Secondly, suppose the given circles intersect each other, and the required circle is to touch their concave sides, its

given radius being not

greater than half the distance apart of their circumferences as measured on the line joining the centres.

Let A, B (figs. 52, 53, 54) be the centres of the circles, joined by the line MN, meeting the circumferences in M and N. Take M C, N D each equal to the length of the given radius; from A as a centre, with

Join the points A, P, B, Q, producing A P, A Q, BP, B Q, to meet the circles in F, G, H, and E. Then a circle drawn

Gothic arch struck from centres A, B, in which the circle of radius equal to MC or N D is inscribed.

61. To describe a circle through two given points and touching a given circle.-First, let the given points be outside the given circle. Let O (fig. 55) be the centre of

the given circle, P and Q the given points.

Through P and Q draw an indefinite right-line FPQ; bisect P Q in A, and draw A B perpendicular thereto. Take any point B, on the perpendicular A B, as a centre, and describe a

circle that will cut the given circle in any two points C, D, and also pass through P and Q. Draw the chord CD, and produce it to meet the line through FPQ, in E. Find EF a mean proportional (17) between EP and EQ, and from E as a centre, with EF as radius, describe a circle. cutting the given circle in H; join OH, and produce O H to meet A B in J. Then a circle drawn from J as a centre, with J H as radius, will touch the given circle and also pass through the given points.

Secondly, let the given points be within the circumference of the given circle.

Let O (fig. 56) be the centre of the given circle, P, Q, the given points. Draw through P and Q an indefinite

right-line; and bisect PQ in A. Draw A B perpendicular to PQ, and from any point B therein describe a circle that shall cut the given circle in any points C and D, and also pass through the given points P and Q. Join CD, and produce DC to meet PQ in E. Find EF a mean proportional (17) between EP and EQ, and from E as a

centre, with E F as radius, describe a circle cutting the given one in H; draw HO cutting AB in the point I; then a circle drawn from I as a centre, and with IH as radius, will touch the given one and also pass through the given points.

62. To draw geometrical tracery in a Gothic arch.Let BDA (fig. 57) be a Gothic arch springing from the Fig. 57.

line A B, joining the centres E, F, from which the arcs BD, AD, are struck. perpendicular thereto.

Bisect A B in C, and draw CD

Take BM, A K equal to each other and less than half A B; make M H, KG each equal to AF or B E. From G and H as centres, and with G K, HM as radius, describe the arcs MN, KL. It is required to draw a circle which shall touch all four of the arcs, and have its centre on C D.

Take BI equal to BE, and from E as a centre, with EI as radius, describe a circle; draw a circle (61) passing through G and H, and touching the last-drawn circle in the point R; this circle will have its centre at O on the

line CD. Draw the lines RO, HO, cutting the given arcs in T and P; and from O as a centre, with OP or OT as radius, draw a circle, which will touch all four of the given arcs as required.

Let the given arch ADB (fig. 58) be equilateral, A

being the centre of the arc BD, and B that of the arc AD; and suppose the line A B divided into three equal parts at K and M. Let G and H be the centres of the arcs MN, KL, having their radius equal to A B. Take C the centre of A B, and draw the vertical line CD. Measure M X equal to one-sixth of M B or to one-third of M C, and from A as a centre, with A X as radius, describe a circle cutting CD in O. Draw HPO cutting K L in P, and join

Fig. 59.

A O, producing AO to meet BD in T. Then a circle drawn from O as a centre, and with OT or OP for radius, will touch all the four arcs as required.

Let the given arch A D B (fig. 59) be equilateral, and also the inner arches AFC, BEC; AB being bisected in C. Draw the vertical

CD; bisect BC in X, and from A as a centre, with AX