FRO
332
PRO
angle of the pentagon was found by dividing a circle into fiveequal parts, and joining the chords of two adjoining parts, asab, B C, Figure 4, No. 1. Then, to obtain the verticalangle, draw A c as a base ; bisect the angle abc, which willalso bisect the straight line A c in E ; make E F equal to thehalf of b c, No. 2, and draw F G parallel to B C, cutting b d atG ; make e g, No. 2, equal to E G, No 1 ; and in No. 2 joinG b and G C, which will give the vertical angle bgc.
Problem XIX. To project the dodecaedron.
Figure 6.—Find ab c de, the orthography of one of theplanes, by Problem VII; find the angle which two contigu-ous planes of the solid make with each other, by ProblemXVIII.; find the intersection of each of the planes adjoiningto the plane whose projection is found, by Problem XIV., andthus three of the arrises of the solid will be projected ; then,to project the planes adjoining the extremities of the arrises,or edges, ab, a e, ah, suppose that contiguous to a b, producebe to meet the intersecting line x z in o, and produce bf tomeet XT in p ; join o,p ; then, having the projection of theangle c bf and the projected lengths, c b or bf of one of thesides, and the intersecting line, op, of the plane of that side,find the orthography, cb f ml, of the whole figure, by ProblemVIII. In like manner, find the orthography of the figuresghiqr, k std c, and the representation of the solid will becompleted.
Problem XX.— To divide the representation of a circleinto parts, representing the original circle divided into equalparts.
Plate VI. Figure 1.—Let A b c d be the original circle,and let abed be its orthographic representation. Supposethe first part of the original circle to commence at the givenpoint, E ; let e F, f G, G ii, &c. be the equal parts in the origi-nal circle, and let o be its centre ; join eo,fo,GO,ho, &c.;let the diameter, b d, be perpendicular to the intersectingline, IN, and let A c be parallel to i N ; then b d and a c beingthe representations of bd and A c, draw b a and b a, and letb a cut" the radii e o, f o, go, ho, &c. in the points k l m n,&c. : draw k k, l l, M m, N n, &e. cutting the representativechord, ab, athlmn, & c. Let o be the representative centre;draw ok, ol, o m, on, &c., and produce ok, ol,o m, o n, &c.to meet the circumference of the representative circle in e,fg, h, &c.: then will e ffg, g h, &c. be the representations ofthe equal parts e f, f g, g h, &c. of the original circle : andif the representative radii, eo,fo, go, ho, See. be producedto meet the opposite parts of the curve at e',f, g', h’, See.,e f,f g,g' h, &c. will also represent equal parts.
In the same manner, if b c, and its representation, b c, bejoined, the other two representative quadrants may bedivided, so as to represent equal parts in the original quad-rants remaining; and thus the whole representative circlewill be orthographically divided into parts, representing thewhole of the original circle divided into equal parts.
To find the orthographical representation of a water-wheel.
Figure 2.—Let No. 1 be the face of the wheel, or itssection, perpendicular to the axis: let i N be the intersectingline : draw the ellipsis abed, representing the originalcircle, abcd, divided into equal parts, as in the precedingproblem : through the points of division draw the portionsof the radii, as intercepted between the orthographic circum-ferences : draw the ellipsis efg, which will give the ortho-graphy of the opposite outer circle : then draw the edges ofthe boards perpendicular to the intersecting line, i N, so asto be terminated by the orthographic circles efg and c da ;then, two of the sides of the float-boards being thus found,the other two may be completed by parallel lines.
The ring which supports the float-boards is represented in
con
tai*-
the same manner as the orthographic circles, win 0 * 1 ^ qlisthe representations of the float-boards themselves- a( eradiating pieces, to which the float-boards are attacmade to tend to the centre, on the side that adjoins t aCC oiH'boards, and the opposite face is tapered off; and, to ofplish this regularly, it is made to touch the represena circle in the same plane. , nUf 1 .
To project a straight line in a plane, supposing 1 F e ofwhich contains the line to be wrapped round the s Ja cylinder.
Plate VII. Figure 1.—Let abcd be a cylmde ^ t |,atgraphically projected ; and let b a be continued to >
ag may be equal to the semi-circumference of the c ^ 0 g inalso let f e be the line to be projected, the point E e( j t0the line a d, and the position of e f to A G being sU Pf
be given. • ntdist aI, f
Continue the axis of the cylinder to any conveniei a xi3beyond the base a b : at any convenient point in ^ th ethus produced, describe a circle equal to the rad \ iv ;<]e d 10
_1: „ j j a i, _ j:_j. . __n _i a u ; t*i v g j
f,-om
dr»' v
cylinder ; draw the diameter HJ parallel to ab: aS ,
semi-circumference H I j into any number of equal P a ^ Jalso divide a g into the same number of equal p aI
the points of d’ • •
fines parallel to thp°^J- H tb p 8emi ' c,reum ference b lJ > in-jection A b c d. ’S of the cylinder through the P
dicular tootle snidTnT ° f divis, ' on A o, draw lines perp*
Points as ag ; from the * t0 Cut *he Iine E F in 03 ’f'nes
Parallel to ab, to “ U L P p tS ° f section “ e f, d 'TJ’£axis ; then the line e f f/° rmer hues drawn parallel torequired. F of contrary flexure, is the project’
Projected Sf thHurf th ° r Concave line de, Fig** fjthe line d e. ace op a cylinder upon a plane
Figure d.—Fet UP t n tfle sur f ace of a cone. Q f
the cone on a plane n. tlie orthographical project* 011development of ,i lf / )ara . e t0 the axis, and let a b E he ^this development w i]F I , < , i)ecte< ^ surface : then any hue if’' jBthe following mainTr ; 6 Pr ° jeCted from the conic surface
»ito any number*of S f“ lc ! rcJe BDC > and divide the ar0B ( j, earc b dc, dr“w q “ al pnrts i f rom the divisions of "
“any points of section !. rpe " dleular t0 B c > t0 meet fromthe points of section ■ a ‘\ t lere are divisions in live,
the points of sectin ™ 1 le base B c > draw lines to A > t)
draw straight lines tn m t,le base BE oi ’ the de 'i°d to beProjected ; from ft ■' to cut the Jine B F iDtend ,t arc*from the centre a t P ° nits ot section in be, describ « „
f the number i n A E in as man L P° ifltS f on in * *
draw lines parallel to nn ! evei 7 P 0IIit of s l PC tiveIf ;through the corr^ B ,?’ to cut each former line respec ^
the dark line and ^” dlng P °‘ nts ’ draw a curve, as si \io p0
the conic surfaced * ^ be the projection of the h”
to any giV^’Ta^M r 16 P r °j ection ma y be fouad ’ %% < l,e
Project,“fa of fg forin * as ia Fi & ure 4 > ” here/3
depend des 'S ns , hut mechanical ^‘fbe
architect enmr, 10 ^ rap h’oal representation ; it en* 1 0 j,jectintended* to be ^ ° r rnec hanic, to construct 1greatly facilit-ir ^ x ® cute d. As drawings of every . ^ 0 Jies,so, in drawino- 6 a knowledge of the formation 0 t \i° dsof represent,n g n>ac unes > it is necessary to know the>surfaces as tl ^ S ^ stem oP Pttes upon cylindr’O ] jpe s
represented 1“ knowl ^ge will determine whether^
In drawl ^ pamPe ^ to each other, or tend to a p ppics° dlama S machinery, as in other kinds of ortbog*P