Babbage at Cambridge & in the Royal Society
Charles Babbage, 1791-1871.
“An Essay towards the calculus of functions,”
Philosophical Transactions of the Royal Society, 105 (1815): 389-423.
-At Peterhouse, Cambridge (1811-1814), Babbage was a brilliant undergraduate mathematician, though he was barred from competing for honors following official  suspicions about his religious views.   His early mathematical papers in the Philosophical Transactions led to his election as a Fellow of the Royal Society in 1816, when he was still only twenty-five.  Later, he was elected to Sir Isaac Newton’s former chair, as Lucasian Professor of Mathematics (1828-1839), though never he never delivered a lecture.

The Analytical Society
Herschel, John Frederick William, 1792-1871.
“On equations of differences and their application to the determination of functions from given conditions,” extracted from Memoirs of the Analytical Society, Cambridge: J. Smith, 1813.  Separately bound in nineteenth century calf.  South Carolina College Library Collection.
–In 1812, while still an undergraduate,  Babbage founded this Cambridge-based society for mathematical research, with his friend John Herschel, son of the astronomer Sir William Herschel.  Herschel and Babbage both contributed to the Society's Memoirs.   In the 1820's, Herschel became an influential ally for Babbage’s work on the difference engine.

Elopement and Applied Mathematics
Charles Babbage, “An Examination of Some Questions Connected to Games of Chance,” Transactions of the Royal Society of Edinburgh, 9 (1821): 153-177.
And: Charles Babbage, A comparative view of the various institutions for the assurance of lives. London, J. Mawman, 1826. Modern calf.
–Soon after graduation in 1814, Babbage eloped with his fiancée, Georgina Whitmore, quarreling with disapproving father, a wealthy banker.  Though he still enjoyed a substantial private income, and inherited more on his father’s death, in the early years of his marriage he undertook a number of paid projects in applied mathematics, as shown here in his work for a gambling syndicate (their proposed system was not going to be a good investment) and for the new financial industry of life insurance.  He also worked for banking firms and railway companies.  Georgina died quite suddenly in 1827. 

The Mechanics of Astronomy and Logarithmic Tables
Charles Babbage, “On a New Zenith Micrometer,”
Memoirs of the Astronomical Society, 2 (1826):
101-103.
And: Charles Babbage,
Table of the logarithms of the natural numbers, from 1 to 108000. Stereotyped.
3rd ed.  London: Printed for the Hungarian Academy of Sciences, 1834.  Rebound.
–Babbage’s involvement with the recently-founded Astronomical Society led to an interest in the mechanics of precision instruments and in the mathematical tables on which astronomers (and ocean-going navigation) depended.  He first published his revised Tables of Logarithms in 1826, involving laborious multiple rechecking for mistakes either in the mathematics or in the printing.  The stereotyping of the printed text was in part a strategy to reduce the chance of introducing fresh misprints in later printings.