बौधायन-शुल्बसूत्र-रहस्यम्

अन्वय अथ अतः: इसके अनन्तर | याज्ञिकान् वेद्यादीन्: यज्ञ सम्बन्धी वेदियों को | मानतः परिमातुम्: मापदण्डों से नापने हेतु | शुल्बस्य प्रयोगम्: शुल्ब (रस्सी) के प्रयोग की | व्याख्यास्यामः: हम व्याख्या करते हैं।

English Hereafter we shall explain the practical application of the rope (śulba) for measuring altars and sacrificial spaces with exact proportions.

English 15 aṅgulas constitute a Pada; 24 aṅgulas make an Aratni (cubit); 120 aṅgulas make a Puruṣa (height of a man), which is the principal scaling module for Agnicayana altars.

अन्वय दीर्घचतुरस्रस्य: आयत (Rectangle) की | अक्ष्णया रज्जु:: विकर्ण रस्सी (Diagonal Rope) | पार्श्वमानी तिर्यग्मानी च: लम्बाई (Flank) तथा चौड़ाई (Horizontal side) | यत् पृथग्भूते कुरुतः: जो अलग-अलग वर्ग-क्षेत्रफल उत्पन्न करती हैं | तदुभयं करोति: वह विकर्ण रस्सी उन दोनों के योग के बराबर अकेला वर्ग बनाती है।

English The diagonal rope of a rectangle produces both areas which the flank (length) and the horizontal (breadth) sides produce separately.

\[ c^2 = a^2 + b^2 \]

English The diagonal rope of a square produces an area twice as large as the original square.

\[ d^2 = 2 a^2 \implies d = a\sqrt{2} \]

वैदिक पायथागोरीय त्रिक

• (३, ४, ५): \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\)
• (५, १२, १३): \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\)
• (८, १५, १७): \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\)
• (१२, ३५, ३७): \(12^2 + 35^2 = 144 + 1225 = 1369 = 37^2\)
• (१५, ३६, ३९): \(15^2 + 36^2 = 225 + 1296 = 1521 = 39^2\)

English The ropes for rectangles of sides (3, 4), (5, 12), (8, 15), and (12, 35) produce diagonals of 5, 13, 17, and 37 respectively. These exact integer triples were used to lay out right-angled altar corners without angle-measuring devices.

अन्वय समस्य द्विकरणी: वर्ग के विकर्ण (\(\sqrt{2}\)) हेतु | प्रमाणं तृतीयेन वर्धयेत्: भुजा-प्रमाण को उसके एक-तिहाई से बढ़ाएं (\(1 + \frac{1}{3}\)) | तच्च चतुर्थेन: फिर उस तिहाई के चौथाई भाग को जोड़ें (\(+ \frac{1}{3 \times 4}\)) | आत्मचतुस्त्रिंशोनेन सविशेषः: तथा उस चौथाई के ३४वें भाग को घटा दें (\(- \frac{1}{3 \times 4 \times 34}\))। यह सविशेष (विशिष्ट मान) प्राप्त होता है।

\[ \sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot 4 \cdot 34} = \frac{577}{408} \approx 1.41421568627\dots \]

गणितीय परिशुद्धता आधुनिक मान \(\sqrt{2} \approx 1.41421356237\dots\)। बौधायन का सन्निकटन दशमलव के ५ स्थानों तक पूर्णतः शुद्ध है (त्रुटि केवल \(0.0000021\))।

English The rope measuring \(\sqrt{3}\) (Trikaraṇī) is obtained by taking the rectangle of sides 1 and \(\sqrt{2}\). Its diagonal gives \(\sqrt{1^2 + (\sqrt{2})^2} = \sqrt{3} \approx 1.73205\).

वर्ग से वृत्त बनाना (Circling the Square)

English If you wish to transform a square into a circle: stretch half the diagonal from the center towards the east. One-third of the segment extending outside the square is added to half the side to form the radius of the desired circle.

\[ r = \frac{a}{2} + \frac{1}{3}\left(\frac{a\sqrt{2}}{2} - \frac{a}{2}\right) = \frac{a}{6}\left(2 + \sqrt{2}\right) \]

महावेदि मापन

English The Mahāvedi is a isosceles trapezoid: western back side = 30 prakramas, eastern front side = 24 prakramas, distance along east-west axis = 36 prakramas. Constructed using (5, 12, 13) and (12, 35, 37) triples.

श्येनचिति (Falcon/Hawk Altar)

English He who desires heaven shall construct the falcon-shaped altar (Śyena-citi). Total area = \(7\frac{1}{2}\) square puruṣas, built in 5 layers of 200 bricks each (1000 bricks total), preserving total area across complex polygonal components.

Python Verification Engine

# Python 3 Verification Suite for Baudhayana Sulba Sutra
import math

def test_baudhayana_math():
    # 1. Baudhayana Root 2 Formula: 1 + 1/3 + 1/(3*4) - 1/(3*4*34)
    b_root2 = 1 + (1/3) + (1/(3*4)) - (1/(3*4*34))
    true_root2 = math.sqrt(2)
    frac_root2 = 577 / 408
    
    print(f"[√2] Baudhayana Fraction : 577/408 = {frac_root2:.10f}")
    print(f"[√2] True math.sqrt(2)  : {true_root2:.10f}")
    print(f"[√2] Absolute Error      : {abs(b_root2 - true_root2):.10f}")
    
    # 2. Pythagorean Triples Check
    triples = [(3,4,5), (5,12,13), (8,15,17), (12,35,37)]
    for a, b, c in triples:
        assert a**2 + b**2 == c**2, f"Triple failed: {a},{b},{c}"
    print(f"[Triples] All {len(triples)} Vedic Pythagorean Triples verified ✓")

    # 3. Circling the Square Radius: r = (a/6)*(2 + sqrt(2))
    a = 10.0 # square side
    r = (a / 6.0) * (2.0 + math.sqrt(2))
    circle_area = math.pi * (r ** 2)
    square_area = a ** 2
    print(f"[Square to Circle] Square Area: {square_area}, Circle Area: {circle_area:.4f}")

test_baudhayana_math()