We all know what it means to exaggerate. The word comes from Latin:

*ex*= “thoroughly” +

*aggerare* “heap, up” [*ad*– “toward” + *gerere* “carry”].

But like many concepts in English we also have a Greek derived word with similar meaning: “Hyperbole”: Gk huperbole, * huper*, “above” + *bole*, “throw”.

But “hyperbole” is sometimes just used to mean the same as exaggeration, it is often used to identify an intentional literary or speech device which is not meant to be taken literally. Thus here are the common definitions of “hyperbole”:

- Exaggeration
- Intentional Exaggeration
- Intentional Exaggeration not intended to be taken literally

As a communication tool, hyperbole can be used in the following ways:

- To grab attention
- To emphasize a contrast
- To deceive

A hyperbole is effective at contrasting one idea against another — it makes their differences clear albeit with gross exaggeration. This can wake up the listener and help them realize that the speaker is introducing a new paradigm. Whereas if a speaker uses slow, careful, caveat-laden comparisons and descriptions to contrast two concepts, a listener may not really get their point or may get tired of listening. “Hyperbole” is a great rhetorical tool. It makes the contrasting idea easy to remember and often easy to apply. Such is the simple nature of the human mind.

Well that is all great for the mind ready to be moved. Nonetheless, if the listener has no desire to be swayed, they may point out the exaggeration of the hyperbole and focus only on its inaccuracies. They may not forgive the rhetoric. Hyperbole is a rhetoric tool but it disobeys all sorts of logic rules. But when the goal of the communication is victory and not truth, a competitor will choose their weapon appropriately.

Finally, some geometry to explain the picture used in this post. As I said, the etymology of “hyperbole” is:

“Hyperbole”: Gk huperbole . to huper, “above” + bole, “throw”.

When we throw an object, it follows a certain geometric shape — a parabola which is related to a hyperbola. “Hyperbol**a**” has the exact same etymology as hyperbol**e**. While reviewing the definitions of hyperbola and parabola, I found that they and circles and ellipses were simply sections of a cone. But none of the definitions I found were elegant — none put explained the differences in these shapes in clear, yet concise terms. So I will offer Sabio’s elegant definition of Conic Sections below:

Four geometric figures are determined by the intersection of a (non-vertex) plane with the sides (nappes) of a cone. The figure types are determined by the acute angle formed by the plane and the cone’s axis.

**Hyperbola** = 0 degrees (parallel axis) to degree of Nappe Angle

**Parabola** = degree of Nappe Angle

**Ellipse** = degree of Nappe Angle to 90 degrees

**Circle** = 90 degrees (perpendicular to axis)

Note: for simplicity I limited to planes which do not include the vertex of the cone. Otherwises Lines and a Point must be included as possible conic sections.

Math folks, please help me if I have erred. Others, if you have read this far, let me know what you think about the literary tool of hyperbole.

See other “Word!” posts, here.

—Math note—Almost, right. Cones in math are ‘double cones’: imagine two cones touching at their point, end to end, going off to infinity.

Your hyperbola is any section than intersects with both parts of the cone. It doesn’t have to be at 90 degrees, it can be at any angle up to the angle of slope of the sides of the cone.

The other 2D conic sections is the parabola, where the plane doesn’t intersect with the second part of the cone.

If the slope of the cone is x, and the planes don’t pass through the vertex of the cone:

Then

Hyperbola: 0 < angle < x

Parabola: angle = x

Ellipse: x < angle x),

Single line (angle = x)

Pair of lines crossing (angle < x).

If the cone is allowed to have 0 slope (i.e. it is a cylinder), then

you get the set of cylindrical sections:

Circle (angle = 90deg), Ellipse (0 < angle < 90) and at angle = 0

either double parallel lines, or (if the plane just touches the edge

of the cylinder) single line.

Thus the 8 conic sections: hyperbola, parabola, ellipse, circle, crossing lines, parallel lines, single line and point.

—Math note—Drat, my less than and greater than signs were stripped. Can you recover them, Sabio?

[I have edited Ian’s first comment after an e-mail he sent – Thanx Ian]

-math note 2-

When you throw a ball, it forms a parabola, not a hyperbola. A parabola is basically half a hyperbola, and only has one focal point (a hyperbola has two). Hyperbolas are best thought of as either double parabolas or inside-out ellipses. Mathematically, they more closely resemble the latter.

