### On Spherical Triangles.

I mostly like being a lazy bum. Except that I can't do it. At all. So, in order to avoid boredom, I do several things. Like accidentally screw up my 3D drivers and then fix them (but that only took five minutes, maybe), or read Copernicus, an activity that takes far more time than dealing with my computer. Really, it wouldn't be so bad if the theorems were also explained in nice math terms using sines and cosines, etc. But no, he has to make it difficult. Take this passage, explaining that if you are given all sides of an isosceles triangle and the radius of the circle that circumscribes it, you can find the angles of the triangle.

*"For each of the equal sides is to the third side as half of the diameter is to the side subtending the arc by which the angle comprehended by the equal sides is given according to the table, wherein the [360 degrees] around the centre are equal to four right angles. Then the two angles at the base are given as half of the supplementary angle."*

The accompanying figure sheds some light on this, but it wasn't until I redrew the picture myself that I understood exactly what he was talking about. However, one really (like *really*... okay, maybe it's just me) cool thing is that he calculated a table of sines using chords in a circle. Granted, whereas our "circle" has radius 1, his had radius 100000, and whereas we call it the sine of the angle, he called it "Halves of the chords subtending twice the arcs." Good stuff. Anyways, he made this massive table that covers four pages in this book, three columns a page, of sines from 0 to 90 degrees, increasing by one-sixth degree. I'm impressed. The title of this post comes from the last section of Book One of *On the Revolution of Heavenly Spheres* of the same name. Very good stuff.

Yeah... I'll do my best to *not* blog about Copernicus anymore (or to ever split infinitives). I'm sure Wheeler is sitting there staring at the screen with his mouth open. Or maybe crying like a baby. Or maybe he didn't even read this far because he already moved on. Right. So tomorrow I go to church in Sherman. Maybe *that* will be post-worthy.

Posted by Gallagher at May 8, 2004 11:49 PM