L. van Cenlen devoted no inconsidorable part of his life to

the subject. In 1596* he gave the result to 20 places of deci- mals : this was calculated by finding the perimeters of the inscribed and circumscribed regular polygons of 60 x 2^^ sides, obtained by the repeated use of a theorem of his discovery equivalent to the formula 1 — cos^ = 2 sin^ Jil. I possess a finely executed engraving of him of this date, with the result printed round a circle which is below his portrait. He died in 1610, and by his directions the result to 35 places of decimals (which was as far as he had calculated it) was engraved on his tombstone i* in St Peter's Church, Le3^den. His posthumous arithmetic J: contains the result to 32 places ; this was obtained by calculating the perimeter of a polygon, the number of whose sides is 2'\ i.e. 4,611686,018427,387904. Van Ceulen also com- piled a table of the perimeters of various regular polygons.
Willebrord Snell§, in 1621, obtained from a polygon of 2^*' sides an approximation to 34 places of decimals. This is less than the numbers given by van Ceulen, but Snell's method was so superior that he obtained his 34 places by the use of a polygon from which van Ceulen had obtained only 14 (or perhaps 16) places. Similarly, Snell obtained from a hexagon an approximation as correct as that for which Archimedes had required a polygon of 96 sides, while from a polygon of 96 sides he determined the value of tt correct to seven decimal places instead of the two places obtained by Archimedes. The reason is that Archimedes, having calculated the lengths of the sides of inscribed and circumscribed regular polygons of n sides, assumed that the length of l/nth of the perimeter of the circle was intermediate between them; whereas Snell constructed
• Vanden Circkel, Delf, 1596, fol. 14, p. 1 ; or De Circulo, Leyden, 1619, p. 3.
t The inscription is quoted by Prof, de Haan in the Messenger of Mathematics, 1874, vol. Ill, p. 25.
X De Arithmetische en Geometrische Foudavienten, Leyden, 1615, p. 163; or p. 144 of the Latin translation by W. Snell, published at Leyden in 1615 under the title Fundamenta Arithmetica et Gcometrica. This was reissued, togetlier with a Latin translation of the Vanden Circkel, in 1619, under the title De Circulo; in which see pp. 3, 29 — 32, 92.
§ Cyclometricus, Leaden, 1621, p. 55.
302 THREE GEOMETRICAL PROBLEMS [CH. XII
from the sides of these polj^gons two other lines which gave closer limits for the corresponding arc. His method depends on the theorem 3 sin ^/(2 + cos ^) the aid of which a polygon of n sides gives a value of tt correct to at least the integral part of (4 log n — •2305) places of decimals, which is more than twice the number given by the older rule. Snell's proof of his theorem is incorrect, though the result is true.
Snell also added a table* of the perimeters of all regular inscribed and circumscribed polygons, the number of whose sides is 10 X 2" where ii is not greater than 19 and not less than 3. Most of these were quoted from van Ceulen, but some were recalculated. This list has proved useful in refuting circle- squares. A similar list was given by James Gregory f.
In 1630 Grienberger J, by the aid of Snell's theorem, carried the approximation to 39 places of decimals. He was the last mathematician who adopted the classical method of finding the perimeters of inscribed and circumscribed polygons. Closer approximations serve no useful purpose. Proofs of the theorems used by Snell and other calculators in applying this method were given by Huygens in a work§ which may be taken as closing the history of this method.
In 1656 Wallisll proved that
TT 2.2.4.4.6.6 ...
Ji X.O.O.O.0.4.1 ...
and quoted a proposition given a few years earlier by Viscount Brouncker to the effect that
7r_ P 32 5^
4~ ^2+ 2 + 2 + ...,
* It is quoted by Montucla, ed. 1831, p. 70.
t Vera Circuli et Hyperbolae Quadratura, prop. 29, quoted by Huj'gens, Opera Varia, Leyden, 1724, p. 447.
X Elementa Trigonometrica, Home, 1630, end of preface.
§ De Gircula Magnitudine Inventa, 1654 ; Opera Varia, pp. 851 — 387. The proofs are given in G. Pirie's Geometrical Methods of Approximating to the Value of IT, London, 1877, pp. 21—23.
