Exploring the Methods of Differential Calculus through the ...

Biographies
Sir Isaac Newton was born prematurely on Christmas, 1642. He went to Cambridge University,
where his focus was mainly in chemistry. He also studied algebra and analytic geometry. He
worked under Isaac Barrow, who discovered infinitesimal calculus. In 1665 there was an
outbreak of the Great Plague which caused Newton to return home to Woolsthrope, England
(Cooke, 1997). He remained there for two years and while Cambridge was closed he studied his
own mathematics and physics research. It was during this time that he created the Binomial
Theorem. Newton died in 1727, at the age of 85, in England (Struik, 1948).
Gottfried Wilhelm von Leibniz was born in Leipzig, Germany in 1646. He was a Roman
Catholic and his religion was a large part of his life (Struik, 1948). At fifteen, Leibniz entered the
University of Leipzig, where he studied law, following in the footsteps of Descartes, Fermat and
Viète (Cooke, 1997). When he graduated at the age of 20 he was deemed too young to receive
his doctorate in law. After college he served as a diplomat for the Elector of Mainz. He
eventually became the diplomat for the Electors of Hanover, where he worked for four decades.
It was during a mission to Paris in 1672 that Leibniz first became interested in mathematics. The
following year he went to London, where he met members of the royal society, Henry Oldenburg
and James Collins (Cooke, 1997). Oldenburg and Collins played major roles in the calculus
controversy. It was said that they told Leibniz of Newton’s advances in calculus (Rickey, 1994).
He discovered his calculus between 1673 and 1676. He was influenced by Huygens and the
studies of Pascal and Descartes. Leibniz died in 1716, in Germany (Struik, 1948).
Results
There are three basic curves that could be the solution to the Brachistochrone problem: a concave
up curve, a straight line, and a concave down curve. The first candidate is the concave up curve;
the second candidate is the straight line; the third candidate is the concave down curve. These
candidate ramps are graphed in Figure 5.
Candidate Ramp One:
Candidate Ramp Two:
Candidate Ramp Three:
(3)