Jain Sahitya Me Kshetra Ganit - Book Summary

This document is a chapter from a larger work, likely an academic or scholarly text on Jain literature and its contributions to various fields. This particular chapter, titled "Jain Sahitya me Kshetra Ganit" (Geometry in Jain Literature) by Dr. Mukutbiharilal Agarwal, focuses on the significant presence and development of geometrical concepts within Jain scriptures and texts.

Here's a comprehensive summary of the content:

Introduction to Geometry in Jain Literature:

The chapter begins by acknowledging that the origin of geometry in India can be traced back to the Shulva Sutras (around 3000 BCE), which dealt with the construction of sacrificial altars and mentioned various geometric shapes like squares, rhombuses, trapezoids, rectangles, right triangles, and isosceles right triangles.
It notes that hints of geometry are also found in Vedic tradition, particularly in texts like Vedanga Jyotisha.
However, the core argument is that Jain texts contain substantial material related to geometry, particularly in the context of Jain philosophy.
These Jain texts describe the structure of the universe (loka) and extensively use geometric figures in detailing the sun, moon, stars, continents, and oceans.
Key Jain scriptures mentioned for their extensive use of geometry include the Upangas: Suryaprajnapti, Chandraprajnapti, and Jambudvipaprajnapti. Other important texts are Tiloyapannatti, the Dhavala commentary on Shatkhandagama, Gommatasara, and Trilokyasara with their commentaries.
The chapter emphasizes that this geometrical content is crucial for understanding the development of ancient Indian mathematics. It highlights that Shatkhandagama even has a significant section dedicated to geometry titled "Kshetra Ganit."
Independent mathematical works by Jain scholars are also recognized for their importance and comprehensive contemplation of geometry. Prominent examples are Mahaviracharya's Ganitasarasangraha (850 CE) and Umaswati's Kshetrasamas.
The chapter points out that the importance of geometry is evident in the Sutrakritanga, which refers to it as "Ganita-Saroja" (the lotus of mathematics).

Types of Geometric Figures Discussed:

The chapter then delves into specific geometric figures mentioned and elaborated upon in Jain literature:

Quadrilaterals: Suryaprajnapti (300 BCE) mentions eight types of quadrilaterals:
- Samachaturasra (Square)
- Vishama Chaturasra (Irregular quadrilateral)
- Samachatuskona (Rectangle)
- Vishama Chatuskona (Rhombus/Parallelogram)
- Samachakrawal (Circle)
- Vishama Chakrawal (Ellipse)
- Chakrärdhachakrawal (Semi-ellipse)
- Chakrakara (Sphere segment/Sphere)
- Professor Weber's interpretations of these terms are also provided.
Figures in Bhagavati Sutra and Anuyogadwarasutra: These texts mention five types of figures:
- Trisra (Triangle)
- Chatusra (Quadrilateral)
- Ayata (Rectangle)
- Vritta (Circle)
- Parimandala (Ellipse)
Solid Figures (Ghana): The Anuyogadwarasutra discusses two categories of shapes: pratara (plane) and ghana (solid).
- Ghana Trisra (Triangular prism)
- Ghana Chatusra (Cube)
- Ghana Ayata (Rectangular prism/Cuboid)
- Ghana Vritta (Sphere)
- Ghana Parimandala (Elliptical cylinder)
Ring Shapes (Valaya): Jain texts also mention circular, triangular, and quadrilateral rings, referred to as Valayavritta, Valayatrisra, and Valayachatusra, respectively.
Triangles in Ganitasarasangraha: Mahaviracharya's Ganitasarasangraha discusses triangles based on their sides:
- Samatribhuja (Equilateral triangle)
- Dvisam Tribhuja (Isosceles triangle)
- Vishama Tribhuja (Scalene triangle)
- It notes that while triangles are classified by sides, classification by angles isn't explicitly detailed, though right triangles are discussed.
Quadrilaterals in Ganitasarasangraha: This text describes five types of quadrilaterals:
- Samachaturasra (Square)
- Dvidvi Samachaturasra (Rectangle)
- Dvisamachaturasra (Trapezoid with two equal non-parallel sides)
- Trisamachaturasra (Trapezoid with three equal sides)
- Vishama Chaturasra (General quadrilateral)
Curvilinear Figures in Ganitasarasangraha: Mahaviracharya describes eight types of curvilinear figures:
- Samavritta (Circle)
- Ardhavritta (Semicircle)
- Ayatavritta (Ellipse)
- Kambuka-vritta (Conical shape)
- Nimnavritta (Concave circular area, like a fire altar)
- Uchhavritta (Convex circular area, like a tortoise shell)
- Bahishchakrawal Vritta (Outer circular segment)
- Antahchakrawal Vritta (Inner circular segment)
Other Shapes: The Ganitasarasangraha also mentions:
- Hastidanta Kshetra (Elephant tusk shape)
- Yavakara Kshetra (Barley grain shape)
- Murjaka-kara Kshetra (Mridanga/drum shape)
- Panavaka-kara Kshetra (Panava drum shape)
- Vajrakara Kshetra (Diamond/rhombus shape)
- Ubhayishedha Kshetra (A shape defined by exclusion of two areas)
- Ek Nishedha Kshetra (A shape defined by exclusion of one area)
- Areas bounded by three and four mutually touching circles.

