It took mathematicians centuries to determine how to work with curves.

During the era of the Ancient Greeks, the pre-eminent goal when it came to curved shapes was to “square the circle.” This is just a strange way of saying “find the exact area of a circle.” What the Greeks were saying was this: if you can find the exact area of a circle, then you will be able to express that area in the form of a square with equal area. The Greeks thought very, very geometrically. They didn’t really “do” equations, especially before the time of Archimedes. They did shapes… they compared them and divided them and otherwise manipulated them.

Note that when Ancient Greeks talk of squaring the circle, it is “the” circle they are talking about, not “a” circle… The Greeks sought for mathematical truths which would be true for ALL circles (or for ALL squares, or for ALL triangles, etc…). They strove to discover universal truths, not the particular truths about some particular shape.

Greek mathematicians did not express areas in terms of square-feet or what-have-you. That would have been beneath their dignity. People who got their hands dirty –like bricklayers and carpenters– they were the ones who used those sorts measurements. Typically, a Greek mathematician (and by “mathematician” the Ancient Greeks mostly meant “geometrist”) would speak only of generic “units” of lengths.

As Boyer tells us in his book, The History Of The Calculus And Its Conceptual Development, a Greek mathematician would not have even known what to make of the question, What is the area of this pentagon?– other than offering to tell you that its area would be less than, greater than, or equal to some other shape’s area. Greek mathematicians trafficked in shape congruence, not equations. The use of numbers was kept to the barest minimum.

One of the most ancient methods of “squaring the circle”– that is, finding the area of a circle– was that of inscribing a polygon inside the circle.

The Greeks knew how to find the area of a polygon. So, they ingeniously came up with a method of inscribing inside a circle a many-sided polygon which would approximate the same area of the circle, and then they would solve for the area of the polygon.

Now if you picture a polygon with only a few sides, like say, a pentagon or octogon, then you can see in your imagination that there will be quite a lot of circle area still falling outside the perimeter of the polygon. But what those clever Greeks learned to do was to bisect the angles of their inscribed polygon and thereby double the sides of their polygon. Then, they’d repeat the process, and repeat it and repeat it. Eventually, after enough side-doublings, they would succeed in pushing perimeter of the polygon up very, very close to the circle’s circumference.

When the polygon had so many sides that its perimeter was practically indistinguishable from the circumference of the circle, this was called “exhausting the circle,” and it does indeed sound exhausting.

However, the Greeks knew that, even if the polygon became so many-sided that it looked to the human eye indistinguishable from the circle, there would still be some area of the circle left-over, remaining outside the polygon. And unlike my family, the Greeks didn’t like left-overs.

I should mention that, though we talk of exhausting the CIRCLE, this same method will work for semi-circles as well. This means that it’s applicable to shapes which contain curves as part of their perimeters. What one does to find the areas of these “mixed” shapes is to find the area of the uncurved part of the shape and then separately find the area of the area bound by the curve, and then add them together. Thus, the “exhausting the circle” method is actually used quite often for semi-circles, and you will frequently read of mathematicians or geometrists speaking of finding the area “beneath the curve.”

When the brilliant Greek mathematician Archimedes wanted to find the area “beneath the curve,” he would form his inscribed polygon by combining triangles. He was able to find the sum of this series of triangles to infinity, adding together the areas of all the triangles and then tossing in the left-over part as well. I’m not sure how Archimedes arrived at the number to use for the left-over part, but he did say that by using smaller and smaller triangles, “the remainder can be made as small as desired.” And he knew something else as well… that as the triangles grew more and more numerous, their sum came closer and closer to some particular value. This value, said Archimedes, could be neither less than the area of the circle, nor more. Perhaps this was all Archimedes needed to know in order to determine how much area was “left-over.”

Using this method brought Archimedes very close to something like the modern-day LIMIT (thought it won’t be called by that name for a long time yet). The main difference between the work of Archimedes and the modern day Limit is that the Limit has no “left-over” part you have lump into it at the end. The Limit IS the sum, it IS the area beneath the curve.

Nicholas Of Cusa approached the problem slightly differently, philosophically speaking. He accused the circle of being nothing more than the polygon with the largest number of sides.

Kepler, following Cusa, said that – since a circle is a polygon with an infinite number of sides– we can think of the circle as a immense group of skinny triangles… triangles with their tops all meeting at the center of the circle, and with their bases comprising the circumference of the circle. We can then find the area of the circle by finding the area of the triangles which form it. The calculation would go something like this…

Since we know the heights of all the triangles making-up the circle (the radius of the circle), and since we know that the summation of all their bases is equal to the circumference of the circle, we can take the area formula for triangles ( 1/2 base * height) and, substituting, re-write it as…

Area = 1/2 circumference * radius

And we don’t have to just take Kepler’s word for it. Every middle school student knows (I hope) that the formula for circumference is: 2*pi*r. We can substitute this formula for “circumference” in Kepler’s equation above…

if Area = 1/2 [circumference] * r, then…

A = 1/2 [2*pi*r] * r, simplifying to…

A = pi*r^2

…which we smart moderns know is the area of a circle (or as the Ancient Greeks might prefer it stated: the area of THE circle).

Another way in which the area of a circle can be found is to find the area of the largest polygon which could be drawn inside the circle (like the Ancient Greeks did), but then also find the area of the SMALLEST polygon which could be drawn outside the circle. The area of the circle could then be said to lie between these two polygon-areas.

Boyer credits Antiphon and Bryson with coming up with the two-polygons approach. Bryson went so far as to say that the circle-area would be equal to the AVERAGE of the two polygons, though I think he guessed this from intution and did not derive the rule via some rigorous proof.

Valerio, even without taking the average, bragged that he could use the inscribed/circumscribed polygons method to find the area of a circle so exactly that (in Boyer’s words): “the approximating inscribed and circumscribed figures could be made to differ by less than any magnitude.” In the words, Valerio claimed he could find the area of a circle as exactly as anyone cared to know.

By the way, the granddaddy of geometry, Euclid, was content using only the single, inscribed polygon method for determing circle area. Perhaps because, with the two-polygon method, the notion that the area would be the average of the two polygons was without a solid proof.

Today, a series of polygons with more and more sides which arrive closer and closer to the value of the circle containing them would be known as an Infinite Series. Boyer states that it was Gregory Of St. Vincent who first explicitly spoke of the Infinite Series and of how it defines itself by summing toward a magnitude which he called the “limit” of the series. Thus, we have one of the first, if not the first, mention of the modern term Limit.

Boyer also informs us that the method of inscribing polygons changed “after the introduction of analytical geometry” by Descartes and his generation [basically, “analytical geometry” is the geometry we know today, in which equations represent curves, areas, shapes, etc are represented by equations]. After Analytical Geometry hits the scene, “it became customary,” writes Boyer, “in order to find the area of a curvilinear figure, to substitute for the series of approximating polygons a sequence of sums of approximating rectangles.”

If you can imagine the change here, mathematicians go from drawing a single many-sided polygon beneath the whole curve, to the drawing of a series vertical rectangles subdividing the area beneath the curve. This is how the Integral of Calculus is usually taught today at the visualization stage of instruction. The base of each rectangle represents “x plus the change-in-x.” The smaller the change-in-x becomes, the narrower these rectangles shrink.