history of mathematics - National STEM Centre

Calculus
JPractiee exercises
this chapter are on
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122
The problem with Newton's basis for calculus, although it worked, was that it was
not clear whether you could actually let the augments vanish. If they did vanish,
how could they have a ratio? Leibniz avoided this problem to some extent by using
his infinitely small increments djcand dy, which never became zero, but which were
so small that they could be ignored with respect to the quantities x and y.
The dispute over Newton's vanishing augments, he called them 'evanescent
augments', raged on for many years after his death. It was only resolved in the 19th
century when Cauchy developed the idea of a limit.
Reflecting on Chapter 9
What you should know
» that calculus provides a general method for solving problems that previously
required diverse special techniques
some early methods for finding tangents
examples from the work of each of Archimedes, Fermat and Descartes
some earlier methods for finding areas enclosed by curves
an example from the work of each of Archimedes and Cavalieri
the distinctive approach taken by Newton, his notation, and some examples
the logical difficulties inherent in his approach
the distinctive approach taken by Leibniz, his notation, and some examples
the logical difficulties inherent in his approach.
Preparing for your next review
• Reflect on the 'What you should know' list for this chapter. Be ready for a
discussion on any of the points.
• Answer the following check question.
1 Fermat had determined the area under any curve of the form y = xk and, in the
1640s, had been able to construct the tangent to such a curve. Use any sources at
your disposal to discover why it is that calculus is not considered 'invented' until
the time of Newton and Leibniz.