Edinburgh New Philosophical Journal (1839 Edinburgh New Philosophical Journal edinburghnewphil28edin Year: 1839 - 1840 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 s
Edinburgh New Philosophical Journal (1839 Edinburgh New Philosophical Journal edinburghnewphil28edin Year: 1839 - 1840 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.
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