To divide a sum into parts proportional to given numbers. Of questions which can be solved by two arbitrary and successive hypotheses made on the desired result. _Of the square and of the square root. Of the cube and of the cube root._ Formation of the square and the cube of the sum of two numbers.--Rules for extracting the square root and the cube root of a whole number.--If this root is not entire, it cannot be exactly expressed by any number, and is called incommensurable. Square and cube of a fraction.--Extraction of the square root and cube root of vulgar fractions. Any number being given, either directly, or by a series of operations which permit only an approximation to its value by means of decimals, how to extract the square root or cube root of that number, to within any decimal unit. _Of the proportions called geometrical._ In every proportion the product of the extremes is equal to the product of the means.--Reciprocal proportion.--Knowing three terms of a proportion to find the fourth.--Geometrical mean of two numbers.--How the order of the terms of a proportion can be inverted without disturbing the proportion. When two proportions have a common ratio, the two other ratios form a proportion. In any proportion, each antecedent may be increased or diminished by its consequent without destroying the proportion. When the corresponding terms of several proportions are multiplied together, the four products form a new proportion.--The same powers or the same roots of four numbers in proportion form a new proportion. In a series of equal ratios, the sum of any number of antecedents and the sum of their consequents are still in the same ratio. II. GEOMETRY Some knowledge of Geometry is, next to arithmetic, most indispensable to every one, and yet very few possess even its first principles. This is the fault of the common system of instruction. We do not pay sufficient regard to the natural notions about straight lines, angles, parallels, circles, etc., which the young have acquired by looking around them, and which their minds have unconsciously considered before making them a regular study. We thus waste time in giving a dogmatic form to truths which the mind seizes directly.