2 + cos [theta]

It is readily shown that the latter gives the best approximation to
[theta]; but, while the former requires for its application a knowledge
of the trigonometrical ratios of only one angle (in other words, the
ratios of the sides of only one right-angled triangle), the latter
requires the same for two angles, [theta] and (1/3)[theta]. Grienberger,
using Snell's method, calculated the ratio correct to 39 fractional
places.[15] C. Huygens, in his _De Circuli Magnitudine Inventa_, 1654,
proved the propositions of Snell, giving at the same time a number of
other interesting theorems, for example, two inequalities which may be
written as follows[16]--

4chd [theta] + sin [theta] 1
chd [theta] + --------------------------- . --- (chd [theta] - sin [theta]) >
2chd [theta] + 3sin [theta] 3

1
[theta] > chd [theta] + --- (chd [theta] - sin [theta]).
3
[Illustration: FIG. 11.]

[Illustration: FIG. 12.]

As might be expected, a fresh view of the matter was taken by Rene
Descartes. The problem he set himself was the exact converse of that of
Archimedes. A given straight line being viewed as equal in length to the
circumference of a circle, he sought to find the diameter of the circle.
His construction is as follows (see fig. 12). Take AB equal to one-fourth
of the given line; on AB describe a square ABCD; join AC; in AC produced
find, by a known process, a point C1 such that, when C1B1 is drawn
perpendicular to AB produced and C1D1 perpendicular to BC produced, the
rectangle BC1 will be equal to 1/4ABCD; by the same process find a point C2
such that the rectangle B1C2 will be equal to 1/4BC1; and so on _ad
infinitum_. The diameter sought is the straight line from A to the
limiting position of the series of B's, say the straight line AB[oo]. As
in the case of the process of Archimedes, we may direct our attention
either to the infinite series of geometrical operations or to the
corresponding infinite series of arithmetical operations. Denoting the
number of units in AB by 1/4c, we can express BB1, B1B2, ... in terms of
1/4c, and the identity AB[oo] = AB + BB1 + B1B2 + ... gives us at once an
expression for the diameter in terms of the circumference by means of an
infinite series.[17] The proof of the correctness of the construction is
seen to be involved in the following theorem, w