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° a e t >ar a ]’ at ! 0ns ’ ^ le object is represented upon two planes,Ag t[j e ®‘» an d the other perpendicular to the horizon.

?*lly jjj aces of the different parts of a machine are gene-* S al »o tn ^ rtlca l planes parallel to each other, the elevationXs, or t ° n a vert * ca ^ pl ane parallel to the faces of the, th e e ° greatest number of faces, in order to facili-• Xect *tion, as this position not only gives the altitude^Udg Q j. r s _> but is also convenient in ascertaining the mag-Xea . boes in every direction in these planes, and thetl l ^ e Wo t ^ le bnes form with each other ; for if the facesi re Can k Stan< ^ at oblique angles with the projecting plane,t a Verti i onl y one direction, namely, that of measuring; e line > in which the true dimensions can be ascer-tk^cd' bv Cr ^ ot; bcr will be fore-shortened, and the angles^orijji y 4 be lines of representation will also vary from1) ^ the & an §^ es of the object.

aXes of wheels are sometimes obliquely situate, ittk Cev fi ,!, Cessar y to represent them in various positions.

hinds of representations are sections through

^rizon^! P arts ( as through the axis of a wheel) of plane. I'hoij^h a Projections, and elevations.jXne TO a ,P^ an an d elevation are sufficient to construct any* cooip r , bh Proper attention, yet from the greater facilityb ei * S| Jr es , en ding the nature of the design, and applying the^ l *cul ar i °. Practice, two elevations are sometimes given,a P r eaag s y lri a complex machine. One of these elevationsig 6 - ' v °rl{ he P a rts in the longest horizontal dimension ofbre’-j 1 ^ *be other represents the parts comprehendedthg r ta > or horizontal extension,f difjj„ Presentation of a machine, the wheel-work is the(^8 0 f *■ of any part, except when the whole of the sur-a big wheel are planes, with two or more faces paralleli Cu W tri Jecting plane, and all the other surfaces perpen-

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„, e as in this case the boundaries of the parallell he a P re sented by figures similar to the original faces,°r ci rol e other surfaces are projected into straight

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rig ■ S ' . Lhis is so very simple, as not to require anyi if the wheel, or its faces, have an oblique

‘h7 8sar y to P ro j ect ' n g plane, several considerations areW;.? u gh tbg , ect a true representation. A ground sectionlion % w aXls pf the wheel, placed at the required angle,»J* ,°f the er ? ect i°n of the plane of elevation, and an eleva-cl)(i e d. . cl on a plane parallel to the faces, will bepbjg^e pro- . 1 . 3 likewise necessary to observe, that, in thisit*th ate ren Ctl011 ’ fbo parallel arrises of any original° r 'gii\ 1 Seilt;e ^ by parallel lines, and all equal distancesOrj I'll th e a object are represented by other distances equalof ? la al pl atl Se ^p s ,5 also, all straight lines whatever in theof ^ art a r eg e > ,i v ided into parts, are represented by a seriesi H r *y lie e p c hvely in the same proportion. The proportiontlie^'ori of \° lts original) as radius to the cosine of thethe J^ibcsem, ■ ot 'ig' n al line and the plane of projection.PUp, ar >e 0 f at }on of an original circle is an ellipsis ; and as'Otgji e rep re g °-l ect i°n is vertical, the elevation of an ellipsisj.® at cd with its greater axis perpendicular to the

—Given the seat and altitude of a line on‘h anqi * ntersec tion of the plane with another plane,Pi, of .if to find the sun’s seat on such plane, as also

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^ ' 6 j a. ^^ Wre 1-— Let a c, No. 1 and 2, be the seatta ke o„ intersection of the last named plane.

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® pet-j,. y P°’ nt ) C; draw C D perpendicular to A c ;? X p4end CUl , ar t0 A B > cuttin S a b at b. In No. 2,P Xr e C B G p cular to BC; make CH equal to c d, and^ 0 |_ 1 j^'aw to mlk, No. 3, the inclination of the

, Cl perpendicular to b g ; make b e equal4 Q

to B g, and join ae; then ae will be the seat of the sun’srays on the plane, as required. Draw E f perpendicularto A E ; make e f equal to G h, and join a f; then e A F willbe the angle of altitude of the line on the same plane.

In No. 1, the two planes are at right angles to each other;in this case, the problem may be constructed without thequadrilateral B c h G ; thus, having drawn c e perpen-dicular to a B, cutting A b at B ; make b e equal to c d,and complete the triangle A e f, as in No. 2 ; then a ewill be the seat of the line, and eaf the angle of altitude,as before.

This may easily be conceived, by supposing, that upon thetriangle abc, No. 2, the triangle acd, and the quadrilateralB C h G, are raised perpendicularly ; then C H and C D willcoincide, and the point H will coincide with the point D; turnthe triangle A b e upon A b, and the side b e, being perpen-dicular to A b, will describe a plane upon B e, perpendicularto the plane abc; and because bc and be are in the samestraight line, b c will be in that plane ; and because theplane bcbg, is supposed to be raised perpendicular to theplane abc upon b c, the plane supposed to be describedby b e will be in the same plane with bciig, when raisedperpendicularly to the plane ABC; therefore, move theplane abe till b e coincide with b g, supposed to be raisedwith the plane bchg; the planes acd, bchg, and abe,being thus supposed to be raised, turn Air round a e, till E ffalls upon g h ; then the point f will fall upon h ; and thestraight line a f will also be coincident with ad; forthe triangle a e f will be at a right angle with thetriangle a e b ; therefore e f will be at right angles to theplane abe.

Pkoblem XXII. — Given the seat and altitude of the sun'srays on a plane, and the angle which the seat of a lineparallel to the plane of projection makes with the seat ofthe sun’s rays; to determine the inclination of the planeof shade obstructed by the original line towards the plane ofprojection.

Figure 2. — Let A c be the seat of the sun’s rays given uponthe plane of representation ; make cad equal to the altitudeof the sun’s rays; draw C d perpendicular to a c ; make cabequal to the angle made by the seat of the sun’s rays withthe seat of the original line, which occasions a plane of shade;from C draw c b perpendicular to a b, and c e perpendicularto C b ; make C E equal to C D, and join b e ; then cbs willbe the inclination of the plane required. Produce c b to f ;make b f equal to b e, and join a f ; then let the trianglescad, cbe, b A F, be turned up, and af and ad will coin-cide ; and the triangle cbe will measure the inclination ofthe plane a bf to C b a. This problem is of the utmost usein finding the shadows of parallel lines to planes in variouspositions ; as also in finding the shadows of cylinders wherethe axis is parallel to the plane of projection ; and in deter-mining the points and lines of light and shade in cylindersand cylindric rings.

Pkoblem XXIII. — Given the seat and altitude of a line,and the seat and altitude of the sun’s rays ; to determine theshadow of the line on that plane.

Figure 3.—Let he be the seat of the line; make theangle, ehi, equal to the altitude of the line, viz., of abc,No. 1; e being any point in be; draw ei perpendicularto h e ; draw e f parallel to the seat of the sun’s rays ; anddraw E G perpendicular to e f ; make e G equal to E i ; makethe angle e g f equal to the complement of the sun’s altitude,and join H f, which will be the shadow of the linerequired.

This problem will be of great use in finding the shadowsof objects upon inclined planes.