Aristophanes introduces into his comedy of the Birds a geometer who is going to make a square circle. Plutarch asserts that Anaxagoras employed himself upon this problem in prison. Hippocrates of Chios actually found the way to make a rectilinear space equal to certain circular spaces, and is reported to have attempted the general problem. There is evidence enough that it acquired an early celebrity, and it may be doubted whether the researches of Euclid in incommensurables [IRRATIONAL QUANTITIES] had not some reference to a supposition that the circle and its diameter might possibly be discovered to belong to a particular class of these quantities. Archimedes, in his book on the mensuration of the circle, is the first who made any approach even to a practical determination of the question. By inscribing and circum scribing polygons of 96 sides in and about the circle, be demonstrates that the excess of the circumference over three times the diameter must be less than 10-70ths of the diameter, and greater than 10-71st parts. His limits are perfectly correct, and even tolerably close. According to him, a circle of 4270 feet diameter would have a circum ference lying between 15,610 and 15,620 feet, the truth being that such a circle would have a circumference of 15,6134 feet very nearly. This measure of Archimedes gives 3.14296 for the approximate value of sr, the ratio of the circumference to the diameter; several of the Greeks are said to have made further approximation, but their results are not preserved.
Among the Hindus [Vies GAcITA, in B10˘. Div.] are found the ratios of 3927 to 1250, and also that of the square root of 10 to 1. The first gives a=3.1416 exactly, and is considerably more correct than that of Archimedes; the second gives 3.1623, and is much less exact. The date of the first result is not known ; but all agree that the writings in which it is found are anterior to any European improve ment on the measure of Archimedes. The ratio given by Ptolemy, in the Syntaxis, is 3'141552, not quite so correct as 31416, but so near to it that those who doubt of the antiquity of Hindu science will probably suppose the 31416 above mentioned to be a version of Ptolemy's measure.
This subject began to be reconsidered in the 16th century, in the middle of which were calculated the tables [TABLES] of Rhcticus, the celebrated Copernican, from which the value of ir might easily have been calculated to eight decimals, but it does not appear that this was done. Purbach used the ratio of 377 to 120, or 3141667, not so exact as Ptolemy's. Regiomontanus slightly corrected the limits of Archimedes, but Peter Meting, father of Adrian (to whom it is often attributed, merely because Adrian records and delivers it), and of James (to whom the invention of the telescope has been given), made a decided Improvement. He gives the ratio of 355 to 113, or 314159292, which is correct to the sixth decimal inclusive. Nothing more precise could be desired for practical purposes, Insomuch that a circle of 113 in diameter may be reckoned as one of 355 in circumference, which, though a little too great, does not give the circumference wrongly by BO much as one foot In 1900 miles. On looking, however, at the
account which Adrian Methis gives of his father's investigation, we find that the extreme closeness of the approximation is only a piece of good fortune. As it is very curious that this should never have been noticed, we quote the whole passage :—" Parens meus Illustrium D.D. Ordinum Confcedaratarum Belem Provinciarum Geometra, in libello einem conscripsit adversum quadraturam circuli Simonis h Quercu demonstravit proportionem peripheriai ad seam diametrum ease mine rem quam hoc est W, majorem vcro quam SA, hoc est 3'1, quarum proportionum intermedia existit sive fa. Qum quidem intermedia proportio aliquantulum existit major, quam ea, quam invent Mr. Lu dolph i1 Collen, cujus Lunen differentia est minus quam -,,„)." The book of Peter 3Ietius was never published, that we can find ; and we doubt if Adrian Metins would have published his father's result, It it had not come so very near to the then recent result of Van Cculen.
It seems that the elder 3Ietins, finding too great, and too small, took the mean of the numerators for a numerator, and the mean of the denominators for a denominator, and presumed that the result was nearer the truth than either limit: a presumption to which he had no right whatever, except that of trusting to the chapter of accidents.
Matins lived in the latter half of the 16th century, as also did Vieta, who gave a still more accurate though not so elegant a measure. He was the first who exhibited a series of arithmetical operations by which a mere calculator might carry on the process to any extent, and gave the following result : The circle whose diameter is ten thousand million of parts, has a circumference greater than 31,415,926,535 of those parts, and less than 31,415,926,537. Other approximations rapidly followed : Adrianus Romanna calculated the perimeter of an inscribed polygon of 1073741924 sides, by means of which he found for the ratio 3.141592653589793; but his contemporary Ludolph van Ceulen, by calculating the chords of successive arcs, each of which is the half of the preceding, found the perimeter of a polygon of 36993489147419103232 sides, and obtained 36 figures of the ratio 314159, &c., presently given to a still greater length. So far the method of calculating by means of inscribed polygons, though Victa had reduced it to routine, had received no material simplification. This was given by Snell, who found some propositions (afterwards demonstrated by Huyghens) which very much abridge the labour. He found a result as correct as that of Archimedes, by means of a simple hexagon ; making the 96-sided polygon of Archimedes give seven decimals correctly, instead of three. He also calculated the ratio to 55 decimal places, and by means of a polygon of only 5242830 sides. Huyghens introduced some new theorems of the same species as those of Snell.