# Circle

**CIRCLE** (from the Lat. *circulus*, the diminutive of *circus*, a ring; the cognate Gr. word is , generally used in the form ), a plane curve definable as the locus of a point which moves so that its distance from a fixed point
is constant.

The form of a circle is familiar to all; and we proceed to define certain lines, points, etc., which constantly occur in studying its geometry. The fixed
point in the preceding definition is termed the "centre" (C in fig. 1); the constant distance, *e.g.* CG, the "radius." The curve
itself is sometimes termed the "circumference." Any line through the centre and terminated at both extremities by the curve, *e.g.* AB, is a
"diameter"; any other line similarly terminated, *e.g.* EF, a "chord." Any line drawn from an external point to cut the circle in
two points, *e.g.* DEF, is termed a "secant"; if it touches the circle, *e.g.* DG, it is a "tangent." Any portion of the
circumference terminated by two points, *e.g.* AD (fig. 2), is termed an "arc"; and the plane figure enclosed by a chord and arc, *e.g.*
ABD, is termed a "segment"; if the chord be a diameter, the segment is termed a "semicircle." The figure included by two radii and an
arc is a "sector," *e.g.* ECF (fig. 2). "Concentric circles" are, as the name obviously shows, circles having the same centre; the
figure enclosed by the circumferences of two concentric circles is an "annulus" (fig. 3), and of two non-concentric circles a "lune,"
the shaded portions in fig. 4; the clear figure is sometimes termed a "lens."

The circle was undoubtedly known to the early civilizations, its simplicity specially recommending it as an object for study. Euclid defines it (Book I. def.
15) as a "plane figure enclosed by one line, all the straight lines drawn to which from one point within the figure are equal to one another." In
the succeeding three definitions the centre, diameter and the semicircle are defined, while the third postulate of the same book demands the possibility of
describing a circle for every "centre" and "distance." Having employed the circle for the construction and demonstration of several
propositions in Books I. and II. Euclid devotes his third book entirely to theorems and problems relating to the circle, and certain lines and angles, which he
defines in introducing the propositions. The fourth book deals with the circle in its relations to inscribed and circumscribed triangles, quadrilaterals and
regular polygons. Reference should be made to the article Geometry: *Euclidean*, for a detailed summary of the Euclidean treatment, and the elementary
properties of the circle.

*Analytical Geometry of the Circle.*

In the article Geometry: *Analytical*, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is x [2] + y [2] + 2gx + 2fy + c = 0, *i.e.* in the general equation of the second degree the co-efficients of x [2] and y [2] are equal, and of xy zero. The co-ordinates of its
centre are -g/c, -f/c; and its radius is (g [2] + f [2] - c). The equations to the
chord, tangent and normal are readily derived by the ordinary methods.

Consider the two circles: -

x [2] + y [2] + 2gx + 2fy + c = 0, x [2] + y [2] + 2g'x + 2f'y + c' = 0.

Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2(g - g')x + 2(f - f')y + c - c' = 0, the radical axis, with the circles, and the other two where the lines x + y = (x + iy) (x - iy) = 0 (where i = √-1) intersect the circles. The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.

The equation x + y = 0 denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at
infinity along these lines, and since the terms x + y occur in the equation of every circle, it is seen that all circles pass through two fixed points at
infinity. The introduction of these lines and points constitutes a striking achievement in geometry, and from their association with circles they have been
named the "circular lines" and "circular points." Other names for the circular lines are "circulars" or "isotropic
lines." Since the equation to a circle of zero radius is x + y = 0, *i.e.* identical with the circular lines, it follows that this circle consists
of a real point and the two imaginary lines; conversely, the circular lines are both a pair of lines and a circle. A further deduction from the principle of
continuity follows by considering the intersections of concentric circles. The equations to such circles may be expressed in the form x + y = α, x +
y = β. These equations show that the circles touch where they intersect the lines x + y = 0, *i.e.* concentric circles have double contact at the
circular points, the chord of contact being the line at infinity.

In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms;
consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry. In
this article the equations to the more important circles - the circumscribed, inscribed, escribed, self-conjugate - will be given; reference should be
made to the article Triangle for the consideration of other circles (nine-point, Brocard, Lemoine, etc.); while in the article Geometry: *Analytical*,
the principles of the different systems are discussed.

