It is readily shown that the latter gives the best approximation to θ; 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, θ and 1⁄3θ.
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]—
| chd θ + | 4 chd θ + sin θ | . 1⁄3(chd θ - sin θ) > θ > chd θ + 1⁄3(chd θ - sin θ). |
| 2 chd θ + 3 sin θ |
As might be expected, a fresh view of the matter was taken by René 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 ¼ABCD; by the same process find a point C2 such that the rectangle B1C2 will be equal to ¼BC1; 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∞. 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 ¼c, we can express BB1, B1B2, ... in terms of ¼c, and the identity AB∞ = 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, which serves likewise to throw new light on the subject:—AB being any straight line whatever, and the above construction being made, then AB is the diameter of the circle circumscribed by the square ABCD (self-evident), AB1 is the diameter of the circle circumscribed by the regular 8-gon having the same perimeter as the square, AB2 is the diameter of the circle circumscribed by the regular 16-gon having the same perimeter as the square, and so on. Essentially, therefore, Descartes’s process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.[18]
In 1655 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and algebra playing an important part. In a very curious manner, by viewing the circle y = (1 - x²)½ as a member of the series of curves y = (1 - x²)¹, y = (1 - x²)², &c., he was led to the proposition that four times the reciprocal of the ratio of the circumference to the diameter, i.e. 4⁄π;, is equal to the infinite product
| 3 · 3 · 5 · 5 · 7 · 7 · 9 ... | ; |
| 2 · 4 · 4 · 6 · 6 · 8 · 8 ... |
and, the result having been communicated to Lord Brounker, the latter discovered the equally curious equivalent continued fraction
| 1 + | 1² | 3² | 5² | 7² | ... | |||
| 2 | + | 2 | + | 2 | + | 2 |
The work of Wallis had evidently an important influence on the next notable personality in the history of the subject, James Gregory, who lived during the period when the higher algebraic analysis was coming into power, and whose genius helped materially to develop it. He had, however, in a certain sense one eye fixed on the past and the other towards the future. His first contribution[19] was a variation of the method of Archimedes. The latter, as we know, calculated the perimeters of successive polygons, passing from one polygon to another of double the number of sides; in a similar manner Gregory calculated the areas. The general theorems which enabled him to do this, after a start had been made, are