Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c13.54

c13.54

The next instance, which will be here noticed, is the general solution of indeterminate problems of the first degree. It was first given among moderns by Bachet de Meziriac in 1624.⁸³ Having shown how the solution of equations of the form ax-by = c is reduced to ax-by = ±1, he proceeds to resolve this equation: and prescribes the same operation on a and b as to find the greatest common divisor. He names the residues c, d, e, f, & c. and the last remainder is necessarily unity: a and b being prime to each other. By retracing the steps from e ± 1 or f ± 1 (according as the number of remainders is even or odd) e ± 1 = ε, ed ± 1 / e = δ, δc μ 1 / d = γ, γb ± 1 / c = β, βα μ 1 / b = a or f ± 1 = ζ, ζe μ 1 / f = ε, ed ± 1δ / e = , & c.