cancellation 57
History of Calculus
The study of calculus begins with the study of motion, a topic
that has fascinated and befuddled scholars since the time of
antiquity. The first recorded work of note in this direction
dates back to the Greek scholars P
YTHAGORAS
(ca. 569–475
B
.
C
.
E
) and Z
ENO OF
E
LEA
(ca. 500
B
.
C
.
E
.), and their followers,
who put forward the notion of an
INFINITESIMAL
as one possi-
ble means for explaining the nature of physical change.
Motion could thus possibly be understood as the aggregate
effect of a collection of infinitely small changes. Zeno, how-
ever, was very much aware of fundamental difficulties with
this approach and its assumption that space and time are
consequently each continuous and thus infinitely divisible.
Through a series of ingenious logical arguments, Zeno rea-
soned that this cannot be the case. At the same time, Zeno
presented convincing reasoning to show that the reverse
position, that space is composed of fundamental indivisible
units, also cannot hold. The contradictory issues proposed by
Zeno were not properly resolved for well over two millennia.
The concept of the infinitesimal also arose in the
ancient Greek study of area and volume. Scholars of the
schools of
PLATO
(428–348
B
.
C
.
E
.) and of E
UDOXUS OF
C
NIDUS
(ca. 370
B
.
C
.
E
.) developed a “method of exhaustion,” which
attempted to compute the area or volume of a curved figure
by confining it between two known quantities, both of which
can be made to resemble the desired object with any pre-
scribed degree of accuracy. For example, one can sand-
wich a circle between two n-gons, one inscribed and one
circumscribed. As one can readily compute the area of a
regular n-gon, the formula for the area of a circle follows by
taking larger and larger values of n. (See
AREA
.) The figure
of a regular n-gon as ngrows differs from that of a true cir-
cle only by an infinitesimal amount. A
RCHIMEDES OF
S
YRACUSE
(287–212
B
.
C
.
E
.) applied this method to compute the area of a
section of a
PARABOLA
, and 600 years later, P
APPUS OF
A
LEXANDRIA
(ca. 300–350
C
.
E
.) computed the volume of a
SOLID OF REVOLUTION
via this technique. Although successful
in computing the areas and volumes of a select collection of
geometric objects, scholars had no general techniques that
allowed for the development of a general theory of area and
volume. Each individual calculation for a single specific
example was hailed as a great achievement in its own right.
The resurgence of scientific investigation in the mid-
1600s led European scholars to push the method of exhaus-
tion beyond the point where Archimedes and Pappus had
left it. J
OHANNES
K
EPLER
(1571–1630) extended the use of
infinitesimals to solve
OPTIMIZATION
problems. (He also
developed new mathematical methods for computing the
volume of wine barrels.) Others worked on the problem of
finding tangents to curves, an important practical problem
in the grinding of lenses, and the problem of finding areas of
irregular figures. In 1635, Italian mathematician B
ONAVEN
-
TURA
C
AVALIERI
wrote the first textbook on what we would
call integration methods. He described a general “method
of indivisibles” useful for computing volumes. The principle
today is called C
AVALIERI
’
S PRINCIPLE
.
French mathematician Gilles Personne de Roberval
(1602–75) was the first to link the study of motion to geome-
try. He realized that the tangent line to a geometric curve
could be interpreted as the instantaneous direction of
motion of a point traveling along that curve. Philosopher
and mathematician R
ENÉ
D
ESCARTES
(1596–1650) developed
general techniques for finding the formula for the tangent
line to a curve at a given point. This technique was later
picked up by P
IERRE DE
F
ERMAT
(1601–65), who used the
study of tangents to solve maxima and minima problems in
much the same way we solve such problems today. As a
separate area of study, Fermat also developed techniques
of integral calculus to find areas between curves and
lengths of arcs of curves, which were later developed fur-
ther by B
LAISE
P
ASCAL
(1623–62) and English mathematicians
J
OHN
W
ALLIS
(1616–1703) and I
SAAC
B
ARROW
(1630–77).
At the same time scholars, including Wallis, began
studying
SERIES
and
INFINITE PRODUCT
s. Scottish mathemati-
cian J
AMES
G
REGORY
(1638–75) developed techniques for
expressing trigonometric functions as infinite sums, thereby
discovering T
AYLOR SERIES
40 years before B
ROOK
T
AYLOR
(1685–1731) independently developed the same results.
By the mid-1600s, certainly, all the pieces of calculus
were in place. Yet scholars at the time did not realize that all
the varied problems being studied belonged to one unified
whole, namely, that the techniques used to solve tangent
problems could be used to solve area problems, and vice
versa. A fundamental breakthrough came in the 1670s
when, independently, G
OTTFRIED
W
ILHELM
L
EIBNIZ
(1646–1716)
of Germany and S
IR
I
SAAC
N
EWTON
(1642–1727) of England
discovered an inverse relationship between the “tangent
problem” and the “area problem.” The discovery of the
FUN
-
DAMENTAL THEOREM OF CALCULUS
brought together the dis-
parate topics being studied, provided a beautiful and
natural perspective on the subject as a whole, and allowed
scholars to make significant advances in solving geometric
and physical problems with spectacular success. Despite
the content of knowledge that had been established up until
that time, it is the discovery of the fundamental theorem of
calculus that represents the discovery of calculus.
Newton approached calculus through a concept of
“flowing entities.” He called any quantity being studied a
“fluent” and its rate of change a
FLUXION
. Records show that
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