The history of Calculus can be traced back as early as the 5th Century B.C.E. in Ancient Greece. The first major discovery was made by a Greek scholar named Hippasus. Not much is known about this philosopher who studied under Pythagoras; but, he has sometimes been credited with the discovery of irrational numbers.
\begin{center}
\includegraphics[width = 6 cm]{triangle.png}
\end{center}
He described his reasoning as:
\begin{enumerate}
\item Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by $c:b$.
\item Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).
\item By the Pythagorean theorem: $c^2 = a^2+b^2 = b^2+b^2 …show more content…
Archimedes was also an inventor, scientist, physicist, and an engineer that created many inventions, principles, and plans that had different purposes. One principle he thought of was known as Archimedes’ Principle. He discovered this principle when he was taking a bath and noticed that the water level rose when he got into the tub. From his excitement, he rushed outside naked yelling “Eureka!”, for the principle was then used to determine if jewelry was made entire out of gold or had other elements mixed in to make it cheaper to create. An invention he created was known as Archimedes’ Screw. He invented it for King Hiero II’s ship Syracusia in order to removed bilge water that would leak through the ship’s large hull. The screw consisted of a revolving screw-shaped blade within a cylinder that would be hand-turned to raise water from a low altitude to a higher altitude. It was most commonly used in irrigation canals to obtain water from a lower plain. Another invention created was known as the Claw of Archimedes. It consisted of a large, metal hook that is suspended by a crane-like arm on shore. It was used to sink ships by dropping the heavy hook onto the ship and swinging it up, hoping to sink the ship. It was proven to be a workable device in 2005 during a television documentary called “Superweapons of the Ancient …show more content…
He showed the solution as an infinite geometric series with the common ratio $\frac{1}{4}$.
\begin{center}
\includegraphics[width = 9 cm]{infinitegeometricseries.png}
\end{center}
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series:
$\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + ...$
which sums to $\frac{1}{3}$.
Sadly, Archimedes’ life came to an end in 212 BC, for he was slain by a roman soldier who thought the mathematical instruments Archimedes was carrying were valuable. The roman general Marcus Claudius Marcellus was enraged by this news since he specifically ordered his troops to not harm Archimedes, for he was a valuable scientific asset. His last words were reported as “do not disturb my circles” to the roman soldier, for he was studying the circles in a mathematical drawing when he was
\begin{center}
\includegraphics[width = 6 cm]{triangle.png}
\end{center}
He described his reasoning as:
\begin{enumerate}
\item Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by $c:b$.
\item Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).
\item By the Pythagorean theorem: $c^2 = a^2+b^2 = b^2+b^2 …show more content…
Archimedes was also an inventor, scientist, physicist, and an engineer that created many inventions, principles, and plans that had different purposes. One principle he thought of was known as Archimedes’ Principle. He discovered this principle when he was taking a bath and noticed that the water level rose when he got into the tub. From his excitement, he rushed outside naked yelling “Eureka!”, for the principle was then used to determine if jewelry was made entire out of gold or had other elements mixed in to make it cheaper to create. An invention he created was known as Archimedes’ Screw. He invented it for King Hiero II’s ship Syracusia in order to removed bilge water that would leak through the ship’s large hull. The screw consisted of a revolving screw-shaped blade within a cylinder that would be hand-turned to raise water from a low altitude to a higher altitude. It was most commonly used in irrigation canals to obtain water from a lower plain. Another invention created was known as the Claw of Archimedes. It consisted of a large, metal hook that is suspended by a crane-like arm on shore. It was used to sink ships by dropping the heavy hook onto the ship and swinging it up, hoping to sink the ship. It was proven to be a workable device in 2005 during a television documentary called “Superweapons of the Ancient …show more content…
He showed the solution as an infinite geometric series with the common ratio $\frac{1}{4}$.
\begin{center}
\includegraphics[width = 9 cm]{infinitegeometricseries.png}
\end{center}
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series:
$\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + ...$
which sums to $\frac{1}{3}$.
Sadly, Archimedes’ life came to an end in 212 BC, for he was slain by a roman soldier who thought the mathematical instruments Archimedes was carrying were valuable. The roman general Marcus Claudius Marcellus was enraged by this news since he specifically ordered his troops to not harm Archimedes, for he was a valuable scientific asset. His last words were reported as “do not disturb my circles” to the roman soldier, for he was studying the circles in a mathematical drawing when he was