The quadrature of the circle, the square root of two, and the right-angled triangle, by William Alexander Myers.

INTRODUCTION. 25
gave the value of the circle, or the circumference to within a few millionths. Adrianus Romanus carried the approximation to 17 figures.
But all that is far below what was done by Ludolph Van Ceulen, and
which he published in his book de Circulo et adscriptis, of which
Snellius published a Latin translation, at Leyden, in 1619. Ceulen,
assisted by Petrus Cornelius, found with inconceivable labor a ratio
of 32 decimals; see V. II, p. 6. Snellius found the means of shortening this calculation by some very ingenious theorems, and if he did
not excel Van Ceulen he verified his result, which he put beyond attack. His discoveries on this subject are found in the book entitled
Willebrordi Snelli Cyclometricus de Circuli dimensione, etc. Descartes
also found a geometrical construction which, carried to infinity, would
give the circular circumference, and from which he could easily deduce an expression in the form of a series. (See his Opera posthuma.)
Gregpoire de Saint-Vincent is one of those who are most distinguished in this field; true, he claimed incorrectly to have found the
quadrature of the circle and of the hyperbola, but the failure in this
respect was preceded by so great a number of beautiful geometrical
discoveries, deduced with much elegance according to the method of
the ancients, that it would hâve been unjust to have placed him among
the paralogists we have mentioned. He announced, in 1647, his discoveries in a book entitled,: Opus Geometricumr quadratzirae Circuli et
Sectionern Coni libris, X, Comprehensum. All the beautiful things conitained in this book are admired; only the conclusion is impugned.
Gregoire de Saint-Vincent lost himself in the maze of his proofs which
he calls prolortionalities, and which he introduces in his speculations.
It was the subject of quite a lively quarrel between his disciples on the
one hand, and his adversaries on the other, Huygens, Mersenne, and
Leotand, from 1652 to 1664.
If that skillful geometrician had not been mistaken, it would only
have followed from his investigations that the quadrature of thc circle
depends upon logarithms, and consequently on that of the hyperbola.
That would still be a handsome discovery, but it did not even have
that advantage. This furnished Huygens the occasion of divers investigations on this subject. He demonstrated several new and curious
theorems on the quadrature of the circle: Theoremata de quadratura
hyper, ellipsis et Circuli, 1651; De Circuli MIagnitude inventa, 1654.
He gave several methods of approaching his quadrature much shorter