294 
[763 
763. 
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvn. (1881), 
pp. 258—276.] 
A sum of 2 squares multiplied by a sum of 2 squares is a sum of 2 squares; a 
sum of 4 squares multiplied by a sum of 4 squares is a sum of 4 squares; a sum 
of 8 squares multiplied by a sum of 8 squares is a sum of 8 squares; but a sum 
of 16 squares multiplied by a sum of 16 squares is not a sum of 16 squares. These 
theorems were considered in the paper, Young, “ On an extension of a theorem of Euler, 
with a determination of the limit beyond which it fails,” Trans. R. I. A., t. xxi. (1848), 
pp. 311—341; and the later history of the question is given in the paper by Mr S. 
Roberts, “ On the Impossibility of the general Extension of Eulers Theorem &c.,” Quart. 
Math. Jour. t. xvi. (1879), pp. 159—170; as regards the 16-question, it has been 
throughout assumed that there is only one type of synthematic arrangement (what this 
means will appear presently); but as regards this type, it is, I think, well shown that 
the signs cannot be determined. It will appear in the sequel, that there are in fact 
four types (the last three of them possibly equivalent) of synthematic arrangement; and 
for a complete proof, it is necessary to show in regard to each of these types that the 
signs cannot be determined. The existence of the four types has not (so far as I am 
aware) been hitherto noticed; and it hence follows, that no complete proof of the 
non-existence of the 16-square theorem has hitherto been given. 
For the 2 squares the theorem is of course 
(Vi + Vi) = Ui2/i + x -iyf + ( x dJ-2 ~ oOiVif- 
For the 4 squares (for which the nature of the theorem is better seen) it is 
Ui 2 + X Z + X Z + x i) {yi + yi + y-? + yf) = (x x y x + x 2 y 2 + x 3 y s + x 4 y 4 ) 2 
+ Ui2/ 2 - v-dJi + x z y 4 ~ x ±yf 
+ Ui2/s - x zV\ ~ 
+ Ui2/4 - + x -iy-i ~ x Ahf ;