when the I-T department starts scrutinizing these cases should make an interesting story. The bigger question is will the government act against its own?

How ‘Only’ May Change your Sentence

(270 words, about 3 minutes reading time)

James J Kilpatrick, an ultra-conservative, a fiery supporter of racial segregation and a strong voice against the Civil Rights Movement is an unusual columnist to be remembered. Only for his skill of being a grammarian, I read a few of his columns, and amply use the tips as a reviewer of the documents. The column on long sentences made a mark, and the one about usage of ‘the’ rang a bell.

But what stuck with me was a usage of ‘only’ and how it can change your sentence. So here goes the example.

A simple English sentence - Jack hit John in the nose.

Check how the meaning changes by placing an ‘only’ in the sentence. As the ‘only’ moves from left to right, the meaning differs every time.

Only Jack hit John in the nose.

May be there were other people in the room. But none hit John, except for Jack. Or maybe others slapped, or kicked, but not hit. Or maybe others hit him in the ear, jaw, back or stomach, but not nose.

Jack only hit John in the nose.

May be, Jack also carried a revolver, or a machete, but he did not use that. A full force of the fist landing on John’s nose was enough.

Jack hit only John in the nose.

Jack may have spared Jason, Jerald, Jeremy – only John was subject of his angst.

Jack hit John only in the nose.

Jack was so focused, that other body parts didn’t matter. A bloody nose was sufficient!

The trick? Place ‘only’ as close to the subject, on which it is acting, as possible.