38. To draw a volute for the capital of the Ionic column.

square; di

vide each into six equal parts, which are the centres from which the volute is to be described; thus, from 1, with the distance 1 B, describe the curve BD; from 2, with 2 D, describe DE; from 3, with 3 E, describe E A, &c., approaching the centre by degrees until the volute is completed as required.

39. To draw a scroll for the termination of a hand rail, &c.

Let A B equal the given breadth, draw A E perpendicular to A B, which divide into eleven equal parts, and A E equal to one of them; join BE, bisect AB in C and BE in F, make CD H equal to C F, and draw D G perpendicular to A B; from F, with the radius FE, or F B, describe an arc cutting

R

P

B

A E

K

T

G

DG at G; draw GH perpendicular to B E, cutting BE at O (or centre of the scroll); draw the diagonals

DOK, IOL, perpendicular to DOK; draw IK parallel to BA, KL parallel to ID, &c., meeting the diagonals; from D as a centre, with the distance D B, describe the arc BG; from I as a centre, with the distance IG, describe G E; from K, with the distance K E, describe KH, &c., proceeding in the same manner until the outside of the scroll is completed; make BR equal to the breadth of the rail; then from D, with the distance D R, describe the arc RS; from I, and distance IS, describe ST; and from T, with K T, describe TU, which completes the scroll as required.

40. To find the various angles and proper dimensions of materials whereby to construct any figure whose form is the frustum of a proper or inverted pyramid, as hipped roofs, mill hoppers, &c., &c.

Let A B C D be the given dimensions of plan for a roof, the height RT also being given; draw the diagonal A R, meeting the top or ridge Rs on plan; from R, at right angles with A R and equal to the required

D

A

T

R

с

m

n

B

height, draw the line RT, then T A, equal the length of the struts or corners of the roof; from A, with the distance A T, describe an arc Tl, continue the diagonal A R until it cuts the arc Tl, through which, and parallel with the ridge R s, draw the line m n, which determines the required breadth for each side of the roof; from A, meeting the line m n, draw the line A o, or proper angle for the end of each board by which the roof might require to be covered; and the angle at T is what the boards require to be made in the direction of their thickness, when the corners or angles require to be mitred.

41. To describe the proper form of flat plate, by which to construct any given frustum of a cone.

Let A B C D represent the required frustum, continue the lines AD and B C until they meet at E; then from E as centre, with the radius E C, describe the arc CH; also from E with the radius EB, describe the arc BI; make BI equal in length to twice AG B, draw the line EI, and BCIH is the form of plate as required.

A

D

E

H

B

G

GEOMETRY APPLIED TO MECHANICS.

42. To delineate a vee-threaded screw, the pitch and diameter of the screw being given. (See Plate A, fig. 1.)

Upon the end of the line, or vertical centre of the screw A B, describe the semicircles CD, cd, the one being equal to the greatest diameter of the screw, and the other the lesser diameter, or diameter at the