Tuesday, October 17, 2017

Vedic Mathematics of India, & Ancient Indian Mathematics, & Some Of India's Greatest Mathematicians

Sri Bharati Krishna Tirthaji Maharaj : Shankaracharya of Govardhan Math Puri

Vedic Mathematics is the name given to an ancient Indian system of calculation, which was rediscovered from the Immortal Hindu Scriptures Vedas between 1911 and 1918, by His Holiness Sri Bharati Krishna Tirthaji Maharaj (1884-1960), Shankaracharya (Pontiff) of Govardhan Math, Puri. According to Tirthaji, the whole of Vedic Mathematics is based on sixteen Sutras, or Word-Formulae. For example, "Vertically and Crosswise" is one of these Sutras. These formulae are intended to describe the way the mind naturally works, and are therefore supposed to be a great help in directing the student to the appropriate method of solution. The Western Scholars say, none of these Sutras has ever been found in Vedic literature, nor are its methods consistent with known Mathematical Knowledge from the Vedic Era.

Perhaps the most striking feature of Tirthaji's system is its coherence. The whole system is interrelated and unified. The general multiplication method, for example, is easily reversed to allow one-line divisions, and the simple squaring method can be reversed to give one-line square roots. And, these are all easily understood. This unifying quality is very satisfying, it makes arithmetic easy and enjoyable, and it encourages innovation.

Difficult mathematical problems and huge sums can often be solved immediately by Tirthaji's methods. These striking and beautiful methods are a part of a system of mathematics, which Tirthaji claims to be far more methodical than the modern system. 

Vedic Mathematics is said to manifest the coherent and unified structure of arithmetic, and its methods are complementary, direct and easy.

The simplicity of Tirthaji system means that calculations can be carried out mentally, though the methods can also be written down. There are many advantages in using a flexible, mental system. Students can invent their own methods; they are not limited to one method. This leads to more creative, interested and intelligent students.

Interest in the Tirthaji's system is growing in education, where mathematics teachers are looking for something better, and finding the Vedic system to be the answer. Research is being carried out in many areas, including the effects learning the Tirthaji system has on children; developing new, powerful but easy applications of these Sutras in arithmetic and algebra.

The real beauty and effectiveness of the Tirthaji's system cannot be fully appreciated without practising the system. One can then see why its enthusiasts claim that it is the most refined and efficient calculating system known.

Vedic Mathematics refers to a technique of calculation based on a set of 16 Sutras, or aphorisms, as algorithms and their Upa-Sutras or corollaries derived from these Sutras. Its enthusiasts advance the claim that any mathematical problem can be solved mentally with these Sutras.

Swami Bharati Krishna Tirthajee Maharaj of Govardhan Mutt, Puri wrote a book on Vedic Mathematics, and first published it in 1965. It contains a list of ancient Indian Mental Calculation Techniques claimed to be based on the sacred Hindu Scriptures Vedas of  India.  Its characterization as Vedic Mathematics has been criticized by many academicians.

Although the book was first published in 1965, Tirthaji had been propagating the techniques all over the globe since much earlier, through lectures and classes. He wrote the book in 1957. It was published in 1965, five years after his death, as a book of 367 pages in forty chapters. Reprints were made in 1975 and 1978. Several reprints have been made since the 1990s.

Tirthaji claimed that he found the Sutras after years of studying the Vedas & deep meditation. However, the Vedas do not contain any of the Vedic Mathematics Sutras, mentioned in this book. Tirthaji’s description of Mathematics as a Vedic Science is most commonly criticised on the basis that, thus far, none of the Sūtras can be found anywhere in Vedic literature. When challenged by Professor K.S. Shukla to point out the Sutras in question in the Parishishta of the Atharvaveda, Shukla reported that Tirthaji said the sixteen Sutras were not included in standard editions of the Parishishta, and that they occurred in his own Parishishta and not any other.

Academicians feel that Tirthaji had liberally interpreted three-word Sanskrit phrases to associate them with arithmetic.

Some of them have pointed out that while Tirthaji's methods were not unique, they may have been invented by him independently, as Tirthaji held an MA in mathematics, & also that the term Vedic Mathematics is incorrect, and there are other texts that can be used to teach a correct account of the Indian Mathematics during the Vedic period.

Proponents of Vedic Mathematics however argue that the methods are not merely mathematical tricks and that there is an underlying psychology because the aphorisms describe personal approaches to problem-solving. As pedagogic tools, the methods are useful because they invite students to deal with strategies.

According to Scholars, Indian Mathematics  emerged in the Indian subcontinent from 1200 BC until the end of the 18th century. In the classical period of Indian Mathematics (400 AD to 1200 AD), important contributions were made by Mathematicians & Astronomers like Aryabhata, Brahmagupta, Bhaskara II & Varahamihira. The decimal number system in use today, was first recorded in Indian Mathematics . Indian Mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of Sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries ) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form.

Hitopadesha : Ancient Indian Fables Of Worldly Wisdom & Morals On Political Affairs In Simple Elegant Sanskrit Language

Hitopadesha (Sanskrit: हितोपदेश) meaning "Beneficial Advice" ) is an Indian text in Sanskrit language consisting of enjoyable stories with animal and human characters. It incorporates maxims, worldly wisdom and morals on political affairs in a simple, elegant language. This Hindu text has been very popular, widely translated into many Indian languages, as well as languages found in Southeast Asia, Middle East and Europe.

