CIRCLE
, a plane figure, bounded by a curve line which returns into itself, called its circumference, and which is every where equally distant from a point within, called its centre.
The circumference or periphery itself is called the circle, though improperly, as that name denotes the space contained within the circumference.
A circle is described with a pair of compasses, fixing one foot in the centre, and turning the other round to trace out the circumference.
The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and marked °; each degree into 60 minutes or primes, marked ; each minute into 60 seconds, marked ″; and so on. So 24° 12′ 15″ 20‴, is 24 degrees 12 minutes 15 seconds and 20 thirds.
Circles have many curious properties, some of the most important of which are these:
1. The circle is the most capacious of all plain figures, or contains the greatest area within the same perimeter, or has the least perimeter about the same area; being the limit and last of all regular polygons, having the number of its sides infinite.
2. The area of a circle is always less than the area of any regular polygon circumscribed about it, and its circumference always less than the perimeter of the polygon. But on the other hand, its area is always greater than that of its inscribed polygon, and its circumference greater than the perimeter of the said inscribed polygon. However, the area and perimeter of the circle approach always nearer and nearer to those of the two polygons, as the number of the sides of these is greater; the circle being always limited between the two polygons.
3. The area of a circle is equal to that of a triangle whose base is equal to the circumference, and perpendicular equal to the radius. And therefore the area of the circle is found by drawing half the circumference into half the diameter, or the whole circumference into the whole diameter, and taking the 4th part of the product. Demonstrated by Euclid.
4. Circles, like other similar plane figures, are to one another, as the squares of their diameters. And the area of the circle is to the square of the diameter, as 11 to 14 nearly, as proved by Archimedes; or as .7854 to 1 more nearly; or still more nearly as .7853981633,9744830961,5660845819,8757210492, 9234984377,6455243736,1480769541,0157155224, 9657008706,3355292669,9553702162,8318076661, 7734611 + to 1; as it has been found by modern mathematicians.
In Wallis's Arithmetic of Infinites are contained the first infinite series for expressing the ratio of a circle to the square of its diameter: viz,
1st, The circle is to the square of its diameter, | or as , by Sir I. Newton; or as , by Gregory and Leibnitz; and a great many other forms of series have been invented by different authors, to express the same ratio between the circle and circumscribed square.
5. The circumferences of circles are to one another, as their diameters, or radii. And as the areas of circles are proportional to the rectangles of their radii and circumferences; therefore the quadrature of the circle will be effected by the rectification of its circumference; that is, if the true length of the circumference could be found, the true area could be found also. But whilst several mathematicians have endeavoured to determine the true area and circumference, others have even doubted of the possibility of the same. Of this latter opinion is Dr. Isaac Barrow: towards the end of his 15th Mathematical Lecture he says, he is of opinion, that the radius and circumference of a circle, are lines of such a nature, as to be not only incommensurable in length and square, but even in length, square, cube, biquadrate, and all other powers to infinity: for, continues he, the side of the inscribed square is incommensurable to the radius, and the square of the side of the inscribed octagon is incommensurable to the square of the radius; and consequently the square of the octagonal perimeter is incommensurable to the square of the radius: and thus the ambits of all regular polygons, inscribed in a circle, may have their superior powers incommensurate with the co-ordinate powers of the radius; from whence the last polygon, that is, the circle itself, seems to have its periphery incommensurate with the radius. Which, if true, will put a final stop to the quadrature of the circle, since the ratio of the circumference to the radius is altogether inexplicable from the nature of the thing, and consequently the problem requiring the explication of such a ratio is impossible to be solved, or rather it requires that for its solution which is impossible to be apprehended. But, concludes he, this great mystery cannot be explained in a few words: But if time and opportunity had permitted, I would have endeavoured to produce many things for the explication and confirmation of this conjecture. Sir Isaac Newton too, in book 1 of his Principia, has attempted to demonstrate the impossibility of the general quadrature of oval figures, by the description of a spiral, and the impossibility of determining, by a finite equation, the intersections of that oval and spiral, which must be the case, if the oval be quadrable. And several other authors have attempted to demonstrate the impossibility of the general quadrature of the circle by any means whatever. On the other hand, many authors not only believe in the possibility of the quadrature of the circle, but some have even pretended to have discovered the same, and have published to the world their pretended discoveries: of which no one has rendered himself more remarkable than our countryman Mr. Hobbes, though a great scholar, and of excellent understanding in other matters. See Quadrature.
