![]() So we have a perfectly algebraic way to calculate complex geometric problems. ![]() Ultimately, if we add an infinite number of infinitely small rectangles, we’ll get “infinitely close” - which is to say, our approximation is no longer an approximation, it’s the actual curve. As we add more and more rectangles, we’ll get closer and closer to the truth. Now what if we add a few more rectangles? We’ll still be off, but the more rectangles we add, the closer we’ll get to approximating the shape under the curve. What if instead of one, we used two or three rectangles of different heights? Well, it’s still not really good, but it’s somewhat closer to the truth.ĪLSO READ: Inside the 'Gates of Hell': Turkmenistan's 50-year inextinguishable fire pit Well, as you’d imagine, it’s not very accurate. But what if it’s just a random curve? Well let’s start by being gross about it: let’s approximate the area underneath the curve by a rectangle. ![]() If the curve happens to be a half-circle or some other well-known feature, we’re in luck - we’ve got the formula for that. Let’s say you want to calculate the area under a curve, like Leibniz did. But before that, let’s look at what really makes calculus tick: and that’s infinitesimals. The two are inverse operations (somewhat like addition and subtraction, but not quite), and we’ll get to them in a bit. There are two different types of calculus (differential calculus and integral calculus). Many mathematicians from other parts of the world made contributions to calculus, including a foundational treatise by Italian mathematician Bonaventura Cavalieri - but it wasn’t until Leibniz and Newton that we truly had calculus. Independently, Chinese mathematician Liu Hui discovered the method of exhaustion (which calculates the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the original shape) seven centuries later, around 300 AD. The Ancient Greece mathematician Euxodus discussed the method of exhaustion in the 4th century BC, and Archimedes later built on these methods, developing calculations which resemble calculus. Some mentions of ways to calculate volume and area, one of the first goals of integral calculus, appear in the Egyptian Moscow papyrus from over 3,800 years ago. Many ideas in calculus appeared in Ancient Greece, and even before that. ![]() Truth be told, Newton and Leibniz weren’t the first to dabble with calculus, though they were the first to develop it as a rigorous system. Both men were titans of science and mathematics, and both made valuable contributions that are useful to this day. Honestly, it matters little to us who discovered it. Leibniz, who was supported by scientists from the rest of Europe, was more interested in calculating the area under a curve, a challenging problem for mathematicians at the time. Newton, who received the support of British scientists, needed calculus for studying the planets’ movement across the sky, which was important both scientifically (as the field of astronomy was taking off at the time) and practically because it was important for the navigation of ships. The two worked separately and had a bit of rivalry between them, even ending up arguing over who deserves credit for calculus. Calculus was discovered (or invented) in the late 17th century by Sir Isaac Newton in England and Gottfried Leibniz. Generally speaking, mathematicians speak of discovering a proof or a method, and this also seems to be the case with calculus.
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