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Point 6 .P r ;’j ect ' on be made by drawing straight lines fromMallei’l' 1 18 ca ^ ec * a perspective representation ; but if byl fj le hnes, it is called an orthographical representation.

’p]„ .P r °J e ctions of points, lines, and plane figures, are foundPo 8it io“'g *-be originals in a given original plane, in a certainPtoj eeti to the intersection of such plane with the plane ofa fe e ?. n : and thence the projections of the faces of solidsthei r ^ discovered, either by finding the intersection oftepiw ne ® with the plane of projection, or by finding thehiie s u ation of one of the planes, and the representation ofto >r*l lng an y 8 iven angle with the original of the planetb e P^sented ; and thus the seat, inclination, and lengthhas j ta 0l '’ginal line must be known. Each of these methodsNned h ma g e ' hct that in which the representations are“fiair, ’ Ending the intersections of the different planesfheir J| ea °h of the solid angles, having the inclinations of'P itg ne ®' ' s the most natural, as well as the most universal®hject s PP ’ ca tion ; though the other method, where the^Pce, ^ 0 he represented are prisms, is more expeditious;^h°] e he representation of one of the ends be found, theon e tnay be also obtained by finding the representation8i <k s . the arrises, or edges, of the planes which form theh'e rL r> ' n tl>e case of a cylinder or cylindroid, by findingP^ntj „ sent ation of the axis, then drawing lines from all

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l; : b^gg the base, or from a sufficient number of points inMeting t ’| P ar allel and equal in length to the axis, and com-lv ill C g le other end equal and similar to the one projected,• ^ v erv - 6 th e representation of the whole solid,jh any I- s °hfi) whose properties are known, may be projected' fi* 0 (l ,® n position to the plane of projection; and no otherw s Pac e ° T? suc h ready means of projecting points or lines.pere t'h have a relation to a given or original plane,

®' v eti ; n 6 °riginal plane and the plane of projection arela ^ P os >tion.

^ 0f e c . e fi°ctrine of shadows, nothing can be more easy, or'Pie; tb Vetl '. ent: ’ than the methods furnished by this prin-,Wly Un ' v crsality of the rules is such, that they applytl “re P ers P ft ctive and to orthographical representations,*e w. Jhuch more convenient than the method of finding

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s had e " n » l* n es of their planes and those of the planes

Pplle<j ( ? n 1—The plane on which the object is represented,^ IWl e plane of projection.

“graphical projection, the rays by which the pro-

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e J>lan e , lme< l> are here understood to be perpendicular tok&nitio P r °jection.

c .jlte l, or ‘ n ^—If the projection be made on a plane parallelN th e „? n ’ or °n a plane representing the horizon, it isane °f Me object.

tiM^alar f ^ the projection be made on a plane per-

y, Plan e .° horizon, or on a plane representing a ver-

o j}’$nitj 0 14 ' s ca lled the elevation of the object.to i* e > is ” ^—The _pZa«e of position of an inclined line andpf Plane* 10 ^ 161, I^ ane P ass ing along the line perpendicular

° a ffie ^—^lie inclination of a line to a plane, is theyfiie Plane of position comprehended between the line

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th e *—A he plane on which any object is given, is

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to s u ati 0r , . •"—A point on a plane is said to be given, when

y'^e fly ! s known in respect of some given line, relatively

' n that plane.

- The position of a line to a plane is said toseat of the line is given upon the plane,

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angle which the line makes with its seat

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Definition 9.—In the representation of figures, in planesinclined to the plane of projection, the situation of the figure,in respect to the intersection of its plane with the plane ofprojection, is supposed to be known. The intersectionis therefore given on the plane of projection, and the spaceon the one side is to be considered as the original plane,while that on the other is the plane of projection.

Axiom. —If any point, line, or plane, coincide with theplane of projection, that point, line, or plane, so coincident,is both the original and the projection of that point, line,or plane.

Proposition I.—The projection of a point is in a straightline drawn from the original point perpendicular to theintersection.

For, suppose a plane to pass through the original point,perpendicular to the primary plane and to the plane of pro-jection, its intersection with each of them will be perpen-dicular to that of the primary plane and the plane of projec-tion ; but lines perpendicular to the same straight line, drawnfrom the same point, are in the same straight line, and there-fore the projection of a point is in a straight line drawnfrom the original point perpendicular to the plane ofprojection.

Proposition II.—The orthography, or projection of a straightline, is also a straight line. Thus, let a plane pass throughthe original straight line perpendicular to, and intersectingthe plane of projection ; here, as the intersection of one planewith another is a straight line, the orthographic representa-tion must be so likewise.

Proposition III.—All original parallel straight lines arerepresented by parallel straight lines ; because the intersec-tion of one plane with another is a straight line ; and, whena plane cuts several parallel planes, the intersections of theseplanes with the cutting plane, are parallels.

Proposition IY.—The orthographical representation of aline parallel to the plane of projection, is a line equal andparallel to its original.

Proposition V.—The orthographical representation of aline parallel to the intersection, is a line equal and parallelto the original.

Proposition YI.—The orthographical representation of aplane figure parallel to the plane of projection, is a figureequal and similar to the original.

Proposition VII.—All planes perpendicular to the planeof projection, are described as straight lines.

Proposition VIII.—All straight lines in the primary planeperpendicular to the intersection, have their representationsalso perpendicular to the intersection.

Proposition IX.—Every straight line in a plane perpen-dicular to the plane of projection, is represented by the inter-section, or a part of the intersection, of that plane.

Proposition X.—The intersecting point of every line inany original plane is that point in the intersecting line of theoriginal plane, which is cut by producing the original line tothe intersecting line of the plane.

Proposition XI.—If, through any point in the originalplane, a plane be drawn perpendicular to the intersection, soas to meet, or intersect, the plane of projection, the represen-tation of the point will be in the line of section where it is jcut by a perpendicular to such line, in the plane perpen- jdicular to the intersection. j

Proposition XII.—If two points in the intersection of oneplane with another be found, or known, the intersecting lineof the first plane in the second, is the line which joins thetwo points. _ j

Proposition XIII.—If the intersecting lines of two planes, jinclined to each other, be given, or found, upon a third plane, j