The approximate quadrature of the circle however, or the determination of the ratio between the diameter and the circumference, is what the mathematicians of all ages have successfully attempted, and with different degrees of accuracy, according to the improved state of the science. Archimedes, in his book de Dimensione Circuli, first gave a near value of that ratio in small numbers, being that of 7 to 22, which are still used as very convenient numbers for this purpose in common measurements. Other and nearer ratios have since been successively assigned, but in larger numbers,
From such methods as the foregoing, a variety of series have been discovered for the length of the circumference of a circle, such as the following, viz, If the diameter be 1, the circumference c will be variously expressed thus, | And many other series might here be added. See my Mensuration in several places; also my paper on such series in the Philos. Trans. 1776; Euler's Introductio in Analysin Infinitorum; and many other authors.
6. Some of the more remarkable properties of the circle, are as follow.
If two lines AB, CD cut the circle, and intersect within it, the angle of intersection E is measured by half the sum of the intercepted arcs AC, DB.
But if the lines intersect without the circle, the angle E is measured by half the difference of the intercepted arcs AC, DB.
7. The angle at the centre of a circle is double the angle at the circumference, standing on the same arc; and all angles in the same segment are equal. Also the angle at the centre is measured by the arc it stands upon, and the angle at the circumference by half the same arc.
8. If the chords FG, HI cross at right angles, the sums of the opposite arcs are equal; viz .
9. If one side NO of a trapezium inscribed in a circle be continued out, the external angle, LOP will be equal to the opposite internal angle M.
10. An angle, as RQS, formed by a tangent QR and chord QS, is measured by half the arc of the chord QS, and is equal to any angle T formed in the opposite arc QTS.
11. If VW be a diameter, and XYZ a chord perpendicular to it; then is XZ or ZY a mean proportional between the segments YZ, ZW. So that if d denote the diameter VW, x the absciss VZ, and y the ordinate ZX; then is ; which is called the equation to the circle.
The chord VX is a mean proportional between the diameter VW and the versed sine VZ; and the chord WX is a mean proportional between the diameter and the versed sine WZ; also each versed sine is proportional to the square of the corresponding chord; viz VZ : WZ :: VX2 : WX2.
12. When two lines cut the circle, whether they intersect within the circle, or without it, as in the two figures to article 6, the segments between the common intersection and the two points where each line cuts the curve, are reciprocally proportional, and their rectan- gles are equal; viz, EA : EC :: ED : EB, or .
13. In a trapezium inscribed in a circle, the rectangle of the two diagonals is equal to the sum of the two rectangles of the two pairs of opposite sides; viz, .
14. If any chords EF, EG, drawn from the same point E in the circumference, be cut by any other line HI, the rectangles will be all equal which are made of each chord and the part intercepted by this line, viz, .
15. In a circle whose centre is N and radius NO, if two points M, P, in the radius produced, be so placed that the three NM, NO, MP, be in continued proportion; then if from the points M and P lines be drawn to any, or every point in the circumference, as Q; these lines will be always in the given ratio of MO to PO; viz, MQ : PQ :: MO : PO.
16. If VW, be two points in the diameter, equidistant from the centre T; and if two lines be drawn from these to any point X in the circumference; the sum of their squares will be equal to the sum of the squares of the segments of the diameter made by either point; viz, .
