Royal Society. 63 



1 + 2 + 3 + 4 + 5= 15 



2 + 3 + 4 + 5 + 6= 20 



3 + 4 + 5 + 6 + 7= 25 



4 + 5 + 6 + 7 + 8= 30 



5 + 6 + 7 + 8 + 9= 35 



125=5 3 



The sum of all the numhers contained in a square thus formed 

 is equal to the cube of the number which occupies the upper right- 

 hand and lower left-hand corners. The sum of the numbers in either 

 of the diagonals is the corresponding square, and in the case of the 

 odd numbers the sum of the middle horizontal or vertical line is also 

 the square. 



This last-mentioned relation was pointed out by Lichtenberg*, 

 who stated the theorem thus : — If a be a whole number, and A be the 

 sum of all the natural numbers from 1 to a, then : 



« 3 =A + (A + fl) + (A + 2a) + (A + 3a) + (A+[«-l]«). 



G. 



Every cube k 3 above 1 is the sum of an arithmetical progression 

 of n terms, the first term of which is (n — 2)-, and the difference=8. 



+ 8 = 2 3 



1 + 9 + 17 =3 3 



4 + 12 + 20 + 28 =4 3 



9 + 17 + 25 + 33 + 41 =5 3 



16 + 24 + 32 + 40 + 4S + 56 =6 3 



25 + 33 + 41+49 + 57 + 65 + 73 ....=7 3 

 Each progression of this triangle, consisting of an uneven number 

 of terms, contains two consecutive odd square numbers. 



An uninterrupted arithmetical progression commencing with unity 

 and proceeding by the constant addition of 8, arranged in a trian- 

 gular form, presents some curious results. 1st. The first terms of 

 each line are the squares of the odd numbers in their regular sequence. 

 2nd. The sum of all the numbers in any two adjacent lines is the 

 cube of an odd number. 



1 \ . . . . = 3 3 



9 + !7 ....= 5 3 



25 + 33 + 41 ' _ ?3 



49 + 57 + 65 + 73 1 !..'.' as 9' 



81 + 89 + 97+105 + 113 ..!" " _ u * 

 121 + 129 + 137 + 145 + 153 + 161 

 It is evident from the preceding arrangement that 



( o„ +1) = =1 + 8 (if"). 

 * (i. C. Lichtenbcrg's Vermischle Schriften, vol. ix. p. 359. Guttingen, 180G. 



