Lilavati of Bhaskaracarya (Set of 2 Volumes)

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The present work provides the explanations and transliteration, in English prose order which makes sense in English translation. Some of the features of the book are: It starts with various of methods of the eight arithmetical operations and zero as an infinitesimal. The linear equations in multiple variables, quadratic equations, combinatorials and The Rules of Three, Five, Seven, Nine and Eleven. Arithmetic and Geometric series.

The following examples are of special interest:

Solve for x: 3(x.0 +(x/2).0)/0 = 63.

Formulae for the sum of integers, their square, cubes and higher powers.

To determine 2 numbers, given any two of the following seven: their product, sum, difference, sum of their squares, differences of their squares, sum of their cubes and the differences cubes.

One of them, a tough one at that, is to find a and b given their difference and sum of their cubes.

Demonstration of proofs as given in the Buddhivilasini are explained along with Sanskrit text and are compared with modern methods, where ever possible.

The author born in 1929, a physics graduate (1950) from Govt. Arts and Science College, Anantapur (A.P), did M.Sc. in Applied and Pure Mathematics from Pune (1958) and Ph.D from Shivaji University, Kolhapur (1979). He retired as professor of Mathematics and Dean of Science Faculty, Walchand College of Arts and Science, Solapur (1964-89), Maharashtra. He has published several text books and research papers in Mathematics. He presented a paper on The Relevance of Bhaskara's Methods in the Present Context at the International Congress of Mathematicians (ICM), at Hyderabad, India (August 2010). This is was published (2011) in extended form of a monograph entitled Bhaskara and Pingala - Relevance of their Mathematics in the present context by Lambert Academic Publishing, Germany. Earlier he worked in the Physics Depts of Voorhee's College, Vellore (T.N.) (1951-53) and Govt. Arts and Science College, Kadapa (A.P.) (1953-56) and worked as a Research Assistant in the Bihar Institute of Hydraulics and Allied Research, Khagaul, Patna (1959-61), and was a professor of Mathematics in Mudhoji College, Phaltan (Maharastra) (1961-64). He was the Principal of Hirachand Nemchand College of Commerce (1972-73), Solapur. At present he is translating the bhaskaracarya's Bijaganitam into English.

This is a translation of Bhaskara's Lilavati [8], Part I containing Arithmetic and Algebra. The second part of Lilavati is published recently in 2014. It contains Geometry, some Algebra and Linear Indeterminate Equations. There have been several English translations of Bhaskaracarya’s Lilavati. In most cases these translations do not give either the Sanskrit text or the Roman transliteration (for the benefit of those who are not familiar with Devanagari script), nor have they given the copious explanations contained in the commentaries by learned commentators such as Ganesadaivajna [8] (1545 A.D.) and others. Moreover (most of) these translations explain the methods in modern terms and give the modern proofs. This is not in the spirit of the traditional methods which the reader is entitled to know. A comparison with the modern methods is more educative if the traditional methods are explained, as given in the commentaries.

No word by word translations are provided by earlier translators. The word by word translations throw light on how dexterously words are used to replace the numerals and the mathematical formalism to suit the poetic form in which entire mathematics was presented. One of the reasons why translators in general are not in favour of word by word meanings may be that the prose order being in Sanskrit syntax, the English translation often makes no sense. The present work, Lilavati part I follows the method of presentation of Part II and gives the Sanskrit text in Devanagari along with the standard Roman transliteration, and word (phrase) by word (phrase) translation of Sanskrit in English prose order.

There is a general impression that the ancient Indian mathematics is only algorithmic in its treatment and devoid of any demonstration of construction or proof. This is far from truth. It is true that the poetic form in which mathematics is expressed has the drawback that it does not admit symbols and the rigour essential to mathematics. These short comings are mitigated to a very great extent by providing commentaries (on these works) which contain the much needed demonstrations of proofs. Hence, wherever necessary this translation is followed by explanations of the Buddhivilasini commentary of Ganesadaivajna, which define the technical terms and demonstrate the logic behind the algorithms.

The difficulties encountered in the word by word translation of the Sanskrit prose order is to a great extent eliminated by presenting the prose order according to the English syntax. This may necessitate splitting Sanskrit compound words to suit the English syntax. The cases such as dative, possessive, locative etc. in English may not exactly correspond to their Sanskrit counterpart. Moreover the poetic beauty and the rhythm in the Sanskrit slokas cannot percolate through the prosaic translation, in a foreign language like English. These are some of the short comings in this approach. The Sanskrit lovers may not approve of this. The main purpose of this effort is to make the prose order meaningful in English which is the medium of translation of this work. Wherever possible the methods of Bhaskaracarya are compared with modern methods. However, the emphasis is on the traditional methods.

Bhaskaracarya’s Lilavati is a part of the monumental work, which, as Bhaskaracarya himself acknowledges, is based upon earlier works of a galaxy of mathematicians like Aryabhata, Varahamihira, Sridhara, Padmanabha, Mahavira, Brahmagupta and others. This great work is a treatise consisting of four major works.

