Early mathematical work focused on functional equations, contributing to the field of analysis and laying the groundwork for his later theoretical work.

Babbage's early mathematical work led to his election as a Fellow of the Royal Society, a prestigious scientific organization. This reflected early recognition of his intellectual abilities.

Babbage was a key figure in founding the Astronomical Society (later the Royal Astronomical Society). This demonstrates his commitment to advancing scientific knowledge and facilitating collaboration.

Babbage made contributions to actuarial science and statistics, applying mathematical principles to problems in insurance and population analysis. His work laid the groundwork for improved actuarial calculations.

Babbage first publicly describes the Difference Engine, a mechanical calculator designed to automate the calculation of polynomial functions. The initial purpose was to automate the creation of logarithmic tables. This idea propelled his efforts in mechanical computation.

Babbage held the prestigious Lucasian Professorship of Mathematics at Cambridge University, though he reportedly never delivered a single lecture.

Babbage published 'Reflections on the Decline of Science in England, and on Some of its Causes', a critical assessment of the state of British science and a call for reform. It generated considerable debate and controversy, influencing the direction of scientific policy.

Babbage began conceiving of the Analytical Engine, a more general-purpose mechanical computer that could be programmed using punched cards, inspired by the Jacquard loom used in weaving. This represented a significant leap in his thinking about computation.