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geometry) and his notation for exponents (1637), which we use today.
Fermat (about 1640) knew how to find the slope of the tangent to the
graph of a polynomial using thelimit of theslopeof a chord, and
found maxima and minima by examining the slope of the chord near
the point. There were in fact a host of special methods for dealing with
special curves.
Both Newton (when he was away from Cambridgefor fear of the
plague) in 1665 and 1666, and Leibniz in Paris in 1675, working
independently, systematically established the product, quotient and
chain rules and discovered the inverse relationship between slopes of
tangents and areas under curves. Having discovered the Binomial
Theorem for rational index, Newton was strongly motivated by what he
could do by differentiating and integrating power series term by term.
Newton considered his variables as real entities which were changing
with time, and he expressed the derivative of the 'fluent'
x
, as the
'fluxion'
x
. Leibniz had originally been motivated by the use of
differences in summing series (as in qns 5.3 and 5.4). In this notation,
dx
originally meant the difference between
x
and its next value, and
dy
/
dx
was originally a ratio of infinitesimals. Leibniz' great discovery was of
the results he could generate using the characteristic triangle with sides
dx
,
dy
and
ds
. Leibniz'
dy
/
dx
was expressed by Newton in the form
y
/
x
.
Thefirst calculus topic was published in 1696. This was
Analyse des
infiniments petits
by theMarquis del'Ho ˆ pital. It was repeatedly
reprinted during the eighteenth century. The topic consists of lectures
on the differential calculus which Johann Bernoulli gave to de l'Hoˆ pital
in 1691, on Leibniz' work. A little of Newton's calculus appeared in his
Principia
(1687), much morewas in theappendix to his
Opticks
(1704),
but his original exposition
De Analysi
(written in 1669 and circulated to
friends) was not published until 1736, after his death. Both Newton and
Leibniz originally thought of limits in terms of infinitesimals, but later
in life tried to avoid them. Newton spoke of prime and ultimate ratios
(the slope of the chord and tangent respectively). Leibniz eventually
came to speak of infinitesimals as convenient fictions.
In Bishop Berkeley's tract of 1734, the apparently self-contradictory
natureof limits was highlighted. In aMrming that, as
x
tends to
a
,
(
x
a
)/(
x
a
) tends to 2
a
,
x
is not equal to
a
at thebeginning of the
computation, and this allows the division to be performed, and then
x
is put equal to
a
at theend of thecomputation to obtain theanswr.
This paradox was not resolved during the eighteenth century, but it did
not discourage mathematicians in their work. Indeed, an immediate
chronological successor to Bishop Berkeley was L. Euler, the most
ingenious manipulator of infinitesimals there has ever been. Berkeley's
criticisms were not forgotten and in 1764 d'Alembert offered the useful
