Jain Sahitya Me Ganitik Sanketan

Here is a comprehensive summary in English of the Jain text "Jain Sahitya me Ganitik Sanketan" by Dr. Mukutbiharilal Agarwal, based on the provided pages:

Book Title: Jain Sahitya me Ganitik Sanketan (Mathematical Notations in Jain Literature) Author: Dr. Mukutbiharilal Agarwal Publisher: Z_Jain_Divakar_Smruti_Granth_012021.pdf Catalog Link: https://jainqq.org/explore/210936/1

Core Argument:

This article, by the renowned scholar and author Dr. Agarwal, sheds new light on how ancient Jain scholars analyzed subjects like the soul and the supreme soul using the language of mathematics. It specifically focuses on the mathematical notations and methods employed in Jain literature, highlighting "Ganitanuyog" (mathematical discourse) as a distinct and important branch of Jain philosophy.

Key Themes and Content:

The article emphasizes the fundamental role of symbols (संकेत) in human expression, suggesting that language itself originated from these symbols. It then establishes the crucial importance of mathematics in human life, from the earliest moments of counting. The author's primary goal is to explore the development and use of mathematical notations (गाणितिक संकेतन) that simplify mathematical operations and represent quantities. These notations are defined as signs used to express mathematical actions, denote mathematical quantities, or specify mathematical entities. The article uses examples like the colon (:) for division and inequality signs to illustrate this point.

Specific Notations and their Evolution:

The article delves into various mathematical operations and the specific notations used in Jain texts, often drawing parallels with other ancient Indian mathematical traditions:

• Addition (जोड़ने के लिए):

  • The "Bakhshali Manuscript" used a notation where a line was placed beneath the digits to be added, without a dividing line.
  • The Jain text "Tiloypannatti" (2nd century CE) uses the word "Dhan" (धण) for addition, reflecting the ancient use of this word for wealth and accumulation.
  • Pt. Todarmal, in his "Arthasandrishti," uses the symbol 'Log Log (अं)' for addition, and a vertical line for the addition of fractions.

• Subtraction (घटाने के लिए):

  • The "Bakhshali Manuscript" used a '+' symbol placed after the digit to be subtracted (e.g., 20 - 3 written as "20 3+").
  • Some Jain texts also use this '+' symbol, but placed above the digit being subtracted. Acharya Virsen's "Dhawala" (9th century CE) provides an example of this.
  • Professor Lachhmichand Jain suggests the '+' symbol for subtraction originated from the Brahmi script, specifically from the letter 'Ri' in the word 'Rin' (ऋण - debt/minus).
  • Jain texts also employ a '2' symbol placed after the number from which another is to be subtracted. For example, $1$ $lakh$ - $1$ $lakh$ is shown as $l$ $\underline{2}$ $1$.
  • "Arthasandrishti" uses a similar notation, demonstrating the subtraction of 1 lakh from a larger sum.
  • "Triloksar" (10th century CE) uses a notation where the number to be subtracted is written above the original number, followed by a dot-like appendage (e.g., $200 - 2$ is written as $\underset{2}{200}$ .
  • Another subtraction notation used is a dot below the number, with the subtrahend below the dot (e.g., $200$ with $2$ subtracted below it).
  • The symbol '•' is also seen for subtraction, as in "1 crore - 1" written as $1$ $k$
    • $1$.
  • The symbolic first letter of 'Rin' (ऋण), which is 'Ri' (रि) or 'Rin' (रिण), is also used, placed after the number being subtracted. "Tiloypannatti" has many examples of this.
  • The symbol 'रि०' is used for subtraction, as in "14-3 rio 100000 | 3", meaning 1 lakh yojanas less than the flagstaff of Saudharma Viman from the upper part of the middle world.

• Multiplication (गुणा के लिए):

  • The "Bakhshali Manuscript" uses the symbol 'Gu' (गु), the initial letter of "Guna" (गुण) or "Gunita" (गुणित).
  • "Tiloypannatti" uses a vertical line (|) for multiplication.
  • "Arthasandrishti" and "Triloksar" also use this vertical line symbol for multiplication.

• Division (भाग के लिए):

  • The "Bakhshali Manuscript" uses the symbol 'Bha' (भा), the initial letter of "Bhag" (भाग) or "Bhajita" (भाजित).
  • The text mentions that in ancient Jain literature, fractions were represented without a horizontal line between the numerator and denominator.
  • "Tiloypannatti" shows the volume of a cylinder calculated as $\frac{19}{24}$ written as "19 24".
  • "Triloksar" illustrates division with a remainder. For example, dividing 8164 by 64 yields 128 with a remainder of 2, written as "128 | 2".

• Zero (शून्य का प्रयोग):

  • The use of '0' initially began not as a numerical digit but as a placeholder for an empty space.
  • Ancient practice involved leaving a space between digits, which could lead to ambiguity (e.g., "4 6" could mean 46 or 406). To resolve this, a dot or a slightly wider space was used, which gradually evolved into the modern '0'.
  • An inscription on a Jain statue from Gopnath Ji's temple in Agra, dated Samvat 1506, is written as "15 ॥ 06", indicating this transitional phase.

• Squaring (वर्ग के लिए):

  • The symbol 'Va' (व) is used after a number to indicate squaring. For example, the square of 'Ja Ju A' would be 'Ja Ju A v'.
  • Other notations for higher powers are also mentioned: 'Gha' (घ) for cube, 'Va-Va' (व-व) for the fourth power (square-square), and so on.

• Vargit-Samvargit (वगित-संवगित):

  • This refers to raising a number to a power equal to its own value (e.g., 5 raised to the power of 5).
  • Special symbols are used for this in Jain texts. The first application is shown as $n$. The second application involves reapplying the process to the result, denoted as $n'$. For instance, the second vargit-samvargit of 2 is $(2')^2$, and its third vargit-samvargit is $(256)^2$.

• Square Root (वर्गमूल के लिए):

  • "Tiloypannatti" and "Arthasandrishti" use the symbol 'M' (म) for square root.
  • In "Tiloypannatti," "Mu°" (मू०) is shown for square root.
  • Pt. Todarmal's "Arthasandrishti" uses "ke Mu" (के मू) for the first square root and "ke Mu₂" (के मू₂) for the square root of the square root.
  • The "Bakhshali Manuscript" also uses "Mu°" (मु०).

Conclusion:

The article by Dr. Mukutbiharilal Agarwal demonstrates the sophisticated and systematic use of mathematical notations within Jain literature. It showcases how Jain scholars, through these symbols, developed and expressed complex mathematical concepts, contributing significantly to the history of mathematics in India. The research highlights the continuous evolution of these notations, often influenced by and influencing broader Indian mathematical traditions.