Bhaskara II – Invention of great mathematician of 12th century

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The great Bhaskara II or Bhaskaracharya invented the systematic use of decimal numbers system 600 years before the calculus of Newton and Leibnitz , gave the theory of infinity ,he had worked at the differential coefficient and had solved the problem   67 x2 + 1 = Y2 ( y square ) , he had given a deep concept of gravity and also gave his significant contribution in astronomy .

His leading work on the treatise  “ Siddharth Shiromani “  which comprises of four Grantha ( text-books ): Lilavati , bija-ganit ,Graha- Ganit , and Golaadhyay ( study of sphere ) were the most renowned work in the field of Mathematics and astronomy .

Bhaskaracharya , also known as Bhaskara or Bhaskara II , is one of those names from ancient India which got added with those brains which culminated themselves to their highest efficiency through working on their inner consciousness and using it in the form of formulating new formulas in mathematics and reaching to its highest peak in the field of astronomy . Another big name which came after him in 19th century was the great Srinivasa Ramaujan .

Many great scholars like Gangadhar , Ganeshdaivagya , Suryadas , Laxmidas ,Munishwar ,Ramakrishna had written their commentaries on his written book Lilavati . The commentary Kriyakramakari by two great mathematicians Shankar and Narayan in 16th century is the most extensive exposition of the Lilavati .

In 1587 AD , on the order of king Akbar , Lilavati was translated into Persian language by Abdul Faizi . Then in 1816 , J. Tayler and in 1817 , Henry Thomas Celebrooke translated Lilavati into English .

Let us find the deep rooted answer of this fined questions ” what are the inventions of great Bhaskaracharya and why did he become so famous worldwide ? ” Before that let us have some background knowledge of him .

When we turn our eyes in the field of Mathematics , we find numerous names added with names of Bhaskaracharya like Aryabhatta , BrahmaGupta , Varahmihira and many more .These names carry the great work of Mathematics and are the pioneer of many advanced formulas of mathematics at least half millennia before the European mathematicians came into light .

Bhaskaracharya was widely famous for his nearly absolute calculation of the time taken by the earth to orbit the sun in 365. 2588 days which is just 3.5 minutes different from what today’s advanced technology has found out .

 Now let us dive deep into the  life of Bhaskaracharya and his great achievements and contribution in the field of mathematics and astronomy .

Who was Bhaskaracharya ?

Bhaskaracharya was born in a Deshashtha brahmin family of Sandilya gotra in Bijapur , Karnataka in 1114 CE . His father Sri Maheshwar Upadhyaya was follower of Vaishnavism and was a great astrologer and intellect person who always supported his son in his interest . Bhaskaracharya gave the information of his birth in the question study of  “ The Goladhyaya “ which in English called spherical trigonometry .  He was a genius since childhood .

Bhaskaracharya became the head of the geometrical observatory in Ujjain in the 12th century which was then the leading mathematical centre for Bharat and where great mathematicians like VarahMihir and Brahmagupta were also the head of  the geometrical observatory in their times. 

Lilavati by Bhaskaracharya

Bhaskaracharya and Lilavati story

Bhaskaracharya  had a daughter named Lilavati who was very close to him . On the name of her, he wrote one of his famous Granth (text-book) of mathematics and named it as Lilavati . 

According to some sources it has been said that Lilavati didn’t have the destiny to get married . So , to make her feel happy and to live life with dignity , he named  one of his famous granth as Lilavati . Her daughter too became a famous mathematician but there is not much information received from the history about her personal or social life.

There is a controversy floating around the name of Lilavati .According to some historians ,  Lilavati was the name of  his wife who was very unhappy of not having any child in her life . So to reduce her pain and having her name known to the world ,  he named one of his famous Granth of mathematics as Lilavati . 

But the Tamrapatra which had been received from Nashik region told that Bhaskaracharya had a great grandson and granddaughter too which made it clear that Lilavati was the name of his daughter not of his wife. 

Bhaskaracharya great works

The greatness of Bhaskaracharya could be seen from his six great works which were Lilavati , Golaadhayaaya , bijaganita , Ganitadhaayaya , Karanakutuhal and Grahaganit. 

His most famous work was on Siddhant Shiromani which became very popular . He used to follow some of the very eminent gurus and great mathematicians  in his time but the way the great rishi “ Brahmagupta “   and his great works on mathematics showed him the path for discovering next level mathematics and astronomy was far more advanced and accurate than other ones .  

The popularity of the text book Lilavati could be predicted by the multiple commentaries written on it . There are over hundreds of commentaries on his book “ Lilavati ” which were written by many prominent persons .

