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of all the points of the right line A b, upon the plane l m n o,is called the projection of the right line a b upon this plane.

“ As two points are sufficient for determining the positionof a right line ; it is only necessary, in constructing the pro-jection of a right line, to construct the projections of two ofits points, and the line which they describe will be therequired projection.

“ Hence it follows, that if the proposed right line be per-pendicular to the plane of projection, its projection will bereduced to a single point, which will be that in which it fallsupon the plane.

“ Figure 2.—The projections a b , a' b', of the same indefi-nite right line, A b, being given upon two planes, not parallel,L M n o, l m p Q, this right line is determined ; for, if fromone of the projections, ab, we imagine a plane perpendiculartoutso, the position of this plane being known, it wouldnecessarily pass through the right line A B. The position ofthis right line, which is found at once upon both the knownplanes, consequently at their mutual intersection, is thereforeabsolutely determined.

“ What we have just stated is to be understood as inde-pendent of the planes of projection, and takes place, whatevermay be the angle formed by the two planes. But if the angleformed by the two planes of projection be very obtuse, thatformed by their perpendiculars will be very acute: verytrifling mistakes in this respect, will, in practice, lead to verygrave errors in determining the position of the right line.To obviate this cause of inaccuracy, at least in the absenceof better means, it is usual to have the planes of projectionperpendicular to each other : besides which, as most artists,who use projections, are familiar with the position of a hori-zontal plane and the direction of a plumb-line, they gene-rally represent one plane of projection as horizontal, and theother vertical.

‘ The necessity of representing, in drawings, the two pro-jections upon the same sheet, and in larger operations uponthe same area, has farther determined artists to represent thevertical plane as turning and folding down, as upon a hinge,at its intersection with the horizontal plane, so as that thetwo may form but one plane, upon which they construct theirprojections.

“ The vertical projection is thus, in fact, traced upon ahorizontal plane, and it must ever be kept in mind, that itmust be corrected and put in its place, by turning it one-fourth of a revolution round its intersection with the hori-zontal plane : to do which accurately, care must be taken totrace tbis intersection very plainly upon the design.

“ Thus, in Figure 2, the projection a b' of the right lineA B, could not be drawn upon a real vertical plane; but if weconceive the plane to be turned about the right line l m, so asto bring it in contact at lip' q', we shall readily execute thevertical projection a! b'.

“ Besides the facilities of execution presented by this dis-position, it possesses the additional advantage of abridging thelabour of projections. Thus, suppose the points a, a', to bethe horizontal and vertical projections of the point A, theplane indicated by the right lines a a, a o', will be perpen-dicular to the two planes of projection at the same time,because it passes along the right lines perpendicular to them:consequently, it will be perpendicular to their common inter-section l m ; and the right lines a C, a c, according to whichthey cut these two planes, will be themselves perpendicularto Jj M.

“ Now, if the vertical plane be turned about l m, as upona hinge, the right line a c, still continues perpendicular toi. m ; and the case is still the same, when the vertical plane,laid down, assumes the position c a". Thus the two right

lines a c, c a", passing both by the point C, and beingperpendicular to l m, are in prolongation to each otn ’ o ^gicase is similar with the right lines b d, d b' 1 , as to evel Xjne<ipoint, as b. Hence it follows, that when we have o ^the horizontal projection of a point, the projection ^same point upon the vertical plane, supposed to be la* y V Q-will be in the right line drawn along by the horizonjection perpendicularly to the intersection, L M, otplanes of projection, and this reciprocally.

“ This result very frequently occurs in practice. 2,

“ We have hitherto considered the right line, A do

in which case we should only have to |jy

as indefiniteits direction

, but we must now consider it as termin'*"' oU rthe points a b, which will bring us to take its extent in ^calculation. We shall, therefore, examine how thisdeduced from a knowledge of its two projections. _ oO

“When a right line is parallel to one of the P^^jeC*which it is projected, its length is equal to that of it® g 0 tlition on the plane ; for the line and its projection, b el0 ^ pro-tend nated at two points perpendicular at the P^^gtwee 11jection, are parallel to each other, and comprised t j 0 [iparallels. In this particular case, therefore, thebeing given, the length of the right line, which is eq uis also given. , ane 9 of

“ A right line is always parallel to one of the tw° P. a jq e I 10projection, when its projection upon the second is P athe first of its planes. oD t"'°

“ If the right line be at the same time obliquf U P. 0 f it 3planes, its length will be greater than that ot eit i ^ ve r 1projections ; but the true length may be obtained 1simple operation. , - tg give 0

“ Figure 2 —Let ab be the right line ; ab, 0 f tb®

projections : to find its true length. From en^ pilingextremities of the right line A, in the vertical P ia t t jU t*from it, imagine a horizontal line, a e, stretching tre n>itfmeet in e the vertical line falling from the other w iiie°at b ; this will give the rectangular triangle, ijno-th Of ^ Imust be constructed in order to obtain the len 0right line ab, which is its hypothenuse. In tm g jj e a 0 ’’independently of the right angle, we know the - g tbwhich is equal to the given projection, a b. ^j zoD tal h* 1 ,vertical plane, we draw from the point a, the h°° ^

a' e, which is the projection of a e, it will cut p o1line b 'd at the point e, which will be the project! 011 0 aIJ g c o( \e. Thus b' e will be the vertical projection of B a g C ert al ° e 9 ysequently of an equal length with it. Having .y y,e e a 'by these means, the two sides of the triangle, it j v e 1to construct the triangle, whose hypothenuse w I

length of ab. afl> Dlt ^all

“ Figure 2, being drawn in perspective, has n ^y e sb.^constructions done in the manner of projection 9 - jjlhere give the construction of this first questiosimplicity. -ed t0 ^rt^

“ Figure 3.—The right line l m being supp°“^ j,intersection of the two planes of projection, an . t iiea" b", the given projections of a right line ; to ^gniteof tbis line. Draw through the point a , the m ; 0 tzontal h e, which will cut the line b b", at t ie F p ra vVupon it measure the length of a b, from e to r jgbthypothenuse h b", and its length will be that o ^

required. . . a rec tang uI fV^

“ As both the planes of projection are * r,\a& e5 ’ u jfc.

operation, which is performed upon one of * g ar re 3 gjjt

be also done upon the other, and will yield ®.f 1 oe jceN®>|,t 0 <|

“ From what has been said, the reader ' V1 ^ ten 0 , 1 ^ of

whenever we have the two projections of a ? v the a P Jby plane surfaces, by rectilinear angles, and j

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