105 



however, we arrange a rectangle we shall find that this multi- 

 plication will give us the area'. 



We can now go a step further. We see that each square 

 inch is an inch long and an inch high, that is, each side of it is 

 an inch long. So it is evident that when we put four of them 

 in a row the bottom line of the row is four lineal inches long, 

 and that when we put three rows in height the side line is three 

 inches high. And as this lineal measure of length and height 

 represents the rectangle of Figure 1, whose area was twelve 

 square inches, we find out these two things : — First, that as 

 four lineal inches multiplied by three lineal inches give this 

 area of twelve square inches, square measure is the product of 

 the simple multiplication of two lineal measures ; and secondly, 

 that we need not count the rows of square inches in a rectangle, 

 nor the number of them in each row, but by simply measuring 

 with a rule the length and height of it, and multiplying these 

 together, we shall get the area of the rectangle. ' Thus we 

 have proved the correctness of the rule that multiplying the 

 length by the height will give the area of a rectangle. 



We have also illustrated some other things. For example : 

 if the rectangle be a square like Figure 2, it is evident that in 

 multiplying the length by the height we were multiplying a 

 number by itself— three by three. As this produces a square, 

 multiplying a number by itself is called squaring it, and the 

 product- of a number multiplied by itself is called its square. 

 Thus, nine is the square of three, twenty-five of five, and so 

 on. Again, as a square of an area of nine inches is based upon 

 a line of three inches, and is, as it were, grown upon it, three is 

 called the square root of nine, and in like manner five is the 

 square root of twenty-five, and so on. 



To revert to the rule for obtaining the area of a rectangle. 

 We proved its truth by applying square inches, which for the 

 present we have assumed as our standard measure, actually to 

 the rectangle. But as all our standards for measuring areas 

 are squares— square inches, square feet, square miles, and so 

 on — it is evident that we cannot in like manner prove the 

 correctness of a rule for measuring areas that are not rec- 

 tangular. For instance, we cannot cover Figure 3 with square 

 inches ; either some parts of the figure will be uncovered, or 

 some parts of the square inches unoccupied. How, then, can 

 we be sure that we are right when we say that such an 

 irregular figure contains so many square inches ? 



Let us take a, rectangle like Figure 4, and call it rectangle 

 A. If it is eight inches long and four inches high, we know 

 that its area is 8 X 4 = 32 square inches. Then let us divide 

 it into a square and two triangles by these models arranged as 

 in Figure 5 ; it is evident that the base of the square is four 



