Apollonius drew a parabolic with a fortuitous point P and its symmetric axis. This axis goes through the minima T. He drew a line from P, that formed a right angle with the axis in point Q. A new point, R, was set on the extension of the axis where TR=TQ. The line through R and P is the tangent to the parabolic.
The Greek defined the tangent of a curve as a straight line, which only has one point in common with the curve. This is correct if we look at a curve of second degree equations, but a curve of a third degree equation has a tangent that will touch two pointsof the curve.
René Descartes, a French philosopher, who lived in the 17th century was the first one to develop the construction of a tangent further. Descartes developed the co-ordinate system, that he used in his method of drawing a tangent. Pierre de Fermat also developed his own way of constructing a tangent, but he used what we today call limits. In the following years mathematicians saw that their problem of deciding areas was closely connected to the problems with the tangent that both Descartes and Fermat had tried to solve.

Pierre de Fermat

René Descartes

The first problems occurring in the history of the calculus were concerned with the computation of areas, volumes, and lengths of arcs. Antiphon the Sophist (approximatly 430 B.C) gave one of the earliest important contributions to the problem of squaring the circle. His method was to double the number of sides of a regular polygon inscribed in a circle. Then the difference in area between the circle and the polygon would at last be exhausted. A square can be constructed equal in area to any given polygon. Then it will be possible to construct a square equal to the circle. This argument met a lot of criticism because Antiphon`s process could never use up the whole area of the circle. However, Antiphon`s bold pronouncement contained the term of the famous Greek method of exhaustion.

Archimedes (approximatly 250 B.C) is said to be the greatest scientist before Newton, and Archimedes was able to apply the method of exhaustion, which is the early form of integration. While Antiphon used squares and circles, Archimedes used parabolic segments and triangles. But Archimedes based his calculations on the same principle as Antiphon.

The theory of integration received very little stimulus after Archimedes's remarkable achievements until relatively modern times. Archimedes's works reached Western Europe through a translation of a ninth-century copy of his manuscripts in about 1450. Two early writers of modern times (Simon Stevin (1548-1620) and Luca Valerio (1552-1618)) used methods comparable to those of Archimedes. They tried to avoid doubling the number of sides in the method of exhaustion. Instead, they made a direct passage to the limit of the area of a parabolic segment.

Isaac Newton was born in Woolsthorpe hamlet on Christmas Day in 1642. When Newton was a boy, he didn't show any signs of a special talent in mathematics. When he was eighteen years of age, he was allowed to enter Trinity College, Cambridge. It was not until this stage in his schooling that his attention came to be directed to mathematics. And from reading mathematics, he turned to creating it.

When Newton was between the age of 23 and 25, he made three important discoveries:
The first discovery led to a theory, which tells us that white light consist of several beams of coloured light. The second was the differential and integral calculation, and this discovery is fundamental of the third, the gravitation law.

From the late summer of 1665 until late the summer of 1667, Cambridge University closed down for a period between mid-March to mid-June in 1666. It has been generally reported that it was in 1665, during the first year of this closure, and while living at home in Woolsthorpe, Newton developed his calculus. But recent research has shown that this account is a myth and that these discoveries were not made until he was at Cambridge in 1666 during the Univeristy's brief, temporary reopening. Newton returned to Cambridge in 1667, and he started to lecture on mathematics. His first lecture was on optics. His theory of colours was criticised, and he saw it as a personal criticism. He decided to stop publishing his theories.

If the astronaut Edmond Hally had not encouraged Newton, and if Hally had not given him economical support, Newton would never have published "Philosophiae Naturalis Principa Mathematica". This work consists of three different parts, but it is often called just Principa. Hally saw Newton's manuscript, and he realized its tremendous importance. He made Newton send his work to the Royal Society, and in the middle of 1687 the work caused an enormous impression throughout Europe.

The last part of his life was made unhappy by the unfortunate controversy with Leibniz. He died in 1727 when he was eighty-four years old, and he was buried in Westminister Abbey.

