CalGeo: Teaching Calculus using dynamic geometric tools
Outcome 2.1.2 “Historical elements of teaching calculus”
HISTORICAL ELEMENTS OF TEACHING CALCULUS
(I)
Introduction to infinite processes
The Greeks geometricians assumed on infinite grounds that simple curvilinear figures such as
circles or ellipses, have areas that are geometric magnitudes of the same type as areas of
polygonal figures. For their calculation they used two natural properties
i)
(monotonicity):
If A is contained in B , then E(A)< E(B), where E(T) denotes the area of T,.
ii)
(additivity):
if      and    then E(Γ) = E(A)+ E(B)
These properties played an essential role, as due to them Eudoxus was led later in a method of
calculation of areas curvilinear figures that is today known as “method of exhaustion”. The
principle of exhaustion is formulated in the proposition I in X book of the elements of Euclid :
“Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude
greater than its half, and from that which is left a magnitude greater than its half , and if this
process be repeated continually there will be left some magnitude which will be less than
lesser magnitude set out.”
This method is of great importance since it is origin of infinite procedures.
If we want to describe the method in modern terminology, we consider a magnitude α We
abstract from α, a magnitude α1>
a
and from the remainder we substract a magnitude α2
2
greater than the half of the former and so on.
We consider the sequence (sn )n with s1  a1 , s2  a1  a2 ,..., sn  a1  a2  ...  an , e.t.c., then
the remainder corresponding to the n-nth step is a-sn. In virtue of the exhaustion principle for
any ε>0 there is a n0  N such that
a  sn   for every n  n0 which is equivalent to lim sn  a
n
i
Application: Organization of activities leading to the introduction
i)
in the concept of limit, via measurement of circle’ s area, and
ii)
in the integral, with the quadrature of the parabola.
(II)
The fundamental theorem of calculus
Medieval investigations, and the subsequent work of Galileo suggested that the motion of a
point, along a straight line with varying velocity be represented by means of a graph of its
velocity versus time. Indivisibles considerations then indicated that the total distance travelled
by the point would equal the area under the velocity-time curve. This led to an embryonic
formulation of the fundamental theorem of calculus (Torricelli). Barrow gave a geometric
proof of the fundamental theorem of calculus for monotonous and positive functions. Newton
in 1666 discusses the computation of areas by means of andidifferention. This is the first
Historical appearance of the fundamental theorem in the explicit form
dE
y
dx
where E denotes the area under curve y=f (x), providing basis for a algorithmic approach to
the computation of areas.
Application: Organization of activities leading to the introduction to the fundamental theorem
of calculus.
(III) The problem of tangent line, the derivative and the local extreme values.
One of the main mathematical problem at the beginning of 17th century was the determination
of the tangent line. Interest in this problem stemmed from more than one source. It was a
problem of pure geometry, and it was of great importance for scientific application. Optics
was one of the major scientific pursuits of the seventeenth century; the design of lenses was a
direct interest to Fermat, Descartes, Huygens and Newton. To study the passage of light
through a lens, one must know the angle at which the ray strikes the lens in order to apply the
law of refraction. The significant angle is that between the ray and the normal to the curve, the
ii
normal being the perpendicular to the tangent. Hence the problem was to find either the
normal or the tangent.
Roberval thought a curve as the locus of a point moving under the action of two velocities.
While the notion of a tangent as a line having the direction of the resultant velocity was more
complicated than the Greek definition of a line touching a curve, this newer concept applied
to many curves for which the older one failed. It was also valuable because it linked pure
geometry and dynamics, which before Galileo’s work had been regarded as essential distinct.
On the other hand, this notion of a tangent was objectionable on mathematical grounds,
because it had the definition of tangents on physical concepts. Many curves arose in situations
having nothing to do with notion and the definition of tangent was accordingly inapplicable
hence other methods of finding tangents grained favor.
In Geometry the tangents lines of two curves, in their common point, determined the angle of
the curves. Descartes devised a method of constructing tangent lines that was algebraic rather
than infinitesimal in charachter. His method of finding the tangent line to the curve y=f(x) at
the point P(x, f(x)) involved list locating the point Q(x0,0) of intersection with the
perpendicular through Q to the normal line. P. Fermat, circa 1637, developed a new method in
which , for the first time in the history of calculus, the idea of the small change of a variable
was presented. He allowed a minimal increase to the variable and after the calculation he left
increase to disappear. Fermat observed that the method of drawing the tangent line could be
applied to finding of the extreme values which are searched in points x such that (in modern
terminology) f’(x)=0
Application: Organization of activities leading to the introduction of concepts of tangent,
derivative and extreme values.
(IV)
The continuity concept
During 18th century the approximate solution of equations used the theorem of Intermediate
values as a basic tool. Many mathematicians considered it (erroneously) as characterization
of continuity. In 1817 Bolzano claimed that the theorem required analytic proof. Bolzano and
Cauchy gave the strict definition of continuity, while Weierstrass in 1869 in his courses in
Berlin, after clarified the distinction between continuous and differentiable fluxion function,
first proved that every continuous function is bounded and takes extreme values.
Application: Organization of activities about the continuity concept.
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