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330

PRO

the point of concourse of the two intersecting lines will bea point in the intersection of the two first planes.

Proposition XIV.—If two points be given in the projec-tion of a straight line, the whole line is given in position.

Problem I. — Given a point on a plane, inclined to theplane of projection, to find the orthography, or projection ofthe point.

Plate I. Figure 1.—Let in be the given intersection ofthe inclined plane ; and let A be the given point thereon ;draw A a perpendicular to the intersection, I n, cutting it in b ;

, make a b c equal to the inclination of the original plane tothe plane of projection ; make b c equal to b a, and draw C aperpendicular to A a ; then will a be the projection of thepoint A.

Demonstration. —Conceive the triangle BBC to be turnedup on the base B a, perpendicular to the plane of projection ;also conceive the original plane, with the point a, to be turnedon the line i n, as a hinge ; and the line b a will always bein a vertical plane, whose intersection is a a; and since b cis equal to b a, and b c is also in the same vertical plane whenturned up, BA may be made to coincide with B c ; thereforelet b a coincide with b c; and, because b c is equal to b a,

! the point a will fall upon c : but, because c a is perpendicularI to B a, and the plane a b c is by supposition perpendicular to! the plane of projection, therefore a c is perpendicular to theplane of projection ; consequently a is the projection of the| point A.

Corollary. —Hence the projection of a line may be obtainedby finding the projection of two points in that line, and thenjoining the two projected points.i Problem II.— To find the projection of a straight line inthe original plane.

! Figure 2.—Let a b be the given straight line; producei A b to its intersecting point, c ; by Problem I. find a, theJ , representation of A; join a c ; draw b b perpendicular toj I i N, cutting a c at b ; then a b is the projection of the line: required.

| i Problem III.— To find the orthographical projection of an\ angle.

j j Figures 3 and 4.—Let abc be the given angle ; find b,

; j the projection of the point b ; produce a b and c b to theirj intersecting points, d and e ; join e b and i> b, and produce| them to a and c, if necessary, as in Figure 3 ( Figure 4 doesj j not require it) ; then a be is the projection of the anglej i required.

: I Problem IV.— The indefinite projection of a straight line! being given; to find the projection of any original point: therein.

Figure 2. —Let a c be the indefinite projection of A b, andlet it be required to find the projection of the point b, in theoriginal line a b. Draw b b perpendicular to i n, cutting a cin b; and Swill be the projeclion of the point b.

Problem V.— To find the projection of a given triangle.

Figure 5. —Let a b c be the triangle given ; find the pro-jection, a Sc, of the angle A b 0 ; by Problem III. and byProblem IV. find the points a and c, the projections of thepoints a and c ; join a c, and a b c will be the projectionrequired.

Problem VI.— To find the projection of a parallelogram.

First, when the sides are obliquely situate.

I Figure 6. Let a b c d be the given parallelogram ; find abc,

! the representation of the angle abc (being any one of thethree angles that is most convenient) by Problem III. andthe representations a and c, of the points a and c, by ProblemIV.: draw a d parallel to S c, and c d parallel to S a ; andthe parallelogram abed will be the representation of theoriginal parallelogram abcp.

2. When the parallelogram is a rectangle, with 1parallel and perpendicular to the intersecting line- . «oiPlate II. Fiqure I.—Find the representation of 1 ,j.

c, d in one of the sides, c d perpendicular to the in

ters eCl

jots;ti o«’

ti o<>

die

by Problem I. ; draw d a and c S parallel to the ’ nt ^ n tl> ein; draw b a parallel to d d, cutting eb at S; , ecta .i)gl erectangle abed will be the representation of the *a b c d, as required

3. When two of the sides are parallel and twoFigure 2. —Find the representation a, of the _ p' n0 jnt>Problem I. ; produce the side A b to its intersecting P^ ^ p et'join the intersecting point e and the point a ; dl j a " r ( jra" r ^pendicular to the intersecting line, cutting e a at b> pelparallel to i n ; draw c c parallel to b b ; draw cd P a l)cd>b a, and a d parallel to be; and the parallelogr a ®^ o g rl i[Sf

, oblifi 11 ®' b y'oint/: j;

will be the representation of the original p araA B CD, as required.

Note. Both of these might have been — „

problem ; they are thus particularized, in order to

. oB 0

included t) ,e

reader a clear view of the subject. * a rej

’ Problem VII.— To find the representation ojpentagon. ; n t A .’

Figure 3.—Find a, the representation of the p ^ noiri*

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W

seC nd g tl‘ >e ° lpr °.

and join each of those points with the point a V roblc 10jections, b and e, of the points B and e, by F 1

Problem I.; produce A E and a b to their intei'Se^'^" j e pr

py

produce the diagonals, a c and a d, to their in ^j P)points, and find the representations, c and d, of c gen t»h 011Problem IV.; join e d; and abed will be the reprrequired. ^ the t r> j

Problem VIII.— Given the intersecting line, on 0 ject ejection of one of the angles of a pentagon, arl \,. , re ,length of one of the sides; to project the wholefiS

Fiqure 4.—Let b a e be the projection of one o , a e

produce a b ^ b a

1 — F a >r, „efl'

li>t

of a pentagon, say the most remote . _

F and G, their intersecting points ; then, U P°P 0 f a P 1the segment, fag, of a circle, to contain the an ” er pendi aUtagon, that is, of two right angles ; draw a A P . | e t tf ® ^p G, to cut the segment in A ; join A F and A o r G, n

projected side T * ’ JO,,i ~ - —

t,n S af at ts ; u n J ’ dra fi h b perpendicular to r«>Abc ® e ; and project thf ,f escribe ‘be regular P e " ‘^ ePreceding example hev P urts > bc,cd,de, as J"

Problem IX _ r n # ,

Plate III Figure Paction of a circle. ^

the diameter c e no,.,, ’ ,. aw the diameter b v paiall >, ,

a > the projection of $ en ^ lc . u ^ v to the intersecting lino* ^

Parallel to the inteLltji^T A ; and draw the diaia6 0 fUPomt b ; j 0 j a be w g ,' ne ; bnd b, the projection d gjt

ate; then, with hole P roduce it to f; join bt, CL J t1 ’ f.jnoraxis, ae, describe tl jnajor axis, ab, and half the n .Problem X T* ^ ‘PS’s, b ede, the projection reqt\

Be i) e n, of a circle '^ n< ^ the projection of the teg

Figure 2 —««TJ * c

Parallel to in . J™ U f h tile centre, A, draw the diameter, j

the axis a c and ^, ete ‘be semi-circumference, CP ’ ( j,e

chord b e, to meet th! S - in the last P robiem ’ ^

representation of the " Uers ? ct,n g b'ne, in, in " to bi

then , with the semi J . mt H•' J oin f ff, and produce '. g , gfso much of it as m a f ,S u 0 and a ^> describe an ei ,P jeh,will be the rem-o ^ necessary; and the part,required. P dentation of the segment, sen

Problem XI - , . s ell‘P s ‘ s '

bcdeb- » ti „ ' .7 10 F rl d the projection of , ,Aer

at A. BD «^CE being the two axes cutting each ot

axe »f then 3 ’^"l* h ®. ^presentations, bd andJJ’ f t ue

th c diameters, b d and c e, descr