412 



Example I. The following values are the cubes of numbers 

 from 8-0 to 8-5. Find the cube of 8-23. 



8-5 614-125 



Let U Q = (8-2) 3 ; then (8-23) 3 will be denoted by u . s 

 and w . 3 = (1 + A)' 3 w 



= U Q + 0-3Aw - 0-105A 2 w + 0-0595A 3 w 



where Aw , A 2 w , and A 3 w are the successive differences running 

 diagonally downwards from u . 

 Then (8-23) 3 = 551-368 + 0-3 x 20-419 - 0-105 x 0-498 + 0-0595 x 0-006 



= 557-442 



Or taking U Q = (8-3) 3 , then (8-23) 3 will be denoted by w_ . 7 

 and w- 07 = (l-8)' 7 w 



= M - 0-7Sw - 0-105S 2 M ~ 0-0455S 3 w 



where 8w , 8 2 w , and 8 3 w are the successive differences running 

 diagonally upwards from u . 

 Then(8-23) 3 = 571-787 - 0'7 x 20-419 - 0-105 x 0-492 - 0-0455 x 0-006 



= 557-442 



Example 2. The foUowing values are the tangents of angles 

 from 70 to 77. Find tan 70 36' and tan 76 36'. 



