circle ratio equal circumference diameter regular archimedes circumscribed process
SQUARING (or QIIADRATITRE) OF THE CIRCLE is the problem of finding a square equal in area to a given circle. Like all problems, it may be increased in difficulty by the imposition of restrictions ; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the " squaring " of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the Euclidean geometry.
Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, KiicAov u&piicris, prop. 1), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary Euclidean construction ; also, it is evident that, conversely, if a square equal in area to the circle could be obtained, it would be possible to draw a straight line equal to the circumference. Rectification and quadrature of the circle have thus been, since the time of Archimedes at least, practically identical problems. Again, since the circumferences of circles are proportional to their diameters - a proposition assumed to be true from the dawn almost of practical geometry - the rectification of the circle is seen to be transformable into finding the ratio of the circumference to the diameter. This correlative numerical problem and the two purely geometrical problems are inseparably connected historically.
Probably the earliest value for the ratio was 3. It was so among the Jews (1 Kings vii. 23, 26), the Babylonians (Oppert, Journ. Asiatique, August 1872, October 1874), the Chinese (Biot, Journ. Asiatique, June 1841), and probably also the Greeks. Among the ancient Egyptians, as would appear from a calculation in the Rhind papyrus, the number (1)4, i.e., 3.16 ..., was at one time in use.' The first attempts to solve the purely geometrical problem appear to have been made by the Greeks (Anaxagora.s, d:c.),2 one of whom, Hippocrates,3 doubtless raised hopes of a solution by his quadrature of the so-called rneniscoz. As for Euclid, it is sufficient to recall the facts that the original author of prop. 8 of book iv. had strict proof of the ratio being <4, and the author of prop. 15 of the i ratio being >3, and to direct attention to the importance Eisenlohr, Bin math. Ilandbuch d. alien Acgypter, fibers. u. erkkirt, Leipsic, 1877 ; Rodet, Bull. de la Sic. Math. de France, vi. pp. 139 -149.
of book x. on incommensurables and props. 2 and 16 of book xii., viz., that " circles are to one another as the squares on their diameters" and that "in the greater of two concentric circles a regular 2n-gon can be inscribed which shall not meet the circumference of the less," however nearly equal the circles may be. With Archimedes (287-212 n.c.) a notable advance was made. Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular n-gons, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle. As a matter of fact, he started with a semi-side AB of a circumscribed regular hexagon meeting the circle in B (see fig. 1), joined A and B with 0 the centre, bisected the ..AOB by OD, so that BD became the semi-side of a circumscribed regular 12-gon ; then as AB_: BO : OA : : 1 : : 2 he sought an approximation to NiS and found that AB : BO >153 : 265. Next he applied his theorem1 BO + OA : AB : : OB : BD to calculate BD; from this in turn he calculated the semi-sides of the circumscribed A regular 24-gon, 48-gon, and 96-gon, and D regular 96-gon that perimeter : diameter Fig 1.
< 34.: 1. In a quite analogous manner he proved for the inscribed regular 96-gon that perimeter : diameter > 4;1. : 1. The conclusion from these therefore was that the ratio of circumference to diameter is <34- and >344. This is a most notable piece of work ; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.2 No advance of any importance was made upon the achievement of Archimedes until after the revival of learning. His immediate successors may have used his method to attain a greater degree of accuracy, but there is very little evidence pointing in this direction. Ptolemy (fl. 127151), in the Great Syntaxis, gives 3.141552 as the ratio 3 ; - and the Hindus (c. 500 A.D.), who were very probably indebted to the Greeks, used 62832/20000, that is, the now familiar 3.1416.4 It was not until the 15th century that attention in Europe began to be once more directed to the subject, and after the resuscitation a considerable length of time elapsed before any progress was made. The first advance in accuracy was due to a certain Adrian, son of Anthony, a native of Metz (1527), and father of the better-known Adrian Metius of Alkmaar. In refutation of Duchesne (Van der Eycke) he showed that the ratio was < 3.-iltv,Tu and > 3N-, and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 3,1,°, or '11g, which, being equal to 3.1415929..., is correct to the sixth fractional place.5 The next to advance the calculation was Viete (De Viette, Vieta), the greatest mathematician of his age. By finding the perimeter of the inscribed and that of the circumscribed regular polygon of 393216 (i.e., 6 x 21°) sides, he proved that the ratio was >3.1415926535 and <3.1415926537, so that its value became known (in 1579) correctly to 10 fractional places. The theorem for angle-bisection which Viete used was not that of Archimedes, but that which would now appear in the form 1 - cos 0 = 2 sine 0. With Viete, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical ; indeed, the Universales Inspections, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon Mathematicus, a table of sines, tangents, and secants.° Further, in comparing the labours of Archimedes and Viete, the effect of increased power of symbolical expression is very noticeable. Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a "rule" in words; in the 16th century it could be condensed into a "formula." Accordingly, we find in Viete a formula for the ratio of diameter to circumference, viz., the interminate product 7 - Vi• 1+ 11 NA • 1 A ± F 1NA - • • From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy : the mere arithmetician's art and length of days were the only requisites. Thus in connexion with the subject a genus of workers became possible who may be styled " ir-computers," - a name which, if it connotes anything uncomplimentary, does so because of the almost entirely fruitless character of their labours. Passing over Adriaan van Roomen (Adrianus Romanus) of Louvain, who published the value of the ratio correct to 15 places in his Idea Mathematica (1593),8 we come to the notable computer Ludolph van Ceulen (d. 1610), a native of Germany, long resident in Holland. His book, Van den. Circkel (Delf, 1596), gave the ratio correct to 20 places, but he continued his calculations as long as he lived, and his best result was published on his tombstone in St Peter's church, Leyden. The inscription, which is not known to be now in existence,° is in part as follows :- . . . Qui in vita sua multo labore circumferentiae circuli proximam rationem ad diametrum invenit sequentemturn circuli circumferentia plus est 314159265358979323846264338327950289 (1"In 100000000000000000000000000000000000 ...