## Trilinear co-ordinates

The equation to the circumcircle assumes the simple form aβγ + bγα + cαβ = 0, the centre being cos A, cos B, cos C. The inscribed circle is cos A √(α) cos B √(β) + cos C √(γ) = 0, with centre α = β = γ; while the escribed circle opposite the angle A is cos A √(-α) + sin B √(β) + sin C √(γ) = 0, with centre -α = β = γ. The self-conjugate circle is α sin 2A + β sin 2B + γ sin 2C = 0, or the equivalent form a cosA α + b cos B β + c cos C γ = 0, the centre being sec A, sec B, sec C.

The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points. Consider the equation

aβγ + bγα + Cαβ + (lα + mβ + nγ) (aα + bβ + cγ) = 0 (1).

This obviously represents a conic intersecting the circle aβγ + bγα + cαβ = 0 in points on the common chords lα +
mβ + nγ = 0, aα + bβ + cγ = 0. The line lα + mβ + nγ is the radical axis, and since aα + bβ + cγ =
0 is the line infinity, it is obvious that equation (1) represents a conic passing through the circular points, *i.e.* a circle. If we compare (1) with the
general equation of the second degree uα + vβ + wγ + 2u'βγ + 2v'γα + 2w'αβ = 0, it is
readily seen that for this equation to represent a circle we must have

-kabc = vc + wb - 2u'bc = wa + uc - 2v'ca = ub + va - 2w'ab.

## Areal co-ordinates

The corresponding equations in areal co-ordinates are readily derived by substituting x/a, y/b, z/c for α, β, γ respectively in the trilinear equations. The circumcircle is thus seen to be ayz + bzx + cxy = 0, with centre sin 2A, sin 2B, sin 2C; the inscribed circle is √(x cot A) + √(y cot B) + √(z cot C) = 0, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is √(-x cot A) + √(y tan B) + √(z tan C)=0, with centre - sin A, sin B, sin C; and the self-conjugate circle is x cot A + y cot B + z cot C = 0, with centre tan A, tan B, tan C. Since in areal co-ordinates the line infinity is represented by the equation x + y + z = 0 it is seen that every circle is of the form ayz + bzx + cxy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux + vy + wz + 2u'yz + 2v'zx + 2w'xy = 0, we obtain as the condition for the general equation of the second degree to represent a circle: -

(v + w - 2u')/a = (w + u - 2v')/b = (u + v - 2w')/c.

## Tangential co-ordinates

In tangential (p, q, r) co-ordinates the inscribed circle has for its equation (s - a)qr + (s - b)rp + (s - c)pq = 0, s being equal to (a + b + c); an
alternative form is qr cot A + rp cot B + pq cot C = 0; the centre is ap + bq + cr = 0, or p sin A + q sin B + r sin C = 0. The escribed circle opposite the angle A is -sqr + (s - c)rp + (s - b)pq = 0 or -qr cot A + rp tan B + pq tan C = 0, with centre -ap +
bq + cr = 0. The circumcircle is a √(p) + b √(q) + c √(r) = 0, the centre being p sin 2A + q sin 2B + r sin 2C = 0. The general equation to a
circle in this system of co-ordinates is deduced as follows: If ρ be the radius and lp + mq + nr = 0 the centre, we have ρ = (lp1 - mq1 + nr1/(l + m + n), in which p1, q1,
r1 is a line distant ρ from the point lp + mq + nr = 0. Making this equation homogeneous
by the relation Σa(p - q) (p - r) = 4Δ (see Geometry: *Analytical*), which is generally written {ap, bq, cr} = 4Δ, we obtain {ap, bq,
cr}ρ = 4Δ {(lp + mq + nr)/(l + m + n)}, the accents being dropped, and p, q, r regarded as current co-ordinates. This equation, which may be more
conveniently written {ap, bq, cr} = (λp + μq + νr), obviously represents a circle, the centre being λp + μq + νr = 0, and radius
2Δ/(λ + μ + ν). If we make λ = μ = ν = 0, ρ is infinite, and we obtain {ap, bq, cr} = 0 as the equation to the circular
points.

*Systems of Circles.*

*Centres and Circle of Similitude.* - The "centres of similitude" of two circles may be defined as the intersections of the common
tangents to the two circles, the direct common tangents giving rise to the "external centre," the transverse tangents to the "internal
centre." It may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided
externally and internally in the ratio of their radii.

The circle on the line joining the internal and external centres of similitude as diameter is named the "circle of similitude." It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.

With a system of three circles it is readily seen that there are six centres of similitude, viz. two for each pair of circles, and it may be shown that these lie three by three on four lines, named the "axes of similitude." The collinear centres are the three sets of one external and two internal centres, and the three external centres.

