as early as 1670, and then called his attention to Mercator's work on the rectification of the parabola. While in London, Leibniz exhibited to the Royal Society his arithmetical machine, which was similar to Pascal's, but more efficient and perfect. After his return to Paris, he had the leisure to study mathematics more systematically. With indomitable energy he set about removing his ignorance of higher mathematics. Huygens was his principal master. He studied the geometric works of Descartes, Honorarius Fabri, Gregory St. Vincent, and Pascal. A careful study of infinite series led him to the discovery of the following expression for the ratio of the circumference to the diameter of the circle:—
etc.
This elegant series was found in the same way as Mercator's on the hyperbola. Huygens was highly pleased with it and urged him on to new investigations. Leibniz entered into a detailed study of the quadrature of curves and thereby became intimately acquainted with the higher mathematics. Among the papers of Leibniz is still found a manuscript on quadratures, written before he left Paris in 1676, but which was never printed by him. The more important parts of it were embodied in articles published later in the Acta Eruditorum.
In the study of Cartesian geometry the attention of Leibniz was drawn early to the direct and inverse problems of tangents. The direct problem had been solved by Descartes for the simplest curves only; while the inverse had completely transcended the power of his analysis. Leibniz investigated both problems for any curve; he constructed what he called the triangulum characteristicum—an infinitely small triangle between the infinitely small part of the curve coinciding with the tangent, and the differences of the ordinates and abscissas.
