Introduction 3By 1670 the idea of a limit had been conceived; integration had been de-
fined; many integrals had been calculated to find the areas under curves, the
volumes of solids, and the arc lengths of curves; differentiation had been de-
fined; tangents to many curves had been effected; many minima and maxima
problems had been solved; and the relationship between integration and differ-
entiation had been discovered and proved. What remained to be done? And
why should Isaac Newton and Gottfried Wilhelm Leibniz be given credit for
inventing the calculus? The answers to these question are: A general symbol-
ism for integration and differentiation needed to be invented and strictly for-
mal rules, independent of geometric meaning, for analytic operations needed
to be discovered. Working independently of each other, Newton and Leibniz
both developed the required symbolism and rules for operation. Newton's
"fluxional calculus" was invented as early as 1665, but he did not publish his
work until 1687. Leibniz, on the other hand, formulated his "differential cal-
culus" about 1676, ten years later than Newton, but published his results in
1684, thus provoking a bitter priority dispute. It is noteworthy that Leibniz's
notation is superior to Newton's, and it is Leibniz's notation which we use
today.In 1661 , at the age of eighteen, Isaac Newton was admitted to Trinity Col-
lege in Cambridge. The Great Plague of 1664-65 (a bubonic plague) closed the
university in 1665, and Newton returned home. During the next two years,
1665-1667, Newton discovered the binomial theorem, invented differential cal-
culus (his fl.uxional calculus), proved that white light is composed of all the
spectral colors, and began work on what would later evolve into the universal
law of gravitation. In 1670- 71 , Isaac Newton wrote his Methodus fiuxionum et
serierum infinitorum, but it was not published until 1736 - nine years after his
death. Newton's approach to differential calculus was a physical one. He con-
sidered a curve to be generated by the continuous motion of a point. He call ed
a quantity which changes with respect to time a fluent (a fl.owing quantity).
And the rate of change of a fluent with respect to time he called a fiuxion of
the fluent. If a fluent is represented by y , then the fluxion of the fluent y is
represented by y. The fl.uxion of y is denoted by y and so on. The fluent of y
was denoted by [1LJ or y. Thus, to Newton y was the derivative of y and [1L] or
y was the integral. Newton considered two different types of problems. The
first problem, which is equivalent to differentiation, is to find a relation con-
necting fluents and their fl.uxions given some relation connecting the fluents.
The inverse problem, which is equivalent to solving a differential equation, is
to find a relation between fluents alone given a relation between the fluents
and their fl.uxions. Using his method of fl.uxions, Newton determined tangents
to curves, maxima and minima, points of inflection, curvature, and convexity
and concavity of curves; he calculated numerous quadratures; and he com-
puted the arc length of many curves. Newton was the first to systematically
use results of differentiation to obtain antiderivatives- that is, Newton used
his results from differentiation problems to solve integration problems.