380 Trans. Acad. Sci. of St. Louis. 
case marked by small dots in the middle. The resulting 
areas are 2’, 3°, and 47. This may be the way in whieh 
the Greek geometers arrived at the results in Table I of 
this paper. 
If the area marked 2? in Fig. 1 is multiplied by 2, 
the resulting area will be half the area of the square 
marked 4°, or 2*. If this area is again multiplied by 2, 
the resulting area will be equal to that of the square 
marked 4. It will be 2%. It has long been known that 
2*==4?. At the bottom of this plate is a square area, 
having within it a rectangular area whose sides have a 
length 4and1. 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 47. If this area be again multi- 
plied by 4, the resulting area will be 4°. 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 wil! 
have sides 1 and 4. Its area will be 22. 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°. Continuing the operation we shall find that the 
area of the square marked 4° is also 2°. By reference to 
the diagram above it will appear that the resulting area 
is also 8?. 
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° 
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°. 
Added 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. 
