Let us suppose again that we have thus recognized that the equation has a root comprised between 2.3 and 2.4; we will approximate still nearer by substituting intermediate numbers, differing by 0.01, and employing the course just prescribed. As soon as the third differences can be neglected, the calculation will be finished at once, by the consideration of an equation of the second degree; or, if it is preferred to continue the approximations till the second differences in their turn may be neglected, the calculation will then be finished by a simple proportion. When, in a transcendental equation _f_(X) = 0, we have substituted in _f_(X) equidistant numbers, sufficiently near to each other to allow the differences of the results to be neglected, commencing with a certain order, the 4th, for example, we may, within certain limits of _x_, replace the transcendental function by an algebraic and entire function of _x_, and thus reduce the search for the roots of _f_(X) = 0 to the preceding theory. Whether the proposed equation be algebraic or transcendental, we can thus, when we have obtained one root of it with a suitable degree of exactness, continue the approximation by the method of Newton. PROGRAMME OF ALGEBRA. _Algebraic calculation._ Addition and subtraction of polynomials.--Reduction of similar terms. Multiplication of monomials.--Use of exponents.--Multiplication of polynomials. Rule of the signs.--To arrange a polynomial.--Homogeneous polynomials. Division of monomials. Exponent _zero_.--Division of polynomials. How to know if the operation will not terminate.--Division of polynomials when the dividend contains a letter which is not found in the divisor. _Equations of the first degree._ Resolution of numerical equations of the first degree with one or several unknown quantities by the method of substitution.--Verification of the values of the unknown quantities and of the degree of their exactness. Of cases of impossibility or of indetermination. Interpretation of negative values.--Use and calculation of negative quantities. Investigation of general formulas for obtaining the values of the unknown quantities in a system of equations of the first degree with two or three unknown quantities.--Method of Bezout.--Complete discussion of these formulas for the case of two unknown quantities.--Symbols m/o and o/o. Discussion of three equations with three unknown quantities, in which the terms independent of the unknown quantities are null. _Equations of the second degree with one unknown quantity._ Calculus of radicals of the second degree. Resolution of an equation of the second degree with one unknown quantity.--Double solution.--Imaginary values. When, in the equation _ax^{2} + bx + c = 0_, _a_ converges towards 0, one of the roots increases indefinitely.--Numerical calculation of the two roots, when _a_ is very small.