In her notes accompanying a translation about the Analytical Engine, Ada Lovelace discussed computational procedures. She also proposed that a machine able to handle symbolic relationships might work with things beyond numbers.

Her notes included steps for calculating Bernoulli numbers. Because the Analytical Engine remained a design at the time, these discussions belong to the early history of ideas about computing.

Babbage's lecture, Menabrea's French transcription, Lovelace's translation and Notes A-G

In 1840, Babbage was invited to give a seminar at the University of Turin about his Analytical Engine. Luigi Menabrea, a young Italian engineer and the future Prime Minister of Italy, transcribed Babbage's lecture into French, and this transcript was subsequently published in the Bibliothèque universelle de Genève in October 1842. Babbage's friend Charles Wheatstone commissioned Lovelace to translate Menabrea's paper into English.

During a nine-month period in 1842–43, Lovelace translated Menabrea's article. She augmented the paper with seven notes, A to G, about three times longer than the translation. The translation and notes were then published in the September 1843 edition of Taylor's Scientific Memoirs under her initials AAL.

Explaining the Analytical Engine's function was a difficult task; many other scientists did not grasp the concept and the British establishment had shown little interest in it. Lovelace's notes even had to explain how the Analytical Engine differed from the original Difference Engine. Her work was well received at the time; the scientist Michael Faraday described himself as a supporter of her writing.

First published computer program

The notes are important in the early history of computers, especially since Note G described, in complete detail, a method for calculating a sequence of Bernoulli numbers using the Analytical Engine, which might have run correctly had it ever been built. Though Babbage's personal notes from 1837 to 1840 contain the first programs for the engine, the algorithm in Note G is often called the first published computer program. The engine was never completed and so the program was never tested.

In 1953, more than a century after her death, Ada Lovelace's notes on Babbage's Analytical Engine were republished as an appendix to B. V. Bowden's Faster than Thought: A Symposium on Digital Computing Machines. The engine has now been recognised as an early model for a computer and her notes as a description of a computer and software.

Insight into potential of computing devices

In her notes, Ada Lovelace emphasised the difference between the Analytical Engine and previous calculating machines, particularly its ability to be programmed to solve problems of any complexity. She realised the potential of the device extended far beyond mere number crunching.

This analysis was an important development from previous ideas about the capabilities of computing devices and anticipated the implications of modern computing one hundred years before they were realised. Walter Isaacson ascribes Ada's insight regarding the application of computing to any process based on logical symbols to an observation about textiles: "When she saw some mechanical looms that used punchcards to direct the weaving of beautiful patterns, it reminded her of how Babbage's engine used punched cards to make calculations." This insight is seen as significant by writers such as Betty Toole and Benjamin Woolley, as well as the programmer John Graham-Cumming, whose project Plan 28 has the aim of constructing the first complete Analytical Engine.

This objection has been the subject of much debate and rebuttal, including in the paper that originally coined it.

Education

From 1832, when she was seventeen, her mathematical abilities began to emerge, and her interest in mathematics dominated the majority of her adult life. Her mother's obsession with rooting out any of the insanity of which she accused Byron was one of the reasons that Ada was taught mathematics from an early age. She was privately educated in mathematics and science by William Frend, William King, and Mary Somerville, the noted 19th-century researcher and scientific author. In the 1840s, the mathematician Augustus De Morgan extended her "much help in her mathematical studies" including study of advanced calculus topics including the "numbers of Bernoulli" (that formed her celebrated algorithm for Babbage's Analytical Engine). In a letter to Lady Byron, De Morgan suggested that Ada's skill in mathematics might lead her to become "an original mathematical investigator, perhaps of first-rate eminence".