380 Trans. Acad. Sci. of St. Louis. 



case marked by small dots in the middle. The resulting 

 areas are 2 2 , 3 2 , and 4 2 . This may be the way in which 

 the Greek geometers arrived at the results in Table I of 

 this paper. 



If the area marked 2 2 in Fig. 1 is multiplied by 2, 

 the resulting area will be half the area of the square 

 marked 4 2 , or 2 4 . If this area is again multiplied by 2, 

 the resulting area will be equal to that of the square 

 marked 4 2 . It will be 2 4 . It has long been known that 

 2 4 _4 2 . j^ fae bottom of this plate is a square area, 

 having within it a rectangular area whose sides have a 

 length 4 and 1. If this area be multiplied by 4, the result- 

 ing area will fill the lower left-hand quarter of the square. 

 The resulting area will be 4 2 . If this area be again multi- 

 plied by 4, the resulting area will be 4 3 . If we start with 

 the lower half of the rectangular area having sides 2 

 and 1, and multiply this area by 2, the resulting area will 

 have sides 1 and 4. Its area will be 2 2 . Multiplying 

 again by 2, the resulting area will be one-half the area of 

 the square in the lower left-hand corner of this diagram, 

 or 2 3 . Continuing the operation we shall find that the 

 area of the square marked 4 3 is also 2 6 . By reference to 

 the diagram above it will appear that the resulting area 

 is also 8 2 . 



Small cubes, the six faces of each having unit area, may 

 be placed on the 64 squares in the two lower diagrams of 

 Fig. 1. The volume upon each large square will be 8 2 

 units. If the cubes resting on the 48 unit areas in the 

 two upper quarters and the lower right quarter of the 

 large squares be superposed on the other quarter, the vol- 

 umes of the two cubes resulting will be 4 3 . 



Adaed Feb. 25, 1920. 



It will be observed that if the numeral 1 be dropped 

 from the exponents in Eqs. (4) and (5), these equations 

 become equations (1) and (2), the value a becoming unity. 



