3 sin [theta]
tan (1/3)[theta] + 2sin (1/3)[theta] > [theta] > --------------- (Snell).
2 + cos [theta]
It is readily shown that the latter gives the best approximation to
[theta]; 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, [theta] and (1/3)[theta]. Grienberger,
using Snell's method, calculated the ratio correct to 39 fractional
places.[15] 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[16]--
4chd [theta] + sin [theta] 1
chd [theta] + --------------------------- . --- (chd [theta] - sin [theta]) >
2chd [theta] + 3sin [theta] 3
1
[theta] > chd [theta] + --- (chd [theta] - sin [theta]).
3
[Illustration: FIG. 11.]
[Illustration: FIG. 12.]
As might be expected, a fresh view of the matter was taken by Rene
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 1/4ABCD; by the same process find a point C2
such that the rectangle B1C2 will be equal to 1/4BC1; 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[oo]. 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 1/4c, we can express BB1, B1B2, ... in terms of
1/4c, and the identity AB[oo] = AB + BB1 + B1B2 + ... gives us at once an
expression for the diameter in terms of the circumference by means of an
infinite series.[17] 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), AB1 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.[18]
In 1655 appeared the _Arithmetica Infinitorum_ of John Wallis, where
numerous problems of quadrature are dealt with, the curves being now
represented in Cartesian co-ordinates, and algebra playing an important
part. In a very curious manner, by viewing the circle y = (1 - x squared)^1/2
as a member of the series of curves y = (1 - x squared)1, y = (1 - x
squared) squared, &c., he was led to the proposition that four times the
reciprocal of the ratio of the circumference to the diameter, i.e. 4/[pi],
is equal to the infinite product
3 . 3 . 5 . 5 . 7 . 7 . 9 ...
-----------------------------;
2 . 4 . 4 . 6 . 6 . 8 . 8 ...
and, the result having been communicated to Lord Brounker, the latter
discovered the equally curious equivalent continued fraction
1 squared 3 squared 5 squared 7 squared
1 + --- --- --- --- ...
2 + 2 + 2 + 2
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 contribution[19] 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
_____
A2n = \/AnA'n (Snell's _Cyclom._),
2 An A'n 2 A'n A2n
A'2n = ---------- or ----------- (Gregory),
An + A'2n A'n + A2n
where An, A'n are the areas of the inscribed and the circumscribed
regular n-gons respectively. He also gave approximate rectifications of
circular arcs after the manner of Huygens; and, what is very notable, he
made an ingenious and, according to J.E. Montucla, successful attempt to
show that quadrature of the circle by a Euclidean construction was
impossible.[20] Besides all this, however, and far beyond it in
importance, was his use of infinite series. This merit he shares with
his contemporaries N. Mercator, Sir I. Newton and G.W. Leibnitz, and the
exact dates of discovery are a little uncertain. As far as the
circle-squaring functions are concerned, it would seem that Gregory was
the first (in 1670) to make known the series for the arc in terms of the
tangent, the series for the tangent in terms of the arc, and the secant
in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little
treatise in manuscript containing the series for the arc in terms of the
sine, for the sine in terms of the arc, and for the cosine in terms of
the arc. These discoveries formed an epoch in the history of
mathematics generally, and had, of course, a marked influence on after
investigations regarding circle-quadrature. Even among the mere
computers the series
[theta] = tan - (1/3) tan^3 [theta] + (1/5) tan^5 [theta] - ...,
specially known as Gregory's series, has ever since been a necessity of
their calling.
The calculator's work having now become easier and more mechanical,
calculation went on apace. In 1699 Abraham Sharp, on the suggestion of
Edmund Halley, took Gregory's series, and, putting tan [theta] = (1/3)
sqrt(3), found the ratio equal to
__ / 1 1 1 \
\/12 ( 1 - ----- + ------ - ------ + ... ),
\ 3 . 3 5 . 3 squared 7 . 3 cubed /
from which he calculated it correct to 71 fractional places.[21] About
the same time John Machin calculated it correct to 100 places, and, what
was of more importance, gave for the ratio the rapidly converging
expression
16 / 1 1 1 \
-- ( ---- + ----- - ----- + ... ) -
5 \ 3.5 squared 5.5^4 7.5^6 /
4 / 1 1 \
--- ( 1 - ------ + ------- - ... ),
239 \ 3.239 squared 5.239^4 /
which long remained without explanation.[22] Fautet de Lagny, still
using tan 30 deg., advanced to the 127th place.[23]
Leonhard Euler took up the subject several times during his life,
effecting mainly improvements in the theory of the various series.[24]
With him, apparently, began the usage of denoting by [pi] the ratio of
the circumference to the diameter.[25]
The most important publication, however, on the subject in the 18th
century was a paper by J.H. Lambert,[26] read before the Berlin Academy
in 1761, in which he demonstrated the irrationality of [pi]. The general
test of irrationality which he established is that, if
a1 a2 a2
-- -- -- ...
b1 +- b2 +- b3 +-
be an interminate continued fraction, a1, a2, ..., b1, b2 ... be
integers, a1/b1, a2/b2, ... be proper fractions, and the value of every
one of the interminate continued fractions
a1 a2
-- , -- , ... be < 1,
b1 +- ... b2 +- ...
then the given continued fraction represents an irrational quantity. If
this be applied to the right-hand side of the identity
m m m squared m squared
tan --- = --- ---- ---- ...
n n - 3n - 5n
it follows that the tangent of every arc commensurable with the radius
is irrational, so that, as a particular case, an arc of 45 deg., having
its tangent rational, must be incommensurable with the radius; that is to
say, [pi]/4 is an incommensurable number.[27]
This incontestable result had no effect, apparently, in repressing the
[pi]-computers. G. von Vega in 1789, using series like Machin's, viz.
