Jain Sahitya Me Sankhya Tatha Sankalnadisuchak Sanket

Here's a comprehensive summary of the Jain text "Jain Sahitya me Sankhya tatha Sankalnadisuchak Sanket" by Mukutbiharilal Agarwal:

The article "Jain Literature: Symbols Indicating Numbers and Operations" by Dr. Mukutbiharilal Agarwal highlights the significant contributions of Jain scholars to the field of mathematics, particularly in their understanding and representation of numbers and mathematical operations.

Key Points:

• Vastness of Jain Literature: Jain literature is presented as a vast repository of knowledge, with its scholars making significant advancements in various fields, including mathematics.
• Jain Mathematical Prowess: Jain mathematicians demonstrated exceptional intellect and understanding in mathematics, even in ancient times. Their work often differs from modern mathematics in surprising yet logically grounded ways.
• Definition of "Number" in Jainism:
  • While modern linguistics defines "number" as that by which things are counted, and dictionaries agree, Jain scholars had a distinct view.
  • One is Not Considered a Number: A key distinction made in Jain literature is that "one" (एक) is not considered a number in the context of counting or mathematical operations.
  • Reasoning: The rationale provided is that seeing a single object (like a pot) only confirms its presence, not its numerical quantity. Furthermore, in practices like donation, individuals don't typically count single items. This might be due to the influence of proper practice or the understanding that counting "one" denotes littleness.
  • Counting Begins from Two: Consequently, the concept of a number in Jain mathematics begins with "two."
  • "Kriti" (Square) Concept: The text mentions that numbers from three onwards are called "Kriti" ( कृति). A "Kriti" is defined by a unique Jain mathematical characteristic: if you subtract the square root from its square and then square the remainder, the result is larger than the original square. This definition is not found in non-Jain texts.
• Enormous Numbers in Jain Literature:
  • Jain texts like Sthanangasutra, Jambudvipa Prajnapti, Anuyogadvārasūtra, and Jīvasamāsa mention units of time measurement that involve incredibly large numbers.
  • These units are presented in a cascading fashion, where each subsequent unit is 84 lakh (8.4 million) times larger than the previous one.
  • The largest unit mentioned, Shirshaprahelika, can extend to numbers with 194 or even 250 digits. This demonstrates the Jain scholars' capacity to conceptualize and work with vast quantities.
• Numeral Systems and Script:
  • Ancient Jain Agamas (dating back to the 3rd-4th century BCE) list eighteen scripts, including numeral and mathematical scripts.
  • This suggests that different forms of numerical writing were used for different purposes. For instance, numerical scripts might have been used for inscriptions, while mathematical scripts were for calculations.
  • Jain manuscripts show the use of Brahmi numerals, and the article includes references to tables detailing the early forms of Jain numerals and their evolution based on various Jain manuscripts.
• Symbols for Operations (Addition, Subtraction, Multiplication, Division):
  • The article traces the history of mathematical symbols, noting that modern symbols like +, -, x, :, and = were introduced much later than their potential precursors in Indian mathematical traditions.
  • Addition:
    • The Vakshali Manuscript uses the first letter of "yuta" (युत) as an indicator for addition, placed at the end of the numbers being added (e.g., 4 यु 9).
    • The Jain text Tiloyapannatti uses the word "chhan" (छण) for addition, possibly derived from "ghan" (धन) meaning wealth.
    • Later texts like Arthasandrishti use a vertical line | to indicate addition, especially with fractions (e.g., 1 | 1 means 1 + 1).
  • Subtraction:
    • The Vakshali Manuscript uses a + symbol after the number to be subtracted (e.g., 20+3 means 20-3).
    • Some Jain texts place this + symbol above the number being subtracted.
    • Tiloyapannatti, Trilokasara, and Arthasandrishti also use a 0 symbol for subtraction (e.g., 200 - 2 is written as 200 followed by a symbol resembling 0 or a dash).
    • Arthasandrishti also employs U and other symbols for subtraction.
  • Multiplication:
    • The Vakshali Manuscript uses the initial letter "gu" (गु) of "guna" (गुण) or "gunita" (गुणित) for multiplication.
    • Tiloyapannatti uses a vertical line | for multiplication.
    • Arthasandrishti and Trilokasara also use this vertical line for multiplication.
  • Division:
    • The Vakshali Manuscript uses the initial letter "bha" (भा) of "bhaga" (भाग) or "bhajita" (भाजित) for division.
    • Ancient Jain literature did not use a line between the numerator and denominator for fractions; instead, they used specific notations.
    • The text describes how remainders in division were represented, which differed from modern methods.
• Zero (0):
  • The article explains that the symbol for zero did not initially start as a numeral itself but as a placeholder for an empty position.
  • Ancient texts would leave a space where zero is now used. To avoid confusion (e.g., between 46 and 406), a dot or a slightly larger space was introduced, eventually evolving into the modern 0.
  • Examples of this usage are found in ancient Jain texts and temples.
• Symbols for Powers:
  • The letter v (व) is used for squaring a number, placed after the number.
  • Symbols for higher powers are also described: dha (ध) for cube, v-v for fourth power, v-dha-dha for fifth, and so on.
• Symbols for "Vargit Samvargit" (Iterated Squaring):
  • This refers to raising a number to its own power.
  • Specific symbols are used: n] for the first iteration, n]] for the second, and n]]] for the third, indicating repeated application of the operation.
• Square Root:
  • Tiloyapannatti and Arthasandrishti use the abbreviation mū (मू) for square root.
  • Arthasandrishti further uses mū1 for the first square root and mū2 for the square root of the square root.
  • The symbol mū is placed at the end of the number.
  • The Vakshali Manuscript also uses mū.
  • Bhaskara II (1150 CE) used the letter k (क) from "karani" (करणी) for square root, placed before the number.
• Symbols for Special Numbers:
  • Trilokasara and Arthasandrishti use symbols like 2 for sankhyata (countable), 2 for pr Sankhyata (greatly countable), and kh (ख) for ananta (infinite).

Conclusion:

The author concludes that Jain Acharyas made a valuable effort to enrich mathematics through their detailed and profound study of numbers and operational symbols. They recognized the crucial importance of numbers and symbols for mathematical insight. The originality, practicality, attractiveness, and simplicity of these contributions are highly commendable and cannot be forgotten.