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surface, of which the horizontal projection is given, to drawa tangent plane to that surface.
“ Solution. — Figure 12. Let a b, a b, represent the hori-zontal and vertical projections of the given right line, towhich the generator of the cylindric surface must be parallel;let epd be the given curve in the horizontal plane, on whichthe generator must constantly rest, and which may be con-sidered as the outline of the cylindric surface ; lastly, let Cbe the given horizontal projection of the point supposed onthe cylindric surface, from which the tangent plane mustbe drawn.
“ Next, from the supposed point on the surface, whosehorizontal projection is in c, imagine the generating rightline in the position it would have, if it passed through thatpoint: this generator, being a straight line, will be its owntangent, and consequently one of the two right lines by whichthe position of the tangent plane will be determined ; also,it will be parallel to the given right line : its two projections,therefore, will be respectively parallel to a b and a b: then,if from the point c, an indefinite line be drawn, parallel toAB, asEP, we shall have the horizontal projection of thegenerator. To obtain its vertical projection, we must sup-pose the generator to be produced upon the cylindric surface,till it meet the horizontal plane, which it can only do in apoint, that will be at once on the projection e f, and on thecurve epd, and consequently the intersection of these twolines : thus the point will be determined by producing e ptill it cut some part of the curve epd.
“ Two cases here present themselves : either the line E pwill cut the outline of the cylinder in a single point, or it willcut it in several points. In examining these two cases sepa-rately, we shall suppose, in the first instance, that to what-ever length the line e f may be produced, it shall cut thecurve epd only in the single point d.
“ The point d being the trace of the generator, if it beprojected on the vertical plane, by means of the perpendicular
D d, and if from the point d, the line d f be drawn parallelto a b, we shall obtain the vertical projection of the generator.We are thus in possession of the two projections of one ofthe right lines, through which the required tangent planemust pass. The vertical projection of the point of contactought to be on the line C c', drawn from the given point c,perpendicularly tOLM; it should also be on d f; consequently,it will be in the intersecting point, c, of those two lines.
“ If the line e f cut the trace, epd, of the cylindric sur-face in several points, as d and e, we must proceed for eachin the same manner as that just directed for the point n,considered by itself, and we shall obtain the vertical projec-tions, dj', ef, of as many generating lines, and the verticalprojections, c, c', of as many points of contact, as thereare points of intersection between the line ef and thetrace epd.
“In the instance of Figure 12, the trace of the cylindricsurface in the circumference of a circle, which has the pro-perty of being cut by a right line in two points ; so that thevertical line elevated from the given point c, must meet thesurface twice; first, in a point whose vertical projection is c,through which the generator passes when it touches uponthe point d ; and. secondly, in another point, whose verticalprojection is c, through which the generator passes when itrests on the point e of the trace. These two points, althoughthey have the same horizontal projection, are neverthelessvery distinct, and should each have a particular corres-pondent tangent plane. Indeed, for each ol the two pointsof contact, we must find the second right line, by whichthe position of the tangent plane is to be determined.Were we strictly to follow the general method, by consider-
ing the trace as a second generating line, it would be n ^ tb eto conceive it as passing successively through eac gl jchpoints of contact, and to construct a tangent for eac ^ aC ]\
points ; but in the present case of cylindric surfaces, _ th emore simple process may be pursued. For exaD3 ^ iro ug^'plane tangent to the point c, c, touches the surface, ^ fr 0 " 1out its extent, of the generating right line which pas ^ s ucbthis point ; it touches it, then, in D, which is a p°' n j, tl' egenerating line, and ought therefore to pass ° „ingf
tangent to the trace at the point D. By parity of re ^ ^gtigwe shall find that the plane tangent C, c‘, ought to P a ^ g ' t"’ 0the tangent to the trace in e. Therefore, ifpoints d, e, we draw the trace of the two tangent p ‘ \v' e
E G, produced till they cut the lineL M in two points ’^geldshall have, in the horizontal plane, the traces of the ttplanes. pla^ 9
“ It only remains to discover the traces of the sa ( ] ie s«
on the vertical plane ; and having already for ° ne li» ve
traces the point K, and for the other, the point G >only to determine a single point for each of them. f a gg e ?
“ With this view, operating for the first of the ttv^ jn
planes, imagine the point to be constructed, t0 _ oU <rbwhich a horizontal line drawn upon the plane t> ® s \i&
point of contact, would meet the vertical plane > ^ froffl
have the horizontal projection of such line, by dp'^ b
the point C, a line parallel to the trace D K> w %v e nw
produced till it meet the line L m in the point I ; a ” 0 ;,it £ ?obtain its vertical projection by drawing from the P 0 f tban indefinite horizontal line. The point of coinci ® ^ ce ty)vertical plane with the horizontal, will be foundupon the vertical line i i and the horizontal line c ^be at the point, i, of their intersection ; therefore,^ tr» ce .| lirthe points i and K a line be drawn, it will gi ve a s ‘the first plane tangent to the vertical plane. ^ j (g O' 11 ^
process for the second tangent plane, we shall ^on the vertical plane, by drawing from the point ^ pro^jj^parallel to the horizontal trace E G, which may j ]jrm (0till it cut l M in the point it, upon which the ve .^ 0])ta l h 11 ® fl dmay be raised ; from the point c draw a h° uZ w |,;ch, mcut the vertical line h h, in the point h, fi o£tl g j jU H ah'from the point G, by drawing the line G h, vve ;<j
the trace required. . t 0 f 9 *L
“ Second Question. —From an imaginary P°.' ven) to asurface, of which the horizontal projection is § .
a tangent plane to such surface. from
“ The solution of this question differs only of
ceding, in having the generating right line, i ^ tl |(J * # r 1 i
always parallel to itself, passing constantly ^ u nn e f e g fo f jwhose two projections are given. We thinK ^ aP) ii ^ jto enlarge in this place, and leave the reader j J0 st 9 ’ j
himself, with the assistance of Figure 13, sho 1
need of such aid. _ , nt of a S '
“ Third Question. From an imaginary P° 1)0 ,|Z
revolving upon a vertical axis, and given ’” r ^ lC e. .^rd 9 'projection, to draw a plane tangent to such - n h o1 jf
“ Solution. — Figure 14. Let a be the ?^ ct j 0 n, p jj-a"' 1 ’projection of the axis, a d its vertical P 1 °J^ plane ^ t h 9the given generating curve, considered in j eC tion j
from the axis, and G the given horizonta P !
point of contact. , a %\s, f 1
“ If from the point of contact, and from ^ t |ie ,aplane be conceived, whose projection wou reV oh'' v ^A )( o' , ihorizontal line AG, such plane must cut t ie B[]gg jng ■’
in a curve, which will become the genera o , * ert icalthe point of contact: if from the point G, c<m s< o ,g0 9 :
conceived, it will meet the generating clirV .®[ 1 w ill h e j
the surface, in one or several points, w