LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig, Germany, 1 July 1646; d. Hannover, Germany, 14 November 1716), mathematics, philosophy, metaphysics.

    (1666); several additions are presented in the Hypothesis physica nova (1671).

In accord with the encyclopedic approach popular at the time, Leibniz limited himself primarily to methods and results and considered demonstrations nonessential and unimportant. His effort to mechanize computation led him to work on a calculating machine which would perform all four fundamental operations of arithmetic.

Leibniz was occupied with diplomatic tasks in Paris from the spring of 1672, but he continued the studies (begun in 1666) on the arithmetic triangle which had appeared on the title page of Apianus' Arithmetic (1527) and which was well known in the sixteenth century; Leibniz was still unaware, however, of Pascal's treatise of 1665. He also studied the array of differences of the number sequences, and discovered both fundamental rules of the calculus of finite differences of sequences with a finite number of members. He revealed this in conversation with Huygens, who challenged his visitor to produce the summation of reciprocal triangular numbers and, therefore, of a sequence with infinitely many members. Leibniz succeeded in this task at the end of 1672 and summed further sequences of reciprocal polygonal numbers and, following the work of Grégoire de St.-Vincent (1647), the geometric sequence, through transition to the difference sequence.

As a member of a delegation from Mainz, Leibniz traveled to London in the spring of 1673 to take part in the unsuccessful peace negotiations between England, France, and the Netherlands. He was received by Oldenburg at the Royal Society, where he demonstrated an unfinished model of his calculating machine. Through Robert Boyle he met John Pell, who was familiar with the entire algebraic literature of the time. Pell discussed with Leibniz his successes in calculus of differences and immediately referred him to several relevant works of which Leibniz was not aware—including Mercator's Logarithmotechnia (1668), in which the logarithmic series is determined through prior division, and Barrow's Lectiones opticae (1669) and Lectiones geometricae (1670). (Barrow's works were published in 1672 under the single title Lectiones opticae et geometricae.)

Leibniz became a member of the Royal Society upon application, but he had seriously damaged his scientific reputation through thoughtless pronouncements on the array of differences, and still more through his rash promise of soon producing a working model of the calculating machine. Leibniz could not fully develop the calculator's principle of design until 1674, by which time he could take advantage of the invention of direct drive, the tachometer, and the stepped drum.

Through Oldenburg, Leibniz received hints, phrased in general terms, of Newton's and Gregory's results in infinitesimal mathematics; but he was still a novice and therefore could not comprehend the significance of what had been communicated to him. Huygens referred him to the relevant literature on infinitesimals in mathematics and Leibniz became passionately interested in the subject. Following the lead of Pascal's Lettres de “A. Delonville” [= Pascal] contenant quelques unes de ses inventions de géométrie (1659), by 1673 he had mastered the characteristic triangle and had found, by means of a transmutation—that is, of the integral transformation, discovered through affine geometry,

1/2?x 0 [y(?) - x dy/dx(?)] ? d?,

for the determination of a segment of a plane curve—a method developed on a purely geometrical basis by means of which he could uniformly derive all the previously stated theorems on quadratures.

Leibniz' most important new result, which he communicated in 1674 to Huygens, Oldenburg, and his friends in Paris, were the arithmetical quadrature of the circle, including the arc tangent series, which had been achieved in a manner corresponding to Mercator's series division, and the elementary quadrature of a cycloidal segment (presented in print in 1678 in a form that concealed the method). Referred by Jacques Ozanam to problems of indeterminate analysis that can be solved algebraically, Leibniz also achieved success in this area by simplifying methods, as in the essay on

x + y + z = p2, x2 + y2 + z2 = q4

(to be solved in natural numbers). Furthermore, a casual note indicates that at this time he was already concerned with dyadic arithmetic.

The announcement of new publications on algebra provoked Leibniz to undertake a thorough review of the pertinent technical literature, in particular the Latin translation of Descartes's Géométrie (1637), published by Frans van Schooten (1659-1661) with commentaries and further studies written by Descartes's followers.

His efforts culminated in four results obtained in 1675: a more suitable manner of expressing the indices (ik in lieu of aik, for instance); the determination of symmetric functions and especially of sums of powers of the solutions of algebraic equations; the construction of equations of higher degree that can be represented by means of compound radicals; and ingenious attempts to solve higher equations algorithmically by means of radicals,