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._

Cartesian co-ordinates.

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)^½. 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 x² + y² = (x + iy) (x - iy) = 0 (where i = sqrt -1) intersect the circles. The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.

The equation x² + y² = 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 x² + y² 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 x² + y² = 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 x² + y² = [alpha]², x² + y² = [beta]². These equations show that the circles touch where they intersect the lines x² + y² = 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, &c.); 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 a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0, the centre being cos A, cos B, cos C. The inscribed circle is cos ½A sqrt([alpha]) cos ½B sqrt([beta]) + cos ½C sqrt([gamma]) = 0, with centre [alpha] = [beta] = [gamma]; while the escribed circle opposite the angle A is cos ½A sqrt(-[alpha]) + sin ½B sqrt([beta]) + sin ½C sqrt([gamma]) = 0, with centre -[alpha] = [beta] = [gamma]. The self-conjugate circle is [alpha]² sin 2A + [beta]² sin 2B + [gamma]² sin 2C = 0, or the equivalent form a cos A [alpha]² + b cos B [beta]² + c cos C [gamma]² = 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

a[beta][gamma] + b[gamma][alpha] + C[alpha][beta] + (l[alpha] + m[beta] + n[gamma]) (a[alpha] + b[beta] + c[gamma]) = 0 (1).

This obviously represents a conic intersecting the circle a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0 in points on the common chords l[alpha] + m[beta] + n[gamma] = 0, a[alpha] + b[beta] + c[gamma] = 0. The line l[alpha] + m[beta] + n[gamma] is the radical axis, and since a[alpha] + b[beta] + c[gamma] = 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 u[alpha]² + v[beta]² + w[gamma]² + 2u'[beta][gamma] + 2v'[gamma][alpha] + 2w'[alpha][beta] = 0, it is readily seen that for this equation to represent a circle we must have

-kabc = vc² + wb² - 2u'bc = wa² + uc² - 2v'ca = ub² + va² - 2w'ab.

Areal co-ordinates.

The corresponding equations in areal co-ordinates are readily derived by substituting x/a, y/b, z/c for [alpha], [beta], [gamma] respectively in the trilinear equations. The circumcircle is thus seen to be a²yz + b²zx + c²xy = 0, with centre sin 2A, sin 2B, sin 2C; the inscribed circle is sqrt(x cot ½A) + sqrt(y cot ½B) + sqrt(z cot ½C) = 0, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is sqrt(-x cot ½A) + sqrt(y tan ½B) + sqrt(z tan ½C)=0, with centre - sin A, sin B, sin C; and the self-conjugate circle is x² cot A + y² cot B + z² 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 a²yz + b²zx + c²xy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux² + vy² + wz² + 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')/a² = (w + u - 2v')/b² = (u + v - 2w')/c².

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 ½(a + b + c); an alternative form is qr cot ½A + rp cot ½B + pq cot ½C = 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 ½A + rp tan ½B + pq tan ½C = 0, with centre -ap + bq + cr = 0. The circumcircle is a sqrt(p) + b sqrt(q) + c sqrt(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 [rho] be the radius and lp + mq + nr = 0 the centre, we have [rho] = (lp1 + mq1 + nr1)/(l + m + n), in which p1, q1, r1 is a line distant [rho] from the point lp + mq + nr = 0. Making this equation homogeneous by the relation [Sigma]a²(p - q) (p - r) = 4[Delta]² (see GEOMETRY: _Analytical_), which is generally written {ap, bq, cr}² = 4[Delta]², we obtain {ap, bq, cr}²[rho]² = 4[Delta]²{(lp + mq + nr)/(l + m + n)}², the accents being dropped, and p, q, r regarded as current co-ordinates. This equation, which may be more conveniently written {ap, bq, cr}² = ([lambda]p + [mu]q + [nu]r)², obviously represents a circle, the centre being [lambda]p + [mu]q + [nu]r = 0, and radius 2[Delta]/([lambda] + [mu] + [nu]). If we make [lambda] = [mu] = [nu] = 0, [rho] is infinite, and we obtain {ap, bq, cr}² = 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 = PC² = PD², 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.

[Illustration: Fig. 5]

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, AB¹ be the tangents from any point on the line AX. Then by Euc. i. 47, AB² = AO² - OB² = AX² + OX² + OP²; and OX² = OD² - DX² = OP² + PD² - DX². Therefore AB² = AX² - DX² + PD². Similarly AB'² = AX² - DX² + DP'². 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 x² + y² + 2ax ± k² = 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 h² ± k², 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 k², 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 [pi], 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 2[pi]r and [pi]r². The exact geometrical evaluation of the second quantity, viz. [pi]r², 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 [pi], is given below (_Squaring of the Circle_).

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

+--------------+-----------+-----------+ | | Number. | Logarithm.| +--------------+-----------+-----------+ | [pi] | 3.1415927 | 0.4971499 | | 2 [pi] | 6.2831858 | 0.7981799 | | 4 [pi] |12.5663706 | 1.0992099 | | (1/2) [pi] | 1.5707963 | 0.1961199 | | (1/3) [pi] | 1.0471976 | 0.0200286 | | (1/4) [pi] | 0.7853982 | 1.8950899 | | (1/6) [pi] | 0.5235988 | 1.7189986 | | (1/8) [pi] | 0.3926991 | 1.5940599 | | (1/12) [pi] | 0.2617994 | 1.4179686 | | (4/3) [pi] | 4.1887902 | 0.6220886 | | | | | | [pi] | | | | ------ | 0.0174533 | 2.2418774 | | 180 | | | | | | | | 1 | | | | ------ | 0.3183099 | 1.5028501 | | [pi] | | | | | | | | 4 | | | | ------ | 1.2732395 | 0.1049101 | | [pi] | | | | | | | | 1 | | | | ------ | 0.0795775 | 2.9097901 | | 4 [pi] | | | | | | | | 180 | | | | ------ |57.2957795 | 1.7581226 | | [pi] | | | | | | | | [pi]² | 9.8696044 | 0.9942997 | | | | | | 1 | | | | -------- | 0.0168869 | 2.2275490 | | 6 [pi]² | | | | | | | | _____ | | | | \/ [pi] | 1.7724539 | 0.2485750 | | | | | | _____ | | | | \³/ [pi] | 1.4645919 | 0.1657166 | | | | | | | | | | 1 | | | | -------- | | | | _____ | 0.5641896 | 1.7514251 | | \/ [pi] | | | | | | | | 2 | | | | -------- | | | | _____ | 1.1283792 | 0.0524551 | | \/ [pi] | | | | | | | | 1 | | | | ---------- | | | | _____ | 0.2820948 | 1.4503951 | | 2 \/ [pi] | | | | | | | | _____ | | | | / 6 | | | | \³/ ---- | 1.2407010 | 0.0936671 | | V [pi] | | | | | | | | ______ | | | | / 3 | | | | \³/ ------- | 0.6203505 | 1.7926371 | | V 4 [pi] | | | | | | | | log e [pi] | 1.1447299 | 0.0587030 | +--------------+-----------+-----------+

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 = 2[pi]a. Area = [pi]a².

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 ½[theta]. Area of sector = ½a²[theta] = ½ radius × 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 = ½a²([theta] - sin [theta]) = ½a²[theta] -¼c² cot ½[theta] = ½a² - ½c sqrt(a² - ¼c²). If h be given, we can use c² + 4h² = 8ah, 2h = c tan ¼[theta] to determine [theta].

Approximate formulae are:--Area = (1/15)(6c + 8c2)h; = (2/3) sqrt(c² + (8/5)h²)·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 metrêsis], 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