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Jim, you asked:
"Were infinite series used in the initial creation of tables?"

No, not the early tables.

One way to generate tables is to use the trigonometric identities for sums and 
differences of angles. Consider this: we know some exact values, e.g. sine 
and cosine of 0�, 30�, 45�, 60�, 90�. If you have arbitrarily good values for 
the square root of two and the square root of three, then you have 
arbitrarily good values for the trig functions of those angles. Now we can 
also calculate the sine of one degree to fairly good accuracy using a variety 
of methods (see below). That's our anchor. And the cosine of one degree is 
sqrt(1-sin^2(1)) so we calculate that one by hand as accurately as possible. 
Then how do we get the sine and cosine of 2 degrees? Use the identities:
sin(a+b) = sin(a)*cos(b) + sin(b)*cos(a),
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), 
which in this case would be
sin(2�) = sin(1�)*cos(1�) + sin(1�)*cos(1�),
cos(2�) = cos(1�)*cos(1�) - sin(1�)*sin(1�).

And then the sine and cosine of 3 degrees? Just build on the previous result:
sin(3�) = sin(2�)*cos(1�) + sin(1�)*cos(2�),
cos(3�) = cos(2�)*cos(1�) - sin(2�)*sin(1�).
And so on... 

And note that you really only need to calculate from 0 to 45 degrees. After 
that, you just re-label the table using sin(x) = cos(90-x) etc.

Next we look up all of these values in our table of logarithms (let's assume 
someone has already provided us with that) and then we can get a table of 
tangents rather quickly from tan(a) = inverselog[logsin(a)-logcos(a)].

These are the general sorts of calculations that were used. There are lots of 
tricks for speeding things up and reducing error. For example, we can get 
exact values for the sine and cosine of 15 degrees using half-angle 
identities. And from there, the sine/cosine of 7.5 degrees. Then going back 
to the sum identities, we get "exact" values for any angle that is a multiple 
of 7.5 degrees. 

Finally, up at the top I mentioned that we need the sine of 1 degree. At some 
level of accuracy the sine of a small angle is equal to the arc. That is, 
sin(x degrees) = x*pi/180, for a small angle. For a low accuracy table, you 
could assume that this holds for x=1�. For a higher accuracy table, you would 
assume that this holds for x=1'. Then you build up arcminute by arcminute 
until you get a better value for the sine of 1 degree and proceed again as 
above.

-FER



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