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which a circle is first drawn and divided into any number of equal parts, in this case 1 to 12. A tangent is then drawn at O equal in length to the circumference of the circle and similarly divided. If further tangents are drawn at each point of the circumference progressively increasing in length according to the number of divisions on the first tangent, i.e. from point 1 one part, point 2 two parts, etc., they will yield points for the free-hand drawing of the curve.

Spirals are shown in 348:2, 3. In 2 the containing circle is divided into any number of equal sectors (eight are shown). One radius (radius vector) is then divided into the same number of equal parts, and concentric circles described through each division. The spiral is then unwound from point O at the centre of the circle, gaining one division as it travels through each sector. If necessary the number of divisions in the radius vector can be doubled, i.e. 16, and the curve will then travel twice round the containing circle to form a double spiral; while the spiral need not start from the centre of the circle but from any point on the radius sector provided the divisions 0,1,2,3,4, 5, etc., are marked off from the commencement of the spiral and not from the centre of the containing circle.

Spirals of constant pitch built up of quadrants are constructed as shown in 348:3. A square 1234 is first drawn and a perpendicular X erected equal in height to eight times the side of the square. The arcs are then drawn from centre 1 with radius one part (one side of the square), centre 2, two parts, centre 3, three parts, centre 4, four parts, centre 5, five parts, etc.

Entasis

Tapered shafts or columns with straight sides give the optical illusion of curving inwards, therefore classical architecture gave a slight outward swelling (entasis) to counteract the tendency, and to convey an impression of the weight-bearing function of the column. Various methods are used of which 349 gives one of the simplest and most satisfactory, for the curve must be subtle or the effect becomes exaggerated. The height of the column AB is first drawn, also the top EF and bottom diameters CD. Semicircles with centres A and

349 Entasis radii AC and BE are then drawn, and perpendiculars erected from the smaller semicircle to cut the larger semicircle at 3. The arc C3 is then divided and from points 1, 2 perpendiculars are erected to cut corresponding divisions in the height of the column, yielding points through which the curve CE can be drawn.

Three-centred

Four-centred

Trefoil

Ogee

Three-centred

Four-centred

Trefoil

Ogee

350 Gothic details: arches and tracery
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