17. If a line FE perpendicular to the diameter AB, meet any other chord CD in the point E; then is .
18. If upon the diameter GH of a circle there be formed a rectangle GHKI, whose breadth GI or HK is equal to GL or HL, the chord of a quadrant, or side of the inscribed square; then if from I and K lines be drawn to any point M in the circle GMH, they will cut the diameter GH in such a manner that .
19. If the arcs PQ, QR, RS, &c, be equal, and there be drawn the chords PQ, PR, PS, PT, &c, then it | will be PQ : PR :: PR : PQ + PS :: PS : PR + PT :: PT : PS + PV, &c.
20. The centre of a circle being O, and P a point in the radius, or in the radius produced; if the circumference be divided into as many equal parts AB, BC, CD, &c, as there are units in 2n, and lines be drawn from P to all the points of division; then shall the continual product of all the alternate lines viz PA X PC X PE &c be = rn - xn when P is within the circle, or = xn - rn when P is without the circle; and the product of the rest of the lines, viz PB X PD + PF &c = rn + xn: where r = AO the radius, and x = OP the distance of P from the centre.
21. A circle may thus be divided into any number of parts that shall be equal to one another both in area and perimeter. Divide the diameter QR into the same number of equal parts at the points S, T, V, &c; then, on one side of the diameter describe semicircles on the diameters QS, QT, QV, and on the other side of it describe semicircles on RV, RT, RS; so shall the parts 1 7, 3 5, 5 3, 7 1 be all equal, both in area and perimeter. See my Tracts, pa. 93.
22. To describe a Circle either about or within a given Regular Polygon. Bisect two of its angles, or two of its sides, with perpendiculars, and the intersection of the bisecting lines will, in either case, be the centre of the circles.
Parallel, or Concentric Circles, are such as are equally distant from each other in every point of their peripheries; or that have the same centre. As, on the other hand, those are called the eccentric circles, that have not the same point for their centres.
The Quadrature of the Circle, is the manner o sdeseribing, or assigning, a square, whose surface shall be perfectly equal to that of a circle. This problem has exercised the geometricians of all ages, but it is now generally given up as a problem impossible to be effected, by most persons that have any just claim to that rank. Des Cartes insists on the impossibility of it, for this reason, that a right line and a circle being of different natures, there can be no strict proportion between them. Dr. Barrow shews the strongest probability of the same thing; and not only that the diameter and circumference themselves, but that all powers of them to infinity, are incommensurate.
The Emperor Charles V offered a reward of 100,000 crowns to any person who should resolve this celebrated problem: and the States of Holland also proposed a reward for the same thing. See Quadrature.
Circles of the Higher Orders, are curves in which WYm : YZm :: YZ : YX, or WYm : YZm :: YZn : YXn.
When m and n are each equal to 1, then WY : YZ :: YZ : YX, which is the property of the common circle.
Put WY = x, YZ = y, WX = a; then is YX - a - x, and the above proportions become , and , the equations to curves of this kind.
Curves defined by this equation will be ovals when m is an odd number. Thus suppose m = 1, then the equation becomes y2 = x. ―(a-x) or ax-x2, the equation to the common circle. And if m = 3, it becomes y4 = x3. ―(a-x) or ax3-x4, which denotes a curve of this form AB.
But when m denotes an even number, the curve will have two insinite legs. So if m = 2, the equation becomes y3 = x2. ―(a-x) or ax2-x3, for a circle of the 2d order, and which defines one of Newton's defective hyperbolas, being his 37th species of curves, whose asymptote is the right line EF, making an angle of 40 degrees with the absciss AB.
Circle of Curvature, or circle of equi-curvature, is that circle which has the same curvature with a given curve at a certain point; or that circle whose radius is equal to the radius of curvature of the given curve at that point.
Circles of the Sphere, are such as cut the mundane sphere, and have their circumference in its surface.
These circles are either fixed or moveable.
The latter are those whose peripheries are in the | moveable or revolving surface; and which therefore move or turn with it; as the meridians, &c. The former, having their periphery in the immoveable surface, do not revolve; as the ecliptic, equator, and its parallels.
