Viete / Quadrature by parts,trigonometry

A spectrum of people have made a challenge to solve number . Viete(1540-1603), one of these challenger, first succeeded in representing it by a precise equality while it was an infinite product. If a square has a unit circle (1 in diameter) circumscribed about it; an equilateral octagon does by doubling the number of the former sides; an equilateral hexadecagon does by doubling the number of the former sides; and so on, then what he deduced is:

Viete researched trigonometry analytically.
Since the decimal fraction of the decadal system was undeveloped at that point in time, Viete went around studying hypotenuse by setting it equal to 100,000 in order to avoid decimal fraction.

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If we make , in this expression, then we will get:

Since then, this expression came into widespread use in Europe as a means to simplify multiplications in astronomic evaluations.
For instance, first of all, the value of 100,000 is assigned to . And then, an angular measurement table where is changed from 1 to 100,000, and another angular measurement table where is changed from 1 to 100,000 are prepared. It would be a 6-digit calculation in this case but, at that time in Europe, calculations using 12-digit and 15-digit were also done.

Let's take 78 x 62 as a calculation example.
First thing to do is to find carrying 78 as from the entries in the table and, likewise, find carrying 62 as . The next step is to find and from the entries in the table. We can obtain the value of by adding the aforesaid two values and then dividing the sum by 2.

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