The resolution of inequalities of the first degree with one or more unknown quantities, was added to equations of the first degree some years ago. We do not retain that addition. The equations of the second degree, like the first, must be very carefully given. In dwelling on the case where the coefficient of _x_^{2} converges towards zero, it will be remarked that, when the coefficient is very small, the ordinary formula would give one of the roots by the difference of two numbers almost equal; so that sufficient exactness could not be obtained without much labor. It must be shown how that inconvenience may be avoided. It is common to meet with expressions of which the maximum or the minimum can be determined by the consideration of an equation of the second degree. We retain the study of them, especially for the benefit of those who will not have the opportunity of advancing to the general theory of maxima and minima. The theory of the algebraic calculation of imaginary quantities, given _à priori_, may, on the contrary, be set aside without inconvenience. It is enough that the pupils know that the different powers of √-1 continually reproduce in turn one of these four values, ±1, ±√-1. We will say as much of the calculation of the algebraic values of radicals, which is of no use. The calculation of their _arithmetical_ values will alone be demanded. In this connection will be taught the notation of fractional exponents and that of negative exponents. The theory of numbers has taken by degrees a disproportionate development in the examinations for admission; it is of no use in practice, and, besides, constitutes in the pure mathematics a science apart. The theory of continued fractions at first seems more useful. It is employed in the resolution of algebraic equations, and in that of the exponential equation _a_^{_x_}=_b_. But these methods are entirely unsuited to practice, and we therefore omit this theory. The theory of series, on the contrary, claims some farther developments. Series are continually met with in practice; they give the best solutions of many questions, and it is indispensable to know in what circumstances they can be safely employed.