INTRODUCTION

In the latter part of the last century, two rather different types of symbolic calculus emerged onto the mathematical scene, both with English roots. These were the umbral calculus, or “representative notation”, of John Blissard (1803-1875), and the operator calculus of Oliver Heaviside (1850-1925). The umbral calculus was intended to simplify and extend various types of algebraic manipulation in classical analysis, whilst the operator calculus was designed to facilitate the solution of certain kinds of differential equation. For many years, the only contact between the two seems to have been in cases of mistaken identity!

Typical and instructive examples of the early pioneering work can be found in Blissard (1861) and Heaviside (1893), although we should note that the term “umbra” was coined by J.J. Sylvester (1851), and apparantly first used in this connection by E.T. Bell (1929).

Remarkably, neither of the two inventors was a professional mathematician, Blissard being a country vicar and Heaviside an electrical engineer. Perhaps this helps to explain the strikingly similar histories of their respective methods.

Both at first appeared to be a success. But then the mathematical powers of the day began to level accusations of insufficient rigour, and the rejection that followed was sometimes accompanied by considerable disdain. The theories then lay dormant for a while, with only a few authors attempting to revive them for their own purposes. For example the umbral calculus was championed by Bell (1927), and the operator calculus by B. Van der Pol (1929). As a further indignity, other mathematicians who did find such symbolic methods useful often contrived to attribute them to incorrect sources!