Lilavati (slate arithmetic), Bijaganitam (algebra), Goladhyayah (spherical trigonometry) and Grahaganitam (planetary mathematics). Lilavati consists of arithmetic, algebra and geometry (mensuration) and it continued to be the most popular and the main text book of mathematics for over seven centuries. The first translations of Lilavati and algebra were done during Akbar's and Jahangir's reigns (16th and 17th centuries) respectively. These translations were in Persian! They were later translated into English and other foreign languages and into Indian local languages during the last three centuries. This shows the popularity of Bhaskara the mathematician.

Lilavati and most of all books on science and mathematics were, as was customary in those days, entirely written in poetic form - unthinkable to a western mind. A mathematician or a scientist was also necessarily a poet. Lilavati is the best example of poetic beauty in a work, essentially mathematical in nature. It is amazing how a work in a subject like mathematics could ever be in poetic form. But a poetic form has restrictions. The authors had to sacrifice the mathematical format of symbolism and rigour and rather be satisfied with algorithmic style, leaving the demonstrations of proofs and the details of the workings of the examples to the wisdom of the eminent commentators and teachers. Bhaskara himself has written a commentary on his own work (astronomical part of) Siddhantasiromani. There are many commentaries on Lilavati, such as Kriyakramakari, Buddhivilasini, Lilavativivaranam etc.etc.

The present translation and explanations are based on the commentaries, Buddhiviliisini by Ganesadaivajna and to some extent, Lilavativivaranam of Srimahidhara as given in Bhaskaracarya viracita Lilavati (in Sanskrit) and Vimala commentary on (kuttaka) indeterminate first degree equations Bijaganita [13]. Wherever necessary the Sanskrit text and the translation of the commentaries of Buddhivilasini and Lilavativivaranam are provided along with references.

This is the elder sister of an already born girl! Siddhantasiromani of Bhaskara (or to be precise Bhaskaracarya II) consists of the four topics: Arithmetic, Algebra, Astronomy (Grahaganitam), and Spherical trigonometry (Goladhyaya). Professor A. B. Padmanabha Rao has translated its more widely known part Lilavati, which discusses arithmetic and algebra. He divided the work in two volumes. The second volume of the Lilavatiis already in the hands of readers. The present one is the first volume which consists of 134 slokas of the total 272 in Lilavati.

For non Indian readers it is necessary to mention that Lilavati is a feminine name and as per a legend she was Bhaskara's daughter. One may find it, therefore, a little interesting to note that a 12th century Indian mathematician addressed his important mathematical work to a girl, while European universities were prohibiting female students to study mathematics even in 19th century.

Professor Rao has done great service to Indian mathematics and mathematical community of the world by translating the Lilavati.

If there is one contribution of singular importance from India to the world of science, then it is undoubtedly Indian number system. The whole world today counts in .Indian numerals. Unfortunately, this fact is rarely acknowledged outside India. Indian arithmetic which was introduced to Europeans by Arab traders was, understandably at that time, considered as Arabian contribution. But strangely enough, this lack of information persists in some quarters even today. I have come across a technical report on solving quadratic equations, published as recently as in the year 2005 in an American university. The report mentions contributions of the Greeks, Persians and Arabs, but not of Indians. And this is when the celebrated Persian poet-mathematician Omar Khayyam, whose contribution is mentioned in the report, compares zifrenadan ('innocent' zero) coming from Hind with the beauty spot on the cheek of his beloved, in one of his rubaiyats. It is relatively very recent phenomenon that the phrase 'Arabic numerals' is being replaced by 'Indo Arabic numerals'.

European mathematicians greeted Indian arithmetic with a mixture of excitement and skepticism: the joyous excitement because they found a goldmine in the Indian decimal system and skepticism because there was no surety of purity of the gold. The grand project of converting physics into mathematics, initiated by Galilee and carried forward with great gusto by Newton and other great natural philosophers from seventeenth century onwards, needed unrestricted use of real numbers which only Indian numerical system was capable of providing. But European mathematicians brought up in the Greek tradition of rigour found Indian arithmetic without any foundation. This was potentially very dangerous situation, especially when they observed cracks even in well-founded edifice of Euclidean geometry. It is clearly demonstrable that 2 + 3 = 3 + 2, or 300 + 500 = 500 +300 are true statements. But why the equality a + b = b + a should hold true for all a, b? Similar doubts were expressed about other properties of real numbers which European physicists were using without bothering for proper justification.

This reference volume contains indexes, glossaries, descriptions of the flowers and other information.

There are three indexes giving the location of the flowers in Part 1: an Index of the Mother's Significances, an Index of Botanical Names and an Index of Common Names. The user should note that in all these indexes the reference numbers are those of the flowers as they are arranged in Part 1, not the page numbers of the book.

Two glossaries explain technical terms used in the book. A Glossary of Botanical Terms defines words that occur in the Descriptions of the Flowers. A Glossary of Philosophical and Psychological Terms provides definitions of the Sanskrit and other words that are found in the quotations of Sri Aurobindo and the Mother.

A brief section, The Symbolism of Colours, explains the meaning of various colours and their relationship to the Mother's flower- significances.

Descriptions of the Flowers, the largest section of the book, gives the full botanical name and family of each flower and indicates its size, shape, colour, lifespan, leaf-type, etc.

Note on the Texts and Photographs, the concluding section, identifies the sources of the quotations of Sri Aurobindo and the Mother and provides general information on the texts and photographs.