## An essay towards the calculus of functions

*“It can be said with some assurance that no mathematician prior to Babbage had treated the calculus of functions in such a systematic way ... Babbage must be given full credit as the inventor of a distinct and important branch of mathematics” (Dubbey).*

*"My subsequent inquiries have produced several new methods of solving functional equations containing only one variable quantity and much more complicated, and have convinced me of the importance of the Calculus, particularly as an instrument of discovery in the more difficult branches of analysis. Nor is it only in the recesses of this abstract science that its advantages will be felt... it may be applied to every branch of natural philosophy, where the object is to discover by calculation from the results of experiment the laws which regulate the action of the ultimate particles of bodies..."-Babbage*

FIRST EDITIONS, journal issues, of Babbage's two most important contributions to mathematics, essentially founding the general theory of the calculus of functions.

Babbage began work on the calculus of functions as early as 1809 and continued during his studies at Cambridge. In 1815, at just 23 years of age, Babbage published the results of his work in two long papers in the *Philosophical Transactions*. "Babbage believed that his new scheme would serve as a generalized calculus to include all problems capable of analytical formulation, and it is possible to see here a hint of the inspiration for his concept of the Analytical Engine. While the work on the engines and his other scientific, social and political activities caused him virtually to abandon mathematical research at the age of thirty, the calculus of functions was the area he often yearned to continue. In fact the calculus of functions was not taken up by other workers, and it is the aspect of Babbage's mathematical work that modern mathematicians find the most fascinating" (Dubbey, *The Mathematical Work of Charles Babbage*). Many years later, in his *Passages from the Life of a Philosopher*, Babbage referred to the calculus of functions as his "earliest step" and "one to which I would willingly recur if other demands on my time permitted."

An essay towards the calculus of functions [Part I]. In Philosophical Transactions, Vol. 105, Part II, 1815. London: W. Bulmer & Co., 1815, pp.389-423. WITH: An essay towards the calculus of functions, Part II, pp. 179-256. In Philosophical Transactions, Vol. 106, Part II, 1816. London: W. Bulmer & Co., 1816. Two complete journal issues. Quarto, modern blue wrappers, uncut and largely unopened. Text extremely clean. Fine copies of both parts. Rare.

Price: $3,000 .