How were trig tables made before computers or calculators?

How were trig tables made before computers or calculators?  I've read before about how early mathematicians made logarithm tables.  The methods are heavy on tedium, low on cleverness.  I've also read how some mechanical devices were contrived to estimate trig functions (sine, cosine, and tangent), but the precision of these devices was quite low: a couple of digits, three at the very most.  But I knew that mathematicians were using much higher precision trig functions hundreds of years ago.  How did they do it?

Here's an explanation of how they got tables at one degree intervals.  That's enough for many kinds of work, but I know there were tables at much smaller intervals (down to 10s of arc-seconds, I believe) that filled entire books.  I don't fully understand the math being described here, but I'm guessing that method for getting to one degree intervals can be extended to smaller intervals.  What jumps out at me about the method is that it involves some clever hoop-jumping, along with an awe-inspiring amount of tedium as well...