Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/292

268 must express each of these dimensions in terms of the same unit; we may say that the rectangle measures 36 inches by 18 inches, and contains 36 times 18 square inches, that is, 648 square inches.

Thus universally, if one side of a rectangle contain a unit of length an exact number of times, and if an adjacent side of the rectangle also contain the same unit of length an exact number of times, the product of these numbers will be the number of square units contained in the rectangle.

Next suppose we have a square, and let its side be 5 inches in length. Then, by our rule, the area of the square is 5 times 5 square inches, that is 25 square inches. Now the number 25 is called in Arithmetic the square of the number 5. And universally, if a straight line contain a unit of length an exact number of times, the area of the square described on the straight line is denoted by the square of the number which denotes the length of the straight line.

Thus we see that there is in general a connexion between the product of two numbers and the rectangle contained by two straight lines, and in particular a connexion between the square of a number and the square on a straight line; and in consequence of this connexion the first ten propositions in Euclid's Second Book correspond to propositions in Arithmetic and Algebra.

The student will perceive that we speak of the square described on a straight line, when we refer to the geometrical figure, and of the square of a number when we refer to Arithmetic. The editors of Euclid generally use the words "square described upon" in 1. 47 and I. 48, and afterwards speak of the square of a straight line. Euclid himself retains throughout the same form of expression, and we have imitated him.

Some editors of Euclid have added Arithmetical or Algebraical demonstrations of the propositions in the second book, founded on the connexion we have explained. We have thought it unnecessary to do this, because the student who is acquainted with the elements of Arithmetic and Algebra will find no difficulty in supplying such demonstrations himself, ''so far as they are usually given. We say so far as they are usually given, because these demonstrations usually imply that the sides of rectangles can always be expressed exactly in terms of some unit of length; whereas the student will find hereafter that this is not the case, owing to the existence of what are technically called incommensurable'' magnitudes. We do not enter on this subject,