“application of areas”

Question

As a word nerd, it was thrilling to learn that the word *parable* is related (metaphorically and etymologically) to the Latin word *parabola* + the Greek *parabole.* Online Etymology Dictionary explains that the sense of “juxtaposition” owes to the fact that the curve is “produced by ‘application’ of a given area to a given straight line.” So I’m digging around to learn more about the so-called *application of areas* and like, there’s a reason geometry was one of my least favorite subjects to study. It ought not be!

So, tell me — what is the application of areas? In what sense can an area be “applied”? To what? How the hell is this related to cones (and, as I’m seeing, squares/rectangles)?

In: 2

Answer 1

First off, I’m really not sure what you’re getting it. Can you provide the source language on the application of areas?

since you specifically mentioned cones – there is a *whole branch* of math that revolves are something called “conic sections”. Without getting into it too deeply. If you imagine looking at a cone from the top down, you’d have a circle. You could then use math to describe the “shape” of a circle (the line traced by spinning at a fixed length from a single, central point – the center and the radius).

Then imagine slicing the cone on a bit of slant, you’d have an ellipse, aka an “oval” (oval is not a proper geometry term).

[If you look at these images](https://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php) you’ll see that shapes called “hyperbolas” (also related to a literary term – hyperbole!)

Anywho, the point here is that from a math point of view, the equations you’d use to describe curving lines all go back to the equations you’d use to describe sections cut through a cone. It’s obviously pretty hard math to describe in a paragraph but it’s all insanely useful techniques for people who study both math and physics, for example, concepts like acceleration, all go back to curved lines, which go back to conic sections, which go back to basic geometry you learned in 8th grade, just harder.