The Invention Of Calculus

In the 17th century Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) quite independently invented a new branch of mathematics later called Calculus. This new branch attracted a wide scale attack on its foundations by many mathematicians and philosophers. The definition of infinitely small or infinitesimal had the properties of sometimes being zero and non-zero the other times. This was quite rightly found to be a strange phenomenon. George Berkeley one of the critics discussed in this essay referred to the infinitesimals as "ghosts of departed quantities" [1].

Calculus is the branch of mathematics that includes derivatives, integral and later on limits was added to it also. Differential equations are also an offshoot of calculus which is now considered as a separate discipline. But we will be concentrating on the early inventions and discoveries made in derivatives and integrals, and later on also discuss how limits came about and helped move calculus forward.

The problems that calculus was invented to resolve mainly involve finding the area of a curved graph, the slope of the tangent of a curve and finding the rate of change of one object or value with respect to the other. Before calculus there wasn't a clear way of resolving these problems. Many great mathematicians and philosophers tried to solve these problems and were successful in solving some of them. For example Rene Descartes [2], with his Descartes' Method was able to find the circle tangent to a curve then calculating the tangent to that circle and working backwards to finding the value of the tangent at a given point on the curve. Here is an example to further illustrate what is meant by infinitesimals being zero and non-zero at the same time, which were mentioned earlier.

Derivatives are one way of finding the gradient or tangent of a graph at a certain point using calculus but using the formula for finding the gradient of a tangent, the following formula is used

(difference in y)/(difference in x)=dy/dx =(y_(n+1)-y_n)/(x_(n+1)-x_n ) (i)

If we take the graph y=x^2 to have y-axis distance to be (x+o)^2-x^2 and the x-axis to have distance o. Where o is a very small number.

Using these values to find the derivative we get the formula [5]

dy/dx=(y_(n+1)-y_n)/(x_(n+1)-x_n )=((x+o)^2-x^2)/o (ii)

Lets assume that o is a small, but not zero number.

(x^2+2ox+o^2-x^2)/o=(2ox+o^2)/o=2x+o (iii)

Let o=0 then dy/dx=2x, this means that the value of y at point x^2 is 2x. This is the usual answer we get from differentiating y=x^2. But many critics, including George W. Berkeley, have opposed this method as o is set to be both zero and non-zero.