August i r, 1923] 



NA TURE 



195 



meaning of such terms as air-foot^ margin^ margin 

 parallel line^ carto-photo-field^ parameter parallel^ and 

 so on. By taking measurements from the horizon on 

 the photograph and from the " margin " on the 

 reference plane (the margin being the intersection of 

 the reference plane with a plane through the nodal 

 point parallel to the plate), the invariable relation is 

 obtained B.lp = plh, where f = Y sec 0, F being the 

 focal length, 9 the tilt of the optical axis measured 

 downward from the horizontal, h the distance measured 

 to any point in the photograph from the horizon, and 

 H the distance from the " margin " to the projection 

 of that point in the reference plane, these distances 

 being measured in the principal plane. Such distances 

 have thus the reciprocal relation that if one set, say 

 in the photo plane, is expressed as an arithmetical 

 series, the other set in the reference plane will be 

 expressed as a harmonical series. 



Along the line of the intersection of the photo plane 

 with the reference plane all magnitudes have, of course, 

 the same value ; and it also results from the perfect 

 similarity of position of the two planes that, at the 

 point on their intersection where it is cut by the 

 principal plane (the vertical plane containing the 

 optical axis), angles on the reference plane are correctly 

 represented on the photograph. This point, which is 

 sometimes known as the " isocentre," is called by 

 Mr. Gordon the " field centre," and, as he remarks, 

 this property of the identity in the two fields of any 

 angle located in the field centre is the fundamental 

 law of the perspective of angular magnitudes. The 

 field centre is thus an appropriate origin for polar 

 co-ordinates. 



Let us now imagine the photograph to be hinged 

 along the line of its intersection with the reference (or 

 map) plane, and let it be turned round on this axis 

 until it is in the map plane. The hinge (parameter 

 parallel) is a line on which all lengths are truly repre- 

 sented in the photograph, and the field centre is a 

 point in this line at which angles are truly represented. 

 Distances measured at right angles to the hinge are 

 connected by the expression }i/p = p/h. For distances 

 measured parallel to the hinge, we have Y/y = p/h 

 where Y is the ordinate of a point on the map, y that 

 of a point on the photograph; or Y/y = {X+p)/p, 

 where X is the abscissa on the map plane, measured 

 at right angles to the hinge. 



To make use of these expressions we must fix on the 

 photograph the position of this hinge line, which is 

 parallel to the line of the horizon ; and to do this we 

 must draw the horizon. The distance between the 

 hinge line and horizon is p. To fix the horizon, Mr. 

 Gordon rediscovered, in the course of his investigation, 

 ;i solution which he afterwards found had been given 

 NO. 2806, vor,. 1 12] 



by Brook Taylor, of Taylor's Theorem, two hundred 

 years ago. Thus, let there be three points in a line 

 in the reference plane (or cartographic field), and let 

 the known length of one segment be a and of the other 

 b, the line lying in any direction. Let A and B be the 

 lengths of the representations of these segments in the 

 photograph. Then the distance, V, from the inter- 

 mediate point of the three, on the photograph, 

 measured along the given line, to the horizon, is 

 (a + b)AB/{aB-bA). This gives one point on the 

 horizon and a second divided Hne will give a second 

 point, so that the horizon can be drawn on the photo- 

 graph. 



Mr. Gordon also points out that it is possible, in a 



CO»/STMCTI0NOrA CROmOPlAM 

 rpOKAN OBLIQUE morOCMP/1 THKLU 

 AT THE BRITISH MUSEUM mrriC 

 ?8r^ MARCH I92i BYMRJV^GOflDcnlS'i£rftOO 



ALTITUDe. orCAMCfiA STKTIOV lerriOm. 



SCALE 



3t)"37vS 



HOmZDN 



Y.RWC/PAL WmSHINC POINT. 

 e-3T'ST 



OniCAL CCNTfiL- 



tMD/K POIffT. 



similar way, to identify the nadir point on a photo- 

 graph, by making use of a vertical line on which three 

 points have been marked at known distances from 

 each other. From the nadir point, a line drawn 

 through the optical centre, at a distance 2F/sin 2d 

 from the nadir point, gives the position of the principal 

 vanishing point. 



The accompanying illustration will serve to give 

 an idea of the lines made use of by Mr. Gordon in 

 constructing a plan from an oblique photograph. The 

 method used was not precisely that which would be 

 employed in survey work, but the diagram indicates 

 the general principle. The height of the nodal point 

 of the lens was 16 ft. 10 in. above the floor ; the focal 



