PROCEEDINGS. 
581 
Mr. Martin gave a proof of the proposition that a series of n 
squares whose sum is a square can always he found. The ap¬ 
plication of the formulas expressing this proof was illustrated 
for the cases in which n= 2 and n— 3, and the geometrical con¬ 
structions corresponding to these and the general case were 
pointed out. 
When the number of squares to be found is large, the tenta¬ 
tive process explained by Mr. Martin in his communication on 
wth-power numbers whose sum is an nth power, at the 35th 
meeting of the Section, is found to be more convenient. By this 
process many series were derived. A few of these are as follows : 
1. The sum of the squares of the natural numbers from 1 to 
24, both inclusive, is 70 2 . This result is of historical interest in 
being the only known series of squares of consecutive natural 
numbers, from 1 upward, whose sum is a square. It is the 
solution of a prize problem proposed in the Ladies’ Diary in 
1792. 
2. Several series of 50 squares whose sum is 231 2 . 
3. Several series of 1,000 squares whose sum is a square. 
4. One series of 999,995 squares whose sum is a square. 
5. Two series of 1,000,000 squares whose sum is a square. 
6. Several series of squares of consecutive numbers whose sum 
is a square. 
38th Meeting. February 22, 1888. 
The Chairman presided. 
Present, nineteen members and guests. 
The minutes of the 37th meeting were read and approved. 
Mr. G. W. Hill read a paper on The Interior Constitution of 
the Earth as Respects Density. 
This paper was published inextenso in vol. iv, No. 1, of Annals 
of Mathematics, University of Virginia, Va., 1888. 
Mr. H. A. Hazen presented a paper on A Failure in the Ap¬ 
plication of the Law of Probabilities. 
