What device did Charles Babbage develop in 1834?

The device that Charles Babbage developed in 1834 is the Analytical Engine. This mechanical computing device is significant in the history of computing as it represents one of the first designs for a general-purpose computer. The Analytical Engine was intended to perform any calculation, based on a series of instructions and data input, and it introduced concepts such as arithmetic operations, control flow through conditional branching, and memory, which laid the groundwork for modern computing.

While the Difference Engine, which Babbage designed earlier, focused on calculating polynomial functions and producing mathematical tables, it was the Analytical Engine that included a broader capability, allowing for more complex operations and programming potential. Babbage’s vision for the Analytical Engine included features such as an integrated memory and the ability to execute sequences of operations, making it a pivotal step toward the computers we use today.

The other options, like the telegraph and calculator, though important inventions, do not relate specifically to Babbage's contributions in the realm of mechanical computation or show the same level of innovation towards general-purpose computing as the Analytical Engine.