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CIRCLE (Latin circuius); a plane figure comprehended under a single line which returns into itself, having a point in the middle, from which all the lines drawn to its circumference are equal. This point is called the centre, and these lines the radii. Although, properly speaking, it is the space included within the periphery or circumference, yet, in the popular use of the word, circle is frequently used for the periphery alone. From the geometrical definition of the circle, it appears that its magnitude is dependent upon the magnitude of its radius or its diameter, i. e., a line which touches two points of the circumference, and passes, at the same time, through the centre, or, which is the same thing, a line equal to twice the length of the radius. The surface of the circle is equal to the product of the circumference and half the radius. If there existed a rational proportion, that is, a proportion to be expressed in whole numbers, of the surface of the circle to a square surface, there would be, at the same time, a rational proportion between the diameter and the circumference. But, from geometrical reasons, no rational proportion of the diameter to the circumference is possible; it can be expressed only by approximation. However, the proportion thus obtained is quite as accurate as is necessary for any purpose in the applied mathematics. Yet there have always been instances, and some of a very late date, of men laboring long and intensely in searching for the square equal to the surface of the circle, and who often believed that they had actually solved the problem. Very recently, the newspapers were full of such a solution by a boy in England, In the approximate proportion, if the diameter is called 1, the circumference willbe equal to 3.1415926535___ Francis Vieta obtained the proportion to this number of figures. Afterwards it was further determined by Adrianus Romanus to 15, by Ludolphus of Cologne (often improperly called von Keuleri) to 35 (from him it is often called the Ludolphic number), by Sharp to 72, by Machin to 100, by Lagny to 126, and lastly, in an Oxford manuscript, it was obtained up to 156 decimals. Archimedes first estimated the proportion of the diameter to the circumference tobe as 7 to 22, or as 1 to 3.142.....; after him, Metius, as 113 to 355, or as 1 to 3.1415929, which is correct to 6 decimals, and sufficiently accurate for most purposes. Every circle is divided into 360 degrees, and by its arcs all angles are measured. The circle, therefore, is one of the most important geometrical figures, and an accurate division of it is requisite for measuring the angles under which distant objects appear (upon which surveying, astronomical observations, &c. rest)a very desirable object, for which many prizes have been offered by learned societies. (See Degree.)Circle, in logic; the fault of an argument that supposes the principle it should prove, and afterwards proves the principle by the thing which it seemed to have proved. The same fault takes place in definitions, when an idea is defined by others which suppose the knowledge of the first. Arguing in a circle is a fault into which men are very liable to fall, particularly in theological discussions.