151 



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. 



,,>. 25 + 33 + 41 ...... __ 7 , 



49 + 57 + 65 + 73 .. .. J ' " _ g3 



81 + 89 + 97+105+113 .. | ' ' =11 a 

 121 + 129+137 + 145 + 153 + 16r ' 



It is evident from the preceding arrangement that 



Thus any triangular number multiplied by 8 with 1 added is equal 

 to the square of an odd number; or, any square of an uneven 

 number minus 1 is divisible by 8, and the quotient is a triangular 

 number. 



3. 

 Of the higher powers I will confine myself to one example. 



H. 



Every fourth power w 4 is the sum of an arithmetical progression 

 of n terms, the first term of which is 2 , and the difference 2 2 . 



1 .............. = 1* 



4+12 ............ = 2 4 



9 + 27+45 .......... =3* 



16+48 + 80+112 ........ =4 4 



25 + 75 + 125 + 150+225 ...... =5 4 



36+108+180 + 252 + 324 + 396 .... =6^ 



This triangle consists of the progressions in (D.) multiplied re- 

 spectively by n, or of those in (A.) multiplied by w 2 . 