II Arithmetica Infiiiitorum, Oxford, 1656, prop. 191. An analysis of the investigation by Wallis was given by Cayley, Quarterly Journal of Mathematics^ 1839, vol. xxiii, pp. 165—169.
CH. XIl] THREE GEOMETRICAL PROBLEMS 303
but neither of these theorems was used to any large extent for calculation.
Subsequent calculators have relied on converging infinite series, a method that was hardly practicable prior to the in- vention of the calculus, though Descartes* had indicated a geometrical process which was equivalent to the use of such a series. The employment of infinite series was proposed by James Gregory f, who established the theorem that
^ = tan ^ - 1 tan^ e-\-^ tan' (9 - ...,
the result being true only if 6 lies between — {tt and J tt.
The first mathematician to make use of Gregory's series for obtaining an approximation to the value of tt was Abraham Sharp J:, who, in 1699, on the suggestion of Halley, determined it to 72 places of decimals (71 correct). He obtained this value by putting 0=^7r in Gregory's series.
Machin§, earlier than 1706, gave the result to 100 places (all correct). He calculated it by the formula
J TT = 4 tan~^ ^ — tan~^ ■^.
De Lagnyll, in 1719, gave the result to 127 places of decimals (112 correct), calculating it by putting 6 = ^ir in Gregor3^"'s series.
HuttonlF, in 1776, and Euler**, in 1779, suggested the use of
• See Euler's paper in the Novi Commentarii Academiae Scientiarum, St Petersburg, 1763, vol. viii, pp. 157—168.
t See the letter to Collins, dated Feb. 15, 1671, printed in the Coinmercium EpistoUcum, London, 1712, p. 25, and in the Macclesfield Collection, Corre- fpondence of Scientific Men of the Seventeenth Century^ Oxford, 1841, vol. ii. p. 216.
X See Life of A. Sharp by W. Cudworth, London, 1889, p. 170. Sharp's work is given in one of the preliminary discourses (p. 53 et seq.) prefixed to H. Sherwin's Mathematical Tables. The tables were issued at London in 1705: probably the discourses were issued at the same time, though the earliest copies I have seen were printed in 1717.
§ W.Jones's Synopsis Pahnarionim, London, 1706, p. 243; and Maseres, Scriptores Logarithmici, London, 1796, vol. iii, pp. vii— ix, 155 — 164.
il Histoire de VAcad€mie for 1719, Paris, 1721, p. 144.
H Philosophical Transactions ^ 1776, vol. lxvi, pp. 476 — 492.
•* Nova Acta Academiae Scientiarum Petropolitanae for 1793, St Petersburg, 1798, vol. XI, pp. 133 — 149: the memoir was read in 1779.
304 THREE GEOMETRICAL PROBLEMS [CH. XII
the formula Jtt = tan~^ J + tan~^ i or Jtt = 5 tan"^ f + 2 tan~^ ^, but neither carried the approximation as far as had been done previously.
Vega, in 1789*, gave the value of tt to 143 places of decimals (126 correct); and, in l794f, to 140 places (136 correct).
Towards the end of the eighteenth century Baron Zach saw in the Radcliffe Library, Oxford, a manuscript by an unknown author which gives the value of tt to 154 places of decimals (152 coiTect).
In 1837, the result of a calculation of vr to 154 places of decimals (152 correct) was published^.
In 1841 Rutherford§ calculated it to 208 places of decimals (152 correct), using the formula J7r=4 tan~^ -J— tan~^ Y^^j^+tan"^ ■^.
In 1844 Dase || calculated it to 205 places of decimals (200 correct), using the formula ^ir = tan~^ J + tan~^ -J + tan~^ J.
In 1847 Clausen IF carried the approximation to 250 places of decimals (248 correct), calculating it independently by the formulae ^tt = 2 tan~^ J + tan'^ f and ^tt = 4 tan"^ i - tan"^ gfg-
In 1853 Rutherford** carried his former approximation to 440 places of decimals (all correct), and William Shanks pro- longed the approximation to 530 places. In the same year Shanks published an approximation to 607 places ff : and in 1873 he carried the approximation to 707 places of decimals JJ. These were calculated from Machin's formula.
