S
IR
I
SAAC
N
EWTON
(1642–1727) on the same topics.
Leibniz also devoted a considerable amount of effort into
developing a characteristica generalis, a universal lan-
guage, as an attempt to generalize the logical formalism
created by A
RISTOTLE
(384–322
B
.
C
.
E
.). (Logician
G
EORGE
B
OOLE
later followed this goal in the 1800s.)
Leibniz entered the University of Leipzig at age 14,
as was customary at the time, to commence a 2-year
general degree course. Having read the works of Aris-
totle, Leibniz was already beginning initial work on
formalizing and systematizing the process of reason-
ing. He received degrees in law and in philosophy over
the following 6 years, and then began the ambitious
project of collating all human knowledge. He began
with the study of motion and kinematics, and pub-
lished, in 1671, his book Hypothesis physica nova
(New physical hypothesis).
Leibniz traveled to Paris in 1672 and began a study
of physics and mathematics with leading scientists in
the city at that time. Two years later, he had developed
his theory of differential calculus, but was struggling to
find a good system of mathematical notation for the
theory. In an unpublished 1675 manuscript Leibniz had
described the
PRODUCT RULE
for differentiation and
established the rules for differentiating
POLYNOMIAL
s.
Word had reached Newton of the results Leibniz
had developed, and Newton immediately wrote to him
explaining that he had already discovered the theory
a decade earlier. Newton, however, did not provide
details of his work. Leibniz courteously replied to
Newton but, not realizing that correspondences were
delayed by months, Newton suspected Leibniz of
dwelling over his letter, reconstructing the missing
details, and stealing his ideas. Although it is under-
stood today that Leibniz had accomplished his work
completely independently of Newton, a bitter dispute
between the two gentlemen ensued, one that lasted for
decades.
In 1684 Leibniz published his details of differen-
tial calculus in Nova methodus pro maximis et min-
imis (A new method for determining maxima and
minima) after finally establishing an effective system of
notation for his work—the dnotation we use today.
By this time Leibniz had also developed his theory of
integral calculus (along with the familiar ∫dx nota-
tion), and began publishing details of the work in
1686. (Newton wrote of his method of “fluxions” in
1671 but failed to get it published.)
Throughout his life, Leibniz also made significant
contributions to the study of
DIFFERENTIAL EQUA
-
TION
s, the theory of equations and the use of a
DETER
-
MINANT
to solve systems of equations, and
generalizing the
BINOMIAL THEOREM
to more than
two variables. Also, in his quest to collate all human
knowledge, Leibniz wrote significant treatises on
metaphysics and philosophy. He also developed a gen-
eral “law of continuity” for the universe, suggesting
that all that occurs in nature does so in matters of
degree, and argued that “mass times velocity squared”
is a fundamental quantity that is conserved in physical
systems. (This is today called the “law of conservation
of energy.”)
Leibniz died in Hanover, Germany, on November
14, 1716. It is not possible to exaggerate the effect Leib-
niz had on the development of analytical theory in the
centuries that followed him. His choice of notational
system for calculus, for instance, facilitated clear under-
standing of the subject and easy use of its techniques.
Mathematicians today typically use the notation devel-
oped by Leibniz rather than that developed by Newton.
Leibniz’s theorem See
PRODUCT RULE
.
lemma See
THEOREM
.
length The distance along a line, or the distance in
which a figure or solid extends in a certain direction, is
called its length. One can measure two lengths for a
rectangle to give an indication of its size. (The greater
of the two dimensions is usually called its length, and
the smaller its breadth.)
Early units for length were given by parts of the
body. For example, a “cubit” was defined to be the
length of the forearm, measured from the elbow to the
tip of the middle finger (about 19 in.); an “ell,” still
sometimes used for measuring cloth, is the length from
the tip of one’s nose to the end of an outstretched arm
(about 35 in.); a “hand,” used for measuring the heights
of horses, is the width of a man’s hand (about 4 in.); and
a “foot” was defined as the distance paced by one step.
The ancient Romans considered a foot to be the equiva-
lent of 12 thumb-widths, yielding the word inch from
the Latin word unicia meaning one-twelfth. The Romans
also identified 1,000 paces as a milia passuum, leading
length 309