Confining ourselves strictly to the best ascertained steps in the history of the ques tion, we remark that Archimedes proved that the ratio of the diameter to the circumfer ence is greater than 1 to WI, and less than 1 6 to 3. The difference between these two extreme limits is less than of the whole ratio. Archimedes' process depends upon the obvious truth, that the circumference of an inscribed polygon is less, while that of a circumscribed polygon is greater, than that of the circle. His calculations were •extended to regular polygons of 96 sides.
Little more seems to have been done by mathematicians till the end of the 16th c., when P. Metius gave the expression for the ratio of the circumference to the diameter as the fraction VI:, which, in decimals, is true to the seventh significant figure inclu sive. Curiously enough, it happens that this is one of the convergent fractions which express in the lowest possible terms the best approximations to the required number. Metius seems to have employed, with the aid of far superior arithmetical notation, a process similar to that of Archimedes.
Vieta shortly afterward gave the ratio iu a form true to the tenth decimal place, and was the first to give, though of course in infinite terms, an exact formula. Desig nating, as is usual in mathematical works, the ratio of the circumference to the diam eter by 7e, Vista's formula is: 1 1 , 1 , ff r --I- T y T 2 x etc.
Shortly afterward, Adrianus Romanus, by calculating the length of the side of an equilateral inscribed polygon of 1073741824 sides, determined the value of ir to 16 sig nificant figures; and Ludolph von Ceulen, his contemporary, by that of the polygon of 36893488147419103232 sides, arrived (correctly) at 36 significant figures. It is scarcely possible to give, in the present day, an idea of the enormous labor which this mode of procedure entails even when only 8 or 10 figures are sought; and when we con sider that Ludolpli was ignorant of logarithms, we wonder that a lifetime sufficed the attainment of such a result by the method he employed.
The value of Tr was thus determined t-o 3 of its amount, a fraction of which, after Montucla, we shall attempt to give an idea, thus: Suppose a circle whose radius is the distance of the nearest fixed star (250,000 times the earth's distance from the sun), the error in calculating its circumference by Ludolph's result would be so excessively small fraction of the diameter of a human hair as to be utterly invisible, not merely under the most poWerful microscope yet made, but under any which future generations may be' able to construct.
These results were, as we have pointed out, all derived by common arithmetical. operations, based on the obvious truth that the circumference of a circle is They than that of any inscribed, and less than that of circumscribed polygon. They involve none of those more subtle ideas connected with limits, infinitesimals, or differentials; which seem to render more recent results suspected by modern " squarers." If one of that unhappy body would only consider this simple fact, he could hardly have the presumption to publish his 3.125, or whatever it may be, as the accurate value of a quantity which by common arithmetical processes, founded on an obvious geometrical truth, was several centuries ago shown to be greater than 3.14159265358979323846264338327950288, and less than 3.14159265358979323846264338327950289.
We now know, by far simpler processes, its exact value to more than 600 places of decimals; but the above result of Von Ceulcn is much more than sufficient for any possible practical application even in the most delicate calculations in astronomy.
Snellius, Huygens, Gregory .de Saint Vincent, and others, suggested simplifications of the polygon process, which are in reality seine of the approximate expressions derived from modern trigonometry.
In 1668 the celebrated James Gregory gave a demonstration of the impossibility of effecting exactly the quadrature of the circle, which, although objected to by Huyghens, is now received as quite satisfactory.
We may merely advert to the speculations of Fermat, Roberval, Cavalleri, Wallis, Newton, and others as to quadrature in general—their most valuable result was the• invention of the differential and integral calculus by Newton, under the name of. fluxions and fluents. Wallis, however, by an ingenious process of interpolation, showed that 7r2.4.4.6.6.8.8.10.10. etc.
4 3.3.5.5.7.7.9.9.11. etc.