C YL

224

C YL

J

planes of the chord plane and section. In No. 2, draw b dperpendicular to b a, and make b d equal to the distancebetween the parallel lines e f and z m, No. 1, and draw d cparallel to b a. Through any point, d in i k, No. 1, drawn v at right angles to i k ; make d n equal to d c, No. 2, andd v, No. 1, equal to bd, No. 2.. In No. 1, draw n > parallelto f e, meeting ie at o, and join o v. Draw n s and o lparallel toKF, meeting e f at Q and L; produce N q, meetingz m produced at Y, and join L y. Draw any number of linesa a, bb, e c, &c. parallel to o l, cutting the lines e f and i k,at the points a, b, c ; from the points a, b, e, in i k, draw thelines a 1, b2, c 3, &c. parallel to o v, and through the pointsctj b, c, in e f, draw the lines a 1, b2, c 3, &c., parallel toiy. Make all the lines a 1, b2, c 3, &c., from ie, equalto their corresponding distances al, b2, c3, &c. from ef.A curve being drawn through the points 1, 2, 3, &c., theextremities of the ordinates from i k, will give the sectioni v k I of the segment of the cylinder. Or take any pointpoint l in e f, and draw l o parallel to f k, the one side ofthe rectangle, cutting lEato; produce oi. to m, and drawz m parallel to e f, to touch the base in M. In No. 2, drawB D at a right angle with b a ; make b d equal to l m, anddraw d c parallel to b a. In No. 1, draw g h parallel to i k,at the distance D C, No. 2 ; draw O N parallel to e f, cuttingG H at N ; draw N Q parallel to o L, cutting e f at Q ; producenq to meet zm produced at y, and join it; draw n v atright angles to i k, cutting IK at d ; make d v equal to B c,and join o v; then proceed to find the ordinates as before.

The difference between the old and the first method is, thatin the old method, the breadth of -the segment or base of thesolid is limited, in order to find the disecting lines of theordinates, and two diagrams are employed in the construction ;whereas in the first method, one diagram only is used, andthe angle of inclination of the planes of the section and chordbeing made as directed, the measure of the first side of thehypothenusal side of the right-angled triangle formed by it,is arbitrary; by this the other parts are regulated. The readermust observe, that it is only the angles which are required ;the length of the lines is of no other consequence than, ifthey be too long, they will be rather cumbrous, and if tooshort they will not be a sufficient guide for drawing theordinates. This limitation has been the occasion of some notunderstanding the principle, by supposing the point v to bein the curve, which may be either on the one side or theother, according to this method.

A knowledge of the sections of prisms, which includesegmental cylinders, is an acquisition of the first importanceto the carpenter and joiner, and is the very essence of theart of constructing hand-railings and groins.

The method by having the position of the cutting plane, isnot so well adapted to practice, as that of having three givenpoints on the surface of the segmental cylinder.

Under such given data, we have already shown how thesection is to be found for a whole cylinder, by describing thecurve through the extremities of the axis, which are given inposition. We shall now show how the curve is to be obtainedby ordinates in the segmental cylinder from the same data.

In a segment of a cylinder are given three points, one oneach common line of the chord plane, and curved surface,and one upon the intermediate curved surface itself, to findthe section passing through these three points.

Figures 8 and 9.— -In both figures let a b c be the base ofthe solid, and A, b, c, the seats of the given points, draw aj>and c e perpendicular to a c : make A d equal to the heightof the point upon its seat A, and CE equal to the height of thepoint upon its seat c ; also from c e take c f equal to theheight of the intermediate point, from its seat B. Draw f g

parallel to A c, cutting d e in G ; draw G H P ar ® n dto E c, cutting AC at h ; join H b. Produce C t,.‘ r eE D to meet in 1 ; in E 1 , Figure 8, also in El produced ,9, take any point, N, and draw N m perpendicular to ^produce m n to meet 1 c at K, Figure 8 ; in Figurewill cut c 1 , produced in k. In both figures draw K ,-k vjjgpendicular to k c, and 1 l parallel to hb; from 1 , ' V1distance 1 l, describe an arc, cutting n m, at m ; join I,r '

To find any point in the curve of the section

take

an/

ar«

the

draw

point a, in a c, and draw a b parallel to l 1 , cuttinga b c at b, draw a c parallel to c e cutting d e in c; theparallel to 1 m ; make c d equal to a b, and d is a point 1curve. In the same manner as many points may be t°will be sufficient to complete the section. p the

Another method Figure 10, let a b c be the base °^ {e esegment, as before ; also, let A, b, c, be the seats ofpoints, as before ; and fgh their respective heights- , t0AD, bp, C E, each perpendicular to A c, respectively e( lf, g, 11 ; draw abi and dpi, also ace and bee; theo t J*the intersection 1 it. In d e take any point l, and dr $0

perpendicular to d k; produce h l to meet A K at N> jj 3 -

perpendicular to A k, cutting 1 k at o. From K, with ^tance k o, describe an arc, cutting iMatii, and join .inThen, to find any point d, in the curve of the sec *‘ theA c take any point a ; draw a b parallel to K O, m ee c!

arc A B c at b; draw a e parallel to A D, meeting t jjoS

draw c d parallel to k m, and make c d equal to a b; afor any other point. , re lb

The very same description of words applies to ■y^gtvreenexcept that the point L is inDK, produced, instead ot j of

D and k, and also the point N, in A K produced, m ce d'

between A and k. The same is also the case with tuping method. eylin^

Figure 12. To form the edge of the envelope ot a J e i 0 pe 19surface, terminated by a line, so that, when the enV 0 jo'folded upon the cylindric surface, the edge formed ‘ iscide with a plane passing through three given P oin .p aC e,each common line of the chord-plane and curved sui ‘the other in the intermediate curved surface itself „ perLet a b c be the base of the solid ; draw A D a " be poi nt ’pendicular to a c, making a d equal to the height 0 >v boS®whose seat is a, and c e equal to the height of the P° ^ po>° ’seat is C; make c f on c e equal to the height o cA , c 1 ^whose seat is b, and join d e. Draw f g parallel ^ ^ ,

ting e D at G; draw G H parallel to E C, cutting C . ^join h b. Divide the arc abc into any numbei .^gparts, and extend them upon a c produced to I, re5 po0^ 1113points of division at a, b, c, &c., to 1 ; the copoints of 1, 2, 3, &c., on the arc cab. _ u ; r ed, s °? e

To find any point in the line of the envelope re< j )age of 4pose that which corresponds to its seat 1, on thesolid. , . dra^j to

Draw 1 h parallel to b h, cutting AC in - - ra i- , eparallel to c E, cutting e d at h ; also draw cl P P t |,e h ac e, and make ap equal h A, then p is a point obt ai' ie ’required ; and thus the whole line E p q r s tu v K 1 u jred-and this line is the edge of the envelope e c 1K ,- fi,;r st9 . n ,«Cylinder, Sealine. When the axis of a c y' n g Ql .at oblique angles with its base, it is called a scaM 1 ^

cylinder. _ . r a

CYLINDRICAL, something peculiar, similarto a cylinder. k er a se

Cylindrical Ceiling, a ceiling which is el ^

cylinder, or a segment less than a semi-cyli n< i er ' g yro^Cylindrical ceilings are vulgarly called by 80 ^ - s

waggon-headed ceilings. rcA-ef^

When an apartment is sufficiently high, a 3

h;