Little is known about the origin of Hitopadesha. The surviving text is believed to be from the 12th-century. Its oldest manuscript found in Nepal has been dated to the 14th-century, and its content and style has been traced to the ancient Sanskrit treatise called Panchatantra from 100 BCE to 500 CE. Dating the work has been very problematic.

The purpose of creating Hitopadesha has been to encourage proficiency in Sanskrit expression and knowledge of wise behaviour. This is done through the telling of moral stories, in which birds, beasts and humans interact. Interest is maintained through the device of enclosed narratives in which a story is interrupted by an illustrative tale before resuming.

The Hitopadesha is quite similar to the ancient classic Sanskrit text Panchatantra, another widely popular collection of fables with morals. Many scholars consider Hitopadesha to be a version or derivative work of the Panchatantra.

The Hitopadesha is organized into four books, with a preface section called Prastavika. The opening verse expresses reverence to the Hindu God Ganesha and Goddess of Learning Saraswati. There are several versions of the text available today, though the versions are quite similar unlike other ancient and medieval era Hindu texts, wherein the versions vary significantly. The shortest version has 655 verses, while the longest has 749 verses. 

In his own introductory verses, Narayana, who is believed by many to be the author of Hitopadesha, acknowledges and attributes his work to the older text Panchatantra. In his ninth verse, he states that he is indebted to the Panchatantra and 'Another Work'. The latter is unknown but possibly it's Hindu Dharmasastras
Hitopadesha differs by having only four divisions to the Panchatantra's five.

Monday, October 16, 2017

Vishnu Sharma's Panchatantra : Immortal Stories of Good Governance & Practical Wisdom From The Most Ancient Land Of India, Stories That Have Always Enchanted Both Children & Grown-Ups Alike

The Panchatantra ( Sanskrit: पञ्चतन्त्र, "Five Treatises") is an ancient Indian collection of interrelated animal fables in Sanskrit verse and prose, arranged within a frame story. Panchatantra is dated to about 300 BCE, but the fables are in all probability far far more ancient, as the Indians believe. The text's author has been attributed to Vishnu Sharma in some recensions and Vasubhaga in a few others, both of which may be fictitious pen names, according to Western Scholars. In all probability, Panchatantra Stories are based on older oral traditions with animal fables, that are as old & ancient as we are able to imagine.

It is certainly the most frequently translated literary work of India, and these stories are among the most widely known stories in the world. It goes by many names in many cultures. There is a version of Panchatantra in nearly every major language of India, and in addition there are 200 versions of the text in more than 50 languages around the world.

The prelude section of the Panchatantra in most places identifies an octogenarian learned Brahmin named Vishnu Sharma (Sanskrit: Viṣṇuśarman) as its author. He is stated to be teaching the principles of good governance & practical wisdom, to the three dud & utterly ignorant princes of the kingdom of Amarasakti in India. Based on the content and mention of the same name in several texts dated to ancient and medieval centuries, most scholars feel that the name 'Vishnu Sharma' is a fictitious name. Regardless of who the author was, Panchatantra stories are one of its kind in the world full of practical wisdom for children & grown-ups alike.
As already mentioned, the fables of Panchatantra are found in numerous world-languages.

In the Indian tradition, The Panchatantra is a Nītiśāstra. Nīti can be roughly translated as "the wise conduct of life" and a Sāstra is a treatise. Thus Panchatantra is considered a treatise on political science and human conduct.  It draws from the Dharma and Artha Sāstras, quoting them extensively. It is also explained that Nīti represents an admirable attempt to answer the insistent question how to win the utmost possible joy from life in the world, and that Nīti is the harmonious development of the inherent powers of man, a life in which security, prosperity, resolute action, friendship, and good learning are so combined to produce immense joy.


Panchatantra deploy metaphors of anthropomorphized animals with human virtues and vices. According to its own narrative, it illustrates, for the benefit of three ignorant princes, the central Hindu principles of nīti. While nīti is hard to translate, it roughly means prudent worldly conduct, or the wise conduct of life.

Apart from a short introduction, it consists of five parts. Each part contains a main story, which in turn contains several stories inserted into it, as one character narrates a story to another. Often these stories contain further inserted stories. The stories thus operate like a succession of  one narrative opening within another, sometimes three or four even. Besides the stories, the characters also quote various epigrammatic verses to make their point indelible in the minds of readers.

The five books of Panchatantra have their own subtitles.

Khavda Pottery Of Gujarat in West India

'Khavda' pottery is an Art-Form of Khavda in the North Kutch region of Gujarat in West India . While the men do all the throwing in terracotta, the women handle all the surface decoration which in this northern Kutch village of Ludia is primarily in the form of painting.   
The potter gets mud from a specific lake area near village. It is called 'Rann ka mitti'. This soft clay is shaped into a pot on a potter’s wheel and left to dry in shade, then Kumbhar women use red, black, and white clay-based paints to decorate each piece of pottery with distinct designs.   After a while, it is cleaned and put in the sun to dry and then baked in a furnace. The vessels are coated with a thin wash of Geru (Red Colour). The pots of Khavda get their red colour from Geru, a type of soil, and the black and white dots and stripes are also made with natural material.

The Terracotta pots of Khavda are distinct from the ones we may see elsewhere, because of their painting and form, which is a result of repeated cleaning at every stage of the process of their making.
More information on Khavda Pottery of Gujarat, will follow soon. Please stay tuned.

Anyone who requires more information may contact Mala Chandrashekhar at the Email Id :  indianartsandcrafts2008@gmail.com

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