2. _Arc_ and _Sector_.--Data: radius = a; [theta] = circular measure
of angle subtended at centre by arc; c = chord of arc; c2 = chord of
semi-arc; c4 = chord of quarter-arc.
Exact formulae are:--Arc = a[theta], where [theta] may be given
directly, or indirectly by the relation c = 2a sin 1/2[theta]. Area of
sector = 1/2a squared[theta] = 1/2 radius x arc.
Approximate formulae are:--Arc = (1/3)(8c2 - c) (Huygen's formula);
arc = (1/45)(c - 40c2 + 256c4).
3. _Segment._--Data: a, [theta], c, c2, as in (2); h = height of
segment, i.e. distance of mid-point of arc from chord.
Exact formulae are:--Area = 1/2a squared([theta] - sin [theta]) = 1/2a squared[theta]
-1/4c squared cot 1/2[theta] = 1/2a squared - 1/2c sqrt(a squared - 1/4c squared). If h be given, we can use
c squared + 4h squared = 8ah, 2h = c tan 1/4[theta] to determine [theta].
Approximate formulae are:--Area = (1/15)(6c + 8c2)h; = (2/3) sqrt(c squared +
(8/5)h squared).h; = (1/15)(7c + 3[alpha])h, [alpha] being the true length of
the arc.
From these results the mensuration of any figure bounded by circular
arcs and straight lines can be determined, e.g. the area of a _lune_
or _meniscus_ is expressible as the difference or sum of two segments,
and the circumference as the sum of two arcs. (C. E.*)
_Squaring of the Circle._
The problem of finding a square equal in area to a given circle, like
all problems, 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, [Greek: Kyklou metresis], 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
(4/3)^4, i.e. 3.1605, was at one time in use.[1] The first attempts to
solve the purely geometrical problem appear to have been made by the
Greeks (Anaxagoras, &c.)[2], one of whom, Hippocrates, doubtless raised
hopes of a solution by his quadrature of the so-called _meniscoi_ or
_lune_.[3]
[Illustration: Fig. 6.]
[Illustration: Fig. 7.]
[The Greeks were in possession of several relations pertaining to the
quadrature of the lune. The following are among the more interesting. In
fig. 6, ABC is an isosceles triangle right angled at C, ADB is the
semicircle described on AB as diameter, AEB the circular arc described
with centre C and radius CA = CB. It is easily shown that the areas of
the lune ADBEA and the triangle ABC are equal. In fig. 7, ABC is any
triangle right angled at C, semicircles are described on the three
sides, thus forming two lunes AFCDA and CGBEC. The sum of the areas of
these lunes equals the area of the triangle ABC.]
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 ratio being >3, and to direct
attention to the importance 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.
[Illustration: FIG. 8.]
With Archimedes (287-212 B.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. 8), joined A and B
with O the centre, bisected the angle AOB by OD, so that BD became the
semi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1:
sqrt(3):2 he sought an approximation to sqrt(3) and found that AB:BO >
153:265. Next he applied his theorem[4] BO + OA:AB::OB:BD to calculate
BD; from this in turn he calculated the semi-sides of the circumscribed
regular 24-gon, 48-gon and 96-gon, and so finally established for the
circumscribed regular 96-gon that perimeter:diameter < (3-1/7):1. In a
quite analogous manner he proved for the inscribed regular 96-gon that
perimeter:diameter > 3-(10/71):1. The conclusion from these therefore
was that the ratio of circumference to diameter is < 3-1/7 and >
3-(10/71). 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.[5]
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. 127-151),
in the _Great Syntaxis_, gives 3.141552 as the ratio[6]; and the Hindus
(c. A.D. 500), who were very probably indebted to the Greeks, used
62832/20000, that is, the now familiar 3.1416.[7]
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-(17/120) and > 3-(15/106), 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-(16/113) or 355/113,
which, being equal to 3.1415929..., is correct to the sixth fractional
place.[8]
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 X 2^16) 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 [theta] = 2 sin squared 1/2[theta]. 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.[9] 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[10]--
___________________
__________ / ___________
___ / ___ / / ___
1/2 \/ 1/2 . \/ 1/2 + 1/2\/ 1/2 . \/ 1/2 + 1/2 \/ 1/2 + 1/2 \/ 1/2 ...
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 "[pi]-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),[11] 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,[12] 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
1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
et minus
quam 314159265358979323846264338327950289
1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ...
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.[13]
[Illustration: FIG. 9.]
[Illustration: FIG. 10.]
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 = (1/3)AOC.[14] 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:--