Quadrature, the finding of a square equal in area to that of any given figure. No mathematical problem has excited so great interest as the quadrature of the circle, or the determination of a square of the same area. As it is proved that the area of a circle is equal to that of a right-angled triangle, the altitude of which is the radius of the circle and the base its circumference, and as the side of the square of equal surface with the triangle is a mean proportional between the height and half the base of the triangle, the problem would be solved if the circumference could be immediately calculated from the radius which is known. Thus the question of the quadrature of the circle is reduced to finding the proportion between the diameter and circumference. Archimedes undertook the solution of the problem on the principle of calculating the peripheries of two polygons of many sides (as 96), one circumscribed about the circle and the other inscribed, between which must lie the circumference of the circle.
He thus found that the ratio of the diameter to the circumference lay between 1:3 10/70 and 1:3 10/71, and he adopted the former, which is also expressed 7:22. The Hindoos at some early period, certainly before any improvement was made upon this result in Europe, obtained the proportion 1,250 : 3,927, or 3.1416, which is much more exact than that of Archimedes; Ptolemy gives 3.141552, which is not quite so correct. In modern times the first great step in extending this calculation was made by Peter Metius, a Hollander, and was published by his son Adrian Metius. By calculating from polygons of about 1,536 sides he found that the proportion was less than 3 17/120 and greater than 3 15/106; and presuming that the mean of these was nearer the truth than either limit, he happily hit thus by chance on a near approximation, and determined a ratio convenient for practical purposes, and easy to recollect from its terms being made up of successive pairs of the first three odd numbers, viz.: 113 : 355. The error involved in this expression in a circle of 1,900 miles circumference is less than one foot.
Lu-dolph van Ceulen (or Keulen), another Hollander, in 1590, about the same time that Metius made his calculations, extended the calculation to 36 figures, which are engraved upon his tombstone in Leyden. These are 3.1415926535897-9323846264338327950289. 'The last figure is too large, and 8 would be too small. This was obtained by calculating the chords of successive arcs, each one being half of the preceding; for the above result this was carried out so far, that the last arc was one side of a polygon of 36,893,488,147,419,103,232 sides. The method of calculation was greatly simplified by Snell, who carried the computation to 55 decimal places by means of a polygon of only 5,242,880 sides. By other mathematicians the computation was carried on, reaching successively during the last century 75, 100, 128, and 140 places of decimals; and Montucla received from Baron Zach 154 figures, said to have been obtained from a manuscript in the Radcliffe library at Oxford, of the existence of which there is no other evidence.
The figures, however, except the last two, have since been proved correct. (See Montucla, Histoire des recherches sur la quadrature du cercle, 1754.) Notwithstanding that Lambert in 1761, and still later Legendre in his Éléments de géométrie, proved that the ratio of the diameter to the circumference cannot be expressed by any numbers, the wish to satisfy those who still sought the exact expression of this ratio led other mathematicians to continue to add to these figures; and some must have derived a singular gratification in the computation itself and its never terminating result. In May, 1841, a paper was communicated to the royal society by Dr. Rutherford of Woolwich, presenting 208 figures of decimals, of which however 56 were afterward proved to be wrong, so that the series was not really carried beyond the result obtained from the Oxford manuscript. In 1846 200 decimals were correctly made out by Mr. Dase; and the next year 250 by Dr. Clausen of Dorpat. In 1851 Mr. William Shanks of Durham calculated 315 decimals, which Dr. Rutherford verified and extended to 350. Mr. Shanks soon carried these to 527 decimals, of which 411 were confirmed by Dr. Rutherford. Finally in 1853 Mr. Shanks reached the number of 607 decimals, and gave the result in his "Contributions to Mathematics" (London, 1853). - When it was made evident that the arithmetical expression was impossible, it was still hoped by many that the ratio might be determined by geometrical construction; and the bare possibility of this, which a few mathematicians have admitted, has given encouragement to some to seek the solution in this direction.
But this, too, is now generally admitted to be impracticable. - Little benefit has resulted from the vast amount of time and labor that have been expended upon this famous problem. Wallis, investigating it at a time when the nature of the subject was not so well understood, and the investigation was consequently a proper one, was led to the discovery of the binomial theorem; but most of those who have since interested themselves in the question understood too little of the mathematical sciences to avail themselves of any opportunity that might be presented of increasing the means of mathematical research. The academy of sciences at Paris in 1775, and soon after the royal society in London, to discourage this and other similarly futile researches, declined to examine in future any paper pretending to the quadrature of the circle, the trisection of an angle, the duplication of the cube, or the discovery of perpetual motion.