*Coaxal Circles.* - A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line. Consider
the case of two circles, and in the first place suppose them to intersect in two real points A and B. Then by Euclid iii. 36 it is seen that the line joining
the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA·PB = PC =
PD, and therefore PC = PD. Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the
radii. The line AB is termed the "radical axis." A system coaxal with the two given circles is readily constructed by describing circles through the
common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for
diameter, the maximum is the radical axis - considered as a circle of infinite radius. In the case of two non-intersecting circles it may be shown that the
radical axis has the same metrical relations to the line of centres.

There are several methods of constructing the radical axis in this case. One of the simplest is: Let P and P' (fig. 5) be the points of contact of a common tangent; drop perpendiculars PL, P'L', from P and P' to OO', the line joining the centres, then the radical axis bisects LL' (at X) and is perpendicular to OO'. To prove this let AB, AB be the tangents from any point on the line AX. Then by Euc. i. 47, AB = AO - OB = AX + OX + OP; and OX = OD - DX = OP + PD - DX. Therefore AB = AX - DX + PD. Similarly AB' = AX - DX + DP'. Since PD = PD', it follows that AB = AB'.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle. Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.

In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I', where the orthogonal
circle to the system intersects the line of centres; these points are named the "limiting points." In the case of a coaxal system having real points
of intersection the limiting points are imaginary. Analytically, the Cartesian equation to a coaxal system can be written in the form x + y + 2ax ± k = 0,
where a varies from member to member, while k is a constant. The radical axis is x = 0, and it may be shown that the length of the tangent from a point (0, h)
is h ± k, *i.e.* it is independent of a, and therefore of any particular member of the system. The circles intersect in real or imaginary points
according to the lower or upper sign of k, and the limiting points are real for the upper sign and imaginary for the lower sign. The fundamental properties of
coaxal systems may be summarized: -

1. The centres of circles forming a coaxal system are collinear;

2. A coaxal system having real points of intersection has imaginary limiting points;

3. A coaxal system having imaginary points of intersection has real limiting points;

4. Every circle through the limiting points cuts all circles of the system orthogonally;

5. The limiting points are inverse points for every circle of the system.

The theory of centres of similitude and coaxal circles affords elegant demonstrations of the famous problem: To describe a circle to touch three given
circles. This problem, also termed the "Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on
*Contacts* or *Tangencies*; geometrical solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others. The
earliest analytical solution appears to have been given by the princess Elizabeth, a pupil of Descartes and daughter of Frederick V. John Casey, professor of
mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are
reproduced in his *Sequel to Euclid*; an analytical solution by Gergonne is given in Salmon's *Conic Sections*. Here we may notice that there are
eight circles which solve the problem.

*Mensuration of the Circle.*

All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter. This ratio, invariably denoted by
π, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number. Very early in the
history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2πr and πr. The exact
geometrical evaluation of the second quantity, viz. πr, which, in reality, is equivalent to determining a square equal in area to a circle, engaged the
attention of mathematicians for many centuries. The history of these attempts, together with modern contributions to our knowledge of the value and nature of
the number π, is given below (*Squaring of the Circle*).

The following table gives the values of this constant and several expiessions involving it: -

Number. Logarithm. Number. Logarithm.

π 3.1415927 0.4971499 π 9.8696044 0.9942997

2π 6.2831858 0.7981799

4π 12.5663706 1.0992099 1 0.0168869 2.2275490

½π 1.5707963 0.1961199 6π

1/3π 1.0471976 0.0200286 √π 1.7724539 0.2485750

¼π 0.7853982 1.8950899

1/6π 0.5235988 1.7189986 [3]√π 1.4645919 0.1657166

1/8π 0.3926991 1.5940599

1/12π 0.2617994 1.4179686 1 0.5641896 1.7514251

4/3π 4.1887902 0.6220886 √π

π 0.0174533 2.2418774 2
1.1283792 0.0524551

180 √π

1 0.3183099 1.5028501 1 0.2820948 1.4503951

π 2√π

4 1.2732395 0.1049101 [3]√(6/π) 0.2820948
1.4503951

π

1 0.0795775 2.9097901 [3]√(3/4π) 0.6203505 1.7926371

4π

180 57.2957795 1.7581226 loge π 1.1447299 0.0587030

π

Useful fractional approximations are 22/7 and 355/113.

A synopsis of the leading formula connected with the circle will now be given.

1. *Circle.* - Data: radius = a. Circumference = 2πa. Area = πa.

