Decomposition of the trinomial _x^{2} + px + q_ into factors of the first degree.--Relations between the coefficients and the roots of the equation _x^{2} + px + q = 0_. Trinomial equations reducible to the second degree. Of the maxima and minima which can be determined by equations of the second degree. Calculation of the _arithmetical_ values of radicals. Fractional exponents.--Negative exponents. _Of series._ Geometrical progressions.--Summation of the terms. What we call a series.--Convergence and divergence. A geometrical progression is convergent, when the ratio is smaller than unity; diverging, when it is greater. The terms of a series may decrease indefinitely and the series not be converging. A series, all the terms of which are positive, is converging, when the ratio of one term to the preceding one tends towards a _limit_ smaller than unity, in proportion as the index of the rank of that term increases indefinitely.--The series is diverging when this _limit_ is greater than unity. There is uncertainty when it is equal to unity. In general, when the terms of a series decrease indefinitely, and are alternately positive and negative, the series is converging. * * * Combinations, arrangements, and permutations of _m_ letters, when each combination must not contain the same letter twice. Development of the entire and positive powers of a binomial.--General terms. Development of _(a + b √-1)^{m}_. Limit towards which _(1 + 1/m)^{m}_ tends, when _m_ increases indefinitely. Summation of piles of balls. _Of logarithms and of their uses._ All numbers can be produced by forming all the powers of any positive number, greater or less than _one_. General properties of logarithms. When numbers are in geometrical progression, their logarithms are in arithmetical progression. How to pass from one system of logarithms to another system. Calculation of logarithms by means of the series which gives the logarithm of _n + 1_, knowing that of _n_.--Calculation of Napierian logarithms.--To deduce from them those of Briggs. Modulus. Use of logarithms whose base is 10.--Characteristics.--Negative characteristics. Logarithms entirely negative are not used in calculation. A number being given, how to find its logarithm in the tables of Callet. A logarithm being given, how to find the number to which it belongs.--Use of the proportional parts.--Their application to appreciate the exactness for which we can answer. Employment of the sliding rule. Resolution of exponential equations by means of logarithms. Compound interest. Annuities. _Derived functions._ Development of an entire function F(_x + h_) of the binomial (_x + h_).--Derivative of an entire function.--To return from the derivative to the function.