Note G is a computer algorithm traditionally attributed to Ada Lovelace that was designed to calculate Bernoulli numbers using the hypothetical analytical engine designed by Charles Babbage. The algorithm was the last note in a series labelled "A" to "G", which were employed as visual aids to accompany Lovelace's English translation of Luigi Menabrea's 1842 French transcription of Charles Babbage's only lecture on the analytical engine. Her translation, along with her substantial notes on the possibilities of the analytical engine, were published in 1843. In Babbage's 1864 memoir he discusses the creation of the various notes with Lovelace, including note "G". He provided the mathematical formulas for the calculation of the Bernoulli numbers, which Ada converted into a step-by-step table of instructions for the analytical engine; she also spotted a bug for him:
We discussed together the various illustrations that might be introduced: I suggested several, but the selection was entirely her own. So also was the algebraic working out of the different problems, except, indeed, that relating to the numbers of Bernoulli, which I had offered to do to save Lady Lovelace the trouble. This she sent back to me for an amendment, having detected a grave mistake which I had made in the process. Historians like Allan G. Bromley have noted several dozen sample programs prepared by Babbage between 1837 and 1840 (all substantially predating the illustrative notes), though they were never published and were substantially simpler, which has led to, in popular culture, Note G being generally considered to be the first algorithm specifically for a computer, and Lovelace is considered as the first computer programmer. The program described in Note G was not tested in Babbage's or Lovelace's lifetime, as the analytical engine was never built. In the modern era, the algorithm has been tested using modern computing methods and revealed to have a software bug in it due to two variables being swapped in a division operation.
Origin
In 1840, Charles Babbage was invited to give a seminar in Turin on his analytical engine, the only public explanation he ever gave on the engine. During Babbage's lecture, mathematician Luigi Menabrea wrote an account of the engine in French. A friend of Babbage's, Charles Wheatstone, suggested that in order to contribute, Lovelace should translate Menabrea's account. Babbage suggested that she augment the account with appendices, which she compiled at the end of her translation as a series of seven "notes" labelled A-G. Her translation was published in August 1843, in Taylor's Scientific Memoirs, wherein Lovelace's name was signed "A.A.L". In these notes, Lovelace described the capabilities of Babbage's analytical engine if it were to be used for computing, laying out a more ambitious plan for the engine than even Babbage himself had. Lovelace's notes for the article were three times longer than the article itself. In the first notes, she explores beyond the numerical ambitions that Babbage had for the machine, and suggests the machine could take advantage of computation in order to deal with the realms of music, graphics, and language.
Again, it 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. She explains to readers how the analytical engine was separate from Babbage's earlier difference engine, and likens its function to the Jacquard machine, in that it used binary punch cards to denote machine language. In note C, this point is furthered by the fact that simultaneous and iterated actions can be made by the machine, ensuring that any card or collection of cards can be used several times in the solution of a single problem, essentially anticipating modern methods of control flow and looping. These ideas were brought to a head in the final note, G, where Lovelace sought to demonstrate an example of computation. Note G only made use of only the four arithmetical operations: addition, subtraction, multiplication and division, the implementation of Babbage's vision:
Under the impossibility of my here explaining the process through which this end is attained, we must limit ourselves to admitting that the first four operations of arithmetic, that is addition, subtraction, multiplication and division, can be performed in a direct manner through the intervention of the machine. This granted, the machine is thence capable of performing every species of numerical calculation, for all such calculations ultimately resolve themselves into the four operations we have just named.It also uses Babbage's idea of storing information in columns of discs, each denoted by V {\displaystyle V} (for variable) and a subscript number denoting which column is being referred to.
Function The algorithm used a recursive equation to calculate Bernoulli numbers, wherein it used the previous values in an equation to generate the next one. The method ran thus:
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