One of the original uses for math, back at the dawn of mathematics with the Egyptians and Babylonians, was to determine areas. This originally came up for the problem of determining the areas of land for taxation, but by the Classic Greek period there were plenty of other uses for determining areas or volumes, not to mention the abstract interest of just finding solutions to these problems. The Greeks, in particular, found a great number of practical solutions - not just areas of squares or triangles or circles, but the area under parabolas and other, more complicated shapes. But the Greeks never managed a general solution to the problem of finding areas - that would wait until much later, and an important conceptual advance.
The Greek conception of geometry was much like the basic conception of it today: you have a bunch of things - points, lines circles - which you can do a few basic things to, and with this you can prove things about them. This has the advantage of being close to the real-world view of geometry, a sort of "drawing pictures in the sand" where there's nothing deeper to it than the actual lines you draw. But the downside of this is that it is very, very hard to create more complicated shapes. Proving things by this method pretty much requires that you be dealing with straight lines or circles, so more complicated shapes have to be broken down, for instance by taking them as sections of a cone:
And shapes more complicated still - heart shapes, figure-8s, even spirals - are essentially impossible. This wasn't a huge problem for the Greeks, who accomplished quite a lot with what they had already, but eventually people started trying to come up with new ways to deal with geometry. The most successful by far was the invention, around 1640, of analytical geometry by Rene Descartes (1596-1650). In the years since about 1400, the study of arithmetic had developed into a field concerned with equations in a more abstract sense: algebra. While geometry had more or less stagnated since about the time of Archimedes (287-212 BC), algebra had developed numerous new and useful techniques. So, if the techniques of algebra could somehow be applied to geometry, great strides forward could be made. The obvious problem is that algebra deals with numbers and equations, and geometry... doesn't.
Descartes' solution was to lay two lines at right angles, run numbers down each of them, producing the Cartesian plane, and then express the points and lines of regular geometry as pairs of numbers:
So a line might include the points (0,0), (1, 1), (2, 2), and (as a continuous line) also a number of other points (1.00001, 1.00001), (3.14159, 3.14159), and so on. This at first glance might seem a step backward: we've replaced the intuitive definition of a line with an infinite collection of pairs of numbers. But all of the pairs are pairs (x, y) where x and y are solutions to the equation x = y, and Descartes realized that if you treat the equation as the line, you can do anything to it you could do to the regular geometric line. You can find intersections by solving equations together - one of the oldest and best understood fields of algebra - and most simple shapes, conveniently, have simple analytic expressions: lines are of the form ax + by = c for some numbers a, b, c; circles are x^2 + y^2 = r^2 for some number r; figure-eights are (x^2 + y^2)^2 = 2(a^2)(x^2 - y^2) for some number a; and so on. Most importantly for our purposes, it reduces the problem of finding areas to an algebra problem - an algebra problem Descartes didn't know how to solve, admittedly, but he had more techniques to throw at it now. Over the next couple of decades, he and a few others began making progress with analytical geometry.
Analytic geometry also has wide applications to the sciences: pretty much any physical activity that can be expressed as two numbers can be graphed on the Cartesian plane. A particularly common use is measuring distance traveled vs time, for instance for a bouncing ball:
This opens up a new avenue of investigation. It turns out that if you measure the slope of the line just touching to the graph at any given point, that provides the rate of change at that point: on a distance-time graph, this is the speed the object graphed is traveling at. So now we can find instantaneous velocities just by using a graph! The problem now is that finding a line just touching the graph - a tangent line - turns out to be a nontrivial problem as well. In classical geometry, finding a tangent is trivially easy for a few shapes (straight lines, circles, a few conic sections) and effectively impossible for anything else. So Descartes and his successors set to work on that problem too.
In the 1670s and 80s, two men - Isaac Newton (1643-1727) and Gottfried Liebniz (1646-1716) - independently came up with a method for solving both problems. Newton began by trying to find a general method for finding tangents; Liebniz, for finding areas. Both of them discovered in the process that the two problems are related in a surprisingly intimate way, and used this to develop an amazingly powerful tool, the infinitesimal calculus (nowadays so prominent it is usually simply referred to as the calculus). The two of them then promptly got into an amazingly bitter and lengthy fight over which one plagiarized the other (it is generally believed now that they developed it independently).
We've reached the calculus itself now, so we might as well get into the mathematical details. For reasons of simplicity, most calculus courses usually start with the differential half of calculus (the section devoted to tangents), so the meat of our tour will begin there.
interesting
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