Of the geometrical system which pervaded the Greek arithmetic, we have permanently retained only the words square and cube; rectangle was frequently used for product, but rarely at present. These words are the causes of much confusion to students who begin to apply arithmetic to geometry. Thus in algebra the square of a sum is equal to the sum of the squares of the two numbers, together with twice their product. In geometry the square or the sum of two lines is equal to the sum of the squares on the lines, together with twice their rectangle. Those who are not made to see clearly the distinction of these propositions confound them together. A sufficient distinc tion might be made by a little variation in phraseology : speak of the square on a line, and the square of a number. Thus 49 is the square of 7 : erect two perpendiculars each equal to A a at the two extremi ties of A a, and joining their other extremities completes the square on A B. It is already customary to speak of the rectangle whose con tiguous sides are A n and A e, as the rectangle under A n and a e.
, The second book of Euclid is devoted to the properties of the rectangle, as they arise from subdivision into other rectangles. Some persons advocate what is called the arithmetical proof of these propositions, namely, the substitution of the analogous properties of numbers for those of rectangular spaces. This question must he settled in the same manner as that of l'ACOFORTION, awl the remaike in that article apply. If all pairs of lines were commensurable, no objection could be taken against the rigour of the enbetitution ; but unless a theory of incommensurables, and a modification of • the definition of multiplication to suit them, be formally introduced, the method of Euclid is sound, and the substitute for it unsound; though proper enough for the adoption of those who, as explained in the article cited, only wish to become mathematicians to a certain number of decline! places.