This gives the ratio correct to 35 places. Van Ceulen's process was essentially identical with that of Viete. Its numerous root extractions amply justify a stronger expression than " multo labore," especially in an epitaph. In Germany the " Ludolphische Zahl" is still a common name for the ratio."
Up to this point the credit of most that had been done Willebrord Snell of Leyden in his Cyclometria, published in 1621. His Fig. 2.
achievement was a closely approximate geometrical solos Vieta, Opera Math., Leyden, 1646; Marie, Hist. des Sciences Math., iii. p. 27 sq., Paris, 1884.
Mauer, Gesch. d. Math., i., Gottingen, 1796-1800.
2° For minute and lengthy details regarding the quadrature of the circle in the Low Countries, see De Haan, " Bouwstoffen voor de geschiedenis, &c.," in Versl. en Mededed. der K. Akad. van Welenseh., ix., x., xi., xii., Amsterdam ; also his " Notice sur quelques quadrateurs, &c.," in Bull. di Bibliogr. e di Storia delle Sei. Mat. e Fis., vii. pp. 99-144.
tion of the problem of rectification (see fig. 2). ACB being a semicircle whose centre is 0, and AC the arc to be rectified, he produced AB to D, making BD equal to the radius, joined DC, and produced it to meet the tangent at A in E; and then his assertion (not established by him) was that AE was nearly equal to the arc AC, the error being in defect. For the purposes of the calculator a solution erring in excess was also required, and this Snell gave by slightly varying the former construction. Instead of producing AB (see fig. 3) so that BD E' was equal to r, he produced it only so far that, when the extremity D' was A joined with C, the part of Fig. 3.
D'C outside the circle was equal to r; in other words, by a non-Euclidean construction he trisected the angle AOC, for it is readily seen that, since FD' = FO = OC, the angle FOB = lA0C.1 This couplet of constructions is as important from the calculator's point of view as it is interesting geometrically. To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell's. AMC being an arc of a circle (see fig. 4) whose centre is drawn at the middle point of the arc and Fig. 4.
It is readily shown that the latter gives the best approximation to 6; but, while the former requires for its application a knowledge of the trigonometrical ratios of only one angle (in other words, the ratios of the sides of only one right-angled triangle), the latter requires the same for two angles, B and 10. Grienberger, using Snell's method, calculated the ratio correct to 39 fractional places.2 Huygens, in his De Circuli Magnitudine Inventa, 1654, proved the propositions of Snell, giving at the same time a number of other interesting theorems, for example, two inequalities which may be written as follows 3i(chd 0 - sin 0)> 0> chd 0+ i(chd 0 - sin 9).
As might be expected, a fresh view of the matter was taken by Descartes. The problem he set himself was the exact converse of that of Archimedes. A given straight line being viewed as equal in length to the circumference of a circle, he sought to find the diameter of the circle. His construction is as follows (see fig. 5).
Take AB equal to one-fourth of the given line ; on AB describe a square ABCD ; join AC ; in AC produced find, by a known process, a point Cl such that, when 01B1 " is drawn perpendicular to AB pro- Fig. 5.
duced and C,D, perpendicular to BC produced, the rectangle BC, will be equal to )ABCD ; by the same process find a point C2 such that the rectangle B1C2 will be equal to IBC,; and so on ad infinitum. The diameter sought is the It is thus manifest that by his first construction Snell gave an approximate solution of two great problems of antiquity.
straight line from A to the limiting position of the series of B's, say the straight line ABn. As in the case of the process of Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations. Denoting the number of units in AB by c, we can express BB„ B5B2, ... in terms of ic, and the identity ABA = AB + BB, + B1B2 + ... gives us at once an expression for the diameter in terms of the circumference by means of an infinite series.4 The proof of the correctness of the construction is seen to be involved in the following theorem, which serves likewise to throw new light on the subject : - AB being any straight line whatever, and the above construction being made, then AB is the diameter of the circle circumscribed by the square ABCD (self-evident), AB, is the diameter of the circle circumscribed by the regular 8-gon having the same perimeter as the square, AB2 is the diameter of the circle circumscribed by the regular 16-gon having the same perimeter as the square, and so on. Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.5 In 1655 appeared the Ariametica Ltfinitorum of Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian coordinates, and algebra playing an important part. In a very curious manner, by viewing the circle y = (1 - x2)1 as a member of the series of curves y = (1 - x2)1, y = (1 - x2)2, dzc., he was led to the proposition that four times the reciprocal of the ratio of the circumference to the diameter is equal to The work of Wallis had evidently an important influence on the next notable personality in the history of the subject, James Gregory, who lived during the period when the higher algebraic analysis was coming into power, and whose genius helped materially to develop it. He had, however, in a certain sense one eye fixed on the past and the other towards the future. His first contributions was a variation of the method of Archimedes. The latter, as we know, calculated the perimeters of successive polygons, passing from one polygon to another of double the number of sides; in a similar manner Gregory calculated the areas. The general theorems which enabled him to do this, after a start had been made, are A2„= A„A'„ (Snell's Cyclom.),