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.
