The next to advance the calculation was Francisco Vieta. By finding the perimeter of the inscribed and that of the circumscribed regular polygon of 393216 (i.e. 6 × 216) 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 Vieta used was not that of Archimedes, but that which would now appear in the form 1 − cos θ＝2 sin2 12θ. With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, 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.[1] Further, in comparing the labours of Archimedes and Vieta, 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 Vieta a formula for the ratio of diameter to circumference, viz. the interminate product[2]—
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 problem being reduced to an arithmetical computation. Thus in connexion with the subject a genus of workers became possible who may be styled “π-computers or circle-squarers”—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),[3] we come to the notable computer Ludolph van Ceulen (d. 1610), a native of Germany, long resident in Holland. His book, Van den Circkel (Delft, 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, Leiden. The inscription, which is not known to be now in existence,[4] is in part as follows:—
. . . . Qui in vita sua multo labore circumferentiae circuli proximam rationem ad diametrum invenit sequentem—
| quando diameter est 1 | |
| tum circuli circumferentia plus est | |
| quam | 314159265358979323846264338327950288 |
| 100000000000000000000000000000000000 | |
| et minus | |
| quam | 314159265358979323846264338327950289 |
| 100000000000000000000000000000000000 . . . | |
This gives the ratio correct to 35 places. Van Ceulen’s process was essentially identical with that of Vieta. Its numerous root extractions amply justify a stronger expression than “multo labore,” especially in an epitaph. In Germany the “Ludolphische Zahl” (Ludolph’s number) is still a common name for the ratio.[5]
Up to this point the credit of most that had been done may be set down to Archimedes. A new departure, however, was made by Willebrord Snell of Leiden in his Cyclometria, published in 1621. His achievement was a closely approximate geometrical solution of the problem of rectification (see fig. 9): ACB being a semicircle whose centre is O, 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. 10) so that BD was
equal to 𝑟, he produced it only so far that, when the extremity D′ was joined with C, the part D′F 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 = 13AOC.[6] 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. 11) whose centre is O, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC < HK, but > AC. In modern trigonometrical notation the propositions to be compared stand as follows:—
| 2 tan 12θ > θ > 2 sin 12θ | (Archimedes); |
| tan 13θ + 2 sin 13θ > θ > 3 sin θ2 + cos θ | (Snell). |
It is readily shown that the latter gives the best approximation to θ; 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, θ and 13θ.
Grienberger, using Snell’s method, calculated the ratio correct to 39 fractional places.[7] C. 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[8]—
chd θ + 4 chd θ + sin θ2 chd θ + 3 sin θ. 13(chd θ − sin θ) > θ > chd θ + 13(chd θ − sin θ).
As might be expected, a fresh view of the matter was taken by René 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. 12). 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 C1 such that, when C1B1 is drawn perpendicular to AB produced and C1D1 perpendicular to BC produced, the rectangle BC1 will be equal to 14ABCD; by the same process find a point C2 such that the rectangle B1C2 will be equal to 14BC1; and so on ad infinitum. The diameter sought is the straight line from A to the limiting position of the series of B’s, say the straight line AB∞. As in the case of the process of
- ↑ Vieta, Opera math. (Leiden, 1646); Marie, Hist. des sciences math. iii. 27 seq. (Paris, 1884).
- ↑ Klügel, Math. Wörterb. ii. 606, 607.
- ↑ Kästner, Gesch. d. Math. i. (Göttingen, 1796–1800).
- ↑ But see Les Délices de Leide (Leiden, 1712); or de Haan, Mess. of Math. iii. 24-26.
- ↑ 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 Mededeel. der K. Akad. van Wetensch. ix., x., xi., xii. (Amsterdam); also his “Notice sur quelques quadrateurs, &c.,” in Bull. di bibliogr. e di storia delle sci. mat. e fis. vii. 99-144.
- ↑ It is thus manifest that by his first construction Snell gave an approximate solution of two great problems of antiquity.
- ↑ Elementa trigonometrica (Rome, 1630); Glaisher, Messenger of Math. iii. 35 seq.
- ↑ See Kiessling’s edition of the De Circ. Magn. Inv. (Flensburg, 1869); or Pirie’s tract on Geometrical Methods of Approx. to the Value of π (London, 1877).
