Bary centric Perspective. 1 79 



irig of any element, and proportion the tint or elevation to the 

 inverse cube of its distance from the vanishing line, then any 

 portion of the picture will accurately represent (and indeed if we 

 use relief, the volume or weight of such portion will be strictly pro- 

 portional to) the area (or its weight) of the corresponding part in 

 the object plane. Supposing different object planes to be repre- 

 sented in perspective on the same picture plane, with liberty for the 

 position of the eye to vary, it may be shown without difficulty* that 

 if the absolute intensity of tint or relief for any object plane varies 

 as the square of the distance of its trace upon the picture plane 

 from its vanishing line, and as the first power of the distance of 

 the eye from the same line, the ratio between corresponding por- 

 tions of object and picture will be alike for every plane. 



In the corresponding problem for right lines, the relief or tint 

 of any element in the perspective of a given right line must vary 

 as the inverse square of the distance from the vanishing point, 

 and the absolute intensity for different lines must vary as the 

 product of the distance between the trace and the vanishing 

 point into the distance of the eye from that point. In bary- 

 centric perspective we have seen the further consideration of 

 atomic weight enters, so that the density follows the law of the 

 inverse fourth and third powers for planes and lines respectively *, 

 instead of third and second powers as in geometrical perspec- 

 tive; in fact in the geometrical theory the quantities visibly 

 represented correspond to the momentsf in respect to the vanish- 



* For if we take T the trace of an object line, V its vanishing point, and 

 through O (the eye) draw OPp meeting TV in P and the object line in p, 



uTP T» 



Tp the quantity of TP= ^ p y , so that )li=TV^PV=TV . OV; and 



again, if tTt' be the trace of an object plane, V the foot of the perpendicular 

 from O on the vanishing line VT perpendicular to tTt', P a point in VT, 

 and p the point where OP meets the object plane, we have tpt' (the quan- 

 t¥t' 



tity Of tYt')=fXr^Y rpy py, Or 



^_Ty2 1 ty£_ pv=TV 2 . S? . PV=TV 2 . OV. 



The preceding calculations assume the expressions pup, fixfiy applicable to 

 a linear and triangular space, given in a preceding foot-note. 



t And consequently if, in the pictorial representation of any plane sur- 

 face, there is taken a triangular patch of given area, the quantity in the 

 object corresponding thereto will vary inversely as the product of the 

 distances of the three angles of the patch from the vanishing line, — a 

 proposition in perspective which I imagine to be new, and at all events is 

 certainly little known. This may be applied to determine instantaneously 

 the area of an ellipse of which the perspective projection is a circle of radius 

 r, and whose centre is at the distance h from the vanishing line. Writing fi 

 equal to the distance of the vanishing line from the eye, multiplied by the 

 square of its distance from the trace of the ellipse upon the plane of the 



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