Gregory's series and the identities
[pi]/4 = 5tan^{-1} (1/7) + 2tan^{-1} (3/79) (Euler, 1779),
[pi]/4 = tan^{-1} (1/7) + 2tan^{-1} ( 1/3) (Hutton, 1776),
neither of which was nearly so advantageous as several found by Charles
Hutton, calculated [pi] correct to 136 places.[28] This achievement was
anticipated or outdone by an unknown calculator, whose manuscript was
seen in the Radcliffe library, Oxford, by Baron von Zach towards the end
of the century, and contained the ratio correct to 152 places. More
astonishing still have been the deeds of the [pi]-computers of the 19th
century. A condensed record compiled by J.W.L. Glaisher (_Messenger of
Math._ ii. 122) is as follows:--
+-----+------------+-----------------+--------------------------------------------+
| | |No. of fr. digits| |
|Date.| Computer. +--------+--------+ Place of Publication. |
| | | calcd. |correct.| |
+-----+------------+--------+--------+--------------------------------------------+
|1842 | Rutherford | 208 | 152 | _Trans. Roy. Soc._ (London, 1841), p. 283. |
|1844 | Dase | 205 | 200 | _Crelle's Journ._. xxvii. 198. |
|1847 | Clausen | 250 | 248 | _Astron. Nachr._ xxv. col. 207. |
|1853 | Shanks | 318 | 318 | _Proc. Roy. Soc._ (London, 1853), 273. |
|1853 | Rutherford | 440 | 440 | Ibid. |
|1853 | Shanks | 530 | .. | Ibid. |
|1853 | Shanks | 607 | .. | W. Shanks, _Rectification of the Circle_ |
| | | | | (London, 1853). |
|1853 | Richter | 333 | 330 | _Grunert's Archiv_, xxi. 119. |
|1854 | Richter | 400 | 330 | Ibid. xxii. 473. |
|1854 | Richter | 400 | 400 | Ibid. xxiii. 476. |
|1854 | Richter | 500 | 500 | Ibid. xxv. 472. |
|1873 | Shanks | 707 | .. | _Proc. Roy. Soc._ (London), xxi. |
+-----+------------+--------+--------+--------------------------------------------+
By these computers Machin's identity, or identities analogous to it, e.g.
[pi]/4 = tan^{-1} (1/2) + tan^{-1} 1/5 + tan^{-1} 1/8 (Dase, 1844),
[pi]/4 = 4tan^{-1} (1/5) - tan^{-1} 1/70 + tan^{-1} 1/99 (Rutherford),
and Gregory's series were employed.[29]
A much less wise class than the [pi]-computers of modern times are the
pseudo-circle-squarers, or circle-squarers technically so called, that
is to say, persons who, having obtained by illegitimate means a
Euclidean construction for the quadrature or a finitely expressible
value for [pi], insist on using faulty reasoning and defective
mathematics to establish their assertions. Such persons have flourished
at all times in the history of mathematics; but the interest attaching
to them is more psychological than mathematical.[30]
It is of recent years that the most important advances in the theory of
circle-quadrature have been made. In 1873 Charles Hermite proved that
the base [eta] of the Napierian logarithms cannot be a root of a
rational algebraical equation of any degree.[31] To prove the same
proposition regarding [pi] is to prove that a Euclidean construction for
circle-quadrature is impossible. For in such a construction every point
of the figure is obtained by the intersection of two straight lines, a
straight line and a circle, or two circles; and as this implies that,
when a unit of length is introduced, numbers employed, and the problem
transformed into one of algebraic geometry, the equations to be solved
can only be of the first or second degree, it follows that the equation
to which we must be finally led is a rational equation of even degree.
Hermite[32] did not succeed in his attempt on [pi]; but in 1882 F.
Lindemann, following exactly in Hermite's steps, accomplished the
desired result.[33] (See also TRIGONOMETRY.)
REFERENCES.--Besides the various writings mentioned, see for the
history of the subject F. Rudio, _Geschichte des Problems von der
Quadratur des Zirkels_ (1892); M. Cantor, _Geschichte der Mathematik_
(1894-1901); Montucla, _Hist. des. math._ (6 vols., Paris, 1758, 2nd
ed. 1799-1802); Murhard, _Bibliotheca Mathematica_, ii. 106-123
(Leipzig, 1798); Reuss, _Repertorium Comment._ vii. 42-44 (Goettingen,
1808). For a few approximate geometrical solutions, see Leybourn's
_Math. Repository_, vi. 151-154; _Grunert's Archiv_, xii. 98, xlix. 3;
_Nieuw Archief v. Wisk._ iv. 200-204. For experimental determinations
of [pi], dependent on the theory of probability, see _Mess. of Math._
ii. 113, 119; _Casopis pro pistovani math. a fys._ x. 272-275;
_Analyst_, ix. 176. (T. MU.)