सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

325
ie. x² (1397² + 4835²) - 2 × 4835 × 1397 × 5000 x = 1397² (R² - 5000²)
ie. x² (25328834) - 2 × 4835 × 1397 × 5000 x = 1397² (R² - 5000²)
∴ x² - (2 × 4835 × 1397 × 5000 x) / 25328834 = (1397² (R² - 5000²)) / 25328834
∴ x² - 2 × 5000 x × 6754495 / 25328834 = ,,
Converting 675 / 2533 into a continued fraction we have
1/(3 +) 1/(1 +) 1/3 = 4 / 15 so that the equation could be written as
x² - 2 × 5000 (x 4) / 15 = (1397² (R² - 5000²)) / 25328834
Here (5000 × 4) / 15 is symbolized as Ādya so that we have
x² - 2 Ādya x = (1397² (3438² - 5000²)) / 2532883
∴ (x - Ādya)² = Ādya² + (1397² (3438² - 5000²)) / 2532883
= 5000² × 16 / 225 - (5000² × 1397²) / 2532883 + (1397² × 3438²) / 2532883
= 5000² (16 / 225 - 1397² / 2532883) + (1397² × 3438²) / 2532883
Here 16 / 225 - 1397² / 2532883 is approximated to -2 / 337
and (1397² × 3438²) / 2532883 is approximated to 910678 so that we
have
x = Ādya ± √(910678 - (2 s²) / 337) where s is the given sum.
Since the positive sign of the radical is invalid because
H sin δ ≯ R, so the negative sign is taken.