Notes on Bernoulli Numbers
Lovelace's table of operations shows how a mathematical procedure becomes a sequence a machine can follow.
The Bernoulli number example is easy to underestimate. It looks like a piece of mathematics, but it is also a description of a process: initialize values, repeat a calculation, update a running total, and store the result. The table of operations makes the repetition explicit and gives each step a place in the sequence.
What makes the example endure is its clarity. It does not assume that the reader already knows what a program is. Instead, it shows how a problem can be broken into operations that a machine could perform in order. That combination of notation and procedure is one of the earliest signs that programming would become a discipline of its own.