The circles of the sphere are either great or little.
A Great Circle of the Sphere, is that which divides it into two equal parts or hemispheres, having the same centre and diameter with it. As the horizon, meridian, equator, ecliptic, the colures, and the a<*>muths.
A Little, or Lesser Circle of the Sphere, divides the sphere into two unequal parts, having neither the same centre nor diameter with the sphere; its diameter being only some chord of the sphere less than its axis. Such as the parallels of latitude, &c.
Circles of Altitude, or Almucantars, are little circles parallel to the horizon, having their common pole in the zenith, and still diminishing as they approach it. They are so called from their use, which is to shew the altitude of a star above the horizon.
Circles of Declination, are great circles intersecting each other in the poles of the world.
Circle of Dissipation, in Optics. See the article Dissipation.
Diurnal Circles, are parallels to the equinoctial, supposed to be described by the several stars, and other points of the heavens, in their apparent diurnal rotation about the earth.
Circle Equant, in the Ptolomaic Astronomy, is a circle described on the centre of the equant. Its chief use is, to find the variation of the first inequality.
Circles of Excursion, are little circles parallel to the ecliptic, and at such a distance from it, as that the excursions of the planets towards the poles of the ecliptic, may be included within them; being usually fixed at about 10 degrees.
It may here be observed, that all the circles of the sphere, described above, are transferred from the heavens to the earth; and so come to have a place in geography as well as in astronomy: all the points of each circle being conceived as let fall perpendicularly on the surface of the terrestrial globe, and thus tracing out circles perfectly similar to them. So, the terrestrial equator is a circle conceived precisely under the equinoctial line, which is in the heavens: and so of the rest.
Horary Circles, in Dialling, are the lines which shew the hours on dials. These are straight lines on the dials, but called circles as being the projections of the meridians.
Horary Circle, or Hour Circle, on the globe, is a small brazen circle fixed to the north pole, divided into 24 hours, and furnished with an index to point them out, thereby shewing the difference of meridians in time, and serving for the solution of many problems, on the artificial globes.
Circle of Illumination, is that imaginary circle on the surface of the earth, which separates the illuminated side or hemisphere of the earth from the dark side: and all lines passing from the sun to the earth, being physically parallel, are perpendicular to the plane of this circle.
Circles of Latitude, or Secondaries of the Ecliptic, are great circles perpendicular to the plane of the ecliptic, intersecting one another in its poles, and passing through every star and planet, &c.—These are so-called, because they serve to measure the latitude of the stars, which is an arch of one of these circles, intersected between the star and the ecliptic.
Circles of Longitude, are lesser circles parallel to the ecliptic, diminishing more and more as they recede from it, or as they approach the pole of that circle.
They are so called, because the longitudes of the stars are counted upon them.
Circle of Perpetual Apparition, one of the lesser circles parallel to the equator, described by the most northern point of the horizon, as the sphere revolves round by its diurnal motion.—All the stars included within this circle, are continually above the horizon, and so never set.
Circle of Perpetual Occultation, is another lesser circle at a like distance from the equator, but on the other side of it, being described by the most southern point of the horizon, and contains all those stars which never appear in our hemisphere, or which never rise.
All other stars, being contained between these two circles, do alternately rise and set, at certain moments of the diurnal rotation.
Polar Circles, are immoveable circles, parallel to the equator, and at such a distance from the pole as is equal to the greatest declination of the ecliptic, which now is 23° 28′. That next the northern pole is called the arctic, and that next the southern one the antarctic.
Circles of Position, are circles passing through the common intersections of the horizon and meridian, and through any degree of the ecliptic, or the centre of any star, or other point in the heavens; and are used for finding out the situation or position of any star. These are usually six in number, cutting the equinoctial into 12 equal parts, which the astrologers call the Celestial Houses, and hence they are sometimes called Circles of the Celestial Houses.
