Thomas Muir

“ 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.” ”

“ 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 πr2. The exact geometrical evaluation of the second quantity, viz. πr2, which, in reality, is equivalent to determining a square equal in area to a circle, engaged the attention of mathematicians for many centuries. ”

“ 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. ”

“ 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 ”

“ 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 form of a circle is familiar to all; and we proceed to define certain lines, points, &c., 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. ”

“ 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 ”

“ 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. ”

“ 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 x2+y2+2gx+2fy+c=0, i.e. in the general equation of the second degree the co-efficients of x2 and y2 are Cartesian co-ordinates. equal, and of xy zero. The co-ordinates of its centre are –g/c, –f/c; and its radius is (g2+f 2–c) 1/2. The equations to the chord, tangent and normal are readily derived by the ordinary methods. ”