Jinbhadragani Ke Ek Ganitya Sutra Ka Rahasya

Here's a comprehensive summary of the provided Jain text, "जिनभद्रगणि के एक गणितीय सूत्र का रहस्य" (The Mystery of a Mathematical Formula of Jinbhadragani) by Dr. Radhacharan Gupta:

This article, authored by Dr. Radhacharan Gupta, delves into a significant mathematical formula presented by the esteemed Jain Acharya Jinbhadragani Kshamashraman. Jinbhadragani, considered the tenth "Yugapradhan" (epochal leader) among Jains, lived around 600 CE. He is renowned for his "Visheshavasthak Bhashya," a commentary on the Samayikadhyayana section of the Avashyakasutra, completed in Shak 531 (609 CE) and comprising approximately 3600 Prakrit verses. Besides this major work, he is credited with numerous other texts and commentaries, including "Kshetravishayas" or "Brihatkshetravishayas," "Brihatsangrahani," "Jeetkalpa," "Dhyanashatak," "Nishithabhashya," and commentaries on the Pragyapana-sutra and Sharirapada.

The focus of this article is specifically on Jinbhadragani's "Brihatkshetravishayas," a work of 637 verses. The author notes that this text was commented upon by several prominent scholars like Haribhadra, Siddhasuri, Malayagiri, Vijaysingh, Devbhadra, Anandasuri, and Devanand, with Malayagiri's commentary being the only one published (in 1920-21 CE by the Jain Dharma Prasarak Sabha in Bhavnagar).

The core of the article lies in the analysis of a particular mathematical formula that Jinbhadragani cited in his "Brihatkshetravishayas" (Chapter 1, verse 122). This formula is designed to calculate the area of a circular segment formed between two parallel chords of a circle (where the segment is less than a semicircle). The author highlights that this formula is unique and its derivation has been a challenge for scholars until now.

The article presents the formula as follows: Let:

• a be the length of the smaller chord (AB).
• b be the length of the larger parallel chord (CD).
• h be the distance between the two parallel chords (LN).

According to Jinbhadragani's formula, the area (K) of the circular segment is given by: K = [√(a² + b²)] * h .....(1)

The author then proposes a simplified derivation for this formula. He acknowledges that the area of the trapezoid (ABHDCGA) formed by connecting the endpoints of the chords would be: T = ½ (a + b) h .....(2)

However, Jinbhadragani, aware that using the trapezoid's area would yield a result significantly less than the actual circular segment area, sought a formula that would provide a greater area. Formula (1) achieves this. The article explains the rationale behind the formula by comparing it to another known formula.

It references a well-established ancient formula relating a chord (c) and the sagitta (height of the segment, g) of a circle with radius R: 4g (2R - g) = c² .....(4)

This formula was also known to Jinbhadragani (Brihatkshetravishayas, Chapter 1, verse 36). Using this, the length of the chord (EF) exactly in the middle of the two parallel chords (AB and CD), passing through the midpoint (M) of their distance (LN), can be calculated as: (EF) = (a + b) / 2 + h²/ (a + b) .....(5) (Note: The provided text has a slight ambiguity in the formula presentation for EF. The interpretation here is based on standard geometric principles for the midpoint chord length in a segment defined by two parallel chords. The text implies a modification of the simple average.)

The author clarifies that using the length of a chord GH which is intermediate to the chords AB and CD, as implied by Jinbhadragani's approach to find a better approximation, is what leads to his formula. The formula (a + b²)/2 represents an effective average length, and multiplying it by the distance h yields Jinbhadragani's formula (1). This approach demonstrates Jinbhadragani's mathematical ingenuity in finding an intermediate value that provides a more accurate area than the simple trapezoidal approximation.

The article concludes by highlighting the significant role of mathematics in Jain scriptures, particularly in subjects like the division of space (lok-alok), the changes of epochs (yuga-parivartan), and the descriptions of different modes of existence (gati). It points to advanced mathematical concepts found in texts like Dhavala, Tiloyapannatti, Rajavartika, and Trilokasara, citing an example from Bhagavat Bhutabali's Dhavala regarding the extensive numerical system used by Jain Acharyas.

In essence, the article provides an insightful exploration of a unique and historically significant mathematical formula by Jinbhadragani, offering a simplified derivation and placing it within the broader context of Jain mathematical achievements.