![]() These may be paradoxes of thought, but they are not paradoxes in reality. To cover a quarter of the distance, Athena must first cover half that distance, i.e., an eighth of a mile, and so on. To cover half a mile, Athena must first cover half that distance, i.e., a quarter mile. So how will Achilles ever catch the tortoise?Ī similar paradox stated by Zeno, called Dichotomy, states that if a person, let’s call her Athena, wishes to walk one mile, she must first cover half a mile. Achilles must now close this distance, during which time the tortoise will have moved even further. By the time Achilles has closed it, the tortoise has moved some distance further, say, 10 meters. In order to pass the tortoise, Achilles must close that 100 meter gap. Achilles gives the tortoise a head start of, say, 100 meters. Another spacial application of an infinite series is Zeno’s Paradox of Achilles and the tortoise, which goes something like this:Īchilles and a tortoise decide to race. We saw a geometric version above in the method of exhaustion. Paradoxes of infinityĪnother integral (ahem) part of calculus is the infinite series, which is a series of numbers that continues indefinitely. In integral calculus, which is concerned with finding the areas beneath curves (among other things), the principal strategy is to divide these areas up into increasingly infinitesimal slices, find the areas of these slices, and then add them together to find the area of the whole. But it’s not difficult to see how this idea relates to integral calculus. Of course, Archimedes, Liu Hui, and their contemporaries did not invent calculus. Several hundred years later, the Chinese mathematician Liu Hiu used similar methods to calculate π and determine the area of a circle. Archimedes used it to find the area of a circle, to find a very accurate value of π, and for other purposes besides. Taking this idea to its logical conclusion is known as the Method of Exhaustion. And since the pentagon comes closer to approximating the area of the circle (there’s less of area outside the pentagon and inside the circle than in our previous drawing with the square), you’re moving closer to finding the area of the circle. What if instead of drawing a square, you drew a pentagon? Since you can compute the areas of triangles, and a pentagon is a collection of five equal triangles, you can calculate its area. It won’t be particularly accurate (in fact, you’ll be off by more than 50 percent), but you’ll be on the right track.Īn illustration of the Method of Exhaustion as applied to finding the area of a circle. Measure this square and you’ll have an approximation of the area of a circle. One strategy would be to draw a square inside the circle such that the four corners of the square lie on the circumference of the circle. In fact, you don’t know even know a precise value for π. Say you want to find the area a of a circle of radius r. To see why, I’ll cover what the Greeks and Chinese knew about exhaustion and what the Keralites knew of infinite series, and then move forward to the era of Newton and Leibniz. Perhaps none of these statements are mutually exclusive. Or maybe their work was just a natural outgrowth of seeds sown by the Greeks two millennia before. The article describes the development of ‘calculus’ over the years.Calculus was invented in the late 17th century by the Englishman Isaac Newton. In short, calculus is the mathematical study of continuous change. ![]() This happened primarily during the Rigorization stage. But it took a lot of time for mathematicians to justify their methods and put calculus on a sound mathematical foundation. Newton and Leibniz laid the foundation for calculus in the development stage, bringing all the techniques under the umbrella of derivative and integral. In the anticipation stage, mathematicians used techniques which involved infinite processes to find area under curves. Anticipation, Development and Rigorization. The development of calculus can be mainly divided into three periods. While Newton thought Calculus was geometrical, Leibniz took it towards analysis. Newton, on the other hand, used finite velocities x’ and y’ to compute the tangent. Despite knowing the fact that dy/dx gives the tangent, he did not use it as a defining property. He used dx and dy as differences between successive values of such sequences. Newton considered variables changing with time.On the other hand, Leibniz assumed variables x and y as sequences of infinitely close values. They independently developed the foundations for calculus. Isaac Newton and Gottfried Leibniz are two key men behind the discovery of Calculus. ![]()
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