2. *Arc* and *Sector*. - Data: radius = a; θ = 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 = aθ, where θ may be given directly, or indirectly by the relation c = 2a sin θ. Area of sector = aθ = radius × arc.

Approximate formulae are: - Arc = 1/3(8c2 - c) (Huygen's formula); arc = 1/45(c - 40c2 + 256c4).

3. *Segment.* - Data: a, θ, c, c2, as in (2); h = height of segment, *i.e.* distance of mid-point of arc from
chord.

Exact formulae are: - Area = a(θ - sin θ) = aθ - c cot θ = a - c √(a - c). If h be given, we can use c + 4h = 8ah, 2h = c tan θ to determine θ.

Approximate formulae are: - Area = 1/15(6c + 8c2)h; = 2/3 √(c + 8/5h)·h; = 1/15(7c + 3α)h, α 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, , 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, etc.) [2] ,
one of whom, Hippocrates, doubtless raised hopes of a solution by his quadrature of the so-called *meniscoi* or *lune*. [3]

[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.

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: √3:2 he sought an approximation to √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 < 317/120 and > 315/106, 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 316/113 or 335/113, 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 × 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 θ = 2 sin θ. 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] -

√ · √ + √ · √ + √( + √) ...

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), [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

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. [13]

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 = 1/3AOC. [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: -

2 tan θ > θ > 2 sin θ (Archimedes);

tan 1/3θ + 2 sin 1/3θ > θ > 3 sin θ (Snell).

2 + cos θ

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 1/3θ.

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] -

chd θ + 4 chd θ + sin θ .
1/3(chd θ - sin θ) > θ > chd θ + 1/3(chd θ - sin θ).

2 chd θ + 3 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 ABCD; by the same
process find a point C2 such that the rectangle B1C2 will be equal to BC1; 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
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 BB1, B1B2, ... in
terms of c, and the identity AB∞ = 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) as a member
of the series of curves y = (1 - x), y = (1 - x), etc., he was led to the proposition that four times the reciprocal of the ratio of the circumference to
the diameter, *i.e.* 4/π;, 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 + 1 3 5 7 ...

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.*),

A'2n = 2An A'n or 2A'n A2n (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

θ = tan - 1/3 tan [3] θ + 1/5 tan [5] θ - ...,

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 θ = 1/3√3, found the ratio equal to

√12 ( 1 - 1 + 1 - 1 + ... ),

3 · 3 5 · 3 7 · 3

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 -
1 + ... ) - 4 ( 1 -
1 + 1 - ... ),

5 3 · 5 5 · 5 [4] 7 · 5 [6] 239 3 · 239 5 · 239 [4]

which long remained without explanation. [22] Fautet de Lagny, still using tan 30°, 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 π 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 π. The general test of irrationality which he established is that, if

a1 a2 a3
...

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 ...

b1 ± ..., b2 ± ...,

be < 1, then the given continued fraction represents an irrational quantity. If this be applied to the right-hand side of the identity

tan m = m m m ...

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°, having its tangent rational, must be incommensurable with the radius; that is to say, π/4 is an incommensurable number. [27]

This incontestable result had no effect, apparently, in repressing the π-computers. G. von Vega in 1789, using series like Machin's, viz. Gregory's series and the identities

π/4 = 5 tan-1 1/7 + 2 tan-1 3/79 (Euler, 1779),

π/4 = tan-1 1/7 + 2 tan-1 1/3 (Hutton, 1776),

neither of which was nearly so advantageous as several found by Charles Hutton, calculated π 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 π-computers of the 19th century.
A condensed record compiled by J.W.L. Glaisher (*Messenger of Math.* ii. 122) is as follows: -

Date. Computer. No. of

fr. digits

calcd. No. of

fr. digits

correct. Place of Publication.

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.*

π/4 = tan-1 + tan-1 1/5 + tan-1 1/8 (Dase, 1844),

π/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 π-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 π, 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 η of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree. [31] To prove the same proposition regarding π 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 π; 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 (Göttingen, 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 π, dependent on the theory of probability, see *Mess. of Math.* ii. 113, 119; *Casopis pro pïstováni math.
a fys.* x. 272-275; *Analyst*, ix. 176.

(T. MU.)

[1] Eisenlohr, *Ein math. Handbuch d. alten Agypter, übers. u.
erklärt* (Leipzig, 1877); Rodet, *Bull. de la Soc. Math. de France*, vi. pp. 139-149.

