494 



BRIDGE. 



Theory, cians to beware of thinking calculation the surest 

 * v ' mode of eliciting truth. It should be the last thing 

 employed. Nothing is so simple, so perspicuous, as 

 the diagram. Its geometrical properties should be 

 pursued as far as possible. They are not only clear, 

 they are palpable. And in such applications of ma- 

 thematical theory, the whole being a creature of the 

 mind, it seldom admits of an approximating value in 

 any part. 



Let us now return to the geometrical construction. 



The weight of the section C, may be determined 

 in the same way as the foregoing. But surely more 

 simply thus : From c draw cs parallel to wz, that is 

 at right anghs to KO, and make it equal to wz-f-H'5 

 flraw sc at right angles to LO, meeting the vertical 

 c c in c, then c c represents the weight of C. From 

 D, draw DT parallel and equal to s c, draw T d per- 

 pendicular to DO, meeting the vertical D d in </, orf is 

 the weight of D, and so on successively. 



Nay, instead of drawing DT parallel to s c, and T d 

 perpendicular to DO, we may at once draw from s, 

 s d' perpendicular to DO, which will cut off for us 

 cd'=D d, the weight of the section D. It is of no 

 consequence, although the lines of abutment do not 

 all run to the same centre o. 



And thus we obtain a general construction for all 

 Fig. 10. the sections, which turns out abundantly simple, Fig. 

 10 : for, upon any vertical line b' e, if c b be taken to 

 represent the given weight of any section C, and c T 

 be drawn at right angles to co, and b T at right 

 angles to BO, meeting the other in T : then T b repre- 

 sents the pressure against the abutment OB, and TC the 

 pressure against o c, and by drawing T d at right 

 angles to DO, t e to EO, &c. we have the weights of 

 the successive sections represented by cd,de, &c. and 

 the pressure on their lower abutments represented by 

 T d, T c, &c. 



We may carry the same mode of determination to 

 the other side of C, and pass the vertex of the arch. 

 The divisions representing the weights of the sections 

 will run upwards along the indefinite line c b'. The 

 pressures on the abutments will be determined as be- 

 fore. Should the two sides of a section be parallel, 

 the perpendiculars through T upon them will coin- 

 cide ; such a section therefore should have no weight. 

 But should the two lines of abutment diverge to- 

 wards the lower side, the line expressing the weight 

 of that section will return upon the vertical, shewing 

 that such a section requires the reverse of weight, 

 viz. a support from below. The line TV drawn hori- 

 zontally through T exhibits the horizontal pressure, 

 which is uniform through the same equilibrated arch. 

 But it is evidently greater, the less b T and c T are 

 inclined to each other, the weight b c being constant, 

 that is, the smaller the angle of the wedges or sec- 

 tions. It also increases directly as the weight of the 

 section C, &c. The line v c expresses the weight of 

 the semi-arch or perpendicular pressure on each pier ; 

 being the sum of the weights of all the sections in 

 the semi-arch. 



Again, it is obvious that the angles b T c, or c T d, &c. 

 are equal to the angles of the sections BOO, COD, &c. 

 If therefore the weight of any section E be given =de, 

 and the requisite angle of that section be required, 

 every thing else being known, we have only to join 



T e, and the line EO beii.g drawn perpendicular to te, '! ii 

 will exhibit the inclination of the lower abutment of **" "v 

 the section ; die is the angle of that section. And 

 here it matters not where the point E be, that is, how 

 great the base of the section be, provided the weight 

 is equal to de. We also see that while the angles re- 

 main the same, and the weights proportional, it is of no 

 consequence what thecurve passing through the lower 

 edges of the sections, or through their upper edges 

 may be, they may even be straight lines. Accord- 

 ing to this principle, the architect is not confined to 

 given forms of intrados or exlrados ; he may take 

 whatever curve appears rr.tisi beautiful or useful : and 

 what is more, by the proper adjustment of the joints, 

 he may cast the ultimate pressure in any direction 

 which he thinks most conducive to the strength of the 

 edifice. 



The reader will easily perceive, that the segments 

 of the vertical line rapidly increase, as the perpendi- 

 culars to the line of abutment approach the vertical ; 

 that is, as the abutments approach the horizontal 

 line ; and in that position, the last segment becom- 

 ing infinite, it is impossible by mere weight alone to 

 effect the adjustment of the sections. 



Though the geometrical construction we have just 

 given is so simple, that it appears likely to answer 

 every practical purpose; yet it may be proper to ex- 

 press analytically, or rather arithmetically, the values 

 of the several quantities concerned in the investigation. 

 This is attended with no difficulty, as TVC being a 

 right angled triangle, it is obvious that the weight \e 

 of the same arch is the tangent of vre, or of the incli- 

 nation of the lower abutment, when TV the horizontal 

 force is radius ; at the same time also, the pressure 

 Te on the abutment is the secant of the same angle ; 

 and the weight cd of any section is the difference of 

 the tangents of the inclinations of its upper and lower 

 abutments. In like manner a v, the weight of half 

 the key-stone, is to TV the horizontal force as the 

 tangent of half the angle of that section is to the 

 radius; or, as radius is to the cotangent of the same 

 angle. 



We now proceed to shew the application of this Ap 

 investigation to some practical cases, and the first we '' on to 

 shall consider, is that known by the common, though 

 aukward name of \hej1at arch; one with which every PF.ATZ 

 mason is perfectly familiar, though it be seldom no- '- xx5 

 ticed by writers on equilibration. ABiais a stnic- *>' " 

 ture of this kind, adjusted to this equilibrium, and 

 resting on the abutments A a, B6. Its construction is 

 exceedingly simple ; nothing more is necessary than 

 to draw all the joints mM, /L, &c. to one centre C ; 

 and the reason is obvious ; for DK, KL, &c. are the 

 differences of the natural tangents of the inclinations 

 of the abutments, the perpendicular CD being radius; 

 and the same thing is true in the line da, and in every 

 other parallel section. The surface therefore A m, 

 M/, that is, the bulks or weights of the stones, are in 

 the same ratio, and it is that which is required by the 

 above principles. Also, if we assume the line of its 

 base to represent the weight of any stone in the arch, 

 for example, KD for half the keystone ; then the 

 perpendicular CD is the horizontal thrust, drift, or 

 shoot of the arch. By increasing DC, or diminishing 

 it, that is, by drawing the joints to a lower, or u 



