A Comprehensive Overview of the History and Development of Calculus

This essay provides a comprehensive overview of the history of calculus, tracing its development from ancient Greek mathematicians like Archimedes, who employed methods of exhaustion, to the pivotal contributions of Newton and Leibniz. The essay highlights the evolution from integration to differentiation, exploring the works of mathematicians such as Apollonius, Diophantus, and Ibn Al Haitham. It discusses the impact of the Roman Empire's decline and the subsequent resurgence of mathematical interest in the 16th and 17th centuries, with a focus on the contributions of Kepler, Descartes, Fermat, Roberval, Pascal, and others. The essay emphasizes the contributions of Huygens and Gregory, whose work laid the groundwork for Newton and Leibniz's independent development of calculus, ultimately establishing the foundations of modern calculus by integrating prior discoveries.

Summary of The History of Calculus
By Arthur Rosenthal
It’s a basic knowledge among many people that Newton and Leibniz are the cofounders
of the calculus theory. This though is not all and one need to go deeper to understand the
development and history of calculus up to the era of Newton and Leibniz. Currently calculus
begin with differentiation followed by integration but historically it did start with integration.
The problems around calculus integration were first discussed by ancient Greek mathematicians
such as Archimedes. Archimedes applied the method of exhaustion in plane areas and the
inscribed and circumscribed methods in polygons. The method is traced to Eudoxus who applied
it to prove that the volume of pyramid equals one third of the corresponding prism. The method
was later applied by Euclid before being successfully implemented by Archimedes in the 3rd
century B.C. Archimedes was able to predict more accurately the area of a circle, ellipses as well
as sectors of a spiral. Archimedes was also the founder of statics and hydrostatics and applied
the integration method in the principle of the lever to the elementary parts of a figure. He later
determined the area of a parabola segment by applying the ingenious statics method.
Despite his wonderful achievements Archimedes never had successors to continue his
work only Dionysodorus is mentioned thereafter for finding the volume of the torus. During the
time of Archimedes there lived another great mathematician called Apollonius who completed
the Greek theory of conic sections. Upon the decline of the two mathematicians, development of
calculus took a turn. Triggered by the needs of the astronomies anew mathematical branch
termed trigonometry was established and later on the theory of numbers was invented by
Diophantus. The invasion of the Roman empire by the Teutonic was a big dent to the
development of mathematics as the concept receded to the Orient, Byzantium and later to the
Arabian nations. Mathematics later thrived in the 800 -1200 where a mathematician
Mesopotamian Ibn Al Haitham was able to compute the volume of solid.
In the 12th and 13th century the influence of the Orient did see interest in mathematics
reemerge with the 16th century showing signs of great progress in algebra. In this era the
Archimedes work were studied and once again understood. As the 17th century began
Archimedes ideas’ developments were initiated. This is also the era when Galileo established
modern science. Simon Stevin and Luca Valerio did improve the method of exhaustion. Later a

German astronomer called Johannes Kepler did publish books that were aimed at determining
volume of wine casks. He applied the work of Archimedes in his discovery in addition to adding
a few new cases. Its during this period that analytics geometry was invented by Descartes and
later in collaboration with Fermat. These inventions were of great support to the development of
calculus. Fermat calculus achievements were of significant and he was termed the greatest
mathematician in the first part of the 17th century especially in the calculus domain. His work
was reflected in the theory of numbers and also applied tangents to find maxima and minima.
During the 17th century, another approach to finding tangents was invented by Roberval
and Torricelli independently. Later on, Johannes Hudde and Rene Francois did give an explicit
formal rule for finding the extreme and subtangents of algebraic curves. Fermat had great
success in calculus and was the first to prove the power formula of integration. Roberval with the
guidance of Fermat found and proved the same before Cavalieri later on discovered it by himself
and publish it in 1647. Fermat was aware of the relationship of the various problems in
differential calculus although failed to observe the general relation between integration and
differentiation.
Blaise Pascal is a French mathematician who is considered as the master of integration.
Even though Roberval was the first to integrate trigonometric functions, Pascal did more
integrations for both the trigonometric functions and the algebraic functions as well. His work is
responsible for identifying the integration by parts formula. At the same time John Willis added
his contributions by identifying the notion of limits. The limit notion was carefully taken into
account by Pietro Mengoli who applied it to modify the procedure of Luca Valerio. This method
was later adopted by Newton.
The two other mathematicians whose work became the foundation to the inventions by
Newton and Leibniz are Christiaan Huygens who is famous for the introduction of the notion of
evolutes and involutes. The other is James Gregory who did massive work in integration in the
1660’s. Gregory obtained the Newton’s interpolation formula independently of Newton as well
as invented the theory of series. The series theory was applied by Nicolaus Mercator to find the
logarithmic series. The method was later discovered by Newton. Newton and Gregory even
though worked independently did made tremendous progress in infinite series where they both

discovered the binomial series and also many other series for trigonometric and inverse
trigonometric functions.
To the time of Newton and Leibniz several mathematicians have done a lot of work in
regard to integration and differentiation, all that was missing was the creation of calculus which
was done by Newton and Leibniz independently although they applied to a great degree the
previous discoveries.