Lady Ada Lovelace and the analytical engine
By ADITI KAR
AUGUSTA ADA, COUNTESS OF LOVELACE
Lady Ada Lovelace was born Augusta Ada Byron. Her father was the famous Romantic poet Lord Byron, and her mother Lady Annabella Byron nee Millebank was a remarkable woman with a deep interest in studying mathematics. Graced with beauty, rank and an illustrious parentage, Ada was the subject of much public fascination and scrutiny all her life.
The poet and Lady Byron separated within a month of Ada's birth. Lord Byron departed almost immediately for the continent and was never to see his daughter again. Ada grew up solely in her mother's care and Lady Byron was determined that the young girl be trained in mathematics. Mother and daughter met Charles Babbage when Ada was only 17. She was at once interested and intrigued by the inventions of the great man. Nevertheless, her real involvement with Babbage's engines commenced almost seven years later. In the meantime, she married William King, later Lord Lovelace, the first Earl of Somerset and started a family. At about this time, she resumed her training in mathematics under the instruction of Augustus DeMorgan.
In 1842, Charles Babbage was unequivocally refused any further financial support from the government of Britain for the construction of his Difference Engine. At this time of great professional turmoil, Lady Lovelace became the champion of his computing machines. Ada's great contribution to the world of mathematics and computing is an annotated translation of Menabrea's `Notions sur la machine analytique de Charles Babbage'. Ada's translation was accompanied by her own Chapters of Notes and in the final one, she described how the Analytical engine would sequentially compute the Bernoulli numbers (see below).
Ada Lovelace died very young at the age of thirty six. Tragically, the `Sketch of The Analytical Engine' remained her only publication. 

THE ANALYTICAL ENGINE, BERNOULLI NUMBERS AND THE FIRST COMPUTER `PROGRAM'
Babbage's Analytical engine was even grander than his difference engine. We have seen (in Charles Babbage and the Difference Engine) that the difference engine could only perform addition. Babbage designed a machine that could conduct all operations of arithmetic. He was obsessed with reducing the time required to perform computations. His idea was to split the Analytical engine into two parts: the mill and the store. These are the precursors of the central processor and memory of the modern day computer. In its final form the analytical engine was to be programmed to work with punched cards and the output produced on a cleverly designed printing device. Ada's Notes to the translation of Menabrea's memoirs terminated with a stepbystep description of how `the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers.' This `algorithm' is sometimes labelled the world's first computer program. Numbers of Bernoulli The numbers of Bernoulli are some of the most intriguing objects of mathematics. They appear in different guises. One way to define them is using the series for the function . The Bernoulli numbers are written as . Consider The exponential function is given by the series and so substituting this into the denominator of the previous expression, we get precisely the equation Multiplying and collecting coefficients of powers of , one obtains a strategy for calculating the numbers recursively. Indeed, So the first few Bernoulli numbers are and so on. The machine could compute so long as it could store all the previous numbers of Bernoulli.
Something to think about: Imagine that you are Ada Lovelace and design a simple algorithm to tell the Analytical engine how to compute . 
Bernoulli discovered his numbers studying sums of powers of integers:
In some early attempts to solve Fermat's Last Theorem, the Bernoulli numbers made an exciting appearance. Kummer, in 1850 proved Fermat's theorem for `regular primes'. According to Kummer, a prime number p is regular precisely when it does not divide the numerators of the first p3 nonzero Bernoulli numbers (other than 1).

Modern day constructions of Babbage's engines may be found on display at the Science Museum, London. 
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