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KANSAS ACADEMY OF SCIENCE. 



HARMONIC FOEMS.— THEIR CONSTRUCTION REDUCED 



TO A SCIENCE. 



By Beenaed B. Smyth, Topeka. 



PART I. PLANIMETRIC FORMS. 



1, It is possible to so arrange arithmetical series, either regular or varying, in 

 any regular form that, upon addition of the numbers in any stated direction, 

 eaual or proportional sums will be obtained. 



" 2. Trigonal, square, and other figurate series may be arranged in circles, 

 hexagons, and other forms so that addition in various directions will give equal 

 or proportional sums. 



3, Geometric series may be arranged, like arithmetical series, so that con- 

 tinued products in any direction will equal continued products of an equal num- 

 ber of factors in any other direction. 



In arithmetical series, among the forms that have been successfully tried are 

 pquares, rectangles, triangles, hexagons, octagons, four-, five-, six-, and eight- 

 pointed stars, circles, ellipses, cubes, parallelopipeds, prisms, pyramids, cylin 

 ders, ellipsoids, and spheres. 



The principles of construction, so far as relate to plane forms, are here shown: 



SECTION I. SQUARES. 

 Square of Tiuo. 

 The smallest square that may be attempted is one of four places. It gives 

 equal sums only twice — either vertically, horizontally, or diagonally, according 

 to arrangement, thus : 



1 4 



2 3 



1 2 

 3 4 



As to differences, there are three differences of one unit each between two 

 contiguous numbers, namely : between 1 and 2, between 2 and 3, and between 3 

 and 4 ; two differences of two units each, namely, between 1 and 3, and between 

 2 and 4; and one difference of three units, namely, between 1 and 4. These dif- 

 ferences perform an important part in the formation of higher squares, as of 6, 



io, etc. 



As to laying the four numbers, there are twenty-four different ways of laying 

 them, thus : When 1 is placed in the upper left-hand corner, there is a choice of 

 three squares in which 2 may be laid ; 1 and 2 being laid, there is a choice of 

 two squares in which 3 may be placed, and then there is but one place for 4. 

 Multiplying these (3, 2, and 1) together gives 6 as the possible arrangements 

 when 1 is placed in a certain corner ; but there are four corners : Hence there 

 are twenty-four different arrangements by which these four numbers may be 

 laid. 



This square is not harmonic, and is introduced here because it is a factor in 

 the construction of harmonic squares of six, ten, fourteen, etc., places on a side 



