1839.] 



THE CIVIL'ENGINEER AND ARCHITECT'S JOURNAL. 



349 



105 



128 

 If now we assume R — p 



(R—p) = -R-aOS (R—p). 

 141bs., we shall have 



n<- 



■P)' 



14 



(R- 



■P)] 



or r = — (R—p) = -8125 {R—p). 



Thus if we made use of the co-efficient -8203 instead of '8125, we 

 should commit an error of -0078 (R — p) = •10921b. per square inch. 

 In the same manner it may be shewn thiit if we applied the same co- 

 efficient, -8203, when the effective pressure i? — p was ISlbs., the error 

 would be •UOGl (R — p) = 'lOOSlbs. per square inch. It is thus de- 

 monstrated, for condensing engines, that, supposing M. de Pambour's 

 constant co-efficients to be correct, no error worthy of notice would be 

 committed by applying a constant co-efficient to the effective pressure 

 in the cylinder. 



For high pressure engines, the same values are attributed toy and 5 

 as for low pressure condensing engines. Thus, taking R z= 601bs. and 

 p = 151bs. for simplicity, we find 



r=i{(R-p)-^,(R-I»}> 



or r = "SS-jG (R — p) 

 Assuming now r ^= 95, whence r — p 



80, we have 

 r = ^ [(R-p)-l(R-p)}, 



orr= -8041 (R—p). 



Tlie error committed by making use of the co-efficient -8.556 instead 

 of the latter would be -0085 (R — p) = •G8lbs. per square inch, = '01 

 r nearly. In the same manner, by taking R — P ^ 10, it may be 

 shewn that the co-efficient ought to be ■787.1, in which case the error 

 committed by using the first co-efficient would be -OGSl (R — p) = 

 •68 libs. ^ -687 r nearly. This error is too great, even for practical 

 purposes; but it would be easy to determine another co-efficient for 

 the lower pressures, which should be sufficiently accurate, and the 

 method of eo-efficknts would be as correct, and nuich more easy of ap- 

 plication than that proposed in this work. 



An Essay on Jlrithnetical Perspiclire ; in which the representation is 



obtained by computation from the hwwn dimensions and position of 



the object. By C. E. Bernard, C. E. 1839, J. Williams, London. 



Mr. Barnard in this essay has attempted, what we believe has not 



been before done, to make Perspective a Science, and a branch of 



Mathematics. Instead of drawing the lines to vanishing points, he 



)u-oposes to ascertain the relative positions, heights and lengths by 



arithmetical calculation, although the artist may be averse to this 



mode of proceeding, calculation being foreign to his profession, it will 



be found by the engineer and the scientific, a most interesting and 



valuable acquisition. We cannot do better than by letting the author 



explain for himself, for which purpose we shall give some extracts 



from the introduction. 



By the term Arithmetical Perspective, I mean the application of arithmetic 

 to the piu-pose of obtaining the (Umensions and position of the representation 

 of an original object, which application of aritlnnetic amounts to this : when 

 certain geometrical relations are found to exist Ijetween Unes, we substitute 

 the numerical values of those Unes for the lines themselves. Now, as hy far 

 the greater part of the lines necessarj' to the consideration of perspective are 

 imaginary ones, by making use of their values we are thus enabled to desig- 

 nate them, and to draw only sucli as are aiisolutely essential to a complete 

 representation of the original. The oliject, however, of the present treatise 

 is to show liow we may indicate the original lines of an object, as well as the 

 imaginarj' ones, by means of their numerical values ; thus obviating the 

 necessity of drawing a plan and elevation of the object to be represented per- 

 spectively. 



In the description of objects whose forms are geometrical, such as build- 

 ings, hy means of perspectve, it will often l)e the easier mode to ascertain 

 the dimensions and position of the representation, liy computation than l)y 

 construction, according to the usual methods. If, for instance, a draughts- 

 man were asked of what size should a tower, one hundred feet in height, and 

 distant a mile, he shown upon his drawing, he would be obliged to perform 

 sevcr.al operations before the required answer could he given ; the tnith of 

 which would depend altogctlier upon his accuracy in drawing. 



Aritlnnetically, however, tlie result may he obtained witli far greater cor- 

 rectness and dispatch, thus : if the pictvn'e l)e viewed at the distance of a foot, 



100 

 then 5280 ; 100 ; : 1 ; x, or x=t:;^=Q.22 in., the required height of 



tlie representation. But, before detailing the means tiy which we arrived at 

 this answer, some preliminary considerations recpiire om" attention. 



Proposition I. — The size of the image in the eye varies as the size of the 

 object directly, Imt as the distance of the oljject inversely. 



Let the distance vb he constant, then in the triangles avd and qvr, we 

 have by the preceding, oS ; av = qr ; vr. Likewise in the triangles avc 

 and j)vr we have ; av '. ac = vr ; pr; therefore a// ; ac = qr \ /;>•, and 

 alternately ab '. qr = ac '. pr. That is to say, tlic size of the image is in 

 proportion to the size of the object, when the distance remains the same. 



Let the size of the oljject be constant ; then in the triangles avc and pvr 



we liave ac '. cv^pr '. pv, or — = ^. But »» is constant, for it is the 

 pv cv 



radial from v, the pupil, to p at the back of the eye ; therefore p r, the 



nc 



image, varies as — : that is, as ac, the object directly, and as cv, its distance 



CO 



reciprocally. 



We now perceive that olijects vaiy in apparent size according to their dis- 

 tances, because the images of those olyects in the eye actually become larger 

 as the otijects approach, or they decrease in size as the originals recede. 



I have here considered the object to be of but one dimension, as a line. 

 If, however, the oliject be of two dimensions, as a plane, then tlic plane of 

 the image will evidently vary as the plane of the original object directly, and 

 reciprocally as the square of the distance. 



Mr. Barnard commences his instructions by giving some definitions 

 of perspective, he then proceeds to lay down preliminary propositions, 

 for the study of his system of perspective. 



PRELIMINARY PROPOSITIONS. 



Proposition 1. — The size of the image in the eye is proportional to the size 

 of the picture, divided by the distance of the pictine. 



It has already been proved that the size of the image pr is proportional to 

 the size of the object ac divided by the distance cv. Let xz, representing 

 the plane of the picture, be drawn parallel to ac, then the triangles avc and 

 .!■ V : are similar, and therefore the sides about the equal angles proportional. 



But ym is constant, therefore the image j!;r varies, as ocz, the picture di- 

 rectly, but as ,rv, the distance of the picture inversely, which relation is the 

 same as that abeady sliown to exist between the image and the original ob- 

 ject ; therefore, if the representation he drawn, as here supposed, hearing the 

 same proportion to its distance as the ohject does to its distance, we may 

 then dismiss altogether the consideration of the image formed mthin the eye, 

 and confine our .attention exclusively to the ohject and its representation. 



Proposition 2, Case 1. — The representation is- equal to the product of the 

 original ohject into the chstance of the picture, divided by the distance of the 

 oliject. 



In the triangle avc let ac he periicnilicular to vc, and draw;.rfrom z parallel 



to ca. Then we have, by preceding projiositions, ?ie : ac = cz: xz 



ac X vz 



or .rr= 



vc 



Example: — Let ac, equal to 1000 feet, be a vertical line whose perspective 

 representation is required; vc, equal to .'iOOO feet, the distance of the ohject 

 from the point of sight v. Let also the plane .r-, upon «hich the represen- 

 tation of a c is required to he drawn at the distance of 500 feet from v, he 

 parallel to ac; then, to find the height ,rz of the representation, we have 



•*'*= ■ — = 100 feet., the rcquiied height. 



