THE RATIO OF THE DIAMETER TO THE CIRCUMFERENCE.
The proportion of the diameter of a circle to its circumference has never yet been exactly ascertained. Nor can a square or any other right-lined figure be found that shall be equal to a given circle. This is the celebrated problem called the squaring of the circle, which has exercised the abilities of the greatest mathematicians for ages and been the occasion of so many disputes. Several persons of considerable eminence have, at different times, pretended that they had discovered the exact quadrature; but their errors have readily been detected; and it is now generally looked upon as a thing impossible to be done.
But though the relation between the diameter and circumference cannot be accurately expressed in known numbers, it may yet be approximated to any assigned degree of exactness. And in this manner was the problem solved, about two thousand years ago, by the great Archimedes, who discovered the proportion to be nearly as seven to twenty-two. The process by which he effected this may be seen in his book De Dimensione Circuli. The same proportion was also discovered by Philo Gadarensis and Apollonius Pergeus at a still earlier period, as we are informed by Eutocius.
The proportion of Vieta and Metius is that of one hundred and thirteen to three hundred and fifty-five, which is a little more exact than the former. It was derived from the pretended quadrature of a M. Van Eick, which first gave rise to the discovery.
But the first who ascertained this ratio to any great degree of exactness was Van Ceulen, a Dutchman, in his book De Circulo et Adscriptis. He found that if the diameter of a circle was 1, the circumference would be 3·141592653589793238462643383279502884 nearly; which is exactly true to thirty-six places of decimals, and was effected by the continual bisection of an arc of a circle, a method so extremely troublesome and laborious that it must have cost him incredible pains. It is said to have been thought so curious a performance that the numbers were cut on his tombstone in St. Peter’s churchyard, at Leyden.
But since the invention of fluxions, and the summation of infinite series, several methods have been discovered for doing the same thing with much more ease and expedition. Euler and other eminent mathematicians have by these means given a quadrature of the circle which is true to more than one hundred places of decimals,—a proportion so extremely near the truth that, unless the ratio could be completely obtained, we need not wish for a greater degree of accuracy.