history of mathematics - National STEM Centre

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The beginnings of calculus
9.1
Euclid's definition
of tangents
9.2
Archimedes's
spiral
9.3
Fermat
and maxima
9.4
Fermat and I
constructing tangents I
9.5
Descartes's approach
9.6
Using
indivisibles
9.7
Leibniz
and area
9.8
Relating gradients
and integration
9.9
Newton and
differentiation
Calculus was developed to solve two major problems with which many
mathematicians had worked for centuries. One was the problem of finding extreme
values of curves - the problem now called finding maxima and minima. The other
was the problem of quadrature, or finding a systematic way to calculate the areas
enclosed by curves.
Until Leibniz and Newton applied themselves to the problems, several different
geometrical methods existed for .solving these problems in specific cases. There was
no general method which would work for any curve.
The chapter begins with some of these geometrical methods, so that you can see
how difficult they are and realise why there was such a need for a general method. It
goes on to show something of the pioneering work of Fermat, Descartes, Cavalieri,
Leibniz and Newton as they developed their ideas.
Activities 9.1 and 9.2 show how Euclid and Archimedes drew tangents to two very
special curves. In Activities 9.3 to 9.5 you see how Fermat and then Descartes
approached the idea of drawing a tangent to a curve. Activities 9.6 and 9.7 show the
beginnings of the modern ideas and notations for integrals.
Activity 9.8 shows how Leibniz thought of the relation between differentiation and
integration, and Activity 9.9 shows how Newton developed a system for
differentiation.
All the activities are suitable for working in a small group.
Work through the activities in sequence.
Early methods for finding tangents
To find an extreme value, you have to find a tangent to a curve. For the Greeks, this
was a problem of pure geometry. Later, in the 17th century, it became important for
the science of optics studied by Fermat, Descartes, Huygens and Newton, and in the
science of motion studied by Newton and Galileo.
The work that the Greeks were able to do was limited by three factors:
• their unsatisfactory ideas of angle - they tried to include angles in which at least
one of the sides was curved (see Figure 9.1)