It is evident that in [fig. 16] four of the points determine a square, which square we may take as the unit of measurement for areas. But we can also measure areas in another way.
Fig. 16 (1) shows four points determining a square.
But four squares also meet in a point, [fig. 16] (2).
Hence a point at the corner of a square belongs equally to four squares.
Thus we may say that the point value of the square shown is one point, for if we take the square in [fig. 16] (1) it has four points, but each of these belong equally to four other squares. Hence one fourth of each of them belongs to the square (1) in [fig. 16]. Thus the point value of the square is one point.
The result of counting the points is the same as that arrived at by reckoning the square units enclosed.
Hence, if we wish to measure the area of any square we can take the number of points it encloses, count these as one each, and take one-fourth of the number of points at its corners.
Fig. 17.
Now draw a diagonal square as shown in [fig. 17]. It contains one point and the four corners count for one point more; hence its point value is 2. The value is the measure of its area—the size of this square is two of the unit squares.