Wednesday, March 12, 2014

Subscripts and Nested Arrays

  The use of subscripts in mathematics came about as a effort to simplify the terms of a series. In 1827 Thomas Jarrett proposed a system of algebraic notation which included the subscript and superscript notation that is now used with Σ and Π for sums and products. An appendix on the system can be found in a 1829 work by Brooke. Prior to that a sequence of points was often indicated by letters given in alphabetical order and this was a common practice in Geometry. This can be found in the works of ancient mathematicians such as Euclid and Ptolemy. The use of Greek letters to designate a list of stars in a constellation was originated by Bayer in 1603 and refers to the position of stars in the heavens. In Ptolemy's star catalog the order of the stars are in order of longitude and latitude in a constellation. Bayer's notation is a lot in common with the index notation.

  The use of nested arrays in programming came about in the 1970's as an improvement of APL by I. P. Sharp Associates and was used with time sharing computers. This Canadian company also was an early developer of packet systems and email which are now used with cell phones and over the internet. A lot of computer programs use the function notation with parenthesis to designate the components of vectors and matrices such as v(i) and m(i,j) for example. Mathcad follows the mathematics convention of using subscripts.

  Placing objects in a package the way that the spheres and cylinders are nested in a vector for the ball and stick models is similar to the use of clay tokens by the Sumerians. With the use of nested arrays we don't have to just use numbers as the values of the components of an array. So, none of the ideas that I used for the ball and stick models are new except maybe the use of Mathcad to do them.

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