Published books

1. Memoirs of the Analytical Society, Cambridge, 1813. (Written with J. F. W. Herschel.)

2. Translation of S. F. Lacroix's ‘Sur le Calcul Différentiel et Intégral’, Cambridge, 1816. (Written with J. F. W. Herschel and G. Peacock.)

3. Examples of the Solutions of Functional Equations, Cambridge, 1820.

Unpublished books

1. ‘The History of the Origin and Progress of the Calculus of Functions during the years 1809, 1810 … 1817’ (Museum of the History of Science, Oxford).

2. ‘The Philosophy of Analysis’, c. 1821 (British Museum).

Papers

1. ‘An essay towards the calculus of functions, Part I’, Philosophical Transactions, 1815.

2. ‘An essay towards the calculus of functions, Part II’, Philosophical Transactions, 1816.

3. ‘Demonstration of some of Dr. Matthew Stewart's general theorems, to which is added an account of some new properties of the circle’, Journal of Sciences and the Arts, 1817.

4. ‘Observations on the analogy which subsists between the calculus of functions and other branches of analysis’, Philosophical Transactions, 1817.

5. Solutions of some problems by means of the calculus of functions', Journal of Science and the Arts, London, 1817.

6. ‘Note respecting elimination’, Journal of Science and the Arts, London, 1817.

7. ‘An account of Euler's method of solving a problem relating to the Knight's move at chess’, Journal of Science and the Arts, 1817.

8. ‘On some new methods of investigating the sums of several classes of infinite series’, Philosophical Transactions, 1819.

9. ‘Demonstration of a theorem relating to prime numbers.’ Edinburgh Philosophical Journal, 1819.

10. ‘Observations on the notation employed in the calculus of functions’, Transactions of the Cambridge Philosophical Society, 1821.

11. […]

‘The Philosophy of Analysis’ is the title of a set of mathematical essays by Charles Babbage, only one of which was ever published. The remainder are bound together and kept in the British Museum Manuscripts Room as Additional Manuscripts 37202.

It is difficult to determine why these essays of generally high quality and much original thought should have been so neglected by Babbage. If published as a book, which would have been his first one, they would no doubt have had a considerable influence on mathematical thought. Round about the year 1830 was one of the most fruitful periods in the history of mathematics, when revolutionary ideas in algebra and geometry were first put forward and the era of modern mathematics could be said to have begun. Two, in particular, of Babbage's papers would have made a major contribution both to the type of algebra generally known as ‘modern’, to distinguish it from symbolised arithmetic, and to stochastic analysis.

Babbage makes no reference to this proposed work in his Passages from the Life of a Philosopher, or in any other of his writings. We can learn of the fate of this work only from a few letters from his friends. E. F. Bromhead, writing on 7 March 1821, is most enthusiastic:

It is clear that Babbage regarded the question of notation as one of supreme importance in mathematics. Good notation leads to rapid mathematical progress and poor symbolism to stagnation. The history of mathematics shows many examples of the correlation between advances made and the notation used. Babbage believed that in any type of logical reasoning it was essential to take great pains first to derive a notation both simple and comprehensive. Then not only was the immediate working of a problem greatly facilitated but even new and unsuspected results could be suggested. Sometimes a notation could be strong enough to open up new branches of a subject and greatly assist the whole process of mathematical discovery.

Babbage was, as we have seen, very much involved in notational reform in his early career, being born at a time when symbolism was at one of its lowest points in mathematical history. Later, he wrote three long and interesting papers on the subject of notation. These were ‘Observations on the notation employed in the calculus of functions’, Transactions of the Cambridge Philosophical Society, 1821; ‘On the influence of signs in mathematical reasoning’, Transactions of the Cambridge Philosophical Society, 1827, and an article ‘On notation’ for the Edinburgh Encyclopaedia, 1830.

Charles Babbage is an almost legendary figure of the Victorian era, yet relatively little is known about him. No authoritative account of his life and work has yet been published. In the absence of accurate knowledge, he is misrepresented as the eccentric genius, inventing computers which he never completed and quarrelling with almost everyone, especially the organgrinders. Certainly Babbage was a man of highly individual talent, applying his ability with great success to a variety of subjects from economics to ballet, from deciphering to life insurance, from tool-making to astronomy. Indeed it is hard to think of any field of knowledge in which he did not excel, excepting possibly classics, for which he admitted a dislike. As far as his personal life is concerned, there is abundant, scarcely touched material available for studies of his exceptional personality, and when this is thoroughly examined it will almost certainly be discovered that he was a far different person from the one represented by popular misconception.

Primarily, Charles Babbage was a mathematician. In spite of the great variety of interests in other spheres, together with a considerable amount of family and social commitment, there is no doubt that he devoted himself essentially to a study of pure mathematics during the early years of his working life. In these years, as will be shown, his productive work was most original in content, exerting a strong influence on the course of British mathematics. It was recognised and esteemed by some of the greatest contemporary Continental mathematicians and contained many ideas, the value of which was not acknowledged till many years later.