2
1 Historical Background
conclusion is easily derived. Achilles must first pass the 10 meters gap. At that
time the tortoise travels a shorter distance, say 1 meter. Now, before catching up
with the tortoise, Achilles must travel a distance of 1 meter. The tortoise does not
wait and advances another 0.1 meter. If the process is repeated we realize that just
to get to the tortoise, Achilles must travel an infinite number of distances, namely
(10 + 1 +
1
1
+ 2+
10 10
) meters
which cannot be done in a finite time.
Most philosophers and scientists were not comfortable with Zeno’s paradoxes
and did not accept his convictions. Aristotle discarded them as “fallacies” but did
not possess any scientific tools to justify his conviction, as the basic concepts of
calculus, the ‘infinite series’ and the ‘limit’, were yet unknown. Only the
rigorously formulated notion of converging series as presented by the French
mathematician Cauchy, and the theory of infinite sets developed by the German
mathematician Cantor in the 19-th century, finally explained Zeno’s paradoxes to
some satisfaction.
1.2 Calculus – Where Do We Start?
Some people claim that it was Sir Isaac Newton, the 17-th century English
physicist and mathematician who discovered calculus. Others give the credit to the
German philosopher and mathematician Baron Gottfried Wilhelm von Leibniz,
but most believe that both scientists invented calculus independently. Later we
will discuss this particular dispute in length. One fact though is above dispute. All
agree that discovery of calculus took place between the years 1665 and 1675 and
that the first related publication was Leibniz’s article Nova Methodus pro Maximis
et Minimis (“New Method for the Greatest and the Least”) which appeared in Acta
Eruditorum Lipsienium (Proceedings of the Scholars of Leipzig) 1684.
Yet, to think that this unique mathematical field started from scratch in the
second half of the 17-th century is incorrect. Various results closely related to
some of the fundamental problems and ideas of calculus, particularly the integral
calculus, has been known for thousands of years. The ancient Egyptians for
example who were practically involved in almost every important development in
culture and science, knew how to calculate the volume of a pyramid and how to
approximate the area of a circle.
Then came Eudoxus. Born about 400 BC in Cnidus, southwest Asia Minor, this
brilliant Greek studied mathematics and medicine (not an unusual combination for
those days), in a school that competed with Hipocrates’ famous medical school in
the island of Cos. A local rich physician, impressed by the talented youngster paid
Eudoxus's way to Plato’s Academy in Athens. Eudoxus then spent some time in
Egypt where he studied astronomy. Later he was a traveling teacher at northwest
of Asia Minor and finally returned to Athens where he became an accomplished
legislator until his death (about 350 BC).
Pure Mathematical Physics
1.2 Calculus – Where Do We Start?
3
The best mathematician of his day, Eudoxus is remembered for two major
contributions in mathematics which are extensively discussed in the thirteen books
collection Elements by Euclid (second half of 4-th century BC). The first is the
Theory of Proportions which paved the way to the introducing of Irrational
Numbers and is discussed in Book V. The second is the Method of Exhaustion,
presented in Book XII, which was developed to calculate areas and volumes
bounded by curves or surfaces. The general idea of this method was to
approximate an unknown area or a volume using elementary bodies like rectangles
or boxes, whose areas or volumes were already known. At each stage, at least half
(or other constant fraction) of the remaining area must be replaced by new
elementary bodies. This guarantees that the unknown area may be approximated to
any desired degree. It was the first time that the concept of infinitesimally small
quantity was included, over 2000 years before Newton and Leibniz. It is therefore
quite legitimate to consider Eudoxus as the discoverer of the conceptual
philosophy of integral calculus.
The true successor of Eudoxus and his method of exhaustion was Archimedes
of Syracuse (287-212 BC). A first class inventor, the founder of theoretical
mechanics and one of the greatest mathematicians of all times, Archimedes
studied in Alexandria, capital of Egypt and center of the scientific world at the
time. After completing his studies he returned to Syracuse where he remained for
the rest of his life. The Greek biographer Plutarch (about 40-120 AD) claims that
Archimedes despised everything of practical nature and rather preferred pure
mathematical research, whether it was solving a complex problem in geometry or
calculating a new improved approximation to the circumference of a circle. Yet,
when forced by the circumstances, his beloved city under Roman siege, he put his
engineering ingenuity to work and devised war machines for Hieron II, King of
Syracuse, which inflicted heavy casualties on the Roman Navy. When the city
finally surrendered to the Romans, Archimedes, realizing that everything was lost,
went back to his research. Unfortunately there was not much time left for him.
While engaged in drawing his figures on the sand he was stabbed to death by a
Roman soldier.
As stated above, Archimedes adopted and improved the method of exhaustion
originated by Eudoxus and followed by Euclid. One of his applications was to
approximate the area of a unit circle (equals to 2π ) by using inscribed and
circumscribed equilateral polygons to bound the circle from below and above. It is
an early example of integration, which led to approximate values of π . A sample
of his diagram is given in Fig. 1.1.1.
By increasing the number of the vertices of each polygon, the interval where
the exact value of the area is located decreases, and the approximation improved.
At some point Archimedes believed that π equals 10 . However, after using the
method of exhaustion, he found the assumption false and by taking 96 vertices
showed
3
10
1
<π <3
71
7
Pure Mathematical Physics