DG = A and DH=C. From DF cut off DI = B. Join IH, and draw GK parallel to IH, cutting DF in K.

KD is a fourth proportion to the lines A, B, C. That is, DH: DI: DG:DD, or in other words, C B A KD.

The above problem produces a proportional less than the given lines.

To produce a greater proportional, join GI and draw a line HL parallel to GI cutting the line DF in L. Then A B C DL, because Dg DI Dh DL.

## Problem

To set out the cusping in a circular window, say a trefoil cusp. Fig. 79.

Let Acbd (Fig. 79) be a circle with diameters AB, CD at right angles to each other.

From D divide the circumference into 6 equal parts in E, F, C, G, H, and D. Join FH, EG. Join CB cutting FH in I. Through I draw a line IK parallel to AB cutting GE in K. IK is the diameter of one semicircle of the trefoil.

With I as centre and IK as radius, describe an arc cutting DC in M. Join IM and KM. Then the points marked 1, 2, and 3 will be the centres for the arcs of the trefoil, with radii 1 C, 2 H, and 3 E respectively.

## Note

Any number of cusps can be set out in the circle by dividing the circumference into twice the number of parts that there are to be cusps.

## Problem

Given any portion of the circumference of a circle, to find a point that will be within the circumference if produced. It is presumed that the portion of the circumference is too large to use the centre.

Let AB (Fig. 80) be the given portion of the circumference.

Take any convenient point C in this circumference, and join AC, CB, and AB. From B draw BD = AC, making the angle Cbd = to the angle Bca. Then D is the point in the circumference required. As a check, join CD, which should be equal to AB. Fig. 80.

Alternatively, instead of making the angle Cbd = to the angle Bca by the use of an angle-measuring instrument, from the centre B, and with radius = AC, strike an arc. Similarly from C with radius = AB, strike another arc. These arcs will intersect at D.

## Problem

To find the length of the semi-circumference of a circle.

## Note

The ratio of diameter to circumference is denoted by the Greek letter it in all trigonometrical treatises, i.e. Diameter circumference 1 3.14159, etc.

The geometrical method of determining the length of the circumference of a circle (shown in Fig. 81) will be found to give results which are quite accurate enough for practical purposes. Fig. 81.

Describe any circle as Abc, and draw a diameter AC. From A draw AD at right angles to AC and equal in length to 1 1/2 times the diameter. From C draw CE parallel to AD.

With C as centre and with a radius = \ the diameter (that is = to CH) draw an arc cutting the circumference in F, and with F as centre and radius CH draw an arc meeting the first arc at G. Join GH, cutting CE at I. Join ID. Then ID is very approximately equal to the length of the semi-circumference.