1840.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



233 



were not i)iopeily designed. If the stones be obtained in squared 

 blocks from the quarry, there will be a loss on the ends of the stones ; 

 but this, as every builder knows, can be avoided by proper inanac;e- 

 inent in the quarry. And thus, on the whole, the loss of material for 

 the skewed bridge need not exceed to any extent worth naming that 

 for the right one. 



The above statements are true of cylindroid oblique arches, whalerer 

 may be the forms of their principal sectioiis ; they are at variance with 

 the statements and so-called experience of engineers of established 

 reputation: complete demonstrations of them are given in the ap- 

 pendix. They are equivalent to dilferential equations, and require to 

 be integrated in order to give practical results ; these results vary 

 according to the particular form assumetl for the longitudinal section 

 of the vault. I proceed to give a few of these results, commencing, 

 on account of its more frequent occurrence, with the circular arch. 



On investigating the form of the projection of a joint of a circular 

 oblique arch upon a horizontal plane, I arrived at a new curve, to 

 which the name Double Logarithmic has been given. 



FiK. I. 



Having pi ojected the entire semicylinder, of which only a porlion 

 can be used with propriety, let AB, CD, be the sides of the projection, 

 and EF, parallel to the parapet, the plan of one of the lines of pres- 

 sure. Bisect EF at right angles by GUI, and form two logarithmic 

 curves of which AB, CD, may be the asymptotes, EG the common 

 subtangent, their ordinates being parallel to EF. Then draw lines 

 KL parallel to AB, and intercepted between the logarithmics, the 

 middles M of these lines trace out the horizontal projection of one of 

 the joints. The lines AB, CD, are thus asymptotes to the horizontal 

 projection, and this geometrical property illustrates the mechanical 

 impossibility of constructing a semirylindric arcli, without trusting to 

 the cohesion of the mortar. The introduction of the logarithmic curve 

 into investigations concerning bridges, has been of great utility, and 

 the analogy between this curve and the connnon catenary is striking. 

 The catenary is also formed by bisecting the interval between two 

 logarithnacs; but these have a common asymptote with rectangular 

 co-ordinates, while the bisected line is parallel to the ordinate. The 

 computations needed for the delineation of such projections, are by no 

 means tedious ; they may be performed rapidly by help of Napierian 

 logarithms ; but a better method, capable of giving all the projections, 

 will be explained shortly. 



It may be expected, from what has been said of such elevations in 

 general, that the end elevation of a circular oblique arch shall present 

 some interesting peculiarity. The end elevation of a joint ought, in 

 fact, to cross at right angles the circumferences of circles described 

 with equal radius from points lying in a straight line ; now, this is the 

 distinguishing characteristic of the tractory, and that curve must there- 

 fore be exhibited on the end projections of all circular oblique arches. 



On examining the projection of one of the joints upon a vertical 

 plane perpendicular to the parapets, I obtained the genesis of a pecu- 

 liar curve still logarithmic in its nature, and somewhat resembling in 

 its form the superior branch of the conchoid. If we conceive the side 

 elevation of the semicylinder to be traversed by horizontal lines, the 

 distances intercepted on these lines bear to the corresponding distances 

 intercepted by a certain normal curve, the ratio of cotangent of obli- 

 quity to radius. This normal curve, which belongs to an arch with its 

 obliquity 45°, I have named the Companion to the Tractory ; it 

 admits of a very neat mechanical delineation. 



Let a rod AB, equal in length to the radius of the arch, be made to 

 rest upon a smooth board only at the point A, while the extremity B is 

 guided along the line BD ; A will, as is well known, describe the equi- 

 tangential curve or tractory. Suppose that the guide to which the 

 point B (or in an oblique position D), is attached, carries a vertical 

 rule DFE, and that, on that rule, there slides a right angle DFC, one 

 side of which is constrained to pass through C :* then will the point 



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

 half Ab trom the middle of AB to the rule DFK as'indicated by the dutteJ 

 lines. 



Fig. 2. 



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 DCE, while the groove DF is con- 

 tinued to meet it: E then traces out the catenary. Since,//om 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 

 catenary, 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 indi- 

 vidual arch-stones, can be much more readily obtained from the de- 

 lineation of the surface of the centering. Regarding the crown line 

 as the absciss, and the actual lines of pressure as the ordinates (on the 

 curve surface), half the ordinate plus 45°, has its logarithmic tangent 

 ])roportional to the absciss. Having once obtained the log-tangent 

 corresponding to a given distance along the crown line, a simple pro- 

 portion will give that corresponding 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 calculation is but a minute fraction of 

 that expended in the drawing of the plans. By these means, the ac- 

 companying model of the surface of the centering, its development, 

 and various orthographic projections, were completed.* The simple 

 inspection of these, and their comparison with most of the skewed 

 bridges already constructed, will shew in what respects this branch of 

 architecture has hitherto been defective. 



I cannot leave the subject of the circular arch without indicating 

 the extensive and indispensable use of logarithms in the calculations. 

 Napier, when he founded first the rudiments of the tluxional 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 impulse which they have since given to 

 every branch of exact science. Each new mathematical research piles 

 another stone on the monument of Napier. 



Neither can I avoid remarking, that the ingenious speculations of 

 the earlier geometers concerning the various mechanical curves, spe- 

 culations which have been by many regarded as ianeiful and useless, 

 are one by one turning to account in the progress of modern philoso- 

 phy. 



The elliptic arch, being much recommended by the gracefulness of 

 its form, is frequently used. If we view the circular oblique arch 

 from a distant point in the continuation of its axis, it does indeed ap- 

 pear elliptical; but then the ehipse has its major axis directed verti- 

 cally, so that a circular skewed bridge can hardly have a fine appear- 

 ance unless the segment be extremely flat. Let us then inquire into 

 the phases of an elliptic skew. 



The horizontal plan of the joint is still a double logarithmic curve ; 

 and its delineation, including, of course, that for the circular arch, is 

 as follows. 



EF being as before, the plan of one of the lines of pressure, find HQ 

 a third proportional to the horizontal and the vertical semi-axis; 

 through Q dravv Q'E parallel to HG. Describe then logarithmics 

 having E'G for their common subtangent, and having their ordinates 



• These are deposited in the Museum of the Society of Arts of .Scoiian A 



2 I 



