Mathematics in Ancient and Medieval India c8.s3.85

c8.s3.85

first quadrant be divided into n equal divisions each being dx. Let p₁, p₂, p₃_n be the sine chords for parts of the arc x corresponding to parts of the arc represented by dx, 2dx, 3dx... . Let p'₁, p'₂, p'₃... be the corresponding sine differences. Then p'₁ = p₁ = R sin x $p'₁ - p'₂ = R sin x. ((dx)²)/(R²) = p₁ ((dx)²)/(R²)$ $or p'₂ = p'₁ - p₁ ((dx)²)/(R²)$ $= p₁ - p₁ ((dx)²)/(R²)$ $p'₃ = p'₂ - p₂ ((dx)²)/(R²) = p₁ - p₁ ((dx)²)/(R²) - p₂ ((dx)²)/(R²)$ $$ $$ Similarly, $p'_n = p₁ - p₁ ((dx)²)/(R²) - p₂ ((dx)²)/(R²) - - p_n-1 ((dx)²)/(R²)$ $p'₁ + p'₂ + p'_n &= np₁ - [(n-1)p₁ + (n-2)p₂ + / & + 2p_n-2 + p_n-1] ((dx)²)/(R²)$ or p_n = sum of all these sine differences $&= np₁ - [(n-1)p₁ + (n-2)p₂ + ... / & + 2p_n-2 + p_n-1] ((dx)²)/(R²)$ $&= whole arc - [(n-1)p₁ + (n-2)p₂ + / & + p_n-1] ((dx)²)/(R²)$