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,. And that the Height of a Point appearing on the Plane, Line i/ it, the Diftances VL and ^if.draw K^and VSi SJ1 fcSft ^^^J^SSLf^S^SS a ," d * ^?A» *» J-? *>' s'beX Appearance ot
Point from the Plane, to the Aggregate of that Diftance and the Diftance of the Eye.
Jchmgraphic PERSPECTIVE; or the Laws of the pro- jeclion 0/ Plane Figures.
Perfpetlive of a Point.
To exhibit the Appearance h of an objective 'Point, H. (Fig. 2. ) From the given Point, draw a Perpendicular to the fun- damental Line Z> E. From the fundamental T> E cut off 1 K =3 IB; thro' the Point of Sight i-'draw a horizontal Line E P ; and make F5> equal to the Diftance of the Eye S L : Laftly from the Point J to the Point of Sight F draw FI; and from K to the Point of Diftance P, the Line P K. The Interferon A is the Appearance of the objective Point. See Point.
Hence, 1 . Since, the Appearance of the extreme Points of a right Line being given, the Appearance of the whole Line is given ; the Ichnographic Projection of any Rectilinear Fi- gure may be had by this Method. See Rectilinear.
And, 2. Since any Number of Points of a Curve Line may by this means be projected on the PerfpeBive Plane ; the Projection of Curve Lines may likewife be eftected after the fame manner. See Curve.
3. Therefore; this Method will likewife fuffice for Mixtili- near Figures ; and is confequently univerfal.
There are indeed other Methods deliver' d by other Authors, but this is the moil ufual. To conceive its force and effect, it will be proper to illuftrate it with fome Examples.
Perfpetlive of a triangle.
To find the Appearance of a Triangle, ABC, ("Fig. 3.) whofe Bafe A B is parallel to the fundamental Line U) E. To the fundamental Line Ti E draw a Parallel at an Interval equal to the Aititude of the Eye. Affume a fundamental Point K, oppofite to this either directly or obliquely, as the Cafe requires. Transfer the Diitance of the Eye from V to K. From the feveral Angles of the Triangle A CB, let fall Perpendiculars A\, C 2, B 3: fet off thefe Perpendiculars upon the fundamental Line 2) E oppofite to the Point of Diftance K. From 1, 2, 3, draw right Lines to the funda- mental or principal Point Vi, V 2, V 2. From the Points A, B and C of the fundamental Line Ti E draw other right Lines A K, S K, C K, to the Point oi Diftance K.
Since a, b, and are the Appearances of the Points A, B and C c ; The right Lines ca, ab and be, being drawn, ac b will be the Appearance of the Triangle A C B.
After the fame Manner is a Triangle projected on a Plane, where the Vertex C is oppofed to the Eye: All here requir'd, is, that its Situation on the Geometrical Plane be changed, and the Vertex C turn'd towards the fundamental Line 3).
Perfpetlive of a Square.
To exhibit the Appearance of a Square, A B D C (Fig.4. ) feen obliquely, and having one of its Sides A B in the fundamental Line. The Square being view'd obliquely af- ume the principal Point V in the horizontal Line H R, in fuch manner as that a Perpendicular to the fundamental Line may fall without the Side of the Squared B, at leaft, may not biflect it 5 and make V K the Diftance of the Eye. Transfer the Perpendiculars AC and B 2) to the fundamental Line © E ; and draw the right Lines K B, K T>, as alfo AV, V C. Then will /?and 8 be their own Ap- pearances ; and c and d the Appearances of the Points C and ZD. Confequently Ac d Bis the Appearance of the Square JBDC.
If the Square ACBTi fhou'd be at a Diftance from the fundamental Line T> E; which yet rarely happens in Practice ; the Diftances of the Angles A and B muft likewife be tranf- fer'd to the fundamental Line : As is evident from the pre- ceeding Problem. And iince, even the oblique View is not very common ; in what follows, we fhall always fuppofe the Figure to be pofited directly oppofite to the Eye ; unlefs, where the contrary is exprefsly mention'd.
4. To exhibit the Appearance of a Square A B C D (Fig. 5 .) -whofe Diagonal A C is 'Perpendicular to the fundamen- tal Line. Continue the Sides 2) C and C B till they meet the fundamental Line in 1 and 2. From the principal Point V. fet off the Diftance of the Eye to K and L. From K to M and 1 draw right Lines K A and Kl ; and from L to A and 2, the right Lines LA, L 2. The Interferons of thefe Lines will exhibit the Appearance of the Square AH CD view'd Angle-wife.
