Mr Sang on the Constructioii of Oblique Arches. 309 



Having once obtained the log-tangent corresponding to a 

 given distance along the crown line, a simple proportion will 

 give that con-esponding to any other absciss ; the log-tangent 

 corresponding to half the length of an arch-stone having been 

 found, the repeated addition of that quantity to itself will lead 

 to a knowledge of the position of the corner of each stone in 

 the whole structure ; the simplest operations of trigonometry 

 only being needed. Indeed, the labour of the whole calcula- 

 tion is but a minute fraction of that expended in the di'awing 

 of the plans. By these means, the accompanying model of 

 the surface of the centering, its development, and various or- 

 thographic projections, were completed.* The simple inspec- 

 tion of these, and then- comparison with most of the skewed 

 bridges already constructed, will shew in what respects this 

 branch of ai-chitecture has hitherto been defective. 



I cannot leave the subject of the circular arch without in- 

 dicating the extensive and indispensable use of logarithms in 

 the calculations. Napier, when he founded first the rudiments 

 of the fluxional calculus, and thence the logarithmic method, 

 sanguine though he may have been as to the immense value of 

 his discoveries, could never have imagined the prodigious im- 

 pulse which they have since given to every branch of exact 

 science. Each new mathematical research piles another stone 

 on tlie monument of Napier. 



Neither can I avoid remarking, that the ingenious specula- 

 tions of the earlier geometers concerning the various mecha- 

 nical curves, speculations which have been by many regarded 

 as fanciful and useless, are one by one turning to account in 

 the progress of modern philosophy. 



The elliptic arch, being much recommended by the grace- 

 fulness of its form, is frequently used. If we view the circu- 

 lar oblique arch from a distant point in the continuation of its 

 axis, it does indeed appear elliptical ; but then the elhpse has 

 its major axis directed vertically, so that a cu-cular skewed 

 bridge can hardly have a fine appearance unless the segment 

 be extremely flat. Let us then inc^uire into the phases of an 

 elliptic skew. 



* These are deposited in the Museum of the Society of Arts of Scotland. 



