Application to the separation of the roots of an equation of the third degree.--Having the results of the substitution of -1, 0, and +1, to deduce therefrom, by means of differences, those of all other whole numbers, positive or negative.--The progress of the calculation leads of itself to the limits of the roots.--Graphical representation of this method. Substitution of numbers equidistant _by a tenth_, between two consecutive whole numbers, when the inspection of the first results has shown its necessity.--This substitution is effected directly, or by means of new differences deduced from the preceding. How to determine, in continuing the approximation towards a root, at what moment the consideration of the first difference is sufficient to give that root with all desirable exactness, by a simple proportion. The preceding method becomes applicable to the investigation of the roots of a transcendental equation X = 0, when there have been substituted in the first member, numbers equidistant and sufficiently near to allow the differences of the results to be considered as constant, starting from a certain order.--Formulas of interpolation. Having obtained a root of an algebraic or transcendental equation, with a certain degree of approximation, to approximate still farther by the method of Newton. * * * Resolution of two numerical equations of the second degree with two unknown quantities. Decomposition of rational fractions into simple fractions. IV. TRIGONOMETRY.