Algebra and Analytical Engines

In 1827, John Herschel wrote enquiring whether Charles Babbage (1791-1871) [8] wished to be considered for the Savilian chair of mathematics at Oxford University, which was then vacant. Babbage's father had just died and on 12th February Georgina Babbage, his wife, replied saying:

It is seldom that I cannot say a priori what Charles would like or would not like, but with regard to this professorship I cannot judge. Before his father's death he would have taken it, but whether this event will make him more or less desirous of obtaining it, without more knowledge of our affairs I cannot say.

In the event, the appointment of the professorship had already effectively been decided. Babbage did not apply and Baden Powell was offered the Chair. Instead Babbage became Lucasian Professor of Mathematics at Cambridge, a post previously occupied by Sir Isaac Newton. He was influential in introducing the continental European notations of the German mathematician Gottfried Wilhelm Leibniz (1646-1716) into Britain. The Newtonian notation prevalent until then had been detrimental to the progress of mathematics compared to the rest of Europe (e.g., the use of the Newtonian `dot' notation for differential calculus instead of the more powerful Leibniz `d's notation). But if Cambridge mathematics was in a poor state at the time despite its renown, Oxford mathematics was even worse!

By 1836, Babbage was already considering the design for an Algebra or Formula Engine, a generalised version of his more famous Analytical Engines. Ada Lovelace's notes for the paper by Luigi Federico Menabrea on Babbage's work state:

Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraic and analytic. This is an error .... The engine might develop three sets of results ... symbolic results ...; numerical results ...; and algebraical results in literal notation.

The potential symbolic results of Babbage's Analytical Engines are important in connection with their relation to modern computers. Ada Lovelace noted that:

... each circle at the top [of a column of wheels recording decimal digits] is intended to contain the algebraic sign + or -, ... In a similar manner any other purely symbolic results of algebraic processes might be made to appear in these circles.

On 10 July 1836, soon after Babbage had considered a punched-card input system, he recorded the following:

...this day I had a general but only indistinct conception of the possibility of making an engine work out algebraic developments ... I mean without any reference to the value of the letters ... My notion is that as the cards (Jacquards) of the calc. engine direct a series of operations and then recommence with the first so it might perhaps be possible to cause some of the cards to punch others equivalent to any given number of repetitions. But these hole[s] might perhaps be small pieces of formula previously made by the first cards and possibly some mode might be found of arranging such detached parts.

Next year, on 13 December 1837, Babbage noted the following in one of his sketchbooks:

On machinery for the Algebraic Development of Functions

About this date the idea of making a development engine arose with considerable distinctness. It is obvious that if the Calculating Engine could print the substitutions which it makes in an Algebraic form we should arrive at the algebraic development `it can print all those substitutions which are noted in the composition in some of the notations' - it will however be better to construct a new engine for such purposes.

I.e., he was considering the construction of an engine for general algebraic operations and printing formulae resulting from calculations on such a machine. These ideas may be regarded as an extension of the Analytical Engines, but were not developed further by Babbage.

Towards the end of his life Babbage asked Harry Wilmot Buxton to write his biography. An early draft, together with other papers Babbage lent him whilst undertaking this task, are now held as part of the Buxton collection at the Museum of the History of Science in Oxford.