history of mathematics - National STEM Centre

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Calculus
Activity 9.7
Cavalieri was able to apply the method of indivisibles to a variety of curves and
solids and derived the relationship now written as
where n is a positive integer. The abbreviation 'o.l.' was a form of integration
symbol.
Cavalieri was aware that his method was built upon shaky ground; there was no firm
mathematical proof that an area could be created by summing an infinite number of
infinitely thin lines, but the method worked if he was careful, and he was sure that
the problems would eventually be ironed out. Proof that he knew of the problems is
revealed in a letter to his friend Torricelli, to whom he points out this paradox,
which uses the same argument as in Activity 9.6.
Triangle ABC in Figure 9.9 is non-isosceles, and AD is an altitude. Clearly, the area
of the triangle ABD does not equal the area of the triangle ADC.
However, PQ is parallel to BC, and PR is parallel to QS. Thus PR = QS .
But, for every PR there is a unique QS, so o.l. PR = o.l. QS.
This implies that the area of triangle ABD = area of triangle ADC. By the method
of indivisibles then, the two areas are both equal and at the same time unequal.
The work of Leibniz
Leibniz (1646-1716) was born in Leipzig. Although his family were well-to-do,
Leibniz had no guaranteed income and had to work all his life. Between 1672 and
1675, when he was working in Paris as a diplomatic attache for the Elector of
Palatine, he visited London where he met some English mathematicians and was
told of some ideas which Newton had used in developing the calculus. In 1677, after
he had left Paris and was working in Brunswick, Leibniz published his own version
of the calculus.
Subsequently Leibniz tried not to become embroiled in a dispute with Newton over
who had first invented calculus. Although Newton spent considerable time and
energy claiming precedence for the invention of the calculus, it is Leibniz's notation
which we use today.
Leibniz initially used Cavalieri's notation, but in the period from 25th October to
11th November 1675 he invented the notation that we use today. He began much as
Cavalieri did, and, over a period of a few days, improved the notation and also
derived a number of rules just by playing with the symbols.
Leibniz and area
Leibniz started with a diagram similar to Figure 9.10. (Leibniz's diagram was
actually rotated through 90° because it was customary at that time to draw curves
with the origin at the top left.)
The vertical lines are all the same distance apart. Each vertical has length y. The
difference in length between one vertical line and the next is called w. The area
OCD is the sum of all the rectangles xw.