[2] H. Hankel, *Zur Gesch. d. Math. im Alterthum*, etc., chap, v
(Leipzig, 1874); M. Cantor, *Vorlesungen über Gesch. d. Math.* i. (Leipzig, 1880); Tannery, *Mém. de la Soc.*, etc., *à Bordeaux*; Allman, in
*Hermathena*.

[3] Tannery. *Bull. des sc. math.* [2], x. pp. 213-226.

[4] In modern trigonometrical notation, 1 + sec θ:tan θ::1:tan θ.

[5] Tannery, "Sur la mesure du cercle d'Archimède," in
*Mém....Bordeaux*[2], iv. pp. 313-339; Menge, *Des Archimedes Kreismessung* (Coblenz, 1874).

[6] De Morgan, in *Penny Cyclop,* xix. p. 186.

[7] Kern, *Aryabhattíyam* (Leiden, 1874), trans. by Rodet
(Paris,1879).

[8] De Morgan, art. "Quadrature of the Circle," in
*English Cyclop.*; Glaisher, *Mess. of Math.* ii. pp. 119-128, iii. pp. 27-46; de Haan, *Nieuw Archief v. Wisk.* i. pp. 70-86, 206-211.

[9] Vieta, *Opera math.* (Leiden, 1646); Marie, *Hist. des
sciences math.* iii. 27 seq. (Paris, 1884).

[10] Klügel, *Math. Wörterb.* ii. 606, 607.

[11] Kästner, *Gesch. d. Math.* i. (Göttingen, 1796-1800).

[12] But see *Les Délices de Leide* (Leiden, 1712); or de Haan,
*Mess. of Math.* iii. 24-26.

[13] For minute and lengthy details regarding the quadrature of the
circle in the Low Countries, see de Haan, "Bouwstoffen voor de geschiedenis, etc.," in *Versl. en Mededeel. der K. Akad. van Wetensch.* ix.,
x., xi., xii. (Amsterdam); also his "Notice sur quelques quadrateurs, etc.," in *Bull. di bibliogr. e di storia delle sci. mat. e fis.* vii.
99-144.

[14] It is thus manifest that by his first construction Snell gave an approximate solution of two great problems of antiquity.

[15] *Elementa trigonometrica* (Rome, 1630); Glaisher,
*Messenger of Math.* iii. 35 seq.

[16] 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).

[17] See Euler, "Annotationes in locum quendam Cartesii,"
in *Nov. Comm. Acad. Petrop.* viii.

[18] Gergonne, *Annales de math.* vi.

[19] See *Vera Circuli et Hyperbolae Quadratura* (Padua, 1667);
and the *Appendicula* to the same in his *Exercitationes geometricae* (London, 1668).

[20] *Penny Cyclop.* xix. 187.

[21] See Sherwin's *Math. Tables* (London, 1705), p. 59.

[22] See W. Jones, *Synopsis Palmariorum Matheseos* (London,
1706); Maseres, *Scriptores Logarithmici* (London, 1791-1796), iii. 159 seq.; Hutton, *Tracts*, i. 266.

[23] See *Hist. de l'Acad.* (Paris, 1719); 7 appears instead of
8 in the 113th place.

[24] *Comment. Acad. Petrop.* ix., xi.; *Nov. Comm. Ac.
Pet.* xvi.; *Nova Acta Acad. Pet.* xi.

[25] *Introd. in Analysin Infin.* (Lausanne, 1748), chap.
viii.

[26] *Mém. sur quelques propriétés remarquables des quantités
transcendantes, circulaires, et logarithmiques.*

[27] See Legendre, *Eléments de géométrie* (Paris, 1794), note
iv.; Schlömilch, *Handbuch d. algeb. Analysis* (Jena, 1851), chap. xiii.

[28] *Nova Acta Petrop.* ix. 41; *Thesaurus Logarithm.
Completus*, 633.

[29] On the calculations made before Shanks, see Lehmann,
"Beitrag zur Berechnung der Zahl π," in *Grunert's Archiv*, xxi. 121-174.

[30] See Montucla, *Hist. des rech. sur la quad. du cercle*
(Paris, 1754, 2nd ed. 1831); de Morgan, *Budget of Paradoxes* (London, 1872).

[31] "Sur la fonction exponentielle," *Comples
rendus* (Paris), lxxvii. 18, 74, 226, 285.

[32] See *Crelle's Journal*, lxxvi. 342.

[33] See "Uber die Zahl π," in *Math. Ann.* xx.
213.

*Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)*