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

LEIBNIZ: Mathematics

(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,