Leibniz received a doctorate of laws in 1667, a step towards entering the diplomatic service of one of the small states which then made up Germany. Traveling extensively on political missions to France, Holland and England, he was brought into contact with most of the leading mathematicians of the day. Leibniz’s real mathematical education began in the years 1672 to 1676, in Paris, when time between assignments allowed him to study the subject in depth. His version of the calculus seems to have been invented in 1673, but the first account was not formally published until 1684. (This was twenty years prior to the appearance of Newton’s presentation of the calculus in De quadratura curvarum.) Leibniz’s diplomatic career came to an end in 1676 when he reluctantly accepted the position of librarian in the court of Hanover, a post which he held for the remainder of his life. He helped to organize the Berlin Academy of Science in 1700, and became its first President.

The most important aspect of Leibniz’s calculus was a suitable symbolism that allowed the geometric arguments of his predecessors to be translated into operational rules. He proposed the symbol for the sum of areas of infinitely small rectangles; it is the script form of s, the initial letter in summa (sum).. In his new formalism, Leibniz expresses relations such as

He also originated the notation dy/dx, treating it as a quotient of differentials (infinitely small increments of the variable); and used the letter d, standing alone, for differentiation. His led to useful algorithms, such as the product rule:

His formula

indicated the inverse relationship of differentiation and integration.

One of Leibniz’s early contributions is an elegant series for p which is now named after him:

When challenged, as a test of his ability, to calculate the sum of the series

he found that the terms could be transformed into differences by the identity 1/n(n+1) = 1/n - 1/(n+1); the series then became

and, when adjacent terms are canceled, had sum 1.