The mathematician and philosopher Gottfried Leibniz used those and other mathematical observations to promote a new system of mathematics to calculate areas under curves called calculus. Infinitely small building blocks (such as 1/∞) add up to something if enough of them are used. The instantaneous rate that water drains from the tank can be calculated using infinitesimal approximations. Suppose that a large tank holds 1000 gallons of water. Before calculus, mathematicians, scientists, and engineers could use infinitesimal quantities in calculations such as finding the area under a curve, or approximating the rate of change. Infinitesimals are close to zero and retain properties such as angles or slopes. Using infinitesimals in mathematical calculations was banned in Rome in the 1600’s, and denounced from pulpits and in books.
However, many philosophers hated the idea. Slicing a figure into infinitely many thin fragments was very attractive to many mathematicians and scientists, because it solved a number of practical problems. The sum of all those slices would equal the area of the circle or curved figure, even though the area of one slice was infinitesimal. Suppose a circle or curve were made up of infinitely many polygons, like thinner and thinner slices of pie.
In the 17 th century, the astronomer and mathematician Johannes Kepler looked at a different way to compute the area of a circle or curved figure. Greek mathematicians such as Archimedes used the smallest possible indivisibles to find areas of solids. Although it was very controversial in the 17 th and 18 th century Europe, the practical aspects of using infinitesimal quantities in calculations led to advances in science, engineering, and technology, along with the development of calculus. Using infinitesimal quantities to approximate measurement of any item is an ancient way to determine the size and shape of irregular objects.
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