5. To exhibit the Appearance of a Square A B C D Fig. 6. wherein another, I M G H is inferibed ; the Side of the greater, A S, being in the fundamental Line ; and the Diago- nal of the left, Perpendicular to the Fundamental. From the principal Point V, fet off, each way, on the horizontal
l.w Appearance of the Square A CDS. Produce the Side of the inferib'd Square IB, till it meet the fundamental Line in 1; and draw the right Lines Ki, and KM; then will ihgMbetae Reprefentation of the inferib'd Square / H G M.
Hence is eafily conceiv'd the Projection of any Figures in- Icrib d in others.
Perfpetlive of a Pavement.
5 . To projeil a Pavement conjifting offpiare Stones, view'd diretlly. Divide the Side AS transfer 'd to the fundamen- tal Line Fig. 7. into as many equal Parts as there are fquare Stones in one row. From the feveral points of Divifion, draw right Lines to the principal Point V; and from A to the Point of Diftance K, draw a right Line A K; and from B to the other Point of Diftance L, draw another L B. Thro' the Points of the Interferons of the correfponding Lines, draw right Lines V on each Side, to be produced to the right Lines A V, and S. Then will Af g S be the Appearance of the Pavement AFG S.
Perfpetlive of a Circle.
6. To exhibit the Appearance of a Circle. 1. If the Cir- cle be fmall, circumfenbe a Square about it. Draw Diago- nals and Diameters ha and de (Fig. S.J interfering each other at right Angles ; and draw the right Lines/ g and b c parallel to the Diameter d e thro' b and/ ; as alfo thro' c and g draw right Lines meeting the fundamental Line 'D E in the Points 3 and 4. To the principal Point V draw right Lines y 1, V^, f $, ^"2; and to the Points of Diitance L and-K, draw the right Lines X 2 and if 1. LaOiy connect the Points of Interfefiion, a, b, d,f, lo,g, e, c, with the Arches a b, b d, d /, {jc. Thus will a b d f h g e c a, be the Appearance of the Circle.
If the Circle be large, on the Middle of the Fundamental y?»(Fig. 5>.J delciibe a Semicircle; and from the feveral Points' of the Periphery, C, F, G, H, J, ci?c. to the funda- mental Line, let tall perpendiculars 67 1, F 1, G 3. H 4, J 5, £gc. From the Points t, 2, 3, 4, 5, &c. of A B draw right Lines to the principal Point V, as alfo a right Line from B to the Point of Diitance ijand another from^to the Point of Diftance K. Thro' thecommon Interferons, draw right Lines as in the preceeding Problem ; thus fhall we have the Points c, f,g, h, i, which are the Reprefentations of thefe A. C, F, G, if, I, which being connected as before, give the Projection of the Circle.
Hence appears not only how any curvilinear Figure may be projected on a Plane; but alfo how any Pavement, confitling of any kind of Stones, may be delineated in SPerjpeffive.
Hence alfo, appears what Ufe the Squaie is of in 'Perfec- tive, for even in the fecond Cafe we ufe a Square divided in- to certain Areola:, and circumfcribed about the Circle ; tho* it be not delineated on the geometrical Plane in the Diagram.
Pe'fpellive of a regular Pentagon.
7. To reprefent a regular pentagon, having a broad Limb, terminated by Lines parallel thereto: i". From the feveral Angles of the exterior Pentagon A,B,C, C L,E, Fig. 10. to the fundamental Line T S, let tall Perpendiculars Act, B 1, C z t Ml, E 4 ; which, as in the former, transfer to the fundament- al Line. Connect the Points 1, 2, 3, 4 to the principal Point f; and the Points 1, 2, 3, 4 to the Point of Dif- tance K. Thus will the common Interferons reprefent the Appearance of the exterior Pentagon. 2. If now, from the inner Angles G HL I, the Perpendiculars Go, H<j,K6, IT, L 8, be in the like manner let fall ; and the reft be done as in the former ; we (hall have the Reprefentation of the inner Pentagon. The Pentagon A S C T> E, therefore, with its Limb, is reprelented in PerfpeBive.
This Problem is added for the fake of an Inftance of the Proje&ion of a Figure that has a broad Limb, or Edge.
It muft be here obferv'd, that if the Magnitudes of the feve- ral Parts of an Object, be given in Numbers, together with the Height and Diftance of the Eye ; its Figure is to be firft conftructed by a geometrical Scale ;and the fundamental Point with the Point of Diftance 1, to be determin'd by the fame.
Nor is it always neceffary, that the Object be delineated under the fundamental Line , in the Projer on of Squares and Pavements 'tis beft let alone. But where 'tis neceffary, and Space is wanting; draw it a-part ; find the Divifions in it, and transfer 'em to the fundamental Line in the Plane.
Threads being hung in the principal Point, and the Point of Diftance, and ftretch'd to the Points of the Divifions of the fundamental Line ; the common Interferon of the Threads will give the Projection of the fevetal Points without Confufion; a Tiling much to be fear'd from the Multiplicity of Lines to be drawn.
