Angles, Inclination and Slope for Dummies

angle of inclination, slope

It's often said that those who can, do; those who can't, teach – but that was before the information age brought us the internet and content farms. Now, those who can't, write about it in hopes of making money doing so. Accuracy and facts be darned, there are pennies to be collected! Among those content farms, we find good old eHow.com, where mere possession of an English or Journalism degree was presumed to render the holder omniscient – or at least able to copy, reword and paste anything. Unfortunately, that assumption was bogus, as amply demonstrated by eHowian Nicolette Calhoun (BA in English, of course), whom we found holding forth on "The Angle of Inclination of a Slope" at Sciencing.com.

We won't bother to mention that the topic at hand is pretty much nonsense, since "angle of inclination" and "slope" more or less refer to the same quantity: the angle between a line and the X-axis of a graph, or between a surface and a horizontal plane. We'll let Nicolette explain things for you...

"...the angle of inclination is the measure of the space between two lines on a graph. Because lines on a graph are often drawn at a diagonal, this space is usually triangular in shape"

Ummm, yeah: except that Calhoun forgot that a triangle has three sides, not two. Apparently she didn't think she was allowed to use the word "angle" in her definition... Nevertheless, let's forge ahead and see what else Nicolette had to say. Oooh, wait: she actually got around to it! "Because all triangles are measured by their angles, this space between two lines must often be represented by 'angles' of inclination. When the slope of a line can not be measured in the conventional way, we can use the angle of inclination because the angle of inclination and the slope of the line are actually equal..." ...although we are still confused about how, if the two terms are equivalent, you could measure one "in the conventional way" but not the other. We guess Calhoun will be telling us... right after she explains how an angle can be a "space."

But no, Nicolette's not going to tell us. Instead, she's going to explain that

"A slope is a ratio of change from the vertical to the horizontal of a line on a graph..."

Well, no: she's close, but "from the vertical to the horizontal" makes no sense. Slope is the ratio of the change in vertical to the change in horizontal, if you must use those words. So let's see what other math stupidity Calhoun can come up with....

• "On a regular graph, the x- and y-axis bisect each other on the perpendicular..." [we think she meant "intersect"]
• "...inclination is the measure of the positive section of the x-axis (the upper two quadrants of a graph) until it hits a line..." [say what?!]
• "Tangent... measures the angle made by the two lines of a triangle that are not the hypotenuse." [Ummm, no, the tangent doesn't measure an angle, it is a property of the angle.]

Even a blind pig finds the occasional acorn, and Calhoun is no exception. Her final sentence explains that

"The slope of a line and the tangent will always be equal to each other."

That sentence would be demonstrably true if she had said "the tangent of the angle of inclination," but what she said is close enough for government work. Of course, mathematics is not government work, not to mention that by the time the reader had waded through 520 words of unmitigated bull to find the 15 that explain the relationship, said reader would be so confused s/he'd want to throw a Dumbass of the Day award at Calhoun -- and that's what we did.

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