62 Royal Society. 



This, it will be observed, is a triangular arrangement of the uneven 

 numbers in their regular order. 



Every cube is the sum of as many consecutive odd numbers as 

 there are units in the root*. 



The known theorem, that the sum of the cubes of any succession 

 of the natural numbers commencing with unity is equal to the square 

 of the sum of the roots, or, in other words, to the square of the cor- 

 responding triangular number, is an immediate consequence of the 

 above. 



(l 3 +2 3 + 3 3 + 4 3 + k 3 ) = (1 + 2 + 3 + 4.. 



The sum of any series of odd numbers commencing with unity 

 being equal to the square of the number of terms (A.), the sum of 

 the numbers in any triangle formed as above is necessarily equal to 

 the square of a triangular number. It is also easy to see that each 

 cube is the difference between the squares of two consecutive trian- 

 gular numbers ; and, that the difference between the squares of any 

 two triangular numbers whatever is the sum of consecutive cubes. 

 The following equations have been found by ascertaining what dif- 

 ferences of the squares of two triangular numbers are equal to single 

 cubes : — 



3 3 + 4 s + 5 3 = 6 3 



11 3 + 12 3 +13 3 +14 3 =20 3 . 



F. 



Every cube n 3 is the sum of an arithmetical progression of n terms, 

 the first term of which is a triangular number — — , and the dif- 



ference=«. 



I =1 3 



3 + 5 =2 3 



G + 9 + 12 =3 3 



10+14 + 18 + 22 =4 3 



15 + 20 + 25 + 30 + 35 =5 3 



21 + 27 + 33 + 39 + 45 + 51 = 6 3 



2S + 35 + 42 + 49 + 56 + 63 + 70 ....=7 3 



Each number contained in this triangle is itself the sum of an 

 arithmetical progression of n terms. Thus, taking the fifth row for 

 example : — 



* Since the present note was communicated to the Royal Society, I have found 

 that this relation has been already noticed by Count d'Adhemar ( Comptes Jlendus, 

 torn, xxiii. p. 501). Cauchy observes, " quokju'elle puisse, comme on le voit, se 

 deduire dcs principes deja connus, toutefois, elle est assez curieuse ct ties simple." 



