Mahavir Ki Rekha Ganitiya Uppattiya

Here's a comprehensive summary of the provided Jain text, "Mahavir ki Rekha Ganitiya Uppattiya," by Swami Satyaprakash Saraswati, focusing on Jain mathematics and geometric principles as attributed to the Jain mathematician Mahavir:

Introduction and Historical Context:

• The text introduces Mahavir as the most renowned Jain mathematician, known for his work Ganitasarasangraha (Compendium of Mathematical Essence).
• It briefly mentions other notable Jain mathematicians like Abhayadev Suri, Singh Tilak Suri, and Amarsingh Yati.
• The text places Mahavir within the timeline of Indian mathematics, noting his approximate contemporaries like Brahmagupta and Prithudak Swami (both around 850-860 CE).
• It highlights the Jain tradition's emphasis on the independent study of mathematics, acknowledging its utility in astronomy but also recognizing its intrinsic value.

Mahavir's Ganitasarasangraha and its Influence:

• Mahavir's Ganitasarasangraha is described as a valuable resource for mathematics experts.
• Unlike mathematicians like Aryabhata and Bhaskara who studied mathematics through the lens of astronomy, Mahavir's work, along with Sridharacharya's Patiganita and Trishatika, are considered works of pure mathematics.
• The Ganitasarasangraha had a significant influence on later mathematical texts, spreading its fame to South India within 150 years of its creation.
• Its Telugu poetic translation by Pavuluri Mallana and its English translation by M. Rangacharya (with introductions by David Eugene Smith) are mentioned, underscoring its importance.

Geometric Traditions and Mahavir's Contributions:

• The text traces the Indian tradition of geometry back to Vedic times, specifically to the Shulbasutras, which dealt with the construction of altars and described geometric principles for shapes like circles, squares, and triangles.
• Mahavir's Ganitasarasangraha details sixteen types of geometric figures, categorized as:
  • Three types of triangles: Equilateral (sama), isosceles (dvisama), and scalene (vishama).
  • Five types of quadrilaterals: Including equilateral, isosceles, and scalene, along with less common classifications like "equidichostic" and "equitritilateral."
  • Eight types of curvilinear figures (circles): This includes regular circles, semi-circles, ellipses, conchiform shapes, concave and convex circles, and annular regions (inner and outer).

Methods of Calculation:

• Mahavir employed two methods for calculating the areas of these figures:
  • Vyavaharik (Approximate): Practical methods for general use.
  • Sukshma (Accurate): Precise methods for exact calculations.

Quadrilaterals and the Brahmagupta-Mahavir Formula:

• The text discusses the calculation of quadrilateral areas. Both Brahmagupta and Mahavir provided the formula: Area = √s(s-a)(s-b)(s-c)(s-d) where 's' is half the semi-perimeter and a, b, c, d are the sides.
• However, the author points out that this formula is only accurate for cyclic quadrilaterals (those whose vertices lie on a circle).
• Aryabhata II is quoted as strongly stating that without knowing the diagonal, the area and perpendiculars of a quadrilateral cannot be determined, calling those who claim otherwise "fools or demons."
• Mahavir did provide a rule for finding the diagonals of cyclic quadrilaterals, which can be expressed algebraically as √((ac+bd)(ab+cd)/(ad+bc)) or √((ac+bd)(ad+bc)/(ab+cd)).

Circles and Ellipses:

• Circles: Mahavir's approximation for the circumference of a circle is Diameter x √10. This translates to approximately 3.16 times the diameter.
• Ellipses: Mahavir provided formulas for both approximate and accurate calculations of an ellipse's circumference and area. The accurate method involves the major axis ('a') and minor axis ('b') with a formula for circumference approximated as √(6b² + 4a²). The area is then calculated as (circumference * b)/4.
• The text provides an example calculation for an ellipse with a minor axis of 12 and a major axis of 36, showing both the approximate and accurate results.

Conchiform (Conch-shaped) Figures:

• Mahavir also provided methods for calculating the circumference and area of conchiform figures. Both approximate and more accurate (sukshma) methods are described, using the maximum width and the opening of the conch.

Annular Regions (Chakrawal Vritti):

• Mahavir addressed the calculation of the area between two concentric circles. Formulas are given for both "bahihchakrawal vritta" (outer annulus) and "antahchakrawal vritta" (inner annulus), based on the diameter of the inner circle and the width of the annular space. The approximate value of pi (π) used in these examples is 3.

Other Geometric Shapes:

• The Ganitasarasangraha also includes rules for calculating the areas of other shapes, including:
  • Yav (Barley grain shape): Formula provided is (width at ends + width at middle) / 2 * length.
  • Mridanga (Drum shape): Similar formula as Yav.
  • Vajra (Diamond/Lozengelike shape): Formula depends on the maximum width at the ends and the length.

Conclusion:

• The text concludes by emphasizing Mahavir's significant contributions to geometry and mathematics.
• It reiterates that Ganitasarasangraha is an immortal text, comparable to other foundational Indian mathematical works, and should be familiar to every Indian mathematics enthusiast.
• The author highlights that Mahavir's detailed descriptions of circles, triangles, and quadrilaterals are extensive and cannot all be covered in the article.

In essence, the text positions Mahavir as a central figure in Jain mathematics, who significantly contributed to the independent study of geometry and developed precise methods for calculating the areas and perimeters of various shapes, often going beyond the limited scope of earlier works.