Page:A Source Book in Mathematics.djvu/657

 (Translated from the Latin by Professor Evelyn Walker, Hunter College, New York City.)

Gottfried Wilhelm, Freiherr von Leibniz (Leipzig, 1646– Hannover, 1716) ranks with Newton as one of the inventors of the calculus. He was an infant prodigy, teaching himself Latin at the age of eight, and taking his degree in law before the age of twenty-one. In the service of the Elector of Mainz, and later in that of three successive dukes of Braunschweig-Lüneburg, he travelled extensively through England, France, Germany, Holland, Italy, everywhere seeking the acquaintance of prominent scholars. He finally settled at Hannover as librarian to the duke. In 1709 he was made a Baron of the Empire. When, in 1714, the Duke of Hannover crossed to England to become George I., he refused to allow Leibniz to accompany him. This embittered the last years of Leibniz’s life.

His was a most versatile genius. He wrote on mathematics, natural science, history, politics, jurisprudence, economics, philosophy, theology, and philology. He invented a calculating machine that would add, subtract, multiply, divide, and even extract roots.

He was elected to membership in the Royal Society of London (1673), and to foreign membership in the Académie des Sciences (1700). He founded the Akademie der Wissenschaften (1700), and became its president for life. Many of his articles appear in the Acta Eruditorum, the organ of the last named society.

His interest in the calculus must have been aroused while he was visiting England in 1672, where he probably heard from Oldenburg that Newton had some such method. His own development of the subject seems, however, to have been independent of that of Newton, while it shows the influence of both Barrow and Pascal. He never published a work on the calculus, but confined himself to short articles in the Acta Eruditorum, and to piecemeal explanations of his discoveries in letters which he wrote to other mathematicians.

Clearly we are indebted to him for the following contributions to the development of the calculus:


 * 1) He invented a convenient symbolism.
 * 2) He enunciated definite rules of procedure which he called algorithms.
 * 3) He realized and taught that quadratures constitute only a special case of integration; or, as he then called it, the inverse method of tangents.
 * 4) He represented transcendental lines by means of differential equations.

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