Introduction.—Every proposition in the theories of ratio and proportion is true for all descriptions of magnitude. Hence it follows that the proper treatment is the Algebraic. It is, at all events, the easiest and the most satisfactory. Euclid’s proofs of the propositions, in the Theory of Proportion, possess at present none but a historical interest, as no student reads them now. But although his demonstrations are abandoned, his propositions are quoted by every writer, and his nomenclature is universally adopted. For these reasons it appears to us that the best method is to state Euclid’s definitions, explain them, or prove them when necessary, for some are theorems under the guise of definitions, and then supply simple algebraic proofs of his propositions. i. A less magnitude is said to be a part or submultiple of a greater magnitude, when the less measures the greater—that is, when the less is contained a certain number of times exactly in the greater. ii. A greater magnitude is said to be a multiple of a less when the greater is measured by the less—that is, when the greater contains the less a certain number of times exactly. iii. Ratio is the mutual relation of two magnitudes of the same kind with respect to quantity. iv. Magnitudes are said to have a ratio to one another when the less can be multiplied so as to exceed the greater. These definitions require explanation, especially Def. iii., which has the fault of conveying no precise meaning—being, in fact, unintelligible. The following annotations will make them explicit:— 1. If an integer be divided into any number of equal parts, one, or the sum of any number of these parts, is called a fraction. Thus, if the line AB represent the integer, and if it be divided into four equal parts in the points C, D, E, then AC is ; AD, ; AE, . Thus, a fraction is denoted by two numbers parted by a horizontal line; the lower, called the denominator, denotes the number of equal parts into which the integer is divided; and the upper, called the numerator, denotes the number of these equal parts which are taken. Hence it follows, that if the numerator be less than the denominator, the fraction is less than unity. If the numerator be equal to the denominator, the fraction is equal to unity; and if greater than the denominator, it is greater than unity. It is evident that a fraction is an abstract quantity—that is, that its value is independent of the nature of the integer which is divided.