CIRCLE, (front the Latin, eirculus), a plane figure con tained under one line, called the circumference, Which is such that all lines drawn to it, from a certain point within the figure, are equal ; and the point from which the lines may be tints drawn, is called the centre of the circle.
A eirennifiirence may he thus described: if the end of a right line be !placed upon a fixed point, and kept upon that point while the other end is carried progressively fOrward, or round, until it comes to the place whence the motion began, the moveable extremity will thus trace out the circumference of a circle.
In order to obtain the measurement of angles, the circum ferences of all circles are supposed to be divided into 360 equal parts, called degrees ; each degree is supposed to be divided into 60 equal parts, called minutes ; each minute is divided into 60 equal parts, called seconds ; each second is to be divided into 60 equal parts, called thirds ; which are again divided and subdivided ad inginitum. Any denomi nation, whether of degrees, minutes, or seconds, &c. is known by a peculiar character, written over the right-hand figure of that denomin:ition ; thus, °, written over the right-hand figure of a number, shows that number to represent. degrees ; the character thus, ', Written over the right-hand figure of a number, shows the number thus distinguished to represent minutes ; e. g. 130°i2 f' 48" 57"', &e. represents 136 degrees, 24 minutes, -18 seconds, 57 thirds, &e. ; and, as similar arcs are such as are contained under the same, or equal angles, they contain the same number of degrees, &c., the number of parts of the are of a circle, described from the meeting of two lines forming an angle, and comprehended between them, is the true measure of the angle; for the number of parts is still the same, whatever be the radius of the arc, or of the circle, the parts being greater as the radius is greater. The are of the circle being supposed to be divided into 360 equal parts, the radius will he found to be equal to the chord of 60; because the circle contains six equilateral triangles, whose bases are chords to the circle, whose summits meet in the centre, and whose sides are radii to the circle. And since the sixth part of :160°, or of a whole circle, is 60°, the chord of 60 is therefore equal to the radius. The parts of the are may be measured by parts of the radii, which are always supposed to contain the same number: for if there he two arcs described from the angular point of an angle, between the legs, these arcs may be measured in parts of their respective radii.
The circle is the most capacious of all plane figures; that is, it contains the greatest area under equal [Perimeters, or has the least perimeter enclosing the same area.
The area of a circle is equal to the area of a triangle, the base of which is equal to the circumference, and the perpen dicular equal to the radius, and consequently equal to a rectangle, whose breadth is equal to the radius, and the length equal to the semi-circumference.
Circles, like other similar plane figures, are to one another as the squares of their diameters.
The ratio of the diameter of a circle to its circumference, has never been exactly ascertained. Archimedes was the first, in his book De Dimensione Cirenli, who gave the ratio in small numbers, being that of 7 to 22, which is still the most useful for practical purposes. Vista carried the approxi mation to ten places of figures, by means of circumscribed and inscribed polygons of 393,216 sides, showing the ratio to be as 10,000,000,000 to 31,415,926,536 nearly, the circumference being greater than 31,415,92(1,535 but less than 31,415,926,537 Van Gilcn carried the approximate ratio to 36 places of figures ; which number was recalculated and confirmed by Willebrod Snell. Mr. Abraham Sharp extended the ratio to 72 places of figures, which was afterwards extended to 100 places by the ingenious Mr. Machin ; and, lastly, M. De Lagny, in the ifemoires de l'Aead., 1719, has carried this ratio to the amazing extent of 128 places of figures.
In approximating the circumference or area of a circle from the diameter, the first authors had recourse to inscribed and circumscribed polygons ; since it was found that the circumference of the circle was greater than the perimeter of the inscribed polygon, hut less than that of the circum scribing one; and, that when the polygon contained a great Dumber of sides, the circumference of the circle did not differ materially from either, it would be still more nearly equal to the arithmetical mean of the two. And. to give the reader an idea how very near the circumference obtained by this means is to the truth, the circumscribed and inscribed polygons may be taken of such a number of sides, as that their perimeters will be each expressed by any given number of figures of the same value, from unity, either taken indi vidually, or as a whole number, and consequently the circum ference of the circle may be expressed, or carried to any degree of accuracy required.
But the method of obtaining the circumference by this means, being found extremely laborious, other methods, by a series of fractions, have been invented, so as not only to be much more easy in the calculation, but also to show how the terms may be continued at pleasure, by inspection only. Dr. Wallis was the first who expressed the area of a circle, in terms of the diameter, by an infinite series, and showed that, if the square of the diameter was 1, the area would be