431 
Now, if we write the number with the digits in reverse order on a slip of paper 
(S) and place it above the number itself as shown above, we see that the 
digits which occur in the third column of the partial products are the digits 
which come together, or correspond, when the slip of paper (S) is placed so 
that the first digit of the number, when digits are reversed, is placed over 
the third column. This is possible in no other scale, because the product 
of two digits in the binary scale does not give a number of more than one 
place. We can give the following rule for squaring a number written in the 
binary scale: 
Write the number with the digits equally spaced and write the same number 
with the digits reversed on a slip of paper, using the same spacing. Place the 
slip of paper above the number so that the first digit in the reversed order comes 
above the last digit of the number. Move the slip of paper a single space to the 
left each time. Count the correspondences at each step. The number of corres- 
pondences at each step is the number which belongs in that place in the result which 
is immediately beneath the first digit on the slip. Continue this until there are 
no more correspondences. 
It is easily seen that by means of the above rule the process described 
by Lucas can be followed out by counting the correspondences and will lead 
to the result in the lines marked (B) in Figure I, without having to write 
the part (A). 
It would be possible to construct a machine which would have two 
parallel bars in which could be set pins for the places where 1 occurs in the 
number. The pins on one bar would be in reverse order. The bars could 
be turned over and the number of pins striking could be recorded automati- 
cally. At the same time one bar could be moved along one place and be in 
readiness for the next turn. From the machine then would come the data 
for compiling the part (B) of Figure I. Or, a more complicated machine 
could be constructed which would give the part (C) at once. This would so 
shorten the work of testing the Mersenne numbers that it would be possible 
to check the results on all of them again with a reasonable expenditure of 
time. 
Purdue University, 
LaFayette, Indiana. 
