Graduation by piecewise cubic polynomials: A historical review

SummaryThe idea of using overlapping piecewise linear arcs extending over nine successive ordinates px to graduate an observed series pxo is due to Finlaison (1829). The application to cubic arcs together with the mathematical theory of linear compounding of observed values at successive integer x-values was made by (canals of Mars) Schiaparelli (1866). Wittstein’s (1867) summation procedure in lieu of symmetrical compounds of ux-values was extended by Higham (1882) and was followed by extensive practical developments culminating in Vaughan (1933). Separate, but related, methods were initiated by Bohlmann (1899) and Karup (1899). The procedure known as “splines” stems from the foregoing researches but its different philosophy must be recognized. The article concludes with several graduations of the same mortality data.

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