The various terms used in connection with circles are illustrated in 343. Arc, any part of a circumference; chord, any straight line shorter than the diameter terminated by the circumference at both ends; diameter, a chord which passes through the centre of a circle; radius, a straight line drawn from the centre of a circle to the circumference and therefore half the diameter; normal, a straight line drawn from any point on the circumference radial to the centre; tangent, a straight line touching the circumference at any one point at right angles to a normal at that point (343:1). Quadrant, a sector which is a quarter of a circle; semicircle, a segment which is a half of a circle; minor segment, part of a circle contained between a chord and its arc and less than a semicircle;
major segment, part of a circle greater than a semicircle; minor sector, part of a circle contained within two radii and less than a semicircle; major sector, part of a circle greater than a semicircle (343:2). Additionally 343:3 shows concentric circles having the same common centre but different radii, and 343:4 eccentric circles each of which has a different centre.
The circumference of a circle is approximately 3.1416 or 31/7 times the diameter, usually expressed by the Greek letter p (pi). Thus the circumference is pXd (diameter), while the area is pr2 (radius). In practice, the development or length of the circumference can easily be found by dividing it into any number of equal parts which are then plotted as straight lines and measured off, but the method is not strictly accurate as the plotted units are in reality chords of the circle. Alternatively, dividing the diameter into seven equal parts and adding one part to three times the length of the diameter will give a reasonable approximation.
Methods of describing circles to pass through given points, between converging lines, tangential to each other or inscribed within two equal arcs, are used in the full sizing of fan (Gothic) tracery and are illustrated in 344 and 350. Figure 343:5 shows the construction of a circle about the fixed points A, B, C, where AB and BC are each bisected and the intersection of the perpendiculars at O will give the centre of the required circle. Where a circle has to be inscribed within a triangle (343:6), as the circle will be tangential at points B and C which are equidistant from A, a perpendicular erected on B will provide one radius, and the bisection of the angle at A another, while the point at which they intersect at O will thus be the centre of the required circle. If a series of circles has to be inscribed tangential to two converging lines (344:1) then the angle BAC is first bisected and the first circle drawn as already described, after which the angle BOD is bisected and from centres E with radius EB an arc is described to cut AB at F. A line drawn from F parallel to BO will give the centre of the next circle, and centres for succeeding circles are found in like manner. If a circle has to pass through a given point X (344:2) and touch a line AB at a given point Y, then erect CY perpendicular to AB. Join XY and construct the angle DXY equal to
343 Circles, etc. (1)
the angle CYX. At the point of intersection of DX with CY the point E will now form the centre of the required circle as EX and EY have equal angles and are, therefore, also equal.
Where three equal tangential circles have to be inscribed within two equal arcs as in a trefoil to a pointed arch (344:3), an equilateral triangle ABC is first drawn, and three circles with radii equal to half the length of the sides are described with centres A, B, C. A horizontal line is then drawn tangential to the base circles and where the bisectors of the angles BAC, BCA intersect this line will lie the points X, Y which are the centres of the arcs. A method of inscribing any number of equal circles within a given circle is shown in 344:4. First divide the
344 Circles, etc. (2)
circumference of the given circle into the same number of equal parts as the number of circles required, and join each division to the centre. One of the sectors so formed is then bisected at AB and a tangent is drawn passing through B and extended (developed) to meet the sector division AC. If the angle ACB is then bisected and extended to meet AB at D it will give the centre of the first circle with radius DB. A circle is now drawn from centre A with radius AD, and the centres of the remaining circles found by stepping off distance DE from the intersection of the inner circle with the radius of each sector.