The former arithmetical programme included the theory of _progressions_ and _logarithms_; the latter being deduced from the former. But the theory of logarithms is again deduced in algebra from exponents, much the best method. This constitutes an objectionable “_double emploi_.” There is finally no good reason for retaining these theories in arithmetic. The programme retains the questions which can be solved by making two arbitrary and successive hypotheses on the desired result. It is true that these questions can be directly resolved by means of a simple equation of the first degree; but we have considered that, since the resolution of problems by means of hypotheses, constitutes the most fruitful method really used in practice, it is well to accustom students to it the soonest possible. This is the more necessary, because teachers have generally pursued the opposite course, aiming especially to give their pupils direct solutions, without reflecting that the theory of these is usually much more complicated, and that the mind of the learner thus receives a direction exactly contrary to that which it will have to take in the end. “Proportions” remain to be noticed. In most arithmetics problems are resolved first by the method of “reduction to unity,” and then by the theory of proportions. But beside the objection of the “_double emploi_,” it is very certain that the method of reduction to unity presents, in their true light and in a complete and simple manner, all the questions of ratio which are the bases of arithmetical solutions; so that the subsequent introduction of proportions teaches nothing new to the pupils, and only presents the same thing in a more complicated manner. We therefore exclude from our programme of examination the solution of questions of arithmetic, presented under the special form which constitutes the theory of proportions. This special form we would be very careful not to invent, if it had not already been employed. Why not say simply “The ratio of M to N is equal to that of P to Q,” instead of hunting for this other form of enunciating the same idea, “M _is to_ N _as_ P _is to_ Q”? It is in vain to allege the necessities of geometry; if we consider all the questions in which proportions are used, we shall see that the simple consideration of the equality of ratios is equally well adapted to the simplicity of the enunciation and the clearness of the demonstrations. However, since all the old books of geometry make use of proportions, we retain the properties of proportions at the end of our programme; but with this express reserve, that the examiners shall limit themselves to the simple properties which we indicate, and that they shall not demand any application of proportions to the solution of arithmetical problems. PROGRAMME OF ARITHMETIC. Decimal numeration. Addition and subtraction of whole numbers.