Fig. 2. 



308 Mr Sang on (he Consfruction of Oblique Arches. 

 the extremity B is guided along the line BD ; A will, as is well 

 known, describe 

 the equitangen- 

 tial cm*ve or 

 tractory. Sup- 

 pose that the 

 guide to which 

 the point B (or 

 in an oblique 

 position D), is 

 attached, car- 

 ries a vertical 

 rule DFE, and 

 that, on that 

 rule, there slides 

 a right angle 

 DFC, one side 

 of which is con- 

 strained to pass through C :* then will the point F trace 

 the Companion to the Tractory. A very simple addition will 

 convert this instrument into that described by Leslie in his 

 Geometry of Curve Lines, for forming the catenary. A 

 grooved rule has only to be attached, making the right angle 

 DOE, while the groove DF is continued to meet it : E then 

 traces out the catenary. Since, /rom the nature of the figure, 

 ED DF = AB^, it follows, that the companion to the tractory 

 has its ordinates inversely proportional to those of the cate- 

 nary, and that, therefore, it might, with propriety, have been 

 named the inverted catenary. 



All these projections of the joints, and the forms too of the 

 individual arch-stones, can be much more readily obtained 

 from the delineation of the surface of the centering. Regai'd- 

 ing the crown line as the absciss, and the actual lines of pres- 

 sure as the ordinates (on the curve surface), half the ordinate 

 plus 45°, has its logarithmic tangent proportional to the absciss. 



* In practice, it would be more convenient to lay a jointed rod equal to half 

 AB from the middle of AB to the rule DFE as indicated by the dotted 

 lines. 



