D O K, I O L, perpendicular to D OK; draw 1 K parallel to B A, K L parallel to I D, &c., meeting the diagonals; from D as a centre, with the distance D B, describe the arc B G ; from I as a centre, with the distance I G, describe G E; from K, with the distance K E, describe K н, &c., proceeding in the same manner until the outside of the scroll is completed; make B R equal to the breadth of the rail; then from D, with the distance D R, describe the arc R S; from 1, and distance I s, describe s T; and from T, with K T, describe Tu, which completes the scroll as required.

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39. 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 R s on plan; from R, at right angles with A R and equal to the required

A

T

R

m

n

S

B

height, draw the line R T, then T A, equal the length of the struts or corners of the roof; from A, with the distance ▲ T, describe an arc T l, continue the diagonal A R until it cuts the arc T l, 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 0, 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.

40. 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 A D and BC until they meet at E; then from E as centre, with the radius E c, describe the arc C H; also from E, with the radius E B, describe the arc B 1; make в I equal in length to twice A G B, A draw the line E I, and B C I H is the form of plate as required.

D

T

E

G

H

1

Sector from which angles may be obtained.

GEOMETRY APPLIED TO MECHANICS.

41. 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 c D, c d, the one being equal to the greatest diameter of the screw, and the other to the lesser diameter, or diameter at the bottom of the threads; divide each semi-circumference

into any number of equal parts, as c, e, f, g, h, i, D, from which draw lines parallel to the line A B ; divide the lines C F, D F, into equal divisions of half the required pitch or consecutive threads, as t, r, v, &c.; draw the lines t t, r r, v v, &c., parallel with the diameter C D, and subdivide any two connected divisions into the same number of equal parts contained in both semicircles, from which draw lines meeting the vertical lines; then by hand, or otherwise, and through the intersections, draw the waved lines, m, n, p, q, &c., and a thread of the screw is delineated as required.

Note. The same process might be continued throughout the whole length of the screw, but it is much more convenient, when the proper curves are obtained, to form a suitable ruler: lay it in its proper situation upon each division, and draw the lines as required.

42. To delineate a square-threaded screw, the pitch and diameter of screw being given. (See Plate A, fig. 2.,

On the line A B representing the centre of the screw, describe the semicircles C D, c d, equal to the diameters at the tops and bottoms of the threads; divide each semi-circumference into four equal parts, draw lines from each and parallel to the line A B; draw also the lines C F, D F, which divide into the proper required pitch, as a a, b b, c c, &c.; divide any two connected pitches or divisions, as a, b, into four equal parts, from which draw lines parallel to the diameter C D, meeting the vertical lines o, p, q, and forming intersections through which the waved lines s, n, i, s, n, i, or tops of the threads, must be traced by hand or otherwise; draw also the lines x, y, forming intersections through which to trace the curve surface exhibited between and caused by the angular return