Newton's fame was equalled to Albert Einstein's. But Newton himself did not regard his work as anything unique. These are Newton's own words about himself: " I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and the finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Newton developed in 1666 a new form of mathematics. This form was called calculus and it was a modern method. But in spite of this pioneer work, Newton did not introduce his method to European mathematics because he was afraid of criticism. This was his method that resulted in derivation and then integration: Newton considered the curve as the point's orbit that moves. The co-ordinates x and y became functions of the time that he called fluents. The velocity to these fluents was called fluxions. Newton marked the functions of x and y as x' and y'. Let's say that o is a short period. During this period, the point moves a short distance on the curve and that x changes value to and y changes to . Let's then say that we are given a function called y = x^3. After the period o, we have got the co-ordinates and .
We place the co-ordinates into the equation and solve it:


We know that y = x^3 (since the first point is on the graph), and that results in reducing the o:


Since the o (the period) is very short, all the links that consist of o are also very small. We can strike off these links, and we will get:

We can write this as:

This is an expression of the gradient number to the tangent. We recognize this gradient number we would have found by derivation today:

Because the o is infinitely small, the point is not really on the graph but on the tangent. Newton had problems with giving a precise explanation to "infinitely small". This became obscure for a long time, also after his death. But in spite of this obscurity, the mathematical methods that Newton developed were of great importance and use in mathematics, physics and astronomy.

Gottfried Wilhelm Leibniz was born in Leipzig in 1646. He had a great talent as a child, and he taught himself both Latin and Greek. At the age of 20 he took a Phd in law. Leibniz showed great talent in law, theology, philosophy, history, literature and national economy, and it was often said that he was a "whole academy in himself". Leibniz spent most of his life in the diplomatic service, but on a diplomatic mission in 1672 he met Huygens who gave him lessons in mathematics. In the following years he made acquaintance with Oldenburg and others and exhibited a calculating machine for the Royal Society.

Leibniz invented the calculus sometime between 1673 and 1676, but his work is not seen in a consist system until 1677. His work was not published until 1684. He was the first one to use the modern integral sign on 29th October, 1675. It was a long letter S that came from the Latin word summa. He designed the integral sign, ò. Leibniz imagined that a curve consisted of infinitely many and small line segments. When a point moves from one position to the another on a curve, the x-value will change infinitely little as well as the y-value. These values were called differentiations. The size dy/dx is the expression for the tangent's gradient number.

Leibniz derived many of the elementary rules of differentiation. He was a remarkable mathematician and saw why it was so important to introduce new symbols to make calculus more convenient. The rule for finding the nth derivative of the product of two functions is still referred to as Leibniz' rule.

Leibniz died in 1716, but the last years of his life bore the stamp of the controversy with Newton and his supporters. They both accused each other for copying each other's work, but the universal opinion today is that they both discovered calculus independently of the other. Newton's discovery was made first, but Leibniz published his work before Newton. The controversy between the two led to a stop in the exchange of discoveries and ideas between Britain and the European continent that lasted a hundred years after Newton's and Leibniz's death. An exception to the rivalry was the noble tribute Leibniz paid to Newton: "Taking mathematics from the beginning of the world to the time when Newton lived, what he did was much the better half."

Mathematicians after Cauchy have had great use in his work. One of them was Karl Weierstrass who developed the theory of hyperelleptic integrals and published it in "Crelle's Journal" in 1856. Weierstrass was the private teacher of the Russian woman Sonja Kovalevskaja that could not enter the University because she was a woman. She wrote two excellent papers, "Partial differential equations" and "Abelian integrals", during her time as a student for Weierstrass. The first one was a remarkable contribution, which was published in "Crelle's Journal" in 1875. In the other paper on the reduction of abelian integrals into simpler elliptic integrals, she showed her complete command of Weierstrass's theory.

We can call the area of a surface the integral of a function, and the function itself is the derivative of the integral.

The most essential discoveries that were made in developing the integral calculus were made by Newton and Leibniz. They published their work late in the 15th century and early in the 16th century. This period was called "Baroque", but it did not really give any extra "support" to scientists and mathematicians. The idea of integration first arose in its role of a summation process in connection with the finding of areas, volumes and arc lengths.