Bhaskaracharya‘s renowned work echoed around the world and later on translated into many different languages .Those who used to call that algebra came after some outsiders’ work then let them introduce from where they got the knowledge of a great concept in mathematics . It all came from ancient India and later on travelled to many countries through translation in different languages .

Lilavati which presents the fundamentals of arithmetic and geometry in about 270 verses holds significant space in the mind of great mathematicians in later eras . 

These fundamentals mathematics provides a basic popularity among Indian minds of calculating mathematical operations whenever required  and study and practice of astronomy (graha-Ganita) and secondly, domestic calculation .Giving a concept of calculating the measure of grain(rasi-vyavahara),  bricks for walls,  pits and excavation etc .

When we talk about the Bija-Ganita then we find that it’s a sanskrit treatise on algebra.

The signs and symbols which had been used in ancient India for mathematical calculation are somewhat different from today’s symbols  which have been used for calculation in now-a-days .

  1. Instead of using the plus (+) symbol for adding the numbers ,  they used the symbol of separation ( | ) or (|| ) for adding the numbers.
  1. For Subtraction they used the symbol of dot over the quantity to be subtracted .
  1. For multiplication ,  they used to write the numbers together  like what we do today . ( eg : abcd or 1234).
  1. For Division, Hindu mathematicians used the same symbol which we use today ( eg : ½ ).
  1. The quantity which needed to be found was represented as zero in these manuscripts (0) instead of Question mark( ?) which we use today .

Bijaganit of Bhaskaracharya

In bija-ganita, Bhaskaracharya beautifully says “ It would be absurd for a person to ignore algebra while writing about astronomy.  it’s the same as a person ignoring grammar while writing a poetry.”

A person can’t imagine astronomy without algebra . Algebra is the backbone of  astronomy .

Bhaskaracharya contribution in mathematics

It’s not possible to present a full list of his great inventions and contributions with detailed explanation in the field of mathematics and astronomy .So I can only present a few of his works which are used in every walk of the field  .

  • Barter of commodities 
  • Distance between the lamp post and the shadow
  • Kuttaka
  • Unit of weight
  •  Month
  •  sum of capital and interest
  •  square root
  • A special operation involving squares and roots of numbers 
  • Proportion of gold in an alloy
  •  unit of time equal to 24 seconds
  • Operation with sums and differences of squares
  •  quadrilateral of unequal sites
  • Interest on capital invested
  •  height of heaped grain
  • Operation for the determination of the degree of purity of gold
  • Triangle formed by producing the two converging sides of a quadrilateral 
  • Solution of indeterminate Quadratic equation 
  • Geometrical progression
  •  purchase and sale in trade
  • Permutation of numbers 

When we turn towards  bija-ganit , We find below concept : 

  • Affirmative and negative
  •  addition and subtraction
  •  Multiplication
  •  Division
  •  square and the square roots
  • cipher 
  • On surds
  • On quadratic equation
  •  on equation involving rectangles 


According to Bhaskaracharya , the science of Jyotisha  without astronomy is like a garden without Mango tree , a lake without water,And like a woman who parted with her newly married lover.

He is trying to convey to us that there is no existence of Jyotisha sastra  without astronomy .

Graha-Ganita of Bhaskaracharya

Bhaskara  in his own commentary  “ Vasana Bhasha “ that there are 27 nakshatras (sectors ) like :  Ashwini, Bharani, Krittika, Rohini, Mrighasira, Ardra, Punarvasu, Pushya, Ashlesha, Magha etc. which occupy their positions almost at an equal distance along the Zodiac , arranged from west to east.

The circle of stars which is known as the Zodiac lies far behind all the planets . And when the zodiac and all the planets move in westerly direction at the same time , have themselves Individually move towards east direction.

 When we talk about the creation of the universe then we find that Bhaskara was of the belief that time is  eternal with no beginning and no end. He mentions the time when the creation happens and he also writes that it gets extinguished after deluge. 

 In Sanskrit we call the time as “ Kala “ and time is measured in different units. Like : Tatapara , Nimesa,Truti , Kastha,Kala,ghatis,Kshana etc.

Guliga daiva : Kantara stone worshipping powerful tradition

Bhaskaracharya gave his great contribution in

  • Lunar eclipse
  •  solar eclipse
  • Syzygies
  • Latitudes of planets
  • Mention about four Yuga padas
  • Sidereal revolution 
  • Solar month 
  • Astronomical calculation
  •  Adhikamasa
  • Kalpa
  • Units of time and many more 

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