Charles Babbage's difference engine was designed to compute polynomial functions.  Since it could not do multiplication, it in fact calculated nearby values (based on a pre-determined initial value) using the method of finite differences - thus giving it its name, the "difference engine". 
What was remarkable about Babbage was that, in true geek fashion, he actually proposed to build one. 

There was considerable controversy about whether it would work - and it didn't, at least not until after his demise - when it functioned almost flawlessly, complete with the first "computer printer" - also designed by Babbage.

Charles Babbage fame as a geek, despite his failure to produce a working model of his difference engine during his lifetime, may be credited to Ada Byron, Lady Lovelace (daughter of the romantic poet Lord Byron), who emerged as the first geek 'publicist'.

In 1834 Babbage had already begun planning for a new type of calculating machine -- the "Analytical Engine", conjecturing a calculating machine that could not only foresee, but act.

When Babbage reported on his plans for this new "Analytical Engine" at a conference in Turin in 1841, one of the attendees, Luigi Menabrea, was so impressed that he wrote an account of Babbage's at lectures. Ada, then 27, married to the Earl of Lovelace, and the mother of three children under the age of eight, translated Menabrea's article from French into English. Babbage suggested she add her own explanatory notes.

What emerged was "The Sketch of the Analytical Engine", published as an article in 1843, with Ada's notes being twice as long as the original material. It became the definitive work on the subject of what was to eventually become "computing".

Ada's notes were divided into sections. Note A was not simply technical, but philosophical than technical, and it was in Note A that Ada anticipated what we would call a general purpose computer, suited to:

"The Analytical Engine . is not merely adapted for tabulating ... but for developing and tabulating any function whatever.  In fact the engine may be described as being the material expression of any indefinite function of any degree of generality and complexity ... "
-from Note A

In Note A, Ada writes about the Analytical Engine's potential to do anything we are able to instruct it to do -- including, if it were properly provided with rules of harmony and composition, produce "scientific" music.

"Again, it [the Analytical Engine] might act upon other things besides number , were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine ...  Supposing for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent."
- from Note A

There is a poetry in Ada's comparison of the Analytical Engine and the Jaquard loom:

"We may say most aptly that the Analytical Engine weaves algebraic patterns just as the Jacquard-loom weaves flowers and leaves."

Notes B through F delve into the functions and capabilities of the Analytical engine.

Note D is particularly prescient. It sets out the method for calculating the Bernoulli number sequence, and is generally regarded as the first "computer program".

Note G which includes a discussion of the future capabilities of the Analytical Engine, is a remarkable anticipation of the modern day computer:

"The Analytical Engine has no pretensions whatever to originate any thing. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with.   ... It is likely to exert an indirect and reciprocal influence on science itself in another manner. For in so distributing and combining the truths and formulas of analysis  ... the relations and the nature of many subjects in that science are necessarily thrown into new lights, and more profoundly investigated."
- from Note G