The Unit Circle

Truth to tell, I was disappointed to find a really nice write-up on Wikipedia.  I was hoping that this was still arcane knowledge.  As far as I can tell, it is not commonly taught in trigonometry classes.  We use the sine and cosine for problems with vectors; and that seems to be about it.  In point of fact, these constructions are the root and rock of computational trigonometry.  If you draw well and measure carefully, you can get two decimal places, or fractions to eighths.

Words have meaning.
If you ever suffered from a respiratory allergy,
then you know that it affects the "bay" the "sinus" cavities.
What we call the sine of an angle is the half-sine,
the semi-bay.
Similarly, our "tangent"
is the measure of the semi-tangent,
ignoring the reflection
below the part that we care about.

The trigonometric identities
come from the Pythagorean theorem.

In Feynman's Lost Lecture,  the professor allows that we do our maths with algebra and calculus.  We no longer rely on geometric constructions. Feynman had to create his own derivations for Newton's Laws simply because he could not follow Newton's easy claims about conic sections.  Newton used geometry to create the calculus.  However, calculus is such a powerful tool that we stopped learning the geometry that Newton knew. 

You can write out the algebraic statements
but a picture is worth a thousand words.

In Cosmos, Carl Sagan tells that Pythagoras fled from Samos because he could not tolerate the tyrant Polycrates, whom Sagan denigrates for having "started out as something like a caterer."  (Bold though he was, Sagan shared the anti-capitalist mentality.) But in The Ancient Engineers, de Camp tells us that engineers working for Polycrates bored through a mountain, starting at opposite ends, met in the middle, and were not off by a foot in a mile.  Lacking the positional notation of Arabic numerals, of course, they did all their calculations with geometry.

Descartes is credited with uniting algebra and geometry. 
His work reflects the seemingly intractable 
analytic-synthetic dichotomy of philosophers. 
In truth, it should have erased the distinction 
between the logically consistent and the empirically verifiable.
See the works of Gregory Browne on this blog here and here.

Long ago, at Curious Books in East Lansing, I found a old manual for apprentice carpenters that showed sines, cosines, and tangents to 32nds of an inch for triangles of given measurement.  Many times I have regretted not buying it then.  Even before that, when Laurel and I were first married, her father asked us what calculus was all about. We told him.  He asked, "Do you mean like this?" and took out a foot-sized caliper and showed us how he measured stair cases.  Rise over Run.  Ain't no doubt.

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