After my work with Babbage I took on new research: in 1844 a planned mathematical model of how the brain gives rise to thought, electrical experiments learned from Andrew Crosse, a review of Reichenbach's magnetism paper, and by 1851 work on maths and music.
Studying differential calculus, I wrote to De Morgan that the hardest part of early mathematics is recognising that formulae which look utterly unlike can be the same — and compared them to shape-shifting sprites and fairies.
From 1832 I was trained privately in mathematics and science by William Frend, William King and Mary Somerville; in the 1840s Augustus De Morgan took me through advanced calculus, including the Bernoulli numbers that formed my algorithm for the engine.
In June 1833 Mary Somerville introduced me to Charles Babbage, and that month he showed me the prototype of his difference engine. I studied the machine as often as I could; he called me "The Enchantress of Number".
When Taylor's Scientific Memoirs ruled Babbage's unsigned preface must be signed, he asked me to withdraw my paper. I refused, kept my work in print under my own initials, and our collaboration recovered.
Historians still argue how much of Note G was mine. My position: making the engine's abstract operation clear, and driving the Bernoulli computation, is real work in computer science, not publicity.
My method for the Notes: separate the Analytical Engine from the Difference Engine, separate mechanism from logic, then show a worked program. It made a technology most scientists did not grasp readable.
Reviewing Menabrea's paper against an 1834 article on the Difference Engine, I separated a computing machine's mechanism from its logical structure and argued both are equally essential fields of engineering and science research.
In Note G I stated the limit of the Analytical Engine: it does what we know how to order it to do and cannot anticipate new truths. Alan Turing later named this "Lady Lovelace's Objection" in his paper on machine intelligence.
In my 1843 notes I argued the Analytical Engine differed from all earlier calculating machines: programmed, it could solve problems of any complexity, and because numbers can represent letters or musical notes it could manipulate any symbols by rules.
