It only remains now to summarise the mathematical work of Charles Babbage and to formulate an assessment of the importance of his contributions to the subject. We will discover the paradoxical conclusion that Babbage's mathematical achievements were quite considerable, but his immediate influence on mathematical thought and progress was almost negligible. It will be instructive to consider the reasons why this was so.

We have seen that Babbage lived through a period of extreme mediocrity in British mathematics. This meant that he had to acquire most of his knowledge by himself. Even at Cambridge, which was then the centre of any mathematical activity in this country, he quickly discovered that he could learn little from the formal tuition and had instead to devote himself to his own studies. He was fortunate at this time to have as undergraduate contemporaries, J. F. W. Herschel and George Peacock who were probably the only mathematicians in the country capable of working with him. They studied together the works of all the reputable Continental mathematicians, and, as the first chapter of the Memoirs of the Analytical Society indicates, their depth of reading and understanding was considerable by 1813. Although Babbage was strongly influenced by Continental rather than British mathematicians, and his work shows many instances of their ideas, his final output was entirely different from anything produced by his contemporaries. In fact, we find Babbage pursuing independent lines of research, working completely on his own apart from a little assistance from Herschel, with very few to teach or encourage or criticise him.