350 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL, 



[Skptembbr, 



Case 2. 



In the triangle arc, ac and xz are drawn parallel, being l)oth perpendicu- 

 lar to 4i'. By the previous proposition we have r A : a// = r!/ : t/.v 



and 



adding these 



but 



and 



therefore 



dividing each side of the equation by 2 



we get . ?'i 



therefore . .rj 



v/j '. bc = vy \ yx 



2vb '. a6 + 6c = 2vy '. yx + yz 



ab + be is equal to flc 



x + yz is equal to xz 



2 vb 



''!/ 



. ac=vy 



ac X vy 



vb 



Example:— Let no, equal to 1100 feet, be a vertical line, whose repre- 

 sentation is required ; vb, equal to 5000 feet, the distance of the object, and 

 xz, the plane of the jiicture parallel to ac, at the distance of 500 feet from v. 



The for the height of the representation we have xz = 



1100 X SOD 

 5000 



= 110 feet. 



Case 3. 



In the triangle arb let ab and xy be both perpendicular to J'A ; then by 



the previous propositions 



we have . vb '. ab=vy '. xy 



also in the triangles cvb and xvy 



we have . vb '. bc^=vy * yz 



multiplying the first equation by 2, and then subtracting the second, 



weget . vb '. ab — bc = vy : xy — yz 



but . a 4 — 4 c is equal to o e 



and . .ry — y^ is equal to XI 



ac y^ V \i 



therefore . bv : ac = vy '. xz or xx ^=i 



' bv 



Example: — Let ac, equal to 900 feet, be a vertical line, whose represen- 

 tation is required ; !'i the horizontal distance, or distance of the object, equal 

 to 5000 feet, and xy, the plane of the pictiu'e, parallel to a 4 at the distance 



of 500 feet from v. Then xz— — — — ^r — = 90 feet, the required height of 



the representation. 



In each of the foregoing three cases, we observe that the same rule holds 

 gooil ; namely, the beiglit of the rejjrcsentation is always equal to the height 

 of the original, uiultiidied into the distance of the picture, and the product 

 divided by the distance of the object, whether the base of the oliject be level 

 with the point of sight, below it or above it. It is evident that tlie same 

 rule applies to liorizontal lines (lines drawn upon the horizontal plane), for 

 the i)iir))ose of olitaining the widths, liy merely substituting in the above pro- 

 portion the word width instead of height, thus : — the width of tlie repre- 

 sentation is always equal to the width of the original object, multiplied into 

 the distance of the picture, and the product thvided by the distance of the 

 object. Observe, that this proportion for the widths holds good only when 

 the plane of picture and the original plane are parallel. 



If the distance of the picture be taken = o, we have vc '. ac^o '. xz, or 

 ac X fi 

 XX = =0. If the distance of the picture be taken equal to the dis- 



fZ C ^ V o 



tance of the object, we have vc '. ac=^ve '. xx, therefore xx = 



ve 

 = ac, the size of the original; hence the picture xx may have any value 

 whatever between o and the original, acconUng to the distance of the picture. 



Proposition 3.- -If we consider the surfaces of objects, we shall find that, 

 the distance of the picture being constant, the representation varies as the 

 object directly, but as the square of the distance inversely. 



Let the original plane, abed, and the plane of the picture, mvoji, be pa- 

 rallel. \A'c have, upon the vertical plane, — v( : ad=ps I mp 

 and upon the horizonta l plane vl ', ab = vs '. »»« multiplying (i!^)2 ; ad x 

 oi = (»»)! ; mpxmn 



butarfxflAis equal to the surface of the plane abed; also mpxmn is 

 equal to the surface of the representation, or (»<)2 : aicrf = (»;«)'-' ; mnop 



, abed X ()i.s-)^ 



and mnop = ;— — 



{vt)' 



that is, the surface of the rejiresentation is equal to the original surface mul- 

 tijiUcd into the square of the distance of the i)icture, and the product (Uvided 

 liy the square of the distance of the object. Now, if (««)- be constant, then 

 mtiop varies as abed directly, but as (»/)^ inversely. 



Having, I hope, ah'eady, by aid of the very few propositions just given, 

 successfully demonstrated the leading principle, I shall now endeavour to 

 apply the foregoing rules to some of the most obvious and general examples 

 in perspective. 



The author has given full instructions and rules for the application 

 of the system to Parallel perspective, followed by similar directions 

 for Angular perspective. From which we select the following practi- 

 cal example: — 



"Required the perspective representation of a square building, of 

 which the accompanying sketch is a plan. 



"Let the length of a side be 30 feet, and the height 38 feet. Let 

 the length of the radial vg be 40 in., and that of the radial r h 30 in. ; 

 then the distance between gA will be equal to 50 in. 



" Upon the plane of the picture draw gk, and make it equal to 50 in. 

 (See engraving bdow.) 



"Let the radial distance of gbe 120 feet; then the radial distance 

 of h will be 'JO feet. 



"Suppose the distance a a, or tlie distance from a to the radial 

 plane of g, to be bO feet ; then for the distance from a to the radial 

 plane of h we have 70 feet. 



" The height of the eye is 5 feet. To find its representation we 



15 

 have 00 : 5 = 30 : a, or a = — in. = 1-G6 in. below gh For the 



distance of a from h we have 90 : 70 =: 30 : a, or a 



•210 

 'IT 



23-33 in. from h Set off this last distance from h upon gh, and at the 

 distance so set off draw a perpendicular to g h, and make the part 

 below It equal to I-Gtj in. for the point a ; next draw ag and ah. 

 " For the distance of 6 from A we have 120 ; 70 = 30 : b, or b=z 



210 



— — =: 17*5 in. Set off that distance accordingly, to intersect ft A in 6 



and from b draw bg. 



" To find the distance of rf from g we have 150 : 80 = 40 : rf, or=(i 

 320 



-nr= 21'33 in. from g. This distance set off in like manner, from g 

 15 



to meet ag in d, and from d draw dh. 



