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inches, for its height is four inches, and therefore that its area 

 j s 4 x 4 = 16 square inches. It is also evident that the base 

 and height of each of the triangles is four inches. The square 

 and the two triangles together cover the whole of the rectangle 

 A, and therefore the sum of their areas is equal to the area of 

 the rectangle. Now let us take another rectangle, B, equal to 

 A, and the same two triangles, and arrange them therein as 

 represented in Figure 6 with this sloping parallelogram 

 between them ; the sloping parallelogram and the two triangles 

 together cover the whole of the rectangle exactly, just as the 

 square and the triangles did. If we now from the equal 

 rectangles take the equal triangles, the remainders must be 

 equal, that is, the sloping parallelogram is equal to the square. 

 Wherefore the area of the sloping parallelogram is 16 square 

 inches. But the base and vertical height of the sloping 

 parallelogram are each four inches, as the base is the difference 

 between the base of the rectangle and that of one of the triangles, 

 8 — 4=4; and the vertical height of it is the height of the 

 rectangle. And as multiplying this base by this height, 

 four bv four, make 16, it is evident that the area of this sloping 

 parallelogram is to be found by the multiplication of its base 

 by its height. 

 " It is evident that whatever shaped parallelogram we may take, 

 wc can form .a rectangle by applying two equal right-angled 

 triangles to it as shown in the upper part of Figure 7. If we 

 then arrange the two triangles as shown in the lower part of 

 Figure 7, it is evident that we leave a smaller rectangle, that 

 must have a base, height, and area each respectively equal to 

 those of the sloping parallelogram in the upper part of the 

 figure, and that these equal areas are the product of the multi- 

 plication of the equal bases and heights. Consequently we can 

 extend the rule we have proved, and say of any parallelogram, 

 whether rectangular or not, that its area is equal to its base 

 multiplied by its height ; and have shown that we can apply 

 square measure to the measurement of things that are not 

 square. But it is to be noted how square measure asserts 

 itself by demanding that the base and height shall be measured 

 at right angles, or, popularly speaking, squarely to each other. 

 We can extend the principle we have thus established. If 

 two equal triangles, such as the models we have been using, be 

 placed together, so that one side of one coincides with the 

 equal side of the other, and the other equal sides are opposite 

 to each other, it is evident that they form a parallelogram, for 

 the opposite sides are equal. Thus these equal right-angled 

 triangles may be arranged to form either the parallelogram 

 shown in Figure 5, or that shown in Figure 8. And these 

 equal scalene triangles may be arranged to form the parallelo- 



