1825.) — Mathematical Probleni. \ 201 
| For the Monthly Magazine. 
- Sotvrion of gv Important Prosrem in Practical PersPEcTIVE. » 
By Mr. T, S. Davies. 
: Probiem. 
=? draw through a given point a line which shall tend to an inaccessible 
vanishing point. i ; 
Let BD be the horizontal line, and AE the perspective representation of the 
base of the plaine, whose vanishing point H, determined by the intersection of 
BD, AE is inaccessible. Let C be the given point through which it is required 
to draw a line tending to H. 
A 
Construction. : 
I. Through the given point C, draw any line, cutting the lines BD, AE in B and 
A; and parallel to it any line, cutting the same lines in D and E. ; 
Note, For these lines we may, in practice, take the vertical extremities of 
the plan we are delineating—those extremities being of course previously deter- 
mined. The line DE, it is hardly necessary to remark, might be taken on the 
other side of BA; and, in either case, if more convenient, might be the vertical 
edge of the drawing, or even a line beyond that edge. a hae 
_ II, Lay the parallel ruler over A and D, and move it forward to E, Mark F, 
the point where it crosses the line BD. 
MI. Lay the parallel ruler over D, and the given point C; move it forward to 
-F, and mark the point G where it crosses DE. 
IV. Draw CG; which is the line required. 
: Demonstration,’ __ 
By parallels, DF : BD :: DE: BA, and 
DF: BD: DG : BC; and, therefore, 
AB: BC :: DE: DG. . 
Hence CG, BD, and AE tend to the same point H. 
The same ‘coustruction and: demonstration, mutatis mutandis, applies when C 
is situated between the given lines; the point F answering to all the positions of 
C, which are in the line AB; and as this, or something easily reducible to this, 
is generally the case, the operation for the most complex building becomes ex- 
tremely simple. & 
I was led to ‘investigate this problem ‘by some remarks in. Nicholson’s 
“ Course,” on the great difficulty of the operation by every method yet known. 
I discovered several remarkably simple solutions involving the fise of only the 
parallel ruler, and requiring no lines to be drawn, and very few marks to be 
made, which did not necessarily forma part of the previous work. Viewed in 
this light, the method just given claims a preference, as the point F is the only 
one introduced for the purposes of this construction. ‘ 
It will hence appear, that such expensive and operose instruments, as Ni¢chol- 
sons Centrolinead (for. the invention of which, I belieye, that gentleman was re- 
warded by the Sogiety of Arts), are altogether unnecessary in the solution of this 
eat kindred. problems—indeed, of every problem to which that instrument can be 
ed. 
Bristol, July 15, 1825. : 
* 
Monruty Mac. No. 415. 2D 