Units of Measurement:

The Anuyogadwarasutra mentions three units of measurement: Suchyangula, Pratyrangula, and Ghanangula, corresponding to length, area, and volume, respectively.

Circumference and Circle Mathematics (Vritta Ganit):

Terms: The Tattvarthadhigama Sutra Bhashya defines terms like circumference (vritta parikshep), chord (jya), diameter (vishkambh), arrow (ishu), bow (dhanusha), and radius (vishkambh).
Formulas: Several formulas for circle calculations are presented from Tattvarthadhigama Sutra Bhashya, Tiloyapannatti, and Jambudvipaprajnapti, covering:
- Circumference of a circle (using √10 for diameter)
- Area of a circle
- Calculating chords, arrows, and bows.
- Formulas for the slant height of a frustum of a cone.
Approximation of Pi (π): The chapter highlights the different values used for π in Jain texts:
- √10 (used in Suryaprajnapti, Jyotishkarandak, Tattvarthadhigama Sutra Bhashya)
- √10 and 3.16 (in Bhagavati Sutra)
- 3.16 (in Bhagavati Sutra)
- Slightly more than 3 (in Jambudvipaprajnapti, Uttaradhyayana Sutra)
- 355/113 (by Virasenacharya in the 10th century, described as unique and accurate)
- 3 (for approximate calculations, and √10 for precise calculations, by Mahaviracharya)
- 3 (for approximate, and √10 for precise calculations, by Nemichandra)
- (16/9)² (in Trilokyasara)

Area Formulas:

Formulas are provided for the area of:
- Circle (from Tattvarthadhigama Sutra Bhashya, Tiloyapannatti, Ganitasarasangraha)
- Trapezoid
- Annular ring (ring shape)
- Bow-shaped area (chapa-kshetra)
- Conical shape (kambuka-vritta)
- Concave and convex areas.
Approximate vs. Exact Area Calculations: Ganitasarasangraha distinguishes between approximate (sannikat) and precise (sukshma) area calculations for shapes like triangles and quadrilaterals.
- Triangle area formulas (Heron's formula is presented).
- Quadrilateral area formulas (including Brahmagupta's formula for cyclic quadrilaterals and a general formula).
- Formulas for ring shapes (nemikshetr), elephant tusk shapes.
- Approximation for circular area using π ≈ 3 and precise calculation using π ≈ √10.
- Ellipse area and circumference formulas.
- Formulas for conical and cylindrical shapes.
- Formulas for concave and convex areas.
Areas related to touching circles: Formulas are given for the area enclosed by touching circles.

Volume Formulas:

General Volume: Volume = Base Area × Height (for prisms and cylinders).
Specific Volume Formulas:
- Volume of a cube
- Volume of a cuboid (rectangular prism)
- Volume of a cylinder
- Volume of a parallelepiped
- Volume of a cone
Special Volumes from Jain texts:
- Volume of a mat-like shape (vetrasana sadṛśa kshetra).
- Mahaviracharya discussed volume under "Khaat Vyavahara," categorizing them as karmantik, aundra, and sukshma (precise) volumes.
- Formulas for the volume of pits and trenches are discussed.
- Volume of a frustum of a pyramid/cone.
- Volume of a sphere.
- Volume of a pyramid with a triangular base.
Formulas in Gommatasara: Formulas for prisms, cones, and spheres are presented.

Conclusion:

The chapter concludes by reiterating the significant and often overlooked contribution of Jain scholars to the field of geometry, not just in India but globally.
It emphasizes that fundamental elements of mathematics like triangles, squares, quadrilaterals, ellipses, and parabolas owe much to Jain scholars for their origin and development.
The text highlights that Jain scholars simplified complex problems in circle mathematics that are still challenging for modern mathematicians.
Mahaviracharya is particularly praised for his contributions to ellipse study and for presenting formulas similar to Brahmagupta's but with greater clarity.
The unique shapes like yavakara, murjaka-kara, panavaka-kara, and vajrakara are presented as special contributions from Jain scholars.
The importance of the value of π is emphasized, with the accurate value of 355/113 provided by Virasenacharya being a standout contribution that matches modern findings.
In essence, Jain scholars are credited with making the "dry" subject of geometry accessible, understandable, and natural through their comprehensive and insightful work.

This summary covers the main points discussed in the provided text, highlighting the depth and breadth of geometrical knowledge present in Jain literature.