Parabola: y=ax*2* +bx+c

Hyperbola: [(x-h)/a]*2* – [(y-k)/b]*2* =1

Ellipse: [(x-h)/a]*2* + [(y-k)/b]*2* =1

Circle: x*2* + y*2* =c*2*

*2* is suppose to be an exponent, but I can’t do those in this format. Can you fix this?

-end math note-

Good post. I spend much of this last weekend debating and certainly used hyperbole, though mostly because I was speaking impromptu and lacked statistics. It is an effective speaking tool, though it does sometimes backfire. It does lead away from truth, but obvious hyperbole is harmless, as it is known to be an exaggeration.

@ Ian

I rushed out the post this morning. I have amended the definition. The point of the my definition is elegance — I think it all depends on the angle of the plane and no talk about how many cones are hit, whether the figure is closed or open or anything else matters. See if you agree with my adjusted simple definition — thanks for your corrections.

Send me an e-mail of where you want signs in your comment, and I will amend it. Thanx

@ IanI think your definition is mistaken (or is it me)?

If the plane does not contain the vertex, Points and Lines don’t occur.

I wanted to keep the definition simple and thus did not include cylinders (slope = infinity (again, I think you were wrong on that) or planes (slope = 0) (depending on orientation of cone on cartesian grid, of course).

But do you see my point of elegance? When you scan the web (including Wolfram and Wiki), I don’t see the description made as simple as mine.

BUT, is do you thinkCRL correct?

@ CRLYou got the main point of this post — the uses and misuses of hyperbole — which I will use in my next post. Thanks.

But back to the math part that Ian is helping me on. You were right about trajectories being parabolas and not hyperbolas — thanx, I changed it. Also thanx for the formulas (I knew them, of course) but I wanted to capture the definitions visually (geometrically) with elegant, concise wording.

And I think you are wrong that “

A parabola is basically half a hyperbola“.Or am I wrong?

I think the equations change after crossing that angles I listed and thus the two figures are not the same and don’t behave the same in a two dimensional plane (ignoring position or dup

@ Ian

Sorry, dude, I keep trying to correct your comment but WordPress won’t accept certain changes — must be the less than and greater than sides it mistakes for code. Try LaTex.

But simpler than that, are you comfortable with my modified definition?

Well I wouldn’t have been wrong if the comments had not been munged.

If the plane does not contain the vertex and the angle x is greater than 0 and less than 90 degrees, then the only conics are circle, ellipse, parabola and hyperbola. But your modified post is still wrong on when they will occur. Circle (angle is exactly 90deg), Ellipse (between 90 and the slope of the cone), Parabola (exactly the same as the slope of the cone), Hyperbola (less than the slope of the curve). The part about what parts of the cone the plane intersects are completely defined by these angles [giving the initial two assumptions].

Your second point about slope being infinite. Yes you could say that, although your original post talked in terms of angles, rather than ratios, so the angle is definitely not infinite. It is zero, or 90deg, depending on what axis you measure from. When you talk about slope you talk about tangents of angles, so that would be 0 or infinity, depending again where your reference axis is.

Having the description in terms of angles alone is fine, as you have it (i.e. not wanting to talk about where the plane is). But you have to be careful, because you’ve only got there by excluding 4 other perfectly valid conic sections. And to make sure of that you do need to talk about where the plane is, not just its angle.

There’s an excellent book, a classic in mathematics, called “Proofs and Refutations” by Imre Lakatos (I’m remembering, maybe spelled differently) that talks about this kind of issue. It is basically about the linguistic precision of mathematics and how talking about mathematics is as vague a linguistic exercise as talking about anything else.

In the case of conics your explanation hits a particular level of generality well, but it isn’t something that would help you much if you came to do anything with conics. For that you’d want more precision, as long as (with Lakatos) you realise that no degree of precision is absolute.

CRL is right: thrown things follow a parabola, not a hyperbola. I don’t agree that a parabola is half a hyperbola, but I think CRL was talking about the apparent shape, rather than their mathematical properties. Sketched, you’d have a hard time telling the difference for some hyperbolae.

[erm sorry, just reread your modifications, you have put right when each shape will occur]

Others, if you have read this far, let me know what you think about the literary tool of hyperbole.I think too many (a)theist writers use it, and I enjoyed the OP.

It’s wrong mathematically. That’s just what it looks like. The equations and shapes do change because the hyperbola adds a y-squared term.

What’s wrong mathematically. Is my definition wrong?