Development of Calculus

Calculus, being a difficult subject, therefore requires much more than the intuition and genius of one man. It took the work and ideas of many great men to establish the advanced concepts now known as calculus. Of the many mathematicians involved in the discovery of calculus, Gottfried Wilhelm von Liebniz and Sir Isaac Newton were the most important. Together, they established the basic principles of calculus, and, with the help of other mathematicians, it was refined using the concept of the limit. The developemtn of calculus can be thought of as being in three periods; Anticipation, Development, and Rigorization. During the Anticipation, various mathematicians provided the stepping stones to build the concepts of calculus. During the Development, Newton and Liebniz developed the main concepts and prinicples used today. In the Rigorization, various mathematicians used the concept of the limit to give concrete meaning to the principles developed my Leibniz and Newton.

The Anticipation of calculus started way back in the time of ancient Greece, when, at around 450BC, the philosopher Zeno of Elea made this conjecture:

If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB1. Continue this argument to see that A must move through an infinite number of distances and so cannot move.

The philosophers Leocippus of Miletus, Democritus of Abdera, and Antiphon the Sophist all made contributions to the Greek method of exhaustion which was put on a scientific basis by Eudoxus of Cnidus at about 370 BC. The method of exhaustion is named so because one must think of the areas measured as if they are expanding so that they account for more and more of the required area. Archimedies, however, made the most important Greek contributions to calculus. His first important contribution was his proof that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. He constructed an infinite sequence of triangles starting with one of area A and adding further triangles continuously between the existing ones and the parabola to get areas:

A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ...

Therefore, the area of the segment of a parabola is:

A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A.

His is the first documented example of summation of an infinite series. Archimedies' second contribution to calculus was his usage the method of exhaustion to find an approximation to the area of a circle. This is an early example of integration, and it also led to the approximation of .

Other early integrations by Archimedies were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of a hyperboloid of revolution.

The development of calculus went into a long standstill until the 16th century, where mechanics caused mathematicians to study problems such as finding centers of gravity. Luca Valerio published De quadratura parabolae (1606) in Rome which continued the Greek methods of conquering such area problems. Johannes Kepler, during his research in planetary motion, came across a problem. He had to find the area of sectors of an ellipse. His method consisted of thinking of areas as sums of lines, another crude form of integration. Kepler had no time or patience for the rigorous Greek methods and was very lucky to get the correct answer only after making two cancelling errors in this work. The next mathematicians to make contributions were born within three years of each other. They were Pierre de Fermat, Gilles Personne de Roberval, and Bonaventura Fransesco Cavalieri. Cavalieri developed his 'method of indivisibles' from Kepler's attempts at integration. His approach was not rigorous and it is unclear to see how he arrived at his method. It seems that he thought of an area as being made up of lines and then summed his infinite number of 'indivisibles'. Using these methods, he showed that the integral of xn from 0 to a was an+1/(n + 1) by showing the result for a number of values of n and inferring the general result. Roberval considered the same problems, but he was much more rigorous than Cavalieri. He looked at the area between a curve and a line as being made up of an infinite number of infinitely narrow rectangular strips. He applied this to the integral of xm from 0 to 1 which he showed had approximate value:

(0m + 1m + 2m + ... + (n-1)m)/nm+1.

He then figured that this apporached 1/(m + 1) as n approached infinity, thereby calculating the area. Fermat was also very rigorous in his approach, but gave no proofs. He gave the general formulae for the hyperbola and the parabola:

Parabola: y/a = (x/b)2 to (y/a)n = (x/b)m

Hyperbola: y/a = b/x to (y/a)n = (b/x)m.

While examining y/a = (x/b)p, Fermat calculated the sum of rp from r=1 to r=n. He also investgated maximums and minimums by calculating when the tangent to the curve was parallel to the x-axis. He wrote to René Descartes, giving him the method as it is used today; finding the maximums and minimums by calculating when the derivatives were equal to 0. Because of his work, Joseph Louis Lagrange stated that he claims Fermat to be the inventor of calculus.

Descartes created an important method of calculating normals in La Géométrie in 1637 based on double intersection. Forimund de Beaune extended his methods and applied them to tangets where double intersections translate to double roots. Johann van Waveren Hudde discovered an easier method, known as Hudde's Rule, which involves the derivative. Both Descartes and Hudde were important in influencing Sir Isaac Newton. Christiaan Huygens was critical of Cavalieri's proofs saying that one needs a proof that at least convices that a rigorous one can be created. He was a maijor influence on Leibniz and so led to a better approach to calculus. The next maijor step in the development of calculus was provided by Evangelista Torricelli and Issac Barrow. Barrow gave a method of calulating tangents to a curve where the tangent is given as the limit of a chord as the points approach each other. This is known as Barrow's Differential Triangle:

Both Torricelli and Barrow considered the problem of motion with variable speed. The derivative of the distance is velocity and the inverse operation takes one from the velocity to the distance. Therefore an awareness of the inverse of differentiation began to evolve naturally and the idea that the integral and derivative were inverses to each other was familiar to Barrow. In fact, although he never explicitly stated the Funamental Theorem of Calculus, he was working close to the answer. Because of his work, however, he allowed Newton to arrive at the final theorem.

The main development of calculus is mainly attributed to two men, Newton and Leibniz. Although they both created the foundations, they both worked independently and used different methods. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x' and y', which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz used analysis. Leibniz was very conscious of the notation he used while, on the other hand, Newton wrote only more for himself than anyone else. He usually used whatever notation he had thought up at that moment. This was important for later developments. Leibniz's notation was better suited to generalizing calculus to multiple variables and, in addition, it highlighted the operator aspect of the derivative and integral. As a result, most of the notation that is used in Calculus today is thanks to Leibniz. In their development of the calculus both Newton and Leibniz used "infinitesimals", quantities that are infinitely small and yet nonzero. Of course, such infinitesimals do not really exist, but Newton and Leibniz found it convenient and much simpler to use these quantities in their computations and their derivations of results. Although one could not argue with the success of calculus, this concept of infinitesimals bothered mathematicians. Newton encountered problems publishing his works, most of them were not published until years after his writing them. One of the reasons for this was that the publisher of Barrow's works went into bankruptcy, scaring all other publishers from publishing mathematical works.

It took about 100 years for calculus to finally be made rigorous. Eventually, Augustin Louis Cauchy, Karl Theodor Wilhelm Weierstrass, and Georg Friederich Bernhard Riemann reformulated Calculus in terms of limits rather than infinitesimals. Thus the need for these infinitely small (and nonexistent) quantities was removed, and replaced by a notion of quantities being "close" to others. The derivative and the integral were both reformulated in terms of limits. While it may seem like alot of work to create rigorous justifications of computations that seemed to work fine in the first place, this is an important development. By putting Calculus on a logical footing, mathematicians were better able to understand and extend its results, as well as to understand some of the more subtle aspects of the theory. The way calculus is taught is backwards to its development. It begins with the concept of the limit, then with derivative and the integral as defined by Newton and Leibniz, using the new definition as stated with limits. These principles are then used to study the problems that preceded the development of calculus.