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KANSAS ACADEMY OF SCIENCE. 



CUBIC SERIES. 



Terms: 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th, etc. 



666666 6 6 6 Constant. 



(1) 6 12 18 24 30 36 42 48 54 Sextate Series. 



(2) 1 7 19 37 61 91 127 169 217 271 Hexagonal Series. 



(3) 1 8 27 64 125 216 343 512 729 1000 Cubic Series. 



(3) 1 9 36 100 225 441 784 1296 2025 3025 Tower Series. 



The cubic series is regular hexahedral, and differs from the hexahedral series 

 before given, as explained under the head of "trigonal series." 



The "tower" series represents piles of cubes of successively deci easing sizes. 

 The tower series is equal to the square of the trigonal series. 



To show how much the formation of squares and cubes depends on the tri- 

 gonal and tetrahedral series, we will analyze a few of each: 



SQUARES. 



1 = 



1X8 = 



2X8 = J6 



25 = 

 3X8 = _24 



49 = 

 4X8 = ^ 



81 = 

 5X8 = JtO 



121 = 



6X8 = ^ 



169 = 

 7X8 = _56 



225 = 

 8X8 = _64 



289 = 

 9X8= 72 



361 = 

 10X8= 80 



441 = 

 11X8= 88 



529 = 

 12X8= 96 



625 = 



13 X 8 = 104 



729 = 



14 X 8 = 112 



841 = 



15 X 8 = 120 



961 = 



16 X 8 = 128 



0-= 



0X8) + 1= 1'^ 

 _J. 2-= 



1X8) + 1= 3-^ 



_2 4-= 



3X8) + 1= 5'^ 

 _3 6-= 



6X8) + 1= V 

 _4 8— 



10X8) + 1= 9^ 

 __5 10^= 



15X8)+l = ir 

 _6 12-= 



21 X 8) + 1 = 13- 

 7 14-= 



"28 X 8 ) + 1 = 15' 

 _8 16'^= 



"36 X 8 ) + 1 = 17^ 

 __9 18^= 



45 X 8 ) + 1 = 19^ 

 10 20^= 



.55 X 8 ) + 1 = 21^ 

 _11 22^= 



66 X 8 ) + 1 = 23'^ 

 J3 24-= 



78 X 8 ) + 1 = 25'^ 

 JL3 26'^= 



91X8) + 1 = 272 

 14 28^= 



105 X 8 ) + 1 = 29^ 

 _15 30'^= 



120X8) + 1 = 3P 

 32-= 



Here we see the constant 8 multip 



0X8) + ( 0X4)= 

 _0 _4 = ( 0X8) + 4 



0X8) + ( 1X4)= 4 

 J. _12 = ( lX8) + 4 



lX8) + ( 2X4)= 16 

 _2 ^ = ( 2X8) + 4 



3X8) + ( 3X4)= 36 



3 28: 



6X8) + ( 4X4) 

 4 



_=( 3X8) + 4 

 04 

 _ ^ = ( 4X8) + 4 



10X8) + ( 5X4)= 100 

 _5 _44 = ( 5X8) + 4 



15X8) + { 6X4)= 144 

 __6 _52 = ( 6X8)+4 



21X8) + ( 7X4)= 196 

 _7 _60 = ( 7X8) + 4 



28X8) + ( 8X4)= 256 

 _8 _68 = ( 8X8) + 4 



36X8) + ( 9X4)= 324 

 _9 _76 = ( 9X8) + 4 



45 X 8 ) + ( 10 X 4 ) = 400 



10 84 = ( 10 X 8 ) + 4 



55X8) + (11X4; 

 11 



484 

 92 = (11X8) +4 



G6 X 8 ) + ( 12 X 4 ) = 576 



12 100 : 



78X8) + (13X4; 

 13 



_ = (12X8) + 4 



676 



108 = (13X8) + 4 



91X8) + (14X4)= 784 

 J4 116 = (14X8) + 4 



105X8) + (15X4)= 900 



J5 124 = (15X8) + 4 



120X8) + (16X4)=1024 



lied by the successive terms of the trigonal 

 series, plus a certain number which is either constant ( 1 in the odd squares ) or 

 bears a constant relation to the root. 