In 1853 Richter, presumably in ignorance of what had been
* Nova Acta Academiae Scientiarum Petropolitanae for 1790, St Petersburg, 1795, vol. IX, p. 41.
t Thesaurus Logarithmorum {logarithmisch-trigonometrischer Tafeln), Jjei^^zig, 1794, p. 633.
X J. F. Callet's Tables, etc., Precis Elenieataire, Paris, tirage, 1837. I have not verified this reference.
§ Philosophical Transactions, 1841, p. 283.
II Crelle's Journal, 1844, vol. xxvii, p. 198.
H Schumacher, Astronomische Nachrichten, vol. xxv, col. 207.
** Proceedings of the Royal Society, Jan. 20, 1853, vol. vi, pp. 273 — 275.
+t Contributions to Mathematics, W. Shanks, London, 1853, pp. 86—87.
XX Proceedings of the Royal Society, 1872-3, vol. xxi, p. 318; 1873-4, vol. XXII, p. 45.
CH. XIl] THREE GEOMETRICAL PROBLEMS 805
done in England, found the value of tt to 333 places* of decimals (330 correct) ; in 1854 he carried the approximation to 400 places -f ; and in 1855 carried it to 500 placesj.
Of the series and formulae by which these approximations have been calculated, those used by Machin and Dase are perhaps the easiest to employ. Other series which converge rapidly are the following:
TT^l 1 J_ K3 1
6~2"'"2"3.2=='^2.4'5.2^
an(
i = 22 tan- ^ + 2 tan- ^^ - 5 tan- j^^ - 10 tan- ^-j^ ;
the latter of these is due to Mr Escott§.
As to those writers who believe that they have squared the circle their number is legion and, in most cases, their ignorance profound, but their attempts are not worth discussing here. " Only prove to me that it is impossible," said one of them, " and I will set about it immediately " ; and doubtless the statement that the problem is insoluble has attracted much attention to ic.
Among the geometrical ways of approximating to the truth the following is one of the simplest. Inscribe in the given circle a square, and to three times the diameter of the circle add a fifth of a side of the square, the result will differ from the circumference of the circle by less than one-seventeen- thousandth part of it.
An approximate value of it has been obtained experimentally by the theory of probability. On a plane a number of equi- distant parallel straight lines, distance apart a, are ruled ; and a stick of length I, which is less than a, is dropped on the plane. The probability that it will fall so as to lie across one of the lines is 2Z/7ra. If the experiment is repeated many hundreds
* GriinerVs Archiv, vol. xxi, p. 119.
t Ibid., vol. xxm, p. 476: the approximation given in vol. xxii, p. 473, is correct only to 330 places.
X Ibid., vol. XXV, p. 472; and Elbinger Anzeigen, No. 85.
§ VIntermediaire des Mathtmaticiens, Paris, Dec. 1896, vol. in, p. 276.
B. R. 20
306 THREE GEOMETRICAL PROBLEMS [CH. XII
of times, the ratio of the number of favourable cases to the whole number of experiments will be very nearly equal to this fraction: hence the value of tt can be found. In 1855 Mr A. Smith* of Aberdeen made 3204 trials, and deduced 7r = 3*1553. A pupil of Prof. De Morgan*, from 600 trials, deduced TT = 3-137. In 1864 Captain Foxf made 1120 trials with some additional precautions, and obtained as the mean value TT = 3-1419.
Other similar methods of approximating to the value of tt have been indicated. For instance, it is known that if two numbers are written down at random, the probability that they will be prime to each other is 6/7r^ Thus, in one case:^ where each of 50 students wrote down 5 pairs of numbers at random, 154 of the pairs were found to consist of numbers prime to each other. This gives QJir^ = 154/250, from which we get 7r = 312.
* A. De Morgan, Budget of Paradoxes, London, 1872, pp. 171, 172 [quoted from an article by De Morgan published in 1861].
t Messenger of Mathematics, Cambridge, 1873, vol. ii, pp. 113, 114.
X Note on w by R. Chartres, Philosophical Magazine, London, series 6, vol. XXXIX, March, 1904, p. 315.
307