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KANSAS ACADEMY OF SCIEXCE. 



SQUARE SERIES. 



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



2 2 





1 3 



1 4 



1 5 



1 



.5 7 

 9 16 



14 

 5 



9 

 2.5 

 30 55 



2 



11 

 36 



13 

 49 



15 



64 



17 

 81 



91 140 204 285 



19 

 100 

 .385 



(3) 1 6 



14 

 19 44 



30 



85 



55 91 140 204 285 

 146 231 344 489 670 



Alternate Series. 

 Square Series. 

 Pyramidal Series. 



Octahedral Series. 



The units of the hexahedral and pyramidal series, though the same, are dif- 

 ferently arranged. For example: In the fifth pyramid the lower tier contains 

 5 X 5 = 25 units. The second tier, laid in the interstices on top of this, contains 

 4X4 = 16 units; the third tier, 9; the fourth, 4; and on top, 1; making in all, 55 

 units. 



Now compare the fifth hexahedron. At the bottom there is 1 unit. Just 

 above it there are 3 in a triangle. The next layer or tier has 6 in a triangle; the 

 fourth tier has 10; and the fifth, 15. Then they decrease to the top. The sixth 

 tier has 10, as has the fourth; the seventh has 6, like the third; the eighth has 3, 

 the same as the second; and on top is 1, as at the bottom. Thus there are 55 in 

 all, the same as in the pyramid. 



These forms, for school work, may be cut out of cardboard, potatoes, etc., or 

 molded from wax, clay, or other plastic substance, or built up of marbles or 

 wooden balls. A very good series may be made in outline by using wooden tooth- 

 picks, fastening together at the tips with small cubes of potato. In a few days 

 the bits of potato dry hard and hold the sticks firmly in position. 



The method of obtaining the square series from the alternate series, as above, 

 may be represented to the eye thus, in which the light circles show the successive 

 additions: 



There are several other methods of obtaining the square series by addition, 



thvis: 



Secondly. By adding to each term of the trigonal series the preceding 

 term, as: 



1 3 6 10 15 21 28 36 45 55 QQ . . . . 



Add 13 6 10 15 21 28 36 45 55 . . . . 







16 25 36 49 64 81 100 121 



