Multiplication of whole numbers.--Table of Pythagoras.--The product of several whole numbers does not change its value, in whatever order the multiplications are effected.--To multiply a number by the product of several factors, it is sufficient to multiply successively by the factors of the product. Division of whole numbers.--To divide a number by the product of several factors, it is sufficient to divide successively by the factors of the product. Remainders from dividing a whole number by 2, 3, 5, 9, and 11.--Applications to the characters of divisibility by one of those numbers; to the verification of the product of several factors; and to the verification of the quotient of two numbers. Prime numbers. Numbers prime to one another. To find the greatest common divisor of two numbers.--If a number divides a product of two factors, and if it is prime to one of the factors, it divides the other.--To decompose a number into its prime factors.--To determine the smallest number divisible by given numbers. _Vulgar fractions._ A fraction does not alter in value when its two terms are multiplied or divided by the same number. Reduction of a fraction to its simplest expression. Reduction of several fractions to the same denominator. Reduction to the smallest common denominator.--To compare the relative values of several fractions. Addition and subtraction of fractions.--Multiplication. Fractions of fractions.--Division. Calculation of numbers composed of an entire part and a fraction. _Decimal numbers._ Addition and subtraction. Multiplication and division.--How to obtain the product of the quotient to within a unit of any given decimal order. To reduce a vulgar fraction to a decimal fraction.--When the denominator of an irreducible fraction contains other factors than 2 and 5, the fraction cannot be exactly reduced to decimals; and the quotient, which continues indefinitely, is periodical. To find the vulgar fraction which generates a periodical decimal fraction: 1º when the decimal fraction is simply periodical; 2º when it contains a part not periodical. _System of the new measures._ Linear Measures.--Measures of surface.--Measures of volume and capacity.--Measures of weight.--Moneys.--Ratios of the principal foreign measures (England, Germany, United States of America) to the measures of France. _Of ratios. Resolution of problems._ General notions on quantities which vary in the same ratio or in an inverse ratio.--Solution, by the method called _Reduction to unity_, of the simplest questions in which such quantities are considered.--To show the homogeneity of the results which are arrived at; thence to deduce the general rule for writing directly the expression of the required solution. Simple interest.--General formula, the consideration of which furnishes the solution of questions relating to simple interest.--Of discount, as practised in commerce.